{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# The Taylor Series\n",
    "** October 2017 **\n",
    "\n",
    "** Andrew Riberio @ [AndrewRib.com](http://www.andrewrib.com) **\n",
    "\n",
    "In this notebook we will explore the taylor series using sympy. \n",
    "\n",
    "Resources:\n",
    "* https://en.wikipedia.org/wiki/Series_(mathematics)\n",
    "* https://en.wikipedia.org/wiki/Taylor_series"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Libraries"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import sympy as sp\n",
    "import sympy.functions.combinatorial.factorials as fact\n",
    "import matplotlib.pyplot as plt\n",
    "import numpy as np\n",
    "from IPython.display import display\n",
    "from decimal import *\n",
    "\n",
    "# Pretty Latex printing for Sympy with an argument to show the polynomials the way we write them \n",
    "# from small term to larger. May give you problems if you don't have a latex distribution. \n",
    "sp.init_printing(order='rev-lex',use_latex='mathjax')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Taylor Series\n",
    "The taylor series is an intriguing power series that makes use of acending derivitives to approximate a function. \n",
    "Please see the [Taylor Series](https://en.wikipedia.org/wiki/Taylor_series) wiki for a formal definition. Here we will implement the taylor series function and use it to approximate various functions. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def taylor(fn,depth, x=None,a = None, printSteps = True):\n",
    "    # If no x or a is provided, we will be symbolically computing the taylor series. \n",
    "    if(x == None):\n",
    "        x = sp.symbols('x')\n",
    "    if(a == None):\n",
    "        a = sp.symbols('a')\n",
    "    \n",
    "    # The start of the taylor series \n",
    "    result = fn(a)\n",
    "    \n",
    "    if depth == 0:\n",
    "        return result \n",
    "    elif depth > 0:\n",
    "        cummulativeDYDX = fn(a)\n",
    "        for i in range(depth):\n",
    "            cummulativeDYDX = sp.diff(cummulativeDYDX)\n",
    "            \n",
    "            if(printSteps):\n",
    "                print(\"Derivitive of order d^{0}y/dx^[0|: {1}\".format(i+1,cummulativeDYDX))\n",
    "                      \n",
    "            result = result + (cummulativeDYDX  / fact.factorial(i+1))*(x - a)**(i+1)\n",
    "                      \n",
    "        return result\n",
    "    else:\n",
    "        # Malformed case.\n",
    "        return 0"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Consider the beloved Euler Number:  $e^{x}$. It has a most interesting taylor series. Let's show it here expanded to 5 steps. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Derivitive of order d^1y/dx^[0|: exp(a)\n",
      "Derivitive of order d^2y/dx^[0|: exp(a)\n",
      "Derivitive of order d^3y/dx^[0|: exp(a)\n",
      "Derivitive of order d^4y/dx^[0|: exp(a)\n",
      "Derivitive of order d^5y/dx^[0|: exp(a)\n"
     ]
    },
    {
     "data": {
      "text/latex": [
       "$$e^{a} + \\left(x - a\\right) e^{a} + \\frac{e^{a}}{2} \\left(x - a\\right)^{2} + \\frac{e^{a}}{6} \\left(x - a\\right)^{3} + \\frac{e^{a}}{24} \\left(x - a\\right)^{4} + \\frac{e^{a}}{120} \\left(x - a\\right)^{5}$$"
      ],
      "text/plain": [
       "                         2  a          3  a          4  a          5  a\n",
       " a            a   (x - a) ⋅ℯ    (x - a) ⋅ℯ    (x - a) ⋅ℯ    (x - a) ⋅ℯ \n",
       "ℯ  + (x - a)⋅ℯ  + ─────────── + ─────────── + ─────────── + ───────────\n",
       "                       2             6             24           120    "
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "eTaylor5 = taylor(sp.exp,5)\n",
    "eTaylor5 "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As we see, one of the nifty features of the $e^{x}$ is that it is equal to its derivative. Thus, when we chain higher order derivatives of $e^{x}$, we simply get the same thing ( as shown in the printout). This leads to the simplest type of taylor series, as the derivitive term remains constant. \n",
    "\n",
    "So what does this taylor series of $e^{x}$ look like with only 5 terms? We will make use of sympy's symbolic computation abilities to substitute in values for x and a.\n",
    "\n",
    "In the case of a = 0, we get:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$1 + x + \\frac{x^{2}}{2} + \\frac{x^{3}}{6} + \\frac{x^{4}}{24} + \\frac{x^{5}}{120}$$"
      ],
      "text/plain": [
       "         2    3    4     5\n",
       "        x    x    x     x \n",
       "1 + x + ── + ── + ── + ───\n",
       "        2    6    24   120"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x, a = sp.symbols('x a')\n",
    "eTaylor5_0 = eTaylor5.subs(a,0)\n",
    "eTaylor5_0 "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "** Theorem **: When we let a = 0, and x = 1, the taylor series as its number of terms approaches infinity = $e$ and is [the definition of $e$](https://en.wikipedia.org/wiki/Exponential_function)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
