Chebyshev2.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "chebyshev.ipynb was getting too big, so we're starting a new one"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "IPython console for SymPy 1.0.1.dev (Python 3.5.2-64-bit) (ground types: gmpy)\n",
      "\n",
      "These commands were executed:\n",
      ">>> from __future__ import division\n",
      ">>> from sympy import *\n",
      ">>> x, y, z, t = symbols('x y z t')\n",
      ">>> k, m, n = symbols('k m n', integer=True)\n",
      ">>> f, g, h = symbols('f g h', cls=Function)\n",
      ">>> init_printing()\n",
      "\n",
      "Documentation can be found at http://docs.sympy.org/dev\n",
      "\n"
     ]
    }
   ],
   "source": [
    "import sys\n",
    "from sympy import init_session\n",
    "init_session()\n",
    "from mpmath import mpf\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now we try to mechanise the CRAM process"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "115\n"
     ]
    },
    {
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      "text/latex": [
       "$$\\left [ -9.42477796076938, \\quad -6.28318530717959, \\quad -3.14159265358979, \\quad 0, \\quad 3.14159265358979, \\quad 6.28318530717959, \\quad 9.42477796076938, \\quad 12.5663706143592\\right ]$$"
      ],
      "text/plain": [
       "[-9.42477796076938, -6.28318530717959, -3.14159265358979, 0, 3.14159265358979,\n",
       " 6.28318530717959, 9.42477796076938, 12.5663706143592]"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "def bisect_roots(expr, bounds, eps=1e-7, division_cutoff=30, solver='bisect', **kwargs):\n",
    "    \"\"\"\n",
    "    Numerically find all the roots of expr in bounds via bisection\n",
    "    \n",
    "    division_cutoff is the fraction of bounds to check for roots. \n",
    "    One should expect the roots to be no more than |bounds|/division_cutoff apart.\n",
    "    \"\"\"\n",
    "    roots = []\n",
    "    intervals = [bounds]\n",
    "    var = expr.free_symbols.pop()\n",
    "    N = 0\n",
    "    while intervals:\n",
    "        N += 1\n",
    "        #print(intervals)\n",
    "        b = intervals.pop()\n",
    "        try:\n",
    "            root = nsolve(expr, b, solver=solver, **kwargs)\n",
    "        except ValueError:\n",
    "            # No root found\n",
    "            if (b[1] - b[0]) > (bounds[1] - bounds[0])/division_cutoff and sign(expr.evalf(subs={var: b[0]})) == sign(expr.evalf(subs={var: b[1]})):\n",
    "                # The bisect method will only work if the interval has alternate signs.\n",
    "                # If not, it will miss 2 roots, so try again with split intervals.\n",
    "                intervals.append([b[0], (b[0] + b[1])/2])\n",
    "                intervals.append([(b[0] + b[1])/2, b[1]])\n",
    "                continue\n",
    "        else:\n",
    "            if root not in roots:\n",
    "                roots.append(root)\n",
    "                intervals.append([b[0], root - eps])\n",
    "                intervals.append([root + eps, b[1]])\n",
    "    \n",
    "    print(N)\n",
    "    return sorted(roots)\n",
    "\n",
    "bisect_roots(sin(x), [-10, 13])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "def nsolve_intervals(expr, bounds, division=30, warn=False, verbose=False, solver='bisect', **kwargs):\n",
    "    \"\"\"\n",
    "    Divide bounds into division intervals and nsolve in each one\n",
    "    \"\"\"\n",
    "    if verbose:\n",
    "        warn = True\n",
    "    roots = []\n",
    "    L = bounds[1] - bounds[0]\n",
    "    for i in range(division):\n",
    "        interval = [bounds[0] + i*L/division, bounds[0] + (i + 1)*L/division]\n",
    "        try:\n",
    "            if verbose:\n",
    "                print(\"Solving in interval\", interval)\n",
    "            root = nsolve(expr, interval, solver=solver, **kwargs)\n",
    "        except ValueError:\n",
    "            if verbose:\n",
    "                print(\"No solution found\")\n",
    "            continue\n",
    "        else:\n",
    "            if interval[0] < root < interval[1]:\n",
    "                if verbose:\n",
    "                    print(\"Solution found:\", root)\n",
    "                roots.append(root)\n",
    "            else:\n",
    "                if warn:\n",
    "                    print(root, \"is not in\", interval, 'discarding', file=sys.stderr)\n",
    "            \n",
    "    return roots\n",
    "        \n",
    "#nsolve_intervals(D, [-1, 0.99])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
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fLeKmXg6nenTwZpqoQnPvV1KwMIJuAU6M/lbPAg6R3EBGGWbEh2mG1FimhiNd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\n",
      "text/latex": [
       "$$\\left ( \\left ( \\frac{p_{0} + p_{1} x + p_{2} x^{2} + p_{3} x^{3} + p_{4} x^{4} + p_{5} x^{5} + p_{6} x^{6} + p_{7} x^{7} + p_{8} x^{8}}{q_{1} x + q_{2} x^{2} + q_{3} x^{3} + q_{4} x^{4} + q_{5} x^{5} + q_{6} x^{6} + q_{7} x^{7} + q_{8} x^{8} + 1}, \\quad \\left ( p_{0}, \\quad p_{1}, \\quad p_{2}, \\quad p_{3}, \\quad p_{4}, \\quad p_{5}, \\quad p_{6}, \\quad p_{7}, \\quad p_{8}\\right ), \\quad \\left ( q_{1}, \\quad q_{2}, \\quad q_{3}, \\quad q_{4}, \\quad q_{5}, \\quad q_{6}, \\quad q_{7}, \\quad q_{8}\\right )\\right ), \\quad \\left ( \\frac{p_{0} + p_{1} x + p_{2} \\left(2 x^{2} - 1\\right) + p_{3} \\left(4 x^{3} - 3 x\\right) + p_{4} \\left(8 x^{4} - 8 x^{2} + 1\\right) + p_{5} \\left(16 x^{5} - 20 x^{3} + 5 x\\right) + p_{6} \\left(32 x^{6} - 48 x^{4} + 18 x^{2} - 1\\right) + p_{7} \\left(64 x^{7} - 112 x^{5} + 56 x^{3} - 7 x\\right) + p_{8} \\left(128 x^{8} - 256 x^{6} + 160 x^{4} - 32 x^{2} + 1\\right)}{q_{1} x + q_{2} \\left(2 x^{2} - 1\\right) + q_{3} \\left(4 x^{3} - 3 x\\right) + q_{4} \\left(8 x^{4} - 8 x^{2} + 1\\right) + q_{5} \\left(16 x^{5} - 20 x^{3} + 5 x\\right) + q_{6} \\left(32 x^{6} - 48 x^{4} + 18 x^{2} - 1\\right) + q_{7} \\left(64 x^{7} - 112 x^{5} + 56 x^{3} - 7 x\\right) + q_{8} \\left(128 x^{8} - 256 x^{6} + 160 x^{4} - 32 x^{2} + 1\\right) + 1}, \\quad \\left ( p_{0}, \\quad p_{1}, \\quad p_{2}, \\quad p_{3}, \\quad p_{4}, \\quad p_{5}, \\quad p_{6}, \\quad p_{7}, \\quad p_{8}\\right ), \\quad \\left ( q_{1}, \\quad q_{2}, \\quad q_{3}, \\quad q_{4}, \\quad q_{5}, \\quad q_{6}, \\quad q_{7}, \\quad q_{8}\\right )\\right )\\right )$$"
      ],
      "text/plain": [
       "⎛⎛                2       3       4       5       6       7       8           \n",
       "⎜⎜p₀ + p₁⋅x + p₂⋅x  + p₃⋅x  + p₄⋅x  + p₅⋅x  + p₆⋅x  + p₇⋅x  + p₈⋅x            \n",
       "⎜⎜─────────────────────────────────────────────────────────────────, (p₀, p₁, \n",
       "⎜⎜            2       3       4       5       6       7       8               \n",
       "⎝⎝ q₁⋅x + q₂⋅x  + q₃⋅x  + q₄⋅x  + q₅⋅x  + q₆⋅x  + q₇⋅x  + q₈⋅x  + 1           \n",
       "\n",
       "                                                             ⎞  ⎛             \n",
       "                                                             ⎟  ⎜p₀ + p₁⋅x + p\n",
       "p₂, p₃, p₄, p₅, p₆, p₇, p₈), (q₁, q₂, q₃, q₄, q₅, q₆, q₇, q₈)⎟, ⎜─────────────\n",
       "                                                             ⎟  ⎜          ⎛  \n",
       "                                                             ⎠  ⎝q₁⋅x + q₂⋅⎝2⋅\n",
       "\n",
       "  ⎛   2    ⎞      ⎛   3      ⎞      ⎛   4      2    ⎞      ⎛    5       3     \n",
       "₂⋅⎝2⋅x  - 1⎠ + p₃⋅⎝4⋅x  - 3⋅x⎠ + p₄⋅⎝8⋅x  - 8⋅x  + 1⎠ + p₅⋅⎝16⋅x  - 20⋅x  + 5⋅\n",
       "──────────────────────────────────────────────────────────────────────────────\n",
       " 2    ⎞      ⎛   3      ⎞      ⎛   4      2    ⎞      ⎛    5       3      ⎞   \n",
       "x  - 1⎠ + q₃⋅⎝4⋅x  - 3⋅x⎠ + q₄⋅⎝8⋅x  - 8⋅x  + 1⎠ + q₅⋅⎝16⋅x  - 20⋅x  + 5⋅x⎠ + \n",
       "\n",
       " ⎞      ⎛    6       4       2    ⎞      ⎛    7        5       3      ⎞      ⎛\n",
       "x⎠ + p₆⋅⎝32⋅x  - 48⋅x  + 18⋅x  - 1⎠ + p₇⋅⎝64⋅x  - 112⋅x  + 56⋅x  - 7⋅x⎠ + p₈⋅⎝\n",
       "──────────────────────────────────────────────────────────────────────────────\n",
       "   ⎛    6       4       2    ⎞      ⎛    7        5       3      ⎞      ⎛     \n",
       "q₆⋅⎝32⋅x  - 48⋅x  + 18⋅x  - 1⎠ + q₇⋅⎝64⋅x  - 112⋅x  + 56⋅x  - 7⋅x⎠ + q₈⋅⎝128⋅x\n",
       "\n",
       "     8        6        4       2    ⎞                                         \n",
       "128⋅x  - 256⋅x  + 160⋅x  - 32⋅x  + 1⎠                                         \n",
       "─────────────────────────────────────, (p₀, p₁, p₂, p₃, p₄, p₅, p₆, p₇, p₈), (\n",
       "8        6        4       2    ⎞                                              \n",
       "  - 256⋅x  + 160⋅x  - 32⋅x  + 1⎠ + 1                                          \n",
       "\n",
       "                               ⎞⎞\n",
       "                               ⎟⎟\n",
       "q₁, q₂, q₃, q₄, q₅, q₆, q₇, q₈)⎟⎟\n",
       "                               ⎟⎟\n",
       "                               ⎠⎠"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "def general_rat_func(d, x, chebyshev=False):\n",
    "    \"\"\"\n",
    "    Return a general rational function with numerator and denominator degree d\n",
    "    \n",
    "    Returns a tuple, (rational function, numerator coefficients, denominator coefficients)\n",
    "    \n",
    "    The constant coefficient in the denominator is always 1.\n",
    "    \"\"\"\n",
    "    num_coeffs = symbols('p:%s' % (d+1))\n",
    "    den_coeffs = symbols('q:%s' % (d+1))[1:]\n",
    "    if chebyshev:\n",
    "        num = Add(*(Mul(c, chebyshevt(i, x)) for i, c in enumerate(num_coeffs)))\n",
    "        den = Add(*(Mul(c, chebyshevt(i, x)) for i, c in enumerate([1, *den_coeffs])))\n",
    "        rat_func = num/den\n",
    "    else:\n",
    "        rat_func = Poly(reversed(num_coeffs), x)/Poly([*reversed(den_coeffs), 1], x)\n",
    "    return rat_func, num_coeffs, den_coeffs\n",
    "\n",
