notebooks/expr.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# [Goulib](../notebook.ipynb).expr\n",
    "\n",
    "math expressions\n",
    "(see [polynomials](polynomials.ipynb) for more)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "from Goulib.notebook import *\n",
    "from Goulib.expr import *\n",
    "from Goulib.table import Table\n",
    "from math import pi,sin\n",
    "from Goulib.math2 import sqrt"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Expr can be created from:\n",
    "* numbers\n",
    "* text (python-like formula)\n",
    "\n",
    "Expr has LaTeX representation in notebooks"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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       "${243}$"
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       "${243}$"
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       "243"
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     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Expr(3**5) # is evaluated before Expr is created"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "as LaTeX (default) : ${3^5}$"
      ],
      "text/plain": [
       "<IPython.core.display.HTML object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "as math formula : 3^5\n",
      "as python code : 3**5\n",
      "evaluated : 243\n"
     ]
    }
   ],
   "source": [
    "e=Expr(3)**Expr(5) # combine Expr essions\n",
    "h('as LaTeX (default) :',e)\n",
    "print('as math formula :',e)\n",
    "print('as python code :',repr(e))\n",
    "print('evaluated :',e())\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "as LaTeX (default) : ${5^3-3^2}$"
      ],
      "text/plain": [
       "<IPython.core.display.HTML object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "as math formula : 5^3-3^2\n",
      "as python code : 5**3-3**2\n",
      "evaluated : 116\n"
     ]
    }
   ],
   "source": [
    "e=Expr('5**3+(-3^2)') # ^ and ** are both considered as power operator for clarity+compatibility\n",
    "h('as LaTeX (default) :',e)\n",
    "print('as math formula :',e)\n",
    "print('as python code :',repr(e))\n",
    "print('evaluated :',e())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Functions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "an Expr may contain variables : ${x^2}$"
      ],
      "text/plain": [
       "<IPython.core.display.HTML object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
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     "data": {
      "text/html": [
       "and be evaluated as a function : for x=2,  ${x^2}$ = 4"
      ],
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       "<IPython.core.display.HTML object>"
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     "metadata": {},
     "output_type": "display_data"
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   ],
   "source": [
    "e=Expr('x^2')\n",
    "h('an Expr may contain variables :',e)\n",
    "h('and be evaluated as a function : for x=2, ',e,'=',e(x=2))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "all functions defined in math module can be used:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table><tr><td>abs</td><td>abs(x)</td><td>abs(x)</td><td>${\\lvert{x}\\rvert}$</td></tr>\n",
       "<tr><td>acos</td><td>acos(x)</td><td>acos(x)</td><td>${\\arccos\\left(x\\right)}$</td></tr>\n",
       "<tr><td>acosh</td><td>acosh(x)</td><td>acosh(x)</td><td>${\\cosh^{-1}\\left(x\\right)}$</td></tr>\n",
       "<tr><td>asin</td><td>asin(x)</td><td>asin(x)</td><td>${\\arcsin\\left(x\\right)}$</td></tr>\n",
       "<tr><td>asinh</td><td>asinh(x)</td><td>asinh(x)</td><td>${\\sinh^{-1}\\left(x\\right)}$</td></tr>\n",
       "<tr><td>atan</td><td>atan(x)</td><td>atan(x)</td><td>${\\arctan\\left(x\\right)}$</td></tr>\n",
       "<tr><td>atan2</td><td>atan2(x)</td><td>atan2(x)</td><td>${\\atan2\\left(x\\right)}$</td></tr>\n",
       "<tr><td>atanh</td><td>atanh(x)</td><td>atanh(x)</td><td>${\\tanh^{-1}\\left(x\\right)}$</td></tr>\n",
       "<tr><td>ceil</td><td>ceil(x)</td><td>ceil(x)</td><td>${\\left\\lceil{x}\\right\\rceil}$</td></tr>\n",
       "<tr><td>copysign</td><td>copysign(x)</td><td>copysign(x)</td><td>${\\copysign\\left(x\\right)}$</td></tr>\n",
       "<tr><td>cos</td><td>cos(x)</td><td>cos(x)</td><td>${\\cos\\left(x\\right)}$</td></tr>\n",
       "<tr><td>cosh</td><td>cosh(x)</td><td>cosh(x)</td><td>${\\cosh\\left(x\\right)}$</td></tr>\n",
       "<tr><td>degrees</td><td>degrees(x)</td><td>degrees(x)</td><td>${x\\cdot\\frac{360}{2\\pi}}$</td></tr>\n",