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7sN8G+r6hlPK3Sf44yR2N1/FC6rWrDko9Ccr/Gsvr2ID/lOTyJBeVUq5MHQo3Sj3ZwltT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\nA6isswAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x2158c951470>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "xs      = np.arange(-2,4,0.1)\n",
    "ys      = np.exp(xs)\n",
    "yTaylor = []\n",
    "\n",
    "for xi in xs:\n",
    "    yTaylor.append( eTaylor5_0.subs(x,xi) )\n",
    "    \n",
    "plt.figure(figsize=(15,15))\n",
    "plt.plot(xs,ys,color = \"blue\")\n",
    "plt.plot(xs,yTaylor,color=\"red\")\n",
    "\n",
    "plt.title(\"Value of $e$ and Taylor Series Approximation\",fontsize=20)\n",
    "plt.ylabel(\"Y\",fontsize=20)\n",
    "plt.xlabel(\"X\",fontsize=20)\n",
    "\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As we see here, due to only having 5 terms in our taylor series, we get an approximation of $e^x$ with some error. As we increase our number of terms in our taylor series, we increase the approximation precision. \n",
    "\n",
    "Let's see what the value of $e$ is in our taylor series and compare that to what numpy defines $e$ as:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Our approximated e value: 2.716666666666667\n",
      "Numpy's approximated e value: 2.718281828459045\n",
      "Error: 0.0016151617923783057\n"
     ]
    }
   ],
   "source": [
    "res        = eTaylor5_0.subs(x,1)\n",
    "decimalVal = res.p/res.q\n",
    "npE        = np.exp(1)\n",
    "\n",
    "print(\"Our approximated e value: {0}\".format(res.p/res.q) )\n",
    "print(\"Numpy's approximated e value: {0}\".format(npE))\n",
    "print(\"Error: {0}\".format(npE - decimalVal))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's beef up our taylor series to include more terms instead of the measly five we've been dealing with. We will produce 100 taylor series with a = 0, starting with a one termed taylor series to a 100 termed one. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "taylors = []\n",
    "for i in range(1,101):\n",
    "    taylors.append(taylor(sp.exp,i,printSteps=False))\n",
    "    taylors[i-1] = taylors[i-1].subs(a,0)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can pretty print one of the taylor series in the list by:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$1 + x + \\frac{x^{2}}{2} + \\frac{x^{3}}{6} + \\frac{x^{4}}{24} + \\frac{x^{5}}{120}$$"
      ],
      "text/plain": [
       "         2    3    4     5\n",
       "        x    x    x     x \n",
       "1 + x + ── + ── + ── + ───\n",
       "        2    6    24   120"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "taylors[4]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$1 + x$$"
      ],
      "text/plain": [
       "1 + x"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/latex": [
       "$$1 + x + \\frac{x^{2}}{2}$$"
      ],
      "text/plain": [
       "         2\n",
       "        x \n",
       "1 + x + ──\n",
       "        2 "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/latex": [
       "$$1 + x + \\frac{x^{2}}{2} + \\frac{x^{3}}{6}$$"
      ],
      "text/plain": [
       "         2    3\n",
       "        x    x \n",
       "1 + x + ── + ──\n",
       "        2    6 "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/latex": [
       "$$1 + x + \\frac{x^{2}}{2} + \\frac{x^{3}}{6} + \\frac{x^{4}}{24}$$"
      ],
      "text/plain": [
       "         2    3    4\n",
       "        x    x    x \n",
       "1 + x + ── + ── + ──\n",
       "        2    6    24"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/latex": [
       "$$1 + x + \\frac{x^{2}}{2} + \\frac{x^{3}}{6} + \\frac{x^{4}}{24} + \\frac{x^{5}}{120}$$"
      ],
      "text/plain": [
       "         2    3    4     5\n",
       "        x    x    x     x \n",
       "1 + x + ── + ── + ── + ───\n",
       "        2    6    24   120"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/latex": [
       "$$1 + x + \\frac{x^{2}}{2} + \\frac{x^{3}}{6} + \\frac{x^{4}}{24} + \\frac{x^{5}}{120} + \\frac{x^{6}}{720}$$"
      ],
      "text/plain": [
       "         2    3    4     5     6\n",
       "        x    x    x     x     x \n",
       "1 + x + ── + ── + ── + ─── + ───\n",
       "        2    6    24   120   720"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/latex": [
       "$$1 + x + \\frac{x^{2}}{2} + \\frac{x^{3}}{6} + \\frac{x^{4}}{24} + \\frac{x^{5}}{120} + \\frac{x^{6}}{720} + \\frac{x^{7}}{5040}$$"
      ],
      "text/plain": [
       "         2    3    4     5     6     7 \n",
       "        x    x    x     x     x     x  \n",
       "1 + x + ── + ── + ── + ─── + ─── + ────\n",
       "        2    6    24   120   720   5040"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/latex": [
       "$$1 + x + \\frac{x^{2}}{2} + \\frac{x^{3}}{6} + \\frac{x^{4}}{24} + \\frac{x^{5}}{120} + \\frac{x^{6}}{720} + \\frac{x^{7}}{5040} + \\frac{x^{8}}{40320}$$"
      ],
      "text/plain": [
       "         2    3    4     5     6     7       8 \n",
       "        x    x    x     x     x     x       x  \n",