    "general_rat_func(8, x), general_rat_func(8, x, chebyshev=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false,
    "scrolled": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "--------------------------------------------------------------------------------\n",
      "Iteration 0:\n",
      "sol {q8: 0.005463385229468977730916600742375623845455619952601565201040525051058109969563323965009036685360870472784376477376823488848737330928391857700842834654074997547465219088377825871978098214192580994355204805259920831722705520466099948464157135677702579093963916165947660008403217235774288796320172247040501899439680923765037743077591446080076737821800992364048418769345187681068584819497098413791534836516383744577860197658067093174331770270273187727962230631801278602050330594650620026697264234864479122596535167946652618975301853832135046863974760813814122856058895711861968519046031131766389873386568316317416325399805257406662547900325490791315243114844435284091310039666857518017321890777238762030731588318309564230528654477569101455866590324684700799623536739399857644789963621180301373074427282668364628659801241924111209843759361169563850351068560276246821245932851766944220220885685588268259640817222897139531991880441244782317384123683207757931508386417403387328250439321130363092076692605436603, q7: 0.004738895453174642158427477950393199271272242132553242078361853239267182235296343799713538755218281640177994024232805500200143873166383731805670408304538065298421531715247059299962271626163053282909483381440332816454186004451404258751180879032476102674125814266575452904882602819294185540221219834375762035245960591379090526267185080989991424175943130611006089643766021268289100905504230340363342110989535184755520824379451250696668423887467974717599735875046841376140010936255240758439457425747899543148466758491718386664270811968895517693168367346032619617961911211590401985644526962896472372461070659020654567388707753370207117663830189535524584753059195953568238185450922243516399092929614232878181841183103939381804923427111203303511068310245558895192540266214839197124688336295986421597192317809998264006272312465276832400662864505255003099384071731539269995794934600273248321963068889940175914629351380594892477592564770739243703105244624353246445876064423712995840850207671727930655003845147707, q9: 0.001152706284204982612579766956060937974171210474625021373215701768059371720217851202386862247379173250418369776219880624847306895563256062046517336657364247207870190891288499252704423925365367436696055074599041239688451298329208545486077286384986613844039403812596511400425133542389967221302859021769916141893658300630864211695898327100853213889894135673346344268422120456645021716665821250660499621079980152530276446958701065855461732846506704924475041581178985264851091301626847787851659615060848494994831082438640786407460490227032548126282363122424644029518614599011824500361692252877718515331659087234956521555731135008221155912366931948993004493591470339800070601182365166087708996005977863140789765293052978320069757774048171566721484664524962868037116199372941374965146217788366260518339459222647161836856232694501312276028764071886118914188777956335231799029583613586907389898530677504572686624223429665367902170989123409746506998185868761522360208133862568067627450132023070873633256918144395, q1: -1.148352362146599397012814987532118551475378188043441129932852984642568351057193012892767468659156188987162471858484202199551229978754184823514944255346933115937411120581465733675110544732113763796163403979445163308052745339776916961011157027773924716312940356754227723852458077213326611702146870259112037682604557322998613702275518421074684481637566044617094730256385531043195385335585281356062018043998647886477210636062780406636929531870244445871201117445700730017698626213778597574251930002375894611831369036304536320304639107297153356337680111008464006198124499084603714716765240575994075678883840678652382683272835991105650791599167055926372954228203504372794512106779231623928919754784786751622845771178466247883971886901436902273389082071219230513458603535792021450787092000908128552832379286577626263069373176893828456116231512946113975259912218484959960180943288504661203378082094873864760908332793427711452991975320335937050430595201688429465943774177313846385062120887487406948983147241829, p9: -0.009281138272582325322699749889615575027946622091219406920297339651461308861568800631658909243467332195764635300828972369355338014082419874380515351245751281598534801192436155116649450365361422611012328697344088482068734134265458262383541304136123800565514421459394759954073274419788822423954242855487026784269714799038180829386468922255593647513875217214393692659020134713617766379256669029812815427255379316372290462218097024598339689401629782878953765441748148930673332580348699425649310740800620519358177099514809250859863908039908016383345857129645056263514121749081907247437063710533870327473195085827204551870181198659297267490598889271773808506667362319596465430308100092100019058284872798388943669219202368580009071608236710846858276484763908137127404930928397743175869542393393295558278448281763964915990702553667407239377617575410702157684524418180970498781121522160300115192369271602936121489281048174461471315642835593813074368873886498278360195903658921914047052059772337793009241678302804, p10: 0.002685862341724177438460088137734625367797021311579690418985019784912665873416731986104204558192492072608767356602416959967274195248303194600635489386434093862227856202339683560464020863139479061038569613609930111195240279135041252623337857302050718760798498750253559935064345678121529434532246204473852380600263318839720167488217276853229228796350087332910346673024049285866599784066481479896471082561045847666143907663350741692771756638847257744696530097069647534709422117548161137649735314659551027044101082704371397739007787192456959640713402834904168509486379859083007978005233852259302902431873807574789016940354643451295410460406603657060694687490150211469196681304095487571642734252472579577889282523593895976456326045564725923482211865369388721845572638452542616984590305844947548089255317122061525234227649674640919257116444337912128800169643335795744269129658081911169722331824349212677604085605715026448395347060041351171703580266343277568772962980321072307660230007446241837727951315448116, p4: 0.4560098591358681516207660837054733440089498656033322916366741469597964606086747311495748053453504582626516032892901463317744075250007027256768724199287117661517991815316212496793606861159688484683632196180247515632105322470004987292259972274716841189650358461290206930083841141520839679943906369286257255812745645620179268057970424378565365117119861310771401362178061325013050936645538420302873086943932423234471846832490250481269772442736992531613891518012953105939860507966094191167929907603283532345022379416387310734779641425559188808029722926509968195090918847346178581857198506707576891574353890737798687932934911290076061343241103097721483646507280635143530532608854508260664201483947769963159038343691334481154933781962386570823921060577235751105088124787858365642634153850685007009387130824478998450413680490597605932852211064903517141292301894259658100583549367023978743790184097233159956755008042240781068792017741487603678425422006002179753252289625297911336260000260164255810969407013594, p7: -0.06701683181724890568626478527469166386992944664232640673967910369214037039232337800275527202319586546213684652698242775801245639240027059523754154307912712740015425387871755087864624126898596350408578250474727140835052536922299607058381129597590386195722950639760330794264263169253220743690676021864082467286077646918670337532082918805437180309168818440468113282113947337910282119251729609454252160909561663733523433788828639949786535965615971477248499155150475792111455288528821674307605701533973018273981799344972241425901301807181867917705271958976014409803124823917541833583838757510532802090632526139270927758887255094761666631731722352370492358121298074152216728930827364739890230983110955843776642120956893680568768312715234104266729979581298244983247724005842083455613532834363105313274289033768958708539563398905736616116517802357244267530461938705534168736076567753828772832721665011907090209267846426408776201324155640629099646952067142323313806827692428107425679956238494841312624422278172, p8: 0.02695531779474421551111416501486447022916338775842449062488546950512528028946860500110919368837087794607486594565076772061914042596687766129073632400693640639779431521129326802076439559052031286016188454850064547055161591998125718751403472610332164556325484270075309739407067698647824981252313444627317157907939636654779980064232427531003955069833522874629361395418274733109518086268573265505040232427736109076479095779737110715468393777842988541379526428479644193244353406242367523185327937598543645255856505896910731208039095543162134730068313597625012883717058009700693505213954350886002179326534305586526972617481383260144192640609488971852695345384796661398972747512479780546721715972351540241600118115860605268918936893855381730911395399821802364379163230927242305630326374258017185112706951044247718049226390841939800043605463092423555854995013450964089376338630425964318328095717298878319991604293730260619432976593614367004006774466118493865726158083245317034492989146867720594753030715888536, p13: -0.00001643262045945472647461342457779725958210705552736097121614725811728134969607272056326158595968457387551626550827053926611225198902134670047306180591118057026913202200102032179292868837083562887502575026584781325581092121128601694255649468387866894209782569085955836704007674702987506236367540830667811580351804453589133890427466407351707804864105729702382246992983081546145997348067117600462794207493284042756331181846952083829573197853471437502043645848623422441754249101317212329392724642780092143106083407782253885117963673865150195207839018853466922234352674388259326696454047759849789849762761653171404786823882520348029849194659104716207952945060117366598885980671167857780325910641497781826650957230276776563251672431937537381412595125011119678242384159357482340234562532334754783631977310467276782213568884070915057693131533849599338145778159620948457766661525379397689630185252526941727781005549790253611919881961600505250650384656539262045798657757501747888023838782720756075781646747718386, p6: 