       "<tr><td>erf</td><td>erf(x)</td><td>erf(x)</td><td>${\\erf\\left(x\\right)}$</td></tr>\n",
       "<tr><td>erfc</td><td>erfc(x)</td><td>erfc(x)</td><td>${\\erfc\\left(x\\right)}$</td></tr>\n",
       "<tr><td>exp</td><td>exp(x)</td><td>exp(x)</td><td>${e^{x}}$</td></tr>\n",
       "<tr><td>expm1</td><td>expm1(x)</td><td>expm1(x)</td><td>${e^{x}-1}$</td></tr>\n",
       "<tr><td>fabs</td><td>fabs(x)</td><td>fabs(x)</td><td>${\\lvert{x}\\rvert}$</td></tr>\n",
       "<tr><td>factorial</td><td>fact(x)</td><td>x!</td><td>${x!}$</td></tr>\n",
       "<tr><td>factorial2</td><td>'factorialk'</td></tr>\n",
       "<tr><td>floor</td><td>floor(x)</td><td>floor(x)</td><td>${\\left\\lfloor{x}\\right\\rfloor}$</td></tr>\n",
       "<tr><td>fmod</td><td>fmod(x)</td><td>fmod(x)</td><td>${\\fmod\\left(x\\right)}$</td></tr>\n",
       "<tr><td>frexp</td><td>frexp(x)</td><td>frexp(x)</td><td>${\\frexp\\left(x\\right)}$</td></tr>\n",
       "<tr><td>fsum</td><td>fsum(x)</td><td>fsum(x)</td><td>${\\fsum\\left(x\\right)}$</td></tr>\n",
       "<tr><td>gamma</td><td>gamma(x)</td><td>gamma(x)</td><td>${\\Gamma\\left(x\\right)}$</td></tr>\n",
       "<tr><td>gcd</td><td>gcd(x)</td><td>gcd(x)</td><td>${\\gcd\\left(x\\right)}$</td></tr>\n",
       "<tr><td>hypot</td><td>hypot(x)</td><td>hypot(x)</td><td>${\\hypot\\left(x\\right)}$</td></tr>\n",
       "<tr><td>isclose</td><td>isclose(x)</td><td>isclose(x)</td><td>${\\isclose\\left(x\\right)}$</td></tr>\n",
       "<tr><td>isfinite</td><td>isfinite(x)</td><td>isfinite(x)</td><td>${\\isfinite\\left(x\\right)}$</td></tr>\n",
       "<tr><td>isinf</td><td>isinf(x)</td><td>isinf(x)</td><td>${\\isinf\\left(x\\right)}$</td></tr>\n",
       "<tr><td>isnan</td><td>isnan(x)</td><td>isnan(x)</td><td>${\\isnan\\left(x\\right)}$</td></tr>\n",
       "<tr><td>ldexp</td><td>ldexp(x)</td><td>ldexp(x)</td><td>${\\ldexp\\left(x\\right)}$</td></tr>\n",
       "<tr><td>lgamma</td><td>log(abs(gamma(x)))</td><td>log(abs(gamma(x)))</td><td>${\\ln\\lvert\\Gamma\\left({x}\\rvert)\\right)}$</td></tr>\n",
       "<tr><td>log</td><td>log(x)</td><td>log(x)</td><td>${\\ln\\left(x\\right)}$</td></tr>\n",
       "<tr><td>log10</td><td>log10(x)</td><td>log10(x)</td><td>${\\log_{10}\\left(x\\right)}$</td></tr>\n",
       "<tr><td>log1p</td><td>log1p(x)</td><td>log1p(x)</td><td>${\\ln\\left(1-{x}\\rvert)}$</td></tr>\n",
       "<tr><td>log2</td><td>log2(x)</td><td>log2(x)</td><td>${\\log_2\\left(x\\right)}$</td></tr>\n",
       "<tr><td>modf</td><td>modf(x)</td><td>modf(x)</td><td>${\\modf\\left(x\\right)}$</td></tr>\n",
       "<tr><td>pow</td><td>pow(x)</td><td>pow(x)</td><td>${\\pow\\left(x\\right)}$</td></tr>\n",
       "<tr><td>radians</td><td>radians(x)</td><td>radians(x)</td><td>${x\\cdot\\frac{2\\pi}{360}}$</td></tr>\n",
       "<tr><td>remainder</td><td>remainder(x)</td><td>remainder(x)</td><td>${\\remainder\\left(x\\right)}$</td></tr>\n",
       "<tr><td>sin</td><td>sin(x)</td><td>sin(x)</td><td>${\\sin\\left(x\\right)}$</td></tr>\n",
       "<tr><td>sinh</td><td>sinh(x)</td><td>sinh(x)</td><td>${\\sinh\\left(x\\right)}$</td></tr>\n",
       "<tr><td>sqrt</td><td>sqrt(x)</td><td>sqrt(x)</td><td>${\\sqrt{x}}$</td></tr>\n",
       "<tr><td>tan</td><td>tan(x)</td><td>tan(x)</td><td>${\\tan\\left(x\\right)}$</td></tr>\n",
       "<tr><td>tanh</td><td>tanh(x)</td><td>tanh(x)</td><td>${\\tanh\\left(x\\right)}$</td></tr>\n",
       "<tr><td>trunc</td><td>trunc(x)</td><td>trunc(x)</td><td>${\\left\\lfloor{x}\\right\\rfloor}$</td></tr>\n",
       "</table>\n"
      ],
      "text/plain": [
       "Table(len=47,titles=[],data=[['abs', 'abs(x)', 'abs(x)', abs(x)], ['acos', 'acos(x)', 'acos(x)', acos(x)], ['acosh', 'acosh(x)', 'acosh(x)', acosh(x)], ['asin', 'asin(x)', 'asin(x)', asin(x)], ['asinh', 'asinh(x)', 'asinh(x)', asinh(x)]])"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "functions=default_context.functions # functions known by default to Expr\n",
    "t=Table()\n",
    "for n in functions:\n",
    "    f=functions[n][0] # get the function itself\n",
    "    try:\n",
    "        e=Expr(f)\n",
    "        t.append([n,repr(e),str(e),e])\n",
    "    except Exception as e:\n",
    "        t.append([n,e])\n",
    "t"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "as LaTeX (default) : ${\\sqrt{x}}$"
      ],
      "text/plain": [
       "<IPython.core.display.HTML object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "as math formula : sqrt(x)\n",
      "as python code : sqrt(x)\n",
      "evaluated : sqrt(x)\n"
     ]
    }
   ],
   "source": [
    "e=Expr(sqrt) #(Expr(pi))+Expr(1/5)\n",
