one-locus-model-with-mutation.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import ipywidgets as widgets\n",
    "import matplotlib.pyplot as plt\n",
    "#import mpld3\n",
    "import numpy as np\n",
    "import sympy as sym\n",
    "\n",
    "import models\n",
    "import payoffs\n",
    "import plotting\n",
    "import selection_functions\n",
    "import symbolics"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "%matplotlib inline\n",
    "plt.style.use(\"ggplot\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# One-locus model\n",
    "\n",
    "Start from full model. Set $x_2=1-x_1, x_3=x_4=0$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [],
   "source": [
    "x1 = sym.symbols('x1', real=True, positive=True)\n",
    "T, R, P, S = sym.symbols('T, R, P, S', real=True, positive=True)\n",
    "M, epsilon = sym.symbols(\"M, epsilon\", real=True, positive=True)\n",
    "\n",
    "UGA = symbolics.UGA\n",
    "UgA = symbolics.UgA"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Total offspring (fitness)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [],
   "source": [
    "x = np.array([[x1], [1 - x1], [0], [0]])\n",
    "payoff_kernel = np.array([[R, S], [T, P]])\n",
    "W = models.generalized_sexual_selection(x, UGA, UgA, payoff_kernel, M, 0, mutation_rate=epsilon)\n",
    "N, = models.total_offspring(W, x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [],
   "source": [
    "N = sym.simplify(N)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-2.0*M + 2.0*P*UGA(x1)**2 - 4.0*P*UGA(x1) + 2.0*P + 2.0*R*UGA(x1)**2 - 2.0*S*UGA(x1)**2 + 2.0*S*UGA(x1) - 2.0*T*UGA(x1)**2 + 2.0*T*UGA(x1)"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "N"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "After a bit of algebra we arrive at the following expression for total offspring.\n",
    "\n",
    "\\begin{align}\n",
    "   % N(x_1) =& 2.0*P*UGA(x1)**2 - 4.0*P*UGA(x1) + 2.0*P + 2.0*R*UGA(x1)**2 - 2.0*S*UGA(x1)**2 + 2.0*S*UGA(x1) - 2.0*T*UGA(x1)**2 + 2.0*T*UGA(x1)\\\\\n",
    "   % N(x_1) =& 2\\bigg(PU_{GA}(x_1)^2 - 2PU_{GA}(x_1) + P + RU_{GA}(x_1)^2 - SU_{GA}(x_1)^2 + SU_{GA}(x_1) - TU_{GA}(x_1)^2 + TU_{GA}(x_1)\\bigg) \\\\\n",
    "   % N(x_1) =& 2\\bigg(RU_{GA}(x_1)^2 + PU_{GA}(x_1)^2 - TU_{GA}(x_1)^2 - SU_{GA}(x_1)^2 + SU_{GA}(x_1) + TU_{GA}(x_1) - 2PU_{GA}(x_1) + P\\bigg) \\\\\n",
    "    N(x_1) =& 2\\bigg(\\big((R + P) - (T + S)\\big)U_{GA}(x_1)^2 + \\big((T + S) - 2P\\big)U_{GA}(x_1) + P\\bigg) \\\\\n",
    "\\end{align}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Equilibrium total offspring\n",
    "\n",
    "Consider an equilibrium $x^*$ and define the constant equilibrium selection probability as $U_{GA}^* = U_{GA}(x^*)$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "UGA_star = sym.symbols(\"UGA_star\", real=True, nonnegative=True)\n",
    "equilibrium_total_offspring = N.subs({UGA(x1): UGA_star})"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2.0*P*UGA_star**2 - 4.0*P*UGA_star + 2.0*P + 2.0*R*UGA_star**2 - 2.0*S*UGA_star**2 + 2.0*S*UGA_star - 2.0*T*UGA_star**2 + 2.0*T*UGA_star"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "equilibrium_total_offspring"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [],
   "source": [
    "# solve for the interior value of UGA* that maximizes total offspring\n",
    "foc = equilibrium_total_offspring.diff(UGA_star, 1)\n",
    "optimal_UGA_star, = sym.solve(foc, UGA_star)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(P - 0.5*S - 0.5*T)/(P + R - S - T)"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "optimal_UGA_star"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [],
   "source": [
    "_equilibrium_total_offspring = sym.lambdify((UGA_star, T, R, P, S),\n",
    "                                            equilibrium_total_offspring,\n",
    "                                            modules=\"numpy\")\n",
    "\n",
    "def plot_total_offspring(T, R, P, S):\n",
    "    fig, ax = plt.subplots(1, 1, figsize=(10,6))\n",
    "    us = np.linspace(0, 1, 100)\n",
    "    ax.plot(us, _equilibrium_total_offspring(us, T, R, P, S))\n",
    "    ax.set_title(r\"Total number of offspring\", fontsize=25)\n",
    "    ax.set_xlabel(r\"$U_{GA}^*$\", fontsize=25)\n",
    "    ax.set_ylabel(r\"N\", rotation=\"horizontal\", fontsize=25)\n",
    "    plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "a7934285395949edae73196c57099497",
       "version_major": 2,
       "version_minor": 0
      },
      "text/plain": [
       "interactive(children=(FloatSlider(value=25.0, description='$T$'), FloatSlider(value=3.0, description='$R$'), F…"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# sliders used to control the Prisoner's Dilemma Payoffs\n",
    "T_slider = widgets.FloatSlider(value=25, min=0, max=100, step=0.1, description=r\"$T$\")\n",
    "R_slider = widgets.FloatSlider(value=3, min=0, max=100, step=0.1, description=r\"$R$\")\n",
    "P_slider = widgets.FloatSlider(value=2, min=0, max=100, step=0.1, description=r\"$P$\")\n",
    "S_slider = widgets.FloatSlider(value=1, min=0, max=100, step=0.1, description=r\"$S$\")\n",
    "\n",
    "w = widgets.interactive(plot_total_offspring, T=T_slider, R=R_slider, P=P_slider, S=S_slider)\n",
    "display(w)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# The locus of potential equilibria (LPE)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [],
   "source": [
    "(f,), *_ = models.offspring_genotypes_evolution(W, x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-x1 + (x1*((-epsilon + 1)*(2*(-M + R)*UGA(x1) + 2*(-M + S)*(-UGA(x1) + 1))*UGA(x1) + 0.5*(2*(-M + P)*(-UGA(x1) + 1) + 2*(-M + T)*UGA(x1))*(-UGA(x1) + 1)) + (-x1 + 1)*(epsilon*(2*(-M + P)*(-UGA(x1) + 1) + 2*(-M + T)*UGA(x1))*(-UGA(x1) + 1) + 0.5*(2*(-M + R)*UGA(x1) + 2*(-M + S)*(-UGA(x1) + 1))*UGA(x1)))/(x1*(epsilon*(2*(-M + R)*UGA(x1) + 2*(-M + S)*(-UGA(x1) + 1))*UGA(x1) + 0.5*(2*(-M + P)*(-UGA(x1) + 1) + 2*(-M + T)*UGA(x1))*(-UGA(x1) + 1)) + x1*((-epsilon + 1)*(2*(-M + R)*UGA(x1) + 2*(-M + S)*(-UGA(x1) + 1))*UGA(x1) + 0.5*(2*(-M + P)*(-UGA(x1) + 1) + 2*(-M + T)*UGA(x1))*(-UGA(x1) + 1)) + (-x1 + 1)*(epsilon*(2*(-M + P)*(-UGA(x1) + 1) + 2*(-M + T)*UGA(x1))*(-UGA(x1) + 1) + 0.5*(2*(-M + R)*UGA(x1) + 2*(-M + S)*(-UGA(x1) + 1))*UGA(x1)) + (-x1 + 1)*((-epsilon + 1)*(2*(-M + P)*(-UGA(x1) + 1) + 2*(-M + T)*UGA(x1))*(-UGA(x1) + 1) + 0.5*(2*(-M + R)*UGA(x1) + 2*(-M + S)*(-UGA(x1) + 1))*UGA(x1)))"
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {},
   "outputs": [],
   "source": [
    "n, d = sym.fraction(f + x1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "x1*((-epsilon + 1)*(2*R*UGA(x1) + 2*S*(-UGA(x1) + 1))*UGA(x1) + 0.5*(2*P*(-UGA(x1) + 1) + 2*T*UGA(x1))*(-UGA(x1) + 1)) + (-x1 + 1)*(epsilon*(2*P*(-UGA(x1) + 1) + 2*T*UGA(x1))*(-UGA(x1) + 1) + 0.5*(2*R*UGA(x1) + 2*S*(-UGA(x1) + 1))*UGA(x1))"
      ]
     },
     "execution_count": 43,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "x1*((-epsilon + 1)*(2*R*UGA(x1) + 2*S*(-UGA(x1) + 1))*UGA(x1) + 0.5*(2*P*(-UGA(x1) + 1) + 2*T*UGA(x1))*(-UGA(x1) + 1))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "(-x1 + 1)*(epsilon*(2*P*(-UGA(x1) + 1) + 2*T*UGA(x1))*(-UGA(x1) + 1) + 0.5*(2*R*UGA(x1) + 2*S*(-UGA(x1) + 1))*UGA(x1))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2*P*UGA(x1)**2 - 4*P*UGA(x1) + 2*P + 2.0*R*UGA(x1)**2 - 2.0*S*UGA(x1)**2 + 2.0*S*UGA(x1) - 2*T*UGA(x1)**2 + 2*T*UGA(x1)"
      ]
     },
     "execution_count": 42,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sym.expand(d)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "x1*(2*(-epsilon + 1)*(R*UGA(x1) + S*(-UGA(x1) + 1))*UGA(x1) + 1.0*(P*(-UGA(x1) + 1) + T*UGA(x1))*(-UGA(x1) + 1)) - x1*(x1*(2*epsilon*(R*UGA(x1) + S*(-UGA(x1) + 1))*UGA(x1) + 1.0*(P*(-UGA(x1) + 1) + T*UGA(x1))*(-UGA(x1) + 1)) + x1*(2*(-epsilon + 1)*(R*UGA(x1) + S*(-UGA(x1) + 1))*UGA(x1) + 1.0*(P*(-UGA(x1) + 1) + T*UGA(x1))*(-UGA(x1) + 1)) + (-x1 + 1)*(2*epsilon*(P*(-UGA(x1) + 1) + T*UGA(x1))*(-UGA(x1) + 1) + 1.0*(R*UGA(x1) + S*(-UGA(x1) + 1))*UGA(x1)) + (-x1 + 1)*(2*(-epsilon + 1)*(P*(-UGA(x1) + 1) + T*UGA(x1))*(-UGA(x1) + 1) + 1.0*(R*UGA(x1) + S*(-UGA(x1) + 1))*UGA(x1))) + (-x1 + 1)*(2*epsilon*(P*(-UGA(x1) + 1) + T*UGA(x1))*(-UGA(x1) + 1) + 1.0*(R*UGA(x1) + S*(-UGA(x1) + 1))*UGA(x1))"
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\\begin{align}\n",
    "    x_1(2R(-\\epsilon + 1)U(x_1)^2 + 2S(-\\epsilon + 1)U(x_1)(-U(x_1) + 1) + P(-U(x_1) + 1)^2 + TU(x_1)(-U(x_1) + 1)) - x_1(x_1(2\\epsilon(RU(x_1) + S(-U(x_1) + 1))U(x_1) + (P(-U(x_1) + 1) + TU(x_1))(-U(x_1) + 1)) + x_1(2(-\\epsilon + 1)(RU(x_1) + S(-U(x_1) + 1))U(x_1) + (P(-U(x_1) + 1) + TU(x_1))(-U(x_1) + 1)) + (-x_1 + 1)(2\\epsilon(P(-U(x_1) + 1) + TU(x_1))(-U(x_1) + 1) + (RU(x_1) + S(-U(x_1) + 1))U(x_1)) + (-x_1 + 1)(2(-\\epsilon + 1)(P(-U(x_1) + 1) + TU(x_1))(-U(x_1) + 1) + (RU(x_1) + S(-U(x_1) + 1))U(x_1))) + (-x_1 + 1)(2\\epsilon(P(-U(x_1) + 1) + TU(x_1))(-U(x_1) + 1) + (RU(x_1) + S(-U(x_1) + 1))U(x_1))\n",
    "\\end{align}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2*P*UGA(x1)**2 - 4*P*UGA(x1) + 2*P + 2.0*R*UGA(x1)**2 - 2.0*S*UGA(x1)**2 + 2.0*S*UGA(x1) - 2*T*UGA(x1)**2 + 2*T*UGA(x1)"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# denominator of the above expression is just total offspring, N(x1)\n",
    "sym.expand(d)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(-2*P*epsilon*x1*UGA(x1)**2 + 4*P*epsilon*x1*UGA(x1) - 2*P*epsilon*x1 + 2*P*epsilon*UGA(x1)**2 - 4*P*epsilon*UGA(x1) + 2*P*epsilon + 1.0*P*x1*UGA(x1)**2 - 2.0*P*x1*UGA(x1) + 1.0*P*x1 - 2*R*epsilon*x1*UGA(x1)**2 + 1.0*R*x1*UGA(x1)**2 + 1.0*R*UGA(x1)**2 + 2*S*epsilon*x1*UGA(x1)**2 - 2*S*epsilon*x1*UGA(x1) - 1.0*S*x1*UGA(x1)**2 + 1.0*S*x1*UGA(x1) - 1.0*S*UGA(x1)**2 + 1.0*S*UGA(x1) + 2*T*epsilon*x1*UGA(x1)**2 - 2*T*epsilon*x1*UGA(x1) - 2*T*epsilon*UGA(x1)**2 + 2*T*epsilon*UGA(x1) - 1.0*T*x1*UGA(x1)**2 + 1.0*T*x1*UGA(x1) - x1*(2*P*UGA(x1)**2 - 4*P*UGA(x1) + 2*P + 2.0*R*UGA(x1)**2 - 2.0*S*UGA(x1)**2 + 2.0*S*UGA(x1) - 2*T*UGA(x1)**2 + 2*T*UGA(x1)))/(2*P*UGA(x1)**2 - 4*P*UGA(x1) + 2*P + 2.0*R*UGA(x1)**2 - 2.0*S*UGA(x1)**2 + 2.0*S*UGA(x1) - 2*T*UGA(x1)**2 + 2*T*UGA(x1))"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sym.together(sym.expand(f))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\\begin{align}\n",
    "    %\\dot{x}_1 =& \\frac{(-2P\\epsilon x_1U(x_1)^2 + 4P\\epsilon x_1U(x_1) - 2P\\epsilon x_1 + 2P\\epsilon U(x_1)^2 - 4P\\epsilon U(x_1) + 2P\\epsilon + Px_1U(x_1)^2 - 2Px_1U(x_1) + Px_1 - 2R\\epsilon x_1U(x_1)^2 + Rx_1U(x_1)^2 + RU(x_1)^2 + 2S\\epsilon x_1U(x_1)^2 - 2S\\epsilon x_1U(x_1) - Sx_1U(x_1)^2 + Sx_1U(x_1) - SU(x_1)^2 + SU(x_1) + 2T\\epsilon x_1U(x_1)^2 - 2T\\epsilon x_1U(x_1) - 2T\\epsilon U(x_1)^2 + 2T\\epsilon U(x_1) - Tx_1U(x_1)^2 + Tx_1U(x_1) - x_1(2PU(x_1)^2 - 4PU(x_1) + 2P + 2RU(x_1)^2 - 2SU(x_1)^2 + 2SU(x_1) - 2TU(x_1)^2 + 2TU(x_1)))}{(2PU(x_1)^2 - 4PU(x_1) + 2P + 2RU(x_1)^2 - 2SU(x_1)^2 + 2SU(x_1) - 2TU(x_1)^2 + 2TU(x_1))} \\\\\n",
    "    %\\dot{x}_1 =& \\frac{(-2P\\epsilon x_1U(x_1)^2 + 4P\\epsilon x_1U(x_1) - 2P\\epsilon x_1 + 2P\\epsilon U(x_1)^2 - 4P\\epsilon U(x_1) + 2P\\epsilon + Px_1U(x_1)^2 - 2Px_1U(x_1) + Px_1 - 2R\\epsilon x_1U(x_1)^2 + Rx_1U(x_1)^2 + RU(x_1)^2 + 2S\\epsilon x_1U(x_1)^2 - 2S\\epsilon x_1U(x_1) - Sx_1U(x_1)^2 + Sx_1U(x_1) - SU(x_1)^2 + SU(x_1) + 2T\\epsilon x_1U(x_1)^2 - 2T\\epsilon x_1U(x_1) - 2T\\epsilon U(x_1)^2 + 2T\\epsilon U(x_1) - Tx_1U(x_1)^2 + Tx_1U(x_1))}{(2PU(x_1)^2 - 4PU(x_1) + 2P + 2RU(x_1)^2 - 2SU(x_1)^2 + 2SU(x_1) - 2TU(x_1)^2 + 2TU(x_1))} - x_1 \\\\\n",
    "    %\\dot{x}_1 =& \\frac{(-2P\\epsilon x_1U(x_1)^2 + 4P\\epsilon x_1U(x_1) - 2P\\epsilon x_1 - 2R\\epsilon x_1U(x_1)^2 + 2S\\epsilon x_1U(x_1)^2 - 2S\\epsilon x_1U(x_1) + 2T\\epsilon x_1U(x_1)^2 - 2T\\epsilon x_1U(x_1) + 2P\\epsilon U(x_1)^2 - 4P\\epsilon U(x_1) + 2P\\epsilon + Px_1U(x_1)^2 - 2Px_1U(x_1) + Px_1 + Rx_1U(x_1)^2 + RU(x_1)^2 - Sx_1U(x_1)^2 + Sx_1U(x_1) - SU(x_1)^2 + SU(x_1) - 2T\\epsilon U(x_1)^2 + 2T\\epsilon U(x_1) - Tx_1U(x_1)^2 + Tx_1U(x_1))}{(2PU(x_1)^2 - 4PU(x_1) + 2P + 2RU(x_1)^2 - 2SU(x_1)^2 + 2SU(x_1) - 2TU(x_1)^2 + 2TU(x_1))} - x_1\n",
    "    \\dot{x}_1 =& \\frac{(-\\epsilon x_1\\big(2PU(x_1)^2 - 4PU(x_1) + 2P + 2RU(x_1)^2 - 2SU(x_1)^2 + 2SU(x_1) - 2TU(x_1)^2 + 2TU(x_1)\\big) + 2P\\epsilon U(x_1)^2 - 4P\\epsilon U(x_1) + 2P\\epsilon + Px_1U(x_1)^2 - 2Px_1U(x_1) + Px_1 + Rx_1U(x_1)^2 + RU(x_1)^2 - Sx_1U(x_1)^2 + Sx_1U(x_1) - SU(x_1)^2 + SU(x_1) - 2T\\epsilon U(x_1)^2 + 2T\\epsilon U(x_1) - Tx_1U(x_1)^2 + Tx_1U(x_1))}{(2PU(x_1)^2 - 4PU(x_1) + 2P + 2RU(x_1)^2 - 2SU(x_1)^2 + 2SU(x_1) - 2TU(x_1)^2 + 2TU(x_1))} - x_1 \\\\\n",
    "    \\dot{x}_1 =& \\frac{2P\\epsilon U(x_1)^2 - 4P\\epsilon U(x_1) + 2P\\epsilon + Px_1U(x_1)^2 - 2Px_1U(x_1) + Px_1 + Rx_1U(x_1)^2 + RU(x_1)^2 - Sx_1U(x_1)^2 + Sx_1U(x_1) - SU(x_1)^2 + SU(x_1) - 2T\\epsilon U(x_1)^2 + 2T\\epsilon U(x_1) - Tx_1U(x_1)^2 + Tx_1U(x_1)}{(2PU(x_1)^2 - 4PU(x_1) + 2P + 2RU(x_1)^2 - 2SU(x_1)^2 + 2SU(x_1) - 2TU(x_1)^2 + 2TU(x_1))} - (1 + \\epsilon)x_1 \\\\\n",
    "    \\dot{x}_1 =& \\frac{2P\\epsilon U(x_1)^2 - 4P\\epsilon U(x_1) + 2P\\epsilon + Px_1U(x_1)^2 - 2Px_1U(x_1) + Px_1 + Rx_1U(x_1)^2 + RU(x_1)^2 - Sx_1U(x_1)^2 + Sx_1U(x_1) - SU(x_1)^2 + SU(x_1) - 2T\\epsilon U(x_1)^2 + 2T\\epsilon U(x_1) - Tx_1U(x_1)^2 + Tx_1U(x_1)}{N(x_1)} - (1 + \\epsilon)x_1\n",
    "\\end{align}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "S*x1**2*UGA(x1)**2 - S*x1**2*UGA(x1) - S*x1*UGA(x1)**2 + S*x1*UGA(x1) + T*x1**2*UGA(x1)**2 - T*x1**2*UGA(x1) - T*x1*UGA(x1)**2 + T*x1*UGA(x1)"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sym.expand((T + S) * UGA(x1) * (1 - UGA(x1)) * x1 * (1 - x1))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [],
   "source": [
    "# solving for the LPE using Python\n",
    "x1_star, = sym.solve(f, x1, implicit=True)\n",
    "x1_star =  sym.cancel(x1_star.subs({UGA(x1): UGA_star}))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.0*(UGA_star**2*epsilon*(0.5*P - 0.5*T) + UGA_star**2*(0.25*R - 0.25*S) + UGA_star*epsilon*(0.5*M - 1.0*P + 0.5*T) + UGA_star*(-0.25*M + 0.25*S) + epsilon*(-0.5*M + 0.5*P))/((1.0*epsilon + 0.5)*(-0.5*M + 0.5*P + UGA_star**2*(0.5*P + 0.5*R - 0.5*S - 0.5*T) + UGA_star*(-1.0*P + 0.5*S + 0.5*T)))"
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sym.factor(x1_star, UGA_star, epsilon)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.5/(1.0*epsilon + 0.5)"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sym.factor(x1_star.subs({UGA_star: 1}))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Setting the equation of motion equal to zero, rearranging the resulting equation yields the locus of potential equilibria (LPE). The LPE represents, for each possible $x_1$, the value that $U_{GA}$ would have to take in order for that $x_1$ to be an equilbrium. \n",
    "\n",
    "\\begin{align}\n",
    "  L(x^*) \\equiv x_1^* - 2\\left(\\frac{\\big(R - S\\big)U_{GA}^{*2} + \\big(S - M\\big)U_{GA}^* - 2\\epsilon\\big((T - P)U_{GA}^{*2} - (T - 2P)U_{GA}^* - P\\big)}{N(U_{GA}^*)(1 + 2\\epsilon)}\\right) = 0\\\\\n",
    "\\end{align}\n",
    "\n",
    "We already know that $x_1^*=0, U_{GA}^*=0$ and $x_1^*=1, U_{GA}^*=1$ are equilibria, so it follows that $L(0)=0$ and $L(1)=0$.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Differentiate $x_1^*$ with respect to $U_{GA}^*$ yields an expression that will be useful for assessing stability and invadability of an equilibrium."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-x1 + (x1*((-epsilon + 1)*(2*R*UGA(x1) + 2*S*(-UGA(x1) + 1))*UGA(x1) + 0.5*(2*P*(-UGA(x1) + 1) + 2*T*UGA(x1))*(-UGA(x1) + 1)) + (-x1 + 1)*(epsilon*(2*P*(-UGA(x1) + 1) + 2*T*UGA(x1))*(-UGA(x1) + 1) + 0.5*(2*R*UGA(x1) + 2*S*(-UGA(x1) + 1))*UGA(x1)))/(x1*(epsilon*(2*R*UGA(x1) + 2*S*(-UGA(x1) + 1))*UGA(x1) + 0.5*(2*P*(-UGA(x1) + 1) + 2*T*UGA(x1))*(-UGA(x1) + 1)) + x1*((-epsilon + 1)*(2*R*UGA(x1) + 2*S*(-UGA(x1) + 1))*UGA(x1) + 0.5*(2*P*(-UGA(x1) + 1) + 2*T*UGA(x1))*(-UGA(x1) + 1)) + (-x1 + 1)*(epsilon*(2*P*(-UGA(x1) + 1) + 2*T*UGA(x1))*(-UGA(x1) + 1) + 0.5*(2*R*UGA(x1) + 2*S*(-UGA(x1) + 1))*UGA(x1)) + (-x1 + 1)*((-epsilon + 1)*(2*P*(-UGA(x1) + 1) + 2*T*UGA(x1))*(-UGA(x1) + 1) + 0.5*(2*R*UGA(x1) + 2*S*(-UGA(x1) + 1))*UGA(x1)))"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 84,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[1.00000000000000, -(-P + S*d)/(P + R*d**2 - S*d - T*d)]"
      ]
     },
     "execution_count": 84,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "d = sym.symbols('d', positive=True)\n",
    "U = selection_functions.kirkpatrick_selection(x1, d)\n",
    "\n",
    "sym.solve(f.subs({UGA(x1): U, epsilon: 0}), x1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 85,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-1.5*(-0.333333333333333*P*epsilon + x1**3*(0.333333333333333*P*epsilon + 0.166666666666667*P + 0.333333333333333*R*d**2*epsilon + 0.166666666666667*R*d**2 - 0.333333333333333*S*d*epsilon - 0.166666666666667*S*d - 0.333333333333333*T*d*epsilon - 0.166666666666667*T*d) + x1**2*(-1.0*P*epsilon - 0.333333333333333*P - 0.166666666666667*R*d**2 + 0.333333333333333*S*d*epsilon + 0.333333333333333*S*d + 0.666666666666667*T*d*epsilon + 0.166666666666667*T*d) + x1*(1.0*P*epsilon + 0.166666666666667*P - 0.166666666666667*S*d - 0.333333333333333*T*d*epsilon))/(0.5*P + x1**2*(0.5*P + 0.5*R*d**2 - 0.5*S*d - 0.5*T*d) + x1*(-1.0*P + 0.5*S*d + 0.5*T*d))"
      ]
     },
     "execution_count": 85,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sym.factor(f.subs({UGA(x1): U}), x1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 86,
   "metadata": {},
   "outputs": [],
   "source": [
    "n, _ = sym.fraction(sym.factor(f.subs({UGA(x1): U}), x1))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 87,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4.0*(0.5*P + x1**2*(0.5*P + 0.5*R*d**2 - 0.5*S*d - 0.5*T*d) + x1*(-1.0*P + 0.5*S*d + 0.5*T*d))/(x1*(d - 1) + 1)**2"
      ]
     },
     "execution_count": 87,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sym.factor(N.subs({UGA(x1): U}), x1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 88,
   "metadata": {},
   "outputs": [],
   "source": [
    "p = sym.Poly(n, x1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 98,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[(-0.25*P**2*R**2*d**2 + 0.5*P**2*R*T*d - 0.25*P**2*T**2 + 0.5*P*R**2*S*d**3 - P*R*S*T*d**2 + 0.5*P*S*T**2*d - 0.25*R**2*S**2*d**4 + 0.5*R*S**2*T*d**3 - 0.25*S**2*T**2*d**2)/(-2.0*P**3*R + 5.0*P**2*R*S*d + P**2*S*T - 2.0*P*R**3*d**4 + 5.0*P*R**2*T*d**3 - 4.0*P*R*S**2*d**2 - 4.0*P*R*T**2*d**2 - 2.0*P*S**2*T*d + P*T**3*d + R**2*S*T*d**4 + R*S**3*d**3 - 2.0*R*S*T**2*d**3 + S**3*T*d**2 + S*T**3*d**2)]"
      ]
     },
     "execution_count": 98,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sym.solve(sym.factor(p.discriminant(), epsilon).subs({epsilon**2: 0}), epsilon)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 92,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.6875*d*(-1.0*P**2*R**2*d + 0.666666666666667*P*R*S*T*d + 0.148148148148148*P*S**3 + 0.148148148148148*R*T**3*d**2 + 0.037037037037037*S**2*T**2*d)"
      ]
     },
     "execution_count": 92,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sym.factor(sym.expand(1.6875 *(-1.0*P**2*R**2*d**2 + 0.666666666666667*P*R*S*T*d**2 + 0.148148148148148*P*S**3*d + 0.148148148148148*R*T**3*d**3 + 0.037037037037037*S**2*T**2*d**2)))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "metadata": {},
   "outputs": [],
   "source": [
    "roots = sym.solve(n, x1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 64,
   "metadata": {},
   "outputs": [
    {
     "ename": "KeyboardInterrupt",
     "evalue": "",
     "output_type": "error",
     "traceback": [
      "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[0;31mKeyboardInterrupt\u001b[0m                         Traceback (most recent call last)",
      "\u001b[0;32m<ipython-input-64-d97b1eb46418>\u001b[0m in \u001b[0;36m<module>\u001b[0;34m()\u001b[0m\n\u001b[0;32m----> 1\u001b[0;31m \u001b[0msym\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mfactor\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0msym\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mtogether\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mroots\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0;36m0\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mepsilon\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m",
      "\u001b[0;32m~/anaconda3/lib/python3.6/site-packages/sympy/polys/polytools.py\u001b[0m in \u001b[0;36mfactor\u001b[0;34m(f, *gens, **args)\u001b[0m\n\u001b[1;32m   6324\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   6325\u001b[0m     \u001b[0;32mtry\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 6326\u001b[0;31m         \u001b[0;32mreturn\u001b[0m \u001b[0m_generic_factor\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mf\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mgens\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0margs\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mmethod\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0;34m'factor'\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m   6327\u001b[0m     \u001b[0;32mexcept\u001b[0m \u001b[0mPolynomialError\u001b[0m \u001b[0;32mas\u001b[0m \u001b[0mmsg\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   6328\u001b[0m         \u001b[0;32mif\u001b[0m \u001b[0;32mnot\u001b[0m \u001b[0mf\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mis_commutative\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m~/anaconda3/lib/python3.6/site-packages/sympy/polys/polytools.py\u001b[0m in \u001b[0;36m_generic_factor\u001b[0;34m(expr, gens, args, method)\u001b[0m\n\u001b[1;32m   6017\u001b[0m     \u001b[0moptions\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mallowed_flags\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m[\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   6018\u001b[0m     \u001b[0mopt\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0moptions\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mbuild_options\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mgens\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0margs\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 6019\u001b[0;31m     \u001b[0;32mreturn\u001b[0m \u001b[0m_symbolic_factor\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0msympify\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mexpr\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mopt\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mmethod\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m   6020\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   6021\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m~/anaconda3/lib/python3.6/site-packages/sympy/polys/polytools.py\u001b[0m in \u001b[0;36m_symbolic_factor\u001b[0;34m(expr, opt, method)\u001b[0m\n\u001b[1;32m   5962\u001b[0m         \u001b[0;32mif\u001b[0m \u001b[0mhasattr\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mexpr\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0;34m'_eval_factor'\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   5963\u001b[0m             \u001b[0;32mreturn\u001b[0m \u001b[0mexpr\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_eval_factor\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 5964\u001b[0;31m         \u001b[0mcoeff\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mfactors\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0m_symbolic_factor_list\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mtogether\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mexpr\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mopt\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mmethod\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m   5965\u001b[0m         \u001b[0;32mreturn\u001b[0m \u001b[0m_keep_coeff\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mcoeff\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0m_factors_product\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mfactors\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   5966\u001b[0m     \u001b[0;32melif\u001b[0m \u001b[0mhasattr\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mexpr\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m'args'\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m~/anaconda3/lib/python3.6/site-packages/sympy/polys/polytools.py\u001b[0m in \u001b[0;36m_symbolic_factor_list\u001b[0;34m(expr, opt, method)\u001b[0m\n\u001b[1;32m   5924\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   5925\u001b[0m         \u001b[0;32mtry\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 5926\u001b[0;31m             \u001b[0mpoly\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0m_\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0m_poly_from_expr\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mbase\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mopt\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m   5927\u001b[0m         \u001b[0;32mexcept\u001b[0m \u001b[0mPolificationFailed\u001b[0m \u001b[0;32mas\u001b[0m \u001b[0mexc\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   5928\u001b[0m             \u001b[0mfactors\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mappend\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mexc\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mexpr\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mexp\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m~/anaconda3/lib/python3.6/site-packages/sympy/polys/polytools.py\u001b[0m in \u001b[0;36m_poly_from_expr\u001b[0;34m(expr, opt)\u001b[0m\n\u001b[1;32m   4284\u001b[0m         \u001b[0;32mreturn\u001b[0m \u001b[0mpoly\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mopt\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   4285\u001b[0m     \u001b[0;32melif\u001b[0m \u001b[0mopt\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mexpand\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 4286\u001b[0;31m         \u001b[0mexpr\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mexpr\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mexpand\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m   4287\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   4288\u001b[0m     \u001b[0mrep\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mopt\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0m_dict_from_expr\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mexpr\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mopt\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m~/anaconda3/lib/python3.6/site-packages/sympy/core/expr.py\u001b[0m in \u001b[0;36mexpand\u001b[0;34m(self, deep, modulus, power_base, power_exp, mul, log, multinomial, basic, **hints)\u001b[0m\n\u001b[1;32m   3131\u001b[0m             \u001b[0;32mif\u001b[0m \u001b[0mhints\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mget\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m'mul'\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;32mFalse\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   3132\u001b[0m                 expr, _ = Expr._expand_hint(\n\u001b[0;32m-> 3133\u001b[0;31m                     expr, '_eval_expand_mul', deep=deep, **hints)\n\u001b[0m\u001b[1;32m   3134\u001b[0m             \u001b[0;32mif\u001b[0m \u001b[0mhints\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mget\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m'log'\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;32mFalse\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   3135\u001b[0m                 expr, _ = Expr._expand_hint(\n",
      "\u001b[0;32m~/anaconda3/lib/python3.6/site-packages/sympy/core/expr.py\u001b[0m in \u001b[0;36m_expand_hint\u001b[0;34m(expr, hint, deep, **hints)\u001b[0m\n\u001b[1;32m   3055\u001b[0m             \u001b[0msargs\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0;34m[\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   3056\u001b[0m             \u001b[0;32mfor\u001b[0m \u001b[0marg\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mexpr\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 3057\u001b[0;31m                 \u001b[0marg\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0marghit\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mExpr\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_expand_hint\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0marg\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mhint\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m**\u001b[0m\u001b[0mhints\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m   3058\u001b[0m                 \u001b[0mhit\u001b[0m \u001b[0;34m|=\u001b[0m \u001b[0marghit\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   3059\u001b[0m                 \u001b[0msargs\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mappend\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0marg\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m~/anaconda3/lib/python3.6/site-packages/sympy/core/expr.py\u001b[0m in \u001b[0;36m_expand_hint\u001b[0;34m(expr, hint, deep, **hints)\u001b[0m\n\u001b[1;32m   3055\u001b[0m             \u001b[0msargs\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0;34m[\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   3056\u001b[0m             \u001b[0;32mfor\u001b[0m \u001b[0marg\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mexpr\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 3057\u001b[0;31m                 \u001b[0marg\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0marghit\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mExpr\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_expand_hint\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0marg\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mhint\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m**\u001b[0m\u001b[0mhints\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m   3058\u001b[0m                 \u001b[0mhit\u001b[0m \u001b[0;34m|=\u001b[0m \u001b[0marghit\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   3059\u001b[0m                 \u001b[0msargs\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mappend\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0marg\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m~/anaconda3/lib/python3.6/site-packages/sympy/core/expr.py\u001b[0m in \u001b[0;36m_expand_hint\u001b[0;34m(expr, hint, deep, **hints)\u001b[0m\n\u001b[1;32m   3055\u001b[0m             \u001b[0msargs\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0;34m[\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   3056\u001b[0m             \u001b[0;32mfor\u001b[0m \u001b[0marg\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mexpr\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 3057\u001b[0;31m                 \u001b[0marg\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0marghit\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mExpr\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_expand_hint\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0marg\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mhint\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m**\u001b[0m\u001b[0mhints\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m   3058\u001b[0m                 \u001b[0mhit\u001b[0m \u001b[0;34m|=\u001b[0m \u001b[0marghit\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   3059\u001b[0m                 \u001b[0msargs\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mappend\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0marg\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m~/anaconda3/lib/python3.6/site-packages/sympy/core/expr.py\u001b[0m in \u001b[0;36m_expand_hint\u001b[0;34m(expr, hint, deep, **hints)\u001b[0m\n\u001b[1;32m   3055\u001b[0m             \u001b[0msargs\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0;34m[\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   3056\u001b[0m             \u001b[0;32mfor\u001b[0m \u001b[0marg\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mexpr\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 3057\u001b[0;31m                 \u001b[0marg\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0marghit\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mExpr\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_expand_hint\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0marg\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mhint\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m**\u001b[0m\u001b[0mhints\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m   3058\u001b[0m                 \u001b[0mhit\u001b[0m \u001b[0;34m|=\u001b[0m \u001b[0marghit\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   3059\u001b[0m                 \u001b[0msargs\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mappend\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0marg\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m~/anaconda3/lib/python3.6/site-packages/sympy/core/expr.py\u001b[0m in \u001b[0;36m_expand_hint\u001b[0;34m(expr, hint, deep, **hints)\u001b[0m\n\u001b[1;32m   3063\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   3064\u001b[0m         \u001b[0;32mif\u001b[0m \u001b[0mhasattr\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mexpr\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mhint\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 3065\u001b[0;31m             \u001b[0mnewexpr\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mgetattr\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mexpr\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mhint\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m**\u001b[0m\u001b[0mhints\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m   3066\u001b[0m             \u001b[0;32mif\u001b[0m \u001b[0mnewexpr\u001b[0m \u001b[0;34m!=\u001b[0m \u001b[0mexpr\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   3067\u001b[0m                 \u001b[0;32mreturn\u001b[0m \u001b[0;34m(\u001b[0m\u001b[0mnewexpr\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;32mTrue\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m~/anaconda3/lib/python3.6/site-packages/sympy/core/mul.py\u001b[0m in \u001b[0;36m_eval_expand_mul\u001b[0;34m(self, **hints)\u001b[0m\n\u001b[1;32m    882\u001b[0m             \u001b[0;32mif\u001b[0m \u001b[0msums\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    883\u001b[0m                 \u001b[0mdeep\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mhints\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mget\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m\"deep\"\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;32mFalse\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 884\u001b[0;31m                 \u001b[0mterms\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mfunc\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_expandsums\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0msums\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    885\u001b[0m                 \u001b[0margs\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0;34m[\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    886\u001b[0m                 \u001b[0;32mfor\u001b[0m \u001b[0mterm\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mterms\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m~/anaconda3/lib/python3.6/site-packages/sympy/core/mul.py\u001b[0m in \u001b[0;36m_expandsums\u001b[0;34m(sums)\u001b[0m\n\u001b[1;32m    845\u001b[0m         \u001b[0mterms\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0;34m[\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    846\u001b[0m         \u001b[0mleft\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mMul\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_expandsums\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0msums\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0mL\u001b[0m\u001b[0;34m//\u001b[0m\u001b[0;36m2\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 847\u001b[0;31m         \u001b[0mright\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mMul\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_expandsums\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0msums\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0mL\u001b[0m\u001b[0;34m//\u001b[0m\u001b[0;36m2\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    848\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    849\u001b[0m         \u001b[0mterms\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0;34m[\u001b[0m\u001b[0mMul\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0ma\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mb\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;32mfor\u001b[0m \u001b[0ma\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mleft\u001b[0m \u001b[0;32mfor\u001b[0m \u001b[0mb\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mright\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m~/anaconda3/lib/python3.6/site-packages/sympy/core/mul.py\u001b[0m in \u001b[0;36m_expandsums\u001b[0;34m(sums)\u001b[0m\n\u001b[1;32m    847\u001b[0m         \u001b[0mright\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mMul\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_expandsums\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0msums\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0mL\u001b[0m\u001b[0;34m//\u001b[0m\u001b[0;36m2\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    848\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 849\u001b[0;31m         \u001b[0mterms\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0;34m[\u001b[0m\u001b[0mMul\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0ma\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mb\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;32mfor\u001b[0m \u001b[0ma\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mleft\u001b[0m \u001b[0;32mfor\u001b[0m \u001b[0mb\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mright\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    850\u001b[0m         \u001b[0madded\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mAdd\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0mterms\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    851\u001b[0m         \u001b[0;32mreturn\u001b[0m \u001b[0mAdd\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mmake_args\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0madded\u001b[0m\u001b[0;34m)\u001b[0m  \u001b[0;31m# it may have collapsed down to one term\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m~/anaconda3/lib/python3.6/site-packages/sympy/core/mul.py\u001b[0m in \u001b[0;36m<listcomp>\u001b[0;34m(.0)\u001b[0m\n\u001b[1;32m    847\u001b[0m         \u001b[0mright\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mMul\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_expandsums\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0msums\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0mL\u001b[0m\u001b[0;34m//\u001b[0m\u001b[0;36m2\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    848\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 849\u001b[0;31m         \u001b[0mterms\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0;34m[\u001b[0m\u001b[0mMul\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0ma\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mb\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;32mfor\u001b[0m \u001b[0ma\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mleft\u001b[0m \u001b[0;32mfor\u001b[0m \u001b[0mb\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mright\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    850\u001b[0m         \u001b[0madded\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mAdd\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0mterms\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    851\u001b[0m         \u001b[0;32mreturn\u001b[0m \u001b[0mAdd\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mmake_args\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0madded\u001b[0m\u001b[0;34m)\u001b[0m  \u001b[0;31m# it may have collapsed down to one term\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m~/anaconda3/lib/python3.6/site-packages/sympy/core/operations.py\u001b[0m in \u001b[0;36m__new__\u001b[0;34m(cls, *args, **options)\u001b[0m\n\u001b[1;32m     43\u001b[0m             \u001b[0;32mreturn\u001b[0m \u001b[0margs\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0;36m0\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     44\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m---> 45\u001b[0;31m         \u001b[0mc_part\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mnc_part\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0morder_symbols\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mcls\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mflatten\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m     46\u001b[0m         \u001b[0mis_commutative\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0;32mnot\u001b[0m \u001b[0mnc_part\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     47\u001b[0m         \u001b[0mobj\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mcls\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_from_args\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mc_part\u001b[0m \u001b[0;34m+\u001b[0m \u001b[0mnc_part\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mis_commutative\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m~/anaconda3/lib/python3.6/site-packages/sympy/core/mul.py\u001b[0m in \u001b[0;36mflatten\u001b[0;34m(cls, seq)\u001b[0m\n\u001b[1;32m    611\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    612\u001b[0m         \u001b[0;31m# order commutative part canonically\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 613\u001b[0;31m         \u001b[0m_mulsort\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mc_part\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    614\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    615\u001b[0m         \u001b[0;31m# current code expects coeff to be always in slot-0\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m~/anaconda3/lib/python3.6/site-packages/sympy/core/mul.py\u001b[0m in \u001b[0;36m_mulsort\u001b[0;34m(args)\u001b[0m\n\u001b[1;32m     32\u001b[0m \u001b[0;32mdef\u001b[0m \u001b[0m_mulsort\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     33\u001b[0m     \u001b[0;31m# in-place sorting of args\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m---> 34\u001b[0;31m     \u001b[0margs\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0msort\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mkey\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0m_args_sortkey\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m     35\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     36\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m~/anaconda3/lib/python3.6/site-packages/sympy/core/basic.py\u001b[0m in \u001b[0;36mcompare\u001b[0;34m(self, other)\u001b[0m\n\u001b[1;32m    207\u001b[0m             \u001b[0mr\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mBasic\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0mr\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;32mif\u001b[0m \u001b[0misinstance\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mr\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mfrozenset\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;32melse\u001b[0m \u001b[0mr\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    208\u001b[0m             \u001b[0;32mif\u001b[0m \u001b[0misinstance\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0ml\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mBasic\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 209\u001b[0;31m                 \u001b[0mc\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0ml\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mcompare\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mr\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    210\u001b[0m             \u001b[0;32melse\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    211\u001b[0m                 \u001b[0mc\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0;34m(\u001b[0m\u001b[0ml\u001b[0m \u001b[0;34m>\u001b[0m \u001b[0mr\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;34m-\u001b[0m \u001b[0;34m(\u001b[0m\u001b[0ml\u001b[0m \u001b[0;34m<\u001b[0m \u001b[0mr\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m~/anaconda3/lib/python3.6/site-packages/sympy/core/basic.py\u001b[0m in \u001b[0;36mcompare\u001b[0;34m(self, other)\u001b[0m\n\u001b[1;32m    198\u001b[0m             \u001b[0;32mreturn\u001b[0m \u001b[0mc\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    199\u001b[0m         \u001b[0;31m#\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 200\u001b[0;31m         \u001b[0mst\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_hashable_content\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    201\u001b[0m         \u001b[0mot\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mother\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_hashable_content\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    202\u001b[0m         \u001b[0mc\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0;34m(\u001b[0m\u001b[0mlen\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mst\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;34m>\u001b[0m \u001b[0mlen\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mot\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;34m-\u001b[0m \u001b[0;34m(\u001b[0m\u001b[0mlen\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mst\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;34m<\u001b[0m \u001b[0mlen\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mot\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m~/anaconda3/lib/python3.6/site-packages/sympy/core/symbol.py\u001b[0m in \u001b[0;36m_hashable_content\u001b[0;34m(self)\u001b[0m\n\u001b[1;32m    250\u001b[0m     \u001b[0;32mdef\u001b[0m \u001b[0m_hashable_content\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    251\u001b[0m         \u001b[0;31m# Note: user-specified assumptions not hashed, just derived ones\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 252\u001b[0;31m         \u001b[0;32mreturn\u001b[0m \u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mname\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;34m+\u001b[0m \u001b[0mtuple\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0msorted\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0massumptions0\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mitems\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    253\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    254\u001b[0m     \u001b[0;34m@\u001b[0m\u001b[0mproperty\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;31mKeyboardInterrupt\u001b[0m: "