0.1445584213056228328291288587208050607203169934645495625021347464933729891713081430378356323673391423209160377368897206324910128988292395749654493959834314135798357147206245725096772244299259534855911573048804294085583322321911939923153807720366407020386456156399685623727548973362576998079211449139513073157338520206333569918079801941922009646146203584405231881225037089681319696024913417163441583828639228782864077053320497079347895619368234463921710013054207632209301246134676343642771349529736230483459741771919598158646081617785321971610675943335145713056288541406283031549760538432882252130662302487290042370740050929203170056394407954768083616087193490295067916438561672384978102173327797171429479800233606353566745660678771950038910856255171460705966571679119257409953659134810165015777217271021686447619470734088449934358610141525979075067784684876467781467283578817678717641558298084169182924777365452473570645781106417722369160279555376658898125663900855502187436020593992238180251076939977, q12: 0.00001719434486116499827595092367414783079474353940246196794490429446369423891876641904959710422935416757697508290757396642350950972844210073263789502227243773170679583757549720646747814656519958820743123942131583965704778272838526532240314214020509342944839390850958021298090852640467955158839123647292705506292402131864618579779863914591598678310988230568759937315399226629294486323950898409786462860608966853089070012455282690830643994511600902880584740364390057761446529319612199411190538484489006875173657918359040940822578255657675482016807773334991165399764987867639754707654160427444138534935869068250520817829220147870436099789119761864521512624259110336912863976221881919212247704300376161489818450802245406767224343532584789168040212430678288461480828943347412816268526351634191267500796900901671847905333428184444755763078308213128958029503894297271290590521513283248712453246258728691612849816417153929969004491203076338013265044545407890586704707354497129414304065352434305776528581361727772, p2: 0.8922500430729679842410862004401322317616608028934753400562045760888110257669889186610707347106052720336086213559266605309329471852358375927071193256388733787299703216243568599117723743603498176784545660016832630698802879603089420395986018657200174792088152249961728596788289726146674098579805155025070531895789117626546967250390372997380587564243986481601781608727961385287000651500823685735427738235494214121456073932895867703924001898139591241950681028620278809654128102527705845373880034671995695344387198987253839652516790156121832449781822510050610770875104676781433047774189989150294554084459214903794207439986791797444891608888046852104027971209915731979695714750870468925194874272955739187292907955608397455258165470303386685184709024976452377435308763889995115176395823100831198699006601500360354747267632666314491448849995092267540103648511928353743917420897495427866934681369205721538546887371401856074920623793879681933907835472903991402515253548396842780291232984864139345405139568890223, p5: -0.2732486698372423797618186530350415626611059883674898373047846748150681897100718064419410217555612324666339881755352816519095122534037846538690443738591343023090260779107101759758669062886656901149881867596193092208070654467286617193084748010494000907018588484893190275866779606041064737027725893766730502231817657389490305301692662290808540913801758964028748652606562096612333152206524129980154775394013530767539938496094164616108012616900167479397859231378901040150810916608243563490602322909161353292843685432536510013962932867906298512528820852446905383834736133387898560179894915912145826742506223908271966288032663192470565599190741315861869310354147210093020267454566481758081283195161338130453614634049958254797482739025987558131055594801539922900297826544164816646051363906979023631290765810924471596528168159946388482958098731243109803847390693216442453227999598004347041834258663674993296669797509375869074761876991061831944298813351180440306662779603731540694351639365968587051439271497157, epsilon: 0.0000000000000003868104904341161795227152767192343958742874596029170322827195112042801264632250873838621527773099511691875011517651037679382201801532909491605352265020893910946642325741413892302899874659662495855492422483558475009662299803177438704144004581693594718826151498694988657287253631180081814045047270417546136578414596825101266820055970848595299545130366462232519024615512451294139309512460243537747063177059111983629750964829297212657555334940501944395604257256061297500080981025351241674154812603817747589195329157421447256482402731886489907067910498550446833375379661637425592619308839083241881844834663039576642563066072420406534493230233492650729452841602102295102809264906540393505945857133211991309610729537610176134396811427774802378198412187205085197585820549115013093805317870943870734393881452562202163757910029208847786614466442694158278980552319005665197696462957952644565628777482529168435380309399940397987922254207835197332833618231753499030599159475354536951308789825463975860911636518641, q2: 1.218228072668408557455661513110346908877015046982799891557963919864420382982701186007631779569195683810560773204379658757533079562672104772342418251234570767858137512016915565410716026313305640649858797596168725403610799422357964506232832820807528672469202579313754801080819410031130150412239789296401809536690651684027462854782006531078331270728807699462915354679463868929470293625804703092020320449464735712345173711558009888087346189809546778180409823550065689494280531445349981790927258487109049992942466347963698893360073495203404445829270155512826232353472523831509427853028311589413443015652689438247219846221482504131566896121167377248788117721951758210362049695158791351678300187261432845774989218657919096726100997267900830351778305262190970335641555533750686678706461267820733287460337249570287593662367418255699060587747989097704269044491090813827899120872446559765128219342477908656188449363185805314471712776906441623542806076429487845370319800584858456966983198586108189648562232089991, p3: -0.6756212743375994555590731295380233148505425646440454233000742442860690054837555565907497112704403126495217583972478413506852537862652025570305992502381666146358490413256033328361560371828464618318585178981132894864387414187819844115067281219428282885328173962880146373258483815536299271738246212440082701003425817884246951390275487849166317817832125086333395225677311706783837111444591693592142858299258016792323327234183594889220071697781069623661439436911687218465928895292240239692619170234275157266112080253686625508800369298949473043515528765582731807999055153226007988656260270339686410404983120791862643458718781110037268489624275748796066545799284925337425145550523608880583300069715241506106486161959765462965335958715513666781879500717979425595700949931780885142121298168020919137984651598155845146316746530292156972301050489904909286121805923927853937659797069102448395830573552610042696116088264041147253736688348939790720607801237971325403365338958681122264716541648182703522074385057498, p15: -0.00000006775040400385666405861363010748585355385209742640732843749767755223711957984279787794537080564647085266507185678841777136630636094888391521631728873368786642548593043170050868273740224712809035712817464997277557499059306743347457327046501168865663856434019263765558997226471495332408523172133492337110814300081104298371028253115788041833109357084977552173909700027829543228779359192647930151081543207683948158347178812571240509876327293434630547152061763978746483478337884023659608634584743716745365963453521641801490833632199633926336595496937029936506400258897925309952035634090613386874188383302791669490855304730254843287282643922122058137268347260084145828330706924849564031650428581135146929136570387453216857795179432830651196356129232957789401256294599879241186902059035730292786027039586459149324722876557077497232121259830091691225166346678488822357793261748584309521366968202110034921429395531296343541680532087353185641869286417816463261134887470698374461939351131332862162585180988359803, q11: 0.00008685651930694648251018668518110950219199824794295957146594048632805724936717894429699515941398829516864178956097154346993463642764750657392834197716859308894203988641350905349745543452883888635444366499051556326943231958852517841651825021615211275329125802769559927740933660345439462754452632225622137980727087492271215299965230885711711944555479359161525527941815884161313782581725760584695041906979418346079696964974151891220814280982366911457033824818895995079451156349046223905018660290060691160122001796230640588231326108002083361094859513880507360562472233010005120040429242934611148479432086900913586155567312307026865132000548003031223494472066326875083672997131618014509046104935360531729921151298303774302492529473790065570410125747124716126505857826852867004147585843187633791424785838266730532709065146462694171222783055512498539454797762694273572027384104507279250137726286454731928182884138665184023906465442978995053823032811510289698104104456241507037648731859780480292232584498596772, q4: 