{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Simplification"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "from sympy import *\n",
    "x, y, z = symbols('x y z')\n",
    "init_printing()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "For each exercise, fill in the function according to its docstring."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Polynomial/Rational Function Simplification"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In each exercise, apply specific simplification functions to get the desired result."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "def polysimp1(expr):\n",
    "    \"\"\"\n",
    "    >>> polysimp1(cos(x)*sin(x) + cos(x))\n",
    "    (sin(x) + 1)*cos(x)\n",
    "    >>> polysimp1(cos(x)*sin(x) + cos(x) + 1)\n",
    "    (sin(x) + 1)*cos(x) + 1\n",
    "    \"\"\"\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "polysimp1(cos(x)*sin(x) + cos(x))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "polysimp1(cos(x)*sin(x) + cos(x) + 1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "def polysimp2(expr):\n",
    "    \"\"\"\n",
    "    >>> polysimp2((2*x + 1)/(x**2 + x))\n",
    "    1/(x + 1) + 1/x\n",
    "    >>> polysimp2((x**2 + 3*x + 1)/(x**3 + 2*x**2 + x))\n",
    "    1/(x**2 + 2*x + 1) + 1/x\n",
    "    \"\"\"\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "polysimp2((2*x + 1)/(x**2 + x))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "polysimp2((x**2 + 3*x + 1)/(x**3 + 2*x**2 + x))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Powers"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In each exercise, apply specific simplification functions to get the desired result. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "def powersimp1(expr):\n",
    "    \"\"\"\n",
    "    >>> powersimp1(exp(x)*(exp(y) + 1))\n",
    "    exp(x) + exp(x + y)\n",
    "    \"\"\"\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "powersimp1(exp(x)*(exp(y) + 1))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "def powersimp2(expr):\n",
    "    \"\"\"\n",
    "    >>> powersimp2(2**x*x**x)\n",
    "    (2*x)**x\n",
    "    >>> powersimp2(x**x*x**x)\n",
    "    (x**2)**x\n",
    "    \"\"\"\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "powersimp2(2**x*x**x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "powersimp2(x**x*x**x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "def powersimp3(expr):\n",
    "    \"\"\"\n",
    "    >>> a, b, c = symbols('a b c')\n",
    "    >>> powersimp3((a**b)**c)\n",
    "    a**(b*c)\n",
    "    >>> powersimp3((a**b)**(c + 1))\n",
    "    a**(b*c + b)\n",
    "    \"\"\"\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "a, b, c = symbols('a b c')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "powersimp3((a**b)**c)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "powersimp3((a**b)**(c + 1))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Logs"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "def logsimp1(expr):\n",
    "    \"\"\"\n",
    "    >>> a, b = symbols('a b', positive=True)\n",
    "    >>> logsimp1(log(x**y*a**b))\n",
    "    y*log(x) + log(a**b)\n",
    "    >>> logsimp1(log(x*y*a*b))\n",
    "    log(x) + log(y) + log(a*b)\n",
    "    \"\"\"\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "a, b = symbols('a b', positive=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "logsimp1(log(x**y*a**b))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "logsimp1(log(x*y*a*b))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Miscellaneous  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "def miscsimp1(expr):\n",
    "    \"\"\"\n",
    "    >>> miscsimp1(sin(x + y))\n",
    "    2*(-tan(x/2)**2 + 1)*tan(y/2)/((tan(x/2)**2 + 1)*(tan(y/2)**2 + 1)) + 2*(-tan(y/2)**2 + 1)*tan(x/2)/((tan(x/2)**2 + 1)*(tan(y/2)**2 + 1))\n",
    "    \"\"\"\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "miscsimp1(sin(x + y))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "def miscsimp2(expr):\n",
    "    \"\"\"\n",
    "    >>> miscsimp2(gamma(x + 4))\n",
    "    x**4*gamma(x) + 6*x**3*gamma(x) + 11*x**2*gamma(x) + 6*x*gamma(x)\n",
    "    \"\"\"\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "miscsimp2(gamma(x + 4))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Continued Fractions"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "If we do not cover this, see http://asmeurer.github.io/scipy-2014-tutorial/html/tutorial/simplification.html#example-continued-fractions."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "def list_to_frac(l):\n",
    "    expr = Integer(0)\n",
    "    for i in reversed(l[1:]):\n",
    "        expr += i\n",
    "        expr = 1/expr\n",
    "    return l[0] + expr"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "a0, a1, a2, a3, a4 = symbols('a0:5')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Determine the list used to create the continued fraction $$\\frac{a_{0} a_{1} a_{2} a_{3} a_{4} + a_{0} a_{1} a_{2} + a_{0} a_{3} a_{4} + a_{0} + a_{1} a_{2} a_{3} + a_{1} a_{3} a_{4} + a_{1} + a_{3}}{a_{0} a_{1} a_{2} a_{4} + a_{0} a_{4} + a_{1} a_{2} + a_{1} a_{4} + 1}.$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "def continued_frac():\n",
    "    \"\"\"\n",
    "    Determine the original list used to create the fraction.  \n",
    "\n",
    "    Return the original list from this function.\n",
    "\n",
    "    >>> orig_frac = (a0*a1*a2*a3*a4 + a0*a1*a2 + a0*a3*a4 + a0 + a1*a2*a3 + a1*a3*a4 + a1 + a3)/(a0*a1*a2*a4 + a0*a4 + a1*a2 + a1*a4 + 1)\n",
    "    >>> pprint(orig_frac)\n",
    "    a₀⋅a₁⋅a₂⋅a₃⋅a₄ + a₀⋅a₁⋅a₂ + a₀⋅a₃⋅a₄ + a₀ + a₁⋅a₂⋅a₃ + a₁⋅a₃⋅a₄ + a₁ + a₃\n",
    "    ─────────────────────────────────────────────────────────────────────────\n",
    "                     a₀⋅a₁⋅a₂⋅a₄ + a₀⋅a₄ + a₁⋅a₂ + a₁⋅a₄ + 1\n",
    "    >>> cancel(list_to_frac(continued_frac())) == orig_frac\n",
    "    True\n",
    "    \"\"\"\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "orig_frac = (a0*a1*a2*a3*a4 + a0*a1*a2 + a0*a3*a4 + a0 + a1*a2*a3 + a1*a3*a4 + a1 + a3)/(a0*a1*a2*a4 + a0*a4 + a1*a2 + a1*a4 + 1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "orig_frac"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "cancel(list_to_frac(continued_frac())) == orig_frac"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.6.1"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 1
}