     ]
    }
   ],
   "source": [
    "sym.factor(sym.together(roots[0]), epsilon)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 67,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-0.333333333333333*(-3.0*(6.0*P*epsilon + P - S*d - 2.0*T*d*epsilon)/(2.0*P*epsilon + P + 2.0*R*d**2*epsilon + R*d**2 - 2.0*S*d*epsilon - S*d - 2.0*T*d*epsilon - T*d) + (-6.0*P*epsilon - 2.0*P - R*d**2 + 2.0*S*d*epsilon + 2.0*S*d + 4.0*T*d*epsilon + T*d)**2/(2.0*P*epsilon + P + 2.0*R*d**2*epsilon + R*d**2 - 2.0*S*d*epsilon - S*d - 2.0*T*d*epsilon - T*d)**2)/(-27.0*P*epsilon/(2.0*P*epsilon + P + 2.0*R*d**2*epsilon + R*d**2 - 2.0*S*d*epsilon - S*d - 2.0*T*d*epsilon - T*d) + 0.5*(-4.0*(-3.0*(6.0*P*epsilon + P - S*d - 2.0*T*d*epsilon)/(2.0*P*epsilon + P + 2.0*R*d**2*epsilon + R*d**2 - 2.0*S*d*epsilon - S*d - 2.0*T*d*epsilon - T*d) + (-6.0*P*epsilon - 2.0*P - R*d**2 + 2.0*S*d*epsilon + 2.0*S*d + 4.0*T*d*epsilon + T*d)**2/(2.0*P*epsilon + P + 2.0*R*d**2*epsilon + R*d**2 - 2.0*S*d*epsilon - S*d - 2.0*T*d*epsilon - T*d)**2)**3 + (-54.0*P*epsilon/(2.0*P*epsilon + P + 2.0*R*d**2*epsilon + R*d**2 - 2.0*S*d*epsilon - S*d - 2.0*T*d*epsilon - T*d) - 9.0*(6.0*P*epsilon + P - S*d - 2.0*T*d*epsilon)*(-6.0*P*epsilon - 2.0*P - R*d**2 + 2.0*S*d*epsilon + 2.0*S*d + 4.0*T*d*epsilon + T*d)/(2.0*P*epsilon + P + 2.0*R*d**2*epsilon + R*d**2 - 2.0*S*d*epsilon - S*d - 2.0*T*d*epsilon - T*d)**2 + 2.0*(-6.0*P*epsilon - 2.0*P - R*d**2 + 2.0*S*d*epsilon + 2.0*S*d + 4.0*T*d*epsilon + T*d)**3/(2.0*P*epsilon + P + 2.0*R*d**2*epsilon + R*d**2 - 2.0*S*d*epsilon - S*d - 2.0*T*d*epsilon - T*d)**3)**2)**0.5 - 4.5*(6.0*P*epsilon + P - S*d - 2.0*T*d*epsilon)*(-6.0*P*epsilon - 2.0*P - R*d**2 + 2.0*S*d*epsilon + 2.0*S*d + 4.0*T*d*epsilon + T*d)/(2.0*P*epsilon + P + 2.0*R*d**2*epsilon + R*d**2 - 2.0*S*d*epsilon - S*d - 2.0*T*d*epsilon - T*d)**2 + (-6.0*P*epsilon - 2.0*P - R*d**2 + 2.0*S*d*epsilon + 2.0*S*d + 4.0*T*d*epsilon + T*d)**3/(2.0*P*epsilon + P + 2.0*R*d**2*epsilon + R*d**2 - 2.0*S*d*epsilon - S*d - 2.0*T*d*epsilon - T*d)**3)**(1/3) - 0.333333333333333*(-27.0*P*epsilon/(2.0*P*epsilon + P + 2.0*R*d**2*epsilon + R*d**2 - 2.0*S*d*epsilon - S*d - 2.0*T*d*epsilon - T*d) + 0.5*(-4.0*(-3.0*(6.0*P*epsilon + P - S*d - 2.0*T*d*epsilon)/(2.0*P*epsilon + P + 2.0*R*d**2*epsilon + R*d**2 - 2.0*S*d*epsilon - S*d - 2.0*T*d*epsilon - T*d) + (-6.0*P*epsilon - 2.0*P - R*d**2 + 2.0*S*d*epsilon + 2.0*S*d + 4.0*T*d*epsilon + T*d)**2/(2.0*P*epsilon + P + 2.0*R*d**2*epsilon + R*d**2 - 2.0*S*d*epsilon - S*d - 2.0*T*d*epsilon - T*d)**2)**3 + (-54.0*P*epsilon/(2.0*P*epsilon + P + 2.0*R*d**2*epsilon + R*d**2 - 2.0*S*d*epsilon - S*d - 2.0*T*d*epsilon - T*d) - 9.0*(6.0*P*epsilon + P - S*d - 2.0*T*d*epsilon)*(-6.0*P*epsilon - 2.0*P - R*d**2 + 2.0*S*d*epsilon + 2.0*S*d + 4.0*T*d*epsilon + T*d)/(2.0*P*epsilon + P + 2.0*R*d**2*epsilon + R*d**2 - 2.0*S*d*epsilon - S*d - 2.0*T*d*epsilon - T*d)**2 + 2.0*(-6.0*P*epsilon - 2.0*P - R*d**2 + 2.0*S*d*epsilon + 2.0*S*d + 4.0*T*d*epsilon + T*d)**3/(2.0*P*epsilon + P + 2.0*R*d**2*epsilon + R*d**2 - 2.0*S*d*epsilon - S*d - 2.0*T*d*epsilon - T*d)**3)**2)**0.5 - 4.5*(6.0*P*epsilon + P - S*d - 2.0*T*d*epsilon)*(-6.0*P*epsilon - 2.0*P - R*d**2 + 2.0*S*d*epsilon + 2.0*S*d + 4.0*T*d*epsilon + T*d)/(2.0*P*epsilon + P + 2.0*R*d**2*epsilon + R*d**2 - 2.0*S*d*epsilon - S*d - 2.0*T*d*epsilon - T*d)**2 + (-6.0*P*epsilon - 2.0*P - R*d**2 + 2.0*S*d*epsilon + 2.0*S*d + 4.0*T*d*epsilon + T*d)**3/(2.0*P*epsilon + P + 2.0*R*d**2*epsilon + R*d**2 - 2.0*S*d*epsilon - S*d - 2.0*T*d*epsilon - T*d)**3)**(1/3) - 0.333333333333333*(-6.0*P*epsilon - 2.0*P - R*d**2 + 2.0*S*d*epsilon + 2.0*S*d + 4.0*T*d*epsilon + T*d)/(2.0*P*epsilon + P + 2.0*R*d**2*epsilon + R*d**2 - 2.0*S*d*epsilon - S*d - 2.0*T*d*epsilon - T*d)"
      ]
     },
     "execution_count": 67,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "roots[0]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 66,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(-0.333333333333333*((-27.0*P*epsilon*(2.0*P*epsilon + P + 2.0*R*d**2*epsilon + R*d**2 - 2.0*S*d*epsilon - S*d - 2.0*T*d*epsilon - T*d)**2 + 0.5*(-4.0*(-3.0*(6.0*P*epsilon + P - S*d - 2.0*T*d*epsilon)*(2.0*P*epsilon + P + 2.0*R*d**2*epsilon + R*d**2 - 2.0*S*d*epsilon - S*d - 2.0*T*d*epsilon - T*d) + (-6.0*P*epsilon - 2.0*P - R*d**2 + 2.0*S*d*epsilon + 2.0*S*d + 4.0*T*d*epsilon + T*d)**2)**3 + (-54.0*P*epsilon*(2.0*P*epsilon + P + 2.0*R*d**2*epsilon + R*d**2 - 2.0*S*d*epsilon - S*d - 2.0*T*d*epsilon - T*d)**2 - 9.0*(6.0*P*epsilon + P - S*d - 2.0*T*d*epsilon)*(-6.0*P*epsilon - 2.0*P - R*d**2 + 2.0*S*d*epsilon + 2.0*S*d + 4.0*T*d*epsilon + T*d)*(2.0*P*epsilon + P + 2.0*R*d**2*epsilon + R*d**2 - 2.0*S*d*epsilon - S*d - 2.0*T*d*epsilon - T*d) + 2.0*(-6.0*P*epsilon - 2.0*P - R*d**2 + 2.0*S*d*epsilon + 2.0*S*d + 4.0*T*d*epsilon + T*d)**3)**2)**0.5*(2.0*P*epsilon + P + 2.0*R*d**2*epsilon + R*d**2 - 2.0*S*d*epsilon - S*d - 2.0*T*d*epsilon - T*d)**3*Abs(2.0*P*epsilon + P + 2.0*R*d**2*epsilon + R*d**2 - 2.0*S*d*epsilon - S*d - 2.0*T*d*epsilon - T*d)**(-3.0) - 4.5*(6.0*P*epsilon + P - S*d - 2.0*T*d*epsilon)*(-6.0*P*epsilon - 2.0*P - R*d**2 + 2.0*S*d*epsilon + 2.0*S*d + 4.0*T*d*epsilon + T*d)*(2.0*P*epsilon + P + 2.0*R*d**2*epsilon + R*d**2 - 2.0*S*d*epsilon - S*d - 2.0*T*d*epsilon - T*d) + (-6.0*P*epsilon - 2.0*P - R*d**2 + 2.0*S*d*epsilon + 2.0*S*d + 4.0*T*d*epsilon + T*d)**3)/(2.0*P*epsilon + P + 2.0*R*d**2*epsilon + R*d**2 - 2.0*S*d*epsilon - S*d - 2.0*T*d*epsilon - T*d)**3)**(2/3)*(-0.5 + 0.866025403784439*I)*(2.0*P*epsilon + P + 2.0*R*d**2*epsilon + R*d**2 - 2.0*S*d*epsilon - S*d - 2.0*T*d*epsilon - T*d)**2 - 0.333333333333333*((-27.0*P*epsilon*(2.0*P*epsilon + P + 2.0*R*d**2*epsilon + R*d**2 - 2.0*S*d*epsilon - S*d - 2.0*T*d*epsilon - T*d)**2 + 0.5*(-4.0*(-3.0*(6.0*P*epsilon + P - S*d - 2.0*T*d*epsilon)*(2.0*P*epsilon + P + 2.0*R*d**2*epsilon + R*d**2 - 2.0*S*d*epsilon - S*d - 2.0*T*d*epsilon - T*d) + (-6.0*P*epsilon - 2.0*P - R*d**2 + 2.0*S*d*epsilon + 2.0*S*d + 4.0*T*d*epsilon + T*d)**2)**3 + (-54.0*P*epsilon*(2.0*P*epsilon + P + 2.0*R*d**2*epsilon + R*d**2 - 2.0*S*d*epsilon - S*d - 2.0*T*d*epsilon - T*d)**2 - 9.0*(6.0*P*epsilon + P - S*d - 2.0*T*d*epsilon)*(-6.0*P*epsilon - 2.0*P - R*d**2 + 2.0*S*d*epsilon + 2.0*S*d + 4.0*T*d*epsilon + T*d)*(2.0*P*epsilon + P + 2.0*R*d**2*epsilon + R*d**2 - 2.0*S*d*epsilon - S*d - 2.0*T*d*epsilon - T*d) + 2.0*(-6.0*P*epsilon - 2.0*P - R*d**2 + 2.0*S*d*epsilon + 2.0*S*d + 4.0*T*d*epsilon + T*d)**3)**2)**0.5*(2.0*P*epsilon + P + 2.0*R*d**2*epsilon + R*d**2 - 2.0*S*d*epsilon - S*d - 2.0*T*d*epsilon - T*d)**3*Abs(2.0*P*epsilon + P + 2.0*R*d**2*epsilon + R*d**2 - 2.0*S*d*epsilon - S*d - 2.0*T*d*epsilon - T*d)**(-3.0) - 4.5*(6.0*P*epsilon + P - S*d - 2.0*T*d*epsilon)*(-6.0*P*epsilon - 2.0*P - R*d**2 + 2.0*S*d*epsilon + 2.0*S*d + 4.0*T*d*epsilon + T*d)*(2.0*P*epsilon + P + 2.0*R*d**2*epsilon + R*d**2 - 2.0*S*d*epsilon - S*d - 2.0*T*d*epsilon - T*d) + (-6.0*P*epsilon - 2.0*P - R*d**2 + 2.0*S*d*epsilon + 2.0*S*d + 4.0*T*d*epsilon + T*d)**3)/(2.0*P*epsilon + P + 2.0*R*d**2*epsilon + R*d**2 - 2.0*S*d*epsilon - S*d - 2.0*T*d*epsilon - T*d)**3)**(1/3)*(-6.0*P*epsilon - 2.0*P - R*d**2 + 2.0*S*d*epsilon + 2.0*S*d + 4.0*T*d*epsilon + T*d)*(2.0*P*epsilon + P + 2.0*R*d**2*epsilon + R*d**2 - 2.0*S*d*epsilon - S*d - 2.0*T*d*epsilon - T*d) - 0.333333333333333*(-0.5 - 0.866025403784439*I)*(-3.0*(6.0*P*epsilon + P - S*d - 2.0*T*d*epsilon)*(2.0*P*epsilon + P + 2.0*R*d**2*epsilon + R*d**2 - 2.0*S*d*epsilon - S*d - 2.0*T*d*epsilon - T*d) + (-6.0*P*epsilon - 2.0*P - R*d**2 + 2.0*S*d*epsilon + 2.0*S*d + 4.0*T*d*epsilon + T*d)**2))/(((-27.0*P*epsilon*(2.0*P*epsilon + P + 2.0*R*d**2*epsilon + R*d**2 - 2.0*S*d*epsilon - S*d - 2.0*T*d*epsilon - T*d)**2 + 0.5*(-4.0*(-3.0*(6.0*P*epsilon + P - S*d - 2.0*T*d*epsilon)*(2.0*P*epsilon + P + 2.0*R*d**2*epsilon + R*d**2 - 2.0*S*d*epsilon - S*d - 2.0*T*d*epsilon - T*d) + (-6.0*P*epsilon - 2.0*P - R*d**2 + 2.0*S*d*epsilon + 2.0*S*d + 4.0*T*d*epsilon + T*d)**2)**3 + (-54.0*P*epsilon*(2.0*P*epsilon + P + 2.0*R*d**2*epsilon + R*d**2 - 2.0*S*d*epsilon - S*d - 2.0*T*d*epsilon - T*d)**2 - 9.0*(6.0*P*epsilon + P - S*d - 2.0*T*d*epsilon)*(-6.0*P*epsilon - 2.0*P - R*d**2 + 2.0*S*d*epsilon + 2.0*S*d + 4.0*T*d*epsilon + T*d)*(2.0*P*epsilon + P + 2.0*R*d**2*epsilon + R*d**2 - 2.0*S*d*epsilon - S*d - 2.0*T*d*epsilon - T*d) + 2.0*(-6.0*P*epsilon - 2.0*P - R*d**2 + 2.0*S*d*epsilon + 2.0*S*d + 4.0*T*d*epsilon + T*d)**3)**2)**0.5*(2.0*P*epsilon + P + 2.0*R*d**2*epsilon + R*d**2 - 2.0*S*d*epsilon - S*d - 2.0*T*d*epsilon - T*d)**3*Abs(2.0*P*epsilon + P + 2.0*R*d**2*epsilon + R*d**2 - 2.0*S*d*epsilon - S*d - 2.0*T*d*epsilon - T*d)**(-3.0) - 4.5*(6.0*P*epsilon + P - S*d - 2.0*T*d*epsilon)*(-6.0*P*epsilon - 2.0*P - R*d**2 + 2.0*S*d*epsilon + 2.0*S*d + 4.0*T*d*epsilon + T*d)*(2.0*P*epsilon + P + 2.0*R*d**2*epsilon + R*d**2 - 2.0*S*d*epsilon - S*d - 2.0*T*d*epsilon - T*d) + (-6.0*P*epsilon - 2.0*P - R*d**2 + 2.0*S*d*epsilon + 2.0*S*d + 4.0*T*d*epsilon + T*d)**3)/(2.0*P*epsilon + P + 2.0*R*d**2*epsilon + R*d**2 - 2.0*S*d*epsilon - S*d - 2.0*T*d*epsilon - T*d)**3)**(1/3)*(2.0*P*epsilon + P + 2.0*R*d**2*epsilon + R*d**2 - 2.0*S*d*epsilon - S*d - 2.0*T*d*epsilon - T*d)**2)"
      ]
     },
     "execution_count": 66,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sym.expand(roots[1])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "dx1star_dUGA_star = sym.factor(sym.cancel(sym.diff(x1_star, UGA_star, 1)))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.0*(2.0*epsilon - 1.0)*(2.0*P*R*UGA_star**2 - 2.0*P*R*UGA_star - 1.0*P*S*UGA_star**2 + 2.0*P*S*UGA_star - 1.0*P*S - 1.0*R*T*UGA_star**2)/((2.0*epsilon + 1.0)*(1.0*P*UGA_star**2 - 2.0*P*UGA_star + 1.0*P + 1.0*R*UGA_star**2 - 1.0*S*UGA_star**2 + 1.0*S*UGA_star - 1.0*T*UGA_star**2 + 1.0*T*UGA_star)**2)"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "dx1star_dUGA_star"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "optimal_x1_star = sym.factor(sym.cancel(x1_star.subs({UGA_star: optimal_UGA_star})))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.0*(2.0*P**2*R + 4.0*P*R**2*epsilon - 4.0*P*R*S*epsilon - 2.0*P*R*T + 1.0*P*S**2*epsilon - 1.0*P*S**2 - 1.0*P*S*T - 1.0*P*T**2*epsilon - 0.5*R*S**2 - 2.0*R*S*T*epsilon - 2.0*R*T**2*epsilon + 0.5*R*T**2 + 0.5*S**3 + 1.0*S**2*T*epsilon + 1.0*S**2*T + 2.0*S*T**2*epsilon + 0.5*S*T**2 + 1.0*T**3*epsilon)/((1.0*epsilon + 0.5)*(0.5*P + 0.5*R - 0.5*S - 0.5*T)*(2.0*P*R - 0.5*S**2 - 1.0*S*T - 0.5*T**2))"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "optimal_x1_star"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Interactive plot of the locus of potential equilibria"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [],
   "source": [
    "_locus_of_potential_equilibria = sym.lambdify((UGA_star, T, R, P, S, epsilon),\n",
    "                                              x1_star,\n",
    "                                              modules=\"numpy\")\n",
    "\n",
    "\n",
    "def plot_locus_of_potential_equilibria(selection_function, d1, T, R, P, S, epsilon):\n",
    "    assert T > R > P > S\n",
    "    us = np.linspace(0, 1, 100)\n",
    "    xs = _locus_of_potential_equilibria(us, T, R, P, S, epsilon)\n",
    "\n",
    "    fig, ax = plt.subplots(1, 1, figsize=(10, 8))\n",
    "    ax.plot(xs, us, label=\"LPE\")\n",
    "    ax.set_xlabel(r\"$x_1^*$\", fontsize=25)\n",
    "    ax.set_ylabel(r\"$U_{GA}^*$\", rotation=\"horizontal\", fontsize=25)\n",
    "    ax.plot(us, us, 'k--')\n",
    "    \n",
    "    # create the selection functions\n",
    "    if selection_function == \"kirkpatrick\":\n",
    "        UGA = lambda x_A: selection_functions.kirkpatrick_selection(x_A, d1)\n",
    "    elif selection_function == \"seger\":\n",
    "        UGA = lambda x_A: selection_functions.seger_selection(x_A, d1)\n",
    "    elif selection_function == \"wright\":\n",
    "        UGA = lambda x_A: selection_functions.wright_selection(x_A, d1)\n",
    "    else:\n",
    "        valid_funcs = (\"kirkpatrick\", \"seger\", \"wright\")\n",
    "        msg = \"Selection_function must be one of {}, {}, or {}.\".format(*valid_funcs)\n",
    "        raise ValueError(msg)\n",
    "        \n",
    "    # add selection functions for comparison\n",
    "    ax.plot(xs, UGA(xs), label=selection_function)\n",
    "    ax.legend()\n",
    "    \n",
    "    plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "eb7b656cc14f4f73b580dade23b8dcaf",
       "version_major": 2,
       "version_minor": 0
      },
      "text/plain": [
       "interactive(children=(Dropdown(description='$U_{\\\\gamma(j)A}$', options=('kirkpatrick', 'seger', 'wright'), va…"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "#mpld3.enable_notebook()\n",
    "\n",
    "# sliders used to control the Prisoner's Dilemma Payoffs\n",
    "T_slider = widgets.FloatSlider(value=25, min=0, max=100, step=0.1, description=r\"$T$\")\n",
    "R_slider = widgets.FloatSlider(value=3, min=0, max=100, step=0.1, description=r\"$R$\")\n",
    "P_slider = widgets.FloatSlider(value=2, min=0, max=100, step=0.1, description=r\"$P$\")\n",
    "S_slider = widgets.FloatSlider(value=1, min=0, max=100, step=0.1, description=r\"$S$\")\n",
    "\n",
    "# slider used to control metabolic costs\n",
    "e_slider = widgets.FloatSlider(value=0, min=0, max=1, step=1e-3, description=r\"$\\epsilon$\", readout_format=\".3f\")\n",
    "\n",
    "# slider used to control which selection function is being used\n",
    "UGA_slider = widgets.Dropdown(options=[\"kirkpatrick\", \"seger\", \"wright\"], index=0, description=r\"$U_{\\gamma(j)A}$\")\n",
    "\n",
    "# slider that controls the parameters of the selection function\n",
    "d1_slider = widgets.FloatSlider(value=1.0, min=0.0, max=10, step=0.05, description=r\"$d_1$\")\n",
    "\n",
    "w = widgets.interactive(plot_locus_of_potential_equilibria, selection_function=UGA_slider, d1=d1_slider,\n",
    "                        T=T_slider, R=R_slider, P=P_slider, S=S_slider, epsilon=e_slider)\n",
    "display(w)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Stability\n",
    "\n",
    "Necessary and sufficient conditions for stability require that the derivative of the equation of motion with respect to $x_1$ be strictly negative when evaluated at the potential equilibrium."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 111,
   "metadata": {},
   "outputs": [],
   "source": [
    "f_jac = f.diff(x1, 1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 112,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(x1*((-epsilon + 1)*(2*R*UGA(x1) + 2*S*(-UGA(x1) + 1))*UGA(x1) + 0.5*(2*P*(-UGA(x1) + 1) + 2*T*UGA(x1))*(-UGA(x1) + 1)) + (-x1 + 1)*(epsilon*(2*P*(-UGA(x1) + 1) + 2*T*UGA(x1))*(-UGA(x1) + 1) + 0.5*(2*R*UGA(x1) + 2*S*(-UGA(x1) + 1))*UGA(x1)))*(epsilon*(2*P*(-UGA(x1) + 1) + 2*T*UGA(x1))*(-UGA(x1) + 1) - epsilon*(2*R*UGA(x1) + 2*S*(-UGA(x1) + 1))*UGA(x1) - x1*(epsilon*(2*R*UGA(x1) + 2*S*(-UGA(x1) + 1))*U_prime(x1) + epsilon*(2*R*U_prime(x1) - 2*S*U_prime(x1))*UGA(x1) - 0.5*(2*P*(-UGA(x1) + 1) + 2*T*UGA(x1))*U_prime(x1) + (-2*P*U_prime(x1) + 2*T*U_prime(x1))*(-0.5*UGA(x1) + 0.5)) - x1*((-epsilon + 1)*(2*R*UGA(x1) + 2*S*(-UGA(x1) + 1))*U_prime(x1) + (-epsilon + 1)*(2*R*U_prime(x1) - 2*S*U_prime(x1))*UGA(x1) - 0.5*(2*P*(-UGA(x1) + 1) + 2*T*UGA(x1))*U_prime(x1) + (-2*P*U_prime(x1) + 2*T*U_prime(x1))*(-0.5*UGA(x1) + 0.5)) + (-epsilon + 1)*(2*P*(-UGA(x1) + 1) + 2*T*UGA(x1))*(-UGA(x1) + 1) - (-epsilon + 1)*(2*R*UGA(x1) + 2*S*(-UGA(x1) + 1))*UGA(x1) - (-x1 + 1)*(-epsilon*(2*P*(-UGA(x1) + 1) + 2*T*UGA(x1))*U_prime(x1) + epsilon*(-2*P*U_prime(x1) + 2*T*U_prime(x1))*(-UGA(x1) + 1) + (1.0*R*UGA(x1) + 1.0*S*(-UGA(x1) + 1))*U_prime(x1) + (1.0*R*U_prime(x1) - 1.0*S*U_prime(x1))*UGA(x1)) - (-x1 + 1)*(-(-epsilon + 1)*(2*P*(-UGA(x1) + 1) + 2*T*UGA(x1))*U_prime(x1) + (-epsilon + 1)*(-2*P*U_prime(x1) + 2*T*U_prime(x1))*(-UGA(x1) + 1) + (1.0*R*UGA(x1) + 1.0*S*(-UGA(x1) + 1))*U_prime(x1) + (1.0*R*U_prime(x1) - 1.0*S*U_prime(x1))*UGA(x1)) - 1.0*(2*P*(-UGA(x1) + 1) + 2*T*UGA(x1))*(-UGA(x1) + 1) + 1.0*(2*R*UGA(x1) + 2*S*(-UGA(x1) + 1))*UGA(x1))/(x1*(epsilon*(2*R*UGA(x1) + 2*S*(-UGA(x1) + 1))*UGA(x1) + 0.5*(2*P*(-UGA(x1) + 1) + 2*T*UGA(x1))*(-UGA(x1) + 1)) + x1*((-epsilon + 1)*(2*R*UGA(x1) + 2*S*(-UGA(x1) + 1))*UGA(x1) + 0.5*(2*P*(-UGA(x1) + 1) + 2*T*UGA(x1))*(-UGA(x1) + 1)) + (-x1 + 1)*(epsilon*(2*P*(-UGA(x1) + 1) + 2*T*UGA(x1))*(-UGA(x1) + 1) + 0.5*(2*R*UGA(x1) + 2*S*(-UGA(x1) + 1))*UGA(x1)) + (-x1 + 1)*((-epsilon + 1)*(2*P*(-UGA(x1) + 1) + 2*T*UGA(x1))*(-UGA(x1) + 1) + 0.5*(2*R*UGA(x1) + 2*S*(-UGA(x1) + 1))*UGA(x1)))**2 - 1 + (-epsilon*(2*P*(-UGA(x1) + 1) + 2*T*UGA(x1))*(-UGA(x1) + 1) + x1*((-epsilon + 1)*(2*R*UGA(x1) + 2*S*(-UGA(x1) + 1))*U_prime(x1) + (-epsilon + 1)*(2*R*U_prime(x1) - 2*S*U_prime(x1))*UGA(x1) - 0.5*(2*P*(-UGA(x1) + 1) + 2*T*UGA(x1))*U_prime(x1) + (-2*P*U_prime(x1) + 2*T*U_prime(x1))*(-0.5*UGA(x1) + 0.5)) + (-epsilon + 1)*(2*R*UGA(x1) + 2*S*(-UGA(x1) + 1))*UGA(x1) + (-x1 + 1)*(-epsilon*(2*P*(-UGA(x1) + 1) + 2*T*UGA(x1))*U_prime(x1) + epsilon*(-2*P*U_prime(x1) + 2*T*U_prime(x1))*(-UGA(x1) + 1) + (1.0*R*UGA(x1) + 1.0*S*(-UGA(x1) + 1))*U_prime(x1) + (1.0*R*U_prime(x1) - 1.0*S*U_prime(x1))*UGA(x1)) + 0.5*(2*P*(-UGA(x1) + 1) + 2*T*UGA(x1))*(-UGA(x1) + 1) - 0.5*(2*R*UGA(x1) + 2*S*(-UGA(x1) + 1))*UGA(x1))/(x1*(epsilon*(2*R*UGA(x1) + 2*S*(-UGA(x1) + 1))*UGA(x1) + 0.5*(2*P*(-UGA(x1) + 1) + 2*T*UGA(x1))*(-UGA(x1) + 1)) + x1*((-epsilon + 1)*(2*R*UGA(x1) + 2*S*(-UGA(x1) + 1))*UGA(x1) + 0.5*(2*P*(-UGA(x1) + 1) + 2*T*UGA(x1))*(-UGA(x1) + 1)) + (-x1 + 1)*(epsilon*(2*P*(-UGA(x1) + 1) + 2*T*UGA(x1))*(-UGA(x1) + 1) + 0.5*(2*R*UGA(x1) + 2*S*(-UGA(x1) + 1))*UGA(x1)) + (-x1 + 1)*((-epsilon + 1)*(2*P*(-UGA(x1) + 1) + 2*T*UGA(x1))*(-UGA(x1) + 1) + 0.5*(2*R*UGA(x1) + 2*S*(-UGA(x1) + 1))*UGA(x1)))"
      ]
     },
     "execution_count": 112,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f_jac"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 113,
   "metadata": {},
   "outputs": [],
   "source": [
    "UGA_prime_star = sym.symbols(\"UGA_prime_star\", real=True, nonnegative=True)\n",
    "evaluated_f_jac = f_jac.subs({sym.Derivative(UGA(x1), x1): UGA_prime_star, UGA(x1): UGA_star, x1: x1_star})"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "sym.expand(evaluated_f_jac)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [],
   "source": [
    "numerator, denominator = sym.fraction(sym.factor(sym.together(evaluated_f_jac)))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(0.5*P*UGA_star**2 - 1.0*P*UGA_star + 0.5*P + 0.5*R*UGA_star**2 - 0.5*S*UGA_star**2 + 0.5*S*UGA_star - 0.5*T*UGA_star**2 + 0.5*T*UGA_star)**2"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# denominator is positive\n",
    "denominator"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-0.25*P**2*UGA_star**4*epsilon - 0.125*P**2*UGA_star**4 + 1.0*P**2*UGA_star**3*epsilon + 0.5*P**2*UGA_star**3 - 1.5*P**2*UGA_star**2*epsilon - 0.75*P**2*UGA_star**2 + 1.0*P**2*UGA_star*epsilon + 0.5*P**2*UGA_star - 0.25*P**2*epsilon - 0.125*P**2 - 0.5*P*R*UGA_star**4*epsilon - 0.25*P*R*UGA_star**4 + 1.0*P*R*UGA_star**3*epsilon + 0.5*P*R*UGA_star**3 + 0.5*P*R*UGA_star**2*epsilon*U_prime(4.0*P*UGA_star**2*epsilon/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) - 8.0*P*UGA_star*epsilon/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) + 4.0*P*epsilon/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) + 2.0*R*UGA_star**2/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) - 2.0*S*UGA_star**2/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) + 2.0*S*UGA_star/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) - 4.0*T*UGA_star**2*epsilon/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) + 4.0*T*UGA_star*epsilon/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star)) - 0.5*P*R*UGA_star**2*epsilon - 0.25*P*R*UGA_star**2*U_prime(4.0*P*UGA_star**2*epsilon/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) - 8.0*P*UGA_star*epsilon/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) + 4.0*P*epsilon/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) + 2.0*R*UGA_star**2/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) - 2.0*S*UGA_star**2/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) + 2.0*S*UGA_star/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) - 4.0*T*UGA_star**2*epsilon/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) + 4.0*T*UGA_star*epsilon/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star)) - 0.25*P*R*UGA_star**2 - 0.5*P*R*UGA_star*epsilon*U_prime(4.0*P*UGA_star**2*epsilon/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) - 8.0*P*UGA_star*epsilon/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) + 4.0*P*epsilon/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) + 2.0*R*UGA_star**2/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) - 2.0*S*UGA_star**2/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) + 2.0*S*UGA_star/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) - 4.0*T*UGA_star**2*epsilon/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) + 4.0*T*UGA_star*epsilon/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star)) + 0.25*P*R*UGA_star*U_prime(4.0*P*UGA_star**2*epsilon/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) - 8.0*P*UGA_star*epsilon/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) + 4.0*P*epsilon/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) + 2.0*R*UGA_star**2/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) - 2.0*S*UGA_star**2/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) + 2.0*S*UGA_star/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) - 4.0*T*UGA_star**2*epsilon/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) + 4.0*T*UGA_star*epsilon/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star)) + 0.5*P*S*UGA_star**4*epsilon + 0.25*P*S*UGA_star**4 - 1.5*P*S*UGA_star**3*epsilon - 0.75*P*S*UGA_star**3 - 0.25*P*S*UGA_star**2*epsilon*U_prime(4.0*P*UGA_star**2*epsilon/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) - 8.0*P*UGA_star*epsilon/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) + 4.0*P*epsilon/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) + 2.0*R*UGA_star**2/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) - 2.0*S*UGA_star**2/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) + 2.0*S*UGA_star/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) - 4.0*T*UGA_star**2*epsilon/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) + 4.0*T*UGA_star*epsilon/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star)) + 1.5*P*S*UGA_star**2*epsilon + 0.125*P*S*UGA_star**2*U_prime(4.0*P*UGA_star**2*epsilon/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) - 8.0*P*UGA_star*epsilon/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) + 4.0*P*epsilon/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) + 2.0*R*UGA_star**2/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) - 2.0*S*UGA_star**2/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) + 2.0*S*UGA_star/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) - 4.0*T*UGA_star**2*epsilon/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) + 4.0*T*UGA_star*epsilon/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star)) + 0.75*P*S*UGA_star**2 + 0.5*P*S*UGA_star*epsilon*U_prime(4.0*P*UGA_star**2*epsilon/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) - 8.0*P*UGA_star*epsilon/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) + 4.0*P*epsilon/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) + 2.0*R*UGA_star**2/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) - 2.0*S*UGA_star**2/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) + 2.0*S*UGA_star/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) - 4.0*T*UGA_star**2*epsilon/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) + 4.0*T*UGA_star*epsilon/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star)) - 0.5*P*S*UGA_star*epsilon - 0.25*P*S*UGA_star*U_prime(4.0*P*UGA_star**2*epsilon/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) - 8.0*P*UGA_star*epsilon/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) + 4.0*P*epsilon/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) + 2.0*R*UGA_star**2/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) - 2.0*S*UGA_star**2/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) + 2.0*S*UGA_star/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) - 4.0*T*UGA_star**2*epsilon/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) + 4.0*T*UGA_star*epsilon/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star)) - 0.25*P*S*UGA_star - 0.25*P*S*epsilon*U_prime(4.0*P*UGA_star**2*epsilon/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) - 8.0*P*UGA_star*epsilon/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) + 4.0*P*epsilon/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) + 2.0*R*UGA_star**2/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) - 2.0*S*UGA_star**2/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) + 2.0*S*UGA_star/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) - 4.0*T*UGA_star**2*epsilon/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) + 4.0*T*UGA_star*epsilon/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star)) + 0.125*P*S*U_prime(4.0*P*UGA_star**2*epsilon/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) - 8.0*P*UGA_star*epsilon/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) + 4.0*P*epsilon/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) + 2.0*R*UGA_star**2/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) - 2.0*S*UGA_star**2/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) + 2.0*S*UGA_star/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) - 4.0*T*UGA_star**2*epsilon/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) + 4.0*T*UGA_star*epsilon/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star)) + 0.5*P*T*UGA_star**4*epsilon + 0.25*P*T*UGA_star**4 - 1.5*P*T*UGA_star**3*epsilon - 0.75*P*T*UGA_star**3 + 1.5*P*T*UGA_star**2*epsilon + 0.75*P*T*UGA_star**2 - 0.5*P*T*UGA_star*epsilon - 0.25*P*T*UGA_star - 0.25*R**2*UGA_star**4*epsilon - 0.125*R**2*UGA_star**4 + 0.5*R*S*UGA_star**4*epsilon + 0.25*R*S*UGA_star**4 - 0.5*R*S*UGA_star**3*epsilon - 0.25*R*S*UGA_star**3 + 0.5*R*T*UGA_star**4*epsilon + 0.25*R*T*UGA_star**4 - 0.5*R*T*UGA_star**3*epsilon - 0.25*R*T*UGA_star**3 - 0.25*R*T*UGA_star**2*epsilon*U_prime(4.0*P*UGA_star**2*epsilon/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) - 8.0*P*UGA_star*epsilon/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) + 4.0*P*epsilon/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) + 2.0*R*UGA_star**2/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) - 2.0*S*UGA_star**2/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) + 2.0*S*UGA_star/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) - 4.0*T*UGA_star**2*epsilon/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) + 4.0*T*UGA_star*epsilon/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star)) + 0.125*R*T*UGA_star**2*U_prime(4.0*P*UGA_star**2*epsilon/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) - 8.0*P*UGA_star*epsilon/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) + 4.0*P*epsilon/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) + 2.0*R*UGA_star**2/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) - 2.0*S*UGA_star**2/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) + 2.0*S*UGA_star/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) - 4.0*T*UGA_star**2*epsilon/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star) + 4.0*T*UGA_star*epsilon/(4.0*P*UGA_star**2*epsilon + 2.0*P*UGA_star**2 - 8.0*P*UGA_star*epsilon - 4.0*P*UGA_star + 4.0*P*epsilon + 2.0*P + 4.0*R*UGA_star**2*epsilon + 2.0*R*UGA_star**2 - 4.0*S*UGA_star**2*epsilon - 2.0*S*UGA_star**2 + 4.0*S*UGA_star*epsilon + 2.0*S*UGA_star - 4.0*T*UGA_star**2*epsilon - 2.0*T*UGA_star**2 + 4.0*T*UGA_star*epsilon + 2.0*T*UGA_star)) - 0.25*S**2*UGA_star**4*epsilon - 0.125*S**2*UGA_star**4 + 0.5*S**2*UGA_star**3*epsilon + 0.25*S**2*UGA_star**3 - 0.25*S**2*UGA_star**2*epsilon - 0.125*S**2*UGA_star**2 - 0.5*S*T*UGA_star**4*epsilon - 0.25*S*T*UGA_star**4 + 1.0*S*T*UGA_star**3*epsilon + 0.5*S*T*UGA_star**3 - 0.5*S*T*UGA_star**2*epsilon - 0.25*S*T*UGA_star**2 - 0.25*T**2*UGA_star**4*epsilon - 0.125*T**2*UGA_star**4 + 0.5*T**2*UGA_star**3*epsilon + 0.25*T**2*UGA_star**3 - 0.25*T**2*UGA_star**2*epsilon - 0.125*T**2*UGA_star**2"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "numerator"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now we can solve the numerator for $U'_{GA}(x^*)$.  This will allow us to write the condition for which the Jacobian will be negative as an upper bound on $U'_{GA}(x^*)$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "ename": "ValueError",
     "evalue": "not enough values to unpack (expected 1, got 0)",
     "output_type": "error",
     "traceback": [
      "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[0;31mValueError\u001b[0m                                Traceback (most recent call last)",
      "\u001b[0;32m<ipython-input-22-060ae9e35fe8>\u001b[0m in \u001b[0;36m<module>\u001b[0;34m()\u001b[0m\n\u001b[0;32m----> 1\u001b[0;31m \u001b[0mupper_bound\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0msym\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0msolve\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mnumerator\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mUGA_prime_star\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m",
      "\u001b[0;31mValueError\u001b[0m: not enough values to unpack (expected 1, got 0)"
     ]
    }
   ],
   "source": [
    "upper_bound, = sym.solve(numerator, UGA_prime_star)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.0*(2.0*epsilon + 1.0)*(1.0*P*UGA_star**2 - 2.0*P*UGA_star + 1.0*P + 1.0*R*UGA_star**2 - 1.0*S*UGA_star**2 + 1.0*S*UGA_star - 1.0*T*UGA_star**2 + 1.0*T*UGA_star)**2/((2.0*epsilon - 1.0)*(2.0*P*R*UGA_star**2 - 2.0*P*R*UGA_star - 1.0*P*S*UGA_star**2 + 2.0*P*S*UGA_star - 1.0*P*S - 1.0*R*T*UGA_star**2))"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sym.factor(upper_bound)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "After fiddling around with the above expression we arrive at the following condition for a negative Jacobian and a stable interior equilibrium at $0 < x^* < 1$.\n",
    "\n",
    "\\begin{align}\n",
    "    U'_{GA}(x^*) <& \\left(\\frac{1 + 2\\epsilon}{1 - 2\\epsilon}\\right)\\frac{\\bigg(\\big((R + P) - (T + S)\\big)U_{GA}^{*2} + \\big((T + S) - 2P\\big)U_{GA}^* + P\\bigg)^2}{\\big(TR + PS - 2RP\\big)U_{GA}^{*2} + 2P(R - S)U_{GA}^* + PS}\\\\\n",
    "\\end{align}\n",
    "\n",
    "Can also use a graphical approach to check the correctness of the above condition.  The above condition should be the zero contour in a contour plot of the value of the Jacobian as a function of $U_{GA}$ and $U'_{GA}$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# vectorized numerical function created from our symbolic expression\n",
    "_numeric_f_jac = sym.lambdify((UGA_star, UGA_prime_star, T, R, P, S, epsilon), evaluated_f_jac, modules=\"numpy\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def plot_jacobian(T, R, P, S, epsilon):\n",
    "\n",
    "    fig, ax = plt.subplots(1, 1, figsize=(20, 10))\n",
    "    ax.set_ylabel(r\"$U_{GA}^*$\", fontsize=20, rotation=\"horizontal\")\n",
    "    ax.set_xlabel(r\"$U_{GA}^{'*}$\", fontsize=20)\n",
    "    ax.set_title(r\"Negative values of $\\frac{\\partial \\dot{x}_1}{\\partial x_1}$ indicate stability!\", fontsize=25)\n",
    "    ax.grid(\"off\")\n",
    "\n",
    "    equilibrium_selection_probs = np.linspace(0, 1, 100).reshape(-1, 1)\n",
    "    equilibrium_selection_derivs = np.linspace(0, 10, 1000).reshape(1, -1) # sensible UGA'(0) >= 1\n",
    "    Z = _numeric_f_jac(equilibrium_selection_probs, equilibrium_selection_derivs, T, R, P, S, epsilon)\n",
    "    cax = ax.imshow(Z, origin=\"lower\", aspect=\"auto\", vmin=-0.5, vmax=1.0)\n",
    "\n",
    "    levels = np.arange(-0.25, 1.25, 0.25)\n",
    "    contours = ax.contour(Z, levels=levels, colors='w', origin='lower')\n",
    "    ax.clabel(contours, contours.levels, inline=True, fontsize=10)\n",
    "    \n",
    "    # adjust the tick labels\n",
    "    locs, _ = plt.xticks()\n",
    "    plt.xticks(locs[1:], np.linspace(0, 10, locs.size-1))\n",
    "    locs, _ = plt.yticks()\n",
    "    plt.yticks(locs[1:], np.linspace(0, 1, locs.size-1))\n",
    "\n",
    "    plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "6ed1b9492f334d24b6c4c0695a3b5ad8",
       "version_major": 2,
       "version_minor": 0
      },
      "text/plain": [
       "A Jupyter Widget"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "mpld3.disable_notebook()  # don't need interactive plotting for this!\n",
    "\n",
    "# sliders used to control the Prisoner's Dilemma Payoffs\n",
    "T_slider = widgets.FloatSlider(value=25, min=0, max=100, step=0.1, description=r\"$T$\")\n",
    "R_slider = widgets.FloatSlider(value=3, min=0, max=100, step=0.1, description=r\"$R$\")\n",
    "P_slider = widgets.FloatSlider(value=2, min=0, max=100, step=0.1, description=r\"$P$\")\n",
    "S_slider = widgets.FloatSlider(value=1, min=0, max=100, step=0.1, description=r\"$S$\")\n",
    "\n",
    "e_slider = widgets.FloatSlider(value=0, min=0, max=1, step=1e-3, description=r\"$\\epsilon$\", readout_format=\".3f\")\n",
    "\n",
    "w = widgets.interactive(plot_jacobian, T=T_slider, R=R_slider, P=P_slider, S=S_slider, epsilon=e_slider)\n",
    "display(w)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Invadability\n",
    "\n",
    "Restrict the full model to obtain a monomorphic $\\gamma$ equilibrium, add a small amount of individuals carrying the $g$ allele of the $\\gamma$ gene and simulate."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "x2, x3 = sym.symbols(\"x2, x3\", real=True, positive=True)\n",
    "x4 = 1 - x1 - x2 - x3\n",
    "x = np.array([[x1], [x2], [x3], [x4]])\n",
    "W = models.generalized_sexual_selection(x, UGA, UgA, payoff_kernel, mutation_rate=epsilon)\n",
    "f1, f2, f3, _ = models.offspring_genotypes_evolution(W, x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "F = sym.Matrix([f1, f2, f3])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "F_jac = F.jacobian([x1, x2, x3])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "UgA_star = sym.symbols(\"UgA_star\", positive=True, real=True)\n",
    "\n",
    "evaluated_F_jac = (F_jac.subs({x2: 1 - x1, x3: 0})\n",
    "                        .subs({UGA(x1): UGA_star, UgA(x1): UgA_star})\n",
    "                        .doit()\n",
    "                        .subs({sym.Derivative(UGA(x1), x1): UGA_prime_star}))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Finished with element 0,0!\n",
      "Finished with element 0,1!\n",
      "Finished with element 0,2!\n",
      "Finished with element 1,0!\n",
      "Finished with element 1,1!\n",
      "Finished with element 1,2!\n",
      "Finished with element 2,0!\n",
      "Finished with element 2,1!\n",
      "Finished with element 2,2!\n"
     ]
    }
   ],
   "source": [
    "simplified_F_jac = sym.zeros(3, 3)\n",
    "\n",
    "for i in range(3):\n",
    "    for j in range(3):\n",
    "        simplified_F_jac[i, j] = sym.factor(sym.cancel(sym.together(evaluated_F_jac[i, j])))\n",
    "        print(\"Finished with element {},{}!\".format(i,j))\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 94,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "charpoly = simplified_F_jac.charpoly()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [],
   "source": [
    "eigenvals = simplified_F_jac.eigenvals()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 114,
   "metadata": {},
   "outputs": [
    {
     "ename": "NameError",
     "evalue": "name 'eigenvals' is not defined",
     "output_type": "error",
     "traceback": [
      "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[0;31mNameError\u001b[0m                                 Traceback (most recent call last)",
      "\u001b[0;32m<ipython-input-114-335ca9b280b0>\u001b[0m in \u001b[0;36m<module>\u001b[0;34m()\u001b[0m\n\u001b[1;32m      2\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m      3\u001b[0m \u001b[0;32mwith\u001b[0m \u001b[0mopen\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m'non-invadable-eigenvalues.pickle'\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m'wb'\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;32mas\u001b[0m \u001b[0mhandle\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m----> 4\u001b[0;31m     \u001b[0mpickle\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mdump\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0meigenvals\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mhandle\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mprotocol\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mpickle\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mHIGHEST_PROTOCOL\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m",
      "\u001b[0;31mNameError\u001b[0m: name 'eigenvals' is not defined"
     ]
    }
   ],
   "source": [
    "import pickle\n",
    "\n",
    "with open('non-invadable-eigenvalues.pickle', 'wb') as handle:\n",
    "    pickle.dump(eigenvals, handle, protocol=pickle.HIGHEST_PROTOCOL)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "e1, e2, e3 = eigenvals.keys()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {},
   "outputs": [],
   "source": [
    "numerator, denominator = sym.fraction(sym.factor(e1))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2*(P*UGA_star**2 - 2*P*UGA_star + P + R*UGA_star**2 - S*UGA_star**2 + S*UGA_star - T*UGA_star**2 + T*UGA_star)**2"
      ]
     },
     "execution_count": 46,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "denominator"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {},
   "outputs": [],
   "source": [
    "upper_bound, = sym.solve(numerator, UGA_prime_star)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(2*epsilon + 1)*(P*UGA_star**2 - 2*P*UGA_star + P + R*UGA_star**2 - S*UGA_star**2 + S*UGA_star - T*UGA_star**2 + T*UGA_star)**2/((2*epsilon - 1)*(2*P*R*UGA_star**2 - 2*P*R*UGA_star - P*S*UGA_star**2 + 2*P*S*UGA_star - P*S - R*T*UGA_star**2))"
      ]
     },
     "execution_count": 50,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sym.factor(upper_bound)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 95,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-(2*R**2*epsilon + R**2)/(2*R**2)"
      ]
     },
     "execution_count": 95,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "e1.subs({UGA_star: 1, UGA_prime_star: 0})"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 73,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# vectorized numeric repr for the eigenvalue\n",
    "_numeric_e2 = sym.lambdify((UGA_star, UgA_star, T, R, P, S, epsilon),\n",
    "                           e2.subs({x1: x1_star}),\n",
    "                           modules=\"numpy\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 75,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def plot_second_eigenvalue(T, R, P, S, epsilon):\n",
    "\n",
    "    fig, ax = plt.subplots(1, 1, figsize=(10, 10))\n",
    "    ax.set_ylabel(r\"$U_{GA}^*$\", fontsize=20, rotation=\"horizontal\")\n",
    "    ax.set_xlabel(r\"$U_{gA}^{*}$\", fontsize=20)\n",
    "    ax.set_title(r\"Eigenvalue, $e_2$\", fontsize=25)\n",
    "    ax.grid(\"off\")\n",
    "\n",
    "    equilibrium_selection_probs = np.linspace(0, 1, 100)\n",
    "    UgAs, UGAs = np.meshgrid(equilibrium_selection_probs, equilibrium_selection_probs)\n",
    "    Z = _numeric_e2(UGAs, UgAs, T, R, P, S, epsilon)\n",
    "    cax = ax.imshow(Z, origin=\"lower\")\n",
    "    contours = ax.contour(Z, colors='w', origin='lower')\n",
    "    ax.clabel(contours, contours.levels, inline=True, fontsize=10)\n",
    "    \n",
    "    # adjust the tick labels\n",
    "    locs, _ = plt.xticks()\n",
    "    plt.xticks(locs[1:], np.linspace(0, 1, locs.size-1))\n",
    "    locs, _ = plt.yticks()\n",
    "    plt.yticks(locs[1:], np.linspace(0, 1, locs.size-1))\n",
    "\n",
    "    plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 80,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "f12f73414a9c48f985f492a9b5cb74a6",
       "version_major": 2,
       "version_minor": 0
      },
      "text/plain": [
       "A Jupyter Widget"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "mpld3.disable_notebook()  # don't need interactive plotting for this!\n",
    "\n",
    "# sliders used to control the Prisoner's Dilemma Payoffs\n",
    "T_slider = widgets.FloatSlider(value=10, min=0, max=100, step=0.1, description=r\"$T$\")\n",
    "R_slider = widgets.FloatSlider(value=8, min=0, max=100, step=0.1, description=r\"$R$\")\n",
    "P_slider = widgets.FloatSlider(value=6, min=0, max=100, step=0.1, description=r\"$P$\")\n",
    "S_slider = widgets.FloatSlider(value=4, min=0, max=100, step=0.1, description=r\"$S$\")\n",
    "\n",
    "# slider used to control the mutation rate\n",
    "e_slider = widgets.FloatSlider(value=0.0, min=0.0, max=1.0, step=1e-3, description=r\"$\\epsilon$\", readout_format=\".3f\")\n",
    "\n",
    "w = widgets.interactive(plot_second_eigenvalue, T=T_slider, R=R_slider, P=P_slider, S=S_slider, epsilon=e_slider)\n",
    "display(w)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 105,
   "metadata": {},
   "outputs": [],
   "source": [
    "evaluated = (e2.subs({x1: x1_star, UgA_star: x1_star})\n",
    "               .subs({UGA_star: 1, UGA_prime_star: 0}))\n",
    "together = sym.together(evaluated)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 107,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "expanded = sym.expand(together)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 108,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-4.0*P*epsilon**4/(16.0*R*epsilon**3 + 16.0*R*epsilon**2 + 4.0*R*epsilon) + 6.0*P*epsilon**3/(16.0*R*epsilon**3 + 16.0*R*epsilon**2 + 4.0*R*epsilon) - 12.0*R*epsilon**3/(16.0*R*epsilon**3 + 16.0*R*epsilon**2 + 4.0*R*epsilon) - 13.0*R*epsilon**2/(16.0*R*epsilon**3 + 16.0*R*epsilon**2 + 4.0*R*epsilon) - 1.5*R*epsilon/(16.0*R*epsilon**3 + 16.0*R*epsilon**2 + 4.0*R*epsilon) - 2.0*S*epsilon**3/(16.0*R*epsilon**3 + 16.0*R*epsilon**2 + 4.0*R*epsilon) + 3.0*S*epsilon**2/(16.0*R*epsilon**3 + 16.0*R*epsilon**2 + 4.0*R*epsilon) - 2.0*T*epsilon**3/(16.0*R*epsilon**3 + 16.0*R*epsilon**2 + 4.0*R*epsilon) + 3.0*T*epsilon**2/(16.0*R*epsilon**3 + 16.0*R*epsilon**2 + 4.0*R*epsilon) + 0.015625*sqrt(65536.0*P**2*epsilon**8 + 65536.0*P**2*epsilon**7 + 16384.0*P**2*epsilon**6 + 524288.0*P*R*epsilon**8 + 655360.0*P*R*epsilon**7 + 229376.0*P*R*epsilon**6 - 8192.0*P*R*epsilon**4 + 65536.0*P*S*epsilon**7 + 65536.0*P*S*epsilon**6 + 16384.0*P*S*epsilon**5 + 65536.0*P*T*epsilon**7 + 65536.0*P*T*epsilon**6 + 16384.0*P*T*epsilon**5 + 65536.0*R**2*epsilon**6 + 163840.0*R**2*epsilon**5 + 151552.0*R**2*epsilon**4 + 61440.0*R**2*epsilon**3 + 9216.0*R**2*epsilon**2 + 65536.0*R*S*epsilon**6 + 114688.0*R*S*epsilon**5 + 65536.0*R*S*epsilon**4 + 12288.0*R*S*epsilon**3 + 262144.0*R*T*epsilon**7 + 327680.0*R*T*epsilon**6 + 114688.0*R*T*epsilon**5 - 4096.0*R*T*epsilon**3 + 16384.0*S**2*epsilon**6 + 16384.0*S**2*epsilon**5 + 4096.0*S**2*epsilon**4 + 32768.0*S*T*epsilon**6 + 32768.0*S*T*epsilon**5 + 8192.0*S*T*epsilon**4 + 16384.0*T**2*epsilon**6 + 16384.0*T**2*epsilon**5 + 4096.0*T**2*epsilon**4)/(16.0*R*epsilon**3 + 16.0*R*epsilon**2 + 4.0*R*epsilon)"
      ]
     },
     "execution_count": 108,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expanded"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 110,
   "metadata": {},
   "outputs": [],
   "source": [
    "tmp_expr = sym.together(expanded)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 111,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(-4.0*P*epsilon**3 + 6.0*P*epsilon**2 - 12.0*R*epsilon**2 - 13.0*R*epsilon - 1.5*R - 2.0*S*epsilon**2 + 3.0*S*epsilon - 2.0*T*epsilon**2 + 3.0*T*epsilon + 0.015625*sqrt(65536.0*P**2*epsilon**6 + 65536.0*P**2*epsilon**5 + 16384.0*P**2*epsilon**4 + 524288.0*P*R*epsilon**6 + 655360.0*P*R*epsilon**5 + 229376.0*P*R*epsilon**4 - 8192.0*P*R*epsilon**2 + 65536.0*P*S*epsilon**5 + 65536.0*P*S*epsilon**4 + 16384.0*P*S*epsilon**3 + 65536.0*P*T*epsilon**5 + 65536.0*P*T*epsilon**4 + 16384.0*P*T*epsilon**3 + 65536.0*R**2*epsilon**4 + 163840.0*R**2*epsilon**3 + 151552.0*R**2*epsilon**2 + 61440.0*R**2*epsilon + 9216.0*R**2 + 65536.0*R*S*epsilon**4 + 114688.0*R*S*epsilon**3 + 65536.0*R*S*epsilon**2 + 12288.0*R*S*epsilon + 262144.0*R*T*epsilon**5 + 327680.0*R*T*epsilon**4 + 114688.0*R*T*epsilon**3 - 4096.0*R*T*epsilon + 16384.0*S**2*epsilon**4 + 16384.0*S**2*epsilon**3 + 4096.0*S**2*epsilon**2 + 32768.0*S*T*epsilon**4 + 32768.0*S*T*epsilon**3 + 8192.0*S*T*epsilon**2 + 16384.0*T**2*epsilon**4 + 16384.0*T**2*epsilon**3 + 4096.0*T**2*epsilon**2))/(R*(16.0*epsilon**2 + 16.0*epsilon + 4.0))"
      ]
     },
     "execution_count": 111,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "tmp_expr"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 117,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(-13.0*R*epsilon - 1.5*R + 3.0*S*epsilon + 3.0*T*epsilon + 0.015625*sqrt(61440.0*R**2*epsilon + 9216.0*R**2 + 12288.0*R*S*epsilon - 4096.0*R*T*epsilon))/(R*(16.0*epsilon + 4.0))"
      ]
     },
     "execution_count": 117,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "tmp_expr.subs({epsilon**2: 0})"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\\begin{align}\n",
    "   % (-13.0*R*epsilon - 1.5*R + 3.0*S*epsilon + 3.0*T*epsilon + 0.015625*sqrt(61440.0*R**2*epsilon + 9216.0*R**2 + 12288.0*R*S*epsilon - 4096.0*R*T*epsilon))/(R*(16.0*epsilon + 4.0)) < 0 \\\\\n",
    "    \\frac{(-13R\\epsilon - 1.5R + 3S\\epsilon + 3T\\epsilon + 0.015625\\sqrt{61440R^2\\epsilon + 9216R^2 + 12288RS\\epsilon - 4096RT\\epsilon}}{R(16\\epsilon + 4.0)} < 0 \\\\\n",
    "    \\frac{(-13R\\epsilon - \\frac{3}{2}R + 3S\\epsilon + 3T\\epsilon + \\frac{1}{64}\\sqrt{61440R^2\\epsilon + 9216R^2 + 12288RS\\epsilon - 4096RT\\epsilon}}{R(16\\epsilon + 4.0)} < 0 \\\\\n",
    "    \\frac{(-13R\\epsilon - \\frac{3}{2}R + 3S\\epsilon + 3T\\epsilon + \\frac{1}{2}\\sqrt{60R^2\\epsilon + 9R^2 + 12RS\\epsilon - 4RT\\epsilon}}{R(16\\epsilon + 4)} < 0\n",