0.3347990155297382583061263701897614281028626454797541562759132115294526378303565774845627667630912946960256507371754924668914001280843341034066775681900992540163927356930763624266771559554374198860448916073431167437300015084864284906160661304286546684725885813555738390694381036115461689216873675493301990065538494746983770826669599071356195981620196180163067865066855497562593064904215763922454505369718939535719992538003517153169548835186916122630898774437813702090711006309809246847369834951220190853659923717282266342677248068404115540971402356370612573781549779028605495415004551864657687072249348184015787870492037823005735426073674618764795185590363468022313518385565836491451711185417685409847271729447837303314804810829372051270923171506376418703576310925072767347623903325938145205643210666424399189270571417019008579122201145941310281079419199970319866610652905468060185897053298788590924180898831065158732574778657010603311965760119409613849985491105519932952034192894357535766457386329383, q15: 0.00000001152088308371057355017489299273716602569570185283584091732193621188199618434003364543586275064442350295889486313376192442370094280031804719767339896150947094996814367470890959992491868899353772232944607742664621581955229509055912771445372520629349720592344576057599148964871681271028130857869761249297850104659693706719664373306826811464787732226186871312119072437390838110818985797151283537770500750022564917516149707402156893977104244843314596809509018437182678492922689544437168110636043781102986579835991445383506933190633504789569329099184377210762985218841674858348922679693897639246131795584134208856459218618776570234975616619983364037558757320681204762508834457958198995858322784853977982216861345494237644819375846273013331790524535745374050810014222214559061502732046538206674919000881396896744861774100250278737310644103810701073751797122553152549705174372806062700094990004760816108722264691683274382872196983564271049909175622048160433391207714926549005534408194095737005499042638844612, q13: 0.000002410656736546815083724653045054624533620925940711948028156546967760444049305485730831869230984370375783795176338618193720687933965804676989935085582431879340004702028186051244812254596749454815780048532169964413159914380036197595852951664762414039466501970351257640550495866280854225807466996498626617569288252568908465142972184459564864255217168004049003496099385870857297721984839043319380251042275539347876113713230127229938211214450348126934857287761541209440690507923574600570600428411357708254736689899138961756790451701391782624033759074851967223125912790492267104760188785296885366067219695205140053597610420315350862309077100147566842218482200662978341241289816663269191717767046007477225815612237976061826210127384518262178582356703668025691784522965409802472227684261128213024297401584573672729501831079104169311863638741860198077880544286360485969527065809270132175087679779045319396964592279171952479549737857686888455557352368494191082315898903010809506838339696629929076248873182869085, q6: 0.05154013259380542556112211573339870568055062345021659681945188464799359741054370498672382140031113420161042834401302187930086605088393474123191426608305645693968436199350963345571565109061757056590779527677568537780869519189108544349534354440519933013283943704689546101004213449223100431148727269834137153831950901801724299924811929808778068484529569379857662366349560821168406579693693221248085590812330574842271626795729695739628849184922108148605567328458796764468078500432262027593236821555520078748174554192956763605190245907339310927405676260237546938675669600736282258999217177812906492825486844186968566195458684307258514039871173139937674446825809656827682335865443249030741812138893741486133708573896722247165594757680360839943876021901236232834938528150877959876703929000590694579925841789306344822798286511123654077115781992423887628842093358029900995797934119725749203382636708416004699649290085949418882693347270057044406391329559160447732319206640861633490367761781411434312701793732793, p14: 0.000001498404148577785397735830364044853840539893506530486617462741437308992976969311840270744520387842184253082515789166034006114655954120489862132729513661245352381408459752307040347589250235576891775776780596695604304534247755196560072036739196798935291083436420458068920458318102721597021282493460231732736741519243499628610719807459991552708178531237430115963596962429133204856982922515733728220242891289884568417042228082916479135731663019649840321221849236701127932330206937898165432496340356326698354978888412114229481524226220261540735749245015636856588646462094191715715737859592650140505844225634370649896385635867044594219111592972848727440658540769431395482906493470322398502875340762317532821888756400702716084275550198200321930277295392967011341312680840918845582606093723839017447419342878799380859290852012481939825469240665046151341886218164645555493456790696238253506269879525834343961783967765194570793088636038432378438720395326871825719351301385268511515752955186426243181288539656134, q5: -0.03205560840195828791940107062582933788640277083088444071091101585322630668427836156949160044561107777934090906043020261084030639087756283279636765783206274748180373147549806558353222393236268855562722390256194472198626016613655890671860586868034547460229969152220059515742325044269833631654252098366808482753610408730484998441131951583799075762568454459647441679236675295205331400561560729229635785687037381421546843850160435693238081683192616426647085957359897372480537308584423792331000798873402819800155985641938567373827358373766250055432833263542524722019390097543470940510388104345583256049669854516408449238161531846860092711292944271244621203340222203332967423682111674413508722959592409532532322116367984106446774521171516372759318731039070816498062161691177282977636514144998471019347192525051879888464886536328514858244616559801918296692787641253779155535274682525493229513217830635077492481598324417170744554458521861639283682958518337502745528974965816634140620095116071078949225197259186, q10: 0.0004120413594629924041427993828307600158986658960978376576467329812369984338438716174484460268856683954601495701631886604691059526519458799422270106268760121129622118409302840847944904987792607939385382487914817975467026937501651408621973181096104642755550294553282094297350484941094018390090897615326780930318733952780948446852942376213529613789558311742561745617020839567199448227484035645826904418663322512238051260938660039205356812114880192713563723270154041945603985812397367939624426931632299822290979209751482389492221875609367327300308748570755199271909434423946702788612254291612128070920306199333401341078787992329006292134066197209021872660904590527693451718498175060748592744301489223474322575674694899950742578681125578763037427896254464753116463483687813963390788932917362600623464590394477149163946712021197264358695699375498419701114236087383182083801195904936355775347996976320296474609768566959212323966326996218646178532158873261272521646515803068716253522981722443875158327539813326, p12: 0.0001188642419336033887211662956957125414764589533797398998464650491970641334720901861790296638091520798666158437332669158991508636553890672187485488302561490495409335489822862795886636059118345909819083068107187630725634914790617638052036167239708981082565002467262674987542265832156845873994335824632187798158198220730485140831011573531341076271041055122228495452713685725028845757230545188254626419730223887600605809345429382979437960374855571896229785948054658093122781052295210922400350903638408731012834164637917804028211332282823546306522847627558716178158019548658593034620384318830712541149821475966313269519287863451865893407801200899172212899084730133802125157172410018135532794158759030475215844802231471386249692660390333955043345475892496618909468904775037470499270145035197012222748047471006465930300392488700495164432242259545754768372887923189850174794654020964141496032682609593080823639786125374364938362038205305646059098751355560342907194448739774967866434710545583206520358383865110, p11: -0.0006365364936038485433458064305823597788572065845387026752174608606513996879685061683524780184547371886927396167924545511659945927015528962277698799448890039661866926141431126219230163218263895998263423601953564874203088018170304940341917684205211846873419384951125555749905547722252671993628646590279228901148101651727756868040200171373243608399064705600684446898363704949063410615517810609677774981636287133792326057358108092098219163059828143285076671952543885806265015722044005088887798403991527549403525212362407424705529478870982950739391234172732997351177983033254633326534398606875077195171843992858254301304688847530103889407430719763732243091509948946581948987802123169563260707442179394524381017611313985356556883202352826263123673579200977125808369328068816896384438309923088853976104680971428023263484319753713989357756362339293541746781387638282900092017563839289866965205118908283536264763006057548362904886292357608071368062264864889328249781110582191749685927473109679405860091683465747, p0: 0.5564204348698052234397526876482989470340679196501998860487688042529573073532607323266002385285016478933463791223781355314738410328188185311887086519736087401746296600292733738069328117451283815837663706709278651596600062702766239321732275436882677880786080451965905623982946616427791885160255869989923825163226265389818739240930993510759517037663595568468513511075104786173765930771666393679492804694656448635464794042509002080131673649846850698877821384205671628925671793864039293468077004427190491621152300940077257763063307881471973231337013588583400533540035789404702401836189558485719207342379606974768643209009285421974549447521209716757353742798292392231806639669416905487012799805081033829206041117625564754117932584535917361753033852642286459662694490439158889635955074310714358544846224986176712665915334922894228737078785327708332438637636163900578626450382454596784425024097862855521892129020403859632335615631114871227615206763753873040920479833148513737549188958251443739851982219477266, q3: -0.3731134357053760237745654648187879341824338651294727623942049673070920111014721515849780097481932356985116036747760146455654659932114027703042197247747322218258502268605704721479940223510467873623816792097816971893130078404149271374493071257614697063646177844582523115748335890870310723401216059033326130213331970781397931056409933397724991483037737108610388727190510814351440956736580718076102473574411273413877028749890809699410699296419696424494733567475869331772832202461027267399520760928447494477101011843118866091814494043317654062529241731764886239950783862979648191258367045512789745282613436506550877029474125516460039399944409478095706083875587195257404404955656037435266611052547165254788635153334238823962293861489843247777058518120573132986116922170184220587811248492603521295961223970941377956558534191761782395780144090208368649909042554099538104045139712603242976019357849675622978372051308296164042570371869210928874171585791692227802388667778221874960298823256481162032899351571594, q14: 0.0000002347882102004562688274130339771076213295143586354998635514589238188586639198883204729240766905742537332740678178730395052135359225940462615342826433677755559237083301960326461727925064074856079644034304315714936303399012343321986098806722069009225742337038033575210054109911402881225230433588481918108608923099690571254927095838861427473406478129878572281333036557396905623949234778467686231052482408204500980471496421404473719976003462711224626877072195751858859353958887765276735482258704504395869696509836352873374631933201995824856729714865565287129512401961630366269214323014053675565300426910773741828541870358586854559258679087653812328661440901501268090467573813816008465707519276873449818091273880690093925480963743036565590211457925807733908114988669062978755818164889061880574104487483509923295307863923250039241947637421049055015211177199376285804853406472078994267947373417264635275625221635675289589981649343325616066119217959359346360537057058806487529112194227355494121276080635188093, p1: -1.053179350037674803946518885670661514622550175188595866693075723568979659107290123997369424487985376609457893068097481398886370479586640029217524593063238583425216964290613092953327080291116912007655529520875576729330732924545647628554537254433180508514717059322348490438503667140520712780205981652638312392920858209982172116691225271542473218977591120133465831600879066801885752686047958634668917412829082580733577654820643117445185422778198198842368142621270998134173482277570921120033215169066269941390279123101235460360254362729861517750736597670255326575237278835342964863378837770850762609161346449197192106093651729919420026943968401927936748117291892990067010379659101295039889817054231107329020615621989888614654007036846308967150114544765532641187238969983743604108251441608452729420087007390518455250489137522105446510633252251215788635502397201180632151906522750632581800123856361102735557377175952136237273993235107893480774989822452924267148217746304676185956441278911207038983411401973}\n",
      "system.subs(sol) [-1.47976954668070e-695, 1.35819881196768e-695, -1.19439216757163e-695, 1.00640877703610e-695, -8.12589913355924e-696, 6.28694262683339e-696, -4.66031625822048e-696, 3.30818557127089e-696, -2.24616635897871e-696, 1.45474540530115e-696, -8.93260478920015e-697, 5.12721788994528e-697, -2.65285839726661e-697, 1.09762044528062e-697, -1.36530313555404e-698, -4.72639901988211e-698, 9.08323827811874e-698, -1.31273685408595e-697, 1.82718761698760e-697, -2.63919966758718e-697, 4.04725815094436e-697, -6.54259670294478e-697, 1.08878878412023e-696, -1.81413926104755e-696, 2.95480451866273e-696, -4.62270300435377e-696, 6.86573718847399e-696, -9.60939854872084e-696, 1.26175955787283e-695, -1.55018322969946e-695, 1.77936852508406e-695, -1.92325202322386e-695]\n"
     ]
    },
    {
     "data": {
      "image/png": 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Kx9ixvp1nyBD5Wk6fNmZczkRG+v6OVadf9PylJkD/uquqjDsnIJkAwPclgqGh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\nBgFERERBikEAERFRkGIQQEREFKT+P9UyKqe1Zvi2AAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x111af5fd0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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edpvsh5w9Gxg7VtqF/vWvsgf188/l+MEPPpDH//OfMvq+5ZaDuOGG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\nyuRMVApt28ra0/Hj9txvyRKpdG3d2p7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SpEn45ptvnA5NCxERwNKlpZ+O3blTtg+VJRn5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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x11237c128>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "-(-0.00111002261919918758393632571568104822462631276423425766911996194901585296719614440043225695527971177845006453730162143676606556341778651406690414245861274200351516148419298113425796959841694663241118801346515395502064587681683004740846329875150695036623814971617534918610558508979526181243652235138451223581492528812824510926899049071277393663706480272214817336525255959236260321021012343687595320003914693806626360177665167204513813746375632986884539379941027782385309087891843639027869036441055156075945222498579275625818229958802249098780621816298479720861841783608278254151828940609730546702472032938712938173312700495352418838838020047800521004601509218645251370302256735257094560621873318247286973569228033504998116219827497389201098821352780421550183130724421487606203335241405117006267016584546702536259609511157714651074721056222269033125423980360865510084800488805327198076407023370812152699216384759292586893837719194593556386388669504934070433996319922167184414328935757613671795605313287*t**15 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0.01227492678514921797825192234225544266170280760549774637025477785443527046733260259549793911101720317340125196934484815057809126157615505295059132017591292192670849810229089927452745113792984589742716338664813039046274455761057022011012496750017687790455551115639250059639454189749532279834618642621835457938652564274895757901666271225079978539852789702750997378631621945921418840410124889070158022976544673438447240993245525179707991378345697149191144938894705564002164905523526177122301002019902831292398705387203976791264646119638254170725781516809712917419181747561853514332454578298995102387589639676436395119112902282931584296216963357677519387476598318199179596999450888108853555479152490522887691269243455665016238530722365703725283160385918575690803348144880718301270911978568923092925925686312452799931065968625205105024401952805807179273189920477639060239802938358377272336285307563494573493426393247392393698210642683804416999747851773399629292586094811964633704820888720378414111571686305*t**14 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0.06314542857992960620025536563685364440596182657419207184213931193957495972812832192549849250856909542492688150699504844612303828421473665738677062279239783330918290655045555881036852157287918600056351803841831576447411124336441271892966485484887692295930096832509351194825841174763602364497751360646164144690394058853000499215030970495304942484477767840152118671288992165878028933854117402667122623877811337355083670178871358349005016858910384307553750721182684137480495239400888054838091250173300582877723053906439629846360845806146765450881289412681214518676647605779158695606421097058249674494001607869900488680700176487958691655681117738335821495710581163669344338195640788254110327655549970934636169930608551159335914456863037904761869051972619621267618776511667087077505625287886404983077751481915565435804613487626646369536558687125380170887001881051637349427606120992959732452268012712771418376670498031447222984623268689780827526595674173836827174688527451222404089920515125835132391558729685*t**13 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0.2004717237319974771770668453869252356278281099024652025896686982650638987151765922104099645925832484599248660730236752344376493507202669786699583837488980267163521649577041531367369859246827816899531404950835704059904245974814965026715696644417802530433682162480220853612751457844884051998283243929078200349461558559820053156328507663408531714143604494361109604636560683786579516664612834370919599565707886102584163191803442528991145865283217241261700893002792827314697878167357807083113299944495090161944822292934270772834182373349233876004768410356851211770910424006149804886187982560650998439178008892256837683812028550395295195500922266317560231708717902657039462939287906303472418019435122730231358210734843916283687082726575943624922428492761574034811145128568488177060437839584357949663927230411883745279330727839792309998691464067673252882057994525639486895520406509129738557234126799406426091582747130112055971080519526631583084331028066892403681825885874946568661599886302274508754920065539*t**12 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(1, nan)]\n"
     ]
    },
    {
     "ename": "AssertionError",
     "evalue": "16",
     "output_type": "error",
     "traceback": [
      "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[0;31mAssertionError\u001b[0m                            Traceback (most recent call last)",