    "h('as LaTeX (default) :',e)\n",
    "print('as math formula :',e)\n",
    "print('as python code :',repr(e))\n",
    "print('evaluated :',e())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/html": [
       "${3x+2}$"
      ],
      "text/latex": [
       "${3x+2}$"
      ],
      "text/plain": [
       "3*x+2"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "e1=Expr('3*x+2') #a very simple expression defined from text\n",
    "e1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
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      "text/html": [
       "${3x+2}$"
      ],
      "text/latex": [
       "${3x+2}$"
      ],
      "text/plain": [
       "3*x+2"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "e1=Expr(lambda x:3*x+2) #the same expression defined from a lambda function\n",
    "e1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
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      "text/html": [
       "${3x+2}$"
      ],
      "text/latex": [
       "${3x+2}$"
      ],
      "text/plain": [
       "3*x+2"
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     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "def f(x):\n",
    "    return 3*x+2\n",
    "Expr(f) #the same expression defined from a regular (simple...) function"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
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       "${\\sqrt{3x+2}}$"
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      "text/latex": [
       "${\\sqrt{3x+2}}$"
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   "source": [
    "e3=Expr(sqrt)(e1) #Expr can be composed\n",
    "e3"
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     "text": [
      "2.23606797749979\n",
      "1.2533141373155001\n"
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   "source": [
    "print(e3(x=1)) # Expr can be evaluated as a function\n",
    "print(e3((pi-4)/6)) #the x variable is implicit"
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     "execution_count": 14,
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   "source": [
    "e3.plot()  # Expr can be also plotted. Note the X axis is automatically restricted to the definition domain"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
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       "   </g>\n",
       "   <g id=\"text_19\">\n",
       "    <!-- ${\\frac{1}{x}}$ -->\n",
       "    <defs>\n",
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       "     <use transform=\"translate(0 43.965625)scale(0.7)\" xlink:href=\"#DejaVuSans-49\"/>\n",
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       " <defs>\n",
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      ],
      "text/plain": [
       "<IPython.core.display.SVG object>"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Expr('1/x').plot(x=range(-100,100))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## multivariable"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
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      "text/html": [
       "${\\frac{-b+\\sqrt{b^2-4ac}}{2a}}$"
      ],
      "text/latex": [
       "${\\frac{-b+\\sqrt{b^2-4ac}}{2a}}$"
      ],
      "text/plain": [
       "(-b+sqrt(b**2-4*a*c))/(2*a)"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "e=Expr('(-b+sqrt(b^2-4*a*c))/(2*a)') #laTex is rendered with some simple simplifications\n",
    "e"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
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      "text/html": [
       "${\\frac{-b+\\sqrt{b^2-4ac}}{2a}}$"
      ],
      "text/latex": [
       "${\\frac{-b+\\sqrt{b^2-4ac}}{2a}}$"
      ],
      "text/plain": [
       "(-b+sqrt(b**2-4*a*c))/(2*a)"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "e(a=1) # substitution doesn't work yet ..."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## complex"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
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      "text/html": [
       "${e^{ipi}}$"
      ],
      "text/latex": [
       "${e^{ipi}}$"
      ],
      "text/plain": [
       "e**(i*pi)"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "e=Expr(\"e**(i*pi)\")\n",
    "e"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
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     "execution_count": 19,
     "metadata": {},
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   "source": [
    "e() # should be -1 one day..."
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