       "1 + x + ── + ── + ── + ─── + ─── + ──── + ─────\n",
       "        2    6    24   120   720   5040   40320"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/latex": [
       "$$1 + x + \\frac{x^{2}}{2} + \\frac{x^{3}}{6} + \\frac{x^{4}}{24} + \\frac{x^{5}}{120} + \\frac{x^{6}}{720} + \\frac{x^{7}}{5040} + \\frac{x^{8}}{40320} + \\frac{x^{9}}{362880}$$"
      ],
      "text/plain": [
       "         2    3    4     5     6     7       8       9  \n",
       "        x    x    x     x     x     x       x       x   \n",
       "1 + x + ── + ── + ── + ─── + ─── + ──── + ───── + ──────\n",
       "        2    6    24   120   720   5040   40320   362880"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/latex": [
       "$$1 + x + \\frac{x^{2}}{2} + \\frac{x^{3}}{6} + \\frac{x^{4}}{24} + \\frac{x^{5}}{120} + \\frac{x^{6}}{720} + \\frac{x^{7}}{5040} + \\frac{x^{8}}{40320} + \\frac{x^{9}}{362880} + \\frac{x^{10}}{3628800}$$"
      ],
      "text/plain": [
       "         2    3    4     5     6     7       8       9        10  \n",
       "        x    x    x     x     x     x       x       x        x    \n",
       "1 + x + ── + ── + ── + ─── + ─── + ──── + ───── + ────── + ───────\n",
       "        2    6    24   120   720   5040   40320   362880   3628800"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Showing the first ten taylors. There must be a better way to pretty print this. \n",
    "for i in range(0,10):\n",
    "    display(taylors[i])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Series with 0 terms: 2 = 2.0\n",
      "Series with 1 terms: 5/2 = 2.5\n",
      "Series with 2 terms: 8/3 = 2.6666666666666665\n",
      "Series with 3 terms: 65/24 = 2.7083333333333335\n",
      "Series with 4 terms: 163/60 = 2.716666666666667\n",
      "Series with 5 terms: 1957/720 = 2.7180555555555554\n",
      "Series with 6 terms: 685/252 = 2.7182539682539684\n",
      "Series with 7 terms: 109601/40320 = 2.71827876984127\n",
      "Series with 8 terms: 98641/36288 = 2.7182815255731922\n",
      "Series with 9 terms: 9864101/3628800 = 2.7182818011463845\n"
     ]
    }
   ],
   "source": [
    "# Showing the first ten computed rationals given x = 1\n",
    "for i in range(0,10):\n",
    "    res = taylors[i].subs(x,1)\n",
    "    print(\"Series with {0} terms: {1} = {2}\".format(i,res,res.p/res.q))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "If we proceed in this fashion, by showing the value of the taylor series rational by calling res.p/res.q, we will run up against pythons default precision for floating point numbers, and it will seem like we reach a point where adding another term to the taylor series adds no further precision. Let's show this by showing the result of the taylor series with 99 vs 98 terms:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Result 1:  2.718281828459045\n",
      "Result 2:  2.718281828459045\n",
      "Numpy exp: 2.718281828459045\n"
     ]
    }
   ],
   "source": [
    "r1 = taylors[99].subs(x,1)\n",
    "r2 = taylors[98].subs(x,1)\n",
    "\n",
    "print(\"Result 1:  {0}\".format(r1.p/r1.q))\n",
    "print(\"Result 2:  {0}\".format(r2.p/r2.q) )\n",
    "print(\"Numpy exp: {0}\".format(np.exp(1)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As we see, it shows the same value for all results. This is a result of truncation. We can use a higher precision floating point object by using the Decimal class. Let's do the same thing, but use that class. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Result 1:  2.718281828459045235360287471\n",
      "Result 2:  2.718281828459045235360287471\n",
      "Numpy exp: 2.718281828459045090795598298427648842334747314453125\n"
     ]
    }
   ],
   "source": [
    "r1Dec = Decimal(r1.p)/Decimal(r1.q)\n",
    "r2Dec = Decimal(r1.p)/Decimal(r1.q)\n",
    "npE   = Decimal(np.exp(1))\n",
    "\n",
    "print(\"Result 1:  {0}\".format(r1Dec))\n",
    "print(\"Result 2:  {0}\".format(r2Dec))\n",
    "print(\"Numpy exp: {0}\".format(npE))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "At this point, we still don't enough precision to show the difference between the last two taylor series in our list which have 98 and 99 terms, nor can we match numpy's precision. Let's go a little deeper here. We can do arithmetic up to any precision we'd like. This is called arbitrary precision. Sympy uses a library called [mpmath](http://mpmath.org/) behind the scenes to accomplish this. \n",
    "\n",
    "See: http://docs.sympy.org/latest/modules/evalf.html"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "No sub:    2.718281828459045090795598298427648842334747314453125\n",
      "Numpy exp: 2.718281828459045090795598298427648842334747314453125\n",
      "Sub        2.718281828459045235360287471352662497757247093699960\n"
     ]
    }
   ],
   "source": [
    "higherPrecision = sp.N(r1.p/r1.q,52)\n",
    "print(\"No sub:    {0}\".format(higherPrecision))\n",
    "print(\"Numpy exp: {0}\".format(Decimal(np.exp(1))))\n",