      "\u001b[0;32m<ipython-input-5-2b9cfe95a3f6>\u001b[0m in \u001b[0;36m<module>\u001b[0;34m()\u001b[0m\n\u001b[1;32m     64\u001b[0m \u001b[0;31m#D = CRAM_exp2()\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     65\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m---> 66\u001b[0;31m \u001b[0mrat_func\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mCRAM_exp2\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;36m15\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;36m1000\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m     67\u001b[0m \u001b[0mprint\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mrat_func\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     68\u001b[0m \u001b[0mplot\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mrat_func\u001b[0m \u001b[0;34m-\u001b[0m \u001b[0mexp\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m-\u001b[0m\u001b[0mt\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m(\u001b[0m\u001b[0mt\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;36m0\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;36m100\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/Users/aaronmeurer/Documents/python/sympy/sympy/sympy/utilities/decorator.py\u001b[0m in \u001b[0;36mfunc_wrapper\u001b[0;34m(*args, **kwargs)\u001b[0m\n\u001b[1;32m     89\u001b[0m         \u001b[0mdps\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mmpmath\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mmp\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mdps\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     90\u001b[0m         \u001b[0;32mtry\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m---> 91\u001b[0;31m             \u001b[0;32mreturn\u001b[0m \u001b[0mfunc\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[0mkwargs\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m     92\u001b[0m         \u001b[0;32mfinally\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     93\u001b[0m             \u001b[0mmpmath\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mmp\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mdps\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mdps\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m<ipython-input-5-2b9cfe95a3f6>\u001b[0m in \u001b[0;36mCRAM_exp2\u001b[0;34m(degree, prec, max_loops)\u001b[0m\n\u001b[1;32m     42\u001b[0m         \u001b[0mprint\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m'D'\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mD\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     43\u001b[0m         \u001b[0mprint\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m'[(i, D.subs(t, i)) for i in points]'\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m[\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mi\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mD\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0msubs\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mt\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mi\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;32mfor\u001b[0m \u001b[0mi\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mpoints\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m---> 44\u001b[0;31m         \u001b[0;32massert\u001b[0m \u001b[0mlen\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mpoints\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;34m==\u001b[0m \u001b[0;36m2\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mdegree\u001b[0m \u001b[0;34m+\u001b[0m \u001b[0;36m1\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mlen\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mpoints\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m     45\u001b[0m         \u001b[0mEvals\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0;34m[\u001b[0m\u001b[0mE\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mevalf\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mprec\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0msubs\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0;34m{\u001b[0m\u001b[0;34m**\u001b[0m\u001b[0msol\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mt\u001b[0m\u001b[0;34m:\u001b[0m \u001b[0mpoint\u001b[0m\u001b[0;34m}\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;32mfor\u001b[0m \u001b[0mpoint\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mpoints\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m-\u001b[0m\u001b[0;36m1\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[0mr\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mevalf\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mprec\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0msubs\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0;34m{\u001b[0m\u001b[0;34m**\u001b[0m\u001b[0msol\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mt\u001b[0m\u001b[0;34m:\u001b[0m \u001b[0;36m1\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     46\u001b[0m         \u001b[0mprint\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m'Evals'\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mEvals\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;31mAssertionError\u001b[0m: 16"
     ]
    }
   ],
   "source": [
    "from sympy.utilities.decorator import conserve_mpmath_dps\n",
    "\n",
    "@conserve_mpmath_dps\n",
    "def CRAM_exp2(degree, prec=128, *, max_loops=10):\n",
    "    #mpmath.mp.dps = prec\n",
    "\n",
    "\n",
    "    epsilon = symbols(\"epsilon\")\n",
    "    i = symbols(\"i\")\n",
    "    #c = S(1)/2\n",
    "    #c = 10\n",
    "    c = degree*0.6\n",
    "    \n",
    "    r, num_coeffs, den_coeffs = general_rat_func(degree, t, chebyshev=True)\n",
    "    E = exp(c*(t + 1)/(t - 1)) - r\n",
    "    expr = E + (-1)**i*epsilon\n",
    "    expr = expr*r.as_numer_denom()[1]\n",
    "    expr = simplify(expr)\n",
    "\n",
    "    points = [chebyshevt_root(2*(degree + 1) + 1, 2*(degree + 1) - j) for j in range(1, 2*(degree + 1) + 1)]\n",
    "    points = [i.evalf(prec) for i in points]\n",
    "    for iteration in range(max_loops):\n",
    "        print('-'*80)\n",
    "        print(\"Iteration %s:\" % iteration)\n",
    "        system = Tuple(*[expr.subs({i: j, t: points[j]}) for j in range(2*degree + 1)])\n",
    "        system = system + Tuple(expr.replace(exp, lambda i: 0).subs({i: 2*degree + 1, t: 1}))\n",
    "        #print(system)\n",
    "        #print([*num_coeffs, *den_coeffs, epsilon])\n",
    "        sol = dict(zip([*num_coeffs, *den_coeffs, epsilon], nsolve(system, [*num_coeffs, *den_coeffs, epsilon], [*[1]*(2*(degree + 1) - 1), 0], prec=prec)))\n",
    "        print('sol', sol)\n",
    "        print('system.subs(sol)', [i.evalf() for i in system.subs(sol)])\n",
    "        D = diff(E.subs(sol), t)\n",
    "        plot(E.subs(sol), (t, -1, 0.999), adaptive=False, nb_of_points=1000)\n",
    "        plot(E.subs(sol), (t, -0.5, 0.5), adaptive=False, nb_of_points=1000)\n",
    "        #plot(E.subs(sol), (t, 0.9, 1))\n",
    "        # we can't use 1 because of the singularity\n",
    "        print(E.subs(sol))\n",
    "        points = [-1, *nsolve_intervals(D, [-1, 0.999999], maxsteps=10000, prec=prec, division=200, warn=True, tol=10**-(2*prec)), 1]\n",
    "        #points2 = [-1, *nsolve_intervals(D, [-1, 0.999999], maxsteps=300, prec=prec, solver='illinois', verify=True), 1]\n",
    "        #print('points', points)\n",
    "        #print('points2', points2)\n",
    "        print('D', D)\n",
    "        print('[(i, D.subs(t, i)) for i in points]', [(i, D.subs(t, i)) for i in points])\n",
    "        assert len(points) == 2*(degree + 1), len(points)\n",
    "        Evals = [E.evalf(prec, subs={**sol, t: point}) for point in points[:-1]] + [-r.evalf(prec, subs={**sol, t: 1})]\n",
    "        print('Evals', Evals)\n",
    "        maxmin = N(max(map(abs, Evals)) - min(map(abs, Evals)))\n",
    "        print('max - min', maxmin)\n",
    "        print('epsilon', N(sol[epsilon]))\n",
    "        if maxmin < 10**-prec:\n",
    "            print(\"Converged in\", iteration + 1, \"iterations.\")\n",
    "            break\n",
    "    else:\n",
    "        print(\"!!!WARNING: DID NOT CONVERGE AFTER\", max_loops, \"ITERATIONS!!!\", file=sys.stderr)\n",
    "        \n",
    "    #print(sol)\n",
    "    #sol = {i: Rational(str(sol[i])) for i in sol}\n",
    "    #print(sol)\n",
    "    inv = solve(-c*(t + 1)/(t - 1) - y, t)[0].subs(y, t)\n",
    "    n, d = together(r.subs(sol).subs(t, inv)).as_numer_denom() # simplify/cancel here will add degree to the numerator and denominator\n",
    "    rat_func = (Poly(n)/Poly(d).TC())/(Poly(d)/Poly(d).TC())\n",
    "    return rat_func.evalf(prec)\n",
    "\n",
    "#D = CRAM_exp2()\n",
    "\n",
    "rat_func = CRAM_exp2(15, 1000)\n",
    "print(rat_func)\n",
    "plot(rat_func - exp(-t), (t, 0, 100))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "!say done"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false,
    "scrolled": true
   },
   "outputs": [],
   "source": [
    "correct_rat_func2 = (9.92641e-1 + -1.88332e-1*t + 4.21096e-3*t**2)/(1 + 6.69295e-1*t + 5.72258e-1*t**2)\n",
    "plot(correct_rat_func2 - exp(-t), (t, 0, 100))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false,
    "scrolled": false
   },
   "outputs": [],
   "source": [
    "#plot(rat_func - correct_rat_func2, (t, 0, 100))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "plot((correct_rat_func2 - exp(-t))/correct_rat_func2, (t, 20, 100))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false,
    "scrolled": true
   },
   "outputs": [],
   "source": [
    "nsolve(D, (-1, -0.39506511254484362 - 1e-20), solver='bisect', tol=1e-20)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "#import mpmath\n",
    "#mpmath.findroot?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "plot(sin(x), (x, -10, 13))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "nsolve(sin(x), [-10, -0.1], solver='bisect')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "bisect_roots(D, [-1, 0.9], solver='bisect')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "plot(D, (t, -1, 1))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "nsolve(D, [mpf('0.2069106899549174') + 1e-4, 0.999], solver='ridder')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "intervals = [[i/30, (i+1)/30] for i in range(-30, 29)]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "def catch_ValueError(f):\n",
    "    try:\n",
    "        return f()\n",
    "    except ValueError:\n",
    "        pass"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "[catch_ValueError(lambda: nsolve(D, b, solver='bisect')) for b in intervals]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "sin(x).evalf(subs={x: mpf('9.9999999999999995e-8')})"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "nsolve(D, (-1, 0), solver='bisect')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "D"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "import mpmath\n",