    "\\end{align}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 177,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(192.0*P**5*UGA_star**10*epsilon - 1920.0*P**5*UGA_star**9*epsilon + 8544.0*P**5*UGA_star**8*epsilon - 22272.0*P**5*UGA_star**7*epsilon + 37632.0*P**5*UGA_star**6*epsilon - 43008.0*P**5*UGA_star**5*epsilon + 33600.0*P**5*UGA_star**4*epsilon - 17664.0*P**5*UGA_star**3*epsilon + 5952.0*P**5*UGA_star**2*epsilon - 1152.0*P**5*UGA_star*epsilon + 96.0*P**5*epsilon + 864.0*P**4*R*UGA_star**10*epsilon + 80.0*P**4*R*UGA_star**10 - 6912.0*P**4*R*UGA_star**9*epsilon - 640.0*P**4*R*UGA_star**9 + 24224.0*P**4*R*UGA_star**8*epsilon + 2192.0*P**4*R*UGA_star**8 - 48576.0*P**4*R*UGA_star**7*epsilon - 4192.0*P**4*R*UGA_star**7 + 60960.0*P**4*R*UGA_star**6*epsilon + 4880.0*P**4*R*UGA_star**6 - 49024.0*P**4*R*UGA_star**5*epsilon - 3520.0*P**4*R*UGA_star**5 + 24672.0*P**4*R*UGA_star**4*epsilon + 1520.0*P**4*R*UGA_star**4 - 7104.0*P**4*R*UGA_star**3*epsilon - 352.0*P**4*R*UGA_star**3 + 896.0*P**4*R*UGA_star**2*epsilon + 32.0*P**4*R*UGA_star**2 - 864.0*P**4*S*UGA_star**10*epsilon - 80.0*P**4*S*UGA_star**10 + 7776.0*P**4*S*UGA_star**9*epsilon + 720.0*P**4*S*UGA_star**9 - 31136.0*P**4*S*UGA_star**8*epsilon - 2832.0*P**4*S*UGA_star**8 + 72800.0*P**4*S*UGA_star**7*epsilon + 6384.0*P**4*S*UGA_star**7 - 109536.0*P**4*S*UGA_star**6*epsilon - 9072.0*P**4*S*UGA_star**6 + 109984.0*P**4*S*UGA_star**5*epsilon + 8400.0*P**4*S*UGA_star**5 - 73696.0*P**4*S*UGA_star**4*epsilon - 5040.0*P**4*S*UGA_star**4 + 31776.0*P**4*S*UGA_star**3*epsilon + 1872.0*P**4*S*UGA_star**3 - 8000.0*P**4*S*UGA_star**2*epsilon - 384.0*P**4*S*UGA_star**2 + 896.0*P**4*S*UGA_star*epsilon + 32.0*P**4*S*UGA_star - 960.0*P**4*T*UGA_star**10*epsilon + 8640.0*P**4*T*UGA_star**9*epsilon - 34176.0*P**4*T*UGA_star**8*epsilon + 77952.0*P**4*T*UGA_star**7*epsilon - 112896.0*P**4*T*UGA_star**6*epsilon + 107520.0*P**4*T*UGA_star**5*epsilon - 67200.0*P**4*T*UGA_star**4*epsilon + 26496.0*P**4*T*UGA_star**3*epsilon - 5952.0*P**4*T*UGA_star**2*epsilon + 576.0*P**4*T*UGA_star*epsilon + 1632.0*P**3*R**2*UGA_star**10*epsilon + 240.0*P**3*R**2*UGA_star**10 - 9792.0*P**3*R**2*UGA_star**9*epsilon - 1440.0*P**3*R**2*UGA_star**9 + 24192.0*P**3*R**2*UGA_star**8*epsilon + 3600.0*P**3*R**2*UGA_star**8 - 31488.0*P**3*R**2*UGA_star**7*epsilon - 4800.0*P**3*R**2*UGA_star**7 + 22752.0*P**3*R**2*UGA_star**6*epsilon + 3600.0*P**3*R**2*UGA_star**6 - 8640.0*P**3*R**2*UGA_star**5*epsilon - 1440.0*P**3*R**2*UGA_star**5 + 1344.0*P**3*R**2*UGA_star**4*epsilon + 240.0*P**3*R**2*UGA_star**4 - 3264.0*P**3*R*S*UGA_star**10*epsilon - 480.0*P**3*R*S*UGA_star**10 + 22848.0*P**3*R*S*UGA_star**9*epsilon + 3360.0*P**3*R*S*UGA_star**9 - 68160.0*P**3*R*S*UGA_star**8*epsilon - 10080.0*P**3*R*S*UGA_star**8 + 112512.0*P**3*R*S*UGA_star**7*epsilon + 16800.0*P**3*R*S*UGA_star**7 - 111360.0*P**3*R*S*UGA_star**6*epsilon - 16800.0*P**3*R*S*UGA_star**6 + 66624.0*P**3*R*S*UGA_star**5*epsilon + 10080.0*P**3*R*S*UGA_star**5 - 22848.0*P**3*R*S*UGA_star**4*epsilon - 3360.0*P**3*R*S*UGA_star**4 + 3840.0*P**3*R*S*UGA_star**3*epsilon + 480.0*P**3*R*S*UGA_star**3 - 192.0*P**3*R*S*UGA_star**2*epsilon - 3456.0*P**3*R*T*UGA_star**10*epsilon - 320.0*P**3*R*T*UGA_star**10 + 24192.0*P**3*R*T*UGA_star**9*epsilon + 2240.0*P**3*R*T*UGA_star**9 - 72864.0*P**3*R*T*UGA_star**8*epsilon - 6576.0*P**3*R*T*UGA_star**8 + 122592.0*P**3*R*T*UGA_star**7*epsilon + 10480.0*P**3*R*T*UGA_star**7 - 124800.0*P**3*R*T*UGA_star**6*epsilon - 9760.0*P**3*R*T*UGA_star**6 + 77376.0*P**3*R*T*UGA_star**5*epsilon + 5280.0*P**3*R*T*UGA_star**5 - 27552.0*P**3*R*T*UGA_star**4*epsilon - 1520.0*P**3*R*T*UGA_star**4 + 4704.0*P**3*R*T*UGA_star**3*epsilon + 176.0*P**3*R*T*UGA_star**3 - 192.0*P**3*R*T*UGA_star**2*epsilon + 1632.0*P**3*S**2*UGA_star**10*epsilon + 240.0*P**3*S**2*UGA_star**10 - 13056.0*P**3*S**2*UGA_star**9*epsilon - 1920.0*P**3*S**2*UGA_star**9 + 45600.0*P**3*S**2*UGA_star**8*epsilon + 6720.0*P**3*S**2*UGA_star**8 - 91008.0*P**3*S**2*UGA_star**7*epsilon - 13440.0*P**3*S**2*UGA_star**7 + 113952.0*P**3*S**2*UGA_star**6*epsilon + 16800.0*P**3*S**2*UGA_star**6 - 92352.0*P**3*S**2*UGA_star**5*epsilon - 13440.0*P**3*S**2*UGA_star**5 + 48096.0*P**3*S**2*UGA_star**4*epsilon + 6720.0*P**3*S**2*UGA_star**4 - 15360.0*P**3*S**2*UGA_star**3*epsilon - 1920.0*P**3*S**2*UGA_star**3 + 2688.0*P**3*S**2*UGA_star**2*epsilon + 240.0*P**3*S**2*UGA_star**2 - 192.0*P**3*S**2*UGA_star*epsilon + 3456.0*P**3*S*T*UGA_star**10*epsilon + 320.0*P**3*S*T*UGA_star**10 - 27648.0*P**3*S*T*UGA_star**9*epsilon - 2560.0*P**3*S*T*UGA_star**9 + 97056.0*P**3*S*T*UGA_star**8*epsilon + 8816.0*P**3*S*T*UGA_star**8 - 195456.0*P**3*S*T*UGA_star**7*epsilon - 17056.0*P**3*S*T*UGA_star**7 + 247392.0*P**3*S*T*UGA_star**6*epsilon + 20240.0*P**3*S*T*UGA_star**6 - 202176.0*P**3*S*T*UGA_star**5*epsilon - 15040.0*P**3*S*T*UGA_star**5 + 104928.0*P**3*S*T*UGA_star**4*epsilon + 6800.0*P**3*S*T*UGA_star**4 - 32256.0*P**3*S*T*UGA_star**3*epsilon - 1696.0*P**3*S*T*UGA_star**3 + 4896.0*P**3*S*T*UGA_star**2*epsilon + 176.0*P**3*S*T*UGA_star**2 - 192.0*P**3*S*T*UGA_star*epsilon + 1920.0*P**3*T**2*UGA_star**10*epsilon - 15360.0*P**3*T**2*UGA_star**9*epsilon + 53184.0*P**3*T**2*UGA_star**8*epsilon - 104064.0*P**3*T**2*UGA_star**7*epsilon + 125760.0*P**3*T**2*UGA_star**6*epsilon - 96000.0*P**3*T**2*UGA_star**5*epsilon + 45120.0*P**3*T**2*UGA_star**4*epsilon - 11904.0*P**3*T**2*UGA_star**3*epsilon + 1344.0*P**3*T**2*UGA_star**2*epsilon + 1632.0*P**2*R**3*UGA_star**10*epsilon + 240.0*P**2*R**3*UGA_star**10 - 6528.0*P**2*R**3*UGA_star**9*epsilon - 960.0*P**2*R**3*UGA_star**9 + 9504.0*P**2*R**3*UGA_star**8*epsilon + 1440.0*P**2*R**3*UGA_star**8 - 5376.0*P**2*R**3*UGA_star**7*epsilon - 960.0*P**2*R**3*UGA_star**7 - 96.0*P**2*R**3*UGA_star**6*epsilon + 240.0*P**2*R**3*UGA_star**6 + 1152.0*P**2*R**3*UGA_star**5*epsilon - 288.0*P**2*R**3*UGA_star**4*epsilon - 4896.0*P**2*R**2*S*UGA_star**10*epsilon - 720.0*P**2*R**2*S*UGA_star**10 + 24480.0*P**2*R**2*S*UGA_star**9*epsilon + 3600.0*P**2*R**2*S*UGA_star**9 - 48352.0*P**2*R**2*S*UGA_star**8*epsilon - 7248.0*P**2*R**2*S*UGA_star**8 + 45952.0*P**2*R**2*S*UGA_star**7*epsilon + 7392.0*P**2*R**2*S*UGA_star**7 - 18528.0*P**2*R**2*S*UGA_star**6*epsilon - 3888.0*P**2*R**2*S*UGA_star**6 - 992.0*P**2*R**2*S*UGA_star**5*epsilon + 912.0*P**2*R**2*S*UGA_star**5 + 2912.0*P**2*R**2*S*UGA_star**4*epsilon - 48.0*P**2*R**2*S*UGA_star**4 - 576.0*P**2*R**2*S*UGA_star**3*epsilon - 4896.0*P**2*R**2*T*UGA_star**10*epsilon - 720.0*P**2*R**2*T*UGA_star**10 + 24480.0*P**2*R**2*T*UGA_star**9*epsilon + 3600.0*P**2*R**2*T*UGA_star**9 - 48352.0*P**2*R**2*T*UGA_star**8*epsilon - 7248.0*P**2*R**2*T*UGA_star**8 + 47104.0*P**2*R**2*T*UGA_star**7*epsilon + 7392.0*P**2*R**2*T*UGA_star**7 - 22560.0*P**2*R**2*T*UGA_star**6*epsilon - 3888.0*P**2*R**2*T*UGA_star**6 + 4192.0*P**2*R**2*T*UGA_star**5*epsilon + 912.0*P**2*R**2*T*UGA_star**5 + 32.0*P**2*R**2*T*UGA_star**4*epsilon - 48.0*P**2*R**2*T*UGA_star**4 + 4896.0*P**2*R*S**2*UGA_star**10*epsilon + 720.0*P**2*R*S**2*UGA_star**10 - 29376.0*P**2*R*S**2*UGA_star**9*epsilon - 4320.0*P**2*R*S**2*UGA_star**9 + 73088.0*P**2*R*S**2*UGA_star**8*epsilon + 10896.0*P**2*R*S**2*UGA_star**8 - 95872.0*P**2*R*S**2*UGA_star**7*epsilon - 14880.0*P**2*R*S**2*UGA_star**7 + 68480.0*P**2*R*S**2*UGA_star**6*epsilon + 11760.0*P**2*R*S**2*UGA_star**6 - 22976.0*P**2*R*S**2*UGA_star**5*epsilon - 5280.0*P**2*R*S**2*UGA_star**5 + 256.0*P**2*R*S**2*UGA_star**4*epsilon + 1200.0*P**2*R*S**2*UGA_star**4 + 1792.0*P**2*R*S**2*UGA_star**3*epsilon - 96.0*P**2*R*S**2*UGA_star**3 - 288.0*P**2*R*S**2*UGA_star**2*epsilon + 9792.0*P**2*R*S*T*UGA_star**10*epsilon + 1440.0*P**2*R*S*T*UGA_star**10 - 58752.0*P**2*R*S*T*UGA_star**9*epsilon - 8640.0*P**2*R*S*T*UGA_star**9 + 146240.0*P**2*R*S*T*UGA_star**8*epsilon + 21696.0*P**2*R*S*T*UGA_star**8 - 193792.0*P**2*R*S*T*UGA_star**7*epsilon - 29280.0*P**2*R*S*T*UGA_star**7 + 145088.0*P**2*R*S*T*UGA_star**6*epsilon + 22560.0*P**2*R*S*T*UGA_star**6 - 59264.0*P**2*R*S*T*UGA_star**5*epsilon - 9600.0*P**2*R*S*T*UGA_star**5 + 11200.0*P**2*R*S*T*UGA_star**4*epsilon + 1920.0*P**2*R*S*T*UGA_star**4 - 512.0*P**2*R*S*T*UGA_star**3*epsilon - 96.0*P**2*R*S*T*UGA_star**3 + 5184.0*P**2*R*T**2*UGA_star**10*epsilon + 480.0*P**2*R*T**2*UGA_star**10 - 31104.0*P**2*R*T**2*UGA_star**9*epsilon - 2880.0*P**2*R*T**2*UGA_star**9 + 78432.0*P**2*R*T**2*UGA_star**8*epsilon + 7056.0*P**2*R*T**2*UGA_star**8 - 106944.0*P**2*R*T**2*UGA_star**7*epsilon - 9024.0*P**2*R*T**2*UGA_star**7 + 84096.0*P**2*R*T**2*UGA_star**6*epsilon + 6336.0*P**2*R*T**2*UGA_star**6 - 37248.0*P**2*R*T**2*UGA_star**5*epsilon - 2304.0*P**2*R*T**2*UGA_star**5 + 8160.0*P**2*R*T**2*UGA_star**4*epsilon + 336.0*P**2*R*T**2*UGA_star**4 - 576.0*P**2*R*T**2*UGA_star**3*epsilon - 1632.0*P**2*S**3*UGA_star**10*epsilon - 240.0*P**2*S**3*UGA_star**10 + 11424.0*P**2*S**3*UGA_star**9*epsilon + 1680.0*P**2*S**3*UGA_star**9 - 34240.0*P**2*S**3*UGA_star**8*epsilon - 5088.0*P**2*S**3*UGA_star**8 + 56928.0*P**2*S**3*UGA_star**7*epsilon + 8688.0*P**2*S**3*UGA_star**7 - 56640.0*P**2*S**3*UGA_star**6*epsilon - 9120.0*P**2*S**3*UGA_star**6 + 33632.0*P**2*S**3*UGA_star**5*epsilon + 6000.0*P**2*S**3*UGA_star**5 - 10944.0*P**2*S**3*UGA_star**4*epsilon - 2400.0*P**2*S**3*UGA_star**4 + 1440.0*P**2*S**3*UGA_star**3*epsilon + 528.0*P**2*S**3*UGA_star**3 + 32.0*P**2*S**3*UGA_star**2*epsilon - 48.0*P**2*S**3*UGA_star**2 - 4896.0*P**2*S**2*T*UGA_star**10*epsilon - 720.0*P**2*S**2*T*UGA_star**10 + 34272.0*P**2*S**2*T*UGA_star**9*epsilon + 5040.0*P**2*S**2*T*UGA_star**9 - 102784.0*P**2*S**2*T*UGA_star**8*epsilon - 15168.0*P**2*S**2*T*UGA_star**8 + 171744.0*P**2*S**2*T*UGA_star**7*epsilon + 25488.0*P**2*S**2*T*UGA_star**7 - 173760.0*P**2*S**2*T*UGA_star**6*epsilon - 25920.0*P**2*S**2*T*UGA_star**6 + 107936.0*P**2*S**2*T*UGA_star**5*epsilon + 16080.0*P**2*S**2*T*UGA_star**5 - 39552.0*P**2*S**2*T*UGA_star**4*epsilon - 5760.0*P**2*S**2*T*UGA_star**4 + 7584.0*P**2*S**2*T*UGA_star**3*epsilon + 1008.0*P**2*S**2*T*UGA_star**3 - 544.0*P**2*S**2*T*UGA_star**2*epsilon - 48.0*P**2*S**2*T*UGA_star**2 - 5184.0*P**2*S*T**2*UGA_star**10*epsilon - 480.0*P**2*S*T**2*UGA_star**10 + 36288.0*P**2*S*T**2*UGA_star**9*epsilon + 3360.0*P**2*S*T**2*UGA_star**9 - 109536.0*P**2*S*T**2*UGA_star**8*epsilon - 9936.0*P**2*S*T**2*UGA_star**8 + 185376.0*P**2*S*T**2*UGA_star**7*epsilon + 16080.0*P**2*S*T**2*UGA_star**7 - 191040.0*P**2*S*T**2*UGA_star**6*epsilon - 15360.0*P**2*S*T**2*UGA_star**6 + 121344.0*P**2*S*T**2*UGA_star**5*epsilon + 8640.0*P**2*S*T**2*UGA_star**5 - 45408.0*P**2*S*T**2*UGA_star**4*epsilon - 2640.0*P**2*S*T**2*UGA_star**4 + 8736.0*P**2*S*T**2*UGA_star**3*epsilon + 336.0*P**2*S*T**2*UGA_star**3 - 576.0*P**2*S*T**2*UGA_star**2*epsilon - 1920.0*P**2*T**3*UGA_star**10*epsilon + 13440.0*P**2*T**3*UGA_star**9*epsilon - 39936.0*P**2*T**3*UGA_star**8*epsilon + 65280.0*P**2*T**3*UGA_star**7*epsilon - 63360.0*P**2*T**3*UGA_star**6*epsilon + 36480.0*P**2*T**3*UGA_star**5*epsilon - 11520.0*P**2*T**3*UGA_star**4*epsilon + 1536.0*P**2*T**3*UGA_star**3*epsilon + 864.0*P*R**4*UGA_star**10*epsilon + 80.0*P*R**4*UGA_star**10 - 1728.0*P*R**4*UGA_star**9*epsilon - 160.0*P*R**4*UGA_star**9 + 896.0*P*R**4*UGA_star**8*epsilon + 32.0*P*R**4*UGA_star**8 - 64.0*P*R**4*UGA_star**7*epsilon + 96.0*P*R**4*UGA_star**7 + 32.0*P*R**4*UGA_star**6*epsilon - 48.0*P*R**4*UGA_star**6 - 3456.0*P*R**3*S*UGA_star**10*epsilon - 320.0*P*R**3*S*UGA_star**10 + 10368.0*P*R**3*S*UGA_star**9*epsilon + 960.0*P*R**3*S*UGA_star**9 - 10656.0*P*R**3*S*UGA_star**8*epsilon - 816.0*P*R**3*S*UGA_star**8 + 4128.0*P*R**3*S*UGA_star**7*epsilon - 112.0*P*R**3*S*UGA_star**7 - 480.0*P*R**3*S*UGA_star**6*epsilon + 432.0*P*R**3*S*UGA_star**6 + 96.0*P*R**3*S*UGA_star**5*epsilon - 144.0*P*R**3*S*UGA_star**5 - 3264.0*P*R**3*T*UGA_star**10*epsilon - 480.0*P*R**3*T*UGA_star**10 + 9792.0*P*R**3*T*UGA_star**9*epsilon + 1440.0*P*R**3*T*UGA_star**9 - 9408.0*P*R**3*T*UGA_star**8*epsilon - 1440.0*P*R**3*T*UGA_star**8 + 1920.0*P*R**3*T*UGA_star**7*epsilon + 480.0*P*R**3*T*UGA_star**7 + 1536.0*P*R**3*T*UGA_star**6*epsilon - 576.0*P*R**3*T*UGA_star**5*epsilon + 5184.0*P*R**2*S**2*UGA_star**10*epsilon + 480.0*P*R**2*S**2*UGA_star**10 - 20736.0*P*R**2*S**2*UGA_star**9*epsilon - 1920.0*P*R**2*S**2*UGA_star**9 + 31776.0*P*R**2*S**2*UGA_star**8*epsilon + 2736.0*P*R**2*S**2*UGA_star**8 - 22848.0*P*R**2*S**2*UGA_star**7*epsilon - 1344.0*P*R**2*S**2*UGA_star**7 + 7488.0*P*R**2*S**2*UGA_star**6*epsilon - 384.0*P*R**2*S**2*UGA_star**6 - 960.0*P*R**2*S**2*UGA_star**5*epsilon + 576.0*P*R**2*S**2*UGA_star**5 + 96.0*P*R**2*S**2*UGA_star**4*epsilon - 144.0*P*R**2*S**2*UGA_star**4 + 9792.0*P*R**2*S*T*UGA_star**10*epsilon + 1440.0*P*R**2*S*T*UGA_star**10 - 39168.0*P*R**2*S*T*UGA_star**9*epsilon - 5760.0*P*R**2*S*T*UGA_star**9 + 58112.0*P*R**2*S*T*UGA_star**8*epsilon + 8736.0*P*R**2*S*T*UGA_star**8 - 36096.0*P*R**2*S*T*UGA_star**7*epsilon - 6048.0*P*R**2*S*T*UGA_star**7 + 4416.0*P*R**2*S*T*UGA_star**6*epsilon + 1728.0*P*R**2*S*T*UGA_star**6 + 4096.0*P*R**2*S*T*UGA_star**5*epsilon - 96.0*P*R**2*S*T*UGA_star**5 - 1152.0*P*R**2*S*T*UGA_star**4*epsilon + 4896.0*P*R**2*T**2*UGA_star**10*epsilon + 720.0*P*R**2*T**2*UGA_star**10 - 19584.0*P*R**2*T**2*UGA_star**9*epsilon - 2880.0*P*R**2*T**2*UGA_star**9 + 29024.0*P*R**2*T**2*UGA_star**8*epsilon + 4416.0*P*R**2*T**2*UGA_star**8 - 18816.0*P*R**2*T**2*UGA_star**7*epsilon - 3168.0*P*R**2*T**2*UGA_star**7 + 4416.0*P*R**2*T**2*UGA_star**6*epsilon + 1008.0*P*R**2*T**2*UGA_star**6 + 64.0*P*R**2*T**2*UGA_star**5*epsilon - 96.0*P*R**2*T**2*UGA_star**5 - 3456.0*P*R*S**3*UGA_star**10*epsilon - 320.0*P*R*S**3*UGA_star**10 + 17280.0*P*R*S**3*UGA_star**9*epsilon + 1600.0*P*R*S**3*UGA_star**9 - 35168.0*P*R*S**3*UGA_star**8*epsilon - 3152.0*P*R*S**3*UGA_star**8 + 37024.0*P*R*S**3*UGA_star**7*epsilon + 2960.0*P*R*S**3*UGA_star**7 - 21056.0*P*R*S**3*UGA_star**6*epsilon - 1120.0*P*R*S**3*UGA_star**6 + 6080.0*P*R*S**3*UGA_star**5*epsilon - 160.0*P*R*S**3*UGA_star**5 - 736.0*P*R*S**3*UGA_star**4*epsilon + 240.0*P*R*S**3*UGA_star**4 + 32.0*P*R*S**3*UGA_star**3*epsilon - 48.0*P*R*S**3*UGA_star**3 - 9792.0*P*R*S**2*T*UGA_star**10*epsilon - 1440.0*P*R*S**2*T*UGA_star**10 + 48960.0*P*R*S**2*T*UGA_star**9*epsilon + 7200.0*P*R*S**2*T*UGA_star**9 - 97792.0*P*R*S**2*T*UGA_star**8*epsilon - 14592.0*P*R*S**2*T*UGA_star**8 + 96832.0*P*R*S**2*T*UGA_star**7*epsilon + 15168.0*P*R*S**2*T*UGA_star**7 - 45888.0*P*R*S**2*T*UGA_star**6*epsilon - 8352.0*P*R*S**2*T*UGA_star**6 + 5824.0*P*R*S**2*T*UGA_star**5*epsilon + 2208.0*P*R*S**2*T*UGA_star**5 + 2432.0*P*R*S**2*T*UGA_star**4*epsilon - 192.0*P*R*S**2*T*UGA_star**4 - 576.0*P*R*S**2*T*UGA_star**3*epsilon - 9792.0*P*R*S*T**2*UGA_star**10*epsilon - 1440.0*P*R*S*T**2*UGA_star**10 + 48960.0*P*R*S*T**2*UGA_star**9*epsilon + 7200.0*P*R*S*T**2*UGA_star**9 - 97792.0*P*R*S*T**2*UGA_star**8*epsilon - 14592.0*P*R*S*T**2*UGA_star**8 + 97984.0*P*R*S*T**2*UGA_star**7*epsilon + 15168.0*P*R*S*T**2*UGA_star**7 - 49920.0*P*R*S*T**2*UGA_star**6*epsilon - 8352.0*P*R*S*T**2*UGA_star**6 + 11008.0*P*R*S*T**2*UGA_star**5*epsilon + 2208.0*P*R*S*T**2*UGA_star**5 - 448.0*P*R*S*T**2*UGA_star**4*epsilon - 192.0*P*R*S*T**2*UGA_star**4 - 3456.0*P*R*T**3*UGA_star**10*epsilon - 320.0*P*R*T**3*UGA_star**10 + 17280.0*P*R*T**3*UGA_star**9*epsilon + 1600.0*P*R*T**3*UGA_star**9 - 35168.0*P*R*T**3*UGA_star**8*epsilon - 3152.0*P*R*T**3*UGA_star**8 + 36960.0*P*R*T**3*UGA_star**7*epsilon + 3056.0*P*R*T**3*UGA_star**7 - 20832.0*P*R*T**3*UGA_star**6*epsilon - 1456.0*P*R*T**3*UGA_star**6 + 5792.0*P*R*T**3*UGA_star**5*epsilon + 272.0*P*R*T**3*UGA_star**5 - 576.0*P*R*T**3*UGA_star**4*epsilon + 864.0*P*S**4*UGA_star**10*epsilon + 80.0*P*S**4*UGA_star**10 - 5184.0*P*S**4*UGA_star**9*epsilon - 480.0*P*S**4*UGA_star**9 + 13152.0*P*S**4*UGA_star**8*epsilon + 1200.0*P*S**4*UGA_star**8 - 18240.0*P*S**4*UGA_star**7*epsilon - 1600.0*P*S**4*UGA_star**7 + 14880.0*P*S**4*UGA_star**6*epsilon + 1200.0*P*S**4*UGA_star**6 - 7104.0*P*S**4*UGA_star**5*epsilon - 480.0*P*S**4*UGA_star**5 + 1824.0*P*S**4*UGA_star**4*epsilon + 80.0*P*S**4*UGA_star**4 - 192.0*P*S**4*UGA_star**3*epsilon + 3264.0*P*S**3*T*UGA_star**10*epsilon + 480.0*P*S**3*T*UGA_star**10 - 19584.0*P*S**3*T*UGA_star**9*epsilon - 2880.0*P*S**3*T*UGA_star**9 + 49088.0*P*S**3*T*UGA_star**8*epsilon + 7296.0*P*S**3*T*UGA_star**8 - 65920.0*P*S**3*T*UGA_star**7*epsilon - 10080.0*P*S**3*T*UGA_star**7 + 50240.0*P*S**3*T*UGA_star**6*epsilon + 8160.0*P*S**3*T*UGA_star**6 - 20864.0*P*S**3*T*UGA_star**5*epsilon - 3840.0*P*S**3*T*UGA_star**5 + 3904.0*P*S**3*T*UGA_star**4*epsilon + 960.0*P*S**3*T*UGA_star**4 - 128.0*P*S**3*T*UGA_star**3*epsilon - 96.0*P*S**3*T*UGA_star**3 + 4896.0*P*S**2*T**2*UGA_star**10*epsilon + 720.0*P*S**2*T**2*UGA_star**10 - 29376.0*P*S**2*T**2*UGA_star**9*epsilon - 4320.0*P*S**2*T**2*UGA_star**9 + 73664.0*P*S**2*T**2*UGA_star**8*epsilon + 10896.0*P*S**2*T**2*UGA_star**8 - 99328.0*P*S**2*T**2*UGA_star**7*epsilon - 14880.0*P*S**2*T**2*UGA_star**7 + 76832.0*P*S**2*T**2*UGA_star**6*epsilon + 11760.0*P*S**2*T**2*UGA_star**6 - 33344.0*P*S**2*T**2*UGA_star**5*epsilon - 5280.0*P*S**2*T**2*UGA_star**5 + 7168.0*P*S**2*T**2*UGA_star**4*epsilon + 1200.0*P*S**2*T**2*UGA_star**4 - 512.0*P*S**2*T**2*UGA_star**3*epsilon - 96.0*P*S**2*T**2*UGA_star**3 + 3456.0*P*S*T**3*UGA_star**10*epsilon + 320.0*P*S*T**3*UGA_star**10 - 20736.0*P*S*T**3*UGA_star**9*epsilon - 1920.0*P*S*T**3*UGA_star**9 + 52448.0*P*S*T**3*UGA_star**8*epsilon + 4752.0*P*S*T**3*UGA_star**8 - 72128.0*P*S*T**3*UGA_star**7*epsilon - 6208.0*P*S*T**3*UGA_star**7 + 57792.0*P*S*T**3*UGA_star**6*epsilon + 4512.0*P*S*T**3*UGA_star**6 - 26624.0*P*S*T**3*UGA_star**5*epsilon - 1728.0*P*S*T**3*UGA_star**5 + 6368.0*P*S*T**3*UGA_star**4*epsilon + 272.0*P*S*T**3*UGA_star**4 - 576.0*P*S*T**3*UGA_star**3*epsilon + 960.0*P*T**4*UGA_star**10*epsilon - 5760.0*P*T**4*UGA_star**9*epsilon + 14304.0*P*T**4*UGA_star**8*epsilon - 18816.0*P*T**4*UGA_star**7*epsilon + 13824.0*P*T**4*UGA_star**6*epsilon - 5376.0*P*T**4*UGA_star**5*epsilon + 864.0*P*T**4*UGA_star**4*epsilon + 192.0*R**5*UGA_star**10*epsilon - 96.0*R**5*UGA_star**8*epsilon - 960.0*R**4*S*UGA_star**10*epsilon + 960.0*R**4*S*UGA_star**9*epsilon + 384.0*R**4*S*UGA_star**8*epsilon - 384.0*R**4*S*UGA_star**7*epsilon - 864.0*R**4*T*UGA_star**10*epsilon - 80.0*R**4*T*UGA_star**10 + 864.0*R**4*T*UGA_star**9*epsilon + 80.0*R**4*T*UGA_star**9 - 32.0*R**4*T*UGA_star**8*epsilon + 48.0*R**4*T*UGA_star**8 + 32.0*R**4*T*UGA_star**7*epsilon - 48.0*R**4*T*UGA_star**7 + 1920.0*R**3*S**2*UGA_star**10*epsilon - 3840.0*R**3*S**2*UGA_star**9*epsilon + 1344.0*R**3*S**2*UGA_star**8*epsilon + 1152.0*R**3*S**2*UGA_star**7*epsilon - 576.0*R**3*S**2*UGA_star**6*epsilon + 3456.0*R**3*S*T*UGA_star**10*epsilon + 320.0*R**3*S*T*UGA_star**10 - 6912.0*R**3*S*T*UGA_star**9*epsilon - 640.0*R**3*S*T*UGA_star**9 + 3744.0*R**3*S*T*UGA_star**8*epsilon + 176.0*R**3*S*T*UGA_star**8 - 384.0*R**3*S*T*UGA_star**7*epsilon + 288.0*R**3*S*T*UGA_star**7 + 96.0*R**3*S*T*UGA_star**6*epsilon - 144.0*R**3*S*T*UGA_star**6 + 1632.0*R**3*T**2*UGA_star**10*epsilon + 240.0*R**3*T**2*UGA_star**10 - 3264.0*R**3*T**2*UGA_star**9*epsilon - 480.0*R**3*T**2*UGA_star**9 + 1536.0*R**3*T**2*UGA_star**8*epsilon + 240.0*R**3*T**2*UGA_star**8 + 384.0*R**3*T**2*UGA_star**7*epsilon - 288.0*R**3*T**2*UGA_star**6*epsilon - 1920.0*R**2*S**3*UGA_star**10*epsilon + 5760.0*R**2*S**3*UGA_star**9*epsilon - 5376.0*R**2*S**3*UGA_star**8*epsilon + 768.0*R**2*S**3*UGA_star**7*epsilon + 1152.0*R**2*S**3*UGA_star**6*epsilon - 384.0*R**2*S**3*UGA_star**5*epsilon - 5184.0*R**2*S**2*T*UGA_star**10*epsilon - 480.0*R**2*S**2*T*UGA_star**10 + 15552.0*R**2*S**2*T*UGA_star**9*epsilon + 1440.0*R**2*S**2*T*UGA_star**9 - 16224.0*R**2*S**2*T*UGA_star**8*epsilon - 1296.0*R**2*S**2*T*UGA_star**8 + 6624.0*R**2*S**2*T*UGA_star**7*epsilon + 48.0*R**2*S**2*T*UGA_star**7 - 864.0*R**2*S**2*T*UGA_star**6*epsilon + 432.0*R**2*S**2*T*UGA_star**6 + 96.0*R**2*S**2*T*UGA_star**5*epsilon - 144.0*R**2*S**2*T*UGA_star**5 - 4896.0*R**2*S*T**2*UGA_star**10*epsilon - 720.0*R**2*S*T**2*UGA_star**10 + 14688.0*R**2*S*T**2*UGA_star**9*epsilon + 2160.0*R**2*S*T**2*UGA_star**9 - 14656.0*R**2*S*T**2*UGA_star**8*epsilon - 2208.0*R**2*S*T**2*UGA_star**8 + 4256.0*R**2*S*T**2*UGA_star**7*epsilon + 816.0*R**2*S*T**2*UGA_star**7 + 1184.0*R**2*S*T**2*UGA_star**6*epsilon - 48.0*R**2*S*T**2*UGA_star**6 - 576.0*R**2*S*T**2*UGA_star**5*epsilon - 1632.0*R**2*T**3*UGA_star**10*epsilon - 240.0*R**2*T**3*UGA_star**10 + 4896.0*R**2*T**3*UGA_star**9*epsilon + 720.0*R**2*T**3*UGA_star**9 - 4864.0*R**2*T**3*UGA_star**8*epsilon - 768.0*R**2*T**3*UGA_star**8 + 1568.0*R**2*T**3*UGA_star**7*epsilon + 336.0*R**2*T**3*UGA_star**7 + 32.0*R**2*T**3*UGA_star**6*epsilon - 48.0*R**2*T**3*UGA_star**6 + 960.0*R*S**4*UGA_star**10*epsilon - 3840.0*R*S**4*UGA_star**9*epsilon + 5664.0*R*S**4*UGA_star**8*epsilon - 3456.0*R*S**4*UGA_star**7*epsilon + 384.0*R*S**4*UGA_star**6*epsilon + 384.0*R*S**4*UGA_star**5*epsilon - 96.0*R*S**4*UGA_star**4*epsilon + 3456.0*R*S**3*T*UGA_star**10*epsilon + 320.0*R*S**3*T*UGA_star**10 - 13824.0*R*S**3*T*UGA_star**9*epsilon - 1280.0*R*S**3*T*UGA_star**9 + 21344.0*R*S**3*T*UGA_star**8*epsilon + 1872.0*R*S**3*T*UGA_star**8 - 15680.0*R*S**3*T*UGA_star**7*epsilon - 1088.0*R*S**3*T*UGA_star**7 + 5376.0*R*S**3*T*UGA_star**6*epsilon + 32.0*R*S**3*T*UGA_star**6 - 704.0*R*S**3*T*UGA_star**5*epsilon + 192.0*R*S**3*T*UGA_star**5 + 32.0*R*S**3*T*UGA_star**4*epsilon - 48.0*R*S**3*T*UGA_star**4 + 4896.0*R*S**2*T**2*UGA_star**10*epsilon + 720.0*R*S**2*T**2*UGA_star**10 - 19584.0*R*S**2*T**2*UGA_star**9*epsilon - 2880.0*R*S**2*T**2*UGA_star**9 + 29600.0*R*S**2*T**2*UGA_star**8*epsilon + 4416.0*R*S**2*T**2*UGA_star**8 - 19968.0*R*S**2*T**2*UGA_star**7*epsilon - 3168.0*R*S**2*T**2*UGA_star**7 + 4704.0*R*S**2*T**2*UGA_star**6*epsilon + 1008.0*R*S**2*T**2*UGA_star**6 + 640.0*R*S**2*T**2*UGA_star**5*epsilon - 96.0*R*S**2*T**2*UGA_star**5 - 288.0*R*S**2*T**2*UGA_star**4*epsilon + 3264.0*R*S*T**3*UGA_star**10*epsilon + 480.0*R*S*T**3*UGA_star**10 - 13056.0*R*S*T**3*UGA_star**9*epsilon - 1920.0*R*S*T**3*UGA_star**9 + 19712.0*R*S*T**3*UGA_star**8*epsilon + 2976.0*R*S*T**3*UGA_star**8 - 13440.0*R*S*T**3*UGA_star**7*epsilon - 2208.0*R*S*T**3*UGA_star**7 + 3648.0*R*S*T**3*UGA_star**6*epsilon + 768.0*R*S*T**3*UGA_star**6 - 128.0*R*S*T**3*UGA_star**5*epsilon - 96.0*R*S*T**3*UGA_star**5 + 864.0*R*T**4*UGA_star**10*epsilon + 80.0*R*T**4*UGA_star**10 - 3456.0*R*T**4*UGA_star**9*epsilon - 320.0*R*T**4*UGA_star**9 + 5376.0*R*T**4*UGA_star**8*epsilon + 480.0*R*T**4*UGA_star**8 - 4032.0*R*T**4*UGA_star**7*epsilon - 320.0*R*T**4*UGA_star**7 + 1440.0*R*T**4*UGA_star**6*epsilon + 80.0*R*T**4*UGA_star**6 - 192.0*R*T**4*UGA_star**5*epsilon - 192.0*S**5*UGA_star**10*epsilon + 960.0*S**5*UGA_star**9*epsilon - 1920.0*S**5*UGA_star**8*epsilon + 1920.0*S**5*UGA_star**7*epsilon - 960.0*S**5*UGA_star**6*epsilon + 192.0*S**5*UGA_star**5*epsilon - 864.0*S**4*T*UGA_star**10*epsilon - 80.0*S**4*T*UGA_star**10 + 4320.0*S**4*T*UGA_star**9*epsilon + 400.0*S**4*T*UGA_star**9 - 8832.0*S**4*T*UGA_star**8*epsilon - 800.0*S**4*T*UGA_star**8 + 9408.0*S**4*T*UGA_star**7*epsilon + 800.0*S**4*T*UGA_star**7 - 5472.0*S**4*T*UGA_star**6*epsilon - 400.0*S**4*T*UGA_star**6 + 1632.0*S**4*T*UGA_star**5*epsilon + 80.0*S**4*T*UGA_star**5 - 192.0*S**4*T*UGA_star**4*epsilon - 1632.0*S**3*T**2*UGA_star**10*epsilon - 240.0*S**3*T**2*UGA_star**10 + 8160.0*S**3*T**2*UGA_star**9*epsilon + 1200.0*S**3*T**2*UGA_star**9 - 16480.0*S**3*T**2*UGA_star**8*epsilon - 2448.0*S**3*T**2*UGA_star**8 + 16960.0*S**3*T**2*UGA_star**7*epsilon + 2592.0*S**3*T**2*UGA_star**7 - 9120.0*S**3*T**2*UGA_star**6*epsilon - 1488.0*S**3*T**2*UGA_star**6 + 2272.0*S**3*T**2*UGA_star**5*epsilon + 432.0*S**3*T**2*UGA_star**5 - 160.0*S**3*T**2*UGA_star**4*epsilon - 48.0*S**3*T**2*UGA_star**4 - 1632.0*S**2*T**3*UGA_star**10*epsilon - 240.0*S**2*T**3*UGA_star**10 + 8160.0*S**2*T**3*UGA_star**9*epsilon + 1200.0*S**2*T**3*UGA_star**9 - 16480.0*S**2*T**3*UGA_star**8*epsilon - 2448.0*S**2*T**3*UGA_star**8 + 16960.0*S**2*T**3*UGA_star**7*epsilon + 2592.0*S**2*T**3*UGA_star**7 - 9120.0*S**2*T**3*UGA_star**6*epsilon - 1488.0*S**2*T**3*UGA_star**6 + 2272.0*S**2*T**3*UGA_star**5*epsilon + 432.0*S**2*T**3*UGA_star**5 - 160.0*S**2*T**3*UGA_star**4*epsilon - 48.0*S**2*T**3*UGA_star**4 - 864.0*S*T**4*UGA_star**10*epsilon - 80.0*S*T**4*UGA_star**10 + 4320.0*S*T**4*UGA_star**9*epsilon + 400.0*S*T**4*UGA_star**9 - 8832.0*S*T**4*UGA_star**8*epsilon - 800.0*S*T**4*UGA_star**8 + 9408.0*S*T**4*UGA_star**7*epsilon + 800.0*S*T**4*UGA_star**7 - 5472.0*S*T**4*UGA_star**6*epsilon - 400.0*S*T**4*UGA_star**6 + 1632.0*S*T**4*UGA_star**5*epsilon + 80.0*S*T**4*UGA_star**5 - 192.0*S*T**4*UGA_star**4*epsilon - 192.0*T**5*UGA_star**10*epsilon + 960.0*T**5*UGA_star**9*epsilon - 1920.0*T**5*UGA_star**8*epsilon + 1920.0*T**5*UGA_star**7*epsilon - 960.0*T**5*UGA_star**6*epsilon + 192.0*T**5*UGA_star**5*epsilon - sqrt(2048.0*P**9*R*UGA_star**20*epsilon - 36864.0*P**9*R*UGA_star**19*epsilon + 316416.0*P**9*R*UGA_star**18*epsilon - 1720320.0*P**9*R*UGA_star**17*epsilon + 6636544.0*P**9*R*UGA_star**16*epsilon - 19281920.0*P**9*R*UGA_star**15*epsilon + 43703296.0*P**9*R*UGA_star**14*epsilon - 78966784.0*P**9*R*UGA_star**13*epsilon + 115241984.0*P**9*R*UGA_star**12*epsilon - 136767488.0*P**9*R*UGA_star**11*epsilon + 132228096.0*P**9*R*UGA_star**10*epsilon - 103833600.0*P**9*R*UGA_star**9*epsilon + 65694720.0*P**9*R*UGA_star**8*epsilon - 33015808.0*P**9*R*UGA_star**7*epsilon + 12882944.0*P**9*R*UGA_star**6*epsilon - 3764224.0*P**9*R*UGA_star**5*epsilon + 775168.0*P**9*R*UGA_star**4*epsilon - 100352.0*P**9*R*UGA_star**3*epsilon + 6144.0*P**9*R*UGA_star**2*epsilon - 2048.0*P**9*S*UGA_star**20*epsilon + 38912.0*P**9*S*UGA_star**19*epsilon - 353280.0*P**9*S*UGA_star**18*epsilon + 2036736.0*P**9*S*UGA_star**17*epsilon - 8356864.0*P**9*S*UGA_star**16*epsilon + 25918464.0*P**9*S*UGA_star**15*epsilon - 62985216.0*P**9*S*UGA_star**14*epsilon + 122670080.0*P**9*S*UGA_star**13*epsilon - 194208768.0*P**9*S*UGA_star**12*epsilon + 252009472.0*P**9*S*UGA_star**11*epsilon - 268995584.0*P**9*S*UGA_star**10*epsilon + 236061696.0*P**9*S*UGA_star**9*epsilon - 169528320.0*P**9*S*UGA_star**8*epsilon + 98710528.0*P**9*S*UGA_star**7*epsilon - 45898752.0*P**9*S*UGA_star**6*epsilon + 16647168.0*P**9*S*UGA_star**5*epsilon - 4539392.0*P**9*S*UGA_star**4*epsilon + 875520.0*P**9*S*UGA_star**3*epsilon - 106496.0*P**9*S*UGA_star**2*epsilon + 6144.0*P**9*S*UGA_star*epsilon + 13312.0*P**8*R**2*UGA_star**20*epsilon + 256.0*P**8*R**2*UGA_star**20 - 212992.0*P**8*R**2*UGA_star**19*epsilon - 4096.0*P**8*R**2*UGA_star**19 + 1612800.0*P**8*R**2*UGA_star**18*epsilon + 31232.0*P**8*R**2*UGA_star**18 - 7669760.0*P**8*R**2*UGA_star**17*epsilon - 150528.0*P**8*R**2*UGA_star**17 + 25626624.0*P**8*R**2*UGA_star**16*epsilon + 512768.0*P**8*R**2*UGA_star**16 - 63750144.0*P**8*R**2*UGA_star**15*epsilon - 1307648.0*P**8*R**2*UGA_star**15 + 122045440.0*P**8*R**2*UGA_star**14*epsilon + 2579456.0*P**8*R**2*UGA_star**14 - 183265280.0*P**8*R**2*UGA_star**13*epsilon - 4009984.0*P**8*R**2*UGA_star**13 + 217958400.0*P**8*R**2*UGA_star**12*epsilon + 4958976.0*P**8*R**2*UGA_star**12 - 205815808.0*P**8*R**2*UGA_star**11*epsilon - 4888576.0*P**8*R**2*UGA_star**11 + 153674752.0*P**8*R**2*UGA_star**10*epsilon + 3824128.0*P**8*R**2*UGA_star**10 - 89708544.0*P**8*R**2*UGA_star**9*epsilon - 2345984.0*P**8*R**2*UGA_star**9 + 40110080.0*P**8*R**2*UGA_star**8*epsilon + 1105152.0*P**8*R**2*UGA_star**8 - 13271040.0*P**8*R**2*UGA_star**7*epsilon - 386048.0*P**8*R**2*UGA_star**7 + 3062784.0*P**8*R**2*UGA_star**6*epsilon + 94208.0*P**8*R**2*UGA_star**6 - 440320.0*P**8*R**2*UGA_star**5*epsilon - 14336.0*P**8*R**2*UGA_star**5 + 29696.0*P**8*R**2*UGA_star**4*epsilon + 1024.0*P**8*R**2*UGA_star**4 - 26624.0*P**8*R*S*UGA_star**20*epsilon - 512.0*P**8*R*S*UGA_star**20 + 452608.0*P**8*R*S*UGA_star**19*epsilon + 8704.0*P**8*R*S*UGA_star**19 - 3651584.0*P**8*R*S*UGA_star**18*epsilon - 70656.0*P**8*R*S*UGA_star**18 + 18565120.0*P**8*R*S*UGA_star**17*epsilon + 363520.0*P**8*R*S*UGA_star**17 - 66592768.0*P**8*R*S*UGA_star**16*epsilon - 1326592.0*P**8*R*S*UGA_star**16 + 178753536.0*P**8*R*S*UGA_star**15*epsilon + 3640832.0*P**8*R*S*UGA_star**15 - 371591168.0*P**8*R*S*UGA_star**14*epsilon - 7774208.0*P**8*R*S*UGA_star**14 + 610621440.0*P**8*R*S*UGA_star**13*epsilon + 13178880.0*P**8*R*S*UGA_star**13 - 802447360.0*P**8*R*S*UGA_star**12*epsilon - 17937920.0*P**8*R*S*UGA_star**12 + 847548416.0*P**8*R*S*UGA_star**11*epsilon + 19695104.0*P**8*R*S*UGA_star**11 - 718981120.0*P**8*R*S*UGA_star**10*epsilon - 17425408.0*P**8*R*S*UGA_star**10 + 486766592.0*P**8*R*S*UGA_star**9*epsilon + 12340224.0*P**8*R*S*UGA_star**9 - 259637248.0*P**8*R*S*UGA_star**8*epsilon - 6902272.0*P**8*R*S*UGA_star**8 + 106762240.0*P**8*R*S*UGA_star**7*epsilon + 2982400.0*P**8*R*S*UGA_star**7 - 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12897280.0*P**7*R**2*T*UGA_star**14 + 815817728.0*P**7*R**2*T*UGA_star**13*epsilon + 18044928.0*P**7*R**2*T*UGA_star**13 - 858012672.0*P**7*R**2*T*UGA_star**12*epsilon - 19835904.0*P**7*R**2*T*UGA_star**12 + 703864832.0*P**7*R**2*T*UGA_star**11*epsilon + 17110016.0*P**7*R**2*T*UGA_star**11 - 445885440.0*P**7*R**2*T*UGA_star**10*epsilon - 11472384.0*P**7*R**2*T*UGA_star**10 + 213638144.0*P**7*R**2*T*UGA_star**9*epsilon + 5864960.0*P**7*R**2*T*UGA_star**9 - 74599424.0*P**7*R**2*T*UGA_star**8*epsilon - 2210304.0*P**7*R**2*T*UGA_star**8 + 17739776.0*P**7*R**2*T*UGA_star**7*epsilon + 579072.0*P**7*R**2*T*UGA_star**7 - 2487296.0*P**7*R**2*T*UGA_star**6*epsilon - 94208.0*P**7*R**2*T*UGA_star**6 + 125952.0*P**7*R**2*T*UGA_star**5*epsilon + 7168.0*P**7*R**2*T*UGA_star**5 + 7168.0*P**7*R**2*T*UGA_star**4*epsilon + 116736.0*P**7*R*S**2*UGA_star**20*epsilon + 4608.0*P**7*R*S**2*UGA_star**20 - 1867776.0*P**7*R*S**2*UGA_star**19*epsilon - 73728.0*P**7*R*S**2*UGA_star**19 + 14094336.0*P**7*R*S**2*UGA_star**18*epsilon + 557568.0*P**7*R*S**2*UGA_star**18 - 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2703360.0*P**7*R*S**2*UGA_star**5*epsilon - 138240.0*P**7*R*S**2*UGA_star**5 + 132096.0*P**7*R*S**2*UGA_star**4*epsilon + 9216.0*P**7*R*S**2*UGA_star**4 + 6144.0*P**7*R*S**2*UGA_star**3*epsilon + 212992.0*P**7*R*S*T*UGA_star**20*epsilon + 4096.0*P**7*R*S*T*UGA_star**20 - 3407872.0*P**7*R*S*T*UGA_star**19*epsilon - 65536.0*P**7*R*S*T*UGA_star**19 + 25765888.0*P**7*R*S*T*UGA_star**18*epsilon + 498688.0*P**7*R*S*T*UGA_star**18 - 122163200.0*P**7*R*S*T*UGA_star**17*epsilon - 2394112.0*P**7*R*S*T*UGA_star**17 + 406360064.0*P**7*R*S*T*UGA_star**16*epsilon + 8110080.0*P**7*R*S*T*UGA_star**16 - 1004963840.0*P**7*R*S*T*UGA_star**15*epsilon - 20537344.0*P**7*R*S*T*UGA_star**15 + 1910044672.0*P**7*R*S*T*UGA_star**14*epsilon + 40178688.0*P**7*R*S*T*UGA_star**14 - 2843484160.0*P**7*R*S*T*UGA_star**13*epsilon - 61884416.0*P**7*R*S*T*UGA_star**13 + 3347660800.0*P**7*R*S*T*UGA_star**12*epsilon + 75761664.0*P**7*R*S*T*UGA_star**12 - 3123755008.0*P**7*R*S*T*UGA_star**11*epsilon - 73891840.0*P**7*R*S*T*UGA_star**11 + 2299500544.0*P**7*R*S*T*UGA_star**10*epsilon + 57164800.0*P**7*R*S*T*UGA_star**10 - 1319047168.0*P**7*R*S*T*UGA_star**9*epsilon - 34674688.0*P**7*R*S*T*UGA_star**9 + 576475136.0*P**7*R*S*T*UGA_star**8*epsilon + 16150528.0*P**7*R*S*T*UGA_star**8 - 184678400.0*P**7*R*S*T*UGA_star**7*epsilon - 5578752.0*P**7*R*S*T*UGA_star**7 + 40454144.0*P**7*R*S*T*UGA_star**6*epsilon + 1346560.0*P**7*R*S*T*UGA_star**6 - 5226496.0*P**7*R*S*T*UGA_star**5*epsilon - 202752.0*P**7*R*S*T*UGA_star**5 + 237568.0*P**7*R*S*T*UGA_star**4*epsilon + 14336.0*P**7*R*S*T*UGA_star**4 + 14336.0*P**7*R*S*T*UGA_star**3*epsilon + 73728.0*P**7*R*T**2*UGA_star**20*epsilon - 1179648.0*P**7*R*T**2*UGA_star**19*epsilon + 8933376.0*P**7*R*T**2*UGA_star**18*epsilon - 42491904.0*P**7*R*T**2*UGA_star**17*epsilon + 142033920.0*P**7*R*T**2*UGA_star**16*epsilon - 353611776.0*P**7*R*T**2*UGA_star**15*epsilon + 677935104.0*P**7*R*T**2*UGA_star**14*epsilon - 1020383232.0*P**7*R*T**2*UGA_star**13*epsilon + 1217829888.0*P**7*R*T**2*UGA_star**12*epsilon - 1155686400.0*P**7*R*T**2*UGA_star**11*epsilon + 868589568.0*P**7*R*T**2*UGA_star**10*epsilon - 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98023424.0*P**6*R**3*T*UGA_star**10*epsilon - 4234752.0*P**6*R**3*T*UGA_star**10 + 26959872.0*P**6*R**3*T*UGA_star**9*epsilon + 1171456.0*P**6*R**3*T*UGA_star**9 - 4262912.0*P**6*R**3*T*UGA_star**8*epsilon - 184320.0*P**6*R**3*T*UGA_star**8 + 218112.0*P**6*R**3*T*UGA_star**7*epsilon + 8704.0*P**6*R**3*T*UGA_star**7 + 20480.0*P**6*R**3*T*UGA_star**6*epsilon + 1024.0*P**6*R**3*T*UGA_star**6 + 411648.0*P**6*R**2*S**2*UGA_star**20*epsilon + 23040.0*P**6*R**2*S**2*UGA_star**20 - 5763072.0*P**6*R**2*S**2*UGA_star**19*epsilon - 322560.0*P**6*R**2*S**2*UGA_star**19 + 37592064.0*P**6*R**2*S**2*UGA_star**18*epsilon + 2104320.0*P**6*R**2*S**2*UGA_star**18 - 151412736.0*P**6*R**2*S**2*UGA_star**17*epsilon - 8477184.0*P**6*R**2*S**2*UGA_star**17 + 420639744.0*P**6*R**2*S**2*UGA_star**16*epsilon + 23549952.0*P**6*R**2*S**2*UGA_star**16 - 852486144.0*P**6*R**2*S**2*UGA_star**15*epsilon - 47697408.0*P**6*R**2*S**2*UGA_star**15 + 1299538944.0*P**6*R**2*S**2*UGA_star**14*epsilon + 72568320.0*P**6*R**2*S**2*UGA_star**14 - 1513678848.0*P**6*R**2*S**2*UGA_star**13*epsilon - 84142080.0*P**6*R**2*S**2*UGA_star**13 + 1354079232.0*P**6*R**2*S**2*UGA_star**12*epsilon + 74562048.0*P**6*R**2*S**2*UGA_star**12 - 926711808.0*P**6*R**2*S**2*UGA_star**11*epsilon - 50079744.0*P**6*R**2*S**2*UGA_star**11 + 479069184.0*P**6*R**2*S**2*UGA_star**10*epsilon + 24938496.0*P**6*R**2*S**2*UGA_star**10 - 182956032.0*P**6*R**2*S**2*UGA_star**9*epsilon - 8808960.0*P**6*R**2*S**2*UGA_star**9 + 50064384.0*P**6*R**2*S**2*UGA_star**8*epsilon + 2012160.0*P**6*R**2*S**2*UGA_star**8 - 9578496.0*P**6*R**2*S**2*UGA_star**7*epsilon - 228864.0*P**6*R**2*S**2*UGA_star**7 + 1342464.0*P**6*R**2*S**2*UGA_star**6*epsilon - 4608.0*P**6*R**2*S**2*UGA_star**6 - 165888.0*P**6*R**2*S**2*UGA_star**5*epsilon + 3072.0*P**6*R**2*S**2*UGA_star**5 + 15360.0*P**6*R**2*S**2*UGA_star**4*epsilon + 817152.0*P**6*R**2*S*T*UGA_star**20*epsilon + 32256.0*P**6*R**2*S*T*UGA_star**20 - 11440128.0*P**6*R**2*S*T*UGA_star**19*epsilon - 451584.0*P**6*R**2*S*T*UGA_star**19 + 74821632.0*P**6*R**2*S*T*UGA_star**18*epsilon + 2961408.0*P**6*R**2*S*T*UGA_star**18 - 302917632.0*P**6*R**2*S*T*UGA_star**17*epsilon - 12052992.0*P**6*R**2*S*T*UGA_star**17 + 847739904.0*P**6*R**2*S*T*UGA_star**16*epsilon + 33991680.0*P**6*R**2*S*T*UGA_star**16 - 1733855232.0*P**6*R**2*S*T*UGA_star**15*epsilon - 70202880.0*P**6*R**2*S*T*UGA_star**15 + 2670560256.0*P**6*R**2*S*T*UGA_star**14*epsilon + 109367808.0*P**6*R**2*S*T*UGA_star**14 - 3143522304.0*P**6*R**2*S*T*UGA_star**13*epsilon - 130369536.0*P**6*R**2*S*T*UGA_star**13 + 2837458944.0*P**6*R**2*S*T*UGA_star**12*epsilon + 119268864.0*P**6*R**2*S*T*UGA_star**12 - 1950056448.0*P**6*R**2*S*T*UGA_star**11*epsilon - 83128320.0*P**6*R**2*S*T*UGA_star**11 + 1000593408.0*P**6*R**2*S*T*UGA_star**10*epsilon + 43287552.0*P**6*R**2*S*T*UGA_star**10 - 368867328.0*P**6*R**2*S*T*UGA_star**9*epsilon - 16218624.0*P**6*R**2*S*T*UGA_star**9 + 90627072.0*P**6*R**2*S*T*UGA_star**8*epsilon + 4067328.0*P**6*R**2*S*T*UGA_star**8 - 12429312.0*P**6*R**2*S*T*UGA_star**7*epsilon - 579072.0*P**6*R**2*S*T*UGA_star**7 + 390144.0*P**6*R**2*S*T*UGA_star**6*epsilon + 23040.0*P**6*R**2*S*T*UGA_star**6 + 79872.0*P**6*R**2*S*T*UGA_star**5*epsilon + 3072.0*P**6*R**2*S*T*UGA_star**5 + 372736.0*P**6*R**2*T**2*UGA_star**20*epsilon + 7168.0*P**6*R**2*T**2*UGA_star**20 - 5218304.0*P**6*R**2*T**2*UGA_star**19*epsilon - 100352.0*P**6*R**2*T**2*UGA_star**19 + 34212864.0*P**6*R**2*T**2*UGA_star**18*epsilon + 663040.0*P**6*R**2*T**2*UGA_star**18 - 139173888.0*P**6*R**2*T**2*UGA_star**17*epsilon - 2738176.0*P**6*R**2*T**2*UGA_star**17 + 392158208.0*P**6*R**2*T**2*UGA_star**16*epsilon + 7888640.0*P**6*R**2*T**2*UGA_star**16 - 808931328.0*P**6*R**2*T**2*UGA_star**15*epsilon - 16754176.0*P**6*R**2*T**2*UGA_star**15 + 1258170368.0*P**6*R**2*T**2*UGA_star**14*epsilon + 27020544.0*P**6*R**2*T**2*UGA_star**14 - 1496598528.0*P**6*R**2*T**2*UGA_star**13*epsilon - 33576960.0*P**6*R**2*T**2*UGA_star**13 + 1365313536.0*P**6*R**2*T**2*UGA_star**12*epsilon + 32266752.0*P**6*R**2*T**2*UGA_star**12 - 947838976.0*P**6*R**2*T**2*UGA_star**11*epsilon - 23833600.0*P**6*R**2*T**2*UGA_star**11 + 490585088.0*P**6*R**2*T**2*UGA_star**10*epsilon + 13303808.0*P**6*R**2*T**2*UGA_star**10 - 181899264.0*P**6*R**2*T**2*UGA_star**9*epsilon - 5435392.0*P**6*R**2*T**2*UGA_star**9 + 44657664.0*P**6*R**2*T**2*UGA_star**8*epsilon + 1534720.0*P**6*R**2*T**2*UGA_star**8 - 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16407040.0*P**6*S**4*UGA_star**9 + 128380928.0*P**6*S**4*UGA_star**8*epsilon + 6329856.0*P**6*S**4*UGA_star**8 - 37515264.0*P**6*S**4*UGA_star**7*epsilon - 1644544.0*P**6*S**4*UGA_star**7 + 7088128.0*P**6*S**4*UGA_star**6*epsilon + 221184.0*P**6*S**4*UGA_star**6 - 636928.0*P**6*S**4*UGA_star**5*epsilon + 10752.0*P**6*S**4*UGA_star**5 - 25600.0*P**6*S**4*UGA_star**4*epsilon - 8960.0*P**6*S**4*UGA_star**4 + 8192.0*P**6*S**4*UGA_star**3*epsilon + 1024.0*P**6*S**4*UGA_star**3 + 272384.0*P**6*S**3*T*UGA_star**20*epsilon + 10752.0*P**6*S**3*T*UGA_star**20 - 4358144.0*P**6*S**3*T*UGA_star**19*epsilon - 172032.0*P**6*S**3*T*UGA_star**19 + 32839680.0*P**6*S**3*T*UGA_star**18*epsilon + 1298944.0*P**6*S**3*T*UGA_star**18 - 154667008.0*P**6*S**3*T*UGA_star**17*epsilon - 6142464.0*P**6*S**3*T*UGA_star**17 + 509453312.0*P**6*S**3*T*UGA_star**16*epsilon + 20353024.0*P**6*S**3*T*UGA_star**16 - 1243924480.0*P**6*S**3*T*UGA_star**15*epsilon - 50079744.0*P**6*S**3*T*UGA_star**15 + 2327711744.0*P**6*S**3*T*UGA_star**14*epsilon + 94588416.0*P**6*S**3*T*UGA_star**14 - 3402651648.0*P**6*S**3*T*UGA_star**13*epsilon - 139769344.0*P**6*S**3*T*UGA_star**13 + 3922902016.0*P**6*S**3*T*UGA_star**12*epsilon + 163125248.0*P**6*S**3*T*UGA_star**12 - 3573684224.0*P**6*S**3*T*UGA_star**11*epsilon - 150678528.0*P**6*S**3*T*UGA_star**11 + 2558302208.0*P**6*S**3*T*UGA_star**10*epsilon + 109604352.0*P**6*S**3*T*UGA_star**10 - 1418950656.0*P**6*S**3*T*UGA_star**9*epsilon - 61974016.0*P**6*S**3*T*UGA_star**9 + 593837056.0*P**6*S**3*T*UGA_star**8*epsilon + 26597376.0*P**6*S**3*T*UGA_star**8 - 178731008.0*P**6*S**3*T*UGA_star**7*epsilon - 8310784.0*P**6*S**3*T*UGA_star**7 + 35110912.0*P**6*S**3*T*UGA_star**6*epsilon + 1749504.0*P**6*S**3*T*UGA_star**6 - 3418112.0*P**6*S**3*T*UGA_star**5*epsilon - 207360.0*P**6*S**3*T*UGA_star**5 - 82944.0*P**6*S**3*T*UGA_star**4*epsilon + 5632.0*P**6*S**3*T*UGA_star**4 + 38912.0*P**6*S**3*T*UGA_star**3*epsilon + 1024.0*P**6*S**3*T*UGA_star**3 + 372736.0*P**6*S**2*T**2*UGA_star**20*epsilon + 7168.0*P**6*S**2*T**2*UGA_star**20 - 5963776.0*P**6*S**2*T**2*UGA_star**19*epsilon - 114688.0*P**6*S**2*T**2*UGA_star**19 + 45022208.0*P**6*S**2*T**2*UGA_star**18*epsilon + 870912.0*P**6*S**2*T**2*UGA_star**18 - 212817920.0*P**6*S**2*T**2*UGA_star**17*epsilon - 4164608.0*P**6*S**2*T**2*UGA_star**17 + 704718848.0*P**6*S**2*T**2*UGA_star**16*epsilon + 14028032.0*P**6*S**2*T**2*UGA_star**16 - 1732421632.0*P**6*S**2*T**2*UGA_star**15*epsilon - 35269632.0*P**6*S**2*T**2*UGA_star**15 + 3268191232.0*P**6*S**2*T**2*UGA_star**14*epsilon + 68417536.0*P**6*S**2*T**2*UGA_star**14 - 4821870592.0*P**6*S**2*T**2*UGA_star**13*epsilon - 104372224.0*P**6*S**2*T**2*UGA_star**13 + 5616680960.0*P**6*S**2*T**2*UGA_star**12*epsilon + 126441216.0*P**6*S**2*T**2*UGA_star**12 - 5175064576.0*P**6*S**2*T**2*UGA_star**11*epsilon - 121944064.0*P**6*S**2*T**2*UGA_star**11 + 3751576576.0*P**6*S**2*T**2*UGA_star**10*epsilon + 93237760.0*P**6*S**2*T**2*UGA_star**10 - 2110908416.0*P**6*S**2*T**2*UGA_star**9*epsilon - 55876608.0*P**6*S**2*T**2*UGA_star**9 + 899041280.0*P**6*S**2*T**2*UGA_star**8*epsilon + 25709312.0*P**6*S**2*T**2*UGA_star**8 - 277207040.0*P**6*S**2*T**2*UGA_star**7*epsilon - 8772608.0*P**6*S**2*T**2*UGA_star**7 + 56777728.0*P**6*S**2*T**2*UGA_star**6*epsilon + 2092032.0*P**6*S**2*T**2*UGA_star**6 - 6215680.0*P**6*S**2*T**2*UGA_star**5*epsilon - 311296.0*P**6*S**2*T**2*UGA_star**5 + 40960.0*P**6*S**2*T**2*UGA_star**4*epsilon + 21760.0*P**6*S**2*T**2*UGA_star**4 + 47104.0*P**6*S**2*T**2*UGA_star**3*epsilon + 172032.0*P**6*S*T**3*UGA_star**20*epsilon - 2752512.0*P**6*S*T**3*UGA_star**19*epsilon + 20815872.0*P**6*S*T**3*UGA_star**18*epsilon - 98746368.0*P**6*S*T**3*UGA_star**17*epsilon + 328788992.0*P**6*S*T**3*UGA_star**16*epsilon - 814485504.0*P**6*S*T**3*UGA_star**15*epsilon + 1552201728.0*P**6*S*T**3*UGA_star**14*epsilon - 2320338944.0*P**6*S*T**3*UGA_star**13*epsilon + 2748404736.0*P**6*S*T**3*UGA_star**12*epsilon - 2586845184.0*P**6*S*T**3*UGA_star**11*epsilon + 1927360512.0*P**6*S*T**3*UGA_star**10*epsilon - 1124229120.0*P**6*S*T**3*UGA_star**9*epsilon + 503043072.0*P**6*S*T**3*UGA_star**8*epsilon - 166842368.0*P**6*S*T**3*UGA_star**7*epsilon + 38664192.0*P**6*S*T**3*UGA_star**6*epsilon - 5591040.0*P**6*S*T**3*UGA_star**5*epsilon + 379904.0*P**6*S*T**3*UGA_star**4*epsilon + 81920.0*P**5*R**5*UGA_star**20*epsilon + 5120.0*P**5*R**5*UGA_star**20 - 819200.0*P**5*R**5*UGA_star**19*epsilon - 51200.0*P**5*R**5*UGA_star**19 + 3704832.0*P**5*R**5*UGA_star**18*epsilon + 231424.0*P**5*R**5*UGA_star**18 - 9996288.0*P**5*R**5*UGA_star**17*epsilon - 623616.0*P**5*R**5*UGA_star**17 + 17878016.0*P**5*R**5*UGA_star**16*epsilon + 1113088.0*P**5*R**5*UGA_star**16 - 22282240.0*P**5*R**5*UGA_star**15*epsilon - 1384448.0*P**5*R**5*UGA_star**15 + 19841024.0*P**5*R**5*UGA_star**14*epsilon + 1232896.0*P**5*R**5*UGA_star**14 - 12783616.0*P**5*R**5*UGA_star**13*epsilon - 800768.0*P**5*R**5*UGA_star**13 + 6023168.0*P**5*R**5*UGA_star**12*epsilon + 388096.0*P**5*R**5*UGA_star**12 - 2113536.0*P**5*R**5*UGA_star**11*epsilon - 145408.0*P**5*R**5*UGA_star**11 + 563200.0*P**5*R**5*UGA_star**10*epsilon + 43008.0*P**5*R**5*UGA_star**10 - 108544.0*P**5*R**5*UGA_star**9*epsilon - 9216.0*P**5*R**5*UGA_star**9 + 11264.0*P**5*R**5*UGA_star**8*epsilon + 1024.0*P**5*R**5*UGA_star**8 - 409600.0*P**5*R**4*S*UGA_star**20*epsilon - 25600.0*P**5*R**4*S*UGA_star**20 + 4505600.0*P**5*R**4*S*UGA_star**19*epsilon + 281600.0*P**5*R**4*S*UGA_star**19 - 22588416.0*P**5*R**4*S*UGA_star**18*epsilon - 1411072.0*P**5*R**4*S*UGA_star**18 + 68178944.0*P**5*R**4*S*UGA_star**17*epsilon + 4254208.0*P**5*R**4*S*UGA_star**17 - 137843712.0*P**5*R**4*S*UGA_star**16*epsilon - 8585728.0*P**5*R**4*S*UGA_star**16 + 196517888.0*P**5*R**4*S*UGA_star**15*epsilon + 12214272.0*P**5*R**4*S*UGA_star**15 - 202627072.0*P**5*R**4*S*UGA_star**14*epsilon - 12576768.0*P**5*R**4*S*UGA_star**14 + 152729600.0*P**5*R**4*S*UGA_star**13*epsilon + 9501696.0*P**5*R**4*S*UGA_star**13 - 84428800.0*P**5*R**4*S*UGA_star**12*epsilon - 5320704.0*P**5*R**4*S*UGA_star**12 + 34394112.0*P**5*R**4*S*UGA_star**11*epsilon + 2249728.0*P**5*R**4*S*UGA_star**11 - 10533888.0*P**5*R**4*S*UGA_star**10*epsilon - 745472.0*P**5*R**4*S*UGA_star**10 + 2511872.0*P**5*R**4*S*UGA_star**9*epsilon + 199168.0*P**5*R**4*S*UGA_star**9 - 451584.0*P**5*R**4*S*UGA_star**8*epsilon - 39424.0*P**5*R**4*S*UGA_star**8 + 45056.0*P**5*R**4*S*UGA_star**7*epsilon + 4096.0*P**5*R**4*S*UGA_star**7 - 411648.0*P**5*R**4*T*UGA_star**20*epsilon - 23040.0*P**5*R**4*T*UGA_star**20 + 4528128.0*P**5*R**4*T*UGA_star**19*epsilon + 253440.0*P**5*R**4*T*UGA_star**19 - 22738944.0*P**5*R**4*T*UGA_star**18*epsilon - 1273344.0*P**5*R**4*T*UGA_star**18 + 68788224.0*P**5*R**4*T*UGA_star**17*epsilon + 3855360.0*P**5*R**4*T*UGA_star**17 - 139238400.0*P**5*R**4*T*UGA_star**16*epsilon - 7810560.0*P**5*R**4*T*UGA_star**16 + 197974016.0*P**5*R**4*T*UGA_star**15*epsilon + 11105280.0*P**5*R**4*T*UGA_star**15 - 201834496.0*P**5*R**4*T*UGA_star**14*epsilon - 11289600.0*P**5*R**4*T*UGA_star**14 + 147857408.0*P**5*R**4*T*UGA_star**13*epsilon + 8183808.0*P**5*R**4*T*UGA_star**13 - 76853248.0*P**5*R**4*T*UGA_star**12*epsilon - 4124160.0*P**5*R**4*T*UGA_star**12 + 27699200.0*P**5*R**4*T*UGA_star**11*epsilon + 1359360.0*P**5*R**4*T*UGA_star**11 - 6842368.0*P**5*R**4*T*UGA_star**10*epsilon - 253440.0*P**5*R**4*T*UGA_star**10 + 1261568.0*P**5*R**4*T*UGA_star**9*epsilon + 15360.0*P**5*R**4*T*UGA_star**9 - 216064.0*P**5*R**4*T*UGA_star**8*epsilon + 1536.0*P**5*R**4*T*UGA_star**8 + 26624.0*P**5*R**4*T*UGA_star**7*epsilon + 819200.0*P**5*R**3*S**2*UGA_star**20*epsilon + 51200.0*P**5*R**3*S**2*UGA_star**20 - 9830400.0*P**5*R**3*S**2*UGA_star**19*epsilon - 614400.0*P**5*R**3*S**2*UGA_star**19 + 54124544.0*P**5*R**3*S**2*UGA_star**18*epsilon + 3381248.0*P**5*R**3*S**2*UGA_star**18 - 180817920.0*P**5*R**3*S**2*UGA_star**17*epsilon - 11284480.0*P**5*R**3*S**2*UGA_star**17 + 408338432.0*P**5*R**3*S**2*UGA_star**16*epsilon + 25442304.0*P**5*R**3*S**2*UGA_star**16 - 657121280.0*P**5*R**3*S**2*UGA_star**15*epsilon - 40857600.0*P**5*R**3*S**2*UGA_star**15 + 773836800.0*P**5*R**3*S**2*UGA_star**14*epsilon + 48015360.0*P**5*R**3*S**2*UGA_star**14 - 674193408.0*P**5*R**3*S**2*UGA_star**13*epsilon - 41803776.0*P**5*R**3*S**2*UGA_star**13 + 434749440.0*P**5*R**3*S**2*UGA_star**12*epsilon + 27064320.0*P**5*R**3*S**2*UGA_star**12 - 206372864.0*P**5*R**3*S**2*UGA_star**11*epsilon - 13058048.0*P**5*R**3*S**2*UGA_star**11 + 72007680.0*P**5*R**3*S**2*UGA_star**10*epsilon + 4761600.0*P**5*R**3*S**2*UGA_star**10 - 18944000.0*P**5*R**3*S**2*UGA_star**9*epsilon - 1372160.0*P**5*R**3*S**2*UGA_star**9 + 4030464.0*P**5*R**3*S**2*UGA_star**8*epsilon + 329728.0*P**5*R**3*S**2*UGA_star**8 - 696320.0*P**5*R**3*S**2*UGA_star**7*epsilon - 61440.0*P**5*R**3*S**2*UGA_star**7 + 69632.0*P**5*R**3*S**2*UGA_star**6*epsilon + 6144.0*P**5*R**3*S**2*UGA_star**6 + 1646592.0*P**5*R**3*S*T*UGA_star**20*epsilon + 92160.0*P**5*R**3*S*T*UGA_star**20 - 19759104.0*P**5*R**3*S*T*UGA_star**19*epsilon - 1105920.0*P**5*R**3*S*T*UGA_star**19 + 109025280.0*P**5*R**3*S*T*UGA_star**18*epsilon + 6104064.0*P**5*R**3*S*T*UGA_star**18 - 365629440.0*P**5*R**3*S*T*UGA_star**17*epsilon - 20481024.0*P**5*R**3*S*T*UGA_star**17 + 829646848.0*P**5*R**3*S*T*UGA_star**16*epsilon + 46492160.0*P**5*R**3*S*T*UGA_star**16 - 1341159424.0*P**5*R**3*S*T*UGA_star**15*epsilon - 75133440.0*P**5*R**3*S*T*UGA_star**15 + 1582921728.0*P**5*R**3*S*T*UGA_star**14*epsilon + 88473600.0*P**5*R**3*S*T*UGA_star**14 - 1374044160.0*P**5*R**3*S*T*UGA_star**13*epsilon - 76259328.0*P**5*R**3*S*T*UGA_star**13 + 871403520.0*P**5*R**3*S*T*UGA_star**12*epsilon + 47490048.0*P**5*R**3*S*T*UGA_star**12 - 395759616.0*P**5*R**3*S*T*UGA_star**11*epsilon - 20597760.0*P**5*R**3*S*T*UGA_star**11 + 124741632.0*P**5*R**3*S*T*UGA_star**10*epsilon + 5729280.0*P**5*R**3*S*T*UGA_star**10 - 26701824.0*P**5*R**3*S*T*UGA_star**9*epsilon - 814080.0*P**5*R**3*S*T*UGA_star**9 + 4270080.0*P**5*R**3*S*T*UGA_star**8*epsilon - 1536.0*P**5*R**3*S*T*UGA_star**8 - 686080.0*P**5*R**3*S*T*UGA_star**7*epsilon + 11776.0*P**5*R**3*S*T*UGA_star**7 + 83968.0*P**5*R**3*S*T*UGA_star**6*epsilon + 817152.0*P**5*R**3*T**2*UGA_star**20*epsilon + 32256.0*P**5*R**3*T**2*UGA_star**20 - 9805824.0*P**5*R**3*T**2*UGA_star**19*epsilon - 387072.0*P**5*R**3*T**2*UGA_star**19 + 54251520.0*P**5*R**3*T**2*UGA_star**18*epsilon + 2148864.0*P**5*R**3*T**2*UGA_star**18 - 182857728.0*P**5*R**3*T**2*UGA_star**17*epsilon - 7292928.0*P**5*R**3*T**2*UGA_star**17 + 417783808.0*P**5*R**3*T**2*UGA_star**16*epsilon + 16832000.0*P**5*R**3*T**2*UGA_star**16 - 680720384.0*P**5*R**3*T**2*UGA_star**15*epsilon - 27781632.0*P**5*R**3*T**2*UGA_star**15 + 809453568.0*P**5*R**3*T**2*UGA_star**14*epsilon + 33537024.0*P**5*R**3*T**2*UGA_star**14 - 705576960.0*P**5*R**3*T**2*UGA_star**13*epsilon - 29718528.0*P**5*R**3*T**2*UGA_star**13 + 445003776.0*P**5*R**3*T**2*UGA_star**12*epsilon + 19063296.0*P**5*R**3*T**2*UGA_star**12 - 195833856.0*P**5*R**3*T**2*UGA_star**11*epsilon - 8524800.0*P**5*R**3*T**2*UGA_star**11 + 55468032.0*P**5*R**3*T**2*UGA_star**10*epsilon + 2443776.0*P**5*R**3*T**2*UGA_star**10 - 8146944.0*P**5*R**3*T**2*UGA_star**9*epsilon - 356352.0*P**5*R**3*T**2*UGA_star**9 + 61440.0*P**5*R**3*T**2*UGA_star**8*epsilon - 1536.0*P**5*R**3*T**2*UGA_star**8 + 100352.0*P**5*R**3*T**2*UGA_star**7*epsilon + 5632.0*P**5*R**3*T**2*UGA_star**7 + 2048.0*P**5*R**3*T**2*UGA_star**6*epsilon - 819200.0*P**5*R**2*S**3*UGA_star**20*epsilon - 51200.0*P**5*R**2*S**3*UGA_star**20 + 10649600.0*P**5*R**2*S**3*UGA_star**19*epsilon + 665600.0*P**5*R**2*S**3*UGA_star**19 - 63891456.0*P**5*R**2*S**3*UGA_star**18*epsilon - 3991552.0*P**5*R**2*S**3*UGA_star**18 + 234162176.0*P**5*R**2*S**3*UGA_star**17*epsilon + 14615552.0*P**5*R**2*S**3*UGA_star**17 - 584734720.0*P**5*R**2*S**3*UGA_star**16*epsilon - 36443136.0*P**5*R**2*S**3*UGA_star**16 + 1050173440.0*P**5*R**2*S**3*UGA_star**15*epsilon + 65319936.0*P**5*R**2*S**3*UGA_star**15 - 1395015680.0*P**5*R**2*S**3*UGA_star**14*epsilon - 86563840.0*P**5*R**2*S**3*UGA_star**14 + 1387395072.0*P**5*R**2*S**3*UGA_star**13*epsilon + 85899264.0*P**5*R**2*S**3*UGA_star**13 - 1033531392.0*P**5*R**2*S**3*UGA_star**12*epsilon - 63934464.0*P**5*R**2*S**3*UGA_star**12 + 571228160.0*P**5*R**2*S**3*UGA_star**11*epsilon + 35459072.0*P**5*R**2*S**3*UGA_star**11 - 230205440.0*P**5*R**2*S**3*UGA_star**10*epsilon - 14508032.0*P**5*R**2*S**3*UGA_star**10 + 66631680.0*P**5*R**2*S**3*UGA_star**9*epsilon + 4387840.0*P**5*R**2*S**3*UGA_star**9 - 14268416.0*P**5*R**2*S**3*UGA_star**8*epsilon - 1037312.0*P**5*R**2*S**3*UGA_star**8 + 2646016.0*P**5*R**2*S**3*UGA_star**7*epsilon + 218112.0*P**5*R**2*S**3*UGA_star**7 - 471040.0*P**5*R**2*S**3*UGA_star**6*epsilon - 39936.0*P**5*R**2*S**3*UGA_star**6 + 51200.0*P**5*R**2*S**3*UGA_star**5*epsilon + 4096.0*P**5*R**2*S**3*UGA_star**5 - 2469888.0*P**5*R**2*S**2*T*UGA_star**20*epsilon - 138240.0*P**5*R**2*S**2*T*UGA_star**20 + 32108544.0*P**5*R**2*S**2*T*UGA_star**19*epsilon + 1797120.0*P**5*R**2*S**2*T*UGA_star**19 - 193112064.0*P**5*R**2*S**2*T*UGA_star**18*epsilon - 10810368.0*P**5*R**2*S**2*T*UGA_star**18 + 711198720.0*P**5*R**2*S**2*T*UGA_star**17*epsilon + 39822336.0*P**5*R**2*S**2*T*UGA_star**17 - 1788506112.0*P**5*R**2*S**2*T*UGA_star**16*epsilon - 100151808.0*P**5*R**2*S**2*T*UGA_star**16 + 3240984576.0*P**5*R**2*S**2*T*UGA_star**15*epsilon + 181386240.0*P**5*R**2*S**2*T*UGA_star**15 - 4350117888.0*P**5*R**2*S**2*T*UGA_star**14*epsilon - 242956800.0*P**5*R**2*S**2*T*UGA_star**14 + 4373895168.0*P**5*R**2*S**2*T*UGA_star**13*epsilon + 242989056.0*P**5*R**2*S**2*T*UGA_star**13 - 3289927680.0*P**5*R**2*S**2*T*UGA_star**12*epsilon - 180559872.0*P**5*R**2*S**2*T*UGA_star**12 + 1825910784.0*P**5*R**2*S**2*T*UGA_star**11*epsilon + 97514496.0*P**5*R**2*S**2*T*UGA_star**11 - 726939648.0*P**5*R**2*S**2*T*UGA_star**10*epsilon - 36403200.0*P**5*R**2*S**2*T*UGA_star**10 + 198457344.0*P**5*R**2*S**2*T*UGA_star**9*epsilon + 8340480.0*P**5*R**2*S**2*T*UGA_star**9 - 35426304.0*P**5*R**2*S**2*T*UGA_star**8*epsilon - 741888.0*P**5*R**2*S**2*T*UGA_star**8 + 4546560.0*P**5*R**2*S**2*T*UGA_star**7*epsilon - 113664.0*P**5*R**2*S**2*T*UGA_star**7 - 694272.0*P**5*R**2*S**2*T*UGA_star**6*epsilon + 26112.0*P**5*R**2*S**2*T*UGA_star**6 + 92160.0*P**5*R**2*S**2*T*UGA_star**5*epsilon - 2451456.0*P**5*R**2*S*T**2*UGA_star**20*epsilon - 96768.0*P**5*R**2*S*T**2*UGA_star**20 + 31868928.0*P**5*R**2*S*T**2*UGA_star**19*epsilon + 1257984.0*P**5*R**2*S*T**2*UGA_star**19 - 192172032.0*P**5*R**2*S*T**2*UGA_star**18*epsilon - 7607808.0*P**5*R**2*S*T**2*UGA_star**18 + 711327744.0*P**5*R**2*S*T**2*UGA_star**17*epsilon + 28325376.0*P**5*R**2*S*T**2*UGA_star**17 - 1801878528.0*P**5*R**2*S*T**2*UGA_star**16*epsilon - 72374784.0*P**5*R**2*S*T**2*UGA_star**16 + 3295051776.0*P**5*R**2*S*T**2*UGA_star**15*epsilon + 133840896.0*P**5*R**2*S*T**2*UGA_star**15 - 4468448256.0*P**5*R**2*S*T**2*UGA_star**14*epsilon - 183955968.0*P**5*R**2*S*T**2*UGA_star**14 + 4539561984.0*P**5*R**2*S*T**2*UGA_star**13*epsilon + 189766656.0*P**5*R**2*S*T**2*UGA_star**13 - 3442065408.0*P**5*R**2*S*T**2*UGA_star**12*epsilon - 146345472.0*P**5*R**2*S*T**2*UGA_star**12 + 1910900736.0*P**5*R**2*S*T**2*UGA_star**11*epsilon + 82764288.0*P**5*R**2*S*T**2*UGA_star**11 - 744228864.0*P**5*R**2*S*T**2*UGA_star**10*epsilon - 32905728.0*P**5*R**2*S*T**2*UGA_star**10 + 185315328.0*P**5*R**2*S*T**2*UGA_star**9*epsilon + 8400384.0*P**5*R**2*S*T**2*UGA_star**9 - 22551552.0*P**5*R**2*S*T**2*UGA_star**8*epsilon - 1064448.0*P**5*R**2*S*T**2*UGA_star**8 - 577536.0*P**5*R**2*S*T**2*UGA_star**7*epsilon - 21504.0*P**5*R**2*S*T**2*UGA_star**7 + 340992.0*P**5*R**2*S*T**2*UGA_star**6*epsilon + 16896.0*P**5*R**2*S*T**2*UGA_star**6 + 6144.0*P**5*R**2*S*T**2*UGA_star**5*epsilon - 745472.0*P**5*R**2*T**3*UGA_star**20*epsilon - 14336.0*P**5*R**2*T**3*UGA_star**20 + 9691136.0*P**5*R**2*T**3*UGA_star**19*epsilon + 186368.0*P**5*R**2*T**3*UGA_star**19 - 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14806016.0*P**4*R**5*T*UGA_star**18*epsilon - 924672.0*P**4*R**5*T*UGA_star**18 + 34898944.0*P**4*R**5*T*UGA_star**17*epsilon + 2175488.0*P**4*R**5*T*UGA_star**17 - 53414912.0*P**4*R**5*T*UGA_star**16*epsilon - 3319808.0*P**4*R**5*T*UGA_star**16 + 55525376.0*P**4*R**5*T*UGA_star**15*epsilon + 3440640.0*P**4*R**5*T*UGA_star**15 - 39954432.0*P**4*R**5*T*UGA_star**14*epsilon - 2480128.0*P**4*R**5*T*UGA_star**14 + 20064256.0*P**4*R**5*T*UGA_star**13*epsilon + 1272832.0*P**4*R**5*T*UGA_star**13 - 7112704.0*P**4*R**5*T*UGA_star**12*epsilon - 488448.0*P**4*R**5*T*UGA_star**12 + 1810432.0*P**4*R**5*T*UGA_star**11*epsilon + 150528.0*P**4*R**5*T*UGA_star**11 - 306176.0*P**4*R**5*T*UGA_star**10*epsilon - 35840.0*P**4*R**5*T*UGA_star**10 + 13312.0*P**4*R**5*T*UGA_star**9*epsilon + 4608.0*P**4*R**5*T*UGA_star**9 + 5120.0*P**4*R**5*T*UGA_star**8*epsilon + 1029120.0*P**4*R**4*S**2*UGA_star**20*epsilon + 57600.0*P**4*R**4*S**2*UGA_star**20 - 10291200.0*P**4*R**4*S**2*UGA_star**19*epsilon - 576000.0*P**4*R**4*S**2*UGA_star**19 + 46417920.0*P**4*R**4*S**2*UGA_star**18*epsilon + 2599680.0*P**4*R**4*S**2*UGA_star**18 - 124692480.0*P**4*R**4*S**2*UGA_star**17*epsilon - 6996480.0*P**4*R**4*S**2*UGA_star**17 + 222096384.0*P**4*R**4*S**2*UGA_star**16*epsilon + 12510976.0*P**4*R**4*S**2*UGA_star**16 - 276938752.0*P**4*R**4*S**2*UGA_star**15*epsilon - 15715328.0*P**4*R**4*S**2*UGA_star**15 + 249884672.0*P**4*R**4*S**2*UGA_star**14*epsilon + 14361088.0*P**4*R**4*S**2*UGA_star**14 - 167563264.0*P**4*R**4*S**2*UGA_star**13*epsilon - 9829376.0*P**4*R**4*S**2*UGA_star**13 + 85806080.0*P**4*R**4*S**2*UGA_star**12*epsilon + 5190400.0*P**4*R**4*S**2*UGA_star**12 - 34146304.0*P**4*R**4*S**2*UGA_star**11*epsilon - 2157056.0*P**4*R**4*S**2*UGA_star**11 + 10176512.0*P**4*R**4*S**2*UGA_star**10*epsilon + 686848.0*P**4*R**4*S**2*UGA_star**10 - 1902592.0*P**4*R**4*S**2*UGA_star**9*epsilon - 147968.0*P**4*R**4*S**2*UGA_star**9 + 103424.0*P**4*R**4*S**2*UGA_star**8*epsilon + 15616.0*P**4*R**4*S**2*UGA_star**8 + 20480.0*P**4*R**4*S**2*UGA_star**7*epsilon + 2048000.0*P**4*R**4*S*T*UGA_star**20*epsilon + 128000.0*P**4*R**4*S*T*UGA_star**20 - 20480000.0*P**4*R**4*S*T*UGA_star**19*epsilon - 1280000.0*P**4*R**4*S*T*UGA_star**19 + 92303360.0*P**4*R**4*S*T*UGA_star**18*epsilon + 5765120.0*P**4*R**4*S*T*UGA_star**18 - 247050240.0*P**4*R**4*S*T*UGA_star**17*epsilon - 15406080.0*P**4*R**4*S*T*UGA_star**17 + 435419136.0*P**4*R**4*S*T*UGA_star**16*epsilon + 27083264.0*P**4*R**4*S*T*UGA_star**16 - 529563648.0*P**4*R**4*S*T*UGA_star**15*epsilon - 32837632.0*P**4*R**4*S*T*UGA_star**15 + 453111808.0*P**4*R**4*S*T*UGA_star**14*epsilon + 28055552.0*P**4*R**4*S*T*UGA_star**14 - 273657856.0*P**4*R**4*S*T*UGA_star**13*epsilon - 17065984.0*P**4*R**4*S*T*UGA_star**13 + 116244480.0*P**4*R**4*S*T*UGA_star**12*epsilon + 7516160.0*P**4*R**4*S*T*UGA_star**12 - 34902016.0*P**4*R**4*S*T*UGA_star**11*epsilon - 2525184.0*P**4*R**4*S*T*UGA_star**11 + 7610368.0*P**4*R**4*S*T*UGA_star**10*epsilon + 704512.0*P**4*R**4*S*T*UGA_star**10 - 1138688.0*P**4*R**4*S*T*UGA_star**9*epsilon - 156672.0*P**4*R**4*S*T*UGA_star**9 + 34816.0*P**4*R**4*S*T*UGA_star**8*epsilon + 18944.0*P**4*R**4*S*T*UGA_star**8 + 20480.0*P**4*R**4*S*T*UGA_star**7*epsilon + 1029120.0*P**4*R**4*T**2*UGA_star**20*epsilon + 57600.0*P**4*R**4*T**2*UGA_star**20 - 10291200.0*P**4*R**4*T**2*UGA_star**19*epsilon - 576000.0*P**4*R**4*T**2*UGA_star**19 + 46417920.0*P**4*R**4*T**2*UGA_star**18*epsilon + 2599680.0*P**4*R**4*T**2*UGA_star**18 - 124231680.0*P**4*R**4*T**2*UGA_star**17*epsilon - 6965760.0*P**4*R**4*T**2*UGA_star**17 + 218179584.0*P**4*R**4*T**2*UGA_star**16*epsilon + 12249856.0*P**4*R**4*T**2*UGA_star**16 - 262234112.0*P**4*R**4*T**2*UGA_star**15*epsilon - 14732288.0*P**4*R**4*T**2*UGA_star**15 + 217935872.0*P**4*R**4*T**2*UGA_star**14*epsilon + 12210688.0*P**4*R**4*T**2*UGA_star**14 - 123408384.0*P**4*R**4*T**2*UGA_star**13*epsilon - 6818816.0*P**4*R**4*T**2*UGA_star**13 + 45736960.0*P**4*R**4*T**2*UGA_star**12*epsilon + 2394880.0*P**4*R**4*T**2*UGA_star**12 - 10491904.0*P**4*R**4*T**2*UGA_star**11*epsilon - 436736.0*P**4*R**4*T**2*UGA_star**11 + 1615872.0*P**4*R**4*T**2*UGA_star**10*epsilon + 11008.0*P**4*R**4*T**2*UGA_star**10 - 315392.0*P**4*R**4*T**2*UGA_star**9*epsilon + 5632.0*P**4*R**4*T**2*UGA_star**9 + 57344.0*P**4*R**4*T**2*UGA_star**8*epsilon + 256.0*P**4*R**4*T**2*UGA_star**8 - 1372160.0*P**4*R**3*S**3*UGA_star**20*epsilon - 76800.0*P**4*R**3*S**3*UGA_star**20 + 15093760.0*P**4*R**3*S**3*UGA_star**19*epsilon + 844800.0*P**4*R**3*S**3*UGA_star**19 - 75427840.0*P**4*R**3*S**3*UGA_star**18*epsilon - 4224000.0*P**4*R**3*S**3*UGA_star**18 + 226222080.0*P**4*R**3*S**3*UGA_star**17*epsilon + 12687360.0*P**4*R**3*S**3*UGA_star**17 - 453257216.0*P**4*R**3*S**3*UGA_star**16*epsilon - 25498624.0*P**4*R**3*S**3*UGA_star**16 + 639442944.0*P**4*R**3*S**3*UGA_star**15*epsilon + 36182016.0*P**4*R**3*S**3*UGA_star**15 - 653346816.0*P**4*R**3*S**3*UGA_star**14*epsilon - 37361664.0*P**4*R**3*S**3*UGA_star**14 + 491618304.0*P**4*R**3*S**3*UGA_star**13*epsilon + 28655616.0*P**4*R**3*S**3*UGA_star**13 - 276381696.0*P**4*R**3*S**3*UGA_star**12*epsilon - 16671744.0*P**4*R**3*S**3*UGA_star**12 + 118378496.0*P**4*R**3*S**3*UGA_star**11*epsilon + 7578624.0*P**4*R**3*S**3*UGA_star**11 - 39337984.0*P**4*R**3*S**3*UGA_star**10*epsilon - 2774016.0*P**4*R**3*S**3*UGA_star**10 + 9773056.0*P**4*R**3*S**3*UGA_star**9*epsilon + 804864.0*P**4*R**3*S**3*UGA_star**9 - 1419264.0*P**4*R**3*S**3*UGA_star**8*epsilon - 162816.0*P**4*R**3*S**3*UGA_star**8 - 16384.0*P**4*R**3*S**3*UGA_star**7*epsilon + 16384.0*P**4*R**3*S**3*UGA_star**7 + 30720.0*P**4*R**3*S**3*UGA_star**6*epsilon - 4096000.0*P**4*R**3*S**2*T*UGA_star**20*epsilon - 256000.0*P**4*R**3*S**2*T*UGA_star**20 + 45056000.0*P**4*R**3*S**2*T*UGA_star**19*epsilon + 2816000.0*P**4*R**3*S**2*T*UGA_star**19 - 225249280.0*P**4*R**3*S**2*T*UGA_star**18*epsilon - 14069760.0*P**4*R**3*S**2*T*UGA_star**18 + 675440640.0*P**4*R**3*S**2*T*UGA_star**17*epsilon + 42132480.0*P**4*R**3*S**2*T*UGA_star**17 - 1349699584.0*P**4*R**3*S**2*T*UGA_star**16*epsilon - 84002816.0*P**4*R**3*S**2*T*UGA_star**16 + 1887481856.0*P**4*R**3*S**2*T*UGA_star**15*epsilon + 117126144.0*P**4*R**3*S**2*T*UGA_star**15 - 1886869504.0*P**4*R**3*S**2*T*UGA_star**14*epsilon - 116760576.0*P**4*R**3*S**2*T*UGA_star**14 + 1352953856.0*P**4*R**3*S**2*T*UGA_star**13*epsilon + 83748864.0*P**4*R**3*S**2*T*UGA_star**13 - 688943104.0*P**4*R**3*S**2*T*UGA_star**12*epsilon - 43186176.0*P**4*R**3*S**2*T*UGA_star**12 + 244875264.0*P**4*R**3*S**2*T*UGA_star**11*epsilon + 16162816.0*P**4*R**3*S**2*T*UGA_star**11 - 60512256.0*P**4*R**3*S**2*T*UGA_star**10*epsilon - 4665344.0*P**4*R**3*S**2*T*UGA_star**10 + 11003904.0*P**4*R**3*S**2*T*UGA_star**9*epsilon + 1176576.0*P**4*R**3*S**2*T*UGA_star**9 - 1509376.0*P**4*R**3*S**2*T*UGA_star**8*epsilon - 251904.0*P**4*R**3*S**2*T*UGA_star**8 + 36864.0*P**4*R**3*S**2*T*UGA_star**7*epsilon + 29696.0*P**4*R**3*S**2*T*UGA_star**7 + 30720.0*P**4*R**3*S**2*T*UGA_star**6*epsilon - 4116480.0*P**4*R**3*S*T**2*UGA_star**20*epsilon - 230400.0*P**4*R**3*S*T**2*UGA_star**20 + 45281280.0*P**4*R**3*S*T**2*UGA_star**19*epsilon + 2534400.0*P**4*R**3*S*T**2*UGA_star**19 - 226728960.0*P**4*R**3*S*T**2*UGA_star**18*epsilon - 12695040.0*P**4*R**3*S*T**2*UGA_star**18 + 681507840.0*P**4*R**3*S*T**2*UGA_star**17*epsilon + 38184960.0*P**4*R**3*S*T**2*UGA_star**17 - 1364612096.0*P**4*R**3*S*T**2*UGA_star**16*epsilon - 76511744.0*P**4*R**3*S*T**2*UGA_star**16 + 1907713024.0*P**4*R**3*S*T**2*UGA_star**15*epsilon + 106966016.0*P**4*R**3*S*T**2*UGA_star**15 - 1894946816.0*P**4*R**3*S*T**2*UGA_star**14*epsilon - 106015744.0*P**4*R**3*S*T**2*UGA_star**14 + 1332189184.0*P**4*R**3*S*T**2*UGA_star**13*epsilon + 73895936.0*P**4*R**3*S*T**2*UGA_star**13 - 646117376.0*P**4*R**3*S*T**2*UGA_star**12*epsilon - 34883584.0*P**4*R**3*S*T**2*UGA_star**12 + 205060096.0*P**4*R**3*S*T**2*UGA_star**11*epsilon + 10118144.0*P**4*R**3*S*T**2*UGA_star**11 - 39455744.0*P**4*R**3*S*T**2*UGA_star**10*epsilon - 1302016.0*P**4*R**3*S*T**2*UGA_star**10 + 5057536.0*P**4*R**3*S*T**2*UGA_star**9*epsilon - 96256.0*P**4*R**3*S*T**2*UGA_star**9 - 1025024.0*P**4*R**3*S*T**2*UGA_star**8*epsilon + 34304.0*P**4*R**3*S*T**2*UGA_star**8 + 193536.0*P**4*R**3*S*T**2*UGA_star**7*epsilon + 1024.0*P**4*R**3*S*T**2*UGA_star**7 - 1361920.0*P**4*R**3*T**3*UGA_star**20*epsilon - 53760.0*P**4*R**3*T**3*UGA_star**20 + 14981120.0*P**4*R**3*T**3*UGA_star**19*epsilon + 591360.0*P**4*R**3*T**3*UGA_star**19 - 75202560.0*P**4*R**3*T**3*UGA_star**18*epsilon - 2979840.0*P**4*R**3*T**3*UGA_star**18 + 227112960.0*P**4*R**3*T**3*UGA_star**17*epsilon + 9070080.0*P**4*R**3*T**3*UGA_star**17 - 457656320.0*P**4*R**3*T**3*UGA_star**16*epsilon - 18496000.0*P**4*R**3*T**3*UGA_star**16 + 644341760.0*P**4*R**3*T**3*UGA_star**15*epsilon + 26455040.0*P**4*R**3*T**3*UGA_star**15 - 643840000.0*P**4*R**3*T**3*UGA_star**14*epsilon - 26951680.0*P**4*R**3*T**3*UGA_star**14 + 452802560.0*P**4*R**3*T**3*UGA_star**13*epsilon + 19389440.0*P**4*R**3*T**3*UGA_star**13 - 215756800.0*P**4*R**3*T**3*UGA_star**12*epsilon - 9479680.0*P**4*R**3*T**3*UGA_star**12 + 63160320.0*P**4*R**3*T**3*UGA_star**11*epsilon + 2854400.0*P**4*R**3*T**3*UGA_star**11 - 8202240.0*P**4*R**3*T**3*UGA_star**10*epsilon - 378880.0*P**4*R**3*T**3*UGA_star**10 - 593920.0*P**4*R**3*T**3*UGA_star**9*epsilon - 33280.0*P**4*R**3*T**3*UGA_star**9 + 204800.0*P**4*R**3*T**3*UGA_star**8*epsilon + 12800.0*P**4*R**3*T**3*UGA_star**8 + 10240.0*P**4*R**3*T**3*UGA_star**7*epsilon + 1029120.0*P**4*R**2*S**4*UGA_star**20*epsilon + 57600.0*P**4*R**2*S**4*UGA_star**20 - 12349440.0*P**4*R**2*S**4*UGA_star**19*epsilon - 691200.0*P**4*R**2*S**4*UGA_star**19 + 67752960.0*P**4*R**2*S**4*UGA_star**18*epsilon + 3793920.0*P**4*R**2*S**4*UGA_star**18 - 224655360.0*P**4*R**2*S**4*UGA_star**17*epsilon - 12595200.0*P**4*R**2*S**4*UGA_star**17 + 501325824.0*P**4*R**2*S**4*UGA_star**16*epsilon + 28175616.0*P**4*R**2*S**4*UGA_star**16 - 793292800.0*P**4*R**2*S**4*UGA_star**15*epsilon - 44789760.0*P**4*R**2*S**4*UGA_star**15 + 913653760.0*P**4*R**2*S**4*UGA_star**14*epsilon + 52024320.0*P**4*R**2*S**4*UGA_star**14 - 774144000.0*P**4*R**2*S**4*UGA_star**13*epsilon - 44789760.0*P**4*R**2*S**4*UGA_star**13 + 483087360.0*P**4*R**2*S**4*UGA_star**12*epsilon + 28834560.0*P**4*R**2*S**4*UGA_star**12 - 221442048.0*P**4*R**2*S**4*UGA_star**11*epsilon - 14072832.0*P**4*R**2*S**4*UGA_star**11 + 74880000.0*P**4*R**2*S**4*UGA_star**10*epsilon + 5391360.0*P**4*R**2*S**4*UGA_star**10 - 19077120.0*P**4*R**2*S**4*UGA_star**9*epsilon - 1720320.0*P**4*R**2*S**4*UGA_star**9 + 3594240.0*P**4*R**2*S**4*UGA_star**8*epsilon + 464640.0*P**4*R**2*S**4*UGA_star**8 - 307200.0*P**4*R**2*S**4*UGA_star**7*epsilon - 92160.0*P**4*R**2*S**4*UGA_star**7 - 75776.0*P**4*R**2*S**4*UGA_star**6*epsilon + 9216.0*P**4*R**2*S**4*UGA_star**6 + 20480.0*P**4*R**2*S**4*UGA_star**5*epsilon + 4096000.0*P**4*R**2*S**3*T*UGA_star**20*epsilon + 256000.0*P**4*R**2*S**3*T*UGA_star**20 - 49152000.0*P**4*R**2*S**3*T*UGA_star**19*epsilon - 3072000.0*P**4*R**2*S**3*T*UGA_star**19 + 269987840.0*P**4*R**2*S**3*T*UGA_star**18*epsilon + 16865280.0*P**4*R**2*S**3*T*UGA_star**18 - 897105920.0*P**4*R**2*S**3*T*UGA_star**17*epsilon - 55971840.0*P**4*R**2*S**3*T*UGA_star**17 + 2006644736.0*P**4*R**2*S**3*T*UGA_star**16*epsilon + 124949504.0*P**4*R**2*S**3*T*UGA_star**16 - 3179560960.0*P**4*R**2*S**3*T*UGA_star**15*epsilon - 197437440.0*P**4*R**2*S**3*T*UGA_star**15 + 3653847040.0*P**4*R**2*S**3*T*UGA_star**14*epsilon + 226145280.0*P**4*R**2*S**3*T*UGA_star**14 - 3061985280.0*P**4*R**2*S**3*T*UGA_star**13*epsilon - 188989440.0*P**4*R**2*S**3*T*UGA_star**13 + 1852600320.0*P**4*R**2*S**3*T*UGA_star**12*epsilon + 114462720.0*P**4*R**2*S**3*T*UGA_star**12 - 788119552.0*P**4*R**2*S**3*T*UGA_star**11*epsilon - 49451008.0*P**4*R**2*S**3*T*UGA_star**11 + 225607680.0*P**4*R**2*S**3*T*UGA_star**10*epsilon + 15114240.0*P**4*R**2*S**3*T*UGA_star**10 - 41738240.0*P**4*R**2*S**3*T*UGA_star**9*epsilon - 3502080.0*P**4*R**2*S**3*T*UGA_star**9 + 5642240.0*P**4*R**2*S**3*T*UGA_star**8*epsilon + 783360.0*P**4*R**2*S**3*T*UGA_star**8 - 819200.0*P**4*R**2*S**3*T*UGA_star**7*epsilon - 174080.0*P**4*R**2*S**3*T*UGA_star**7 + 34816.0*P**4*R**2*S**3*T*UGA_star**6*epsilon + 21504.0*P**4*R**2*S**3*T*UGA_star**6 + 20480.0*P**4*R**2*S**3*T*UGA_star**5*epsilon + 6174720.0*P**4*R**2*S**2*T**2*UGA_star**20*epsilon + 345600.0*P**4*R**2*S**2*T**2*UGA_star**20 - 74096640.0*P**4*R**2*S**2*T**2*UGA_star**19*epsilon - 4147200.0*P**4*R**2*S**2*T**2*UGA_star**19 + 407854080.0*P**4*R**2*S**2*T**2*UGA_star**18*epsilon + 22832640.0*P**4*R**2*S**2*T**2*UGA_star**18 - 1360558080.0*P**4*R**2*S**2*T**2*UGA_star**17*epsilon - 76193280.0*P**4*R**2*S**2*T**2*UGA_star**17 + 3059994624.0*P**4*R**2*S**2*T**2*UGA_star**16*epsilon + 171403776.0*P**4*R**2*S**2*T**2*UGA_star**16 - 4880025600.0*P**4*R**2*S**2*T**2*UGA_star**15*epsilon - 273246720.0*P**4*R**2*S**2*T**2*UGA_star**15 + 5644477440.0*P**4*R**2*S**2*T**2*UGA_star**14*epsilon + 315394560.0*P**4*R**2*S**2*T**2*UGA_star**14 - 4752322560.0*P**4*R**2*S**2*T**2*UGA_star**13*epsilon - 263900160.0*P**4*R**2*S**2*T**2*UGA_star**13 + 2871982080.0*P**4*R**2*S**2*T**2*UGA_star**12*epsilon + 156879360.0*P**4*R**2*S**2*T**2*UGA_star**12 - 1201287168.0*P**4*R**2*S**2*T**2*UGA_star**11*epsilon - 62733312.0*P**4*R**2*S**2*T**2*UGA_star**11 + 323527680.0*P**4*R**2*S**2*T**2*UGA_star**10*epsilon + 14584320.0*P**4*R**2*S**2*T**2*UGA_star**10 - 49305600.0*P**4*R**2*S**2*T**2*UGA_star**9*epsilon - 898560.0*P**4*R**2*S**2*T**2*UGA_star**9 + 4423680.0*P**4*R**2*S**2*T**2*UGA_star**8*epsilon - 391680.0*P**4*R**2*S**2*T**2*UGA_star**8 - 1075200.0*P**4*R**2*S**2*T**2*UGA_star**7*epsilon + 69120.0*P**4*R**2*S**2*T**2*UGA_star**7 + 236544.0*P**4*R**2*S**2*T**2*UGA_star**6*epsilon + 1536.0*P**4*R**2*S**2*T**2*UGA_star**6 + 4085760.0*P**4*R**2*S*T**3*UGA_star**20*epsilon + 161280.0*P**4*R**2*S*T**3*UGA_star**20 - 49029120.0*P**4*R**2*S*T**3*UGA_star**19*epsilon - 1935360.0*P**4*R**2*S*T**3*UGA_star**19 + 270551040.0*P**4*R**2*S*T**3*UGA_star**18*epsilon + 10713600.0*P**4*R**2*S*T**3*UGA_star**18 - 906946560.0*P**4*R**2*S*T**3*UGA_star**17*epsilon - 36149760.0*P**4*R**2*S*T**3*UGA_star**17 + 2054246400.0*P**4*R**2*S*T**3*UGA_star**16*epsilon + 82698240.0*P**4*R**2*S*T**3*UGA_star**16 - 3305441280.0*P**4*R**2*S*T**3*UGA_star**15*epsilon - 134853120.0*P**4*R**2*S*T**3*UGA_star**15 + 3862333440.0*P**4*R**2*S*T**3*UGA_star**14*epsilon + 160220160.0*P**4*R**2*S*T**3*UGA_star**14 - 3284766720.0*P**4*R**2*S*T**3*UGA_star**13*epsilon - 139023360.0*P**4*R**2*S*T**3*UGA_star**13 + 1997936640.0*P**4*R**2*S*T**3*UGA_star**12*epsilon + 86607360.0*P**4*R**2*S*T**3*UGA_star**12 - 829009920.0*P**4*R**2*S*T**3*UGA_star**11*epsilon - 37002240.0*P**4*R**2*S*T**3*UGA_star**11 + 208926720.0*P**4*R**2*S*T**3*UGA_star**10*epsilon + 9699840.0*P**4*R**2*S*T**3*UGA_star**10 - 20613120.0*P**4*R**2*S*T**3*UGA_star**9*epsilon - 1036800.0*P**4*R**2*S*T**3*UGA_star**9 - 2949120.0*P**4*R**2*S*T**3*UGA_star**8*epsilon - 138240.0*P**4*R**2*S*T**3*UGA_star**8 + 645120.0*P**4*R**2*S*T**3*UGA_star**7*epsilon + 38400.0*P**4*R**2*S*T**3*UGA_star**7 + 30720.0*P**4*R**2*S*T**3*UGA_star**6*epsilon + 931840.0*P**4*R**2*T**4*UGA_star**20*epsilon + 17920.0*P**4*R**2*T**4*UGA_star**20 - 11182080.0*P**4*R**2*T**4*UGA_star**19*epsilon - 215040.0*P**4*R**2*T**4*UGA_star**19 + 61895680.0*P**4*R**2*T**4*UGA_star**18*epsilon + 1200640.0*P**4*R**2*T**4*UGA_star**18 - 208803840.0*P**4*R**2*T**4*UGA_star**17*epsilon - 4121600.0*P**4*R**2*T**4*UGA_star**17 + 477521920.0*P**4*R**2*T**4*UGA_star**16*epsilon + 9680640.0*P**4*R**2*T**4*UGA_star**16 - 778444800.0*P**4*R**2*T**4*UGA_star**15*epsilon - 16373760.0*P**4*R**2*T**4*UGA_star**15 + 924825600.0*P**4*R**2*T**4*UGA_star**14*epsilon + 20428800.0*P**4*R**2*T**4*UGA_star**14 - 802959360.0*P**4*R**2*T**4*UGA_star**13*epsilon - 18923520.0*P**4*R**2*T**4*UGA_star**13 + 501258240.0*P**4*R**2*T**4*UGA_star**12*epsilon + 12902400.0*P**4*R**2*T**4*UGA_star**12 - 215326720.0*P**4*R**2*T**4*UGA_star**11*epsilon - 6307840.0*P**4*R**2*T**4*UGA_star**11 + 57308160.0*P**4*R**2*T**4*UGA_star**10*epsilon + 2096640.0*P**4*R**2*T**4*UGA_star**10 - 6584320.0*P**4*R**2*T**4*UGA_star**9*epsilon - 424960.0*P**4*R**2*T**4*UGA_star**9 - 645120.0*P**4*R**2*T**4*UGA_star**8*epsilon + 39680.0*P**4*R**2*T**4*UGA_star**8 + 204800.0*P**4*R**2*T**4*UGA_star**7*epsilon - 411648.0*P**4*R*S**5*UGA_star**20*epsilon - 23040.0*P**4*R*S**5*UGA_star**20 + 5351424.0*P**4*R*S**5*UGA_star**19*epsilon + 299520.0*P**4*R*S**5*UGA_star**19 - 31985664.0*P**4*R*S**5*UGA_star**18*epsilon - 1790976.0*P**4*R*S**5*UGA_star**18 + 116275200.0*P**4*R*S**5*UGA_star**17*epsilon + 6517248.0*P**4*R*S**5*UGA_star**17 - 286448640.0*P**4*R*S**5*UGA_star**16*epsilon - 16087552.0*P**4*R*S**5*UGA_star**16 + 504070144.0*P**4*R*S**5*UGA_star**15*epsilon + 28413440.0*P**4*R*S**5*UGA_star**15 - 650067968.0*P**4*R*S**5*UGA_star**14*epsilon - 36889600.0*P**4*R*S**5*UGA_star**14 + 619429888.0*P**4*R*S**5*UGA_star**13*epsilon + 35599872.0*P**4*R*S**5*UGA_star**13 - 433044480.0*P**4*R*S**5*UGA_star**12*epsilon - 25529856.0*P**4*R*S**5*UGA_star**12 + 216606720.0*P**4*R*S**5*UGA_star**11*epsilon + 13499904.0*P**4*R*S**5*UGA_star**11 - 73396224.0*P**4*R*S**5*UGA_star**10*epsilon - 5237760.0*P**4*R*S**5*UGA_star**10 + 14874624.0*P**4*R*S**5*UGA_star**9*epsilon + 1543680.0*P**4*R*S**5*UGA_star**9 - 1127424.0*P**4*R*S**5*UGA_star**8*epsilon - 400896.0*P**4*R*S**5*UGA_star**8 - 184320.0*P**4*R*S**5*UGA_star**7*epsilon + 107008.0*P**4*R*S**5*UGA_star**7 + 81920.0*P**4*R*S**5*UGA_star**6*epsilon - 23552.0*P**4*R*S**5*UGA_star**6 - 28672.0*P**4*R*S**5*UGA_star**5*epsilon + 2560.0*P**4*R*S**5*UGA_star**5 + 5120.0*P**4*R*S**5*UGA_star**4*epsilon - 2048000.0*P**4*R*S**4*T*UGA_star**20*epsilon - 128000.0*P**4*R*S**4*T*UGA_star**20 + 26624000.0*P**4*R*S**4*T*UGA_star**19*epsilon + 1664000.0*P**4*R*S**4*T*UGA_star**19 - 159411200.0*P**4*R*S**4*T*UGA_star**18*epsilon - 9958400.0*P**4*R*S**4*T*UGA_star**18 + 581596160.0*P**4*R*S**4*T*UGA_star**17*epsilon + 36293120.0*P**4*R*S**4*T*UGA_star**17 - 1440840704.0*P**4*R*S**4*T*UGA_star**16*epsilon - 89752576.0*P**4*R*S**4*T*UGA_star**16 + 2555101184.0*P**4*R*S**4*T*UGA_star**15*epsilon + 158754816.0*P**4*R*S**4*T*UGA_star**15 - 3327923200.0*P**4*R*S**4*T*UGA_star**14*epsilon - 206074880.0*P**4*R*S**4*T*UGA_star**14 + 3209589760.0*P**4*R*S**4*T*UGA_star**13*epsilon + 197936640.0*P**4*R*S**4*T*UGA_star**13 - 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414296064.0*P**4*R*T**5*UGA_star**12*epsilon + 243984384.0*P**4*R*T**5*UGA_star**11*epsilon - 103993344.0*P**4*R*T**5*UGA_star**10*epsilon + 30363648.0*P**4*R*T**5*UGA_star**9*epsilon - 5440512.0*P**4*R*T**5*UGA_star**8*epsilon + 451584.0*P**4*R*T**5*UGA_star**7*epsilon + 68608.0*P**4*S**6*UGA_star**20*epsilon + 3840.0*P**4*S**6*UGA_star**20 - 960512.0*P**4*S**6*UGA_star**19*epsilon - 53760.0*P**4*S**6*UGA_star**19 + 6213632.0*P**4*S**6*UGA_star**18*epsilon + 347904.0*P**4*S**6*UGA_star**18 - 24586240.0*P**4*S**6*UGA_star**17*epsilon - 1377792.0*P**4*S**6*UGA_star**17 + 66349056.0*P**4*S**6*UGA_star**16*epsilon + 3724288.0*P**4*S**6*UGA_star**16 - 128804864.0*P**4*S**6*UGA_star**15*epsilon - 7251456.0*P**4*S**6*UGA_star**15 + 184639488.0*P**4*S**6*UGA_star**14*epsilon + 10450176.0*P**4*S**6*UGA_star**14 - 196962304.0*P**4*S**6*UGA_star**13*epsilon - 11258368.0*P**4*S**6*UGA_star**13 + 154767360.0*P**4*S**6*UGA_star**12*epsilon + 9022464.0*P**4*S**6*UGA_star**12 - 86259712.0*P**4*S**6*UGA_star**11*epsilon - 5254656.0*P**4*S**6*UGA_star**11 + 30593024.0*P**4*S**6*UGA_star**10*epsilon + 2103552.0*P**4*S**6*UGA_star**10 - 4044800.0*P**4*S**6*UGA_star**9*epsilon - 502272.0*P**4*S**6*UGA_star**9 - 1968128.0*P**4*S**6*UGA_star**8*epsilon + 36864.0*P**4*S**6*UGA_star**8 + 1198080.0*P**4*S**6*UGA_star**7*epsilon + 9728.0*P**4*S**6*UGA_star**7 - 262144.0*P**4*S**6*UGA_star**6*epsilon + 768.0*P**4*S**6*UGA_star**6 + 18432.0*P**4*S**6*UGA_star**5*epsilon - 1536.0*P**4*S**6*UGA_star**5 + 1024.0*P**4*S**6*UGA_star**4*epsilon + 256.0*P**4*S**6*UGA_star**4 + 409600.0*P**4*S**5*T*UGA_star**20*epsilon + 25600.0*P**4*S**5*T*UGA_star**20 - 5734400.0*P**4*S**5*T*UGA_star**19*epsilon - 358400.0*P**4*S**5*T*UGA_star**19 + 37175296.0*P**4*S**5*T*UGA_star**18*epsilon + 2322432.0*P**4*S**5*T*UGA_star**18 - 147779584.0*P**4*S**5*T*UGA_star**17*epsilon - 9223168.0*P**4*S**5*T*UGA_star**17 + 401891328.0*P**4*S**5*T*UGA_star**16*epsilon + 25042432.0*P**4*S**5*T*UGA_star**16 - 789393408.0*P**4*S**5*T*UGA_star**15*epsilon - 49072128.0*P**4*S**5*T*UGA_star**15 + 1151315968.0*P**4*S**5*T*UGA_star**14*epsilon + 71334912.0*P**4*S**5*T*UGA_star**14 - 1260306432.0*P**4*S**5*T*UGA_star**13*epsilon - 77732864.0*P**4*S**5*T*UGA_star**13 + 1031467008.0*P**4*S**5*T*UGA_star**12*epsilon + 63207936.0*P**4*S**5*T*UGA_star**12 - 617537536.0*P**4*S**5*T*UGA_star**11*epsilon - 37464064.0*P**4*S**5*T*UGA_star**11 + 256188416.0*P**4*S**5*T*UGA_star**10*epsilon + 15251456.0*P**4*S**5*T*UGA_star**10 - 63533056.0*P**4*S**5*T*UGA_star**9*epsilon - 3585024.0*P**4*S**5*T*UGA_star**9 + 3751936.0*P**4*S**5*T*UGA_star**8*epsilon + 82432.0*P**4*S**5*T*UGA_star**8 + 2789376.0*P**4*S**5*T*UGA_star**7*epsilon + 223232.0*P**4*S**5*T*UGA_star**7 - 724992.0*P**4*S**5*T*UGA_star**6*epsilon - 58368.0*P**4*S**5*T*UGA_star**6 + 8192.0*P**4*S**5*T*UGA_star**5*epsilon + 3072.0*P**4*S**5*T*UGA_star**5 + 12288.0*P**4*S**5*T*UGA_star**4*epsilon + 512.0*P**4*S**5*T*UGA_star**4 + 1029120.0*P**4*S**4*T**2*UGA_star**20*epsilon + 57600.0*P**4*S**4*T**2*UGA_star**20 - 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13172736.0*P**3*R**5*S*T*UGA_star**19*epsilon - 737280.0*P**3*R**5*S*T*UGA_star**19 + 46454784.0*P**3*R**5*S*T*UGA_star**18*epsilon + 2601984.0*P**3*R**5*S*T*UGA_star**18 - 95133696.0*P**3*R**5*S*T*UGA_star**17*epsilon - 5342208.0*P**3*R**5*S*T*UGA_star**17 + 125980672.0*P**3*R**5*S*T*UGA_star**16*epsilon + 7115264.0*P**3*R**5*S*T*UGA_star**16 - 114690048.0*P**3*R**5*S*T*UGA_star**15*epsilon - 6540800.0*P**3*R**5*S*T*UGA_star**15 + 75534336.0*P**3*R**5*S*T*UGA_star**14*epsilon + 4354560.0*P**3*R**5*S*T*UGA_star**14 - 37689344.0*P**3*R**5*S*T*UGA_star**13*epsilon - 2174464.0*P**3*R**5*S*T*UGA_star**13 + 14217216.0*P**3*R**5*S*T*UGA_star**12*epsilon + 798208.0*P**3*R**5*S*T*UGA_star**12 - 3520512.0*P**3*R**5*S*T*UGA_star**11*epsilon - 182784.0*P**3*R**5*S*T*UGA_star**11 + 342016.0*P**3*R**5*S*T*UGA_star**10*epsilon + 12800.0*P**3*R**5*S*T*UGA_star**10 + 30720.0*P**3*R**5*S*T*UGA_star**9*epsilon + 2560.0*P**3*R**5*S*T*UGA_star**9 + 819200.0*P**3*R**5*T**2*UGA_star**20*epsilon + 51200.0*P**3*R**5*T**2*UGA_star**20 - 6553600.0*P**3*R**5*T**2*UGA_star**19*epsilon - 409600.0*P**3*R**5*T**2*UGA_star**19 + 22994944.0*P**3*R**5*T**2*UGA_star**18*epsilon + 1435648.0*P**3*R**5*T**2*UGA_star**18 - 46313472.0*P**3*R**5*T**2*UGA_star**17*epsilon - 2883584.0*P**3*R**5*T**2*UGA_star**17 + 58855424.0*P**3*R**5*T**2*UGA_star**16*epsilon + 3647488.0*P**3*R**5*T**2*UGA_star**16 - 48930816.0*P**3*R**5*T**2*UGA_star**15*epsilon - 3018752.0*P**3*R**5*T**2*UGA_star**15 + 26849280.0*P**3*R**5*T**2*UGA_star**14*epsilon + 1669120.0*P**3*R**5*T**2*UGA_star**14 - 9723904.0*P**3*R**5*T**2*UGA_star**13*epsilon - 647168.0*P**3*R**5*T**2*UGA_star**13 + 2322432.0*P**3*R**5*T**2*UGA_star**12*epsilon + 200704.0*P**3*R**5*T**2*UGA_star**12 - 303104.0*P**3*R**5*T**2*UGA_star**11*epsilon - 53248.0*P**3*R**5*T**2*UGA_star**11 - 36864.0*P**3*R**5*T**2*UGA_star**10*epsilon + 8192.0*P**3*R**5*T**2*UGA_star**10 + 20480.0*P**3*R**5*T**2*UGA_star**9*epsilon - 1361920.0*P**3*R**4*S**3*UGA_star**20*epsilon - 53760.0*P**3*R**4*S**3*UGA_star**20 + 12257280.0*P**3*R**4*S**3*UGA_star**19*epsilon + 483840.0*P**3*R**4*S**3*UGA_star**19 - 49326080.0*P**3*R**4*S**3*UGA_star**18*epsilon - 1958400.0*P**3*R**4*S**3*UGA_star**18 + 117350400.0*P**3*R**4*S**3*UGA_star**17*epsilon + 4730880.0*P**3*R**4*S**3*UGA_star**17 - 184524800.0*P**3*R**4*S**3*UGA_star**16*epsilon - 7659520.0*P**3*R**4*S**3*UGA_star**16 + 204472320.0*P**3*R**4*S**3*UGA_star**15*epsilon + 8883200.0*P**3*R**4*S**3*UGA_star**15 - 168048640.0*P**3*R**4*S**3*UGA_star**14*epsilon - 7705600.0*P**3*R**4*S**3*UGA_star**14 + 107591680.0*P**3*R**4*S**3*UGA_star**13*epsilon + 5089280.0*P**3*R**4*S**3*UGA_star**13 - 55480320.0*P**3*R**4*S**3*UGA_star**12*epsilon - 2490880.0*P**3*R**4*S**3*UGA_star**12 + 22732800.0*P**3*R**4*S**3*UGA_star**11*epsilon + 811520.0*P**3*R**4*S**3*UGA_star**11 - 6963200.0*P**3*R**4*S**3*UGA_star**10*epsilon - 125440.0*P**3*R**4*S**3*UGA_star**10 + 1525760.0*P**3*R**4*S**3*UGA_star**9*epsilon - 10240.0*P**3*R**4*S**3*UGA_star**9 - 256000.0*P**3*R**4*S**3*UGA_star**8*epsilon + 5120.0*P**3*R**4*S**3*UGA_star**8 + 30720.0*P**3*R**4*S**3*UGA_star**7*epsilon - 4116480.0*P**3*R**4*S**2*T*UGA_star**20*epsilon - 230400.0*P**3*R**4*S**2*T*UGA_star**20 + 37048320.0*P**3*R**4*S**2*T*UGA_star**19*epsilon + 2073600.0*P**3*R**4*S**2*T*UGA_star**19 - 148515840.0*P**3*R**4*S**2*T*UGA_star**18*epsilon - 8317440.0*P**3*R**4*S**2*T*UGA_star**18 + 349301760.0*P**3*R**4*S**2*T*UGA_star**17*epsilon + 19599360.0*P**3*R**4*S**2*T*UGA_star**17 - 535294976.0*P**3*R**4*S**2*T*UGA_star**16*epsilon - 30162944.0*P**3*R**4*S**2*T*UGA_star**16 + 563510272.0*P**3*R**4*S**2*T*UGA_star**15*epsilon + 32007168.0*P**3*R**4*S**2*T*UGA_star**15 - 422030336.0*P**3*R**4*S**2*T*UGA_star**14*epsilon - 24285184.0*P**3*R**4*S**2*T*UGA_star**14 + 232883200.0*P**3*R**4*S**2*T*UGA_star**13*epsilon + 13634560.0*P**3*R**4*S**2*T*UGA_star**13 - 98155520.0*P**3*R**4*S**2*T*UGA_star**12*epsilon - 5836800.0*P**3*R**4*S**2*T*UGA_star**12 + 31261696.0*P**3*R**4*S**2*T*UGA_star**11*epsilon + 1869824.0*P**3*R**4*S**2*T*UGA_star**11 - 6305792.0*P**3*R**4*S**2*T*UGA_star**10*epsilon - 378368.0*P**3*R**4*S**2*T*UGA_star**10 + 311296.0*P**3*R**4*S**2*T*UGA_star**9*epsilon + 21504.0*P**3*R**4*S**2*T*UGA_star**9 + 102400.0*P**3*R**4*S**2*T*UGA_star**8*epsilon + 5120.0*P**3*R**4*S**2*T*UGA_star**8 - 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219136.0*P**3*R**4*S*T**2*UGA_star**9*epsilon + 34816.0*P**3*R**4*S*T**2*UGA_star**9 + 81920.0*P**3*R**4*S*T**2*UGA_star**8*epsilon - 1372160.0*P**3*R**4*T**3*UGA_star**20*epsilon - 76800.0*P**3*R**4*T**3*UGA_star**20 + 12349440.0*P**3*R**4*T**3*UGA_star**19*epsilon + 691200.0*P**3*R**4*T**3*UGA_star**19 - 49356800.0*P**3*R**4*T**3*UGA_star**18*epsilon - 2764800.0*P**3*R**4*T**3*UGA_star**18 + 114708480.0*P**3*R**4*T**3*UGA_star**17*epsilon + 6435840.0*P**3*R**4*T**3*UGA_star**17 - 170203136.0*P**3*R**4*T**3*UGA_star**16*epsilon - 9570304.0*P**3*R**4*T**3*UGA_star**16 + 166141952.0*P**3*R**4*T**3*UGA_star**15*epsilon + 9361408.0*P**3*R**4*T**3*UGA_star**15 - 105424896.0*P**3*R**4*T**3*UGA_star**14*epsilon - 5935104.0*P**3*R**4*T**3*UGA_star**14 + 40898560.0*P**3*R**4*T**3*UGA_star**13*epsilon + 2263040.0*P**3*R**4*T**3*UGA_star**13 - 8304640.0*P**3*R**4*T**3*UGA_star**12*epsilon - 404480.0*P**3*R**4*T**3*UGA_star**12 + 659456.0*P**3*R**4*T**3*UGA_star**11*epsilon - 9216.0*P**3*R**4*T**3*UGA_star**11 - 161792.0*P**3*R**4*T**3*UGA_star**10*epsilon + 8192.0*P**3*R**4*T**3*UGA_star**10 + 65536.0*P**3*R**4*T**3*UGA_star**9*epsilon + 1024.0*P**3*R**4*T**3*UGA_star**9 + 1361920.0*P**3*R**3*S**4*UGA_star**20*epsilon + 53760.0*P**3*R**3*S**4*UGA_star**20 - 13619200.0*P**3*R**3*S**4*UGA_star**19*epsilon - 537600.0*P**3*R**3*S**4*UGA_star**19 + 61347840.0*P**3*R**3*S**4*UGA_star**18*epsilon + 2432000.0*P**3*R**3*S**4*UGA_star**18 - 164413440.0*P**3*R**3*S**4*UGA_star**17*epsilon - 6589440.0*P**3*R**3*S**4*UGA_star**17 + 292101120.0*P**3*R**3*S**4*UGA_star**16*epsilon + 11952640.0*P**3*R**3*S**4*UGA_star**16 - 363980800.0*P**3*R**3*S**4*UGA_star**15*epsilon - 15406080.0*P**3*R**3*S**4*UGA_star**15 + 330398720.0*P**3*R**3*S**4*UGA_star**14*epsilon + 14653440.0*P**3*R**3*S**4*UGA_star**14 - 226631680.0*P**3*R**3*S**4*UGA_star**13*epsilon - 10536960.0*P**3*R**3*S**4*UGA_star**13 + 122716160.0*P**3*R**3*S**4*UGA_star**12*epsilon + 5752320.0*P**3*R**3*S**4*UGA_star**12 - 54394880.0*P**3*R**3*S**4*UGA_star**11*epsilon - 2288640.0*P**3*R**3*S**4*UGA_star**11 + 19476480.0*P**3*R**3*S**4*UGA_star**10*epsilon + 568320.0*P**3*R**3*S**4*UGA_star**10 - 5201920.0*P**3*R**3*S**4*UGA_star**9*epsilon - 35840.0*P**3*R**3*S**4*UGA_star**9 + 957440.0*P**3*R**3*S**4*UGA_star**8*epsilon - 23040.0*P**3*R**3*S**4*UGA_star**8 - 133120.0*P**3*R**3*S**4*UGA_star**7*epsilon + 5120.0*P**3*R**3*S**4*UGA_star**7 + 15360.0*P**3*R**3*S**4*UGA_star**6*epsilon + 5488640.0*P**3*R**3*S**3*T*UGA_star**20*epsilon + 307200.0*P**3*R**3*S**3*T*UGA_star**20 - 54886400.0*P**3*R**3*S**3*T*UGA_star**19*epsilon - 3072000.0*P**3*R**3*S**3*T*UGA_star**19 + 246681600.0*P**3*R**3*S**3*T*UGA_star**18*epsilon + 13813760.0*P**3*R**3*S**3*T*UGA_star**18 - 656793600.0*P**3*R**3*S**3*T*UGA_star**17*epsilon - 36833280.0*P**3*R**3*S**3*T*UGA_star**17 + 1149947904.0*P**3*R**3*S**3*T*UGA_star**16*epsilon + 64695296.0*P**3*R**3*S**3*T*UGA_star**16 - 1391153152.0*P**3*R**3*S**3*T*UGA_star**15*epsilon - 78752768.0*P**3*R**3*S**3*T*UGA_star**15 + 1192927232.0*P**3*R**3*S**3*T*UGA_star**14*epsilon + 68313088.0*P**3*R**3*S**3*T*UGA_star**14 - 737542144.0*P**3*R**3*S**3*T*UGA_star**13*epsilon - 43114496.0*P**3*R**3*S**3*T*UGA_star**13 + 336158720.0*P**3*R**3*S**3*T*UGA_star**12*epsilon + 20346880.0*P**3*R**3*S**3*T*UGA_star**12 - 116711424.0*P**3*R**3*S**3*T*UGA_star**11*epsilon - 7448576.0*P**3*R**3*S**3*T*UGA_star**11 + 30625792.0*P**3*R**3*S**3*T*UGA_star**10*epsilon + 2111488.0*P**3*R**3*S**3*T*UGA_star**10 - 4759552.0*P**3*R**3*S**3*T*UGA_star**9*epsilon - 391168.0*P**3*R**3*S**3*T*UGA_star**9 - 126976.0*P**3*R**3*S**3*T*UGA_star**8*epsilon + 19456.0*P**3*R**3*S**3*T*UGA_star**8 + 143360.0*P**3*R**3*S**3*T*UGA_star**7*epsilon + 5120.0*P**3*R**3*S**3*T*UGA_star**7 + 8192000.0*P**3*R**3*S**2*T**2*UGA_star**20*epsilon + 512000.0*P**3*R**3*S**2*T**2*UGA_star**20 - 81920000.0*P**3*R**3*S**2*T**2*UGA_star**19*epsilon - 5120000.0*P**3*R**3*S**2*T**2*UGA_star**19 + 367943680.0*P**3*R**3*S**2*T**2*UGA_star**18*epsilon + 22978560.0*P**3*R**3*S**2*T**2*UGA_star**18 - 976773120.0*P**3*R**3*S**2*T**2*UGA_star**17*epsilon - 60887040.0*P**3*R**3*S**2*T**2*UGA_star**17 + 1695707136.0*P**3*R**3*S**2*T**2*UGA_star**16*epsilon + 105365504.0*P**3*R**3*S**2*T**2*UGA_star**16 - 2009530368.0*P**3*R**3*S**2*T**2*UGA_star**15*epsilon - 124321792.0*P**3*R**3*S**2*T**2*UGA_star**15 + 1645191168.0*P**3*R**3*S**2*T**2*UGA_star**14*epsilon + 101384192.0*P**3*R**3*S**2*T**2*UGA_star**14 - 919904256.0*P**3*R**3*S**2*T**2*UGA_star**13*epsilon - 56909824.0*P**3*R**3*S**2*T**2*UGA_star**13 + 339394560.0*P**3*R**3*S**2*T**2*UGA_star**12*epsilon + 21893120.0*P**3*R**3*S**2*T**2*UGA_star**12 - 78012416.0*P**3*R**3*S**2*T**2*UGA_star**11*epsilon - 6119424.0*P**3*R**3*S**2*T**2*UGA_star**11 + 10354688.0*P**3*R**3*S**2*T**2*UGA_star**10*epsilon + 1564672.0*P**3*R**3*S**2*T**2*UGA_star**10 - 360448.0*P**3*R**3*S**2*T**2*UGA_star**9*epsilon - 397312.0*P**3*R**3*S**2*T**2*UGA_star**9 - 405504.0*P**3*R**3*S**2*T**2*UGA_star**8*epsilon + 57344.0*P**3*R**3*S**2*T**2*UGA_star**8 + 122880.0*P**3*R**3*S**2*T**2*UGA_star**7*epsilon + 5488640.0*P**3*R**3*S*T**3*UGA_star**20*epsilon + 307200.0*P**3*R**3*S*T**3*UGA_star**20 - 54886400.0*P**3*R**3*S*T**3*UGA_star**19*epsilon - 3072000.0*P**3*R**3*S*T**3*UGA_star**19 + 246681600.0*P**3*R**3*S*T**3*UGA_star**18*epsilon + 13813760.0*P**3*R**3*S*T**3*UGA_star**18 - 654950400.0*P**3*R**3*S*T**3*UGA_star**17*epsilon - 36710400.0*P**3*R**3*S*T**3*UGA_star**17 + 1134280704.0*P**3*R**3*S*T**3*UGA_star**16*epsilon + 63650816.0*P**3*R**3*S*T**3*UGA_star**16 - 1332457472.0*P**3*R**3*S*T**3*UGA_star**15*epsilon - 74830848.0*P**3*R**3*S*T**3*UGA_star**15 + 1066053632.0*P**3*R**3*S*T**3*UGA_star**14*epsilon + 59788288.0*P**3*R**3*S*T**3*UGA_star**14 - 563933184.0*P**3*R**3*S*T**3*UGA_star**13*epsilon - 31323136.0*P**3*R**3*S*T**3*UGA_star**13 + 181473280.0*P**3*R**3*S*T**3*UGA_star**12*epsilon + 9630720.0*P**3*R**3*S*T**3*UGA_star**12 - 28545024.0*P**3*R**3*S*T**3*UGA_star**11*epsilon - 1104896.0*P**3*R**3*S*T**3*UGA_star**11 + 1114112.0*P**3*R**3*S*T**3*UGA_star**10*epsilon - 197632.0*P**3*R**3*S*T**3*UGA_star**10 - 561152.0*P**3*R**3*S*T**3*UGA_star**9*epsilon + 44032.0*P**3*R**3*S*T**3*UGA_star**9 + 241664.0*P**3*R**3*S*T**3*UGA_star**8*epsilon + 4096.0*P**3*R**3*S*T**3*UGA_star**8 + 1361920.0*P**3*R**3*T**4*UGA_star**20*epsilon + 53760.0*P**3*R**3*T**4*UGA_star**20 - 13619200.0*P**3*R**3*T**4*UGA_star**19*epsilon - 537600.0*P**3*R**3*T**4*UGA_star**19 + 61347840.0*P**3*R**3*T**4*UGA_star**18*epsilon + 2432000.0*P**3*R**3*T**4*UGA_star**18 - 163553280.0*P**3*R**3*T**4*UGA_star**17*epsilon - 6543360.0*P**3*R**3*T**4*UGA_star**17 + 284789760.0*P**3*R**3*T**4*UGA_star**16*epsilon + 11560960.0*P**3*R**3*T**4*UGA_star**16 - 336435200.0*P**3*R**3*T**4*UGA_star**15*epsilon - 13941760.0*P**3*R**3*T**4*UGA_star**15 + 270033920.0*P**3*R**3*T**4*UGA_star**14*epsilon + 11504640.0*P**3*R**3*T**4*UGA_star**14 - 141803520.0*P**3*R**3*T**4*UGA_star**13*epsilon - 6272000.0*P**3*R**3*T**4*UGA_star**13 + 43310080.0*P**3*R**3*T**4*UGA_star**12*epsilon + 2024960.0*P**3*R**3*T**4*UGA_star**12 - 4597760.0*P**3*R**3*T**4*UGA_star**11*epsilon - 245760.0*P**3*R**3*T**4*UGA_star**11 - 1080320.0*P**3*R**3*T**4*UGA_star**10*epsilon - 51200.0*P**3*R**3*T**4*UGA_star**10 + 225280.0*P**3*R**3*T**4*UGA_star**9*epsilon + 15360.0*P**3*R**3*T**4*UGA_star**9 + 20480.0*P**3*R**3*T**4*UGA_star**8*epsilon - 817152.0*P**3*R**2*S**5*UGA_star**20*epsilon - 32256.0*P**3*R**2*S**5*UGA_star**20 + 8988672.0*P**3*R**2*S**5*UGA_star**19*epsilon + 354816.0*P**3*R**2*S**5*UGA_star**19 - 44838912.0*P**3*R**2*S**5*UGA_star**18*epsilon - 1775616.0*P**3*R**2*S**5*UGA_star**18 + 133957632.0*P**3*R**2*S**5*UGA_star**17*epsilon + 5346816.0*P**3*R**2*S**5*UGA_star**17 - 266693632.0*P**3*R**2*S**5*UGA_star**16*epsilon - 10803200.0*P**3*R**2*S**5*UGA_star**16 + 372847616.0*P**3*R**2*S**5*UGA_star**15*epsilon + 15475712.0*P**3*R**2*S**5*UGA_star**15 - 376679424.0*P**3*R**2*S**5*UGA_star**14*epsilon - 16215552.0*P**3*R**2*S**5*UGA_star**14 + 280464384.0*P**3*R**2*S**5*UGA_star**13*epsilon + 12659712.0*P**3*R**2*S**5*UGA_star**13 - 157691904.0*P**3*R**2*S**5*UGA_star**12*epsilon - 7429632.0*P**3*R**2*S**5*UGA_star**12 + 69955584.0*P**3*R**2*S**5*UGA_star**11*epsilon + 3248640.0*P**3*R**2*S**5*UGA_star**11 - 25921536.0*P**3*R**2*S**5*UGA_star**10*epsilon - 992256.0*P**3*R**2*S**5*UGA_star**10 + 8036352.0*P**3*R**2*S**5*UGA_star**9*epsilon + 155136.0*P**3*R**2*S**5*UGA_star**9 - 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629415936.0*P**3*R**2*S**4*T*UGA_star**12*epsilon - 37929984.0*P**3*R**2*S**4*T*UGA_star**12 + 224526336.0*P**3*R**2*S**4*T*UGA_star**11*epsilon + 14644224.0*P**3*R**2*S**4*T*UGA_star**11 - 58665984.0*P**3*R**2*S**4*T*UGA_star**10*epsilon - 4471296.0*P**3*R**2*S**4*T*UGA_star**10 + 11498496.0*P**3*R**2*S**4*T*UGA_star**9*epsilon + 1142784.0*P**3*R**2*S**4*T*UGA_star**9 - 1081344.0*P**3*R**2*S**4*T*UGA_star**8*epsilon - 208896.0*P**3*R**2*S**4*T*UGA_star**8 - 303104.0*P**3*R**2*S**4*T*UGA_star**7*epsilon + 11264.0*P**3*R**2*S**4*T*UGA_star**7 + 92160.0*P**3*R**2*S**4*T*UGA_star**6*epsilon + 2560.0*P**3*R**2*S**4*T*UGA_star**6 - 8192000.0*P**3*R**2*S**3*T**2*UGA_star**20*epsilon - 512000.0*P**3*R**2*S**3*T**2*UGA_star**20 + 90112000.0*P**3*R**2*S**3*T**2*UGA_star**19*epsilon + 5632000.0*P**3*R**2*S**3*T**2*UGA_star**19 - 449228800.0*P**3*R**2*S**3*T**2*UGA_star**18*epsilon - 28057600.0*P**3*R**2*S**3*T**2*UGA_star**18 + 1338183680.0*P**3*R**2*S**3*T**2*UGA_star**17*epsilon + 83445760.0*P**3*R**2*S**3*T**2*UGA_star**17 - 2642079744.0*P**3*R**2*S**3*T**2*UGA_star**16*epsilon - 164304896.0*P**3*R**2*S**3*T**2*UGA_star**16 + 3620970496.0*P**3*R**2*S**3*T**2*UGA_star**15*epsilon + 224292864.0*P**3*R**2*S**3*T**2*UGA_star**15 - 3500580864.0*P**3*R**2*S**3*T**2*UGA_star**14*epsilon - 215801856.0*P**3*R**2*S**3*T**2*UGA_star**14 + 2370625536.0*P**3*R**2*S**3*T**2*UGA_star**13*epsilon + 145649664.0*P**3*R**2*S**3*T**2*UGA_star**13 - 1087954944.0*P**3*R**2*S**3*T**2*UGA_star**12*epsilon - 67362816.0*P**3*R**2*S**3*T**2*UGA_star**12 + 313524224.0*P**3*R**2*S**3*T**2*UGA_star**11*epsilon + 20688896.0*P**3*R**2*S**3*T**2*UGA_star**11 - 47312896.0*P**3*R**2*S**3*T**2*UGA_star**10*epsilon - 4440064.0*P**3*R**2*S**3*T**2*UGA_star**10 + 1675264.0*P**3*R**2*S**3*T**2*UGA_star**9*epsilon + 1017856.0*P**3*R**2*S**3*T**2*UGA_star**9 + 446464.0*P**3*R**2*S**3*T**2*UGA_star**8*epsilon - 292864.0*P**3*R**2*S**3*T**2*UGA_star**8 - 270336.0*P**3*R**2*S**3*T**2*UGA_star**7*epsilon + 45056.0*P**3*R**2*S**3*T**2*UGA_star**7 + 81920.0*P**3*R**2*S**3*T**2*UGA_star**6*epsilon - 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30265344.0*P**3*R**2*S**2*T**3*UGA_star**10*epsilon - 454656.0*P**3*R**2*S**2*T**3*UGA_star**10 - 1265664.0*P**3*R**2*S**2*T**3*UGA_star**9*epsilon - 577536.0*P**3*R**2*S**2*T**3*UGA_star**9 - 651264.0*P**3*R**2*S**2*T**3*UGA_star**8*epsilon + 82944.0*P**3*R**2*S**2*T**3*UGA_star**8 + 331776.0*P**3*R**2*S**2*T**3*UGA_star**7*epsilon + 6144.0*P**3*R**2*S**2*T**3*UGA_star**7 - 4085760.0*P**3*R**2*S*T**4*UGA_star**20*epsilon - 161280.0*P**3*R**2*S*T**4*UGA_star**20 + 44943360.0*P**3*R**2*S*T**4*UGA_star**19*epsilon + 1774080.0*P**3*R**2*S*T**4*UGA_star**19 - 224901120.0*P**3*R**2*S*T**4*UGA_star**18*epsilon - 8908800.0*P**3*R**2*S*T**4*UGA_star**18 + 674703360.0*P**3*R**2*S*T**4*UGA_star**17*epsilon + 26926080.0*P**3*R**2*S*T**4*UGA_star**17 - 1344983040.0*P**3*R**2*S*T**4*UGA_star**16*epsilon - 54312960.0*P**3*R**2*S*T**4*UGA_star**16 + 1863306240.0*P**3*R**2*S*T**4*UGA_star**15*epsilon + 76508160.0*P**3*R**2*S*T**4*UGA_star**15 - 1818117120.0*P**3*R**2*S*T**4*UGA_star**14*epsilon - 76339200.0*P**3*R**2*S*T**4*UGA_star**14 + 1232931840.0*P**3*R**2*S*T**4*UGA_star**13*epsilon + 53329920.0*P**3*R**2*S*T**4*UGA_star**13 - 552115200.0*P**3*R**2*S*T**4*UGA_star**12*epsilon - 24890880.0*P**3*R**2*S*T**4*UGA_star**12 + 141143040.0*P**3*R**2*S*T**4*UGA_star**11*epsilon + 6812160.0*P**3*R**2*S*T**4*UGA_star**11 - 9262080.0*P**3*R**2*S*T**4*UGA_star**10*epsilon - 583680.0*P**3*R**2*S*T**4*UGA_star**10 - 4285440.0*P**3*R**2*S*T**4*UGA_star**9*epsilon - 199680.0*P**3*R**2*S*T**4*UGA_star**9 + 660480.0*P**3*R**2*S*T**4*UGA_star**8*epsilon + 46080.0*P**3*R**2*S*T**4*UGA_star**8 + 61440.0*P**3*R**2*S*T**4*UGA_star**7*epsilon - 745472.0*P**3*R**2*T**5*UGA_star**20*epsilon - 14336.0*P**3*R**2*T**5*UGA_star**20 + 8200192.0*P**3*R**2*T**5*UGA_star**19*epsilon + 157696.0*P**3*R**2*T**5*UGA_star**19 - 41180160.0*P**3*R**2*T**5*UGA_star**18*epsilon - 799232.0*P**3*R**2*T**5*UGA_star**18 + 124472320.0*P**3*R**2*T**5*UGA_star**17*epsilon + 2462208.0*P**3*R**2*T**5*UGA_star**17 - 251126784.0*P**3*R**2*T**5*UGA_star**16*epsilon - 5119488.0*P**3*R**2*T**5*UGA_star**16 + 353974272.0*P**3*R**2*T**5*UGA_star**15*epsilon + 7537152.0*P**3*R**2*T**5*UGA_star**15 - 353783808.0*P**3*R**2*T**5*UGA_star**14*epsilon - 8010240.0*P**3*R**2*T**5*UGA_star**14 + 248156160.0*P**3*R**2*T**5*UGA_star**13*epsilon + 6139392.0*P**3*R**2*T**5*UGA_star**13 - 116981760.0*P**3*R**2*T**5*UGA_star**12*epsilon - 3322368.0*P**3*R**2*T**5*UGA_star**12 + 32958464.0*P**3*R**2*T**5*UGA_star**11*epsilon + 1207808.0*P**3*R**2*T**5*UGA_star**11 - 3404800.0*P**3*R**2*T**5*UGA_star**10*epsilon - 265216.0*P**3*R**2*T**5*UGA_star**10 - 728064.0*P**3*R**2*T**5*UGA_star**9*epsilon + 26624.0*P**3*R**2*T**5*UGA_star**9 + 189440.0*P**3*R**2*T**5*UGA_star**8*epsilon + 272384.0*P**3*R*S**6*UGA_star**20*epsilon + 10752.0*P**3*R*S**6*UGA_star**20 - 3268608.0*P**3*R*S**6*UGA_star**19*epsilon - 129024.0*P**3*R*S**6*UGA_star**19 + 17895424.0*P**3*R*S**6*UGA_star**18*epsilon + 708096.0*P**3*R*S**6*UGA_star**18 - 59052032.0*P**3*R*S**6*UGA_star**17*epsilon - 2350080.0*P**3*R*S**6*UGA_star**17 + 130648064.0*P**3*R*S**6*UGA_star**16*epsilon + 5254144.0*P**3*R*S**6*UGA_star**16 - 203868160.0*P**3*R*S**6*UGA_star**15*epsilon - 8340992.0*P**3*R*S**6*UGA_star**15 + 229728256.0*P**3*R*S**6*UGA_star**14*epsilon + 9653248.0*P**3*R*S**6*UGA_star**14 - 188233728.0*P**3*R*S**6*UGA_star**13*epsilon - 8228352.0*P**3*R*S**6*UGA_star**13 + 111704064.0*P**3*R*S**6*UGA_star**12*epsilon + 5153280.0*P**3*R*S**6*UGA_star**12 - 47771648.0*P**3*R*S**6*UGA_star**11*epsilon - 2331648.0*P**3*R*S**6*UGA_star**11 + 15126528.0*P**3*R*S**6*UGA_star**10*epsilon + 726528.0*P**3*R*S**6*UGA_star**10 - 4046848.0*P**3*R*S**6*UGA_star**9*epsilon - 129024.0*P**3*R*S**6*UGA_star**9 + 1071104.0*P**3*R*S**6*UGA_star**8*epsilon - 6144.0*P**3*R*S**6*UGA_star**8 - 223232.0*P**3*R*S**6*UGA_star**7*epsilon + 12800.0*P**3*R*S**6*UGA_star**7 + 16384.0*P**3*R*S**6*UGA_star**6*epsilon - 4096.0*P**3*R*S**6*UGA_star**6 + 2048.0*P**3*R*S**6*UGA_star**5*epsilon + 512.0*P**3*R*S**6*UGA_star**5 + 1646592.0*P**3*R*S**5*T*UGA_star**20*epsilon + 92160.0*P**3*R*S**5*T*UGA_star**20 - 19759104.0*P**3*R*S**5*T*UGA_star**19*epsilon - 1105920.0*P**3*R*S**5*T*UGA_star**19 + 108140544.0*P**3*R*S**5*T*UGA_star**18*epsilon + 6054912.0*P**3*R*S**5*T*UGA_star**18 - 356450304.0*P**3*R*S**5*T*UGA_star**17*epsilon - 19977216.0*P**3*R*S**5*T*UGA_star**17 + 786587648.0*P**3*R*S**5*T*UGA_star**16*epsilon + 44169728.0*P**3*R*S**5*T*UGA_star**16 - 1220724736.0*P**3*R*S**5*T*UGA_star**15*epsilon - 68802048.0*P**3*R*S**5*T*UGA_star**15 + 1360037888.0*P**3*R*S**5*T*UGA_star**14*epsilon + 77201920.0*P**3*R*S**5*T*UGA_star**14 - 1087899648.0*P**3*R*S**5*T*UGA_star**13*epsilon - 62651904.0*P**3*R*S**5*T*UGA_star**13 + 611770368.0*P**3*R*S**5*T*UGA_star**12*epsilon + 36375552.0*P**3*R*S**5*T*UGA_star**12 - 229380096.0*P**3*R*S**5*T*UGA_star**11*epsilon - 14785536.0*P**3*R*S**5*T*UGA_star**11 + 50620416.0*P**3*R*S**5*T*UGA_star**10*epsilon + 4171776.0*P**3*R*S**5*T*UGA_star**10 - 4214784.0*P**3*R*S**5*T*UGA_star**9*epsilon - 921600.0*P**3*R*S**5*T*UGA_star**9 - 516096.0*P**3*R*S**5*T*UGA_star**8*epsilon + 225792.0*P**3*R*S**5*T*UGA_star**8 + 219136.0*P**3*R*S**5*T*UGA_star**7*epsilon - 52736.0*P**3*R*S**5*T*UGA_star**7 - 100352.0*P**3*R*S**5*T*UGA_star**6*epsilon + 4608.0*P**3*R*S**5*T*UGA_star**6 + 22528.0*P**3*R*S**5*T*UGA_star**5*epsilon + 512.0*P**3*R*S**5*T*UGA_star**5 + 4096000.0*P**3*R*S**4*T**2*UGA_star**20*epsilon + 256000.0*P**3*R*S**4*T**2*UGA_star**20 - 49152000.0*P**3*R*S**4*T**2*UGA_star**19*epsilon - 3072000.0*P**3*R*S**4*T**2*UGA_star**19 + 269352960.0*P**3*R*S**4*T**2*UGA_star**18*epsilon + 16824320.0*P**3*R*S**4*T**2*UGA_star**18 - 890122240.0*P**3*R*S**4*T**2*UGA_star**17*epsilon - 55521280.0*P**3*R*S**4*T**2*UGA_star**17 + 1971675136.0*P**3*R*S**4*T**2*UGA_star**16*epsilon + 122691584.0*P**3*R*S**4*T**2*UGA_star**16 - 3074293760.0*P**3*R*S**4*T**2*UGA_star**15*epsilon - 190627840.0*P**3*R*S**4*T**2*UGA_star**15 + 3442032640.0*P**3*R*S**4*T**2*UGA_star**14*epsilon + 212398080.0*P**3*R*S**4*T**2*UGA_star**14 - 2762526720.0*P**3*R*S**4*T**2*UGA_star**13*epsilon - 169451520.0*P**3*R*S**4*T**2*UGA_star**13 + 1548533760.0*P**3*R*S**4*T**2*UGA_star**12*epsilon + 94464000.0*P**3*R*S**4*T**2*UGA_star**12 - 565706752.0*P**3*R*S**4*T**2*UGA_star**11*epsilon - 34643968.0*P**3*R*S**4*T**2*UGA_star**11 + 110100480.0*P**3*R*S**4*T**2*UGA_star**10*epsilon + 7280640.0*P**3*R*S**4*T**2*UGA_star**10 - 675840.0*P**3*R*S**4*T**2*UGA_star**9*epsilon - 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3879075840.0*P**3*R*S**3*T**3*UGA_star**13*epsilon - 215408640.0*P**3*R*S**3*T**3*UGA_star**13 + 2218844160.0*P**3*R*S**3*T**3*UGA_star**12*epsilon + 121313280.0*P**3*R*S**3*T**3*UGA_star**12 - 836722688.0*P**3*R*S**3*T**3*UGA_star**11*epsilon - 43640832.0*P**3*R*S**3*T**3*UGA_star**11 + 175411200.0*P**3*R*S**3*T**3*UGA_star**10*epsilon + 7403520.0*P**3*R*S**3*T**3*UGA_star**10 - 7331840.0*P**3*R*S**3*T**3*UGA_star**9*epsilon + 829440.0*P**3*R*S**3*T**3*UGA_star**9 - 3297280.0*P**3*R*S**3*T**3*UGA_star**8*epsilon - 599040.0*P**3*R*S**3*T**3*UGA_star**8 - 266240.0*P**3*R*S**3*T**3*UGA_star**7*epsilon + 66560.0*P**3*R*S**3*T**3*UGA_star**7 + 200704.0*P**3*R*S**3*T**3*UGA_star**6*epsilon + 4096.0*P**3*R*S**3*T**3*UGA_star**6 + 4085760.0*P**3*R*S**2*T**4*UGA_star**20*epsilon + 161280.0*P**3*R*S**2*T**4*UGA_star**20 - 49029120.0*P**3*R*S**2*T**4*UGA_star**19*epsilon - 1935360.0*P**3*R*S**2*T**4*UGA_star**19 + 269844480.0*P**3*R*S**2*T**4*UGA_star**18*epsilon + 10682880.0*P**3*R*S**2*T**4*UGA_star**18 - 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51200.0*P**3*S**5*T**2*UGA_star**20 + 10649600.0*P**3*S**5*T**2*UGA_star**19*epsilon + 665600.0*P**3*S**5*T**2*UGA_star**19 - 63637504.0*P**3*S**5*T**2*UGA_star**18*epsilon - 3975168.0*P**3*S**5*T**2*UGA_star**18 + 231114752.0*P**3*S**5*T**2*UGA_star**17*epsilon + 14418944.0*P**3*S**5*T**2*UGA_star**17 - 567953408.0*P**3*S**5*T**2*UGA_star**16*epsilon - 35359744.0*P**3*S**5*T**2*UGA_star**16 + 994078720.0*P**3*S**5*T**2*UGA_star**15*epsilon + 61692928.0*P**3*S**5*T**2*UGA_star**15 - 1268183040.0*P**3*S**5*T**2*UGA_star**14*epsilon - 78341120.0*P**3*S**5*T**2*UGA_star**14 + 1182885888.0*P**3*S**5*T**2*UGA_star**13*epsilon + 72585216.0*P**3*S**5*T**2*UGA_star**13 - 792121344.0*P**3*S**5*T**2*UGA_star**12*epsilon - 48119808.0*P**3*S**5*T**2*UGA_star**12 + 360636416.0*P**3*S**5*T**2*UGA_star**11*epsilon + 21536768.0*P**3*S**5*T**2*UGA_star**11 - 95037440.0*P**3*S**5*T**2*UGA_star**10*epsilon - 5451776.0*P**3*S**5*T**2*UGA_star**10 + 4003840.0*P**3*S**5*T**2*UGA_star**9*epsilon + 107520.0*P**3*S**5*T**2*UGA_star**9 + 5871616.0*P**3*S**5*T**2*UGA_star**8*epsilon + 381952.0*P**3*S**5*T**2*UGA_star**8 - 1527808.0*P**3*S**5*T**2*UGA_star**7*epsilon - 91136.0*P**3*S**5*T**2*UGA_star**7 + 8192.0*P**3*S**5*T**2*UGA_star**6*epsilon - 1024.0*P**3*S**5*T**2*UGA_star**6 + 30720.0*P**3*S**5*T**2*UGA_star**5*epsilon + 2048.0*P**3*S**5*T**2*UGA_star**5 - 1372160.0*P**3*S**4*T**3*UGA_star**20*epsilon - 76800.0*P**3*S**4*T**3*UGA_star**20 + 17838080.0*P**3*S**4*T**3*UGA_star**19*epsilon + 998400.0*P**3*S**4*T**3*UGA_star**19 - 106844160.0*P**3*S**4*T**3*UGA_star**18*epsilon - 5980160.0*P**3*S**4*T**3*UGA_star**18 + 389980160.0*P**3*S**4*T**3*UGA_star**17*epsilon + 21826560.0*P**3*S**4*T**3*UGA_star**17 - 966219776.0*P**3*S**4*T**3*UGA_star**16*epsilon - 54063104.0*P**3*S**4*T**3*UGA_star**16 + 1711886336.0*P**3*S**4*T**3*UGA_star**15*epsilon + 95704064.0*P**3*S**4*T**3*UGA_star**15 - 2223011840.0*P**3*S**4*T**3*UGA_star**14*epsilon - 124011520.0*P**3*S**4*T**3*UGA_star**14 + 2128834560.0*P**3*S**4*T**3*UGA_star**13*epsilon + 118164480.0*P**3*S**4*T**3*UGA_star**13 - 1486233600.0*P**3*S**4*T**3*UGA_star**12*epsilon - 81546240.0*P**3*S**4*T**3*UGA_star**12 + 729485312.0*P**3*S**4*T**3*UGA_star**11*epsilon + 38888448.0*P**3*S**4*T**3*UGA_star**11 - 230162432.0*P**3*S**4*T**3*UGA_star**10*epsilon - 11225088.0*P**3*S**4*T**3*UGA_star**10 + 34652160.0*P**3*S**4*T**3*UGA_star**9*epsilon + 926720.0*P**3*S**4*T**3*UGA_star**9 + 2713600.0*P**3*S**4*T**3*UGA_star**8*epsilon + 583680.0*P**3*S**4*T**3*UGA_star**8 - 1576960.0*P**3*S**4*T**3*UGA_star**7*epsilon - 209920.0*P**3*S**4*T**3*UGA_star**7 - 14336.0*P**3*S**4*T**3*UGA_star**6*epsilon + 19456.0*P**3*S**4*T**3*UGA_star**6 + 45056.0*P**3*S**4*T**3*UGA_star**5*epsilon + 1024.0*P**3*S**4*T**3*UGA_star**5 - 1361920.0*P**3*S**3*T**4*UGA_star**20*epsilon - 53760.0*P**3*S**3*T**4*UGA_star**20 + 17704960.0*P**3*S**3*T**4*UGA_star**19*epsilon + 698880.0*P**3*S**3*T**4*UGA_star**19 - 106291200.0*P**3*S**3*T**4*UGA_star**18*epsilon - 4206080.0*P**3*S**3*T**4*UGA_star**18 + 389816320.0*P**3*S**3*T**4*UGA_star**17*epsilon + 15505920.0*P**3*S**3*T**4*UGA_star**17 - 973066240.0*P**3*S**3*T**4*UGA_star**16*epsilon - 39024640.0*P**3*S**3*T**4*UGA_star**16 + 1742351360.0*P**3*S**3*T**4*UGA_star**15*epsilon + 70686720.0*P**3*S**3*T**4*UGA_star**15 - 2295188480.0*P**3*S**3*T**4*UGA_star**14*epsilon - 94556160.0*P**3*S**3*T**4*UGA_star**14 + 2240471040.0*P**3*S**3*T**4*UGA_star**13*epsilon + 94172160.0*P**3*S**3*T**4*UGA_star**13 - 1605580800.0*P**3*S**3*T**4*UGA_star**12*epsilon - 69296640.0*P**3*S**3*T**4*UGA_star**12 + 818360320.0*P**3*S**3*T**4*UGA_star**11*epsilon + 36641280.0*P**3*S**3*T**4*UGA_star**11 - 274769920.0*P**3*S**3*T**4*UGA_star**10*epsilon - 13032960.0*P**3*S**3*T**4*UGA_star**10 + 48107520.0*P**3*S**3*T**4*UGA_star**9*epsilon + 2593280.0*P**3*S**3*T**4*UGA_star**9 + 1372160.0*P**3*S**3*T**4*UGA_star**8*epsilon - 46080.0*P**3*S**3*T**4*UGA_star**8 - 2155520.0*P**3*S**3*T**4*UGA_star**7*epsilon - 97280.0*P**3*S**3*T**4*UGA_star**7 + 209920.0*P**3*S**3*T**4*UGA_star**6*epsilon + 15360.0*P**3*S**3*T**4*UGA_star**6 + 20480.0*P**3*S**3*T**4*UGA_star**5*epsilon - 745472.0*P**3*S**2*T**5*UGA_star**20*epsilon - 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19163136.0*P**2*R**6*S*T*UGA_star**17*epsilon - 803328.0*P**2*R**6*S*T*UGA_star**17 + 19467264.0*P**2*R**6*S*T*UGA_star**16*epsilon + 881664.0*P**2*R**6*S*T*UGA_star**16 - 14825472.0*P**2*R**6*S*T*UGA_star**15*epsilon - 723456.0*P**2*R**6*S*T*UGA_star**15 + 8727552.0*P**2*R**6*S*T*UGA_star**14*epsilon + 423936.0*P**2*R**6*S*T*UGA_star**14 - 3717120.0*P**2*R**6*S*T*UGA_star**13*epsilon - 148992.0*P**2*R**6*S*T*UGA_star**13 + 1084416.0*P**2*R**6*S*T*UGA_star**12*epsilon + 19968.0*P**2*R**6*S*T*UGA_star**12 - 251904.0*P**2*R**6*S*T*UGA_star**11*epsilon + 1536.0*P**2*R**6*S*T*UGA_star**11 + 46080.0*P**2*R**6*S*T*UGA_star**10*epsilon + 411648.0*P**2*R**6*T**2*UGA_star**20*epsilon + 23040.0*P**2*R**6*T**2*UGA_star**20 - 2469888.0*P**2*R**6*T**2*UGA_star**19*epsilon - 138240.0*P**2*R**6*T**2*UGA_star**19 + 6306816.0*P**2*R**6*T**2*UGA_star**18*epsilon + 353280.0*P**2*R**6*T**2*UGA_star**18 - 9037824.0*P**2*R**6*T**2*UGA_star**17*epsilon - 508416.0*P**2*R**6*T**2*UGA_star**17 + 8226816.0*P**2*R**6*T**2*UGA_star**16*epsilon + 466944.0*P**2*R**6*T**2*UGA_star**16 - 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3702784.0*P**2*R**5*S**3*UGA_star**12*epsilon - 304640.0*P**2*R**5*S**3*UGA_star**12 + 691200.0*P**2*R**5*S**3*UGA_star**11*epsilon + 125440.0*P**2*R**5*S**3*UGA_star**11 - 97280.0*P**2*R**5*S**3*UGA_star**10*epsilon - 35840.0*P**2*R**5*S**3*UGA_star**10 + 20480.0*P**2*R**5*S**3*UGA_star**9*epsilon + 5120.0*P**2*R**5*S**3*UGA_star**9 - 2451456.0*P**2*R**5*S**2*T*UGA_star**20*epsilon - 96768.0*P**2*R**5*S**2*T*UGA_star**20 + 17160192.0*P**2*R**5*S**2*T*UGA_star**19*epsilon + 677376.0*P**2*R**5*S**2*T*UGA_star**19 - 52439040.0*P**2*R**5*S**2*T*UGA_star**18*epsilon - 2092032.0*P**2*R**5*S**2*T*UGA_star**18 + 92841984.0*P**2*R**5*S**2*T*UGA_star**17*epsilon + 3815424.0*P**2*R**5*S**2*T*UGA_star**17 - 107940864.0*P**2*R**5*S**2*T*UGA_star**16*epsilon - 4692480.0*P**2*R**5*S**2*T*UGA_star**16 + 90218496.0*P**2*R**5*S**2*T*UGA_star**15*epsilon + 4220928.0*P**2*R**5*S**2*T*UGA_star**15 - 58186752.0*P**2*R**5*S**2*T*UGA_star**14*epsilon - 2843136.0*P**2*R**5*S**2*T*UGA_star**14 + 29306880.0*P**2*R**5*S**2*T*UGA_star**13*epsilon + 1339392.0*P**2*R**5*S**2*T*UGA_star**13 - 10804224.0*P**2*R**5*S**2*T*UGA_star**12*epsilon - 359424.0*P**2*R**5*S**2*T*UGA_star**12 + 2770944.0*P**2*R**5*S**2*T*UGA_star**11*epsilon + 23040.0*P**2*R**5*S**2*T*UGA_star**11 - 568320.0*P**2*R**5*S**2*T*UGA_star**10*epsilon + 7680.0*P**2*R**5*S**2*T*UGA_star**10 + 92160.0*P**2*R**5*S**2*T*UGA_star**9*epsilon - 2469888.0*P**2*R**5*S*T**2*UGA_star**20*epsilon - 138240.0*P**2*R**5*S*T**2*UGA_star**20 + 17289216.0*P**2*R**5*S*T**2*UGA_star**19*epsilon + 967680.0*P**2*R**5*S*T**2*UGA_star**19 - 52328448.0*P**2*R**5*S*T**2*UGA_star**18*epsilon - 2930688.0*P**2*R**5*S*T**2*UGA_star**18 + 89929728.0*P**2*R**5*S*T**2*UGA_star**17*epsilon + 5050368.0*P**2*R**5*S*T**2*UGA_star**17 - 97701888.0*P**2*R**5*S*T**2*UGA_star**16*epsilon - 5521920.0*P**2*R**5*S*T**2*UGA_star**16 + 71958528.0*P**2*R**5*S*T**2*UGA_star**15*epsilon + 4101120.0*P**2*R**5*S*T**2*UGA_star**15 - 38578176.0*P**2*R**5*S*T**2*UGA_star**14*epsilon - 2188800.0*P**2*R**5*S*T**2*UGA_star**14 + 15710208.0*P**2*R**5*S*T**2*UGA_star**13*epsilon + 841728.0*P**2*R**5*S*T**2*UGA_star**13 - 4313088.0*P**2*R**5*S*T**2*UGA_star**12*epsilon - 188928.0*P**2*R**5*S*T**2*UGA_star**12 + 442368.0*P**2*R**5*S*T**2*UGA_star**11*epsilon + 61440.0*P**2*R**5*S*T**2*UGA_star**10*epsilon + 7680.0*P**2*R**5*S*T**2*UGA_star**10 - 819200.0*P**2*R**5*T**3*UGA_star**20*epsilon - 51200.0*P**2*R**5*T**3*UGA_star**20 + 5734400.0*P**2*R**5*T**3*UGA_star**19*epsilon + 358400.0*P**2*R**5*T**3*UGA_star**19 - 17197056.0*P**2*R**5*T**3*UGA_star**18*epsilon - 1073152.0*P**2*R**5*T**3*UGA_star**18 + 28690432.0*P**2*R**5*T**3*UGA_star**17*epsilon + 1782784.0*P**2*R**5*T**3*UGA_star**17 - 28932096.0*P**2*R**5*T**3*UGA_star**16*epsilon - 1783808.0*P**2*R**5*T**3*UGA_star**16 + 17981440.0*P**2*R**5*T**3*UGA_star**15*epsilon + 1100800.0*P**2*R**5*T**3*UGA_star**15 - 6799360.0*P**2*R**5*T**3*UGA_star**14*epsilon - 430080.0*P**2*R**5*T**3*UGA_star**14 + 1509376.0*P**2*R**5*T**3*UGA_star**13*epsilon + 125952.0*P**2*R**5*T**3*UGA_star**13 - 120832.0*P**2*R**5*T**3*UGA_star**12*epsilon - 36864.0*P**2*R**5*T**3*UGA_star**12 - 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13604864.0*P**2*R**4*S*T**3*UGA_star**15 + 110069760.0*P**2*R**4*S*T**3*UGA_star**14*epsilon + 6773760.0*P**2*R**4*S*T**3*UGA_star**14 - 32604160.0*P**2*R**4*S*T**3*UGA_star**13*epsilon - 2191360.0*P**2*R**4*S*T**3*UGA_star**13 + 5027840.0*P**2*R**4*S*T**3*UGA_star**12*epsilon + 578560.0*P**2*R**4*S*T**3*UGA_star**12 + 98304.0*P**2*R**4*S*T**3*UGA_star**11*epsilon - 169984.0*P**2*R**4*S*T**3*UGA_star**11 - 415744.0*P**2*R**4*S*T**3*UGA_star**10*epsilon + 31744.0*P**2*R**4*S*T**3*UGA_star**10 + 122880.0*P**2*R**4*S*T**3*UGA_star**9*epsilon + 1029120.0*P**2*R**4*T**4*UGA_star**20*epsilon + 57600.0*P**2*R**4*T**4*UGA_star**20 - 8232960.0*P**2*R**4*T**4*UGA_star**19*epsilon - 460800.0*P**2*R**4*T**4*UGA_star**19 + 28646400.0*P**2*R**4*T**4*UGA_star**18*epsilon + 1605120.0*P**2*R**4*T**4*UGA_star**18 - 56340480.0*P**2*R**4*T**4*UGA_star**17*epsilon - 3164160.0*P**2*R**4*T**4*UGA_star**17 + 67866624.0*P**2*R**4*T**4*UGA_star**16*epsilon + 3826176.0*P**2*R**4*T**4*UGA_star**16 - 50253824.0*P**2*R**4*T**4*UGA_star**15*epsilon - 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2614272.0*P**2*R**3*S**5*UGA_star**14 + 57387008.0*P**2*R**3*S**5*UGA_star**13*epsilon + 1999872.0*P**2*R**3*S**5*UGA_star**13 - 27338752.0*P**2*R**3*S**5*UGA_star**12*epsilon - 1225728.0*P**2*R**3*S**5*UGA_star**12 + 9404416.0*P**2*R**3*S**5*UGA_star**11*epsilon + 594944.0*P**2*R**3*S**5*UGA_star**11 - 1985536.0*P**2*R**3*S**5*UGA_star**10*epsilon - 225792.0*P**2*R**3*S**5*UGA_star**10 + 175104.0*P**2*R**3*S**5*UGA_star**9*epsilon + 66048.0*P**2*R**3*S**5*UGA_star**9 - 9216.0*P**2*R**3*S**5*UGA_star**8*epsilon - 13824.0*P**2*R**3*S**5*UGA_star**8 + 6144.0*P**2*R**3*S**5*UGA_star**7*epsilon + 1536.0*P**2*R**3*S**5*UGA_star**7 - 4085760.0*P**2*R**3*S**4*T*UGA_star**20*epsilon - 161280.0*P**2*R**3*S**4*T*UGA_star**20 + 36771840.0*P**2*R**3*S**4*T*UGA_star**19*epsilon + 1451520.0*P**2*R**3*S**4*T*UGA_star**19 - 147271680.0*P**2*R**3*S**4*T*UGA_star**18*epsilon - 5844480.0*P**2*R**3*S**4*T*UGA_star**18 + 345968640.0*P**2*R**3*S**4*T*UGA_star**17*epsilon + 13923840.0*P**2*R**3*S**4*T*UGA_star**17 - 530288640.0*P**2*R**3*S**4*T*UGA_star**16*epsilon - 21934080.0*P**2*R**3*S**4*T*UGA_star**16 + 561239040.0*P**2*R**3*S**4*T*UGA_star**15*epsilon + 24284160.0*P**2*R**3*S**4*T*UGA_star**15 - 428298240.0*P**2*R**3*S**4*T*UGA_star**14*epsilon - 19676160.0*P**2*R**3*S**4*T*UGA_star**14 + 247726080.0*P**2*R**3*S**4*T*UGA_star**13*epsilon + 11934720.0*P**2*R**3*S**4*T*UGA_star**13 - 114616320.0*P**2*R**3*S**4*T*UGA_star**12*epsilon - 5322240.0*P**2*R**3*S**4*T*UGA_star**12 + 42762240.0*P**2*R**3*S**4*T*UGA_star**11*epsilon + 1543680.0*P**2*R**3*S**4*T*UGA_star**11 - 11796480.0*P**2*R**3*S**4*T*UGA_star**10*epsilon - 161280.0*P**2*R**3*S**4*T*UGA_star**10 + 2150400.0*P**2*R**3*S**4*T*UGA_star**9*epsilon - 53760.0*P**2*R**3*S**4*T*UGA_star**9 - 307200.0*P**2*R**3*S**4*T*UGA_star**8*epsilon + 15360.0*P**2*R**3*S**4*T*UGA_star**8 + 46080.0*P**2*R**3*S**4*T*UGA_star**7*epsilon - 8232960.0*P**2*R**3*S**3*T**2*UGA_star**20*epsilon - 460800.0*P**2*R**3*S**3*T**2*UGA_star**20 + 74096640.0*P**2*R**3*S**3*T**2*UGA_star**19*epsilon + 4147200.0*P**2*R**3*S**3*T**2*UGA_star**19 - 295710720.0*P**2*R**3*S**3*T**2*UGA_star**18*epsilon - 16558080.0*P**2*R**3*S**3*T**2*UGA_star**18 + 687575040.0*P**2*R**3*S**3*T**2*UGA_star**17*epsilon + 38553600.0*P**2*R**3*S**3*T**2*UGA_star**17 - 1029820416.0*P**2*R**3*S**3*T**2*UGA_star**16*epsilon - 57928704.0*P**2*R**3*S**3*T**2*UGA_star**16 + 1039429632.0*P**2*R**3*S**3*T**2*UGA_star**15*epsilon + 58841088.0*P**2*R**3*S**3*T**2*UGA_star**15 - 723480576.0*P**2*R**3*S**3*T**2*UGA_star**14*epsilon - 41416704.0*P**2*R**3*S**3*T**2*UGA_star**14 + 355307520.0*P**2*R**3*S**3*T**2*UGA_star**13*epsilon + 20674560.0*P**2*R**3*S**3*T**2*UGA_star**13 - 128716800.0*P**2*R**3*S**3*T**2*UGA_star**12*epsilon - 7587840.0*P**2*R**3*S**3*T**2*UGA_star**12 + 35598336.0*P**2*R**3*S**3*T**2*UGA_star**11*epsilon + 2064384.0*P**2*R**3*S**3*T**2*UGA_star**11 - 6070272.0*P**2*R**3*S**3*T**2*UGA_star**10*epsilon - 319488.0*P**2*R**3*S**3*T**2*UGA_star**10 - 221184.0*P**2*R**3*S**3*T**2*UGA_star**9*epsilon - 24576.0*P**2*R**3*S**3*T**2*UGA_star**9 + 245760.0*P**2*R**3*S**3*T**2*UGA_star**8*epsilon + 15360.0*P**2*R**3*S**3*T**2*UGA_star**8 - 8192000.0*P**2*R**3*S**2*T**3*UGA_star**20*epsilon - 512000.0*P**2*R**3*S**2*T**3*UGA_star**20 + 73728000.0*P**2*R**3*S**2*T**3*UGA_star**19*epsilon + 4608000.0*P**2*R**3*S**2*T**3*UGA_star**19 - 293580800.0*P**2*R**3*S**2*T**3*UGA_star**18*epsilon - 18329600.0*P**2*R**3*S**2*T**3*UGA_star**18 + 677662720.0*P**2*R**3*S**2*T**3*UGA_star**17*epsilon + 42199040.0*P**2*R**3*S**2*T**3*UGA_star**17 - 996675584.0*P**2*R**3*S**2*T**3*UGA_star**16*epsilon - 61771776.0*P**2*R**3*S**2*T**3*UGA_star**16 + 964681728.0*P**2*R**3*S**2*T**3*UGA_star**15*epsilon + 59379712.0*P**2*R**3*S**2*T**3*UGA_star**15 - 610402304.0*P**2*R**3*S**2*T**3*UGA_star**14*epsilon - 37359616.0*P**2*R**3*S**2*T**3*UGA_star**14 + 240517120.0*P**2*R**3*S**2*T**3*UGA_star**13*epsilon + 14991360.0*P**2*R**3*S**2*T**3*UGA_star**13 - 51855360.0*P**2*R**3*S**2*T**3*UGA_star**12*epsilon - 3891200.0*P**2*R**3*S**2*T**3*UGA_star**12 + 3416064.0*P**2*R**3*S**2*T**3*UGA_star**11*epsilon + 925696.0*P**2*R**3*S**2*T**3*UGA_star**11 + 1257472.0*P**2*R**3*S**2*T**3*UGA_star**10*epsilon - 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81920000.0*P**2*R**2*S**3*T**3*UGA_star**19*epsilon - 5120000.0*P**2*R**2*S**3*T**3*UGA_star**19 + 366673920.0*P**2*R**2*S**3*T**3*UGA_star**18*epsilon + 22896640.0*P**2*R**2*S**3*T**3*UGA_star**18 - 965345280.0*P**2*R**2*S**3*T**3*UGA_star**17*epsilon - 60149760.0*P**2*R**2*S**3*T**3*UGA_star**17 + 1649893376.0*P**2*R**2*S**3*T**3*UGA_star**16*epsilon + 102406144.0*P**2*R**2*S**3*T**3*UGA_star**16 - 1902051328.0*P**2*R**2*S**3*T**3*UGA_star**15*epsilon - 117358592.0*P**2*R**2*S**3*T**3*UGA_star**15 + 1482334208.0*P**2*R**2*S**3*T**3*UGA_star**14*epsilon + 90775552.0*P**2*R**2*S**3*T**3*UGA_star**14 - 754180096.0*P**2*R**2*S**3*T**3*UGA_star**13*epsilon - 46014464.0*P**2*R**2*S**3*T**3*UGA_star**13 + 225566720.0*P**2*R**2*S**3*T**3*UGA_star**12*epsilon + 14295040.0*P**2*R**2*S**3*T**3*UGA_star**12 - 26566656.0*P**2*R**2*S**3*T**3*UGA_star**11*epsilon - 2596864.0*P**2*R**2*S**3*T**3*UGA_star**11 - 3940352.0*P**2*R**2*S**3*T**3*UGA_star**10*epsilon + 540672.0*P**2*R**2*S**3*T**3*UGA_star**10 + 1728512.0*P**2*R**2*S**3*T**3*UGA_star**9*epsilon - 233472.0*P**2*R**2*S**3*T**3*UGA_star**9 - 507904.0*P**2*R**2*S**3*T**3*UGA_star**8*epsilon + 47104.0*P**2*R**2*S**3*T**3*UGA_star**8 + 122880.0*P**2*R**2*S**3*T**3*UGA_star**7*epsilon + 6174720.0*P**2*R**2*S**2*T**4*UGA_star**20*epsilon + 345600.0*P**2*R**2*S**2*T**4*UGA_star**20 - 61747200.0*P**2*R**2*S**2*T**4*UGA_star**19*epsilon - 3456000.0*P**2*R**2*S**2*T**4*UGA_star**19 + 276526080.0*P**2*R**2*S**2*T**4*UGA_star**18*epsilon + 15482880.0*P**2*R**2*S**2*T**4*UGA_star**18 - 728248320.0*P**2*R**2*S**2*T**4*UGA_star**17*epsilon - 40803840.0*P**2*R**2*S**2*T**4*UGA_star**17 + 1243167744.0*P**2*R**2*S**2*T**4*UGA_star**16*epsilon + 69720576.0*P**2*R**2*S**2*T**4*UGA_star**16 - 1425487872.0*P**2*R**2*S**2*T**4*UGA_star**15*epsilon - 80014848.0*P**2*R**2*S**2*T**4*UGA_star**15 + 1093865472.0*P**2*R**2*S**2*T**4*UGA_star**14*epsilon + 61383168.0*P**2*R**2*S**2*T**4*UGA_star**14 - 533796864.0*P**2*R**2*S**2*T**4*UGA_star**13*epsilon - 29784576.0*P**2*R**2*S**2*T**4*UGA_star**13 + 140236800.0*P**2*R**2*S**2*T**4*UGA_star**12*epsilon + 7541760.0*P**2*R**2*S**2*T**4*UGA_star**12 - 6027264.0*P**2*R**2*S**2*T**4*UGA_star**11*epsilon - 32256.0*P**2*R**2*S**2*T**4*UGA_star**11 - 4973568.0*P**2*R**2*S**2*T**4*UGA_star**10*epsilon - 440832.0*P**2*R**2*S**2*T**4*UGA_star**10 + 43008.0*P**2*R**2*S**2*T**4*UGA_star**9*epsilon + 49152.0*P**2*R**2*S**2*T**4*UGA_star**9 + 267264.0*P**2*R**2*S**2*T**4*UGA_star**8*epsilon + 9216.0*P**2*R**2*S**2*T**4*UGA_star**8 + 2451456.0*P**2*R**2*S*T**5*UGA_star**20*epsilon + 96768.0*P**2*R**2*S*T**5*UGA_star**20 - 24514560.0*P**2*R**2*S*T**5*UGA_star**19*epsilon - 967680.0*P**2*R**2*S*T**5*UGA_star**19 + 110002176.0*P**2*R**2*S*T**5*UGA_star**18*epsilon + 4359168.0*P**2*R**2*S*T**5*UGA_star**18 - 290838528.0*P**2*R**2*S*T**5*UGA_star**17*epsilon - 11625984.0*P**2*R**2*S*T**5*UGA_star**17 + 499418112.0*P**2*R**2*S*T**5*UGA_star**16*epsilon + 20258304.0*P**2*R**2*S*T**5*UGA_star**16 - 577198080.0*P**2*R**2*S*T**5*UGA_star**15*epsilon - 23944704.0*P**2*R**2*S*T**5*UGA_star**15 + 447249408.0*P**2*R**2*S*T**5*UGA_star**14*epsilon + 19192320.0*P**2*R**2*S*T**5*UGA_star**14 - 220471296.0*P**2*R**2*S*T**5*UGA_star**13*epsilon - 9988608.0*P**2*R**2*S*T**5*UGA_star**13 + 57811968.0*P**2*R**2*S*T**5*UGA_star**12*epsilon + 2924544.0*P**2*R**2*S*T**5*UGA_star**12 - 1247232.0*P**2*R**2*S*T**5*UGA_star**11*epsilon - 198144.0*P**2*R**2*S*T**5*UGA_star**11 - 3111936.0*P**2*R**2*S*T**5*UGA_star**10*epsilon - 136704.0*P**2*R**2*S*T**5*UGA_star**10 + 387072.0*P**2*R**2*S*T**5*UGA_star**9*epsilon + 30720.0*P**2*R**2*S*T**5*UGA_star**9 + 61440.0*P**2*R**2*S*T**5*UGA_star**8*epsilon + 372736.0*P**2*R**2*T**6*UGA_star**20*epsilon + 7168.0*P**2*R**2*T**6*UGA_star**20 - 3727360.0*P**2*R**2*T**6*UGA_star**19*epsilon - 71680.0*P**2*R**2*T**6*UGA_star**19 + 16794624.0*P**2*R**2*T**6*UGA_star**18*epsilon + 326144.0*P**2*R**2*T**6*UGA_star**18 - 44814336.0*P**2*R**2*T**6*UGA_star**17*epsilon - 888832.0*P**2*R**2*T**6*UGA_star**17 + 78171136.0*P**2*R**2*T**6*UGA_star**16*epsilon + 1605888.0*P**2*R**2*T**6*UGA_star**16 - 92602368.0*P**2*R**2*T**6*UGA_star**15*epsilon - 2008576.0*P**2*R**2*T**6*UGA_star**15 + 74591232.0*P**2*R**2*T**6*UGA_star**14*epsilon + 1760000.0*P**2*R**2*T**6*UGA_star**14 - 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336927744.0*P**2*R*S**6*T*UGA_star**14*epsilon - 14418432.0*P**2*R*S**6*T*UGA_star**14 + 227266560.0*P**2*R*S**6*T*UGA_star**13*epsilon + 10266624.0*P**2*R*S**6*T*UGA_star**13 - 106831872.0*P**2*R*S**6*T*UGA_star**12*epsilon - 5193216.0*P**2*R*S**6*T*UGA_star**12 + 35063808.0*P**2*R*S**6*T*UGA_star**11*epsilon + 1801728.0*P**2*R*S**6*T*UGA_star**11 - 8896512.0*P**2*R*S**6*T*UGA_star**10*epsilon - 377856.0*P**2*R*S**6*T*UGA_star**10 + 2230272.0*P**2*R*S**6*T*UGA_star**9*epsilon + 9216.0*P**2*R*S**6*T*UGA_star**9 - 476160.0*P**2*R*S**6*T*UGA_star**8*epsilon + 27648.0*P**2*R*S**6*T*UGA_star**8 + 24576.0*P**2*R*S**6*T*UGA_star**7*epsilon - 10752.0*P**2*R*S**6*T*UGA_star**7 + 9216.0*P**2*R*S**6*T*UGA_star**6*epsilon + 1536.0*P**2*R*S**6*T*UGA_star**6 - 2469888.0*P**2*R*S**5*T**2*UGA_star**20*epsilon - 138240.0*P**2*R*S**5*T**2*UGA_star**20 + 27168768.0*P**2*R*S**5*T**2*UGA_star**19*epsilon + 1520640.0*P**2*R*S**5*T**2*UGA_star**19 - 134977536.0*P**2*R*S**5*T**2*UGA_star**18*epsilon - 7557120.0*P**2*R*S**5*T**2*UGA_star**18 + 398997504.0*P**2*R*S**5*T**2*UGA_star**17*epsilon + 22358016.0*P**2*R*S**5*T**2*UGA_star**17 - 777449472.0*P**2*R*S**5*T**2*UGA_star**16*epsilon - 43642368.0*P**2*R*S**5*T**2*UGA_star**16 + 1043619840.0*P**2*R*S**5*T**2*UGA_star**15*epsilon + 58791936.0*P**2*R*S**5*T**2*UGA_star**15 - 977190912.0*P**2*R*S**5*T**2*UGA_star**14*epsilon - 55448064.0*P**2*R*S**5*T**2*UGA_star**14 + 629231616.0*P**2*R*S**5*T**2*UGA_star**13*epsilon + 36274176.0*P**2*R*S**5*T**2*UGA_star**13 - 265089024.0*P**2*R*S**5*T**2*UGA_star**12*epsilon - 15906816.0*P**2*R*S**5*T**2*UGA_star**12 + 64401408.0*P**2*R*S**5*T**2*UGA_star**11*epsilon + 4396032.0*P**2*R*S**5*T**2*UGA_star**11 - 5732352.0*P**2*R*S**5*T**2*UGA_star**10*epsilon - 746496.0*P**2*R*S**5*T**2*UGA_star**10 - 608256.0*P**2*R*S**5*T**2*UGA_star**9*epsilon + 129024.0*P**2*R*S**5*T**2*UGA_star**9 + 184320.0*P**2*R*S**5*T**2*UGA_star**8*epsilon - 32256.0*P**2*R*S**5*T**2*UGA_star**8 - 122880.0*P**2*R*S**5*T**2*UGA_star**7*epsilon + 36864.0*P**2*R*S**5*T**2*UGA_star**6*epsilon + 1536.0*P**2*R*S**5*T**2*UGA_star**6 - 4096000.0*P**2*R*S**4*T**3*UGA_star**20*epsilon - 256000.0*P**2*R*S**4*T**3*UGA_star**20 + 45056000.0*P**2*R*S**4*T**3*UGA_star**19*epsilon + 2816000.0*P**2*R*S**4*T**3*UGA_star**19 - 223979520.0*P**2*R*S**4*T**3*UGA_star**18*epsilon - 13987840.0*P**2*R*S**4*T**3*UGA_star**18 + 662743040.0*P**2*R*S**4*T**3*UGA_star**17*epsilon + 41313280.0*P**2*R*S**4*T**3*UGA_star**17 - 1292457984.0*P**2*R*S**4*T**3*UGA_star**16*epsilon - 80306176.0*P**2*R*S**4*T**3*UGA_star**16 + 1734189056.0*P**2*R*S**4*T**3*UGA_star**15*epsilon + 107203584.0*P**2*R*S**4*T**3*UGA_star**15 - 1616533504.0*P**2*R*S**4*T**3*UGA_star**14*epsilon - 99188736.0*P**2*R*S**4*T**3*UGA_star**14 + 1024372736.0*P**2*R*S**4*T**3*UGA_star**13*epsilon + 62244864.0*P**2*R*S**4*T**3*UGA_star**13 - 409391104.0*P**2*R*S**4*T**3*UGA_star**12*epsilon - 24692736.0*P**2*R*S**4*T**3*UGA_star**12 + 79601664.0*P**2*R*S**4*T**3*UGA_star**11*epsilon + 5042176.0*P**2*R*S**4*T**3*UGA_star**11 + 5228544.0*P**2*R*S**4*T**3*UGA_star**10*epsilon - 118784.0*P**2*R*S**4*T**3*UGA_star**10 - 5380096.0*P**2*R*S**4*T**3*UGA_star**9*epsilon - 11264.0*P**2*R*S**4*T**3*UGA_star**9 + 681984.0*P**2*R*S**4*T**3*UGA_star**8*epsilon - 77824.0*P**2*R*S**4*T**3*UGA_star**8 - 65536.0*P**2*R*S**4*T**3*UGA_star**7*epsilon + 19456.0*P**2*R*S**4*T**3*UGA_star**7 + 30720.0*P**2*R*S**4*T**3*UGA_star**6*epsilon - 4116480.0*P**2*R*S**3*T**4*UGA_star**20*epsilon - 230400.0*P**2*R*S**3*T**4*UGA_star**20 + 45281280.0*P**2*R*S**3*T**4*UGA_star**19*epsilon + 2534400.0*P**2*R*S**3*T**4*UGA_star**19 - 225408000.0*P**2*R*S**3*T**4*UGA_star**18*epsilon - 12618240.0*P**2*R*S**3*T**4*UGA_star**18 + 668759040.0*P**2*R*S**3*T**4*UGA_star**17*epsilon + 37447680.0*P**2*R*S**3*T**4*UGA_star**17 - 1309295616.0*P**2*R*S**3*T**4*UGA_star**16*epsilon - 73334784.0*P**2*R*S**3*T**4*UGA_star**16 + 1765612544.0*P**2*R*S**3*T**4*UGA_star**15*epsilon + 98881536.0*P**2*R*S**3*T**4*UGA_star**15 - 1655597056.0*P**2*R*S**3*T**4*UGA_star**14*epsilon - 92580864.0*P**2*R*S**3*T**4*UGA_star**14 + 1055934464.0*P**2*R*S**3*T**4*UGA_star**13*epsilon + 58699776.0*P**2*R*S**3*T**4*UGA_star**13 - 424769536.0*P**2*R*S**3*T**4*UGA_star**12*epsilon - 23092224.0*P**2*R*S**3*T**4*UGA_star**12 + 83388416.0*P**2*R*S**3*T**4*UGA_star**11*epsilon + 3984384.0*P**2*R*S**3*T**4*UGA_star**11 + 4545536.0*P**2*R*S**3*T**4*UGA_star**10*epsilon + 689664.0*P**2*R*S**3*T**4*UGA_star**10 - 4455424.0*P**2*R*S**3*T**4*UGA_star**9*epsilon - 423936.0*P**2*R*S**3*T**4*UGA_star**9 - 62464.0*P**2*R*S**3*T**4*UGA_star**8*epsilon + 36864.0*P**2*R*S**3*T**4*UGA_star**8 + 183296.0*P**2*R*S**3*T**4*UGA_star**7*epsilon + 6144.0*P**2*R*S**3*T**4*UGA_star**7 - 2451456.0*P**2*R*S**2*T**5*UGA_star**20*epsilon - 96768.0*P**2*R*S**2*T**5*UGA_star**20 + 26966016.0*P**2*R*S**2*T**5*UGA_star**19*epsilon + 1064448.0*P**2*R*S**2*T**5*UGA_star**19 - 134516736.0*P**2*R*S**2*T**5*UGA_star**18*epsilon - 5326848.0*P**2*R*S**2*T**5*UGA_star**18 + 400840704.0*P**2*R*S**2*T**5*UGA_star**17*epsilon + 15985152.0*P**2*R*S**2*T**5*UGA_star**17 - 790238208.0*P**2*R*S**2*T**5*UGA_star**16*epsilon - 31884288.0*P**2*R*S**2*T**5*UGA_star**16 + 1076468736.0*P**2*R*S**2*T**5*UGA_star**15*epsilon + 44203008.0*P**2*R*S**2*T**5*UGA_star**15 - 1023931392.0*P**2*R*S**2*T**5*UGA_star**14*epsilon - 43137024.0*P**2*R*S**2*T**5*UGA_star**14 + 666688512.0*P**2*R*S**2*T**5*UGA_star**13*epsilon + 29180928.0*P**2*R*S**2*T**5*UGA_star**13 - 276993024.0*P**2*R*S**2*T**5*UGA_star**12*epsilon - 12913152.0*P**2*R*S**2*T**5*UGA_star**12 + 58027008.0*P**2*R*S**2*T**5*UGA_star**11*epsilon + 3122688.0*P**2*R*S**2*T**5*UGA_star**11 + 2380800.0*P**2*R*S**2*T**5*UGA_star**10*epsilon - 61440.0*P**2*R*S**2*T**5*UGA_star**10 - 3646464.0*P**2*R*S**2*T**5*UGA_star**9*epsilon - 167424.0*P**2*R*S**2*T**5*UGA_star**9 + 344064.0*P**2*R*S**2*T**5*UGA_star**8*epsilon + 30720.0*P**2*R*S**2*T**5*UGA_star**8 + 61440.0*P**2*R*S**2*T**5*UGA_star**7*epsilon - 745472.0*P**2*R*S*T**6*UGA_star**20*epsilon - 14336.0*P**2*R*S*T**6*UGA_star**20 + 8200192.0*P**2*R*S*T**6*UGA_star**19*epsilon + 157696.0*P**2*R*S*T**6*UGA_star**19 - 41043968.0*P**2*R*S*T**6*UGA_star**18*epsilon - 795648.0*P**2*R*S*T**6*UGA_star**18 + 123217920.0*P**2*R*S*T**6*UGA_star**17*epsilon + 2429952.0*P**2*R*S*T**6*UGA_star**17 - 245970944.0*P**2*R*S*T**6*UGA_star**16*epsilon - 4989440.0*P**2*R*S*T**6*UGA_star**16 + 341547008.0*P**2*R*S*T**6*UGA_star**15*epsilon + 7228928.0*P**2*R*S*T**6*UGA_star**15 - 334387200.0*P**2*R*S*T**6*UGA_star**14*epsilon - 7537152.0*P**2*R*S*T**6*UGA_star**14 + 227756032.0*P**2*R*S*T**6*UGA_star**13*epsilon + 5651968.0*P**2*R*S*T**6*UGA_star**13 - 102430720.0*P**2*R*S*T**6*UGA_star**12*epsilon - 2985472.0*P**2*R*S*T**6*UGA_star**12 + 26077184.0*P**2*R*S*T**6*UGA_star**11*epsilon + 1057280.0*P**2*R*S*T**6*UGA_star**11 - 1376256.0*P**2*R*S*T**6*UGA_star**10*epsilon - 225792.0*P**2*R*S*T**6*UGA_star**10 - 1052672.0*P**2*R*S*T**6*UGA_star**9*epsilon + 22016.0*P**2*R*S*T**6*UGA_star**9 + 208896.0*P**2*R*S*T**6*UGA_star**8*epsilon - 73728.0*P**2*R*T**7*UGA_star**20*epsilon + 811008.0*P**2*R*T**7*UGA_star**19*epsilon - 4079616.0*P**2*R*T**7*UGA_star**18*epsilon + 12386304.0*P**2*R*T**7*UGA_star**17*epsilon - 25216000.0*P**2*R*T**7*UGA_star**16*epsilon + 36133888.0*P**2*R*T**7*UGA_star**15*epsilon - 37180416.0*P**2*R*T**7*UGA_star**14*epsilon + 27462656.0*P**2*R*T**7*UGA_star**13*epsilon - 14265344.0*P**2*R*T**7*UGA_star**12*epsilon + 4961280.0*P**2*R*T**7*UGA_star**11*epsilon - 1039360.0*P**2*R*T**7*UGA_star**10*epsilon + 99328.0*P**2*R*T**7*UGA_star**9*epsilon + 13312.0*P**2*S**8*UGA_star**20*epsilon + 256.0*P**2*S**8*UGA_star**20 - 159744.0*P**2*S**8*UGA_star**19*epsilon - 3072.0*P**2*S**8*UGA_star**19 + 874496.0*P**2*S**8*UGA_star**18*epsilon + 16896.0*P**2*S**8*UGA_star**18 - 2883584.0*P**2*S**8*UGA_star**17*epsilon - 56320.0*P**2*S**8*UGA_star**17 + 6364160.0*P**2*S**8*UGA_star**16*epsilon + 126720.0*P**2*S**8*UGA_star**16 - 9867264.0*P**2*S**8*UGA_star**15*epsilon - 202752.0*P**2*S**8*UGA_star**15 + 10948608.0*P**2*S**8*UGA_star**14*epsilon + 236544.0*P**2*S**8*UGA_star**14 - 8650752.0*P**2*S**8*UGA_star**13*epsilon - 202752.0*P**2*S**8*UGA_star**13 + 4697088.0*P**2*S**8*UGA_star**12*epsilon + 126720.0*P**2*S**8*UGA_star**12 - 1576960.0*P**2*S**8*UGA_star**11*epsilon - 56320.0*P**2*S**8*UGA_star**11 + 202752.0*P**2*S**8*UGA_star**10*epsilon + 16896.0*P**2*S**8*UGA_star**10 + 65536.0*P**2*S**8*UGA_star**9*epsilon - 3072.0*P**2*S**8*UGA_star**9 - 31744.0*P**2*S**8*UGA_star**8*epsilon + 256.0*P**2*S**8*UGA_star**8 + 4096.0*P**2*S**8*UGA_star**7*epsilon + 116736.0*P**2*S**7*T*UGA_star**20*epsilon + 4608.0*P**2*S**7*T*UGA_star**20 - 1400832.0*P**2*S**7*T*UGA_star**19*epsilon - 55296.0*P**2*S**7*T*UGA_star**19 + 7649280.0*P**2*S**7*T*UGA_star**18*epsilon + 302592.0*P**2*S**7*T*UGA_star**18 - 25073664.0*P**2*S**7*T*UGA_star**17*epsilon - 996864.0*P**2*S**7*T*UGA_star**17 + 54749184.0*P**2*S**7*T*UGA_star**16*epsilon + 2196480.0*P**2*S**7*T*UGA_star**16 - 83392512.0*P**2*S**7*T*UGA_star**15*epsilon - 3396096.0*P**2*S**7*T*UGA_star**15 + 89892864.0*P**2*S**7*T*UGA_star**14*epsilon + 3750912.0*P**2*S**7*T*UGA_star**14 - 67645440.0*P**2*S**7*T*UGA_star**13*epsilon - 2939904.0*P**2*S**7*T*UGA_star**13 + 33527808.0*P**2*S**7*T*UGA_star**12*epsilon + 1571328.0*P**2*S**7*T*UGA_star**12 - 8982528.0*P**2*S**7*T*UGA_star**11*epsilon - 506880.0*P**2*S**7*T*UGA_star**11 - 129024.0*P**2*S**7*T*UGA_star**10*epsilon + 50688.0*P**2*S**7*T*UGA_star**10 + 903168.0*P**2*S**7*T*UGA_star**9*epsilon + 29184.0*P**2*S**7*T*UGA_star**9 - 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6144.0*P*R**9*UGA_star**17*epsilon + 4096.0*P*R**9*UGA_star**16*epsilon - 2048.0*P*R**9*UGA_star**15*epsilon + 1024.0*P*R**9*UGA_star**14*epsilon - 18432.0*P*R**8*S*UGA_star**20*epsilon + 55296.0*P*R**8*S*UGA_star**19*epsilon - 79872.0*P*R**8*S*UGA_star**18*epsilon + 92160.0*P*R**8*S*UGA_star**17*epsilon - 80896.0*P*R**8*S*UGA_star**16*epsilon + 46080.0*P*R**8*S*UGA_star**15*epsilon - 21504.0*P*R**8*S*UGA_star**14*epsilon + 7168.0*P*R**8*S*UGA_star**13*epsilon - 26624.0*P*R**8*T*UGA_star**20*epsilon - 512.0*P*R**8*T*UGA_star**20 + 79872.0*P*R**8*T*UGA_star**19*epsilon + 1536.0*P*R**8*T*UGA_star**19 - 110592.0*P*R**8*T*UGA_star**18*epsilon - 2560.0*P*R**8*T*UGA_star**18 + 118784.0*P*R**8*T*UGA_star**17*epsilon + 3584.0*P*R**8*T*UGA_star**17 - 94208.0*P*R**8*T*UGA_star**16*epsilon - 3584.0*P*R**8*T*UGA_star**16 + 36864.0*P*R**8*T*UGA_star**15*epsilon + 2560.0*P*R**8*T*UGA_star**15 - 6144.0*P*R**8*T*UGA_star**14*epsilon - 1536.0*P*R**8*T*UGA_star**14 + 2048.0*P*R**8*T*UGA_star**13*epsilon + 512.0*P*R**8*T*UGA_star**13 + 73728.0*P*R**7*S**2*UGA_star**20*epsilon - 294912.0*P*R**7*S**2*UGA_star**19*epsilon + 528384.0*P*R**7*S**2*UGA_star**18*epsilon - 638976.0*P*R**7*S**2*UGA_star**17*epsilon + 611328.0*P*R**7*S**2*UGA_star**16*epsilon - 430080.0*P*R**7*S**2*UGA_star**15*epsilon + 215040.0*P*R**7*S**2*UGA_star**14*epsilon - 86016.0*P*R**7*S**2*UGA_star**13*epsilon + 21504.0*P*R**7*S**2*UGA_star**12*epsilon + 212992.0*P*R**7*S*T*UGA_star**20*epsilon + 4096.0*P*R**7*S*T*UGA_star**20 - 851968.0*P*R**7*S*T*UGA_star**19*epsilon - 16384.0*P*R**7*S*T*UGA_star**19 + 1484800.0*P*R**7*S*T*UGA_star**18*epsilon + 31744.0*P*R**7*S*T*UGA_star**18 - 1687552.0*P*R**7*S*T*UGA_star**17*epsilon - 45056.0*P*R**7*S*T*UGA_star**17 + 1484800.0*P*R**7*S*T*UGA_star**16*epsilon + 50176.0*P*R**7*S*T*UGA_star**16 - 882688.0*P*R**7*S*T*UGA_star**15*epsilon - 40960.0*P*R**7*S*T*UGA_star**15 + 270336.0*P*R**7*S*T*UGA_star**14*epsilon + 25600.0*P*R**7*S*T*UGA_star**14 - 43008.0*P*R**7*S*T*UGA_star**13*epsilon - 12288.0*P*R**7*S*T*UGA_star**13 + 12288.0*P*R**7*S*T*UGA_star**12*epsilon + 3072.0*P*R**7*S*T*UGA_star**12 + 116736.0*P*R**7*T**2*UGA_star**20*epsilon + 4608.0*P*R**7*T**2*UGA_star**20 - 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8632320.0*P*R**6*S*T**2*UGA_star**18*epsilon - 348672.0*P*R**6*S*T**2*UGA_star**18 + 10530816.0*P*R**6*S*T**2*UGA_star**17*epsilon + 454656.0*P*R**6*S*T**2*UGA_star**17 - 8918016.0*P*R**6*S*T**2*UGA_star**16*epsilon - 427008.0*P*R**6*S*T**2*UGA_star**16 + 5815296.0*P*R**6*S*T**2*UGA_star**15*epsilon + 296448.0*P*R**6*S*T**2*UGA_star**15 - 2727936.0*P*R**6*S*T**2*UGA_star**14*epsilon - 127488.0*P*R**6*S*T**2*UGA_star**14 + 804864.0*P*R**6*S*T**2*UGA_star**13*epsilon + 21504.0*P*R**6*S*T**2*UGA_star**13 - 187392.0*P*R**6*S*T**2*UGA_star**12*epsilon + 1536.0*P*R**6*S*T**2*UGA_star**12 + 46080.0*P*R**6*S*T**2*UGA_star**11*epsilon - 274432.0*P*R**6*T**3*UGA_star**20*epsilon - 15360.0*P*R**6*T**3*UGA_star**20 + 1372160.0*P*R**6*T**3*UGA_star**19*epsilon + 76800.0*P*R**6*T**3*UGA_star**19 - 2825216.0*P*R**6*T**3*UGA_star**18*epsilon - 158208.0*P*R**6*T**3*UGA_star**18 + 3165184.0*P*R**6*T**3*UGA_star**17*epsilon + 178176.0*P*R**6*T**3*UGA_star**17 - 2245632.0*P*R**6*T**3*UGA_star**16*epsilon - 127488.0*P*R**6*T**3*UGA_star**16 + 1172480.0*P*R**6*T**3*UGA_star**15*epsilon + 64512.0*P*R**6*T**3*UGA_star**15 - 448512.0*P*R**6*T**3*UGA_star**14*epsilon - 19968.0*P*R**6*T**3*UGA_star**14 + 77824.0*P*R**6*T**3*UGA_star**13*epsilon + 6144.0*P*R**6*T**3*UGA_star**12*epsilon + 1536.0*P*R**6*T**3*UGA_star**12 + 258048.0*P*R**5*S**4*UGA_star**20*epsilon - 1548288.0*P*R**5*S**4*UGA_star**19*epsilon + 4085760.0*P*R**5*S**4*UGA_star**18*epsilon - 6451200.0*P*R**5*S**4*UGA_star**17*epsilon + 7132160.0*P*R**5*S**4*UGA_star**16*epsilon - 6064128.0*P*R**5*S**4*UGA_star**15*epsilon + 4021248.0*P*R**5*S**4*UGA_star**14*epsilon - 2007040.0*P*R**5*S**4*UGA_star**13*epsilon + 752640.0*P*R**5*S**4*UGA_star**12*epsilon - 215040.0*P*R**5*S**4*UGA_star**11*epsilon + 35840.0*P*R**5*S**4*UGA_star**10*epsilon + 1490944.0*P*R**5*S**3*T*UGA_star**20*epsilon + 28672.0*P*R**5*S**3*T*UGA_star**20 - 8945664.0*P*R**5*S**3*T*UGA_star**19*epsilon - 172032.0*P*R**5*S**3*T*UGA_star**19 + 23267328.0*P*R**5*S**3*T*UGA_star**18*epsilon + 465920.0*P*R**5*S**3*T*UGA_star**18 - 35409920.0*P*R**5*S**3*T*UGA_star**17*epsilon - 788480.0*P*R**5*S**3*T*UGA_star**17 + 36720640.0*P*R**5*S**3*T*UGA_star**16*epsilon + 977920.0*P*R**5*S**3*T*UGA_star**16 - 28514304.0*P*R**5*S**3*T*UGA_star**15*epsilon - 950272.0*P*R**5*S**3*T*UGA_star**15 + 16451584.0*P*R**5*S**3*T*UGA_star**14*epsilon + 719872.0*P*R**5*S**3*T*UGA_star**14 - 6176768.0*P*R**5*S**3*T*UGA_star**13*epsilon - 419840.0*P*R**5*S**3*T*UGA_star**13 + 1228800.0*P*R**5*S**3*T*UGA_star**12*epsilon + 189440.0*P*R**5*S**3*T*UGA_star**12 - 153600.0*P*R**5*S**3*T*UGA_star**11*epsilon - 61440.0*P*R**5*S**3*T*UGA_star**11 + 40960.0*P*R**5*S**3*T*UGA_star**10*epsilon + 10240.0*P*R**5*S**3*T*UGA_star**10 + 2451456.0*P*R**5*S**2*T**2*UGA_star**20*epsilon + 96768.0*P*R**5*S**2*T**2*UGA_star**20 - 14708736.0*P*R**5*S**2*T**2*UGA_star**19*epsilon - 580608.0*P*R**5*S**2*T**2*UGA_star**19 + 37730304.0*P*R**5*S**2*T**2*UGA_star**18*epsilon + 1511424.0*P*R**5*S**2*T**2*UGA_star**18 - 55111680.0*P*R**5*S**2*T**2*UGA_star**17*epsilon - 2304000.0*P*R**5*S**2*T**2*UGA_star**17 + 52783104.0*P*R**5*S**2*T**2*UGA_star**16*epsilon + 2388480.0*P*R**5*S**2*T**2*UGA_star**16 - 37158912.0*P*R**5*S**2*T**2*UGA_star**15*epsilon - 1832448.0*P*R**5*S**2*T**2*UGA_star**15 + 20336640.0*P*R**5*S**2*T**2*UGA_star**14*epsilon + 1010688.0*P*R**5*S**2*T**2*UGA_star**14 - 8048640.0*P*R**5*S**2*T**2*UGA_star**13*epsilon - 328704.0*P*R**5*S**2*T**2*UGA_star**13 + 2064384.0*P*R**5*S**2*T**2*UGA_star**12*epsilon + 30720.0*P*R**5*S**2*T**2*UGA_star**12 - 430080.0*P*R**5*S**2*T**2*UGA_star**11*epsilon + 7680.0*P*R**5*S**2*T**2*UGA_star**11 + 92160.0*P*R**5*S**2*T**2*UGA_star**10*epsilon + 1646592.0*P*R**5*S*T**3*UGA_star**20*epsilon + 92160.0*P*R**5*S*T**3*UGA_star**20 - 9879552.0*P*R**5*S*T**3*UGA_star**19*epsilon - 552960.0*P*R**5*S*T**3*UGA_star**19 + 24963072.0*P*R**5*S*T**3*UGA_star**18*epsilon + 1397760.0*P*R**5*S*T**3*UGA_star**18 - 34738176.0*P*R**5*S*T**3*UGA_star**17*epsilon - 1950720.0*P*R**5*S*T**3*UGA_star**17 + 29755392.0*P*R**5*S*T**3*UGA_star**16*epsilon + 1681920.0*P*R**5*S*T**3*UGA_star**16 - 17274880.0*P*R**5*S*T**3*UGA_star**15*epsilon - 975360.0*P*R**5*S*T**3*UGA_star**15 + 7542784.0*P*R**5*S*T**3*UGA_star**14*epsilon + 399360.0*P*R**5*S*T**3*UGA_star**14 - 2334720.0*P*R**5*S*T**3*UGA_star**13*epsilon - 92160.0*P*R**5*S*T**3*UGA_star**13 + 268288.0*P*R**5*S*T**3*UGA_star**12*epsilon - 7680.0*P*R**5*S*T**3*UGA_star**12 + 51200.0*P*R**5*S*T**3*UGA_star**11*epsilon + 7680.0*P*R**5*S*T**3*UGA_star**11 + 409600.0*P*R**5*T**4*UGA_star**20*epsilon + 25600.0*P*R**5*T**4*UGA_star**20 - 2457600.0*P*R**5*T**4*UGA_star**19*epsilon - 153600.0*P*R**5*T**4*UGA_star**19 + 6109184.0*P*R**5*T**4*UGA_star**18*epsilon + 380928.0*P*R**5*T**4*UGA_star**18 - 8054784.0*P*R**5*T**4*UGA_star**17*epsilon - 498688.0*P*R**5*T**4*UGA_star**17 + 5977088.0*P*R**5*T**4*UGA_star**16*epsilon + 364544.0*P*R**5*T**4*UGA_star**16 - 2445312.0*P*R**5*T**4*UGA_star**15*epsilon - 147456.0*P*R**5*T**4*UGA_star**15 + 503808.0*P*R**5*T**4*UGA_star**14*epsilon + 36864.0*P*R**5*T**4*UGA_star**14 - 14336.0*P*R**5*T**4*UGA_star**13*epsilon - 11264.0*P*R**5*T**4*UGA_star**13 - 48128.0*P*R**5*T**4*UGA_star**12*epsilon + 3072.0*P*R**5*T**4*UGA_star**12 + 20480.0*P*R**5*T**4*UGA_star**11*epsilon - 258048.0*P*R**4*S**5*UGA_star**20*epsilon + 1806336.0*P*R**4*S**5*UGA_star**19*epsilon - 5591040.0*P*R**4*S**5*UGA_star**18*epsilon + 10235904.0*P*R**4*S**5*UGA_star**17*epsilon - 12665856.0*P*R**4*S**5*UGA_star**16*epsilon + 11590656.0*P*R**4*S**5*UGA_star**15*epsilon - 8279040.0*P*R**4*S**5*UGA_star**14*epsilon + 4623360.0*P*R**4*S**5*UGA_star**13*epsilon - 1956864.0*P*R**4*S**5*UGA_star**12*epsilon + 623616.0*P*R**4*S**5*UGA_star**11*epsilon - 150528.0*P*R**4*S**5*UGA_star**10*epsilon + 21504.0*P*R**4*S**5*UGA_star**9*epsilon - 1863680.0*P*R**4*S**4*T*UGA_star**20*epsilon - 35840.0*P*R**4*S**4*T*UGA_star**20 + 13045760.0*P*R**4*S**4*T*UGA_star**19*epsilon + 250880.0*P*R**4*S**4*T*UGA_star**19 - 39925760.0*P*R**4*S**4*T*UGA_star**18*epsilon - 788480.0*P*R**4*S**4*T*UGA_star**18 + 71034880.0*P*R**4*S**4*T*UGA_star**17*epsilon + 1505280.0*P*R**4*S**4*T*UGA_star**17 - 83415040.0*P*R**4*S**4*T*UGA_star**16*epsilon - 2014720.0*P*R**4*S**4*T*UGA_star**16 + 70512640.0*P*R**4*S**4*T*UGA_star**15*epsilon + 2060800.0*P*R**4*S**4*T*UGA_star**15 - 45178880.0*P*R**4*S**4*T*UGA_star**14*epsilon - 1666560.0*P*R**4*S**4*T*UGA_star**14 + 21340160.0*P*R**4*S**4*T*UGA_star**13*epsilon + 1057280.0*P*R**4*S**4*T*UGA_star**13 - 6471680.0*P*R**4*S**4*T*UGA_star**12*epsilon - 519680.0*P*R**4*S**4*T*UGA_star**12 + 983040.0*P*R**4*S**4*T*UGA_star**11*epsilon + 197120.0*P*R**4*S**4*T*UGA_star**11 - 92160.0*P*R**4*S**4*T*UGA_star**10*epsilon - 53760.0*P*R**4*S**4*T*UGA_star**10 + 30720.0*P*R**4*S**4*T*UGA_star**9*epsilon + 7680.0*P*R**4*S**4*T*UGA_star**9 - 4085760.0*P*R**4*S**3*T**2*UGA_star**20*epsilon - 161280.0*P*R**4*S**3*T**2*UGA_star**20 + 28600320.0*P*R**4*S**3*T**2*UGA_star**19*epsilon + 1128960.0*P*R**4*S**3*T**2*UGA_star**19 - 86691840.0*P*R**4*S**3*T**2*UGA_star**18*epsilon - 3456000.0*P*R**4*S**3*T**2*UGA_star**18 + 150067200.0*P*R**4*S**3*T**2*UGA_star**17*epsilon + 6151680.0*P*R**4*S**3*T**2*UGA_star**17 - 166625280.0*P*R**4*S**3*T**2*UGA_star**16*epsilon - 7219200.0*P*R**4*S**3*T**2*UGA_star**16 + 129116160.0*P*R**4*S**3*T**2*UGA_star**15*epsilon + 6059520.0*P*R**4*S**3*T**2*UGA_star**15 - 75847680.0*P*R**4*S**3*T**2*UGA_star**14*epsilon - 3778560.0*P*R**4*S**3*T**2*UGA_star**14 + 35082240.0*P*R**4*S**3*T**2*UGA_star**13*epsilon + 1651200.0*P*R**4*S**3*T**2*UGA_star**13 - 11827200.0*P*R**4*S**3*T**2*UGA_star**12*epsilon - 391680.0*P*R**4*S**3*T**2*UGA_star**12 + 2580480.0*P*R**4*S**3*T**2*UGA_star**11*epsilon - 460800.0*P*R**4*S**3*T**2*UGA_star**10*epsilon + 15360.0*P*R**4*S**3*T**2*UGA_star**10 + 92160.0*P*R**4*S**3*T**2*UGA_star**9*epsilon - 4116480.0*P*R**4*S**2*T**3*UGA_star**20*epsilon - 230400.0*P*R**4*S**2*T**3*UGA_star**20 + 28815360.0*P*R**4*S**2*T**3*UGA_star**19*epsilon + 1612800.0*P*R**4*S**2*T**3*UGA_star**19 - 86553600.0*P*R**4*S**2*T**3*UGA_star**18*epsilon - 4846080.0*P*R**4*S**2*T**3*UGA_star**18 + 145689600.0*P*R**4*S**2*T**3*UGA_star**17*epsilon + 8171520.0*P*R**4*S**2*T**3*UGA_star**17 - 151474176.0*P*R**4*S**2*T**3*UGA_star**16*epsilon - 8533504.0*P*R**4*S**2*T**3*UGA_star**16 + 102860800.0*P*R**4*S**2*T**3*UGA_star**15*epsilon + 5826560.0*P*R**4*S**2*T**3*UGA_star**15 - 48849920.0*P*R**4*S**2*T**3*UGA_star**14*epsilon - 2736640.0*P*R**4*S**2*T**3*UGA_star**14 + 17628160.0*P*R**4*S**2*T**3*UGA_star**13*epsilon + 901120.0*P*R**4*S**2*T**3*UGA_star**13 - 4372480.0*P*R**4*S**2*T**3*UGA_star**12*epsilon - 151040.0*P*R**4*S**2*T**3*UGA_star**12 + 229376.0*P*R**4*S**2*T**3*UGA_star**11*epsilon - 29696.0*P*R**4*S**2*T**3*UGA_star**11 + 143360.0*P*R**4*S**2*T**3*UGA_star**10*epsilon + 15360.0*P*R**4*S**2*T**3*UGA_star**10 - 2048000.0*P*R**4*S*T**4*UGA_star**20*epsilon - 128000.0*P*R**4*S*T**4*UGA_star**20 + 14336000.0*P*R**4*S*T**4*UGA_star**19*epsilon + 896000.0*P*R**4*S*T**4*UGA_star**19 - 42675200.0*P*R**4*S*T**4*UGA_star**18*epsilon - 2662400.0*P*R**4*S*T**4*UGA_star**18 + 69821440.0*P*R**4*S*T**4*UGA_star**17*epsilon + 4334080.0*P*R**4*S*T**4*UGA_star**17 - 67523584.0*P*R**4*S*T**4*UGA_star**16*epsilon - 4148736.0*P*R**4*S*T**4*UGA_star**16 + 38379520.0*P*R**4*S*T**4*UGA_star**15*epsilon + 2324480.0*P*R**4*S*T**4*UGA_star**15 - 11781120.0*P*R**4*S*T**4*UGA_star**14*epsilon - 732160.0*P*R**4*S*T**4*UGA_star**14 + 1397760.0*P*R**4*S*T**4*UGA_star**13*epsilon + 156160.0*P*R**4*S*T**4*UGA_star**13 + 271360.0*P*R**4*S*T**4*UGA_star**12*epsilon - 53760.0*P*R**4*S*T**4*UGA_star**12 - 260096.0*P*R**4*S*T**4*UGA_star**11*epsilon + 14336.0*P*R**4*S*T**4*UGA_star**11 + 81920.0*P*R**4*S*T**4*UGA_star**10*epsilon - 411648.0*P*R**4*T**5*UGA_star**20*epsilon - 23040.0*P*R**4*T**5*UGA_star**20 + 2881536.0*P*R**4*T**5*UGA_star**19*epsilon + 161280.0*P*R**4*T**5*UGA_star**19 - 8521728.0*P*R**4*T**5*UGA_star**18*epsilon - 477696.0*P*R**4*T**5*UGA_star**18 + 13651968.0*P*R**4*T**5*UGA_star**17*epsilon + 768000.0*P*R**4*T**5*UGA_star**17 - 12481536.0*P*R**4*T**5*UGA_star**16*epsilon - 707584.0*P*R**4*T**5*UGA_star**16 + 6053888.0*P*R**4*T**5*UGA_star**15*epsilon + 350720.0*P*R**4*T**5*UGA_star**15 - 966656.0*P*R**4*T**5*UGA_star**14*epsilon - 64000.0*P*R**4*T**5*UGA_star**14 - 294912.0*P*R**4*T**5*UGA_star**13*epsilon - 11264.0*P*R**4*T**5*UGA_star**13 + 74752.0*P*R**4*T**5*UGA_star**12*epsilon + 2560.0*P*R**4*T**5*UGA_star**12 + 14336.0*P*R**4*T**5*UGA_star**11*epsilon + 1024.0*P*R**4*T**5*UGA_star**11 + 172032.0*P*R**3*S**6*UGA_star**20*epsilon - 1376256.0*P*R**3*S**6*UGA_star**19*epsilon + 4902912.0*P*R**3*S**6*UGA_star**18*epsilon - 10321920.0*P*R**3*S**6*UGA_star**17*epsilon + 14457856.0*P*R**3*S**6*UGA_star**16*epsilon - 14508032.0*P*R**3*S**6*UGA_star**15*epsilon + 11038720.0*P*R**3*S**6*UGA_star**14*epsilon - 6594560.0*P*R**3*S**6*UGA_star**13*epsilon + 3082240.0*P*R**3*S**6*UGA_star**12*epsilon - 1089536.0*P*R**3*S**6*UGA_star**11*epsilon + 286720.0*P*R**3*S**6*UGA_star**10*epsilon - 57344.0*P*R**3*S**6*UGA_star**9*epsilon + 7168.0*P*R**3*S**6*UGA_star**8*epsilon + 1490944.0*P*R**3*S**5*T*UGA_star**20*epsilon + 28672.0*P*R**3*S**5*T*UGA_star**20 - 11927552.0*P*R**3*S**5*T*UGA_star**19*epsilon - 229376.0*P*R**3*S**5*T*UGA_star**19 + 42104832.0*P*R**3*S**5*T*UGA_star**18*epsilon + 824320.0*P*R**3*S**5*T*UGA_star**18 - 86646784.0*P*R**3*S**5*T*UGA_star**17*epsilon - 1777664.0*P*R**3*S**5*T*UGA_star**17 + 116328448.0*P*R**3*S**5*T*UGA_star**16*epsilon + 2612224.0*P*R**3*S**5*T*UGA_star**16 - 109037568.0*P*R**3*S**5*T*UGA_star**15*epsilon - 2834432.0*P*R**3*S**5*T*UGA_star**15 + 75278336.0*P*R**3*S**5*T*UGA_star**14*epsilon + 2394112.0*P*R**3*S**5*T*UGA_star**14 - 39495680.0*P*R**3*S**5*T*UGA_star**13*epsilon - 1605632.0*P*R**3*S**5*T*UGA_star**13 + 15181824.0*P*R**3*S**5*T*UGA_star**12*epsilon + 845824.0*P*R**3*S**5*T*UGA_star**12 - 3627008.0*P*R**3*S**5*T*UGA_star**11*epsilon - 344064.0*P*R**3*S**5*T*UGA_star**11 + 344064.0*P*R**3*S**5*T*UGA_star**10*epsilon + 107520.0*P*R**3*S**5*T*UGA_star**10 - 6144.0*P*R**3*S**5*T*UGA_star**9*epsilon - 24576.0*P*R**3*S**5*T*UGA_star**9 + 12288.0*P*R**3*S**5*T*UGA_star**8*epsilon + 3072.0*P*R**3*S**5*T*UGA_star**8 + 4085760.0*P*R**3*S**4*T**2*UGA_star**20*epsilon + 161280.0*P*R**3*S**4*T**2*UGA_star**20 - 32686080.0*P*R**3*S**4*T**2*UGA_star**19*epsilon - 1290240.0*P*R**3*S**4*T**2*UGA_star**19 + 114585600.0*P*R**3*S**4*T**2*UGA_star**18*epsilon + 4554240.0*P*R**3*S**4*T**2*UGA_star**18 - 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2457600.0*P*R**3*S**3*T**3*UGA_star**19 + 153088000.0*P*R**3*S**3*T**3*UGA_star**18*epsilon + 8570880.0*P*R**3*S**3*T**3*UGA_star**18 - 304168960.0*P*R**3*S**3*T**3*UGA_star**17*epsilon - 17049600.0*P*R**3*S**3*T**3*UGA_star**17 + 378486784.0*P*R**3*S**3*T**3*UGA_star**16*epsilon + 21277696.0*P*R**3*S**3*T**3*UGA_star**16 - 306704384.0*P*R**3*S**3*T**3*UGA_star**15*epsilon - 17335296.0*P*R**3*S**3*T**3*UGA_star**15 + 165724160.0*P*R**3*S**3*T**3*UGA_star**14*epsilon + 9415680.0*P*R**3*S**3*T**3*UGA_star**14 - 62689280.0*P*R**3*S**3*T**3*UGA_star**13*epsilon - 3507200.0*P*R**3*S**3*T**3*UGA_star**13 + 18104320.0*P*R**3*S**3*T**3*UGA_star**12*epsilon + 906240.0*P*R**3*S**3*T**3*UGA_star**12 - 3477504.0*P*R**3*S**3*T**3*UGA_star**11*epsilon - 101376.0*P*R**3*S**3*T**3*UGA_star**11 - 126976.0*P*R**3*S**3*T**3*UGA_star**10*epsilon - 41984.0*P*R**3*S**3*T**3*UGA_star**10 + 184320.0*P*R**3*S**3*T**3*UGA_star**9*epsilon + 15360.0*P*R**3*S**3*T**3*UGA_star**9 + 4096000.0*P*R**3*S**2*T**4*UGA_star**20*epsilon + 256000.0*P*R**3*S**2*T**4*UGA_star**20 - 32768000.0*P*R**3*S**2*T**4*UGA_star**19*epsilon - 2048000.0*P*R**3*S**2*T**4*UGA_star**19 + 113704960.0*P*R**3*S**2*T**4*UGA_star**18*epsilon + 7096320.0*P*R**3*S**2*T**4*UGA_star**18 - 222679040.0*P*R**3*S**2*T**4*UGA_star**17*epsilon - 13844480.0*P*R**3*S**2*T**4*UGA_star**17 + 267431936.0*P*R**3*S**2*T**4*UGA_star**16*epsilon + 16502784.0*P*R**3*S**2*T**4*UGA_star**16 - 199131136.0*P*R**3*S**2*T**4*UGA_star**15*epsilon - 12140544.0*P*R**3*S**2*T**4*UGA_star**15 + 87080960.0*P*R**3*S**2*T**4*UGA_star**14*epsilon + 5263360.0*P*R**3*S**2*T**4*UGA_star**14 - 18247680.0*P*R**3*S**2*T**4*UGA_star**13*epsilon - 1228800.0*P*R**3*S**2*T**4*UGA_star**13 - 184320.0*P*R**3*S**2*T**4*UGA_star**12*epsilon + 215040.0*P*R**3*S**2*T**4*UGA_star**12 + 1040384.0*P*R**3*S**2*T**4*UGA_star**11*epsilon - 98304.0*P*R**3*S**2*T**4*UGA_star**11 - 466944.0*P*R**3*S**2*T**4*UGA_star**10*epsilon + 26624.0*P*R**3*S**2*T**4*UGA_star**10 + 122880.0*P*R**3*S**2*T**4*UGA_star**9*epsilon + 1646592.0*P*R**3*S*T**5*UGA_star**20*epsilon + 92160.0*P*R**3*S*T**5*UGA_star**20 - 13172736.0*P*R**3*S*T**5*UGA_star**19*epsilon - 737280.0*P*R**3*S*T**5*UGA_star**19 + 45570048.0*P*R**3*S*T**5*UGA_star**18*epsilon + 2552832.0*P*R**3*S*T**5*UGA_star**18 - 88387584.0*P*R**3*S*T**5*UGA_star**17*epsilon - 4961280.0*P*R**3*S*T**5*UGA_star**17 + 103602176.0*P*R**3*S*T**5*UGA_star**16*epsilon + 5837312.0*P*R**3*S*T**5*UGA_star**16 - 72595456.0*P*R**3*S*T**5*UGA_star**15*epsilon - 4123136.0*P*R**3*S*T**5*UGA_star**15 + 26587136.0*P*R**3*S*T**5*UGA_star**14*epsilon + 1546240.0*P*R**3*S*T**5*UGA_star**14 - 1875968.0*P*R**3*S*T**5*UGA_star**13*epsilon - 141312.0*P*R**3*S*T**5*UGA_star**13 - 1673216.0*P*R**3*S*T**5*UGA_star**12*epsilon - 79360.0*P*R**3*S*T**5*UGA_star**12 + 231424.0*P*R**3*S*T**5*UGA_star**11*epsilon + 9728.0*P*R**3*S*T**5*UGA_star**11 + 67584.0*P*R**3*S*T**5*UGA_star**10*epsilon + 4096.0*P*R**3*S*T**5*UGA_star**10 + 272384.0*P*R**3*T**6*UGA_star**20*epsilon + 10752.0*P*R**3*T**6*UGA_star**20 - 2179072.0*P*R**3*T**6*UGA_star**19*epsilon - 86016.0*P*R**3*T**6*UGA_star**19 + 7544832.0*P*R**3*T**6*UGA_star**18*epsilon + 299520.0*P*R**3*T**6*UGA_star**18 - 