    "print(\"Sub        {0}\".format(sp.N(taylors[99].subs(x,1),52)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "So this result is a bit tricky. Somewhere in our computation we are truncating results when we computer higherPrecision. It turns out, we must call the Numeric evaluation function sp.N when performing our x substitution, so that we do not truncate results when dividing during the substitution; however, if we symbolically compute the substitution of x, we get a rational approximation that is more in line with standard results ( like arriving at numpy's $e$). Sympy uses different libraries for performing different things. Namely, the numeric evaluation module sp.N uses mpmath to do arbitrary precision arithmetic, and the symbolic computation uses another module. This explains the difference in values, but since much is hidden in the nuts and bolts of the package, I cannot provide a satisfactory answer why at this time. \n",
    "\n",
    "As observed above, in the special case of a = 0, it transforms the series into the definition of $e$. No division or truncation happens during this substitution, so the symbolic solution should equal the numeric one. \n",
    "\n",
    "Here we will print the results of our taylor series' with differing precision to show how it changes the values. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Precision = 50\n",
      "taylors[0]   = 2.0000000000000000000000000000000000000000000000000\n",
      "taylors[1]   = 2.5000000000000000000000000000000000000000000000000\n",
      "taylors[2]   = 2.6666666666666666666666666666666666666666666666667\n",
      "taylors[3]   = 2.7083333333333333333333333333333333333333333333333\n",
      "taylors[4]   = 2.7166666666666666666666666666666666666666666666667\n",
      "taylors[5]   = 2.7180555555555555555555555555555555555555555555556\n",
      "taylors[6]   = 2.7182539682539682539682539682539682539682539682540\n",
      "taylors[7]   = 2.7182787698412698412698412698412698412698412698413\n",
      "taylors[8]   = 2.7182815255731922398589065255731922398589065255732\n",
      "taylors[9]   = 2.7182818011463844797178130511463844797178130511464\n",
      "taylors[10]   = 2.7182818261984928651595318261984928651595318261985\n",
      "taylors[11]   = 2.7182818282861685639463417241195018972796750574528\n",
      "taylors[12]   = 2.7182818284467590023145578701134256689812245367801\n",
      "taylors[13]   = 2.7182818284582297479122875948272773669599066424463\n",
      "taylors[14]   = 2.7182818284589944642854695764748674801584854494907\n",
      "taylors[15]   = 2.7182818284590422590587934503278418622333966249310\n",
      "taylors[16]   = 2.7182818284590450705160477958486050611789796352510\n",
      "taylors[17]   = 2.7182818284590452267081174817108696833426231358244\n",
      "taylors[18]   = 2.7182818284590452349287527283351994002986043726966\n",
      "taylors[19]   = 2.7182818284590452353397844906664158861464034345403\n",
      "taylors[20]   = 2.7182818284590452353593574317298071473772510089138\n",
      "taylors[21]   = 2.7182818284590452353602471108690522047059258986580\n",
      "taylors[22]   = 2.7182818284590452353602857925707585115463030677773\n",
      "taylors[23]   = 2.7182818284590452353602874043083296076646521164906\n",
      "taylors[24]   = 2.7182818284590452353602874687778324515093860784392\n",
      "taylors[25]   = 2.7182818284590452353602874712574287147341835385141\n",
      "taylors[26]   = 2.7182818284590452353602874713492656133721389999984\n",
      "taylors[27]   = 2.7182818284590452353602874713525455026092088379085\n",
      "taylors[28]   = 2.7182818284590452353602874713526586022380733150778\n",
      "taylors[29]   = 2.7182818284590452353602874713526623722257021309835\n",
      "taylors[30]   = 2.7182818284590452353602874713526624938382062863353\n",
      "taylors[31]   = 2.7182818284590452353602874713526624976385970411900\n",
      "taylors[32]   = 2.7182818284590452353602874713526624977537603973977\n",
      "taylors[33]   = 2.7182818284590452353602874713526624977571475549333\n",
      "taylors[34]   = 2.7182818284590452353602874713526624977572443308628\n",
      "taylors[35]   = 2.7182818284590452353602874713526624977572470190831\n",
      "taylors[36]   = 2.7182818284590452353602874713526624977572470917377\n",
      "taylors[37]   = 2.7182818284590452353602874713526624977572470936497\n",
      "taylors[38]   = 2.7182818284590452353602874713526624977572470936987\n",
      "taylors[39]   = 2.7182818284590452353602874713526624977572470936999\n",
      "taylors[40]   = 2.7182818284590452353602874713526624977572470937000\n",
      "taylors[41]   = 2.7182818284590452353602874713526624977572470937000\n",
      "taylors[42]   = 2.7182818284590452353602874713526624977572470937000\n",
      "taylors[43]   = 2.7182818284590452353602874713526624977572470937000\n",
      "taylors[44]   = 2.7182818284590452353602874713526624977572470937000\n",
      "taylors[45]   = 2.7182818284590452353602874713526624977572470937000\n",
      "taylors[46]   = 2.7182818284590452353602874713526624977572470937000\n",
      "taylors[47]   = 2.7182818284590452353602874713526624977572470937000\n",
      "taylors[48]   = 2.7182818284590452353602874713526624977572470937000\n",
      "taylors[49]   = 2.7182818284590452353602874713526624977572470937000\n",
      "taylors[50]   = 2.7182818284590452353602874713526624977572470937000\n",
      "taylors[51]   = 2.7182818284590452353602874713526624977572470937000\n",
      "taylors[52]   = 2.7182818284590452353602874713526624977572470937000\n",
      "taylors[53]   = 2.7182818284590452353602874713526624977572470937000\n",
      "taylors[54]   = 2.7182818284590452353602874713526624977572470937000\n",
      "taylors[55]   = 2.7182818284590452353602874713526624977572470937000\n",
      "taylors[56]   = 2.7182818284590452353602874713526624977572470937000\n",
      "taylors[57]   = 2.7182818284590452353602874713526624977572470937000\n",
      "taylors[58]   = 2.7182818284590452353602874713526624977572470937000\n",
      "taylors[59]   = 2.7182818284590452353602874713526624977572470937000\n",
      "taylors[60]   = 2.7182818284590452353602874713526624977572470937000\n",
      "taylors[61]   = 2.7182818284590452353602874713526624977572470937000\n",
      "taylors[62]   = 2.7182818284590452353602874713526624977572470937000\n",
      "taylors[63]   = 2.7182818284590452353602874713526624977572470937000\n",
      "taylors[64]   = 2.7182818284590452353602874713526624977572470937000\n",
      "taylors[65]   = 2.7182818284590452353602874713526624977572470937000\n",
      "taylors[66]   = 2.7182818284590452353602874713526624977572470937000\n",
      "taylors[67]   = 2.7182818284590452353602874713526624977572470937000\n",
      "taylors[68]   = 2.7182818284590452353602874713526624977572470937000\n",
      "taylors[69]   = 2.7182818284590452353602874713526624977572470937000\n",
      "taylors[70]   = 2.7182818284590452353602874713526624977572470937000\n",
      "taylors[71]   = 2.7182818284590452353602874713526624977572470937000\n",
      "taylors[72]   = 2.7182818284590452353602874713526624977572470937000\n",
      "taylors[73]   = 2.7182818284590452353602874713526624977572470937000\n",
      "taylors[74]   = 2.7182818284590452353602874713526624977572470937000\n",
      "taylors[75]   = 2.7182818284590452353602874713526624977572470937000\n",
      "taylors[76]   = 2.7182818284590452353602874713526624977572470937000\n",
      "taylors[77]   = 2.7182818284590452353602874713526624977572470937000\n",
      "taylors[78]   = 2.7182818284590452353602874713526624977572470937000\n",
      "taylors[79]   = 2.7182818284590452353602874713526624977572470937000\n",
      "taylors[80]   = 2.7182818284590452353602874713526624977572470937000\n",
      "taylors[81]   = 2.7182818284590452353602874713526624977572470937000\n",
      "taylors[82]   = 2.7182818284590452353602874713526624977572470937000\n",
      "taylors[83]   = 2.7182818284590452353602874713526624977572470937000\n",
      "taylors[84]   = 2.7182818284590452353602874713526624977572470937000\n",
      "taylors[85]   = 2.7182818284590452353602874713526624977572470937000\n",
      "taylors[86]   = 2.7182818284590452353602874713526624977572470937000\n",
      "taylors[87]   = 2.7182818284590452353602874713526624977572470937000\n",
      "taylors[88]   = 2.7182818284590452353602874713526624977572470937000\n",
      "taylors[89]   = 2.7182818284590452353602874713526624977572470937000\n",
      "taylors[90]   = 2.7182818284590452353602874713526624977572470937000\n",
      "taylors[91]   = 2.7182818284590452353602874713526624977572470937000\n",
      "taylors[92]   = 2.7182818284590452353602874713526624977572470937000\n",
      "taylors[93]   = 2.7182818284590452353602874713526624977572470937000\n",
      "taylors[94]   = 2.7182818284590452353602874713526624977572470937000\n",
      "taylors[95]   = 2.7182818284590452353602874713526624977572470937000\n",
      "taylors[96]   = 2.7182818284590452353602874713526624977572470937000\n",
      "taylors[97]   = 2.7182818284590452353602874713526624977572470937000\n",
      "taylors[98]   = 2.7182818284590452353602874713526624977572470937000\n",
      "taylors[99]   = 2.7182818284590452353602874713526624977572470937000\n"
     ]
    }
   ],
   "source": [
    "p = 50\n",
    "print(\"Precision = {0}\".format(p))\n",
    "\n",
    "cntr = 0\n",
    "for taylor_i in taylors:\n",
    "    res = sp.N(taylor_i.subs(x,1),p)\n",
    "    print(\"taylors[{0}]   = {1}\".format(cntr,res))\n",
    "    cntr+=1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As we see, at expansion 40 we arrive at the closest value given this precision. After 40 we see that the differences are chopped off. Let's start at expansion 40 with a higher precision.  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Precision = 100\n",
      "taylors[38]   = 2.718281828459045235360287471352662497757247093698703335742056391892310837864596648186030883920472746\n",
      "taylors[39]   = 2.718281828459045235360287471352662497757247093699928953181184777741734377849536405109756445884951071\n",
      "taylors[40]   = 2.718281828459045235360287471352662497757247093699958846289456201786842269068681277229847313249938347\n",
      "taylors[41]   = 2.718281828459045235360287471352662497757247093699959558030129330930773409335803774185087571996723758\n",