14651392.0*P*R**3*T**6*UGA_star**17*epsilon - 589824.0*P*R**3*T**6*UGA_star**17 + 17182720.0*P*R**3*T**6*UGA_star**16*epsilon + 710656.0*P*R**3*T**6*UGA_star**16 - 12007424.0*P*R**3*T**6*UGA_star**15*epsilon - 522752.0*P*R**3*T**6*UGA_star**15 + 4319232.0*P*R**3*T**6*UGA_star**14*epsilon + 211456.0*P*R**3*T**6*UGA_star**14 - 212992.0*P*R**3*T**6*UGA_star**13*epsilon - 25600.0*P*R**3*T**6*UGA_star**13 - 329728.0*P*R**3*T**6*UGA_star**12*epsilon - 11776.0*P*R**3*T**6*UGA_star**12 + 51200.0*P*R**3*T**6*UGA_star**11*epsilon + 3584.0*P*R**3*T**6*UGA_star**11 + 10240.0*P*R**3*T**6*UGA_star**10*epsilon - 73728.0*P*R**2*S**7*UGA_star**20*epsilon + 663552.0*P*R**2*S**7*UGA_star**19*epsilon - 2678784.0*P*R**2*S**7*UGA_star**18*epsilon + 6414336.0*P*R**2*S**7*UGA_star**17*epsilon - 10175488.0*P*R**2*S**7*UGA_star**16*epsilon + 11363328.0*P*R**2*S**7*UGA_star**15*epsilon - 9326592.0*P*R**2*S**7*UGA_star**14*epsilon + 5836800.0*P*R**2*S**7*UGA_star**13*epsilon - 2856960.0*P*R**2*S**7*UGA_star**12*epsilon + 1087488.0*P*R**2*S**7*UGA_star**11*epsilon - 307200.0*P*R**2*S**7*UGA_star**10*epsilon + 61440.0*P*R**2*S**7*UGA_star**9*epsilon - 9216.0*P*R**2*S**7*UGA_star**8*epsilon + 1024.0*P*R**2*S**7*UGA_star**7*epsilon - 745472.0*P*R**2*S**6*T*UGA_star**20*epsilon - 14336.0*P*R**2*S**6*T*UGA_star**20 + 6709248.0*P*R**2*S**6*T*UGA_star**19*epsilon + 129024.0*P*R**2*S**6*T*UGA_star**19 - 26880000.0*P*R**2*S**6*T*UGA_star**18*epsilon - 523264.0*P*R**2*S**6*T*UGA_star**18 + 63178752.0*P*R**2*S**6*T*UGA_star**17*epsilon + 1268736.0*P*R**2*S**6*T*UGA_star**17 - 96819200.0*P*R**2*S**6*T*UGA_star**16*epsilon - 2064896.0*P*R**2*S**6*T*UGA_star**16 + 102082560.0*P*R**2*S**6*T*UGA_star**15*epsilon + 2413056.0*P*R**2*S**6*T*UGA_star**15 - 76713984.0*P*R**2*S**6*T*UGA_star**14*epsilon - 2125824.0*P*R**2*S**6*T*UGA_star**14 + 42405888.0*P*R**2*S**6*T*UGA_star**13*epsilon + 1462272.0*P*R**2*S**6*T*UGA_star**13 - 17633280.0*P*R**2*S**6*T*UGA_star**12*epsilon - 795648.0*P*R**2*S**6*T*UGA_star**12 + 5304320.0*P*R**2*S**6*T*UGA_star**11*epsilon + 336896.0*P*R**2*S**6*T*UGA_star**11 - 903168.0*P*R**2*S**6*T*UGA_star**10*epsilon - 107520.0*P*R**2*S**6*T*UGA_star**10 - 6144.0*P*R**2*S**6*T*UGA_star**9*epsilon + 25600.0*P*R**2*S**6*T*UGA_star**9 + 18432.0*P*R**2*S**6*T*UGA_star**8*epsilon - 4608.0*P*R**2*S**6*T*UGA_star**8 + 2048.0*P*R**2*S**6*T*UGA_star**7*epsilon + 512.0*P*R**2*S**6*T*UGA_star**7 - 2451456.0*P*R**2*S**5*T**2*UGA_star**20*epsilon - 96768.0*P*R**2*S**5*T**2*UGA_star**20 + 22063104.0*P*R**2*S**5*T**2*UGA_star**19*epsilon + 870912.0*P*R**2*S**5*T**2*UGA_star**19 - 87939072.0*P*R**2*S**5*T**2*UGA_star**18*epsilon - 3488256.0*P*R**2*S**5*T**2*UGA_star**18 + 203931648.0*P*R**2*S**5*T**2*UGA_star**17*epsilon + 8193024.0*P*R**2*S**5*T**2*UGA_star**17 - 304241664.0*P*R**2*S**5*T**2*UGA_star**16*epsilon - 12535296.0*P*R**2*S**5*T**2*UGA_star**16 + 305759232.0*P*R**2*S**5*T**2*UGA_star**15*epsilon + 13163520.0*P*R**2*S**5*T**2*UGA_star**15 - 212619264.0*P*R**2*S**5*T**2*UGA_star**14*epsilon - 9784320.0*P*R**2*S**5*T**2*UGA_star**14 + 105971712.0*P*R**2*S**5*T**2*UGA_star**13*epsilon + 5246976.0*P*R**2*S**5*T**2*UGA_star**13 - 40771584.0*P*R**2*S**5*T**2*UGA_star**12*epsilon - 2010624.0*P*R**2*S**5*T**2*UGA_star**12 + 13062144.0*P*R**2*S**5*T**2*UGA_star**11*epsilon + 477696.0*P*R**2*S**5*T**2*UGA_star**11 - 3127296.0*P*R**2*S**5*T**2*UGA_star**10*epsilon - 10752.0*P*R**2*S**5*T**2*UGA_star**10 + 368640.0*P*R**2*S**5*T**2*UGA_star**9*epsilon - 33792.0*P*R**2*S**5*T**2*UGA_star**9 - 15360.0*P*R**2*S**5*T**2*UGA_star**8*epsilon + 7680.0*P*R**2*S**5*T**2*UGA_star**8 + 9216.0*P*R**2*S**5*T**2*UGA_star**7*epsilon - 4116480.0*P*R**2*S**4*T**3*UGA_star**20*epsilon - 230400.0*P*R**2*S**4*T**3*UGA_star**20 + 37048320.0*P*R**2*S**4*T**3*UGA_star**19*epsilon + 2073600.0*P*R**2*S**4*T**3*UGA_star**19 - 147194880.0*P*R**2*S**4*T**3*UGA_star**18*epsilon - 8240640.0*P*R**2*S**4*T**3*UGA_star**18 + 338273280.0*P*R**2*S**4*T**3*UGA_star**17*epsilon + 18954240.0*P*R**2*S**4*T**3*UGA_star**17 - 494601216.0*P*R**2*S**4*T**3*UGA_star**16*epsilon - 27769344.0*P*R**2*S**4*T**3*UGA_star**16 + 476480512.0*P*R**2*S**4*T**3*UGA_star**15*epsilon + 26861568.0*P*R**2*S**4*T**3*UGA_star**15 - 303256576.0*P*R**2*S**4*T**3*UGA_star**14*epsilon - 17224704.0*P*R**2*S**4*T**3*UGA_star**14 + 125721600.0*P*R**2*S**4*T**3*UGA_star**13*epsilon + 7219200.0*P*R**2*S**4*T**3*UGA_star**13 - 34288640.0*P*R**2*S**4*T**3*UGA_star**12*epsilon - 1966080.0*P*R**2*S**4*T**3*UGA_star**12 + 7003136.0*P*R**2*S**4*T**3*UGA_star**11*epsilon + 359424.0*P*R**2*S**4*T**3*UGA_star**11 - 940032.0*P*R**2*S**4*T**3*UGA_star**10*epsilon - 19968.0*P*R**2*S**4*T**3*UGA_star**10 - 241664.0*P*R**2*S**4*T**3*UGA_star**9*epsilon - 24576.0*P*R**2*S**4*T**3*UGA_star**9 + 112640.0*P*R**2*S**4*T**3*UGA_star**8*epsilon + 7680.0*P*R**2*S**4*T**3*UGA_star**8 - 4096000.0*P*R**2*S**3*T**4*UGA_star**20*epsilon - 256000.0*P*R**2*S**3*T**4*UGA_star**20 + 36864000.0*P*R**2*S**3*T**4*UGA_star**19*epsilon + 2304000.0*P*R**2*S**3*T**4*UGA_star**19 - 146155520.0*P*R**2*S**3*T**4*UGA_star**18*epsilon - 9123840.0*P*R**2*S**3*T**4*UGA_star**18 + 333752320.0*P*R**2*S**3*T**4*UGA_star**17*epsilon + 20771840.0*P*R**2*S**3*T**4*UGA_star**17 - 480548864.0*P*R**2*S**3*T**4*UGA_star**16*epsilon - 29735936.0*P*R**2*S**3*T**4*UGA_star**16 + 446701568.0*P*R**2*S**3*T**4*UGA_star**15*epsilon + 27374592.0*P*R**2*S**3*T**4*UGA_star**15 - 260511744.0*P*R**2*S**3*T**4*UGA_star**14*epsilon - 15748096.0*P*R**2*S**3*T**4*UGA_star**14 + 84336640.0*P*R**2*S**3*T**4*UGA_star**13*epsilon + 5094400.0*P*R**2*S**3*T**4*UGA_star**13 - 7874560.0*P*R**2*S**3*T**4*UGA_star**12*epsilon - 691200.0*P*R**2*S**3*T**4*UGA_star**12 - 3424256.0*P*R**2*S**3*T**4*UGA_star**11*epsilon + 70656.0*P*R**2*S**3*T**4*UGA_star**11 + 1196032.0*P*R**2*S**3*T**4*UGA_star**10*epsilon - 84992.0*P*R**2*S**3*T**4*UGA_star**10 - 321536.0*P*R**2*S**3*T**4*UGA_star**9*epsilon + 24576.0*P*R**2*S**3*T**4*UGA_star**9 + 81920.0*P*R**2*S**3*T**4*UGA_star**8*epsilon - 2469888.0*P*R**2*S**2*T**5*UGA_star**20*epsilon - 138240.0*P*R**2*S**2*T**5*UGA_star**20 + 22228992.0*P*R**2*S**2*T**5*UGA_star**19*epsilon + 1244160.0*P*R**2*S**2*T**5*UGA_star**19 - 88049664.0*P*R**2*S**2*T**5*UGA_star**18*epsilon - 4930560.0*P*R**2*S**2*T**5*UGA_star**18 + 200411136.0*P*R**2*S**2*T**5*UGA_star**17*epsilon + 11234304.0*P*R**2*S**2*T**5*UGA_star**17 - 286126080.0*P*R**2*S**2*T**5*UGA_star**16*epsilon - 16068096.0*P*R**2*S**2*T**5*UGA_star**16 + 260579328.0*P*R**2*S**2*T**5*UGA_star**15*epsilon + 14678016.0*P*R**2*S**2*T**5*UGA_star**15 - 144211968.0*P*R**2*S**2*T**5*UGA_star**14*epsilon - 8169984.0*P*R**2*S**2*T**5*UGA_star**14 + 39235584.0*P*R**2*S**2*T**5*UGA_star**13*epsilon + 2257920.0*P*R**2*S**2*T**5*UGA_star**13 + 1204224.0*P*R**2*S**2*T**5*UGA_star**12*epsilon + 47616.0*P*R**2*S**2*T**5*UGA_star**12 - 3133440.0*P*R**2*S**2*T**5*UGA_star**11*epsilon - 175104.0*P*R**2*S**2*T**5*UGA_star**11 + 215040.0*P*R**2*S**2*T**5*UGA_star**10*epsilon + 13824.0*P*R**2*S**2*T**5*UGA_star**10 + 116736.0*P*R**2*S**2*T**5*UGA_star**9*epsilon + 6144.0*P*R**2*S**2*T**5*UGA_star**9 - 817152.0*P*R**2*S*T**6*UGA_star**20*epsilon - 32256.0*P*R**2*S*T**6*UGA_star**20 + 7354368.0*P*R**2*S*T**6*UGA_star**19*epsilon + 290304.0*P*R**2*S*T**6*UGA_star**19 - 29171712.0*P*R**2*S*T**6*UGA_star**18*epsilon - 1156608.0*P*R**2*S*T**6*UGA_star**18 + 66588672.0*P*R**2*S*T**6*UGA_star**17*epsilon + 2668032.0*P*R**2*S*T**6*UGA_star**17 - 95499264.0*P*R**2*S*T**6*UGA_star**16*epsilon - 3901440.0*P*R**2*S*T**6*UGA_star**16 + 87552000.0*P*R**2*S*T**6*UGA_star**15*epsilon + 3700224.0*P*R**2*S*T**6*UGA_star**15 - 48933888.0*P*R**2*S*T**6*UGA_star**14*epsilon - 2202624.0*P*R**2*S*T**6*UGA_star**14 + 13535232.0*P*R**2*S*T**6*UGA_star**13*epsilon + 711168.0*P*R**2*S*T**6*UGA_star**13 + 396288.0*P*R**2*S*T**6*UGA_star**12*epsilon - 41472.0*P*R**2*S*T**6*UGA_star**12 - 1161216.0*P*R**2*S*T**6*UGA_star**11*epsilon - 46080.0*P*R**2*S*T**6*UGA_star**11 + 125952.0*P*R**2*S*T**6*UGA_star**10*epsilon + 10752.0*P*R**2*S*T**6*UGA_star**10 + 30720.0*P*R**2*S*T**6*UGA_star**9*epsilon - 106496.0*P*R**2*T**7*UGA_star**20*epsilon - 2048.0*P*R**2*T**7*UGA_star**20 + 958464.0*P*R**2*T**7*UGA_star**19*epsilon + 18432.0*P*R**2*T**7*UGA_star**19 - 3820544.0*P*R**2*T**7*UGA_star**18*epsilon - 74240.0*P*R**2*T**7*UGA_star**18 + 8823808.0*P*R**2*T**7*UGA_star**17*epsilon + 175616.0*P*R**2*T**7*UGA_star**17 - 12935168.0*P*R**2*T**7*UGA_star**16*epsilon - 268800.0*P*R**2*T**7*UGA_star**16 + 12332032.0*P*R**2*T**7*UGA_star**15*epsilon + 275968.0*P*R**2*T**7*UGA_star**15 - 7430144.0*P*R**2*T**7*UGA_star**14*epsilon - 189952.0*P*R**2*T**7*UGA_star**14 + 2489344.0*P*R**2*T**7*UGA_star**13*epsilon + 84480.0*P*R**2*T**7*UGA_star**13 - 217088.0*P*R**2*T**7*UGA_star**12*epsilon - 22016.0*P*R**2*T**7*UGA_star**12 - 125952.0*P*R**2*T**7*UGA_star**11*epsilon + 2560.0*P*R**2*T**7*UGA_star**11 + 31744.0*P*R**2*T**7*UGA_star**10*epsilon + 18432.0*P*R*S**8*UGA_star**20*epsilon - 184320.0*P*R*S**8*UGA_star**19*epsilon + 832512.0*P*R*S**8*UGA_star**18*epsilon - 2242560.0*P*R*S**8*UGA_star**17*epsilon + 4008960.0*P*R*S**8*UGA_star**16*epsilon - 5013504.0*P*R*S**8*UGA_star**15*epsilon + 4515840.0*P*R*S**8*UGA_star**14*epsilon - 2985984.0*P*R*S**8*UGA_star**13*epsilon + 1474560.0*P*R*S**8*UGA_star**12*epsilon - 552960.0*P*R*S**8*UGA_star**11*epsilon + 156672.0*P*R*S**8*UGA_star**10*epsilon - 30720.0*P*R*S**8*UGA_star**9*epsilon + 3072.0*P*R*S**8*UGA_star**8*epsilon + 212992.0*P*R*S**7*T*UGA_star**20*epsilon + 4096.0*P*R*S**7*T*UGA_star**20 - 2129920.0*P*R*S**7*T*UGA_star**19*epsilon - 40960.0*P*R*S**7*T*UGA_star**19 + 9558016.0*P*R*S**7*T*UGA_star**18*epsilon + 185344.0*P*R*S**7*T*UGA_star**18 - 25350144.0*P*R*S**7*T*UGA_star**17*epsilon - 501760.0*P*R*S**7*T*UGA_star**17 + 44040192.0*P*R*S**7*T*UGA_star**16*epsilon + 906240.0*P*R*S**7*T*UGA_star**16 - 52488192.0*P*R*S**7*T*UGA_star**15*epsilon - 1155072.0*P*R*S**7*T*UGA_star**15 + 43732992.0*P*R*S**7*T*UGA_star**14*epsilon + 1075200.0*P*R*S**7*T*UGA_star**14 - 25559040.0*P*R*S**7*T*UGA_star**13*epsilon - 749568.0*P*R*S**7*T*UGA_star**13 + 10444800.0*P*R*S**7*T*UGA_star**12*epsilon + 399360.0*P*R*S**7*T*UGA_star**12 - 2977792.0*P*R*S**7*T*UGA_star**11*epsilon - 163840.0*P*R*S**7*T*UGA_star**11 + 563200.0*P*R*S**7*T*UGA_star**10*epsilon + 50176.0*P*R*S**7*T*UGA_star**10 - 28672.0*P*R*S**7*T*UGA_star**9*epsilon - 10240.0*P*R*S**7*T*UGA_star**9 - 24576.0*P*R*S**7*T*UGA_star**8*epsilon + 1024.0*P*R*S**7*T*UGA_star**8 + 6144.0*P*R*S**7*T*UGA_star**7*epsilon + 817152.0*P*R*S**6*T**2*UGA_star**20*epsilon + 32256.0*P*R*S**6*T**2*UGA_star**20 - 8171520.0*P*R*S**6*T**2*UGA_star**19*epsilon - 322560.0*P*R*S**6*T**2*UGA_star**19 + 36526080.0*P*R*S**6*T**2*UGA_star**18*epsilon + 1446912.0*P*R*S**6*T**2*UGA_star**18 - 95932416.0*P*R*S**6*T**2*UGA_star**17*epsilon - 3833856.0*P*R*S**6*T**2*UGA_star**17 + 163547136.0*P*R*S**6*T**2*UGA_star**16*epsilon + 6647808.0*P*R*S**6*T**2*UGA_star**16 - 188510208.0*P*R*S**6*T**2*UGA_star**15*epsilon - 7893504.0*P*R*S**6*T**2*UGA_star**15 + 148279296.0*P*R*S**6*T**2*UGA_star**14*epsilon + 6524928.0*P*R*S**6*T**2*UGA_star**14 - 78618624.0*P*R*S**6*T**2*UGA_star**13*epsilon - 3741696.0*P*R*S**6*T**2*UGA_star**13 + 27568128.0*P*R*S**6*T**2*UGA_star**12*epsilon + 1451520.0*P*R*S**6*T**2*UGA_star**12 - 6721536.0*P*R*S**6*T**2*UGA_star**11*epsilon - 350208.0*P*R*S**6*T**2*UGA_star**11 + 1529856.0*P*R*S**6*T**2*UGA_star**10*epsilon + 27648.0*P*R*S**6*T**2*UGA_star**10 - 331776.0*P*R*S**6*T**2*UGA_star**9*epsilon + 18432.0*P*R*S**6*T**2*UGA_star**9 + 6144.0*P*R*S**6*T**2*UGA_star**8*epsilon - 9216.0*P*R*S**6*T**2*UGA_star**8 + 12288.0*P*R*S**6*T**2*UGA_star**7*epsilon + 1536.0*P*R*S**6*T**2*UGA_star**7 + 1646592.0*P*R*S**5*T**3*UGA_star**20*epsilon + 92160.0*P*R*S**5*T**3*UGA_star**20 - 16465920.0*P*R*S**5*T**3*UGA_star**19*epsilon - 921600.0*P*R*S**5*T**3*UGA_star**19 + 73476096.0*P*R*S**5*T**3*UGA_star**18*epsilon + 4113408.0*P*R*S**5*T**3*UGA_star**18 - 192098304.0*P*R*S**5*T**3*UGA_star**17*epsilon - 10761216.0*P*R*S**5*T**3*UGA_star**17 + 324335616.0*P*R*S**5*T**3*UGA_star**16*epsilon + 18194944.0*P*R*S**5*T**3*UGA_star**16 - 366598144.0*P*R*S**5*T**3*UGA_star**15*epsilon - 20625920.0*P*R*S**5*T**3*UGA_star**15 + 276844544.0*P*R*S**5*T**3*UGA_star**14*epsilon + 15671296.0*P*R*S**5*T**3*UGA_star**14 - 133709824.0*P*R*S**5*T**3*UGA_star**13*epsilon - 7675904.0*P*R*S**5*T**3*UGA_star**13 + 36392960.0*P*R*S**5*T**3*UGA_star**12*epsilon + 2176000.0*P*R*S**5*T**3*UGA_star**12 - 3444736.0*P*R*S**5*T**3*UGA_star**11*epsilon - 257024.0*P*R*S**5*T**3*UGA_star**11 - 391168.0*P*R*S**5*T**3*UGA_star**10*epsilon - 5120.0*P*R*S**5*T**3*UGA_star**10 + 45056.0*P*R*S**5*T**3*UGA_star**9*epsilon + 1024.0*P*R*S**5*T**3*UGA_star**9 - 59392.0*P*R*S**5*T**3*UGA_star**8*epsilon - 3584.0*P*R*S**5*T**3*UGA_star**8 + 26624.0*P*R*S**5*T**3*UGA_star**7*epsilon + 1536.0*P*R*S**5*T**3*UGA_star**7 + 2048000.0*P*R*S**4*T**4*UGA_star**20*epsilon + 128000.0*P*R*S**4*T**4*UGA_star**20 - 20480000.0*P*R*S**4*T**4*UGA_star**19*epsilon - 1280000.0*P*R*S**4*T**4*UGA_star**19 + 91351040.0*P*R*S**4*T**4*UGA_star**18*epsilon + 5703680.0*P*R*S**4*T**4*UGA_star**18 - 238479360.0*P*R*S**4*T**4*UGA_star**17*epsilon - 14853120.0*P*R*S**4*T**4*UGA_star**17 + 401058816.0*P*R*S**4*T**4*UGA_star**16*epsilon + 24863744.0*P*R*S**4*T**4*UGA_star**16 - 448954368.0*P*R*S**4*T**4*UGA_star**15*epsilon - 27615232.0*P*R*S**4*T**4*UGA_star**15 + 330969088.0*P*R*S**4*T**4*UGA_star**14*epsilon + 20099072.0*P*R*S**4*T**4*UGA_star**14 - 149364736.0*P*R*S**4*T**4*UGA_star**13*epsilon - 8894464.0*P*R*S**4*T**4*UGA_star**13 + 30873600.0*P*R*S**4*T**4*UGA_star**12*epsilon + 1817600.0*P*R*S**4*T**4*UGA_star**12 + 3682304.0*P*R*S**4*T**4*UGA_star**11*epsilon + 116736.0*P*R*S**4*T**4*UGA_star**11 - 3110912.0*P*R*S**4*T**4*UGA_star**10*epsilon - 63488.0*P*R*S**4*T**4*UGA_star**10 + 428032.0*P*R*S**4*T**4*UGA_star**9*epsilon - 33792.0*P*R*S**4*T**4*UGA_star**9 - 41984.0*P*R*S**4*T**4*UGA_star**8*epsilon + 11264.0*P*R*S**4*T**4*UGA_star**8 + 20480.0*P*R*S**4*T**4*UGA_star**7*epsilon + 1646592.0*P*R*S**3*T**5*UGA_star**20*epsilon + 92160.0*P*R*S**3*T**5*UGA_star**20 - 16465920.0*P*R*S**3*T**5*UGA_star**19*epsilon - 921600.0*P*R*S**3*T**5*UGA_star**19 + 73476096.0*P*R*S**3*T**5*UGA_star**18*epsilon + 4113408.0*P*R*S**3*T**5*UGA_star**18 - 191913984.0*P*R*S**3*T**5*UGA_star**17*epsilon - 10748928.0*P*R*S**3*T**5*UGA_star**17 + 322768896.0*P*R*S**3*T**5*UGA_star**16*epsilon + 18090496.0*P*R*S**3*T**5*UGA_star**16 - 360753152.0*P*R*S**3*T**5*UGA_star**15*epsilon - 20235776.0*P*R*S**3*T**5*UGA_star**15 + 264341504.0*P*R*S**3*T**5*UGA_star**14*epsilon + 14834176.0*P*R*S**3*T**5*UGA_star**14 - 116951040.0*P*R*S**3*T**5*UGA_star**13*epsilon - 6546944.0*P*R*S**3*T**5*UGA_star**13 + 22042624.0*P*R*S**3*T**5*UGA_star**12*epsilon + 1197568.0*P*R*S**3*T**5*UGA_star**12 + 4081664.0*P*R*S**3*T**5*UGA_star**11*epsilon + 269824.0*P*R*S**3*T**5*UGA_star**11 - 2396160.0*P*R*S**3*T**5*UGA_star**10*epsilon - 157184.0*P*R*S**3*T**5*UGA_star**10 + 34816.0*P*R*S**3*T**5*UGA_star**9*epsilon + 8704.0*P*R*S**3*T**5*UGA_star**9 + 88064.0*P*R*S**3*T**5*UGA_star**8*epsilon + 4096.0*P*R*S**3*T**5*UGA_star**8 + 817152.0*P*R*S**2*T**6*UGA_star**20*epsilon + 32256.0*P*R*S**2*T**6*UGA_star**20 - 8171520.0*P*R*S**2*T**6*UGA_star**19*epsilon - 322560.0*P*R*S**2*T**6*UGA_star**19 + 36526080.0*P*R*S**2*T**6*UGA_star**18*epsilon + 1446912.0*P*R*S**2*T**6*UGA_star**18 - 95760384.0*P*R*S**2*T**6*UGA_star**17*epsilon - 3824640.0*P*R*S**2*T**6*UGA_star**17 + 162084864.0*P*R*S**2*T**6*UGA_star**16*epsilon + 6569472.0*P*R*S**2*T**6*UGA_star**16 - 183029760.0*P*R*S**2*T**6*UGA_star**15*epsilon - 7601664.0*P*R*S**2*T**6*UGA_star**15 + 136421376.0*P*R*S**2*T**6*UGA_star**14*epsilon + 5902848.0*P*R*S**2*T**6*UGA_star**14 - 62361600.0*P*R*S**2*T**6*UGA_star**13*epsilon - 2913792.0*P*R*S**2*T**6*UGA_star**13 + 13031424.0*P*R*S**2*T**6*UGA_star**12*epsilon + 752640.0*P*R*S**2*T**6*UGA_star**12 + 1622016.0*P*R*S**2*T**6*UGA_star**11*epsilon + 4608.0*P*R*S**2*T**6*UGA_star**11 - 1308672.0*P*R*S**2*T**6*UGA_star**10*epsilon - 56832.0*P*R*S**2*T**6*UGA_star**10 + 98304.0*P*R*S**2*T**6*UGA_star**9*epsilon + 10752.0*P*R*S**2*T**6*UGA_star**9 + 30720.0*P*R*S**2*T**6*UGA_star**8*epsilon + 212992.0*P*R*S*T**7*UGA_star**20*epsilon + 4096.0*P*R*S*T**7*UGA_star**20 - 2129920.0*P*R*S*T**7*UGA_star**19*epsilon - 40960.0*P*R*S*T**7*UGA_star**19 + 9558016.0*P*R*S*T**7*UGA_star**18*epsilon + 185344.0*P*R*S*T**7*UGA_star**18 - 25288704.0*P*R*S*T**7*UGA_star**17*epsilon - 499712.0*P*R*S*T**7*UGA_star**17 + 43517952.0*P*R*S*T**7*UGA_star**16*epsilon + 888832.0*P*R*S*T**7*UGA_star**16 - 50534400.0*P*R*S*T**7*UGA_star**15*epsilon - 1089536.0*P*R*S*T**7*UGA_star**15 + 39524352.0*P*R*S*T**7*UGA_star**14*epsilon + 931840.0*P*R*S*T**7*UGA_star**14 - 19838976.0*P*R*S*T**7*UGA_star**13*epsilon - 548864.0*P*R*S*T**7*UGA_star**13 + 5412864.0*P*R*S*T**7*UGA_star**12*epsilon + 212992.0*P*R*S*T**7*UGA_star**12 - 182272.0*P*R*S*T**7*UGA_star**11*epsilon - 49152.0*P*R*S*T**7*UGA_star**11 - 315392.0*P*R*S*T**7*UGA_star**10*epsilon + 5120.0*P*R*S*T**7*UGA_star**10 + 63488.0*P*R*S*T**7*UGA_star**9*epsilon + 18432.0*P*R*T**8*UGA_star**20*epsilon - 184320.0*P*R*T**8*UGA_star**19*epsilon + 832512.0*P*R*T**8*UGA_star**18*epsilon - 2236416.0*P*R*T**8*UGA_star**17*epsilon + 3956736.0*P*R*T**8*UGA_star**16*epsilon - 4816896.0*P*R*T**8*UGA_star**15*epsilon + 4085760.0*P*R*T**8*UGA_star**14*epsilon - 2383872.0*P*R*T**8*UGA_star**13*epsilon + 915456.0*P*R*T**8*UGA_star**12*epsilon - 208896.0*P*R*T**8*UGA_star**11*epsilon + 21504.0*P*R*T**8*UGA_star**10*epsilon - 2048.0*P*S**9*UGA_star**20*epsilon + 22528.0*P*S**9*UGA_star**19*epsilon - 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84480.0*P*S**8*T*UGA_star**12 + 481280.0*P*S**8*T*UGA_star**11*epsilon + 28160.0*P*S**8*T*UGA_star**11 + 75776.0*P*S**8*T*UGA_star**10*epsilon - 5632.0*P*S**8*T*UGA_star**10 - 55296.0*P*S**8*T*UGA_star**9*epsilon + 512.0*P*S**8*T*UGA_star**9 + 8192.0*P*S**8*T*UGA_star**8*epsilon - 116736.0*P*S**7*T**2*UGA_star**20*epsilon - 4608.0*P*S**7*T**2*UGA_star**20 + 1284096.0*P*S**7*T**2*UGA_star**19*epsilon + 50688.0*P*S**7*T**2*UGA_star**19 - 6365184.0*P*S**7*T**2*UGA_star**18*epsilon - 251904.0*P*S**7*T**2*UGA_star**18 + 18708480.0*P*S**7*T**2*UGA_star**17*epsilon + 744960.0*P*S**7*T**2*UGA_star**17 - 36040704.0*P*S**7*T**2*UGA_star**16*epsilon - 1451520.0*P*S**7*T**2*UGA_star**16 + 47351808.0*P*S**7*T**2*UGA_star**15*epsilon + 1944576.0*P*S**7*T**2*UGA_star**15 - 42541056.0*P*S**7*T**2*UGA_star**14*epsilon - 1806336.0*P*S**7*T**2*UGA_star**14 + 25104384.0*P*S**7*T**2*UGA_star**13*epsilon + 1133568.0*P*S**7*T**2*UGA_star**13 - 8423424.0*P*S**7*T**2*UGA_star**12*epsilon - 437760.0*P*S**7*T**2*UGA_star**12 + 559104.0*P*S**7*T**2*UGA_star**11*epsilon + 69120.0*P*S**7*T**2*UGA_star**11 + 688128.0*P*S**7*T**2*UGA_star**10*epsilon + 18432.0*P*S**7*T**2*UGA_star**10 - 215040.0*P*S**7*T**2*UGA_star**9*epsilon - 10752.0*P*S**7*T**2*UGA_star**9 + 1536.0*P*S**7*T**2*UGA_star**8 + 6144.0*P*S**7*T**2*UGA_star**7*epsilon - 274432.0*P*S**6*T**3*UGA_star**20*epsilon - 15360.0*P*S**6*T**3*UGA_star**20 + 3018752.0*P*S**6*T**3*UGA_star**19*epsilon + 168960.0*P*S**6*T**3*UGA_star**19 - 14953472.0*P*S**6*T**3*UGA_star**18*epsilon - 837120.0*P*S**6*T**3*UGA_star**18 + 43877376.0*P*S**6*T**3*UGA_star**17*epsilon + 2457600.0*P*S**6*T**3*UGA_star**17 - 84256768.0*P*S**6*T**3*UGA_star**16*epsilon - 4724224.0*P*S**6*T**3*UGA_star**16 + 110063616.0*P*S**6*T**3*UGA_star**15*epsilon + 6183936.0*P*S**6*T**3*UGA_star**15 - 97830912.0*P*S**6*T**3*UGA_star**14*epsilon - 5520384.0*P*S**6*T**3*UGA_star**14 + 56457216.0*P*S**6*T**3*UGA_star**13*epsilon + 3219456.0*P*S**6*T**3*UGA_star**13 - 17756160.0*P*S**6*T**3*UGA_star**12*epsilon - 1050624.0*P*S**6*T**3*UGA_star**12 + 237568.0*P*S**6*T**3*UGA_star**11*epsilon + 52224.0*P*S**6*T**3*UGA_star**11 + 1954816.0*P*S**6*T**3*UGA_star**10*epsilon + 90624.0*P*S**6*T**3*UGA_star**10 - 547840.0*P*S**6*T**3*UGA_star**9*epsilon - 24576.0*P*S**6*T**3*UGA_star**9 - 6144.0*P*S**6*T**3*UGA_star**8*epsilon - 1536.0*P*S**6*T**3*UGA_star**8 + 16384.0*P*S**6*T**3*UGA_star**7*epsilon + 1024.0*P*S**6*T**3*UGA_star**7 - 409600.0*P*S**5*T**4*UGA_star**20*epsilon - 25600.0*P*S**5*T**4*UGA_star**20 + 4505600.0*P*S**5*T**4*UGA_star**19*epsilon + 281600.0*P*S**5*T**4*UGA_star**19 - 22334464.0*P*S**5*T**4*UGA_star**18*epsilon - 1394688.0*P*S**5*T**4*UGA_star**18 + 65639424.0*P*S**5*T**4*UGA_star**17*epsilon + 4090368.0*P*S**5*T**4*UGA_star**17 - 126395392.0*P*S**5*T**4*UGA_star**16*epsilon - 7846400.0*P*S**5*T**4*UGA_star**16 + 165859328.0*P*S**5*T**4*UGA_star**15*epsilon + 10229760.0*P*S**5*T**4*UGA_star**15 - 148559872.0*P*S**5*T**4*UGA_star**14*epsilon - 9062400.0*P*S**5*T**4*UGA_star**14 + 87013376.0*P*S**5*T**4*UGA_star**13*epsilon + 5200896.0*P*S**5*T**4*UGA_star**13 - 28518400.0*P*S**5*T**4*UGA_star**12*epsilon - 1622016.0*P*S**5*T**4*UGA_star**12 + 1339392.0*P*S**5*T**4*UGA_star**11*epsilon + 25600.0*P*S**5*T**4*UGA_star**11 + 2614272.0*P*S**5*T**4*UGA_star**10*epsilon + 163840.0*P*S**5*T**4*UGA_star**10 - 764928.0*P*S**5*T**4*UGA_star**9*epsilon - 38400.0*P*S**5*T**4*UGA_star**9 - 13312.0*P*S**5*T**4*UGA_star**8*epsilon - 4608.0*P*S**5*T**4*UGA_star**8 + 24576.0*P*S**5*T**4*UGA_star**7*epsilon + 2048.0*P*S**5*T**4*UGA_star**7 - 411648.0*P*S**4*T**5*UGA_star**20*epsilon - 23040.0*P*S**4*T**5*UGA_star**20 + 4528128.0*P*S**4*T**5*UGA_star**19*epsilon + 253440.0*P*S**4*T**5*UGA_star**19 - 22474752.0*P*S**4*T**5*UGA_star**18*epsilon - 1257984.0*P*S**4*T**5*UGA_star**18 + 66238464.0*P*S**4*T**5*UGA_star**17*epsilon + 3707904.0*P*S**4*T**5*UGA_star**17 - 128175104.0*P*S**4*T**5*UGA_star**16*epsilon - 7175168.0*P*S**4*T**5*UGA_star**16 + 169553920.0*P*S**4*T**5*UGA_star**15*epsilon + 9488384.0*P*S**4*T**5*UGA_star**15 - 153964544.0*P*S**4*T**5*UGA_star**14*epsilon - 8602624.0*P*S**4*T**5*UGA_star**14 + 92606464.0*P*S**4*T**5*UGA_star**13*epsilon + 5144576.0*P*S**4*T**5*UGA_star**13 - 32583680.0*P*S**4*T**5*UGA_star**12*epsilon - 1765888.0*P*S**4*T**5*UGA_star**12 + 3364864.0*P*S**4*T**5*UGA_star**11*epsilon + 132608.0*P*S**4*T**5*UGA_star**11 + 1957888.0*P*S**4*T**5*UGA_star**10*epsilon + 144896.0*P*S**4*T**5*UGA_star**10 - 641024.0*P*S**4*T**5*UGA_star**9*epsilon - 50176.0*P*S**4*T**5*UGA_star**9 - 23552.0*P*S**4*T**5*UGA_star**8*epsilon + 2048.0*P*S**4*T**5*UGA_star**8 + 24576.0*P*S**4*T**5*UGA_star**7*epsilon + 1024.0*P*S**4*T**5*UGA_star**7 - 272384.0*P*S**3*T**6*UGA_star**20*epsilon - 10752.0*P*S**3*T**6*UGA_star**20 + 2996224.0*P*S**3*T**6*UGA_star**19*epsilon + 118272.0*P*S**3*T**6*UGA_star**19 - 14899200.0*P*S**3*T**6*UGA_star**18*epsilon - 589824.0*P*S**3*T**6*UGA_star**18 + 44095488.0*P*S**3*T**6*UGA_star**17*epsilon + 1757184.0*P*S**3*T**6*UGA_star**17 - 85947392.0*P*S**3*T**6*UGA_star**16*epsilon - 3464704.0*P*S**3*T**6*UGA_star**16 + 115030016.0*P*S**3*T**6*UGA_star**15*epsilon + 4723712.0*P*S**3*T**6*UGA_star**15 - 106455040.0*P*S**3*T**6*UGA_star**14*epsilon - 4501504.0*P*S**3*T**6*UGA_star**14 + 66203648.0*P*S**3*T**6*UGA_star**13*epsilon + 2938880.0*P*S**3*T**6*UGA_star**13 - 25059328.0*P*S**3*T**6*UGA_star**12*epsilon - 1222144.0*P*S**3*T**6*UGA_star**12 + 3745792.0*P*S**3*T**6*UGA_star**11*epsilon + 249344.0*P*S**3*T**6*UGA_star**11 + 1005568.0*P*S**3*T**6*UGA_star**10*epsilon + 20480.0*P*S**3*T**6*UGA_star**10 - 477184.0*P*S**3*T**6*UGA_star**9*epsilon - 22528.0*P*S**3*T**6*UGA_star**9 + 23552.0*P*S**3*T**6*UGA_star**8*epsilon + 3584.0*P*S**3*T**6*UGA_star**8 + 10240.0*P*S**3*T**6*UGA_star**7*epsilon - 106496.0*P*S**2*T**7*UGA_star**20*epsilon - 2048.0*P*S**2*T**7*UGA_star**20 + 1171456.0*P*S**2*T**7*UGA_star**19*epsilon + 22528.0*P*S**2*T**7*UGA_star**19 - 5843968.0*P*S**2*T**7*UGA_star**18*epsilon - 113152.0*P*S**2*T**7*UGA_star**18 + 17423360.0*P*S**2*T**7*UGA_star**17*epsilon + 342528.0*P*S**2*T**7*UGA_star**17 - 34403328.0*P*S**2*T**7*UGA_star**16*epsilon - 694272.0*P*S**2*T**7*UGA_star**16 + 47026176.0*P*S**2*T**7*UGA_star**15*epsilon + 989184.0*P*S**2*T**7*UGA_star**15 - 45029376.0*P*S**2*T**7*UGA_star**14*epsilon - 1010688.0*P*S**2*T**7*UGA_star**14 + 29681664.0*P*S**2*T**7*UGA_star**13*epsilon + 740352.0*P*S**2*T**7*UGA_star**13 - 12625920.0*P*S**2*T**7*UGA_star**12*epsilon - 380928.0*P*S**2*T**7*UGA_star**12 + 2797568.0*P*S**2*T**7*UGA_star**11*epsilon + 131072.0*P*S**2*T**7*UGA_star**11 + 66560.0*P*S**2*T**7*UGA_star**10*epsilon - 27136.0*P*S**2*T**7*UGA_star**10 - 189440.0*P*S**2*T**7*UGA_star**9*epsilon + 2560.0*P*S**2*T**7*UGA_star**9 + 31744.0*P*S**2*T**7*UGA_star**8*epsilon - 18432.0*P*S*T**8*UGA_star**20*epsilon + 202752.0*P*S*T**8*UGA_star**19*epsilon - 1016832.0*P*S*T**8*UGA_star**18*epsilon + 3068928.0*P*S*T**8*UGA_star**17*epsilon - 6193152.0*P*S*T**8*UGA_star**16*epsilon + 8773632.0*P*S*T**8*UGA_star**15*epsilon - 8902656.0*P*S*T**8*UGA_star**14*epsilon + 6469632.0*P*S*T**8*UGA_star**13*epsilon - 3299328.0*P*S*T**8*UGA_star**12*epsilon + 1124352.0*P*S*T**8*UGA_star**11*epsilon - 230400.0*P*S*T**8*UGA_star**10*epsilon + 21504.0*P*S*T**8*UGA_star**9*epsilon - 2048.0*R**9*T*UGA_star**20*epsilon + 2048.0*R**9*T*UGA_star**19*epsilon - 3072.0*R**9*T*UGA_star**18*epsilon + 3072.0*R**9*T*UGA_star**17*epsilon - 1024.0*R**9*T*UGA_star**16*epsilon + 1024.0*R**9*T*UGA_star**15*epsilon + 18432.0*R**8*S*T*UGA_star**20*epsilon - 36864.0*R**8*S*T*UGA_star**19*epsilon + 43008.0*R**8*S*T*UGA_star**18*epsilon - 49152.0*R**8*S*T*UGA_star**17*epsilon + 31744.0*R**8*S*T*UGA_star**16*epsilon - 14336.0*R**8*S*T*UGA_star**15*epsilon + 7168.0*R**8*S*T*UGA_star**14*epsilon + 13312.0*R**8*T**2*UGA_star**20*epsilon + 256.0*R**8*T**2*UGA_star**20 - 26624.0*R**8*T**2*UGA_star**19*epsilon - 512.0*R**8*T**2*UGA_star**19 + 28672.0*R**8*T**2*UGA_star**18*epsilon + 768.0*R**8*T**2*UGA_star**18 - 30720.0*R**8*T**2*UGA_star**17*epsilon - 1024.0*R**8*T**2*UGA_star**17 + 16384.0*R**8*T**2*UGA_star**16*epsilon + 768.0*R**8*T**2*UGA_star**16 - 2048.0*R**8*T**2*UGA_star**15*epsilon - 512.0*R**8*T**2*UGA_star**15 + 1024.0*R**8*T**2*UGA_star**14*epsilon + 256.0*R**8*T**2*UGA_star**14 - 73728.0*R**7*S**2*T*UGA_star**20*epsilon + 221184.0*R**7*S**2*T*UGA_star**19*epsilon - 307200.0*R**7*S**2*T*UGA_star**18*epsilon + 331776.0*R**7*S**2*T*UGA_star**17*epsilon - 279552.0*R**7*S**2*T*UGA_star**16*epsilon + 150528.0*R**7*S**2*T*UGA_star**15*epsilon - 64512.0*R**7*S**2*T*UGA_star**14*epsilon + 21504.0*R**7*S**2*T*UGA_star**13*epsilon - 106496.0*R**7*S*T**2*UGA_star**20*epsilon - 2048.0*R**7*S*T**2*UGA_star**20 + 319488.0*R**7*S*T**2*UGA_star**19*epsilon + 6144.0*R**7*S*T**2*UGA_star**19 - 422912.0*R**7*S*T**2*UGA_star**18*epsilon - 9728.0*R**7*S*T**2*UGA_star**18 + 420864.0*R**7*S*T**2*UGA_star**17*epsilon + 12800.0*R**7*S*T**2*UGA_star**17 - 321536.0*R**7*S*T**2*UGA_star**16*epsilon - 12288.0*R**7*S*T**2*UGA_star**16 + 119808.0*R**7*S*T**2*UGA_star**15*epsilon + 8192.0*R**7*S*T**2*UGA_star**15 - 15360.0*R**7*S*T**2*UGA_star**14*epsilon - 4608.0*R**7*S*T**2*UGA_star**14 + 6144.0*R**7*S*T**2*UGA_star**13*epsilon + 1536.0*R**7*S*T**2*UGA_star**13 - 38912.0*R**7*T**3*UGA_star**20*epsilon - 1536.0*R**7*T**3*UGA_star**20 + 116736.0*R**7*T**3*UGA_star**19*epsilon + 4608.0*R**7*T**3*UGA_star**19 - 145408.0*R**7*T**3*UGA_star**18*epsilon - 6144.0*R**7*T**3*UGA_star**18 + 124928.0*R**7*T**3*UGA_star**17*epsilon + 6144.0*R**7*T**3*UGA_star**17 - 86016.0*R**7*T**3*UGA_star**16*epsilon - 4608.0*R**7*T**3*UGA_star**16 + 31744.0*R**7*T**3*UGA_star**15*epsilon + 1536.0*R**7*T**3*UGA_star**15 - 6144.0*R**7*T**3*UGA_star**14*epsilon + 3072.0*R**7*T**3*UGA_star**13*epsilon + 172032.0*R**6*S**3*T*UGA_star**20*epsilon - 688128.0*R**6*S**3*T*UGA_star**19*epsilon + 1204224.0*R**6*S**3*T*UGA_star**18*epsilon - 1376256.0*R**6*S**3*T*UGA_star**17*epsilon + 1240064.0*R**6*S**3*T*UGA_star**16*epsilon - 831488.0*R**6*S**3*T*UGA_star**15*epsilon + 387072.0*R**6*S**3*T*UGA_star**14*epsilon - 143360.0*R**6*S**3*T*UGA_star**13*epsilon + 35840.0*R**6*S**3*T*UGA_star**12*epsilon + 372736.0*R**6*S**2*T**2*UGA_star**20*epsilon + 7168.0*R**6*S**2*T**2*UGA_star**20 - 1490944.0*R**6*S**2*T**2*UGA_star**19*epsilon - 28672.0*R**6*S**2*T**2*UGA_star**19 + 2530304.0*R**6*S**2*T**2*UGA_star**18*epsilon + 53760.0*R**6*S**2*T**2*UGA_star**18 - 2695168.0*R**6*S**2*T**2*UGA_star**17*epsilon - 71680.0*R**6*S**2*T**2*UGA_star**17 + 2219008.0*R**6*S**2*T**2*UGA_star**16*epsilon + 75520.0*R**6*S**2*T**2*UGA_star**16 - 1267712.0*R**6*S**2*T**2*UGA_star**15*epsilon - 58368.0*R**6*S**2*T**2*UGA_star**15 + 359424.0*R**6*S**2*T**2*UGA_star**14*epsilon + 33792.0*R**6*S**2*T**2*UGA_star**14 - 43008.0*R**6*S**2*T**2*UGA_star**13*epsilon - 15360.0*R**6*S**2*T**2*UGA_star**13 + 15360.0*R**6*S**2*T**2*UGA_star**12*epsilon + 3840.0*R**6*S**2*T**2*UGA_star**12 + 272384.0*R**6*S*T**3*UGA_star**20*epsilon + 10752.0*R**6*S*T**3*UGA_star**20 - 1089536.0*R**6*S*T**3*UGA_star**19*epsilon - 43008.0*R**6*S*T**3*UGA_star**19 + 1787904.0*R**6*S*T**3*UGA_star**18*epsilon + 73216.0*R**6*S*T**3*UGA_star**18 - 1722368.0*R**6*S*T**3*UGA_star**17*epsilon - 78336.0*R**6*S*T**3*UGA_star**17 + 1244160.0*R**6*S*T**3*UGA_star**16*epsilon + 64000.0*R**6*S*T**3*UGA_star**16 - 669696.0*R**6*S*T**3*UGA_star**15*epsilon - 34816.0*R**6*S*T**3*UGA_star**15 + 202752.0*R**6*S*T**3*UGA_star**14*epsilon + 7680.0*R**6*S*T**3*UGA_star**14 - 40960.0*R**6*S*T**3*UGA_star**13*epsilon + 512.0*R**6*S*T**3*UGA_star**13 + 15360.0*R**6*S*T**3*UGA_star**12*epsilon + 68608.0*R**6*T**4*UGA_star**20*epsilon + 3840.0*R**6*T**4*UGA_star**20 - 274432.0*R**6*T**4*UGA_star**19*epsilon - 15360.0*R**6*T**4*UGA_star**19 + 430080.0*R**6*T**4*UGA_star**18*epsilon + 24064.0*R**6*T**4*UGA_star**18 - 354304.0*R**6*T**4*UGA_star**17*epsilon - 19968.0*R**6*T**4*UGA_star**17 + 195584.0*R**6*T**4*UGA_star**16*epsilon + 11008.0*R**6*T**4*UGA_star**16 - 86016.0*R**6*T**4*UGA_star**15*epsilon - 4096.0*R**6*T**4*UGA_star**15 + 18432.0*R**6*T**4*UGA_star**14*epsilon + 2048.0*R**6*T**4*UGA_star**13*epsilon + 512.0*R**6*T**4*UGA_star**13 - 258048.0*R**5*S**4*T*UGA_star**20*epsilon + 1290240.0*R**5*S**4*T*UGA_star**19*epsilon - 2795520.0*R**5*S**4*T*UGA_star**18*epsilon + 3655680.0*R**5*S**4*T*UGA_star**17*epsilon - 3476480.0*R**5*S**4*T*UGA_star**16*epsilon + 2587648.0*R**5*S**4*T*UGA_star**15*epsilon - 1433600.0*R**5*S**4*T*UGA_star**14*epsilon + 573440.0*R**5*S**4*T*UGA_star**13*epsilon - 179200.0*R**5*S**4*T*UGA_star**12*epsilon + 35840.0*R**5*S**4*T*UGA_star**11*epsilon - 745472.0*R**5*S**3*T**2*UGA_star**20*epsilon - 14336.0*R**5*S**3*T**2*UGA_star**20 + 3727360.0*R**5*S**3*T**2*UGA_star**19*epsilon + 71680.0*R**5*S**3*T**2*UGA_star**19 - 7906304.0*R**5*S**3*T**2*UGA_star**18*epsilon - 161280.0*R**5*S**3*T**2*UGA_star**18 + 9798656.0*R**5*S**3*T**2*UGA_star**17*epsilon + 232960.0*R**5*S**3*T**2*UGA_star**17 - 8561664.0*R**5*S**3*T**2*UGA_star**16*epsilon - 256000.0*R**5*S**3*T**2*UGA_star**16 + 5695488.0*R**5*S**3*T**2*UGA_star**15*epsilon + 219136.0*R**5*S**3*T**2*UGA_star**15 - 2530304.0*R**5*S**3*T**2*UGA_star**14*epsilon - 140800.0*R**5*S**3*T**2*UGA_star**14 + 558080.0*R**5*S**3*T**2*UGA_star**13*epsilon + 69120.0*R**5*S**3*T**2*UGA_star**13 - 56320.0*R**5*S**3*T**2*UGA_star**12*epsilon - 25600.0*R**5*S**3*T**2*UGA_star**12 + 20480.0*R**5*S**3*T**2*UGA_star**11*epsilon + 5120.0*R**5*S**3*T**2*UGA_star**11 - 817152.0*R**5*S**2*T**3*UGA_star**20*epsilon - 32256.0*R**5*S**2*T**3*UGA_star**20 + 4085760.0*R**5*S**2*T**3*UGA_star**19*epsilon + 161280.0*R**5*S**2*T**3*UGA_star**19 - 8491008.0*R**5*S**2*T**3*UGA_star**18*epsilon - 342528.0*R**5*S**2*T**3*UGA_star**18 + 9879552.0*R**5*S**2*T**3*UGA_star**17*epsilon + 425472.0*R**5*S**2*T**3*UGA_star**17 - 7699456.0*R**5*S**2*T**3*UGA_star**16*epsilon - 370688.0*R**5*S**2*T**3*UGA_star**16 + 4610048.0*R**5*S**2*T**3*UGA_star**15*epsilon + 240128.0*R**5*S**2*T**3*UGA_star**15 - 2015232.0*R**5*S**2*T**3*UGA_star**14*epsilon - 96768.0*R**5*S**2*T**3*UGA_star**14 + 514048.0*R**5*S**2*T**3*UGA_star**13*epsilon + 12800.0*R**5*S**2*T**3*UGA_star**13 - 97280.0*R**5*S**2*T**3*UGA_star**12*epsilon + 2560.0*R**5*S**2*T**3*UGA_star**12 + 30720.0*R**5*S**2*T**3*UGA_star**11*epsilon - 411648.0*R**5*S*T**4*UGA_star**20*epsilon - 23040.0*R**5*S*T**4*UGA_star**20 + 2058240.0*R**5*S*T**4*UGA_star**19*epsilon + 115200.0*R**5*S*T**4*UGA_star**19 - 4171776.0*R**5*S*T**4*UGA_star**18*epsilon - 233472.0*R**5*S*T**4*UGA_star**18 + 4460544.0*R**5*S*T**4*UGA_star**17*epsilon + 250368.0*R**5*S*T**4*UGA_star**17 - 2870272.0*R**5*S*T**4*UGA_star**16*epsilon - 161792.0*R**5*S*T**4*UGA_star**16 + 1320960.0*R**5*S*T**4*UGA_star**15*epsilon + 71168.0*R**5*S*T**4*UGA_star**15 - 466944.0*R**5*S*T**4*UGA_star**14*epsilon - 18432.0*R**5*S*T**4*UGA_star**14 + 65536.0*R**5*S*T**4*UGA_star**13*epsilon - 2560.0*R**5*S*T**4*UGA_star**13 + 15360.0*R**5*S*T**4*UGA_star**12*epsilon + 2560.0*R**5*S*T**4*UGA_star**12 - 81920.0*R**5*T**5*UGA_star**20*epsilon - 5120.0*R**5*T**5*UGA_star**20 + 409600.0*R**5*T**5*UGA_star**19*epsilon + 25600.0*R**5*T**5*UGA_star**19 - 805888.0*R**5*T**5*UGA_star**18*epsilon - 50176.0*R**5*T**5*UGA_star**18 + 775168.0*R**5*T**5*UGA_star**17*epsilon + 47616.0*R**5*T**5*UGA_star**17 - 363520.0*R**5*T**5*UGA_star**16*epsilon - 21504.0*R**5*T**5*UGA_star**16 + 69632.0*R**5*T**5*UGA_star**15*epsilon + 4096.0*R**5*T**5*UGA_star**15 + 1024.0*R**5*T**5*UGA_star**14*epsilon - 1024.0*R**5*T**5*UGA_star**14 - 9216.0*R**5*T**5*UGA_star**13*epsilon + 512.0*R**5*T**5*UGA_star**13 + 5120.0*R**5*T**5*UGA_star**12*epsilon + 258048.0*R**4*S**5*T*UGA_star**20*epsilon - 1548288.0*R**4*S**5*T*UGA_star**19*epsilon + 4042752.0*R**4*S**5*T*UGA_star**18*epsilon - 6193152.0*R**4*S**5*T*UGA_star**17*epsilon + 6472704.0*R**4*S**5*T*UGA_star**16*epsilon - 5117952.0*R**4*S**5*T*UGA_star**15*epsilon + 3161088.0*R**4*S**5*T*UGA_star**14*epsilon - 1462272.0*R**4*S**5*T*UGA_star**13*epsilon + 494592.0*R**4*S**5*T*UGA_star**12*epsilon - 129024.0*R**4*S**5*T*UGA_star**11*epsilon + 21504.0*R**4*S**5*T*UGA_star**10*epsilon + 931840.0*R**4*S**4*T**2*UGA_star**20*epsilon + 17920.0*R**4*S**4*T**2*UGA_star**20 - 5591040.0*R**4*S**4*T**2*UGA_star**19*epsilon - 107520.0*R**4*S**4*T**2*UGA_star**19 + 14371840.0*R**4*S**4*T**2*UGA_star**18*epsilon + 286720.0*R**4*S**4*T**2*UGA_star**18 - 21145600.0*R**4*S**4*T**2*UGA_star**17*epsilon - 465920.0*R**4*S**4*T**2*UGA_star**17 + 20561920.0*R**4*S**4*T**2*UGA_star**16*epsilon + 541440.0*R**4*S**4*T**2*UGA_star**16 - 14694400.0*R**4*S**4*T**2*UGA_star**15*epsilon - 488960.0*R**4*S**4*T**2*UGA_star**15 + 7895040.0*R**4*S**4*T**2*UGA_star**14*epsilon + 344320.0*R**4*S**4*T**2*UGA_star**14 - 2775040.0*R**4*S**4*T**2*UGA_star**13*epsilon - 184320.0*R**4*S**4*T**2*UGA_star**13 + 460800.0*R**4*S**4*T**2*UGA_star**12*epsilon + 75520.0*R**4*S**4*T**2*UGA_star**12 - 30720.0*R**4*S**4*T**2*UGA_star**11*epsilon - 23040.0*R**4*S**4*T**2*UGA_star**11 + 15360.0*R**4*S**4*T**2*UGA_star**10*epsilon + 3840.0*R**4*S**4*T**2*UGA_star**10 + 1361920.0*R**4*S**3*T**3*UGA_star**20*epsilon + 53760.0*R**4*S**3*T**3*UGA_star**20 - 8171520.0*R**4*S**3*T**3*UGA_star**19*epsilon - 322560.0*R**4*S**3*T**3*UGA_star**19 + 20725760.0*R**4*S**3*T**3*UGA_star**18*epsilon + 829440.0*R**4*S**3*T**3*UGA_star**18 - 29296640.0*R**4*S**3*T**3*UGA_star**17*epsilon - 1221120.0*R**4*S**3*T**3*UGA_star**17 + 26224640.0*R**4*S**3*T**3*UGA_star**16*epsilon + 1185280.0*R**4*S**3*T**3*UGA_star**16 - 16691200.0*R**4*S**3*T**3*UGA_star**15*epsilon - 834560.0*R**4*S**3*T**3*UGA_star**15 + 8284160.0*R**4*S**3*T**3*UGA_star**14*epsilon + 424960.0*R**4*S**3*T**3*UGA_star**14 - 3000320.0*R**4*S**3*T**3*UGA_star**13*epsilon - 125440.0*R**4*S**3*T**3*UGA_star**13 + 634880.0*R**4*S**3*T**3*UGA_star**12*epsilon + 5120.0*R**4*S**3*T**3*UGA_star**12 - 102400.0*R**4*S**3*T**3*UGA_star**11*epsilon + 5120.0*R**4*S**3*T**3*UGA_star**11 + 30720.0*R**4*S**3*T**3*UGA_star**10*epsilon + 1029120.0*R**4*S**2*T**4*UGA_star**20*epsilon + 57600.0*R**4*S**2*T**4*UGA_star**20 - 6174720.0*R**4*S**2*T**4*UGA_star**19*epsilon - 345600.0*R**4*S**2*T**4*UGA_star**19 + 15436800.0*R**4*S**2*T**4*UGA_star**18*epsilon + 864000.0*R**4*S**2*T**4*UGA_star**18 - 20828160.0*R**4*S**2*T**4*UGA_star**17*epsilon - 1167360.0*R**4*S**2*T**4*UGA_star**17 + 16646144.0*R**4*S**2*T**4*UGA_star**16*epsilon + 935936.0*R**4*S**2*T**4*UGA_star**16 - 8513536.0*R**4*S**2*T**4*UGA_star**15*epsilon - 474624.0*R**4*S**2*T**4*UGA_star**15 + 3206144.0*R**4*S**2*T**4*UGA_star**14*epsilon + 161536.0*R**4*S**2*T**4*UGA_star**14 - 905216.0*R**4*S**2*T**4*UGA_star**13*epsilon - 26624.0*R**4*S**2*T**4*UGA_star**13 + 62464.0*R**4*S**2*T**4*UGA_star**12*epsilon - 9984.0*R**4*S**2*T**4*UGA_star**12 + 40960.0*R**4*S**2*T**4*UGA_star**11*epsilon + 5120.0*R**4*S**2*T**4*UGA_star**11 + 409600.0*R**4*S*T**5*UGA_star**20*epsilon + 25600.0*R**4*S*T**5*UGA_star**20 - 2457600.0*R**4*S*T**5*UGA_star**19*epsilon - 153600.0*R**4*S*T**5*UGA_star**19 + 6045696.0*R**4*S*T**5*UGA_star**18*epsilon + 376832.0*R**4*S*T**5*UGA_star**18 - 7737344.0*R**4*S*T**5*UGA_star**17*epsilon - 478208.0*R**4*S*T**5*UGA_star**17 + 5337088.0*R**4*S*T**5*UGA_star**16*epsilon + 323072.0*R**4*S*T**5*UGA_star**16 - 1789952.0*R**4*S*T**5*UGA_star**15*epsilon - 104448.0*R**4*S*T**5*UGA_star**15 + 155648.0*R**4*S*T**5*UGA_star**14*epsilon + 13312.0*R**4*S*T**5*UGA_star**14 + 69632.0*R**4*S*T**5*UGA_star**13*epsilon - 5120.0*R**4*S*T**5*UGA_star**13 - 53248.0*R**4*S*T**5*UGA_star**12*epsilon + 2560.0*R**4*S*T**5*UGA_star**12 + 20480.0*R**4*S*T**5*UGA_star**11*epsilon + 68608.0*R**4*T**6*UGA_star**20*epsilon + 3840.0*R**4*T**6*UGA_star**20 - 411648.0*R**4*T**6*UGA_star**19*epsilon - 23040.0*R**4*T**6*UGA_star**19 + 999424.0*R**4*T**6*UGA_star**18*epsilon + 56064.0*R**4*T**6*UGA_star**18 - 1224704.0*R**4*T**6*UGA_star**17*epsilon - 69120.0*R**4*T**6*UGA_star**17 + 738304.0*R**4*T**6*UGA_star**16*epsilon + 42496.0*R**4*T**6*UGA_star**16 - 129024.0*R**4*T**6*UGA_star**15*epsilon - 8704.0*R**4*T**6*UGA_star**15 - 63488.0*R**4*T**6*UGA_star**14*epsilon - 2304.0*R**4*T**6*UGA_star**14 + 20480.0*R**4*T**6*UGA_star**13*epsilon + 512.0*R**4*T**6*UGA_star**13 + 2048.0*R**4*T**6*UGA_star**12*epsilon + 256.0*R**4*T**6*UGA_star**12 - 172032.0*R**3*S**6*T*UGA_star**20*epsilon + 1204224.0*R**3*S**6*T*UGA_star**19*epsilon - 3698688.0*R**3*S**6*T*UGA_star**18*epsilon + 6623232.0*R**3*S**6*T*UGA_star**17*epsilon - 7834624.0*R**3*S**6*T*UGA_star**16*epsilon + 6673408.0*R**3*S**6*T*UGA_star**15*epsilon - 4365312.0*R**3*S**6*T*UGA_star**14*epsilon + 2229248.0*R**3*S**6*T*UGA_star**13*epsilon - 852992.0*R**3*S**6*T*UGA_star**12*epsilon + 236544.0*R**3*S**6*T*UGA_star**11*epsilon - 50176.0*R**3*S**6*T*UGA_star**10*epsilon + 7168.0*R**3*S**6*T*UGA_star**9*epsilon - 745472.0*R**3*S**5*T**2*UGA_star**20*epsilon - 14336.0*R**3*S**5*T**2*UGA_star**20 + 5218304.0*R**3*S**5*T**2*UGA_star**19*epsilon + 100352.0*R**3*S**5*T**2*UGA_star**19 - 15834112.0*R**3*S**5*T**2*UGA_star**18*epsilon - 311808.0*R**3*S**5*T**2*UGA_star**18 + 27489280.0*R**3*S**5*T**2*UGA_star**17*epsilon + 577024.0*R**3*S**5*T**2*UGA_star**17 - 30674944.0*R**3*S**5*T**2*UGA_star**16*epsilon - 729088.0*R**3*S**5*T**2*UGA_star**16 + 23843840.0*R**3*S**5*T**2*UGA_star**15*epsilon + 688128.0*R**3*S**5*T**2*UGA_star**15 - 13795328.0*R**3*S**5*T**2*UGA_star**14*epsilon - 508928.0*R**3*S**5*T**2*UGA_star**14 + 5952512.0*R**3*S**5*T**2*UGA_star**13*epsilon + 293888.0*R**3*S**5*T**2*UGA_star**13 - 1638400.0*R**3*S**5*T**2*UGA_star**12*epsilon - 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2981888.0*R**2*S**6*T**2*UGA_star**19*epsilon - 57344.0*R**2*S**6*T**2*UGA_star**19 + 10458112.0*R**2*S**6*T**2*UGA_star**18*epsilon + 204288.0*R**2*S**6*T**2*UGA_star**18 - 21131264.0*R**2*S**6*T**2*UGA_star**17*epsilon - 430080.0*R**2*S**6*T**2*UGA_star**17 + 27278336.0*R**2*S**6*T**2*UGA_star**16*epsilon + 602368.0*R**2*S**6*T**2*UGA_star**16 - 23762944.0*R**2*S**6*T**2*UGA_star**15*epsilon - 604160.0*R**2*S**6*T**2*UGA_star**15 + 14594048.0*R**2*S**6*T**2*UGA_star**14*epsilon + 458752.0*R**2*S**6*T**2*UGA_star**14 - 6608896.0*R**2*S**6*T**2*UGA_star**13*epsilon - 272384.0*R**2*S**6*T**2*UGA_star**13 + 2207744.0*R**2*S**6*T**2*UGA_star**12*epsilon + 125440.0*R**2*S**6*T**2*UGA_star**12 - 444416.0*R**2*S**6*T**2*UGA_star**11*epsilon - 43008.0*R**2*S**6*T**2*UGA_star**11 + 7168.0*R**2*S**6*T**2*UGA_star**10*epsilon + 10752.0*R**2*S**6*T**2*UGA_star**10 + 10240.0*R**2*S**6*T**2*UGA_star**9*epsilon - 2048.0*R**2*S**6*T**2*UGA_star**9 + 1024.0*R**2*S**6*T**2*UGA_star**8*epsilon + 256.0*R**2*S**6*T**2*UGA_star**8 + 817152.0*R**2*S**5*T**3*UGA_star**20*epsilon + 32256.0*R**2*S**5*T**3*UGA_star**20 - 6537216.0*R**2*S**5*T**3*UGA_star**19*epsilon - 258048.0*R**2*S**5*T**3*UGA_star**19 + 22775808.0*R**2*S**5*T**3*UGA_star**18*epsilon + 904704.0*R**2*S**5*T**3*UGA_star**18 - 45201408.0*R**2*S**5*T**3*UGA_star**17*epsilon - 1826304.0*R**2*S**5*T**3*UGA_star**17 + 56206336.0*R**2*S**5*T**3*UGA_star**16*epsilon + 2352128.0*R**2*S**5*T**3*UGA_star**16 - 45664256.0*R**2*S**5*T**3*UGA_star**15*epsilon - 2035712.0*R**2*S**5*T**3*UGA_star**15 + 25036800.0*R**2*S**5*T**3*UGA_star**14*epsilon + 1225728.0*R**2*S**5*T**3*UGA_star**14 - 9943040.0*R**2*S**5*T**3*UGA_star**13*epsilon - 523264.0*R**2*S**5*T**3*UGA_star**13 + 3217408.0*R**2*S**5*T**3*UGA_star**12*epsilon + 146944.0*R**2*S**5*T**3*UGA_star**12 - 792576.0*R**2*S**5*T**3*UGA_star**11*epsilon - 12288.0*R**2*S**5*T**3*UGA_star**11 + 77824.0*R**2*S**5*T**3*UGA_star**10*epsilon - 8704.0*R**2*S**5*T**3*UGA_star**10 + 4096.0*R**2*S**5*T**3*UGA_star**9*epsilon + 2560.0*R**2*S**5*T**3*UGA_star**9 + 3072.0*R**2*S**5*T**3*UGA_star**8*epsilon + 1029120.0*R**2*S**4*T**4*UGA_star**20*epsilon + 57600.0*R**2*S**4*T**4*UGA_star**20 - 8232960.0*R**2*S**4*T**4*UGA_star**19*epsilon - 460800.0*R**2*S**4*T**4*UGA_star**19 + 28538880.0*R**2*S**4*T**4*UGA_star**18*epsilon + 1597440.0*R**2*S**4*T**4*UGA_star**18 - 55818240.0*R**2*S**4*T**4*UGA_star**17*epsilon - 3125760.0*R**2*S**4*T**4*UGA_star**17 + 67108864.0*R**2*S**4*T**4*UGA_star**16*epsilon + 3762176.0*R**2*S**4*T**4*UGA_star**16 - 50601984.0*R**2*S**4*T**4*UGA_star**15*epsilon - 2840576.0*R**2*S**4*T**4*UGA_star**15 + 23500800.0*R**2*S**4*T**4*UGA_star**14*epsilon + 1313280.0*R**2*S**4*T**4*UGA_star**14 - 6594560.0*R**2*S**4*T**4*UGA_star**13*epsilon - 348160.0*R**2*S**4*T**4*UGA_star**13 + 1305600.0*R**2*S**4*T**4*UGA_star**12*epsilon + 44800.0*R**2*S**4*T**4*UGA_star**12 - 221184.0*R**2*S**4*T**4*UGA_star**11*epsilon + 6144.0*R**2*S**4*T**4*UGA_star**11 - 45056.0*R**2*S**4*T**4*UGA_star**10*epsilon - 8704.0*R**2*S**4*T**4*UGA_star**10 + 30720.0*R**2*S**4*T**4*UGA_star**9*epsilon + 2560.0*R**2*S**4*T**4*UGA_star**9 + 819200.0*R**2*S**3*T**5*UGA_star**20*epsilon + 51200.0*R**2*S**3*T**5*UGA_star**20 - 6553600.0*R**2*S**3*T**5*UGA_star**19*epsilon - 409600.0*R**2*S**3*T**5*UGA_star**19 + 22614016.0*R**2*S**3*T**5*UGA_star**18*epsilon + 1411072.0*R**2*S**3*T**5*UGA_star**18 - 43646976.0*R**2*S**3*T**5*UGA_star**17*epsilon - 2711552.0*R**2*S**3*T**5*UGA_star**17 + 50825216.0*R**2*S**3*T**5*UGA_star**16*epsilon + 3128320.0*R**2*S**3*T**5*UGA_star**16 - 35414016.0*R**2*S**3*T**5*UGA_star**15*epsilon - 2140160.0*R**2*S**3*T**5*UGA_star**15 + 13056000.0*R**2*S**3*T**5*UGA_star**14*epsilon + 762880.0*R**2*S**3*T**5*UGA_star**14 - 1110016.0*R**2*S**3*T**5*UGA_star**13*epsilon - 69632.0*R**2*S**3*T**5*UGA_star**13 - 804864.0*R**2*S**3*T**5*UGA_star**12*epsilon - 17408.0*R**2*S**3*T**5*UGA_star**12 + 262144.0*R**2*S**3*T**5*UGA_star**11*epsilon - 10240.0*R**2*S**3*T**5*UGA_star**11 - 67584.0*R**2*S**3*T**5*UGA_star**10*epsilon + 5120.0*R**2*S**3*T**5*UGA_star**10 + 20480.0*R**2*S**3*T**5*UGA_star**9*epsilon + 411648.0*R**2*S**2*T**6*UGA_star**20*epsilon + 23040.0*R**2*S**2*T**6*UGA_star**20 - 3293184.0*R**2*S**2*T**6*UGA_star**19*epsilon - 184320.0*R**2*S**2*T**6*UGA_star**19 + 11326464.0*R**2*S**2*T**6*UGA_star**18*epsilon + 634368.0*R**2*S**2*T**6*UGA_star**18 - 21657600.0*R**2*S**2*T**6*UGA_star**17*epsilon - 1214976.0*R**2*S**2*T**6*UGA_star**17 + 24662016.0*R**2*S**2*T**6*UGA_star**16*epsilon + 1388544.0*R**2*S**2*T**6*UGA_star**16 - 16238592.0*R**2*S**2*T**6*UGA_star**15*epsilon - 923136.0*R**2*S**2*T**6*UGA_star**15 + 4921344.0*R**2*S**2*T**6*UGA_star**14*epsilon + 291840.0*R**2*S**2*T**6*UGA_star**14 + 423936.0*R**2*S**2*T**6*UGA_star**13*epsilon + 10752.0*R**2*S**2*T**6*UGA_star**13 - 645120.0*R**2*S**2*T**6*UGA_star**12*epsilon - 29184.0*R**2*S**2*T**6*UGA_star**12 + 67584.0*R**2*S**2*T**6*UGA_star**11*epsilon + 1536.0*R**2*S**2*T**6*UGA_star**11 + 21504.0*R**2*S**2*T**6*UGA_star**10*epsilon + 1536.0*R**2*S**2*T**6*UGA_star**10 + 116736.0*R**2*S*T**7*UGA_star**20*epsilon + 4608.0*R**2*S*T**7*UGA_star**20 - 933888.0*R**2*S*T**7*UGA_star**19*epsilon - 36864.0*R**2*S*T**7*UGA_star**19 + 3213312.0*R**2*S*T**7*UGA_star**18*epsilon + 127488.0*R**2*S*T**7*UGA_star**18 - 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14336.0*R**2*T**8*UGA_star**15 + 229376.0*R**2*T**8*UGA_star**14*epsilon + 7168.0*R**2*T**8*UGA_star**14 - 20480.0*R**2*T**8*UGA_star**13*epsilon - 2048.0*R**2*T**8*UGA_star**13 - 15360.0*R**2*T**8*UGA_star**12*epsilon + 256.0*R**2*T**8*UGA_star**12 + 4096.0*R**2*T**8*UGA_star**11*epsilon - 18432.0*R*S**8*T*UGA_star**20*epsilon + 165888.0*R*S**8*T*UGA_star**19*epsilon - 666624.0*R*S**8*T*UGA_star**18*epsilon + 1575936.0*R*S**8*T*UGA_star**17*epsilon - 2433024.0*R*S**8*T*UGA_star**16*epsilon + 2580480.0*R*S**8*T*UGA_star**15*epsilon - 1935360.0*R*S**8*T*UGA_star**14*epsilon + 1050624.0*R*S**8*T*UGA_star**13*epsilon - 423936.0*R*S**8*T*UGA_star**12*epsilon + 129024.0*R*S**8*T*UGA_star**11*epsilon - 27648.0*R*S**8*T*UGA_star**10*epsilon + 3072.0*R*S**8*T*UGA_star**9*epsilon - 106496.0*R*S**7*T**2*UGA_star**20*epsilon - 2048.0*R*S**7*T**2*UGA_star**20 + 958464.0*R*S**7*T**2*UGA_star**19*epsilon + 18432.0*R*S**7*T**2*UGA_star**19 - 3820544.0*R*S**7*T**2*UGA_star**18*epsilon - 74240.0*R*S**7*T**2*UGA_star**18 + 8854528.0*R*S**7*T**2*UGA_star**17*epsilon + 176640.0*R*S**7*T**2*UGA_star**17 - 13165568.0*R*S**7*T**2*UGA_star**16*epsilon - 276480.0*R*S**7*T**2*UGA_star**16 + 13078528.0*R*S**7*T**2*UGA_star**15*epsilon + 301056.0*R*S**7*T**2*UGA_star**15 - 8787968.0*R*S**7*T**2*UGA_star**14*epsilon - 236544.0*R*S**7*T**2*UGA_star**14 + 3991552.0*R*S**7*T**2*UGA_star**13*epsilon + 138240.0*R*S**7*T**2*UGA_star**13 - 1230848.0*R*S**7*T**2*UGA_star**12*epsilon - 61440.0*R*S**7*T**2*UGA_star**12 + 258048.0*R*S**7*T**2*UGA_star**11*epsilon + 20480.0*R*S**7*T**2*UGA_star**11 - 23552.0*R*S**7*T**2*UGA_star**10*epsilon - 4608.0*R*S**7*T**2*UGA_star**10 - 9216.0*R*S**7*T**2*UGA_star**9*epsilon + 512.0*R*S**7*T**2*UGA_star**9 + 3072.0*R*S**7*T**2*UGA_star**8*epsilon - 272384.0*R*S**6*T**3*UGA_star**20*epsilon - 10752.0*R*S**6*T**3*UGA_star**20 + 2451456.0*R*S**6*T**3*UGA_star**19*epsilon + 96768.0*R*S**6*T**3*UGA_star**19 - 9723904.0*R*S**6*T**3*UGA_star**18*epsilon - 385536.0*R*S**6*T**3*UGA_star**18 + 22253568.0*R*S**6*T**3*UGA_star**17*epsilon + 892416.0*R*S**6*T**3*UGA_star**17 - 32261120.0*R*S**6*T**3*UGA_star**16*epsilon - 1323520.0*R*S**6*T**3*UGA_star**16 + 30566400.0*R*S**6*T**3*UGA_star**15*epsilon + 1307648.0*R*S**6*T**3*UGA_star**15 - 18823168.0*R*S**6*T**3*UGA_star**14*epsilon - 867328.0*R*S**6*T**3*UGA_star**14 + 7297024.0*R*S**6*T**3*UGA_star**13*epsilon + 379904.0*R*S**6*T**3*UGA_star**13 - 1763328.0*R*S**6*T**3*UGA_star**12*epsilon - 103936.0*R*S**6*T**3*UGA_star**12 + 348160.0*R*S**6*T**3*UGA_star**11*epsilon + 12800.0*R*S**6*T**3*UGA_star**11 - 75776.0*R*S**6*T**3*UGA_star**10*epsilon + 3584.0*R*S**6*T**3*UGA_star**10 - 2048.0*R*S**6*T**3*UGA_star**9*epsilon - 2560.0*R*S**6*T**3*UGA_star**9 + 5120.0*R*S**6*T**3*UGA_star**8*epsilon + 512.0*R*S**6*T**3*UGA_star**8 - 411648.0*R*S**5*T**4*UGA_star**20*epsilon - 23040.0*R*S**5*T**4*UGA_star**20 + 3704832.0*R*S**5*T**4*UGA_star**19*epsilon + 207360.0*R*S**5*T**4*UGA_star**19 - 14653440.0*R*S**5*T**4*UGA_star**18*epsilon - 820224.0*R*S**5*T**4*UGA_star**18 + 33275904.0*R*S**5*T**4*UGA_star**17*epsilon + 1863168.0*R*S**5*T**4*UGA_star**17 - 47436800.0*R*S**5*T**4*UGA_star**16*epsilon - 2657792.0*R*S**5*T**4*UGA_star**16 + 43380736.0*R*S**5*T**4*UGA_star**15*epsilon + 2433024.0*R*S**5*T**4*UGA_star**15 - 24657920.0*R*S**5*T**4*UGA_star**14*epsilon - 1383424.0*R*S**5*T**4*UGA_star**14 + 7708672.0*R*S**5*T**4*UGA_star**13*epsilon + 428032.0*R*S**5*T**4*UGA_star**13 - 794624.0*R*S**5*T**4*UGA_star**12*epsilon - 35328.0*R*S**5*T**4*UGA_star**12 - 112640.0*R*S**5*T**4*UGA_star**11*epsilon - 14848.0*R*S**5*T**4*UGA_star**11 - 2048.0*R*S**5*T**4*UGA_star**10*epsilon + 4096.0*R*S**5*T**4*UGA_star**10 - 8192.0*R*S**5*T**4*UGA_star**9*epsilon - 1536.0*R*S**5*T**4*UGA_star**9 + 7168.0*R*S**5*T**4*UGA_star**8*epsilon + 512.0*R*S**5*T**4*UGA_star**8 - 409600.0*R*S**4*T**5*UGA_star**20*epsilon - 25600.0*R*S**4*T**5*UGA_star**20 + 3686400.0*R*S**4*T**5*UGA_star**19*epsilon + 230400.0*R*S**4*T**5*UGA_star**19 - 14552064.0*R*S**4*T**5*UGA_star**18*epsilon - 908288.0*R*S**4*T**5*UGA_star**18 + 32867328.0*R*S**4*T**5*UGA_star**17*epsilon + 2044416.0*R*S**4*T**5*UGA_star**17 - 46283776.0*R*S**4*T**5*UGA_star**16*epsilon - 2859008.0*R*S**4*T**5*UGA_star**16 + 41160704.0*R*S**4*T**5*UGA_star**15*epsilon + 2508800.0*R*S**4*T**5*UGA_star**15 - 21747712.0*R*S**4*T**5*UGA_star**14*epsilon - 1290240.0*R*S**4*T**5*UGA_star**14 + 5126144.0*R*S**4*T**5*UGA_star**13*epsilon + 283648.0*R*S**4*T**5*UGA_star**13 + 714752.0*R*S**4*T**5*UGA_star**12*epsilon + 41984.0*R*S**4*T**5*UGA_star**12 - 651264.0*R*S**4*T**5*UGA_star**11*epsilon - 23552.0*R*S**4*T**5*UGA_star**11 + 91136.0*R*S**4*T**5*UGA_star**10*epsilon - 5120.0*R*S**4*T**5*UGA_star**10 - 7168.0*R*S**4*T**5*UGA_star**9*epsilon + 2560.0*R*S**4*T**5*UGA_star**9 + 5120.0*R*S**4*T**5*UGA_star**8*epsilon - 274432.0*R*S**3*T**6*UGA_star**20*epsilon - 15360.0*R*S**3*T**6*UGA_star**20 + 2469888.0*R*S**3*T**6*UGA_star**19*epsilon + 138240.0*R*S**3*T**6*UGA_star**19 - 9739264.0*R*S**3*T**6*UGA_star**18*epsilon - 545280.0*R*S**3*T**6*UGA_star**18 + 21931008.0*R*S**3*T**6*UGA_star**17*epsilon + 1228800.0*R*S**3*T**6*UGA_star**17 - 30674944.0*R*S**3*T**6*UGA_star**16*epsilon - 1721344.0*R*S**3*T**6*UGA_star**16 + 26865664.0*R*S**3*T**6*UGA_star**15*epsilon + 1512448.0*R*S**3*T**6*UGA_star**15 - 13633536.0*R*S**3*T**6*UGA_star**14*epsilon - 774144.0*R*S**3*T**6*UGA_star**14 + 2668544.0*R*S**3*T**6*UGA_star**13*epsilon + 158720.0*R*S**3*T**6*UGA_star**13 + 827392.0*R*S**3*T**6*UGA_star**12*epsilon + 40960.0*R*S**3*T**6*UGA_star**12 - 475136.0*R*S**3*T**6*UGA_star**11*epsilon - 24576.0*R*S**3*T**6*UGA_star**11 + 17408.0*R*S**3*T**6*UGA_star**10*epsilon + 512.0*R*S**3*T**6*UGA_star**10 + 17408.0*R*S**3*T**6*UGA_star**9*epsilon + 1024.0*R*S**3*T**6*UGA_star**9 - 116736.0*R*S**2*T**7*UGA_star**20*epsilon - 4608.0*R*S**2*T**7*UGA_star**20 + 1050624.0*R*S**2*T**7*UGA_star**19*epsilon + 41472.0*R*S**2*T**7*UGA_star**19 - 4147200.0*R*S**2*T**7*UGA_star**18*epsilon - 164352.0*R*S**2*T**7*UGA_star**18 + 9363456.0*R*S**2*T**7*UGA_star**17*epsilon + 374784.0*R*S**2*T**7*UGA_star**17 - 13166592.0*R*S**2*T**7*UGA_star**16*epsilon - 537600.0*R*S**2*T**7*UGA_star**16 + 11655168.0*R*S**2*T**7*UGA_star**15*epsilon + 494592.0*R*S**2*T**7*UGA_star**15 - 6064128.0*R*S**2*T**7*UGA_star**14*epsilon - 279552.0*R*S**2*T**7*UGA_star**14 + 1320960.0*R*S**2*T**7*UGA_star**13*epsilon + 79872.0*R*S**2*T**7*UGA_star**13 + 282624.0*R*S**2*T**7*UGA_star**12*epsilon + 1536.0*R*S**2*T**7*UGA_star**12 - 196608.0*R*S**2*T**7*UGA_star**11*epsilon - 7680.0*R*S**2*T**7*UGA_star**11 + 12288.0*R*S**2*T**7*UGA_star**10*epsilon + 1536.0*R*S**2*T**7*UGA_star**10 + 6144.0*R*S**2*T**7*UGA_star**9*epsilon - 26624.0*R*S*T**8*UGA_star**20*epsilon - 512.0*R*S*T**8*UGA_star**20 + 239616.0*R*S*T**8*UGA_star**19*epsilon + 4608.0*R*S*T**8*UGA_star**19 - 950272.0*R*S*T**8*UGA_star**18*epsilon - 18432.0*R*S*T**8*UGA_star**18 + 2170880.0*R*S*T**8*UGA_star**17*epsilon + 43008.0*R*S*T**8*UGA_star**17 - 3125248.0*R*S*T**8*UGA_star**16*epsilon - 64512.0*R*S*T**8*UGA_star**16 + 2895872.0*R*S*T**8*UGA_star**15*epsilon + 64512.0*R*S*T**8*UGA_star**15 - 1662976.0*R*S*T**8*UGA_star**14*epsilon - 43008.0*R*S*T**8*UGA_star**14 + 499712.0*R*S*T**8*UGA_star**13*epsilon + 18432.0*R*S*T**8*UGA_star**13 - 10240.0*R*S*T**8*UGA_star**12*epsilon - 4608.0*R*S*T**8*UGA_star**12 - 38912.0*R*S*T**8*UGA_star**11*epsilon + 512.0*R*S*T**8*UGA_star**11 + 8192.0*R*S*T**8*UGA_star**10*epsilon - 2048.0*R*T**9*UGA_star**20*epsilon + 18432.0*R*T**9*UGA_star**19*epsilon - 73728.0*R*T**9*UGA_star**18*epsilon + 172032.0*R*T**9*UGA_star**17*epsilon - 258048.0*R*T**9*UGA_star**16*epsilon + 258048.0*R*T**9*UGA_star**15*epsilon - 172032.0*R*T**9*UGA_star**14*epsilon + 73728.0*R*T**9*UGA_star**13*epsilon - 18432.0*R*T**9*UGA_star**12*epsilon + 2048.0*R*T**9*UGA_star**11*epsilon + 2048.0*S**9*T*UGA_star**20*epsilon - 20480.0*S**9*T*UGA_star**19*epsilon + 92160.0*S**9*T*UGA_star**18*epsilon - 245760.0*S**9*T*UGA_star**17*epsilon + 430080.0*S**9*T*UGA_star**16*epsilon - 516096.0*S**9*T*UGA_star**15*epsilon + 430080.0*S**9*T*UGA_star**14*epsilon - 245760.0*S**9*T*UGA_star**13*epsilon + 92160.0*S**9*T*UGA_star**12*epsilon - 20480.0*S**9*T*UGA_star**11*epsilon + 2048.0*S**9*T*UGA_star**10*epsilon + 13312.0*S**8*T**2*UGA_star**20*epsilon + 256.0*S**8*T**2*UGA_star**20 - 133120.0*S**8*T**2*UGA_star**19*epsilon - 2560.0*S**8*T**2*UGA_star**19 + 594944.0*S**8*T**2*UGA_star**18*epsilon + 11520.0*S**8*T**2*UGA_star**18 - 1560576.0*S**8*T**2*UGA_star**17*epsilon - 30720.0*S**8*T**2*UGA_star**17 + 2648064.0*S**8*T**2*UGA_star**16*epsilon + 53760.0*S**8*T**2*UGA_star**16 - 3010560.0*S**8*T**2*UGA_star**15*epsilon - 64512.0*S**8*T**2*UGA_star**15 + 2279424.0*S**8*T**2*UGA_star**14*epsilon + 53760.0*S**8*T**2*UGA_star**14 - 1081344.0*S**8*T**2*UGA_star**13*epsilon - 30720.0*S**8*T**2*UGA_star**13 + 254976.0*S**8*T**2*UGA_star**12*epsilon + 11520.0*S**8*T**2*UGA_star**12 + 14336.0*S**8*T**2*UGA_star**11*epsilon - 2560.0*S**8*T**2*UGA_star**11 - 23552.0*S**8*T**2*UGA_star**10*epsilon + 256.0*S**8*T**2*UGA_star**10 + 4096.0*S**8*T**2*UGA_star**9*epsilon + 38912.0*S**7*T**3*UGA_star**20*epsilon + 1536.0*S**7*T**3*UGA_star**20 - 389120.0*S**7*T**3*UGA_star**19*epsilon - 15360.0*S**7*T**3*UGA_star**19 + 1732608.0*S**7*T**3*UGA_star**18*epsilon + 68608.0*S**7*T**3*UGA_star**18 - 4503552.0*S**7*T**3*UGA_star**17*epsilon - 179712.0*S**7*T**3*UGA_star**17 + 7510016.0*S**7*T**3*UGA_star**16*epsilon + 304128.0*S**7*T**3*UGA_star**16 - 8273920.0*S**7*T**3*UGA_star**15*epsilon - 344064.0*S**7*T**3*UGA_star**15 + 5906432.0*S**7*T**3*UGA_star**14*epsilon + 258048.0*S**7*T**3*UGA_star**14 - 2461696.0*S**7*T**3*UGA_star**13*epsilon - 119808.0*S**7*T**3*UGA_star**13 + 346112.0*S**7*T**3*UGA_star**12*epsilon + 26112.0*S**7*T**3*UGA_star**12 + 159744.0*S**7*T**3*UGA_star**11*epsilon + 3072.0*S**7*T**3*UGA_star**11 - 69632.0*S**7*T**3*UGA_star**10*epsilon - 3072.0*S**7*T**3*UGA_star**10 + 2048.0*S**7*T**3*UGA_star**9*epsilon + 512.0*S**7*T**3*UGA_star**9 + 2048.0*S**7*T**3*UGA_star**8*epsilon + 68608.0*S**6*T**4*UGA_star**20*epsilon + 3840.0*S**6*T**4*UGA_star**20 - 686080.0*S**6*T**4*UGA_star**19*epsilon - 38400.0*S**6*T**4*UGA_star**19 + 3050496.0*S**6*T**4*UGA_star**18*epsilon + 170752.0*S**6*T**4*UGA_star**18 - 7901184.0*S**6*T**4*UGA_star**17*epsilon - 442368.0*S**6*T**4*UGA_star**17 + 13085696.0*S**6*T**4*UGA_star**16*epsilon + 732928.0*S**6*T**4*UGA_star**16 - 14233600.0*S**6*T**4*UGA_star**15*epsilon - 797696.0*S**6*T**4*UGA_star**15 + 9906176.0*S**6*T**4*UGA_star**14*epsilon + 555520.0*S**6*T**4*UGA_star**14 - 3874816.0*S**6*T**4*UGA_star**13*epsilon - 217088.0*S**6*T**4*UGA_star**13 + 349184.0*S**6*T**4*UGA_star**12*epsilon + 18688.0*S**6*T**4*UGA_star**12 + 354304.0*S**6*T**4*UGA_star**11*epsilon + 20992.0*S**6*T**4*UGA_star**11 - 119808.0*S**6*T**4*UGA_star**10*epsilon - 7424.0*S**6*T**4*UGA_star**10 - 4096.0*S**6*T**4*UGA_star**9*epsilon + 5120.0*S**6*T**4*UGA_star**8*epsilon + 256.0*S**6*T**4*UGA_star**8 + 81920.0*S**5*T**5*UGA_star**20*epsilon + 5120.0*S**5*T**5*UGA_star**20 - 819200.0*S**5*T**5*UGA_star**19*epsilon - 51200.0*S**5*T**5*UGA_star**19 + 3641344.0*S**5*T**5*UGA_star**18*epsilon + 227328.0*S**5*T**5*UGA_star**18 - 9424896.0*S**5*T**5*UGA_star**17*epsilon - 586752.0*S**5*T**5*UGA_star**17 + 15587328.0*S**5*T**5*UGA_star**16*epsilon + 965120.0*S**5*T**5*UGA_star**16 - 16908288.0*S**5*T**5*UGA_star**15*epsilon - 1036288.0*S**5*T**5*UGA_star**15 + 11698176.0*S**5*T**5*UGA_star**14*epsilon + 702464.0*S**5*T**5*UGA_star**14 - 4497408.0*S**5*T**5*UGA_star**13*epsilon - 256000.0*S**5*T**5*UGA_star**13 + 331776.0*S**5*T**5*UGA_star**12*epsilon + 8192.0*S**5*T**5*UGA_star**12 + 458752.0*S**5*T**5*UGA_star**11*epsilon + 30720.0*S**5*T**5*UGA_star**11 - 151552.0*S**5*T**5*UGA_star**10*epsilon - 8192.0*S**5*T**5*UGA_star**10 - 4096.0*S**5*T**5*UGA_star**9*epsilon - 1024.0*S**5*T**5*UGA_star**9 + 6144.0*S**5*T**5*UGA_star**8*epsilon + 512.0*S**5*T**5*UGA_star**8 + 68608.0*S**4*T**6*UGA_star**20*epsilon + 3840.0*S**4*T**6*UGA_star**20 - 686080.0*S**4*T**6*UGA_star**19*epsilon - 38400.0*S**4*T**6*UGA_star**19 + 3050496.0*S**4*T**6*UGA_star**18*epsilon + 170752.0*S**4*T**6*UGA_star**18 - 7901184.0*S**4*T**6*UGA_star**17*epsilon - 442368.0*S**4*T**6*UGA_star**17 + 13085696.0*S**4*T**6*UGA_star**16*epsilon + 732928.0*S**4*T**6*UGA_star**16 - 14233600.0*S**4*T**6*UGA_star**15*epsilon - 797696.0*S**4*T**6*UGA_star**15 + 9906176.0*S**4*T**6*UGA_star**14*epsilon + 555520.0*S**4*T**6*UGA_star**14 - 3874816.0*S**4*T**6*UGA_star**13*epsilon - 217088.0*S**4*T**6*UGA_star**13 + 349184.0*S**4*T**6*UGA_star**12*epsilon + 18688.0*S**4*T**6*UGA_star**12 + 354304.0*S**4*T**6*UGA_star**11*epsilon + 20992.0*S**4*T**6*UGA_star**11 - 119808.0*S**4*T**6*UGA_star**10*epsilon - 7424.0*S**4*T**6*UGA_star**10 - 4096.0*S**4*T**6*UGA_star**9*epsilon + 5120.0*S**4*T**6*UGA_star**8*epsilon + 256.0*S**4*T**6*UGA_star**8 + 38912.0*S**3*T**7*UGA_star**20*epsilon + 1536.0*S**3*T**7*UGA_star**20 - 389120.0*S**3*T**7*UGA_star**19*epsilon - 15360.0*S**3*T**7*UGA_star**19 + 1732608.0*S**3*T**7*UGA_star**18*epsilon + 68608.0*S**3*T**7*UGA_star**18 - 4503552.0*S**3*T**7*UGA_star**17*epsilon - 179712.0*S**3*T**7*UGA_star**17 + 7510016.0*S**3*T**7*UGA_star**16*epsilon + 304128.0*S**3*T**7*UGA_star**16 - 8273920.0*S**3*T**7*UGA_star**15*epsilon - 344064.0*S**3*T**7*UGA_star**15 + 5906432.0*S**3*T**7*UGA_star**14*epsilon + 258048.0*S**3*T**7*UGA_star**14 - 2461696.0*S**3*T**7*UGA_star**13*epsilon - 119808.0*S**3*T**7*UGA_star**13 + 346112.0*S**3*T**7*UGA_star**12*epsilon + 26112.0*S**3*T**7*UGA_star**12 + 159744.0*S**3*T**7*UGA_star**11*epsilon + 3072.0*S**3*T**7*UGA_star**11 - 69632.0*S**3*T**7*UGA_star**10*epsilon - 3072.0*S**3*T**7*UGA_star**10 + 2048.0*S**3*T**7*UGA_star**9*epsilon + 512.0*S**3*T**7*UGA_star**9 + 2048.0*S**3*T**7*UGA_star**8*epsilon + 13312.0*S**2*T**8*UGA_star**20*epsilon + 256.0*S**2*T**8*UGA_star**20 - 133120.0*S**2*T**8*UGA_star**19*epsilon - 2560.0*S**2*T**8*UGA_star**19 + 594944.0*S**2*T**8*UGA_star**18*epsilon + 11520.0*S**2*T**8*UGA_star**18 - 1560576.0*S**2*T**8*UGA_star**17*epsilon - 30720.0*S**2*T**8*UGA_star**17 + 2648064.0*S**2*T**8*UGA_star**16*epsilon + 53760.0*S**2*T**8*UGA_star**16 - 3010560.0*S**2*T**8*UGA_star**15*epsilon - 64512.0*S**2*T**8*UGA_star**15 + 2279424.0*S**2*T**8*UGA_star**14*epsilon + 53760.0*S**2*T**8*UGA_star**14 - 1081344.0*S**2*T**8*UGA_star**13*epsilon - 30720.0*S**2*T**8*UGA_star**13 + 254976.0*S**2*T**8*UGA_star**12*epsilon + 11520.0*S**2*T**8*UGA_star**12 + 14336.0*S**2*T**8*UGA_star**11*epsilon - 2560.0*S**2*T**8*UGA_star**11 - 23552.0*S**2*T**8*UGA_star**10*epsilon + 256.0*S**2*T**8*UGA_star**10 + 4096.0*S**2*T**8*UGA_star**9*epsilon + 2048.0*S*T**9*UGA_star**20*epsilon - 20480.0*S*T**9*UGA_star**19*epsilon + 92160.0*S*T**9*UGA_star**18*epsilon - 245760.0*S*T**9*UGA_star**17*epsilon + 430080.0*S*T**9*UGA_star**16*epsilon - 516096.0*S*T**9*UGA_star**15*epsilon + 430080.0*S*T**9*UGA_star**14*epsilon - 245760.0*S*T**9*UGA_star**13*epsilon + 92160.0*S*T**9*UGA_star**12*epsilon - 20480.0*S*T**9*UGA_star**11*epsilon + 2048.0*S*T**9*UGA_star**10*epsilon))/(-256.0*P**5*UGA_star**10*epsilon + 2560.0*P**5*UGA_star**9*epsilon - 11520.0*P**5*UGA_star**8*epsilon + 30720.0*P**5*UGA_star**7*epsilon - 53760.0*P**5*UGA_star**6*epsilon + 64512.0*P**5*UGA_star**5*epsilon - 53760.0*P**5*UGA_star**4*epsilon + 30720.0*P**5*UGA_star**3*epsilon - 11520.0*P**5*UGA_star**2*epsilon + 2560.0*P**5*UGA_star*epsilon - 256.0*P**5*epsilon - 1280.0*P**4*R*UGA_star**10*epsilon - 128.0*P**4*R*UGA_star**10 + 10240.0*P**4*R*UGA_star**9*epsilon + 1024.0*P**4*R*UGA_star**9 - 35840.0*P**4*R*UGA_star**8*epsilon - 3584.0*P**4*R*UGA_star**8 + 71680.0*P**4*R*UGA_star**7*epsilon + 7168.0*P**4*R*UGA_star**7 - 89600.0*P**4*R*UGA_star**6*epsilon - 8960.0*P**4*R*UGA_star**6 + 71680.0*P**4*R*UGA_star**5*epsilon + 7168.0*P**4*R*UGA_star**5 - 35840.0*P**4*R*UGA_star**4*epsilon - 3584.0*P**4*R*UGA_star**4 + 10240.0*P**4*R*UGA_star**3*epsilon + 1024.0*P**4*R*UGA_star**3 - 1280.0*P**4*R*UGA_star**2*epsilon - 128.0*P**4*R*UGA_star**2 + 1280.0*P**4*S*UGA_star**10*epsilon + 128.0*P**4*S*UGA_star**10 - 11520.0*P**4*S*UGA_star**9*epsilon - 1152.0*P**4*S*UGA_star**9 + 46080.0*P**4*S*UGA_star**8*epsilon + 4608.0*P**4*S*UGA_star**8 - 107520.0*P**4*S*UGA_star**7*epsilon - 10752.0*P**4*S*UGA_star**7 + 161280.0*P**4*S*UGA_star**6*epsilon + 16128.0*P**4*S*UGA_star**6 - 161280.0*P**4*S*UGA_star**5*epsilon - 16128.0*P**4*S*UGA_star**5 + 107520.0*P**4*S*UGA_star**4*epsilon + 10752.0*P**4*S*UGA_star**4 - 46080.0*P**4*S*UGA_star**3*epsilon - 4608.0*P**4*S*UGA_star**3 + 11520.0*P**4*S*UGA_star**2*epsilon + 1152.0*P**4*S*UGA_star**2 - 1280.0*P**4*S*UGA_star*epsilon - 128.0*P**4*S*UGA_star + 1280.0*P**4*T*UGA_star**10*epsilon - 11520.0*P**4*T*UGA_star**9*epsilon + 46080.0*P**4*T*UGA_star**8*epsilon - 107520.0*P**4*T*UGA_star**7*epsilon + 161280.0*P**4*T*UGA_star**6*epsilon - 161280.0*P**4*T*UGA_star**5*epsilon + 107520.0*P**4*T*UGA_star**4*epsilon - 46080.0*P**4*T*UGA_star**3*epsilon + 11520.0*P**4*T*UGA_star**2*epsilon - 1280.0*P**4*T*UGA_star*epsilon - 2560.0*P**3*R**2*UGA_star**10*epsilon - 384.0*P**3*R**2*UGA_star**10 + 15360.0*P**3*R**2*UGA_star**9*epsilon + 2304.0*P**3*R**2*UGA_star**9 - 38400.0*P**3*R**2*UGA_star**8*epsilon - 5760.0*P**3*R**2*UGA_star**8 + 51200.0*P**3*R**2*UGA_star**7*epsilon + 7680.0*P**3*R**2*UGA_star**7 - 38400.0*P**3*R**2*UGA_star**6*epsilon - 5760.0*P**3*R**2*UGA_star**6 + 15360.0*P**3*R**2*UGA_star**5*epsilon + 2304.0*P**3*R**2*UGA_star**5 - 2560.0*P**3*R**2*UGA_star**4*epsilon - 384.0*P**3*R**2*UGA_star**4 + 5120.0*P**3*R*S*UGA_star**10*epsilon + 768.0*P**3*R*S*UGA_star**10 - 35840.0*P**3*R*S*UGA_star**9*epsilon - 5376.0*P**3*R*S*UGA_star**9 + 107520.0*P**3*R*S*UGA_star**8*epsilon + 16128.0*P**3*R*S*UGA_star**8 - 179200.0*P**3*R*S*UGA_star**7*epsilon - 26880.0*P**3*R*S*UGA_star**7 + 179200.0*P**3*R*S*UGA_star**6*epsilon + 26880.0*P**3*R*S*UGA_star**6 - 107520.0*P**3*R*S*UGA_star**5*epsilon - 16128.0*P**3*R*S*UGA_star**5 + 35840.0*P**3*R*S*UGA_star**4*epsilon + 5376.0*P**3*R*S*UGA_star**4 - 5120.0*P**3*R*S*UGA_star**3*epsilon - 768.0*P**3*R*S*UGA_star**3 + 5120.0*P**3*R*T*UGA_star**10*epsilon + 512.0*P**3*R*T*UGA_star**10 - 35840.0*P**3*R*T*UGA_star**9*epsilon - 3584.0*P**3*R*T*UGA_star**9 + 107520.0*P**3*R*T*UGA_star**8*epsilon + 10752.0*P**3*R*T*UGA_star**8 - 179200.0*P**3*R*T*UGA_star**7*epsilon - 17920.0*P**3*R*T*UGA_star**7 + 179200.0*P**3*R*T*UGA_star**6*epsilon + 17920.0*P**3*R*T*UGA_star**6 - 107520.0*P**3*R*T*UGA_star**5*epsilon - 10752.0*P**3*R*T*UGA_star**5 + 35840.0*P**3*R*T*UGA_star**4*epsilon + 3584.0*P**3*R*T*UGA_star**4 - 5120.0*P**3*R*T*UGA_star**3*epsilon - 512.0*P**3*R*T*UGA_star**3 - 2560.0*P**3*S**2*UGA_star**10*epsilon - 384.0*P**3*S**2*UGA_star**10 + 20480.0*P**3*S**2*UGA_star**9*epsilon + 3072.0*P**3*S**2*UGA_star**9 - 71680.0*P**3*S**2*UGA_star**8*epsilon - 10752.0*P**3*S**2*UGA_star**8 + 143360.0*P**3*S**2*UGA_star**7*epsilon + 21504.0*P**3*S**2*UGA_star**7 - 179200.0*P**3*S**2*UGA_star**6*epsilon - 26880.0*P**3*S**2*UGA_star**6 + 143360.0*P**3*S**2*UGA_star**5*epsilon + 21504.0*P**3*S**2*UGA_star**5 - 71680.0*P**3*S**2*UGA_star**4*epsilon - 10752.0*P**3*S**2*UGA_star**4 + 20480.0*P**3*S**2*UGA_star**3*epsilon + 3072.0*P**3*S**2*UGA_star**3 - 2560.0*P**3*S**2*UGA_star**2*epsilon - 384.0*P**3*S**2*UGA_star**2 - 5120.0*P**3*S*T*UGA_star**10*epsilon - 512.0*P**3*S*T*UGA_star**10 + 40960.0*P**3*S*T*UGA_star**9*epsilon + 4096.0*P**3*S*T*UGA_star**9 - 143360.0*P**3*S*T*UGA_star**8*epsilon - 14336.0*P**3*S*T*UGA_star**8 + 286720.0*P**3*S*T*UGA_star**7*epsilon + 28672.0*P**3*S*T*UGA_star**7 - 358400.0*P**3*S*T*UGA_star**6*epsilon - 35840.0*P**3*S*T*UGA_star**6 + 286720.0*P**3*S*T*UGA_star**5*epsilon + 28672.0*P**3*S*T*UGA_star**5 - 143360.0*P**3*S*T*UGA_star**4*epsilon - 14336.0*P**3*S*T*UGA_star**4 + 40960.0*P**3*S*T*UGA_star**3*epsilon + 4096.0*P**3*S*T*UGA_star**3 - 5120.0*P**3*S*T*UGA_star**2*epsilon - 512.0*P**3*S*T*UGA_star**2 - 2560.0*P**3*T**2*UGA_star**10*epsilon + 20480.0*P**3*T**2*UGA_star**9*epsilon - 71680.0*P**3*T**2*UGA_star**8*epsilon + 143360.0*P**3*T**2*UGA_star**7*epsilon - 179200.0*P**3*T**2*UGA_star**6*epsilon + 143360.0*P**3*T**2*UGA_star**5*epsilon - 71680.0*P**3*T**2*UGA_star**4*epsilon + 20480.0*P**3*T**2*UGA_star**3*epsilon - 2560.0*P**3*T**2*UGA_star**2*epsilon - 2560.0*P**2*R**3*UGA_star**10*epsilon - 384.0*P**2*R**3*UGA_star**10 + 10240.0*P**2*R**3*UGA_star**9*epsilon + 1536.0*P**2*R**3*UGA_star**9 - 15360.0*P**2*R**3*UGA_star**8*epsilon - 2304.0*P**2*R**3*UGA_star**8 + 10240.0*P**2*R**3*UGA_star**7*epsilon + 1536.0*P**2*R**3*UGA_star**7 - 2560.0*P**2*R**3*UGA_star**6*epsilon - 384.0*P**2*R**3*UGA_star**6 + 7680.0*P**2*R**2*S*UGA_star**10*epsilon + 1152.0*P**2*R**2*S*UGA_star**10 - 38400.0*P**2*R**2*S*UGA_star**9*epsilon - 5760.0*P**2*R**2*S*UGA_star**9 + 76800.0*P**2*R**2*S*UGA_star**8*epsilon + 11520.0*P**2*R**2*S*UGA_star**8 - 76800.0*P**2*R**2*S*UGA_star**7*epsilon - 11520.0*P**2*R**2*S*UGA_star**7 + 38400.0*P**2*R**2*S*UGA_star**6*epsilon + 5760.0*P**2*R**2*S*UGA_star**6 - 7680.0*P**2*R**2*S*UGA_star**5*epsilon - 1152.0*P**2*R**2*S*UGA_star**5 + 7680.0*P**2*R**2*T*UGA_star**10*epsilon + 1152.0*P**2*R**2*T*UGA_star**10 - 38400.0*P**2*R**2*T*UGA_star**9*epsilon - 5760.0*P**2*R**2*T*UGA_star**9 + 76800.0*P**2*R**2*T*UGA_star**8*epsilon + 11520.0*P**2*R**2*T*UGA_star**8 - 76800.0*P**2*R**2*T*UGA_star**7*epsilon - 11520.0*P**2*R**2*T*UGA_star**7 + 38400.0*P**2*R**2*T*UGA_star**6*epsilon + 5760.0*P**2*R**2*T*UGA_star**6 - 7680.0*P**2*R**2*T*UGA_star**5*epsilon - 1152.0*P**2*R**2*T*UGA_star**5 - 7680.0*P**2*R*S**2*UGA_star**10*epsilon - 1152.0*P**2*R*S**2*UGA_star**10 + 46080.0*P**2*R*S**2*UGA_star**9*epsilon + 6912.0*P**2*R*S**2*UGA_star**9 - 115200.0*P**2*R*S**2*UGA_star**8*epsilon - 17280.0*P**2*R*S**2*UGA_star**8 + 153600.0*P**2*R*S**2*UGA_star**7*epsilon + 23040.0*P**2*R*S**2*UGA_star**7 - 115200.0*P**2*R*S**2*UGA_star**6*epsilon - 17280.0*P**2*R*S**2*UGA_star**6 + 46080.0*P**2*R*S**2*UGA_star**5*epsilon + 6912.0*P**2*R*S**2*UGA_star**5 - 7680.0*P**2*R*S**2*UGA_star**4*epsilon - 1152.0*P**2*R*S**2*UGA_star**4 - 15360.0*P**2*R*S*T*UGA_star**10*epsilon - 2304.0*P**2*R*S*T*UGA_star**10 + 92160.0*P**2*R*S*T*UGA_star**9*epsilon + 13824.0*P**2*R*S*T*UGA_star**9 - 230400.0*P**2*R*S*T*UGA_star**8*epsilon - 34560.0*P**2*R*S*T*UGA_star**8 + 307200.0*P**2*R*S*T*UGA_star**7*epsilon + 46080.0*P**2*R*S*T*UGA_star**7 - 230400.0*P**2*R*S*T*UGA_star**6*epsilon - 34560.0*P**2*R*S*T*UGA_star**6 + 92160.0*P**2*R*S*T*UGA_star**5*epsilon + 13824.0*P**2*R*S*T*UGA_star**5 - 15360.0*P**2*R*S*T*UGA_star**4*epsilon - 2304.0*P**2*R*S*T*UGA_star**4 - 7680.0*P**2*R*T**2*UGA_star**10*epsilon - 768.0*P**2*R*T**2*UGA_star**10 + 46080.0*P**2*R*T**2*UGA_star**9*epsilon + 4608.0*P**2*R*T**2*UGA_star**9 - 115200.0*P**2*R*T**2*UGA_star**8*epsilon - 11520.0*P**2*R*T**2*UGA_star**8 + 153600.0*P**2*R*T**2*UGA_star**7*epsilon + 15360.0*P**2*R*T**2*UGA_star**7 - 115200.0*P**2*R*T**2*UGA_star**6*epsilon - 11520.0*P**2*R*T**2*UGA_star**6 + 46080.0*P**2*R*T**2*UGA_star**5*epsilon + 4608.0*P**2*R*T**2*UGA_star**5 - 7680.0*P**2*R*T**2*UGA_star**4*epsilon - 768.0*P**2*R*T**2*UGA_star**4 + 2560.0*P**2*S**3*UGA_star**10*epsilon + 384.0*P**2*S**3*UGA_star**10 - 17920.0*P**2*S**3*UGA_star**9*epsilon - 2688.0*P**2*S**3*UGA_star**9 + 53760.0*P**2*S**3*UGA_star**8*epsilon + 8064.0*P**2*S**3*UGA_star**8 - 89600.0*P**2*S**3*UGA_star**7*epsilon - 13440.0*P**2*S**3*UGA_star**7 + 89600.0*P**2*S**3*UGA_star**6*epsilon + 13440.0*P**2*S**3*UGA_star**6 - 53760.0*P**2*S**3*UGA_star**5*epsilon - 8064.0*P**2*S**3*UGA_star**5 + 17920.0*P**2*S**3*UGA_star**4*epsilon + 2688.0*P**2*S**3*UGA_star**4 - 2560.0*P**2*S**3*UGA_star**3*epsilon - 384.0*P**2*S**3*UGA_star**3 + 7680.0*P**2*S**2*T*UGA_star**10*epsilon + 1152.0*P**2*S**2*T*UGA_star**10 - 53760.0*P**2*S**2*T*UGA_star**9*epsilon - 8064.0*P**2*S**2*T*UGA_star**9 + 161280.0*P**2*S**2*T*UGA_star**8*epsilon + 24192.0*P**2*S**2*T*UGA_star**8 - 268800.0*P**2*S**2*T*UGA_star**7*epsilon - 40320.0*P**2*S**2*T*UGA_star**7 + 268800.0*P**2*S**2*T*UGA_star**6*epsilon + 40320.0*P**2*S**2*T*UGA_star**6 - 161280.0*P**2*S**2*T*UGA_star**5*epsilon - 24192.0*P**2*S**2*T*UGA_star**5 + 53760.0*P**2*S**2*T*UGA_star**4*epsilon + 8064.0*P**2*S**2*T*UGA_star**4 - 7680.0*P**2*S**2*T*UGA_star**3*epsilon - 1152.0*P**2*S**2*T*UGA_star**3 + 7680.0*P**2*S*T**2*UGA_star**10*epsilon + 768.0*P**2*S*T**2*UGA_star**10 - 53760.0*P**2*S*T**2*UGA_star**9*epsilon - 5376.0*P**2*S*T**2*UGA_star**9 + 161280.0*P**2*S*T**2*UGA_star**8*epsilon + 16128.0*P**2*S*T**2*UGA_star**8 - 268800.0*P**2*S*T**2*UGA_star**7*epsilon - 26880.0*P**2*S*T**2*UGA_star**7 + 268800.0*P**2*S*T**2*UGA_star**6*epsilon + 26880.0*P**2*S*T**2*UGA_star**6 - 161280.0*P**2*S*T**2*UGA_star**5*epsilon - 16128.0*P**2*S*T**2*UGA_star**5 + 53760.0*P**2*S*T**2*UGA_star**4*epsilon + 5376.0*P**2*S*T**2*UGA_star**4 - 7680.0*P**2*S*T**2*UGA_star**3*epsilon - 768.0*P**2*S*T**2*UGA_star**3 + 2560.0*P**2*T**3*UGA_star**10*epsilon - 17920.0*P**2*T**3*UGA_star**9*epsilon + 53760.0*P**2*T**3*UGA_star**8*epsilon - 89600.0*P**2*T**3*UGA_star**7*epsilon + 89600.0*P**2*T**3*UGA_star**6*epsilon - 53760.0*P**2*T**3*UGA_star**5*epsilon + 17920.0*P**2*T**3*UGA_star**4*epsilon - 2560.0*P**2*T**3*UGA_star**3*epsilon - 1280.0*P*R**4*UGA_star**10*epsilon - 128.0*P*R**4*UGA_star**10 + 2560.0*P*R**4*UGA_star**9*epsilon + 256.0*P*R**4*UGA_star**9 - 1280.0*P*R**4*UGA_star**8*epsilon - 128.0*P*R**4*UGA_star**8 + 5120.0*P*R**3*S*UGA_star**10*epsilon + 512.0*P*R**3*S*UGA_star**10 - 15360.0*P*R**3*S*UGA_star**9*epsilon - 1536.0*P*R**3*S*UGA_star**9 + 15360.0*P*R**3*S*UGA_star**8*epsilon + 1536.0*P*R**3*S*UGA_star**8 - 5120.0*P*R**3*S*UGA_star**7*epsilon - 512.0*P*R**3*S*UGA_star**7 + 5120.0*P*R**3*T*UGA_star**10*epsilon + 768.0*P*R**3*T*UGA_star**10 - 15360.0*P*R**3*T*UGA_star**9*epsilon - 2304.0*P*R**3*T*UGA_star**9 + 15360.0*P*R**3*T*UGA_star**8*epsilon + 2304.0*P*R**3*T*UGA_star**8 - 5120.0*P*R**3*T*UGA_star**7*epsilon - 768.0*P*R**3*T*UGA_star**7 - 7680.0*P*R**2*S**2*UGA_star**10*epsilon - 768.0*P*R**2*S**2*UGA_star**10 + 30720.0*P*R**2*S**2*UGA_star**9*epsilon + 3072.0*P*R**2*S**2*UGA_star**9 - 46080.0*P*R**2*S**2*UGA_star**8*epsilon - 4608.0*P*R**2*S**2*UGA_star**8 + 30720.0*P*R**2*S**2*UGA_star**7*epsilon + 3072.0*P*R**2*S**2*UGA_star**7 - 7680.0*P*R**2*S**2*UGA_star**6*epsilon - 768.0*P*R**2*S**2*UGA_star**6 - 15360.0*P*R**2*S*T*UGA_star**10*epsilon - 2304.0*P*R**2*S*T*UGA_star**10 + 61440.0*P*R**2*S*T*UGA_star**9*epsilon + 9216.0*P*R**2*S*T*UGA_star**9 - 92160.0*P*R**2*S*T*UGA_star**8*epsilon - 13824.0*P*R**2*S*T*UGA_star**8 + 61440.0*P*R**2*S*T*UGA_star**7*epsilon + 9216.0*P*R**2*S*T*UGA_star**7 - 15360.0*P*R**2*S*T*UGA_star**6*epsilon - 2304.0*P*R**2*S*T*UGA_star**6 - 7680.0*P*R**2*T**2*UGA_star**10*epsilon - 1152.0*P*R**2*T**2*UGA_star**10 + 30720.0*P*R**2*T**2*UGA_star**9*epsilon + 4608.0*P*R**2*T**2*UGA_star**9 - 46080.0*P*R**2*T**2*UGA_star**8*epsilon - 6912.0*P*R**2*T**2*UGA_star**8 + 30720.0*P*R**2*T**2*UGA_star**7*epsilon + 4608.0*P*R**2*T**2*UGA_star**7 - 7680.0*P*R**2*T**2*UGA_star**6*epsilon - 1152.0*P*R**2*T**2*UGA_star**6 + 5120.0*P*R*S**3*UGA_star**10*epsilon + 512.0*P*R*S**3*UGA_star**10 - 25600.0*P*R*S**3*UGA_star**9*epsilon - 2560.0*P*R*S**3*UGA_star**9 + 51200.0*P*R*S**3*UGA_star**8*epsilon + 5120.0*P*R*S**3*UGA_star**8 - 51200.0*P*R*S**3*UGA_star**7*epsilon - 5120.0*P*R*S**3*UGA_star**7 + 25600.0*P*R*S**3*UGA_star**6*epsilon + 2560.0*P*R*S**3*UGA_star**6 - 5120.0*P*R*S**3*UGA_star**5*epsilon - 512.0*P*R*S**3*UGA_star**5 + 15360.0*P*R*S**2*T*UGA_star**10*epsilon + 2304.0*P*R*S**2*T*UGA_star**10 - 76800.0*P*R*S**2*T*UGA_star**9*epsilon - 11520.0*P*R*S**2*T*UGA_star**9 + 153600.0*P*R*S**2*T*UGA_star**8*epsilon + 23040.0*P*R*S**2*T*UGA_star**8 - 153600.0*P*R*S**2*T*UGA_star**7*epsilon - 23040.0*P*R*S**2*T*UGA_star**7 + 76800.0*P*R*S**2*T*UGA_star**6*epsilon + 11520.0*P*R*S**2*T*UGA_star**6 - 15360.0*P*R*S**2*T*UGA_star**5*epsilon - 2304.0*P*R*S**2*T*UGA_star**5 + 15360.0*P*R*S*T**2*UGA_star**10*epsilon + 2304.0*P*R*S*T**2*UGA_star**10 - 76800.0*P*R*S*T**2*UGA_star**9*epsilon - 11520.0*P*R*S*T**2*UGA_star**9 + 153600.0*P*R*S*T**2*UGA_star**8*epsilon + 23040.0*P*R*S*T**2*UGA_star**8 - 153600.0*P*R*S*T**2*UGA_star**7*epsilon - 23040.0*P*R*S*T**2*UGA_star**7 + 76800.0*P*R*S*T**2*UGA_star**6*epsilon + 11520.0*P*R*S*T**2*UGA_star**6 - 15360.0*P*R*S*T**2*UGA_star**5*epsilon - 2304.0*P*R*S*T**2*UGA_star**5 + 5120.0*P*R*T**3*UGA_star**10*epsilon + 512.0*P*R*T**3*UGA_star**10 - 25600.0*P*R*T**3*UGA_star**9*epsilon - 2560.0*P*R*T**3*UGA_star**9 + 51200.0*P*R*T**3*UGA_star**8*epsilon + 5120.0*P*R*T**3*UGA_star**8 - 51200.0*P*R*T**3*UGA_star**7*epsilon - 5120.0*P*R*T**3*UGA_star**7 + 25600.0*P*R*T**3*UGA_star**6*epsilon + 2560.0*P*R*T**3*UGA_star**6 - 5120.0*P*R*T**3*UGA_star**5*epsilon - 512.0*P*R*T**3*UGA_star**5 - 1280.0*P*S**4*UGA_star**10*epsilon - 128.0*P*S**4*UGA_star**10 + 7680.0*P*S**4*UGA_star**9*epsilon + 768.0*P*S**4*UGA_star**9 - 19200.0*P*S**4*UGA_star**8*epsilon - 1920.0*P*S**4*UGA_star**8 + 25600.0*P*S**4*UGA_star**7*epsilon + 2560.0*P*S**4*UGA_star**7 - 19200.0*P*S**4*UGA_star**6*epsilon - 1920.0*P*S**4*UGA_star**6 + 7680.0*P*S**4*UGA_star**5*epsilon + 768.0*P*S**4*UGA_star**5 - 1280.0*P*S**4*UGA_star**4*epsilon - 128.0*P*S**4*UGA_star**4 - 5120.0*P*S**3*T*UGA_star**10*epsilon - 768.0*P*S**3*T*UGA_star**10 + 30720.0*P*S**3*T*UGA_star**9*epsilon + 4608.0*P*S**3*T*UGA_star**9 - 76800.0*P*S**3*T*UGA_star**8*epsilon - 11520.0*P*S**3*T*UGA_star**8 + 102400.0*P*S**3*T*UGA_star**7*epsilon + 15360.0*P*S**3*T*UGA_star**7 - 76800.0*P*S**3*T*UGA_star**6*epsilon - 11520.0*P*S**3*T*UGA_star**6 + 30720.0*P*S**3*T*UGA_star**5*epsilon + 4608.0*P*S**3*T*UGA_star**5 - 5120.0*P*S**3*T*UGA_star**4*epsilon - 768.0*P*S**3*T*UGA_star**4 - 7680.0*P*S**2*T**2*UGA_star**10*epsilon - 1152.0*P*S**2*T**2*UGA_star**10 + 46080.0*P*S**2*T**2*UGA_star**9*epsilon + 6912.0*P*S**2*T**2*UGA_star**9 - 115200.0*P*S**2*T**2*UGA_star**8*epsilon - 17280.0*P*S**2*T**2*UGA_star**8 + 153600.0*P*S**2*T**2*UGA_star**7*epsilon + 23040.0*P*S**2*T**2*UGA_star**7 - 115200.0*P*S**2*T**2*UGA_star**6*epsilon - 17280.0*P*S**2*T**2*UGA_star**6 + 46080.0*P*S**2*T**2*UGA_star**5*epsilon + 6912.0*P*S**2*T**2*UGA_star**5 - 7680.0*P*S**2*T**2*UGA_star**4*epsilon - 1152.0*P*S**2*T**2*UGA_star**4 - 5120.0*P*S*T**3*UGA_star**10*epsilon - 512.0*P*S*T**3*UGA_star**10 + 30720.0*P*S*T**3*UGA_star**9*epsilon + 3072.0*P*S*T**3*UGA_star**9 - 76800.0*P*S*T**3*UGA_star**8*epsilon - 7680.0*P*S*T**3*UGA_star**8 + 102400.0*P*S*T**3*UGA_star**7*epsilon + 10240.0*P*S*T**3*UGA_star**7 - 76800.0*P*S*T**3*UGA_star**6*epsilon - 7680.0*P*S*T**3*UGA_star**6 + 30720.0*P*S*T**3*UGA_star**5*epsilon + 3072.0*P*S*T**3*UGA_star**5 - 5120.0*P*S*T**3*UGA_star**4*epsilon - 512.0*P*S*T**3*UGA_star**4 - 1280.0*P*T**4*UGA_star**10*epsilon + 7680.0*P*T**4*UGA_star**9*epsilon - 19200.0*P*T**4*UGA_star**8*epsilon + 25600.0*P*T**4*UGA_star**7*epsilon - 19200.0*P*T**4*UGA_star**6*epsilon + 7680.0*P*T**4*UGA_star**5*epsilon - 1280.0*P*T**4*UGA_star**4*epsilon - 256.0*R**5*UGA_star**10*epsilon + 1280.0*R**4*S*UGA_star**10*epsilon - 1280.0*R**4*S*UGA_star**9*epsilon + 1280.0*R**4*T*UGA_star**10*epsilon + 128.0*R**4*T*UGA_star**10 - 1280.0*R**4*T*UGA_star**9*epsilon - 128.0*R**4*T*UGA_star**9 - 2560.0*R**3*S**2*UGA_star**10*epsilon + 5120.0*R**3*S**2*UGA_star**9*epsilon - 2560.0*R**3*S**2*UGA_star**8*epsilon - 5120.0*R**3*S*T*UGA_star**10*epsilon - 512.0*R**3*S*T*UGA_star**10 + 10240.0*R**3*S*T*UGA_star**9*epsilon + 1024.0*R**3*S*T*UGA_star**9 - 5120.0*R**3*S*T*UGA_star**8*epsilon - 512.0*R**3*S*T*UGA_star**8 - 2560.0*R**3*T**2*UGA_star**10*epsilon - 384.0*R**3*T**2*UGA_star**10 + 5120.0*R**3*T**2*UGA_star**9*epsilon + 768.0*R**3*T**2*UGA_star**9 - 2560.0*R**3*T**2*UGA_star**8*epsilon - 384.0*R**3*T**2*UGA_star**8 + 2560.0*R**2*S**3*UGA_star**10*epsilon - 7680.0*R**2*S**3*UGA_star**9*epsilon + 7680.0*R**2*S**3*UGA_star**8*epsilon - 2560.0*R**2*S**3*UGA_star**7*epsilon + 7680.0*R**2*S**2*T*UGA_star**10*epsilon + 768.0*R**2*S**2*T*UGA_star**10 - 23040.0*R**2*S**2*T*UGA_star**9*epsilon - 2304.0*R**2*S**2*T*UGA_star**9 + 23040.0*R**2*S**2*T*UGA_star**8*epsilon + 2304.0*R**2*S**2*T*UGA_star**8 - 7680.0*R**2*S**2*T*UGA_star**7*epsilon - 768.0*R**2*S**2*T*UGA_star**7 + 7680.0*R**2*S*T**2*UGA_star**10*epsilon + 1152.0*R**2*S*T**2*UGA_star**10 - 23040.0*R**2*S*T**2*UGA_star**9*epsilon - 3456.0*R**2*S*T**2*UGA_star**9 + 23040.0*R**2*S*T**2*UGA_star**8*epsilon + 3456.0*R**2*S*T**2*UGA_star**8 - 7680.0*R**2*S*T**2*UGA_star**7*epsilon - 1152.0*R**2*S*T**2*UGA_star**7 + 2560.0*R**2*T**3*UGA_star**10*epsilon + 384.0*R**2*T**3*UGA_star**10 - 7680.0*R**2*T**3*UGA_star**9*epsilon - 1152.0*R**2*T**3*UGA_star**9 + 7680.0*R**2*T**3*UGA_star**8*epsilon + 1152.0*R**2*T**3*UGA_star**8 - 2560.0*R**2*T**3*UGA_star**7*epsilon - 384.0*R**2*T**3*UGA_star**7 - 1280.0*R*S**4*UGA_star**10*epsilon + 5120.0*R*S**4*UGA_star**9*epsilon - 7680.0*R*S**4*UGA_star**8*epsilon + 5120.0*R*S**4*UGA_star**7*epsilon - 1280.0*R*S**4*UGA_star**6*epsilon - 5120.0*R*S**3*T*UGA_star**10*epsilon - 512.0*R*S**3*T*UGA_star**10 + 20480.0*R*S**3*T*UGA_star**9*epsilon + 2048.0*R*S**3*T*UGA_star**9 - 30720.0*R*S**3*T*UGA_star**8*epsilon - 3072.0*R*S**3*T*UGA_star**8 + 20480.0*R*S**3*T*UGA_star**7*epsilon + 2048.0*R*S**3*T*UGA_star**7 - 5120.0*R*S**3*T*UGA_star**6*epsilon - 512.0*R*S**3*T*UGA_star**6 - 7680.0*R*S**2*T**2*UGA_star**10*epsilon - 1152.0*R*S**2*T**2*UGA_star**10 + 30720.0*R*S**2*T**2*UGA_star**9*epsilon + 4608.0*R*S**2*T**2*UGA_star**9 - 46080.0*R*S**2*T**2*UGA_star**8*epsilon - 6912.0*R*S**2*T**2*UGA_star**8 + 30720.0*R*S**2*T**2*UGA_star**7*epsilon + 4608.0*R*S**2*T**2*UGA_star**7 - 7680.0*R*S**2*T**2*UGA_star**6*epsilon - 1152.0*R*S**2*T**2*UGA_star**6 - 5120.0*R*S*T**3*UGA_star**10*epsilon - 768.0*R*S*T**3*UGA_star**10 + 20480.0*R*S*T**3*UGA_star**9*epsilon + 3072.0*R*S*T**3*UGA_star**9 - 30720.0*R*S*T**3*UGA_star**8*epsilon - 4608.0*R*S*T**3*UGA_star**8 + 20480.0*R*S*T**3*UGA_star**7*epsilon + 3072.0*R*S*T**3*UGA_star**7 - 5120.0*R*S*T**3*UGA_star**6*epsilon - 768.0*R*S*T**3*UGA_star**6 - 1280.0*R*T**4*UGA_star**10*epsilon - 128.0*R*T**4*UGA_star**10 + 5120.0*R*T**4*UGA_star**9*epsilon + 512.0*R*T**4*UGA_star**9 - 7680.0*R*T**4*UGA_star**8*epsilon - 768.0*R*T**4*UGA_star**8 + 5120.0*R*T**4*UGA_star**7*epsilon + 512.0*R*T**4*UGA_star**7 - 1280.0*R*T**4*UGA_star**6*epsilon - 128.0*R*T**4*UGA_star**6 + 256.0*S**5*UGA_star**10*epsilon - 1280.0*S**5*UGA_star**9*epsilon + 2560.0*S**5*UGA_star**8*epsilon - 2560.0*S**5*UGA_star**7*epsilon + 1280.0*S**5*UGA_star**6*epsilon - 256.0*S**5*UGA_star**5*epsilon + 1280.0*S**4*T*UGA_star**10*epsilon + 128.0*S**4*T*UGA_star**10 - 6400.0*S**4*T*UGA_star**9*epsilon - 640.0*S**4*T*UGA_star**9 + 12800.0*S**4*T*UGA_star**8*epsilon + 1280.0*S**4*T*UGA_star**8 - 12800.0*S**4*T*UGA_star**7*epsilon - 1280.0*S**4*T*UGA_star**7 + 6400.0*S**4*T*UGA_star**6*epsilon + 640.0*S**4*T*UGA_star**6 - 1280.0*S**4*T*UGA_star**5*epsilon - 128.0*S**4*T*UGA_star**5 + 2560.0*S**3*T**2*UGA_star**10*epsilon + 384.0*S**3*T**2*UGA_star**10 - 12800.0*S**3*T**2*UGA_star**9*epsilon - 1920.0*S**3*T**2*UGA_star**9 + 25600.0*S**3*T**2*UGA_star**8*epsilon + 3840.0*S**3*T**2*UGA_star**8 - 25600.0*S**3*T**2*UGA_star**7*epsilon - 3840.0*S**3*T**2*UGA_star**7 + 12800.0*S**3*T**2*UGA_star**6*epsilon + 1920.0*S**3*T**2*UGA_star**6 - 2560.0*S**3*T**2*UGA_star**5*epsilon - 384.0*S**3*T**2*UGA_star**5 + 2560.0*S**2*T**3*UGA_star**10*epsilon + 384.0*S**2*T**3*UGA_star**10 - 12800.0*S**2*T**3*UGA_star**9*epsilon - 1920.0*S**2*T**3*UGA_star**9 + 25600.0*S**2*T**3*UGA_star**8*epsilon + 3840.0*S**2*T**3*UGA_star**8 - 25600.0*S**2*T**3*UGA_star**7*epsilon - 3840.0*S**2*T**3*UGA_star**7 + 12800.0*S**2*T**3*UGA_star**6*epsilon + 1920.0*S**2*T**3*UGA_star**6 - 2560.0*S**2*T**3*UGA_star**5*epsilon - 384.0*S**2*T**3*UGA_star**5 + 1280.0*S*T**4*UGA_star**10*epsilon + 128.0*S*T**4*UGA_star**10 - 6400.0*S*T**4*UGA_star**9*epsilon - 640.0*S*T**4*UGA_star**9 + 12800.0*S*T**4*UGA_star**8*epsilon + 1280.0*S*T**4*UGA_star**8 - 12800.0*S*T**4*UGA_star**7*epsilon - 1280.0*S*T**4*UGA_star**7 + 6400.0*S*T**4*UGA_star**6*epsilon + 640.0*S*T**4*UGA_star**6 - 1280.0*S*T**4*UGA_star**5*epsilon - 128.0*S*T**4*UGA_star**5 + 256.0*T**5*UGA_star**10*epsilon - 1280.0*T**5*UGA_star**9*epsilon + 2560.0*T**5*UGA_star**8*epsilon - 2560.0*T**5*UGA_star**7*epsilon + 1280.0*T**5*UGA_star**6*epsilon - 256.0*T**5*UGA_star**5*epsilon)"
      ]
     },
     "execution_count": 177,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sym.together(sym.expand(sym.together(e2.subs({x1: x1_star, UgA_star: x1_star})))).subs({epsilon**2: 0})"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 60,
   "metadata": {},
   "outputs": [],
   "source": [
    "numerator, denominator = sym.fraction(sym.factor(e2))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 61,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "8*x1*(x1 - 1)*(P*UGA_star**2 - 2*P*UGA_star + P + R*UGA_star**2 - S*UGA_star**2 + S*UGA_star - T*UGA_star**2 + T*UGA_star)"
      ]
     },
     "execution_count": 61,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "denominator"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 77,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# vectorized numeric repr for the eigenvalue\n",
    "_numeric_e3 = sym.lambdify((UGA_star, UgA_star, T, R, P, S, epsilon),\n",
    "                           e3.subs({x1: x1_star}),\n",
    "                           modules=\"numpy\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 78,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def plot_third_eigenvalue(T, R, P, S, epsilon):\n",
    "\n",
    "    fig, ax = plt.subplots(1, 1, figsize=(10, 10))\n",
    "    ax.set_ylabel(r\"$U_{GA}^*$\", fontsize=20, rotation=\"horizontal\")\n",
    "    ax.set_xlabel(r\"$U_{gA}^{*}$\", fontsize=20)\n",
    "    ax.set_title(r\"Eigenvalue, $e_3$\", fontsize=25)\n",
    "    ax.grid(\"off\")\n",
    "\n",
    "    equilibrium_selection_probs = np.linspace(0, 1, 100)\n",
    "    UgAs, UGAs = np.meshgrid(equilibrium_selection_probs, equilibrium_selection_probs)\n",
    "    Z = _numeric_e3(UGAs, UgAs, T, R, P, S, epsilon)\n",
    "    cax = ax.imshow(Z, origin=\"lower\")\n",
    "    contours = ax.contour(Z, colors='w', origin='lower')\n",
    "    ax.clabel(contours, contours.levels, inline=True, fontsize=10)\n",
    "    \n",
    "    # adjust the tick labels\n",
    "    locs, _ = plt.xticks()\n",
    "    plt.xticks(locs[1:], np.linspace(0, 1, locs.size-1))\n",
    "    locs, _ = plt.yticks()\n",
    "    plt.yticks(locs[1:], np.linspace(0, 1, locs.size-1))\n",
    "\n",
    "    plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 79,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "32df2ee80b4f4d9d8b03c96b215f1e76",
       "version_major": 2,
       "version_minor": 0
      },
      "text/plain": [
       "A Jupyter Widget"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "mpld3.disable_notebook()  # don't need interactive plotting for this!\n",
    "\n",
    "# sliders used to control the Prisoner's Dilemma Payoffs\n",
    "T_slider = widgets.FloatSlider(value=25, min=0, max=100, step=0.1, description=r\"$T$\")\n",
    "R_slider = widgets.FloatSlider(value=3, min=0, max=100, step=0.1, description=r\"$R$\")\n",
    "P_slider = widgets.FloatSlider(value=2, min=0, max=100, step=0.1, description=r\"$P$\")\n",
    "S_slider = widgets.FloatSlider(value=1, min=0, max=100, step=0.1, description=r\"$S$\")\n",
    "\n",
    "# slider used to control the mutation rate\n",
    "e_slider = widgets.FloatSlider(value=0.0, min=0.0, max=1.0, step=1e-3, description=r\"$\\epsilon$\", readout_format=\".3f\")\n",
    "\n",
    "w = widgets.interactive(plot_third_eigenvalue, T=T_slider, R=R_slider, P=P_slider, S=S_slider, epsilon=e_slider)\n",
    "display(w)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 158,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "False"
      ]
     },
     "execution_count": 158,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "e2 == e3"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 164,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(-13.0*R*epsilon - 1.5*R + 3.0*S*epsilon + 3.0*T*epsilon + 0.015625*sqrt(61440.0*R**2*epsilon + 9216.0*R**2 + 12288.0*R*S*epsilon - 4096.0*R*T*epsilon))/(R*(16.0*epsilon + 4.0))"
      ]
     },
     "execution_count": 164,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sym.simplify(e2.subs({x1: x1_star, UgA_star: x1_star}).subs({UGA_star: 1, UGA_prime_star: 0})).subs({epsilon**2: 0})"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 165,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(-13.0*R*epsilon - 1.5*R + 3.0*S*epsilon + 3.0*T*epsilon - 0.015625*sqrt(61440.0*R**2*epsilon + 9216.0*R**2 + 12288.0*R*S*epsilon - 4096.0*R*T*epsilon))/(R*(16.0*epsilon + 4.0))"
      ]
     },
     "execution_count": 165,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sym.simplify(e3.subs({x1: x1_star, UgA_star: x1_star}).subs({UGA_star: 1, UGA_prime_star: 0})).subs({epsilon**2: 0})"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 153,
   "metadata": {},
   "outputs": [],
   "source": [
    "evaluated = (e3.subs({x1: x1_star, UgA_star: x1_star})\n",
    "               .subs({UGA_star: 1, UGA_prime_star: 0}))\n",
    "together = sym.together(evaluated)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 154,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "expanded = sym.expand(together)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "tmp_expr = sym.together(expanded)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 155,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(-13.0*R*epsilon - 1.5*R + 3.0*S*epsilon + 3.0*T*epsilon + 0.015625*sqrt(61440.0*R**2*epsilon + 9216.0*R**2 + 12288.0*R*S*epsilon - 4096.0*R*T*epsilon))/(R*(16.0*epsilon + 4.0))"
      ]
     },
     "execution_count": 155,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "tmp_expr.subs({epsilon**2: 0})"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "(-13.0*R*epsilon - 1.5*R + 3.0*S*epsilon + 3.0*T*epsilon + 0.015625*sqrt(61440.0*R**2*epsilon + 9216.0*R**2 + 12288.0*R*S*epsilon - 4096.0*R*T*epsilon))/(R*(16.0*epsilon + 4.0))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "First eigenvalue can be written as a hyperbola in $U_{gA}^*,U_{GA}^*$ plane.\n",
    "\n",
    "$$ e_1 = \\left(\\frac{\\big((R + P) - (T + S)\\big)U_{gA}^{*2} + \\big((T + S) - 2P\\big)U_{gA}^*}{N^*}\\right) - \\left(\\frac{\\big((R + P) - (T + S)\\big)U_{GA}^{*2} + \\big((T + S) - 2P\\big)U_{GA}^*}{N^*}\\right) $$\n",
    "\n",
    "### Completing the square\n",
    "\n",
    "To write this hyperbola in standard form we need to [complete the square](https://en.wikipedia.org/wiki/Completing_the_square). Completing the square for the quadratic polynomial in $U_{GA}(x^*)$ yields the following.\n",
    "\n",
    "\\begin{align}\n",
    "    \\big((R + P) - (T + S)\\big)\\left(U_{GA}^* - \\bar{U}_{GA}^*\\right)^2 - \\frac{1}{2}\\big(\\bar{N}^* - 2P\\big) \\\\\n",
    "\\end{align}\n",
    "\n",
    "where $\\bar{U}_{GA}^*$ is the value of $U_{GA}^*$ that maximizes total offspring $N$ derived above. Completing the square for the quadratic polynomial in $U_{gA}(x^*)$ yields a similar expression.\n",
    "\n",
    "\\begin{align}\n",
    "    \\big((R + P) - (T + S)\\big)\\left(U_{gA}^* - \\bar{U}_{GA}^*\\right)^2 - \\frac{1}{2}\\big(\\bar{N}^* - 2P\\big) \\\\\n",
    "\\end{align}\n",
    "\n",
    "Substituting these results into the expression for the eigenvalue yields the following.\n",
    "\n",
    "$$ e_1 = \\left(\\frac{\\big(U_{gA}^* - \\bar{U}_{GA}^*\\big)^{2}}{\\frac{N^*}{\\big((R + P) - (T + S)\\big)}}\\right) - \\left(\\frac{\\big(U_{GA}^* - \\bar{U}_{GA}^*\\big)^{2}}{\\frac{N^*}{\\big((R + P) - (T + S)\\big)}}\\right) $$\n",
    "\n",
    "\n",
    "### Conditions for negative eigenvalue\n",
    "\n",
    "Non-invadability requires that this eigenvalue is strictly negative.\n",
    "\n",
    "$$ \\left(\\frac{\\big(U_{gA}^* - \\bar{U}_{GA}^*\\big)^{2}}{\\frac{N^*}{\\big((R + P) - (T + S)\\big)}}\\right) - \\left(\\frac{\\big(U_{GA}^* - \\bar{U}_{GA}^*\\big)^{2}}{\\frac{N^*}{\\big((R + P) - (T + S)\\big)}}\\right) < 0$$\n",
    "\n",
    "#### Case: (R + P) < (T + S)\n",
    "In this case the inequality simplifies to the following.\n",
    "\n",
    "$$ \\big(U_{gA}^* - \\bar{U}_{GA}^*\\big)^{2} - \\big(U_{GA}^* - \\bar{U}_{GA}^*\\big)^{2} > 0 $$\n",
    "\n",
    "This implies that we have four sub-cases to consider.\n",
    "\n",
    "\\begin{align}\n",
    "    \\big(U_{gA}^* - \\bar{U}_{GA}^*\\big) >& \\big(U_{GA}^* - \\bar{U}_{GA}^*\\big) \\\\\n",
    "    \\big(U_{gA}^* - \\bar{U}_{GA}^*\\big) >& -\\big(U_{GA}^* - \\bar{U}_{GA}^*\\big) \\\\\n",
    "    -\\big(U_{gA}^* - \\bar{U}_{GA}^*\\big) >& \\big(U_{GA}^* - \\bar{U}_{GA}^*\\big) \\\\\n",
    "    -\\big(U_{gA}^* - \\bar{U}_{GA}^*\\big) >& -\\big(U_{GA}^* - \\bar{U}_{GA}^*\\big)\n",
    "\\end{align}\n",
    "\n",
    "These four sub-cases define the following two regions of the $U_{gA}^*,U_{GA}^*$ plane where the eigenvalue is negative.\n",
    "\n",
    "\\begin{align}\n",
    "   & 2\\big(\\bar{U}_{GA}^* - U_{gA}^*\\big) < U_{GA}^* - U_{gA}^* < 0\\\\\n",
    "   & 0 < U_{GA}^* - U_{gA}^* < 2\\big(\\bar{U}_{GA}^* - U_{gA}\\big)\n",
    "\\end{align}\n",
    "\n",
    "#### Case: (R + P) > (T + S)\n",
    "In this case the inequality simplifies to the following.\n",
    "\n",
    "$$ \\big(U_{gA}^* - \\bar{U}_{GA}^*\\big)^{2} - \\big(U_{GA}^* - \\bar{U}_{GA}^*\\big)^{2} < 0 $$\n",
    "\n",
    "This implies that we again have four sub-cases to consider.\n",
    "\n",
    "\\begin{align}\n",
    "    \\big(U_{gA}^* - \\bar{U}_{GA}^*\\big) <& \\big(U_{GA}^* - \\bar{U}_{GA}^*\\big) \\\\\n",
    "    \\big(U_{gA}^* - \\bar{U}_{GA}^*\\big) <& -\\big(U_{GA}^* - \\bar{U}_{GA}^*\\big) \\\\\n",
    "    -\\big(U_{gA}^* - \\bar{U}_{GA}^*\\big) <& \\big(U_{GA}^* - \\bar{U}_{GA}^*\\big) \\\\\n",
    "    -\\big(U_{gA}^* - \\bar{U}_{GA}^*\\big) <& -\\big(U_{GA}^* - \\bar{U}_{GA}^*\\big)\n",
    "\\end{align}\n",
    "\n",
    "These four sub-cases define the following two regions of the $U_{gA}^*,U_{GA}^*$ plane where the eigenvalue is negative.\n",
    "\n",
    "\\begin{align}\n",
    "   & 2\\big(\\bar{U}_{GA}^* - U_{gA}^*\\big) < U_{GA}^* - U_{gA}^* < 0\\\\\n",
    "   & 0 < U_{GA}^* - U_{gA}^* < 2\\big(\\bar{U}_{GA}^* - U_{gA}\\big)\n",
    "\\end{align}\n",
    "\n",
    "\n",
    "### Asymptotes \n",
    "\n",
    "Asymptotes of this hyperbola can be defined as follows.\n",
    "\n",
    "$$ U_{GA}^* - \\bar{U}_{GA}^* = \\pm\\big(U_{gA}^* - \\bar{U}_{GA}^*\\big) $$\n",
    "\n",
    "The above can be written as linear equations in $U_{gA}^*,U_{GA}^*$ plane.\n",
    "\n",
    "\\begin{align}\n",
    "    U_{GA}^* =& U_{gA}^* \\\\\n",
    "    U_{GA}^* =& 2\\bar{U}_{GA}^* - U_{gA}^*\n",
    "\\end{align}\n",
    "\n",
    "Note that the asymptotes of this hyperbola correspond to the locus of points for which the value of the first eigenvalue is exactly zero.\n",
    "\n",
    "#### Discussion\n",
    "\n",
    "Suppose $U_{GA}^* = \\bar{U}_{GA}^*$, then both of the sets of inequalities will hold for any value of $U_{gA}^*$. This implies that the eigenvalue will be negative for any value of $U_{gA}^*$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 83,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# vectorized numeric repr for the eigenvalue\n",
    "_numeric_e1 = sym.lambdify((UGA_star, UGA_prime_star, T, R, P, S, M), e1, modules=\"numpy\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 84,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def plot_first_eigenvalue(T, R, P, S, M):\n",
    "\n",
    "    fig, ax = plt.subplots(1, 1, figsize=(10, 10))\n",
    "    ax.set_ylabel(r\"$U_{GA}^*$\", fontsize=20, rotation=\"horizontal\")\n",
    "    ax.set_xlabel(r\"$U_{GA}^{'*}$\", fontsize=20)\n",
    "    ax.set_title(r\"Eigenvalue, $e_1$\", fontsize=25)\n",
    "    ax.grid(\"off\")\n",
    "\n",
    "    equilibrium_selection_probs = np.linspace(0, 1, 100).reshape(-1, 1)\n",
    "    equilibrium_selection_derivs = np.linspace(0, 10, 1000).reshape(1, -1) # sensible UGA'(0) >= 1\n",
    "    Z = _numeric_e1(equilibrium_selection_probs, equilibrium_selection_derivs, T, R, P, S, M)\n",
    "    cax = ax.imshow(Z, origin=\"lower\", aspect=\"auto\", vmin=-0.5, vmax=1.0)\n",
    "\n",
    "    levels = np.arange(-0.25, 1.25, 0.25)\n",
    "    contours = ax.contour(Z, levels=levels, colors='w', origin='lower')\n",
    "    ax.clabel(contours, contours.levels, inline=True, fontsize=10)\n",
    "    \n",
    "    # adjust the tick labels\n",
    "    locs, _ = plt.xticks()\n",
    "    plt.xticks(locs[1:], np.linspace(0, 10, locs.size-1))\n",
    "    locs, _ = plt.yticks()\n",
    "    plt.yticks(locs[1:], np.linspace(0, 1, locs.size-1))\n",
    "\n",
    "    plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 85,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "31c7e73c14f147b5b4a90a12dbfa23cf",
       "version_major": 2,
       "version_minor": 0
      },
      "text/plain": [
       "A Jupyter Widget"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "mpld3.disable_notebook()  # don't need interactive plotting for this!\n",
    "\n",
    "# sliders used to control the Prisoner's Dilemma Payoffs\n",
    "T_slider = widgets.FloatSlider(value=25, min=0, max=100, step=0.1, description=r\"$T$\")\n",
    "R_slider = widgets.FloatSlider(value=3, min=0, max=100, step=0.1, description=r\"$R$\")\n",
    "P_slider = widgets.FloatSlider(value=2, min=0, max=100, step=0.1, description=r\"$P$\")\n",
    "S_slider = widgets.FloatSlider(value=1, min=0, max=100, step=0.1, description=r\"$S$\")\n",
    "\n",