      "taylors[42]   = 2.718281828459045235360287471352662497757247093699959574582238008352725296318760111323581531502462954\n",
      "taylors[43]   = 2.718281828459045235360287471352662497757247093699959574958422296475951475568372755349456394218502481\n",
      "taylors[44]   = 2.718281828459045235360287471352662497757247093699959574966781947323134279551697480772253613389970026\n",
      "taylors[45]   = 2.718281828459045235360287471352662497757247093699959574966963678863290427464378453064053552937175842\n",
      "taylors[46]   = 2.718281828459045235360287471352662497757247093699959574966967545491804388058265282261751423991371711\n",
      "taylors[47]   = 2.718281828459045235360287471352662497757247093699959574966967626046565095570637924536703462971667458\n",
      "taylors[48]   = 2.718281828459045235360287471352662497757247093699959574966967627690539803887216958052518810705959208\n",
      "taylors[49]   = 2.718281828459045235360287471352662497757247093699959574966967627723419298053548538722835117660645043\n",
      "taylors[50]   = 2.718281828459045235360287471352662497757247093699959574966967627724063994017594255990880535444070256\n",
      "taylors[51]   = 2.718281828459045235360287471352662497757247093699959574966967627724076392016902827476804485786059202\n",
      "taylors[52]   = 2.718281828459045235360287471352662497757247093699959574966967627724076625941418083542576635792511824\n",
      "taylors[53]   = 2.718281828459045235360287471352662497757247093699959574966967627724076630273353551247498342274112798\n",
      "taylors[54]   = 2.718281828459045235360287471352662497757247093699959574966967627724076630352116014296678736937414634\n",
      "taylors[55]   = 2.718281828459045235360287471352662497757247093699959574966967627724076630353522486851128386842116452\n",
      "taylors[56]   = 2.718281828459045235360287471352662497757247093699959574966967627724076630353547161808223994735181397\n",
      "taylors[57]   = 2.718281828459045235360287471352662497757247093699959574966967627724076630353547587238518746595406654\n",
      "taylors[58]   = 2.718281828459045235360287471352662497757247093699959574966967627724076630353547594449201708491342676\n",
      "taylors[59]   = 2.718281828459045235360287471352662497757247093699959574966967627724076630353547594569379757856274943\n",
      "taylors[60]   = 2.718281828459045235360287471352662497757247093699959574966967627724076630353547594571349889813077111\n",
      "taylors[61]   = 2.718281828459045235360287471352662497757247093699959574966967627724076630353547594571381666134961017\n",
      "taylors[62]   = 2.718281828459045235360287471352662497757247093699959574966967627724076630353547594571382170521022666\n",
      "taylors[63]   = 2.718281828459045235360287471352662497757247093699959574966967627724076630353547594571382178402054879\n",
      "taylors[64]   = 2.718281828459045235360287471352662497757247093699959574966967627724076630353547594571382178523301529\n",
      "taylors[65]   = 2.718281828459045235360287471352662497757247093699959574966967627724076630353547594571382178525138599\n",
      "taylors[66]   = 2.718281828459045235360287471352662497757247093699959574966967627724076630353547594571382178525166018\n",
      "taylors[67]   = 2.718281828459045235360287471352662497757247093699959574966967627724076630353547594571382178525166421\n",
      "taylors[68]   = 2.718281828459045235360287471352662497757247093699959574966967627724076630353547594571382178525166427\n",
      "taylors[69]   = 2.718281828459045235360287471352662497757247093699959574966967627724076630353547594571382178525166427\n",
      "taylors[70]   = 2.718281828459045235360287471352662497757247093699959574966967627724076630353547594571382178525166427\n",
      "taylors[71]   = 2.718281828459045235360287471352662497757247093699959574966967627724076630353547594571382178525166427\n",
      "taylors[72]   = 2.718281828459045235360287471352662497757247093699959574966967627724076630353547594571382178525166427\n",
      "taylors[73]   = 2.718281828459045235360287471352662497757247093699959574966967627724076630353547594571382178525166427\n",
      "taylors[74]   = 2.718281828459045235360287471352662497757247093699959574966967627724076630353547594571382178525166427\n",
      "taylors[75]   = 2.718281828459045235360287471352662497757247093699959574966967627724076630353547594571382178525166427\n",
      "taylors[76]   = 2.718281828459045235360287471352662497757247093699959574966967627724076630353547594571382178525166427\n",
      "taylors[77]   = 2.718281828459045235360287471352662497757247093699959574966967627724076630353547594571382178525166427\n",
      "taylors[78]   = 2.718281828459045235360287471352662497757247093699959574966967627724076630353547594571382178525166427\n",
      "taylors[79]   = 2.718281828459045235360287471352662497757247093699959574966967627724076630353547594571382178525166427\n",
      "taylors[80]   = 2.718281828459045235360287471352662497757247093699959574966967627724076630353547594571382178525166427\n",
      "taylors[81]   = 2.718281828459045235360287471352662497757247093699959574966967627724076630353547594571382178525166427\n",
      "taylors[82]   = 2.718281828459045235360287471352662497757247093699959574966967627724076630353547594571382178525166427\n",
      "taylors[83]   = 2.718281828459045235360287471352662497757247093699959574966967627724076630353547594571382178525166427\n",
      "taylors[84]   = 2.718281828459045235360287471352662497757247093699959574966967627724076630353547594571382178525166427\n",
      "taylors[85]   = 2.718281828459045235360287471352662497757247093699959574966967627724076630353547594571382178525166427\n",
      "taylors[86]   = 2.718281828459045235360287471352662497757247093699959574966967627724076630353547594571382178525166427\n",
      "taylors[87]   = 2.718281828459045235360287471352662497757247093699959574966967627724076630353547594571382178525166427\n",
      "taylors[88]   = 2.718281828459045235360287471352662497757247093699959574966967627724076630353547594571382178525166427\n",
      "taylors[89]   = 2.718281828459045235360287471352662497757247093699959574966967627724076630353547594571382178525166427\n",
      "taylors[90]   = 2.718281828459045235360287471352662497757247093699959574966967627724076630353547594571382178525166427\n",
      "taylors[91]   = 2.718281828459045235360287471352662497757247093699959574966967627724076630353547594571382178525166427\n",
      "taylors[92]   = 2.718281828459045235360287471352662497757247093699959574966967627724076630353547594571382178525166427\n",
      "taylors[93]   = 2.718281828459045235360287471352662497757247093699959574966967627724076630353547594571382178525166427\n",
      "taylors[94]   = 2.718281828459045235360287471352662497757247093699959574966967627724076630353547594571382178525166427\n",
      "taylors[95]   = 2.718281828459045235360287471352662497757247093699959574966967627724076630353547594571382178525166427\n",
      "taylors[96]   = 2.718281828459045235360287471352662497757247093699959574966967627724076630353547594571382178525166427\n",
      "taylors[97]   = 2.718281828459045235360287471352662497757247093699959574966967627724076630353547594571382178525166427\n",
      "taylors[98]   = 2.718281828459045235360287471352662497757247093699959574966967627724076630353547594571382178525166427\n",
      "taylors[99]   = 2.718281828459045235360287471352662497757247093699959574966967627724076630353547594571382178525166427\n"
     ]
    }
   ],
   "source": [
    "p = 100\n",
    "print(\"Precision = {0}\".format(p))\n",
    "\n",
    "cntr = 38\n",
    "for taylor_i in taylors[38:100]:\n",
    "    res = sp.N(taylor_i.subs(x,1),p)\n",
    "    print(\"taylors[{0}]   = {1}\".format(cntr,res))\n",
    "    cntr+=1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The trend: the higher the precision, the more we can approximate without truncation, but the precision changes the overall value of the computation."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Feel free to use this section to experiment with values"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# Define the taylor series with depth = your value. \n",
    "depth = 18\n",
    "t = taylor(sp.exp,depth,printSteps = False)\n",
    "t2 = t.subs(a,0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Result:    2.7182818284590452267081174817108696833426231358244\n",
      "Numpy exp: 2.718281828459045090795598298427648842334747314453125\n"
     ]
    }
   ],
   "source": [
    "# Perform the symbolic substitution with precision = p. \n",
    "p = 50\n",
    "r = sp.N(t2.subs(x,1),p)\n",
    "\n",
    "npE = Decimal(np.exp(1))\n",
    "print(\"Result:    {0}\".format(r))\n",
    "print(\"Numpy exp: {0}\".format(npE))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Plotting: \n"
     ]
    },
    {
     "data": {
      "text/latex": [
       "$$1 + x$$"
      ],
      "text/plain": [
       "1 + x"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Plotting: \n"
     ]
    },
    {
     "data": {
      "text/latex": [
       "$$1 + x + \\frac{x^{2}}{2}$$"
      ],
      "text/plain": [
       "         2\n",
       "        x \n",
       "1 + x + ──\n",
       "        2 "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Plotting: \n"
     ]
    },
    {
     "data": {
      "text/latex": [
       "$$1 + x + \\frac{x^{2}}{2} + \\frac{x^{3}}{6}$$"
      ],
      "text/plain": [
       "         2    3\n",
       "        x    x \n",
       "1 + x + ── + ──\n",
       "        2    6 "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Plotting: \n"
     ]
    },
    {
     "data": {
      "text/latex": [
       "$$1 + x + \\frac{x^{2}}{2} + \\frac{x^{3}}{6} + \\frac{x^{4}}{24}$$"
      ],
      "text/plain": [
       "         2    3    4\n",
       "        x    x    x \n",
       "1 + x + ── + ── + ──\n",
       "        2    6    24"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Plotting: \n"
     ]
    },
    {
     "data": {
      "text/latex": [
       "$$1 + x + \\frac{x^{2}}{2} + \\frac{x^{3}}{6} + \\frac{x^{4}}{24} + \\frac{x^{5}}{120}$$"
      ],
      "text/plain": [
       "         2    3    4     5\n",
       "        x    x    x     x \n",
       "1 + x + ── + ── + ── + ───\n",