    "M_slider = widgets.FloatSlider(value=1, min=0, max=100, step=0.1, description=r\"$M$\")\n",
    "\n",
    "w = widgets.interactive(plot_first_eigenvalue, T=T_slider, R=R_slider, P=P_slider, S=S_slider, M=M_slider)\n",
    "display(w)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "UGA_prime_star, = sym.solve(numerator, UGA_prime_star)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-(P*UGA_star**2 - 2*P*UGA_star + P + R*UGA_star**2 - S*UGA_star**2 + S*UGA_star - T*UGA_star**2 + T*UGA_star)**2/(2*P*R*UGA_star**2 - 2*P*R*UGA_star - P*S*UGA_star**2 + 2*P*S*UGA_star - P*S - R*T*UGA_star**2)"
      ]
     },
     "execution_count": 52,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sym.factor(UGA_prime_star)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "After a bit of fiddling, we arrive at the same stability condition derived above."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\\begin{align}\n",
    "   % U'_{GA}(x^*) < -\\frac{(PU_{GA}(x^*)^2 - 2PU_{GA}(x^*) + P + RU_{GA}(x^*)^2 - SU_{GA}(x^*)^2 + SU_{GA}(x^*) - TU_{GA}(x^*)^2 + TU_{GA}(x^*))^2}{(2PRU_{GA}(x^*)^2 - 2PRU_{GA}(x^*) - PSU_{GA}(x^*)^2 + 2PSU_{GA}(x^*) - PS - RTU_{GA}(x^*)^2)} \\\\\n",
    "    U'_{GA}(x^*) < \\frac{\\bigg(\\big((R + P) - (T + S)\\big)U_{GA}(x^*)^2 + \\big((T + S) - 2P\\big)U_{GA}(x^*) + P\\bigg)^2}{\\big(R(T-2P) + PS\\big)U_{GA}(x^*)^2 + 2P(R - S)U_{GA}(x^*) + PS}\n",
    "\\end{align}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 72,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(-3*M + 3*P*UGA_star**2 - 6*P*UGA_star - P*UgA_star**2 + 2*P*UgA_star + 2*P + 3*R*UGA_star**2 - R*UgA_star**2 - 3*S*UGA_star**2 + 3*S*UGA_star + S*UgA_star**2 - S*UgA_star - 3*T*UGA_star**2 + 3*T*UGA_star + T*UgA_star**2 - T*UgA_star + m)/(4*M - 4*P*UGA_star**2 + 8*P*UGA_star - 4*P - 4*R*UGA_star**2 + 4*S*UGA_star**2 - 4*S*UGA_star + 4*T*UGA_star**2 - 4*T*UGA_star)"
      ]
     },
     "execution_count": 72,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "e2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$ e_2 = \\left(\\frac{\\big((R + P) - (T + S)\\big)U_{gA}^{*2} + \\big((T + S) - 2P\\big)U_{gA}^* + P}{4N^*}\\right) - \\frac{3}{4} $$\n",
    "\n",
    "### Completing the square\n",
    "\n",
    "Completing the square for the quadratic polynomial in $U_{gA}(x^*)$ yields the following expression.\n",
    "\n",
    "\\begin{align}\n",
    "    \\big((R + P) - (T + S)\\big)\\left(U_{gA}^* - \\bar{U}_{GA}^*\\right)^2 + \\frac{1}{2}\\bar{N}^* \\\\\n",
    "\\end{align}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Finally we need to find conditions under which the third eigenvalue is strictly negative."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 78,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# vectorized numeric repr for the eigenvalue\n",
    "_numeric_e2 = sym.lambdify((UGA_star, UgA_star, T, R, P, S, M, m), e2, modules=\"numpy\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 79,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def plot_second_eigenvalue(T, R, P, S, M, m):\n",
    "\n",
    "    fig, ax = plt.subplots(1, 1, figsize=(10, 10))\n",
    "    ax.set_ylabel(r\"$U_{GA}^*$\", fontsize=20, rotation=\"horizontal\")\n",
    "    ax.set_xlabel(r\"$U_{gA}^{*}$\", fontsize=20)\n",
    "    ax.set_title(r\"Eigenvalue, $e_2$\", fontsize=25)\n",
    "    ax.grid(\"off\")\n",
    "\n",
    "    equilibrium_selection_probs = np.linspace(0, 1, 100)\n",
    "    UgAs, UGAs = np.meshgrid(equilibrium_selection_probs, equilibrium_selection_probs)\n",
    "    Z = _numeric_e2(UGAs, UgAs, T, R, P, S, M, m)\n",
    "    cax = ax.imshow(Z, origin=\"lower\")\n",
    "    contours = ax.contour(Z, colors='w', origin='lower')\n",
    "    ax.clabel(contours, contours.levels, inline=True, fontsize=10)\n",
    "    \n",
    "    # adjust the tick labels\n",
    "    locs, _ = plt.xticks()\n",
    "    plt.xticks(locs[1:], np.linspace(0, 1, locs.size-1))\n",
    "    locs, _ = plt.yticks()\n",
    "    plt.yticks(locs[1:], np.linspace(0, 1, locs.size-1))\n",
    "\n",
    "    plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 80,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "6b8b65bf9d8e48fa90b4e6e3adc2c68f",
       "version_major": 2,
       "version_minor": 0
      },
      "text/plain": [
       "A Jupyter Widget"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "mpld3.disable_notebook()  # don't need interactive plotting for this!\n",
    "\n",
    "# sliders used to control the Prisoner's Dilemma Payoffs\n",
    "T_slider = widgets.FloatSlider(value=25, min=0, max=100, step=0.1, description=r\"$T$\")\n",
    "R_slider = widgets.FloatSlider(value=3, min=0, max=100, step=0.1, description=r\"$R$\")\n",
    "P_slider = widgets.FloatSlider(value=2, min=0, max=100, step=0.1, description=r\"$P$\")\n",
    "S_slider = widgets.FloatSlider(value=1, min=0, max=100, step=0.1, description=r\"$S$\")\n",
    "\n",
    "M_slider = widgets.FloatSlider(value=1, min=0, max=100, step=0.1, description=r\"$M$\")\n",
    "m_slider = widgets.FloatSlider(value=0, min=0, max=100, step=0.1, description=r\"$m$\")\n",
    "\n",
    "w = widgets.interactive(plot_second_eigenvalue, T=T_slider, R=R_slider, P=P_slider, S=S_slider, M=M_slider, m=m_slider)\n",
    "display(w)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 81,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(-M + P*UGA_star**2 - 2*P*UGA_star - P*UgA_star**2 + 2*P*UgA_star + R*UGA_star**2 - R*UgA_star**2 - S*UGA_star**2 + S*UGA_star + S*UgA_star**2 - S*UgA_star - T*UGA_star**2 + T*UGA_star + T*UgA_star**2 - T*UgA_star + m)/(2*M - 2*P*UGA_star**2 + 4*P*UGA_star - 2*P - 2*R*UGA_star**2 + 2*S*UGA_star**2 - 2*S*UGA_star + 2*T*UGA_star**2 - 2*T*UGA_star)"
      ]
     },
     "execution_count": 81,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "e3"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 86,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# vectorized numeric repr for the eigenvalue\n",
    "_numeric_e3 = sym.lambdify((UGA_star, UgA_star, T, R, P, S, M, m), e3, modules=\"numpy\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 89,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def plot_third_eigenvalue(T, R, P, S, M, m):\n",
    "\n",
    "    fig, ax = plt.subplots(1, 1, figsize=(10, 10))\n",
    "    ax.set_ylabel(r\"$U_{GA}^*$\", fontsize=20, rotation=\"horizontal\")\n",
    "    ax.set_xlabel(r\"$U_{gA}^{*}$\", fontsize=20)\n",
    "    ax.set_title(r\"Eigenvalue, $e_3$\", fontsize=25)\n",
    "    ax.grid(\"off\")\n",
    "\n",
    "    equilibrium_selection_probs = np.linspace(0, 1, 100)\n",
    "    UgAs, UGAs = np.meshgrid(equilibrium_selection_probs, equilibrium_selection_probs)\n",
    "    Z = _numeric_e3(UGAs, UgAs, T, R, P, S, M, m)\n",
    "    cax = ax.imshow(Z, origin=\"lower\")\n",
    "    contours = ax.contour(Z, colors='w', origin='lower')\n",
    "    ax.clabel(contours, contours.levels, inline=True, fontsize=10)\n",
    "    \n",
    "    # adjust the tick labels\n",
    "    locs, _ = plt.xticks()\n",
    "    plt.xticks(locs[1:], np.linspace(0, 1, locs.size-1))\n",
    "    locs, _ = plt.yticks()\n",
    "    plt.yticks(locs[1:], np.linspace(0, 1, locs.size-1))\n",
    "\n",
    "    plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 90,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "40a14cf559fa4ca89dec92de7b9202f9",
       "version_major": 2,
       "version_minor": 0
      },
      "text/plain": [
       "A Jupyter Widget"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "mpld3.disable_notebook()  # don't need interactive plotting for this!\n",
    "\n",
    "# sliders used to control the Prisoner's Dilemma Payoffs\n",
    "T_slider = widgets.FloatSlider(value=25, min=0, max=100, step=0.1, description=r\"$T$\")\n",
    "R_slider = widgets.FloatSlider(value=3, min=0, max=100, step=0.1, description=r\"$R$\")\n",
    "P_slider = widgets.FloatSlider(value=2, min=0, max=100, step=0.1, description=r\"$P$\")\n",
    "S_slider = widgets.FloatSlider(value=1, min=0, max=100, step=0.1, description=r\"$S$\")\n",
    "\n",
    "M_slider = widgets.FloatSlider(value=1, min=0, max=100, step=0.1, description=r\"$M$\")\n",
    "m_slider = widgets.FloatSlider(value=0, min=0, max=100, step=0.1, description=r\"$m$\")\n",
    "\n",
    "w = widgets.interactive(plot_third_eigenvalue, T=T_slider, R=R_slider, P=P_slider, S=S_slider, M=M_slider, m=m_slider)\n",
    "display(w)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## When is the fitness-maximizing interior equilibrium un-invadable by randomista?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 147,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "evaluated_e3 = e3.subs({UGA_star: optimal_UGA_star, UgA_star: optimal_x1_star, m: 0})\n",
    "simplified_e3 = sym.factor(sym.cancel(sym.together(evaluated_e3)))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 148,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "numerator, denominator = sym.fraction(simplified_e3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 151,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-4.0*M**3*P**3 - 12.0*M**3*P**2*R + 12.0*M**3*P**2*S + 12.0*M**3*P**2*T - 12.0*M**3*P*R**2 + 24.0*M**3*P*R*S + 24.0*M**3*P*R*T - 12.0*M**3*P*S**2 - 24.0*M**3*P*S*T - 12.0*M**3*P*T**2 - 4.0*M**3*R**3 + 12.0*M**3*R**2*S + 12.0*M**3*R**2*T - 12.0*M**3*R*S**2 - 24.0*M**3*R*S*T - 12.0*M**3*R*T**2 + 4.0*M**3*S**3 + 12.0*M**3*S**2*T + 12.0*M**3*S*T**2 + 4.0*M**3*T**3 + 8.0*M**2*P**3*R + 16.0*M**2*P**2*R**2 - 16.0*M**2*P**2*R*S - 16.0*M**2*P**2*R*T - 2.0*M**2*P**2*S**2 - 4.0*M**2*P**2*S*T - 2.0*M**2*P**2*T**2 + 8.0*M**2*P*R**3 - 16.0*M**2*P*R**2*S - 16.0*M**2*P*R**2*T + 4.0*M**2*P*R*S**2 + 8.0*M**2*P*R*S*T + 4.0*M**2*P*R*T**2 + 4.0*M**2*P*S**3 + 12.0*M**2*P*S**2*T + 12.0*M**2*P*S*T**2 + 4.0*M**2*P*T**3 - 2.0*M**2*R**2*S**2 - 4.0*M**2*R**2*S*T - 2.0*M**2*R**2*T**2 + 4.0*M**2*R*S**3 + 12.0*M**2*R*S**2*T + 12.0*M**2*R*S*T**2 + 4.0*M**2*R*T**3 - 2.0*M**2*S**4 - 8.0*M**2*S**3*T - 12.0*M**2*S**2*T**2 - 8.0*M**2*S*T**3 - 2.0*M**2*T**4 - 4.0*M*P**3*R**2 - 4.0*M*P**2*R**3 + 4.0*M*P**2*R**2*S + 4.0*M*P**2*R**2*T + 2.0*M*P**2*R*S**2 + 4.0*M*P**2*R*S*T + 2.0*M*P**2*R*T**2 + 2.0*M*P*R**2*S**2 + 4.0*M*P*R**2*S*T + 2.0*M*P*R**2*T**2 - 2.0*M*P*R*S**3 - 6.0*M*P*R*S**2*T - 6.0*M*P*R*S*T**2 - 2.0*M*P*R*T**3 - 0.25*M*P*S**4 - 1.0*M*P*S**3*T - 1.5*M*P*S**2*T**2 - 1.0*M*P*S*T**3 - 0.25*M*P*T**4 - 0.25*M*R*S**4 - 1.0*M*R*S**3*T - 1.5*M*R*S**2*T**2 - 1.0*M*R*S*T**3 - 0.25*M*R*T**4 + 0.25*M*S**5 + 1.25*M*S**4*T + 2.5*M*S**3*T**2 + 2.5*M*S**2*T**3 + 1.25*M*S*T**4 + 0.25*M*T**5 - 1.0*P**2*R**2*S**2 + 2.0*P**2*R**2*S*T - 1.0*P**2*R**2*T**2 + 1.0*P**2*R*S**3 - 1.0*P**2*R*S**2*T - 1.0*P**2*R*S*T**2 + 1.0*P**2*R*T**3 - 0.25*P**2*S**4 + 0.5*P**2*S**2*T**2 - 0.25*P**2*T**4 + 1.0*P*R**2*S**3 - 1.0*P*R**2*S**2*T - 1.0*P*R**2*S*T**2 + 1.0*P*R**2*T**3 - 1.0*P*R*S**4 + 2.0*P*R*S**2*T**2 - 1.0*P*R*T**4 + 0.25*P*S**5 + 0.25*P*S**4*T - 0.5*P*S**3*T**2 - 0.5*P*S**2*T**3 + 0.25*P*S*T**4 + 0.25*P*T**5 - 0.25*R**2*S**4 + 0.5*R**2*S**2*T**2 - 0.25*R**2*T**4 + 0.25*R*S**5 + 0.25*R*S**4*T - 0.5*R*S**3*T**2 - 0.5*R*S**2*T**3 + 0.25*R*S*T**4 + 0.25*R*T**5 - 0.0625*S**6 - 0.125*S**5*T + 0.0625*S**4*T**2 + 0.25*S**3*T**3 + 0.0625*S**2*T**4 - 0.125*S*T**5 - 0.0625*T**6"
      ]
     },
     "execution_count": 151,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "numerator"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\\begin{align}\n",
    "    4\\bigg(-\\big(P^3 + 3P^2R + 3PR^2 + R^3\\big) - \\big(P^3 - 3P^2S + 3PS^2 + S^3\\big) + \\big(-P^3 + 3P^2T - 3PT^2 + T^3\\big) - \\big(R^3 - 3R^2S + 3RS^2 - S^3\\big) + \\big(T^3 - 3RT^2 + 3R^2T - R^3\\big) + 2P^3 + 2R^3 + 8PRS + 8PRT - 8PST - 8RST - 2T^3 + \\big(S^3 + 3S^2T + 3ST^2 + T^3\\big)\\bigg)M^3 + 2\\big(4P^3R + 8P^2R^2 - 8P^2RS - 8P^2RT - P^2S^2 - 2P^2ST - P^2T^2 + 4PR^3 - 8PR^2S - 8PR^2T + 2PRS^2 + 4PRST + 2PRT^2 + 2PS^3 + 6PS^2T + 6PST^2 + 2PT^3 - R^2S^2 - 2R^2ST - R^2T^2 + 2RS^3 + 6RS^2T + 6RST^2 + 2RT^3 - S^4 - 4S^3T - 6S^2T^2 - 4ST^3 - T^4\\big)M^2 + \\big(- 4P^3R^2 - 4P^2R^3 + 4P^2R^2S + 4P^2R^2T + 2P^2RS^2 + 4P^2RST + 2P^2RT^2 + 2PR^2S^2 + 4PR^2ST + 2PR^2T^2 - 2PRS^3 - 6PRS^2T - 6PRST^2 - 2PRT^3 - 0.25PS^4 -PS^3T - 1.5PS^2T^2 - PST^3 - 0.25PT^4 - 0.25RS^4 - RS^3T - 1.5RS^2T^2 - RST^3 - 0.25RT^4 + 0.25S^5 + 1.25S^4T + 2.5S^3T^2 + 2.5S^2T^3 + 1.25ST^4 + 0.25T^5\\big)M + \\big(- P^2R^2S^2 + 2P^2R^2ST - P^2R^2T^2 + P^2RS^3 - P^2RS^2T - P^2RST^2 + P^2RT^3 - 0.25P^2S^4 + 0.5P^2S^2T^2 - 0.25P^2T^4 + PR^2S^3 - PR^2S^2T - PR^2ST^2 + PR^2T^3 - PRS^4 + 2PRS^2T^2 - PRT^4 + 0.25PS^5 + 0.25PS^4T - 0.5PS^3T^2 - 0.5PS^2T^3 + 0.25PST^4 + 0.25PT^5 - 0.25R^2S^4 + 0.5R^2S^2T^2 - 0.25R^2T^4 + 0.25RS^5 + 0.25RS^4T - 0.5RS^3T^2 - 0.5RS^2T^3 + 0.25RST^4 + 0.25RT^5 - 0.0625S^6 - 0.125S^5T + 0.0625S^4T^2 + 0.25S^3T^3 + 0.0625S^2T^4 - 0.125ST^5 - 0.0625T^6\\big) \\\\\n",
    "    4\\bigg(-\\big(R + P\\big)^3 - \\big(P - S\\big)^3 + \\big(T - P\\big)^3 - \\big(R - S\\big)^3 + \\big(T - R\\big)^3 + 2P^3 + 2R^3 + 8PRS + 8PRT - 8PST - 8RST - 2T^3 + \\big(T + S\\big)^3\\bigg)M^3 + 2\\big(4P^3R + 8P^2R^2 - 8P^2RS - 8P^2RT - P^2S^2 - 2P^2ST - P^2T^2 + 4PR^3 - 8PR^2S - 8PR^2T + 2PRS^2 + 4PRST + 2PRT^2 + 2PS^3 + 6PS^2T + 6PST^2 + 2PT^3 - R^2S^2 - 2R^2ST - R^2T^2 + 2RS^3 + 6RS^2T + 6RST^2 + 2RT^3 - S^4 - 4S^3T - 6S^2T^2 - 4ST^3 - T^4\\big)M^2 + \\big(- 4P^3R^2 - 4P^2R^3 + 4P^2R^2S + 4P^2R^2T + 2P^2RS^2 + 4P^2RST + 2P^2RT^2 + 2PR^2S^2 + 4PR^2ST + 2PR^2T^2 - 2PRS^3 - 6PRS^2T - 6PRST^2 - 2PRT^3 - 0.25PS^4 -PS^3T - 1.5PS^2T^2 - PST^3 - 0.25PT^4 - 0.25RS^4 - RS^3T - 1.5RS^2T^2 - RST^3 - 0.25RT^4 + 0.25S^5 + 1.25S^4T + 2.5S^3T^2 + 2.5S^2T^3 + 1.25ST^4 + 0.25T^5\\big)M + \\big(- P^2R^2S^2 + 2P^2R^2ST - P^2R^2T^2 + P^2RS^3 - P^2RS^2T - P^2RST^2 + P^2RT^3 - 0.25P^2S^4 + 0.5P^2S^2T^2 - 0.25P^2T^4 + PR^2S^3 - PR^2S^2T - PR^2ST^2 + PR^2T^3 - PRS^4 + 2PRS^2T^2 - PRT^4 + 0.25PS^5 + 0.25PS^4T - 0.5PS^3T^2 - 0.5PS^2T^3 + 0.25PST^4 + 0.25PT^5 - 0.25R^2S^4 + 0.5R^2S^2T^2 - 0.25R^2T^4 + 0.25RS^5 + 0.25RS^4T - 0.5RS^3T^2 - 0.5RS^2T^3 + 0.25RST^4 + 0.25RT^5 - 0.0625S^6 - 0.125S^5T + 0.0625S^4T^2 + 0.25S^3T^3 + 0.0625S^2T^4 - 0.125ST^5 - 0.0625T^6\\big) \\\\\n",
    "\\end{align}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "(S - R)(S^2 - 2SR + R^2) = S^3 - 3RS^2 + 3SR^2 - R^3"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 158,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-P**3 - 3*P**2*R + 3*P**2*S + 3*P**2*T - 3*P*R**2 + 8*P*R*S + 8*P*R*T - 3*P*S**2 - 8*P*S*T - 3*P*T**2 - R**3 + 3*R**2*S + 3*R**2*T - 3*R*S**2 - 8*R*S*T - 3*R*T**2 + S**3 + 3*S**2*T + 3*S*T**2 + T**3"
      ]
     },
     "execution_count": 158,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sym.factor(-P**3 - 3*P**2*R + 3*P**2*S + 3*P**2*T - 3*P*R**2 + 8*P*R*S + 8*P*R*T - 3*P*S**2 - 8*P*S*T - 3*P*T**2 - R**3 + 3*R**2*S + 3*R**2*T - 3*R*S**2 - 8*R*S*T - 3*R*T**2 + S**3 + 3*S**2*T + 3*S*T**2 + T**3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 164,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "r1, r2, r3 = sym.solve(numerator, M)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "sym.cancel(sym.together(r1))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 149,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(0.5*M*P + 0.5*M*R - 0.5*M*S - 0.5*M*T - 0.5*P*R + 0.125*S**2 + 0.25*S*T + 0.125*T**2)**3"
      ]
     },
     "execution_count": 149,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "denominator"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Denominator will be negative if and only if...\n",
    "\n",
    "\\begin{align}\n",
    "    % 4MP + 4MR - 4MS - 4MT - 4PR + S^2 + 2ST + T^2 < 0 \\\\\n",
    "    % 4M\\big((R + P) - (T + S)\\big) - 4PR + (T + S)^2< 0 \\\\\n",
    "    % 4M\\big((R + P) - (T + S)\\big) < 4RP - (T + S)^2 \\\\\n",
    "    M < \\frac{1}{4}\\frac{4RP - (T + S)^2}{(R + P) - (T + S)} \\\\\n",
    "    M < \\frac{RP - \\left(\\frac{T + S}{2}\\right)^2}{(R + P) - (T + S)} \\\\\n",
    "\\end{align}\n",
    "\n",
    "...since we are looking at interior equilibrium the RHS of the above inequality will be positive."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 139,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "_f = sym.lambdify((T, R, P, S, M), evaluated_e3, modules = \"numpy\")\n",
    "\n",
    "def plot_non_invadable_by_randomista(T, R, P, S):\n",
    "    fig, ax = plt.subplots(1, 1, figsize=(10,6))\n",
    "    Ms = np.linspace(0, min(0.5 * (T + S), P), 100)\n",
    "    ax.plot(Ms, _f(T, R, P, S, Ms))\n",
    "    ax.set_xlabel(r\"$M$\", fontsize=15)\n",
    "    ax.set_ylabel(r\"$e_3$\", rotation=\"horizontal\", fontsize=15)\n",
    "    plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 140,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "a6aaf85998844ff79ddb1c3e570a2981",
       "version_major": 2,
       "version_minor": 0
      },
      "text/plain": [
       "A Jupyter Widget"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# sliders used to control the Prisoner's Dilemma Payoffs\n",
    "T_slider = widgets.FloatSlider(value=10, min=0, max=100, step=0.1, description=r\"$T$\")\n",
    "R_slider = widgets.FloatSlider(value=8, min=0, max=100, step=0.1, description=r\"$R$\")\n",
    "P_slider = widgets.FloatSlider(value=6, min=0, max=100, step=0.1, description=r\"$P$\")\n",
    "S_slider = widgets.FloatSlider(value=4, min=0, max=100, step=0.1, description=r\"$S$\")\n",
    "\n",
    "w = widgets.interactive(plot_non_invadable_by_randomista, T=T_slider, R=R_slider, P=P_slider, S=S_slider)\n",
    "display(w)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 157,
   "metadata": {},
   "outputs": [
    {
     "ename": "KeyboardInterrupt",
     "evalue": "",
     "output_type": "error",
     "traceback": [
      "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[0;31mKeyboardInterrupt\u001b[0m                         Traceback (most recent call last)",
      "\u001b[0;32m<ipython-input-157-fcea7e261b22>\u001b[0m in \u001b[0;36m<module>\u001b[0;34m()\u001b[0m\n\u001b[0;32m----> 1\u001b[0;31m \u001b[0msolutions\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0msym\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0msolve\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0msimplified_e3\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mM\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m",
      "\u001b[0;32m~/anaconda3/lib/python3.6/site-packages/sympy/solvers/solvers.py\u001b[0m in \u001b[0;36msolve\u001b[0;34m(f, *symbols, **flags)\u001b[0m\n\u001b[1;32m   1063\u001b[0m     \u001b[0;31m###########################################################################\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1064\u001b[0m     \u001b[0;32mif\u001b[0m \u001b[0mbare_f\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 1065\u001b[0;31m         \u001b[0msolution\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0m_solve\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mf\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0;36m0\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m*\u001b[0m\u001b[0msymbols\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m**\u001b[0m\u001b[0mflags\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m   1066\u001b[0m     \u001b[0;32melse\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1067\u001b[0m         \u001b[0msolution\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0m_solve_system\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mf\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0msymbols\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m**\u001b[0m\u001b[0mflags\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m~/anaconda3/lib/python3.6/site-packages/sympy/solvers/solvers.py\u001b[0m in \u001b[0;36m_solve\u001b[0;34m(f, *symbols, **flags)\u001b[0m\n\u001b[1;32m   1354\u001b[0m             \u001b[0;31m# in any factor to zero\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1355\u001b[0m             \u001b[0mdens\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0m_simple_dens\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mf\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0msymbols\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 1356\u001b[0;31m             result = [s for s in result if\n\u001b[0m\u001b[1;32m   1357\u001b[0m                 all(not checksol(den, {symbol: s}, **flags) for den in\n\u001b[1;32m   1358\u001b[0m                 dens)]\n",
      "\u001b[0;32m~/anaconda3/lib/python3.6/site-packages/sympy/solvers/solvers.py\u001b[0m in \u001b[0;36m<listcomp>\u001b[0;34m(.0)\u001b[0m\n\u001b[1;32m   1356\u001b[0m             result = [s for s in result if\n\u001b[1;32m   1357\u001b[0m                 all(not checksol(den, {symbol: s}, **flags) for den in\n\u001b[0;32m-> 1358\u001b[0;31m                 dens)]\n\u001b[0m\u001b[1;32m   1359\u001b[0m         \u001b[0;31m# set flags for quick exit at end\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1360\u001b[0m         \u001b[0mcheck\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0;32mFalse\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m~/anaconda3/lib/python3.6/site-packages/sympy/solvers/solvers.py\u001b[0m in \u001b[0;36m<genexpr>\u001b[0;34m(.0)\u001b[0m\n\u001b[1;32m   1355\u001b[0m             \u001b[0mdens\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0m_simple_dens\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mf\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0msymbols\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1356\u001b[0m             result = [s for s in result if\n\u001b[0;32m-> 1357\u001b[0;31m                 all(not checksol(den, {symbol: s}, **flags) for den in\n\u001b[0m\u001b[1;32m   1358\u001b[0m                 dens)]\n\u001b[1;32m   1359\u001b[0m         \u001b[0;31m# set flags for quick exit at end\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m~/anaconda3/lib/python3.6/site-packages/sympy/solvers/solvers.py\u001b[0m in \u001b[0;36mchecksol\u001b[0;34m(f, symbol, sol, **flags)\u001b[0m\n\u001b[1;32m    245\u001b[0m             \u001b[0;32mif\u001b[0m \u001b[0mflags\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mget\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m'simplify'\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;32mTrue\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    246\u001b[0m                 \u001b[0;32mfor\u001b[0m \u001b[0mk\u001b[0m \u001b[0;32min\u001b[0m \u001b[0msol\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 247\u001b[0;31m                     \u001b[0msol\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0mk\u001b[0m\u001b[0;34m]\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0msimplify\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0msol\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0mk\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    248\u001b[0m             \u001b[0;31m# start over without the failed expanded form, possibly\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    249\u001b[0m             \u001b[0;31m# with a simplified solution\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m~/anaconda3/lib/python3.6/site-packages/sympy/simplify/simplify.py\u001b[0m in \u001b[0;36msimplify\u001b[0;34m(expr, ratio, measure, fu)\u001b[0m\n\u001b[1;32m    544\u001b[0m     \u001b[0mexpr\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mbottom_up\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mexpr\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;32mlambda\u001b[0m \u001b[0mw\u001b[0m\u001b[0;34m:\u001b[0m \u001b[0mw\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mnormal\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    545\u001b[0m     \u001b[0mexpr\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mMul\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0mpowsimp\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mexpr\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mas_content_primitive\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 546\u001b[0;31m     \u001b[0m_e\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mcancel\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mexpr\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    547\u001b[0m     \u001b[0mexpr1\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mshorter\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0m_e\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0m_mexpand\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0m_e\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mcancel\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m  \u001b[0;31m# issue 6829\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    548\u001b[0m     \u001b[0mexpr2\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mshorter\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mtogether\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mexpr\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mdeep\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0;32mTrue\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mtogether\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mexpr1\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mdeep\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0;32mTrue\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m~/anaconda3/lib/python3.6/site-packages/sympy/polys/polytools.py\u001b[0m in \u001b[0;36mcancel\u001b[0;34m(f, *gens, **args)\u001b[0m\n\u001b[1;32m   6432\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   6433\u001b[0m     \u001b[0;32mtry\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 6434\u001b[0;31m         \u001b[0;34m(\u001b[0m\u001b[0mF\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mG\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mopt\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mparallel_poly_from_expr\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mp\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mq\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m*\u001b[0m\u001b[0mgens\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m**\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m   6435\u001b[0m     \u001b[0;32mexcept\u001b[0m \u001b[0mPolificationFailed\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   6436\u001b[0m         \u001b[0;32mif\u001b[0m \u001b[0;32mnot\u001b[0m \u001b[0misinstance\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mf\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m(\u001b[0m\u001b[0mtuple\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mTuple\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m~/anaconda3/lib/python3.6/site-packages/sympy/polys/polytools.py\u001b[0m in \u001b[0;36mparallel_poly_from_expr\u001b[0;34m(exprs, *gens, **args)\u001b[0m\n\u001b[1;32m   4264\u001b[0m     \u001b[0;34m\"\"\"Construct polynomials from expressions. \"\"\"\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   4265\u001b[0m     \u001b[0mopt\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0moptions\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mbuild_options\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mgens\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0margs\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 4266\u001b[0;31m     \u001b[0;32mreturn\u001b[0m \u001b[0m_parallel_poly_from_expr\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mexprs\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mopt\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m   4267\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   4268\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m~/anaconda3/lib/python3.6/site-packages/sympy/polys/polytools.py\u001b[0m in \u001b[0;36m_parallel_poly_from_expr\u001b[0;34m(exprs, opt)\u001b[0m\n\u001b[1;32m   4303\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   4304\u001b[0m                 \u001b[0;32mif\u001b[0m \u001b[0mopt\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mexpand\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 4305\u001b[0;31m                     \u001b[0mexpr\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mexpr\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mexpand\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m   4306\u001b[0m         \u001b[0;32melse\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   4307\u001b[0m             \u001b[0mfailed\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0;32mTrue\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m~/anaconda3/lib/python3.6/site-packages/sympy/core/expr.py\u001b[0m in \u001b[0;36mexpand\u001b[0;34m(self, deep, modulus, power_base, power_exp, mul, log, multinomial, basic, **hints)\u001b[0m\n\u001b[1;32m   3049\u001b[0m             \u001b[0;32mif\u001b[0m \u001b[0mhints\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mget\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m'multinomial'\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;32mFalse\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   3050\u001b[0m                 expr, _ = Expr._expand_hint(\n\u001b[0;32m-> 3051\u001b[0;31m                     expr, '_eval_expand_multinomial', deep=deep, **hints)\n\u001b[0m\u001b[1;32m   3052\u001b[0m             \u001b[0;32mif\u001b[0m \u001b[0mhints\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mget\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m'mul'\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;32mFalse\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   3053\u001b[0m                 expr, _ = Expr._expand_hint(\n",
      "\u001b[0;32m~/anaconda3/lib/python3.6/site-packages/sympy/core/expr.py\u001b[0m in \u001b[0;36m_expand_hint\u001b[0;34m(expr, hint, deep, **hints)\u001b[0m\n\u001b[1;32m   2976\u001b[0m             \u001b[0msargs\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0;34m[\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   2977\u001b[0m             \u001b[0;32mfor\u001b[0m \u001b[0marg\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mexpr\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 2978\u001b[0;31m                 \u001b[0marg\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0marghit\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mExpr\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_expand_hint\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0marg\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mhint\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m**\u001b[0m\u001b[0mhints\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m   2979\u001b[0m                 \u001b[0mhit\u001b[0m \u001b[0;34m|=\u001b[0m \u001b[0marghit\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   2980\u001b[0m                 \u001b[0msargs\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mappend\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0marg\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m~/anaconda3/lib/python3.6/site-packages/sympy/core/expr.py\u001b[0m in \u001b[0;36m_expand_hint\u001b[0;34m(expr, hint, deep, **hints)\u001b[0m\n\u001b[1;32m   2976\u001b[0m             \u001b[0msargs\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0;34m[\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   2977\u001b[0m             \u001b[0;32mfor\u001b[0m \u001b[0marg\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mexpr\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 2978\u001b[0;31m                 \u001b[0marg\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0marghit\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mExpr\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_expand_hint\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0marg\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mhint\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m**\u001b[0m\u001b[0mhints\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m   2979\u001b[0m                 \u001b[0mhit\u001b[0m \u001b[0;34m|=\u001b[0m \u001b[0marghit\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   2980\u001b[0m                 \u001b[0msargs\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mappend\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0marg\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m~/anaconda3/lib/python3.6/site-packages/sympy/core/expr.py\u001b[0m in \u001b[0;36m_expand_hint\u001b[0;34m(expr, hint, deep, **hints)\u001b[0m\n\u001b[1;32m   2976\u001b[0m             \u001b[0msargs\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0;34m[\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   2977\u001b[0m             \u001b[0;32mfor\u001b[0m \u001b[0marg\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mexpr\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 2978\u001b[0;31m                 \u001b[0marg\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0marghit\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mExpr\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_expand_hint\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0marg\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mhint\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m**\u001b[0m\u001b[0mhints\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m   2979\u001b[0m                 \u001b[0mhit\u001b[0m \u001b[0;34m|=\u001b[0m \u001b[0marghit\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   2980\u001b[0m                 \u001b[0msargs\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mappend\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0marg\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m~/anaconda3/lib/python3.6/site-packages/sympy/core/expr.py\u001b[0m in \u001b[0;36m_expand_hint\u001b[0;34m(expr, hint, deep, **hints)\u001b[0m\n\u001b[1;32m   2976\u001b[0m             \u001b[0msargs\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0;34m[\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   2977\u001b[0m             \u001b[0;32mfor\u001b[0m \u001b[0marg\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mexpr\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 2978\u001b[0;31m                 \u001b[0marg\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0marghit\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mExpr\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_expand_hint\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0marg\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mhint\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m**\u001b[0m\u001b[0mhints\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m   2979\u001b[0m                 \u001b[0mhit\u001b[0m \u001b[0;34m|=\u001b[0m \u001b[0marghit\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   2980\u001b[0m                 \u001b[0msargs\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mappend\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0marg\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m~/anaconda3/lib/python3.6/site-packages/sympy/core/expr.py\u001b[0m in \u001b[0;36m_expand_hint\u001b[0;34m(expr, hint, deep, **hints)\u001b[0m\n\u001b[1;32m   2976\u001b[0m             \u001b[0msargs\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0;34m[\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   2977\u001b[0m             \u001b[0;32mfor\u001b[0m \u001b[0marg\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mexpr\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 2978\u001b[0;31m                 \u001b[0marg\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0marghit\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mExpr\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_expand_hint\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0marg\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mhint\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m**\u001b[0m\u001b[0mhints\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m   2979\u001b[0m                 \u001b[0mhit\u001b[0m \u001b[0;34m|=\u001b[0m \u001b[0marghit\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   2980\u001b[0m                 \u001b[0msargs\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mappend\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0marg\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m~/anaconda3/lib/python3.6/site-packages/sympy/core/expr.py\u001b[0m in \u001b[0;36m_expand_hint\u001b[0;34m(expr, hint, deep, **hints)\u001b[0m\n\u001b[1;32m   2976\u001b[0m             \u001b[0msargs\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0;34m[\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   2977\u001b[0m             \u001b[0;32mfor\u001b[0m \u001b[0marg\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mexpr\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 2978\u001b[0;31m                 \u001b[0marg\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0marghit\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mExpr\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_expand_hint\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0marg\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mhint\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m**\u001b[0m\u001b[0mhints\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m   2979\u001b[0m                 \u001b[0mhit\u001b[0m \u001b[0;34m|=\u001b[0m \u001b[0marghit\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   2980\u001b[0m                 \u001b[0msargs\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mappend\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0marg\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m~/anaconda3/lib/python3.6/site-packages/sympy/core/expr.py\u001b[0m in \u001b[0;36m_expand_hint\u001b[0;34m(expr, hint, deep, **hints)\u001b[0m\n\u001b[1;32m   2976\u001b[0m             \u001b[0msargs\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0;34m[\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   2977\u001b[0m             \u001b[0;32mfor\u001b[0m \u001b[0marg\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mexpr\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 2978\u001b[0;31m                 \u001b[0marg\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0marghit\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mExpr\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_expand_hint\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0marg\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mhint\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m**\u001b[0m\u001b[0mhints\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m   2979\u001b[0m                 \u001b[0mhit\u001b[0m \u001b[0;34m|=\u001b[0m \u001b[0marghit\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   2980\u001b[0m                 \u001b[0msargs\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mappend\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0marg\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m~/anaconda3/lib/python3.6/site-packages/sympy/core/expr.py\u001b[0m in \u001b[0;36m_expand_hint\u001b[0;34m(expr, hint, deep, **hints)\u001b[0m\n\u001b[1;32m   2976\u001b[0m             \u001b[0msargs\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0;34m[\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   2977\u001b[0m             \u001b[0;32mfor\u001b[0m \u001b[0marg\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mexpr\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 2978\u001b[0;31m                 \u001b[0marg\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0marghit\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mExpr\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_expand_hint\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0marg\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mhint\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m**\u001b[0m\u001b[0mhints\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m   2979\u001b[0m                 \u001b[0mhit\u001b[0m \u001b[0;34m|=\u001b[0m \u001b[0marghit\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   2980\u001b[0m                 \u001b[0msargs\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mappend\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0marg\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m~/anaconda3/lib/python3.6/site-packages/sympy/core/expr.py\u001b[0m in \u001b[0;36m_expand_hint\u001b[0;34m(expr, hint, deep, **hints)\u001b[0m\n\u001b[1;32m   2976\u001b[0m             \u001b[0msargs\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0;34m[\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   2977\u001b[0m             \u001b[0;32mfor\u001b[0m \u001b[0marg\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mexpr\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 2978\u001b[0;31m                 \u001b[0marg\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0marghit\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mExpr\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_expand_hint\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0marg\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mhint\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m**\u001b[0m\u001b[0mhints\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m   2979\u001b[0m                 \u001b[0mhit\u001b[0m \u001b[0;34m|=\u001b[0m \u001b[0marghit\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   2980\u001b[0m                 \u001b[0msargs\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mappend\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0marg\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m~/anaconda3/lib/python3.6/site-packages/sympy/core/expr.py\u001b[0m in \u001b[0;36m_expand_hint\u001b[0;34m(expr, hint, deep, **hints)\u001b[0m\n\u001b[1;32m   2976\u001b[0m             \u001b[0msargs\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0;34m[\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   2977\u001b[0m             \u001b[0;32mfor\u001b[0m \u001b[0marg\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mexpr\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 2978\u001b[0;31m                 \u001b[0marg\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0marghit\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mExpr\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_expand_hint\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0marg\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mhint\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m**\u001b[0m\u001b[0mhints\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m   2979\u001b[0m                 \u001b[0mhit\u001b[0m \u001b[0;34m|=\u001b[0m \u001b[0marghit\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   2980\u001b[0m                 \u001b[0msargs\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mappend\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0marg\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m~/anaconda3/lib/python3.6/site-packages/sympy/core/expr.py\u001b[0m in \u001b[0;36m_expand_hint\u001b[0;34m(expr, hint, deep, **hints)\u001b[0m\n\u001b[1;32m   2976\u001b[0m             \u001b[0msargs\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0;34m[\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   2977\u001b[0m             \u001b[0;32mfor\u001b[0m \u001b[0marg\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mexpr\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 2978\u001b[0;31m                 \u001b[0marg\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0marghit\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mExpr\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_expand_hint\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0marg\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mhint\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m**\u001b[0m\u001b[0mhints\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m   2979\u001b[0m                 \u001b[0mhit\u001b[0m \u001b[0;34m|=\u001b[0m \u001b[0marghit\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   2980\u001b[0m                 \u001b[0msargs\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mappend\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0marg\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m~/anaconda3/lib/python3.6/site-packages/sympy/core/expr.py\u001b[0m in \u001b[0;36m_expand_hint\u001b[0;34m(expr, hint, deep, **hints)\u001b[0m\n\u001b[1;32m   2976\u001b[0m             \u001b[0msargs\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0;34m[\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   2977\u001b[0m             \u001b[0;32mfor\u001b[0m \u001b[0marg\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mexpr\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 2978\u001b[0;31m                 \u001b[0marg\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0marghit\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mExpr\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_expand_hint\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0marg\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mhint\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m**\u001b[0m\u001b[0mhints\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m   2979\u001b[0m                 \u001b[0mhit\u001b[0m \u001b[0;34m|=\u001b[0m \u001b[0marghit\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   2980\u001b[0m                 \u001b[0msargs\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mappend\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0marg\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m~/anaconda3/lib/python3.6/site-packages/sympy/core/expr.py\u001b[0m in \u001b[0;36m_expand_hint\u001b[0;34m(expr, hint, deep, **hints)\u001b[0m\n\u001b[1;32m   2973\u001b[0m         \u001b[0;31m#              |\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   2974\u001b[0m         \u001b[0;31m#              V\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 2975\u001b[0;31m         \u001b[0;32mif\u001b[0m \u001b[0mdeep\u001b[0m \u001b[0;32mand\u001b[0m \u001b[0mgetattr\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mexpr\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m'args'\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;32mand\u001b[0m \u001b[0;32mnot\u001b[0m \u001b[0mexpr\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mis_Atom\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m   2976\u001b[0m             \u001b[0msargs\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0;34m[\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   2977\u001b[0m             \u001b[0;32mfor\u001b[0m \u001b[0marg\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mexpr\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;31mKeyboardInterrupt\u001b[0m: "
     ]
    }
   ],
   "source": [
    "solutions = sym.solve(simplified_e3, M)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
 ],
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