       "        2    6    24   120"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Plotting: \n"
     ]
    },
    {
     "data": {
      "text/latex": [
       "$$1 + x + \\frac{x^{2}}{2} + \\frac{x^{3}}{6} + \\frac{x^{4}}{24} + \\frac{x^{5}}{120} + \\frac{x^{6}}{720}$$"
      ],
      "text/plain": [
       "         2    3    4     5     6\n",
       "        x    x    x     x     x \n",
       "1 + x + ── + ── + ── + ─── + ───\n",
       "        2    6    24   120   720"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Plotting: \n"
     ]
    },
    {
     "data": {
      "text/latex": [
       "$$1 + x + \\frac{x^{2}}{2} + \\frac{x^{3}}{6} + \\frac{x^{4}}{24} + \\frac{x^{5}}{120} + \\frac{x^{6}}{720} + \\frac{x^{7}}{5040}$$"
      ],
      "text/plain": [
       "         2    3    4     5     6     7 \n",
       "        x    x    x     x     x     x  \n",
       "1 + x + ── + ── + ── + ─── + ─── + ────\n",
       "        2    6    24   120   720   5040"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Plotting: \n"
     ]
    },
    {
     "data": {
      "text/latex": [
       "$$1 + x + \\frac{x^{2}}{2} + \\frac{x^{3}}{6} + \\frac{x^{4}}{24} + \\frac{x^{5}}{120} + \\frac{x^{6}}{720} + \\frac{x^{7}}{5040} + \\frac{x^{8}}{40320}$$"
      ],
      "text/plain": [
       "         2    3    4     5     6     7       8 \n",
       "        x    x    x     x     x     x       x  \n",
       "1 + x + ── + ── + ── + ─── + ─── + ──── + ─────\n",
       "        2    6    24   120   720   5040   40320"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Plotting: \n"
     ]
    },
    {
     "data": {
      "text/latex": [
       "$$1 + x + \\frac{x^{2}}{2} + \\frac{x^{3}}{6} + \\frac{x^{4}}{24} + \\frac{x^{5}}{120} + \\frac{x^{6}}{720} + \\frac{x^{7}}{5040} + \\frac{x^{8}}{40320} + \\frac{x^{9}}{362880}$$"
      ],
      "text/plain": [
       "         2    3    4     5     6     7       8       9  \n",
       "        x    x    x     x     x     x       x       x   \n",
       "1 + x + ── + ── + ── + ─── + ─── + ──── + ───── + ──────\n",
       "        2    6    24   120   720   5040   40320   362880"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Plotting: \n"
     ]
    },
    {
     "data": {
      "text/latex": [
       "$$1 + x + \\frac{x^{2}}{2} + \\frac{x^{3}}{6} + \\frac{x^{4}}{24} + \\frac{x^{5}}{120} + \\frac{x^{6}}{720} + \\frac{x^{7}}{5040} + \\frac{x^{8}}{40320} + \\frac{x^{9}}{362880} + \\frac{x^{10}}{3628800}$$"
      ],
      "text/plain": [
       "         2    3    4     5     6     7       8       9        10  \n",
       "        x    x    x     x     x     x       x       x        x    \n",
       "1 + x + ── + ── + ── + ─── + ─── + ──── + ───── + ────── + ───────\n",
       "        2    6    24   120   720   5040   40320   362880   3628800"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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V+/1a8rjP3wGXM8/jACmoPQLMI7XsvdLe/UJacP12YEeM8Y12Dv0eacznb4UQ/kOMsbbJ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\nQoQQCAAAAAAhQggEAAAAgBAhBAIAAABAiBACAQAAA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      "text/plain": [
       "<matplotlib.figure.Figure at 0x2158a0e1550>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Let's tail this off with plotting the first 10 taylors derived above (up to ten units in the series)\n",
    "xs      = np.arange(-2,4,0.1)\n",
    "ys      = np.exp(xs)\n",
    "yTaylor = []\n",
    "\n",
    "plt.figure(figsize=(15,15)),\n",
    "\n",
    "\n",
    "for taylor_i in taylors[0:10]:\n",
    "    print(\"Plotting: \")\n",
    "    display(taylor_i)\n",
    "    for xi in xs:\n",
    "        yTaylor.append( sp.N(taylor_i.subs(x,xi) ) )\n",
    "    \n",
    "    plt.plot(xs,yTaylor,color=\"red\")\n",
    "    yTaylor = []\n",
    "\n",
    "plt.plot(xs,ys,color = \"blue\")\n",
    "\n",
    "plt.title(\"Value of $e$ and Taylor Series Approximation\",fontsize=20)\n",
    "plt.ylabel(\"Y\",fontsize=20)\n",
    "plt.xlabel(\"X\",fontsize=20)\n",
    "\n",
    "plt.show()\n",
    "\n",
    "# Obviously the red lines that get closer to the blue line represent taylor series with more terms. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Our current implementation is a general one and doesn't take into consideration the structure of the series. With a little creativity, it can be clear how we can linearlize the approximation or use dynamic programming/sub-solutions to make it more efficient. \n",
    "\n",
    "I hope this notebook showed you a bit about using the taylor series for numerical function approximation and some considerations of precision and differences between computing a value using sympy's symbolic packages versus the numeric ones. "
   ]
  }
 ],
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  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
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  "language_info": {
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    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
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 },
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