    "try:\n",
    "    mpmath.mp.dps = 128\n",
    "    print(nsolve(D, (-1, 0), solver='bisect', maxsteps=300))\n",
    "finally:\n",
    "    mpmath.mp.dps = 15"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "chebyshevt(2, x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "0.9999999999999999"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "0.99999999999999999"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "exp(-(-t - 1)/(2*t - 2))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "solve((0.57224957904836676579778131095824616229041363386436155632065723326613208320645433529158968043480368460271370103317310738469165646*t**2 + 0.66930154271086975944415360267931286108869186534800442869820775970713699218292282609079303557228276229468949430966691265928683428*t + 1.0))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "[1, 2, 3][:-1]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "plot((t + 1)/(t - 1)*(-1/2), (t, -1, 1))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "solve(y - -(-t - 1)/(2*t - 2), t)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "exp(-(-t - 1)/(2*t - 2)).evalf(subs={t: 0})"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "len('42109959068982')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "@conserve_mpmath_dps\n",
    "def test_increased_dps():\n",
    "    # Issue 8564\n",
    "    import mpmath\n",
    "    mpmath.mp.dps = 128\n",
    "    x = Symbol('x')\n",
    "    e1 = x**2 - pi\n",
    "    q = nsolve(e1, x, 3.0)\n",
    "\n",
    "    assert abs(sqrt(pi).evalf(128) - q) < 1e-128\n",
    "\n",
    "test_increased_dps()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "@conserve_mpmath_dps\n",
    "def test_mpf_subs():\n",
    "    import mpmath\n",
    "    mpmath.mp.dps = 128\n",
    "    a = mpf('1') + mpf('1e-100')\n",
    "    print(a)\n",
    "    print(x.subs(x, a))\n",
    "    print(Float(a))\n",
    "\n",
    "test_mpf_subs()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "@conserve_mpmath_dps\n",
    "def test_increased_dps2():\n",
    "    # Issue 8564\n",
    "    import mpmath\n",
    "    mpmath.mp.dps = 128\n",
    "    x = Symbol('x')\n",
    "    e1 = x**2 - pi\n",
    "    q = nsolve(e1, x, 3.0)\n",
    "    print((x**2).subs(x, q))\n",
    "    print(pi.evalf(128))\n",
    "\n",
    "test_increased_dps2()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "p0, p1, p2, q1, q2, epsilon = symbols(\"p0, p1, p2, q1, q2, epsilon\")\n",
    "r = (p0 + p1*t + p2*t**2)/(1 + q1*t + q2*t**2)\n",
    "mpf = lambda i: Float(i, 128)\n",
    "sol = {q1: mpf('-1.1598158896958144389688907541448290768017556418690225475316113676987123000368036156835800708110657203943382445749046292411027838124'), epsilon: mpf('-0.0073586701695804955713909885597251540212160797660122172060482127633180200310633941837926800633336626154349676612971775593725674790502'), p1: mpf('-1.3420582608178928401767377898250826050312440510776128093723998679786607122879166676244905769597498693757444204486264349423147412592'), q2: mpf('0.54706937851182403348832181267224373295821044378627516587652817178767445252898045643900602804552389487580236216086250141313671452195'), p2: mpf('0.73617861351264467252445532341623038921509966014009367227031009417645097851179620934054146565716699654528814524660644163951335402577'), p0: mpf('0.6087293180014645115203537929174427433501475052521283394857491045045741371996335477254671722188240766010031663167379415577767292914')}\n",
    "other_rat_func =\"(0.0042109959068981902319228224147959051043334784348885130731594832554304543920692286150512328715813585751529478908147308247510818304*t**2 - 0.1883335019892743246297189560004799477752569428797900790843481507588522143198473107264222704579544281038255097909548061885535647*t + 0.99264132983041950442860901144027484597878392023398778279395178723668197996893660581620731993666633738456503233870282244062743252)/(0.57224957904836377040707255710162720198078512465710225748045909842276672878045476079146532782980255617241659394936652703249328588*t**2 + 0.66930154271087159949217750010925437712785181665997133501826390508750116474339450131113420220623360917322211168993495402213680723*t + 1.0)\"\n",
    "\n",
    "n, d = together(r.subs(sol).subs(t, (2*t - 1)/(2*t + 1))).as_numer_denom() # simplify/cancel here will add degree to the numerator and denominator\n",
    "rat_func = (Poly(n)/Poly(d).TC())/(Poly(d)/Poly(d).TC())\n",
    "print(rat_func)\n",
    "print(other_rat_func)\n",
    "del mpf"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "import sys\n",
    "print('test', file=sys.stderr)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "10**-10"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "D = (-2*(t + 1)/(2*t - 2)**2 + 1/(2*t - 2))*exp((t + 1)/(2*t - 2)) - (-0.46319625019273218908*t**13 + 11.076239035694458023*t**12 - 75.38527286719552296*t**11 + 242.00660160608425014*t**10 - 389.01043643289017999*t**9 + 165.53549638768474282*t**8 + 538.42392697795239075*t**7 - 1160.0181368287938552*t**6 + 1064.6133061016735644*t**5 - 435.60064981285300771*t**4 - 47.836952574354747137*t**3 + 134.72356135348252232*t**2 - 56.370239030181725748*t + 8.3057523340557002803)*(0.054548679300092563818*t**14 - 1.3501921589697942721*t**13 + 9.0345240868201266233*t**12 - 27.90433797500592764*t**11 + 40.52259728739627843*t**10 - 2.2008514365998888364*t**9 - 97.728246570230795738*t**8 + 181.14960511639832727*t**7 - 159.50642335868838789*t**6 + 56.243320662736403761*t**5 + 26.808952900880779959*t**4 - 41.915371037748377236*t**3 + 21.829567246295711762*t**2 - 5.6442241022971821787*t + 0.6065306597126334236)/(0.033085446442338013506*t**14 - 0.85201838736111215565*t**13 + 6.28210607226629358*t**12 - 22.000600146007659103*t**11 + 38.901043643289017999*t**10 - 18.392832931964971424*t**9 - 67.302990872244048844*t**8 + 165.71687668982769361*t**7 - 177.43555101694559406*t**6 + 87.120129962570601541*t**5 + 11.959238143588686784*t**4 - 44.907853784494174105*t**3 + 28.185119515090862874*t**2 - 8.3057523340557002803*t + 1)**2 - (0.76368151020129589345*t**13 - 17.552498066607325537*t**12 + 108.41428904184151948*t**11 - 306.94771772506520404*t**10 + 405.2259728739627843*t**9 - 19.807662929398999527*t**8 - 781.82597256184636591*t**7 + 1268.0472358147882909*t**6 - 957.03854015213032735*t**5 + 281.2166033136820188*t**4 + 107.23581160352311984*t**3 - 125.74611311324513171*t**2 + 43.659134492591423524*t - 5.6442241022971821787)/(0.033085446442338013506*t**14 - 0.85201838736111215565*t**13 + 6.28210607226629358*t**12 - 22.000600146007659103*t**11 + 38.901043643289017999*t**10 - 18.392832931964971424*t**9 - 67.302990872244048844*t**8 + 165.71687668982769361*t**7 - 177.43555101694559406*t**6 + 87.120129962570601541*t**5 + 11.959238143588686784*t**4 - 44.907853784494174105*t**3 + 28.185119515090862874*t**2 - 8.3057523340557002803*t + 1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "sols = nsolve_intervals(D, [-1, 0.999999], division=100)\n",
    "print(sols)\n",
    "print(len(sols))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "!say done"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "[D.subs(t, s) for s in sols]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "E = S(\"exp((t + 1)/(2*t - 2)) - (0.041232283944696212177*t**14 + 1.3765734756274427394*t**13 - 11.423548266294669551*t**12 + 34.897781468305479785*t**11 - 48.170553853698286848*t**10 + 13.655351758955469199*t**9 + 46.6129883570929947*t**8 - 56.479130360958931409*t**7 + 5.525098125593180923*t**6 + 31.467377104584776578*t**5 - 18.246029053439227394*t**4 - 5.9943769501854388096*t**3 + 10.340721799222326113*t**2 - 4.2100165484624456618*t + 0.6065306597126334236)/(0.025008644382436209356*t**14 + 0.80992537393279613982*t**13 - 7.7761706067212429946*t**12 + 27.673671566138794968*t**11 - 47.057239733847679512*t**10 + 28.109557440827452275*t**9 + 30.919259146443448213*t**8 - 61.83483011093278413*t**7 + 22.957457767970616141*t**6 + 26.975517392632053618*t**5 - 26.389699317234955361*t**4 - 1.0791378811026364212*t**3 + 11.607824151320777869*t**2 - 5.9411438334495710432*t + 1)\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "plot(E, (t, 0.7, 0.8), adaptive=False, nb_of_points=10000)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "chebyshevt(1, 1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "[*enumerate(['a', 'b', 'c'])]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "system = S(\"Tuple(-p0 + 0.98846832432811139916*p1 - 0.050649168838712711749*p10 - 0.10116832198743217456*p11 + 0.25065253225872053539*p12 - 0.39435585511331859613*p13 + 0.5289640103269624416*p14 - 0.95413925640004885148*p2 + 0.89780453957074165714*p3 - 0.82076344120727632636*p4 + 0.72479278722911995887*p5 - 0.61210598254766284416*p6 + 0.48530196253108102515*p7 - 0.34730525284482028541*p8 + 0.20129852008866007873*p9 + (epsilon + 0.99710456213012649629)*(-0.98846832432811139916*q1 + 0.050649168838712711749*q10 + 0.10116832198743217456*q11 - 0.25065253225872053539*q12 + 0.39435585511331859613*q13 - 0.5289640103269624416*q14 + 0.95413925640004885148*q2 - 0.89780453957074165714*q3 + 0.82076344120727632636*q4 - 0.72479278722911995887*q5 + 0.61210598254766284416*q6 - 0.48530196253108102515*q7 + 0.34730525284482028541*q8 - 0.20129852008866007873*q9 + 1), -p0 + 0.96807711886620430515*p1 + 0.82076344120727632615*p10 - 0.93775213214708045842*p11 + 0.99486932339189514118*p12 - 0.98846832432811142281*p13 + 0.91895781162023063016*p14 - 0.87434661614458211883*p2 + 0.72479278722911995886*p3 - 0.52896401032696245737*p4 + 0.29936312297335795401*p5 - 0.050649168838712712237*p6 - 0.20129852008866007919*p7 + 0.44039415155763430936*p8 - 0.65137248272222220705*p9 + (-epsilon + 0.99192262841900652865)*(-0.96807711886620430515*q1 - 0.82076344120727632615*q10 + 0.93775213214708045842*q11 - 0.99486932339189514118*q12 + 0.98846832432811142281*q13 - 0.91895781162023063016*q14 + 0.87434661614458211883*q2 - 0.72479278722911995886*q3 + 0.52896401032696245737*q4 - 0.29936312297335795401*q5 + 0.050649168838712712237*q6 + 0.20129852008866007919*q7 - 0.44039415155763430936*q8 + 0.65137248272222220705*q9 + 1), -p0 + 0.93775213214708045843*p1 + 0.91895781162023062755*p10 - 0.72479278722911995726*p11 + 0.44039415155763430437*p12 - 0.10116832198743218276*p13 - 0.2506525322587205562*p14 - 0.75875812269279090192*p2 + 0.48530196253108102522*p3 - 0.15142777750457666366*p4 - 0.20129852008866007914*p5 + 0.52896401032696245735*p6 - 0.79077573693769858208*p7 + 0.95413925640004885141*p8 - 0.99871650717105280743*p9 + (epsilon + 0.98406642835075986461)*(-0.93775213214708045843*q1 - 0.91895781162023062755*q10 + 0.72479278722911995726*q11 - 0.44039415155763430437*q12 + 0.10116832198743218276*q13 + 0.2506525322587205562*q14 + 0.75875812269279090192*q2 - 0.48530196253108102522*q3 + 0.15142777750457666366*q4 + 0.20129852008866007914*q5 - 0.52896401032696245735*q6 + 0.79077573693769858208*q7 - 0.95413925640004885141*q8 + 0.99871650717105280743*q9 + 1), -p0 + 0.89780453957074165714*p1 + 0.15142777750457666382*p10 + 0.29936312297335795208*p11 - 0.68896691907568656868*p12 + 0.93775213214708046506*p13 - 0.99486932339189512731*p14 - 0.61210598254766284415*p2 + 0.20129852008866007914*p3 + 0.25065253225872053932*p4 - 0.65137248272222220746*p5 + 0.91895781162023062913*p6 - 0.99871650717105280713*p7 + 0.87434661614458211861*p8 - 0.57126821509479227908*p9 + (-epsilon + 0.97343458335469763991)*(-0.89780453957074165714*q1 - 0.15142777750457666382*q10 - 0.29936312297335795208*q11 + 0.68896691907568656868*q12 - 0.93775213214708046506*q13 + 0.99486932339189512731*q14 + 0.61210598254766284415*q2 - 0.20129852008866007914*q3 - 0.25065253225872053932*q4 + 0.65137248272222220746*q5 - 0.91895781162023062913*q6 + 0.99871650717105280713*q7 - 0.87434661614458211861*q8 + 0.57126821509479227908*q9 + 1), -p0 + 0.84864425749475095046*p1 - 0.7587581226927909016*p10 + 0.98846832432811139721*p11 - 0.91895781162023063016*p12 + 0.57126821509479228581*p13 - 0.050649168838712711749*p14 - 0.44039415155763430952*p2 - 0.10116832198743217778*p3 + 0.61210598254766284414*p4 - 0.93775213214708045842*p5 + 0.97952994125249449394*p6 - 0.72479278722911995886*p7 + 0.25065253225872053918*p8 + 0.29936312297335795414*p9 + (epsilon + 0.9598896419605118508)*(-0.84864425749475095046*q1 + 0.7587581226927909016*q10 - 0.98846832432811139721*q11 + 0.91895781162023063016*q12 - 0.57126821509479228581*q13 + 0.050649168838712711749*q14 + 0.44039415155763430952*q2 + 0.10116832198743217778*q3 - 0.61210598254766284414*q4 + 0.93775213214708045842*q5 - 0.97952994125249449394*q6 + 0.72479278722911995886*q7 - 0.25065253225872053918*q8 - 0.29936312297335795414*q9 + 1), -p0 + 0.79077573693769858208*p1 - 0.95413925640004885152*p10 + 0.57126821509479227959*p11 + 0.050649168838712713483*p12 - 0.65137248272222220499*p13 + 0.97952994125249449528*p14 - 0.25065253225872053932*p2 - 0.3943558551133185801*p3 + 0.87434661614458211883*p4 - 0.98846832432811139917*p5 + 0.6889669190756865678*p6 - 0.10116832198743217778*p7 - 0.52896401032696245732*p8 + 0.93775213214708045842*p9 + (-epsilon + 0.94325632448977341962)*(-0.79077573693769858208*q1 + 0.95413925640004885152*q10 - 0.57126821509479227959*q11 - 0.050649168838712713483*q12 + 0.65137248272222220499*q13 - 0.97952994125249449528*q14 + 0.25065253225872053932*q2 + 0.3943558551133185801*q3 - 0.87434661614458211883*q4 + 0.98846832432811139917*q5 - 0.6889669190756865678*q6 + 0.10116832198743217778*q7 + 0.52896401032696245732*q8 - 0.93775213214708045842*q9 + 1), -p0 + 0.72479278722911995887*p1 - 0.25065253225872053951*p10 - 0.48530196253108102528*p11 + 0.95413925640004885152*p12 - 0.8978045395707416562*p13 + 0.34730525284482029018*p14 - 0.050649168838712712279*p2 - 0.65137248272222220745*p3 + 0.99486932339189514632*p4 - 0.79077573693769858208*p5 + 0.15142777750457666365*p6 + 0.57126821509479227918*p7 - 0.97952994125249449387*p8 + 0.84864425749475095033*p9 + (epsilon + 0.92331964820666129003)*(-0.72479278722911995887*q1 + 0.25065253225872053951*q10 + 0.48530196253108102528*q11 - 0.95413925640004885152*q12 + 0.8978045395707416562*q13 - 0.34730525284482029018*q14 + 0.050649168838712712279*q2 + 0.65137248272222220745*q3 - 0.99486932339189514632*q4 + 0.79077573693769858208*q5 - 0.15142777750457666365*q6 - 0.57126821509479227918*q7 + 0.97952994125249449387*q8 - 0.84864425749475095033*q9 + 1), -p0 + 0.65137248272222220745*p1 + 0.68896691907568656781*p10 - 0.9987165071710528071*p11 + 0.61210598254766284418*p12 + 0.20129852008866007835*p13 - 0.87434661614458211807*p14 + 0.15142777750457666366*p2 - 0.84864425749475095047*p3 + 0.95413925640004885148*p4 - 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0.44039415155763430952*p2 + 0.10116832198743217778*p3 + 0.61210598254766284414*p4 + 0.93775213214708045842*p5 + 0.97952994125249449394*p6 + 0.72479278722911995886*p7 + 0.25065253225872053918*p8 - 0.29936312297335795414*p9 + (epsilon + 0.0022273312261039493487)*(0.84864425749475095046*q1 + 0.7587581226927909016*q10 + 0.98846832432811139721*q11 + 0.91895781162023063016*q12 + 0.57126821509479228581*q13 + 0.050649168838712711749*q14 + 0.44039415155763430952*q2 - 0.10116832198743217778*q3 - 0.61210598254766284414*q4 - 0.93775213214708045842*q5 - 0.97952994125249449394*q6 - 0.72479278722911995886*q7 - 0.25065253225872053918*q8 + 0.29936312297335795414*q9 + 1), -p0 - 0.89780453957074165714*p1 + 0.15142777750457666382*p10 - 0.29936312297335795208*p11 - 0.68896691907568656868*p12 - 0.93775213214708046506*p13 - 0.99486932339189512731*p14 - 0.61210598254766284415*p2 - 0.20129852008866007914*p3 + 0.25065253225872053932*p4 + 0.65137248272222220746*p5 + 0.91895781162023062913*p6 + 0.99871650717105280713*p7 + 0.87434661614458211861*p8 + 0.57126821509479227908*p9 + (-epsilon + 0.000092790113933967817733)*(0.89780453957074165714*q1 - 0.15142777750457666382*q10 + 0.29936312297335795208*q11 + 0.68896691907568656868*q12 + 0.93775213214708046506*q13 + 0.99486932339189512731*q14 + 0.61210598254766284415*q2 + 0.20129852008866007914*q3 - 0.25065253225872053932*q4 - 0.65137248272222220746*q5 - 0.91895781162023062913*q6 - 0.99871650717105280713*q7 - 0.87434661614458211861*q8 - 0.57126821509479227908*q9 + 1), -p0 - 0.93775213214708045843*p1 + 0.91895781162023062755*p10 + 0.72479278722911995726*p11 + 0.44039415155763430437*p12 + 0.10116832198743218276*p13 - 0.2506525322587205562*p14 - 0.75875812269279090192*p2 - 0.48530196253108102522*p3 - 0.15142777750457666366*p4 + 0.20129852008866007914*p5 + 0.52896401032696245735*p6 + 0.79077573693769858208*p7 + 0.95413925640004885141*p8 + 0.99871650717105280743*p9 + (epsilon + 1.7389619963042768938e-7)*(0.93775213214708045843*q1 - 0.91895781162023062755*q10 - 0.72479278722911995726*q11 - 0.44039415155763430437*q12 - 0.10116832198743218276*q13 + 0.2506525322587205562*q14 + 0.75875812269279090192*q2 + 0.48530196253108102522*q3 + 0.15142777750457666366*q4 - 0.20129852008866007914*q5 - 0.52896401032696245735*q6 - 0.79077573693769858208*q7 - 0.95413925640004885141*q8 - 0.99871650717105280743*q9 + 1), -p0 - 0.96807711886620430515*p1 + 0.82076344120727632615*p10 + 0.93775213214708045842*p11 + 0.99486932339189514118*p12 + 0.98846832432811142281*p13 + 0.91895781162023063016*p14 - 0.87434661614458211883*p2 - 0.72479278722911995886*p3 - 0.52896401032696245737*p4 - 0.29936312297335795401*p5 - 0.050649168838712712237*p6 + 0.20129852008866007919*p7 + 0.44039415155763430936*p8 + 0.65137248272222220705*p9 + (-epsilon + 4.0988152678429733251e-14)*(0.96807711886620430515*q1 - 0.82076344120727632615*q10 - 0.93775213214708045842*q11 - 0.99486932339189514118*q12 - 0.98846832432811142281*q13 - 0.91895781162023063016*q14 + 0.87434661614458211883*q2 + 0.72479278722911995886*q3 + 0.52896401032696245737*q4 + 0.29936312297335795401*q5 + 0.050649168838712712237*q6 - 0.20129852008866007919*q7 - 0.44039415155763430936*q8 - 0.65137248272222220705*q9 + 1), -p0 - 0.98846832432811139916*p1 - 0.050649168838712711749*p10 + 0.10116832198743217456*p11 + 0.25065253225872053539*p12 + 0.39435585511331859613*p13 + 0.5289640103269624416*p14 - 0.95413925640004885148*p2 - 0.89780453957074165714*p3 - 0.82076344120727632636*p4 - 0.72479278722911995887*p5 - 0.61210598254766284416*p6 - 0.48530196253108102515*p7 - 0.34730525284482028541*p8 - 0.20129852008866007873*p9 + (epsilon + 3.5986813119484602573e-38)*(0.98846832432811139916*q1 + 0.050649168838712711749*q10 - 0.10116832198743217456*q11 - 0.25065253225872053539*q12 - 0.39435585511331859613*q13 - 0.5289640103269624416*q14 + 0.95413925640004885148*q2 + 0.89780453957074165714*q3 + 0.82076344120727632636*q4 + 0.72479278722911995887*q5 + 0.61210598254766284416*q6 + 0.48530196253108102515*q7 + 0.34730525284482028541*q8 + 0.20129852008866007873*q9 + 1), -epsilon*(q1 + q10 + q11 + q12 + q13 + q14 + q2 + q3 + q4 + q5 + q6 + q7 + q8 + q9 + 1) - p0 - p1 - p10 - p11 - p12 - p13 - p14 - p2 - p3 - p4 - p5 - p6 - p7 - p8 - p9)\")\n",
    "vars = S(\"[p0, p1, p2, p3, p4, p5, p6, p7, p8, p9, p10, p11, p12, p13, p14, q1, q2, q3, q4, q5, q6, q7, q8, q9, q10, q11, q12, q13, q14, epsilon]\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "sol = dict(zip(vars, nsolve(system, vars, [1]*len(vars), solver='secant')))\n",
    "sol"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "print(system.subs(sol))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "plot(exp((t + 1)/(t - 1)), (t, -1, 1))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "plot(exp(10*(t + 1)/(t - 1)), (t, -1, 1))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "r = S(\"(0.0011875196369021855177*t**14 - 0.011420090831167348621*t**13 + 0.050741761684873561064*t**12 - 0.13809510690997622664*t**11 + 0.25723786667078296754*t**10 - 0.34703482945300861262*t**9 + 0.34976856562058063458*t**8 - 0.26762758451401203127*t**7 + 0.15627145956034238656*t**6 - 0.069319897634892213306*t**5 + 0.023002511830763159929*t**4 - 0.0055388208662975187897*t**3 + 0.00091518140765965519386*t**2 - 0.00009291001238164753822*t + 4.3738099008807505585e-6)/(0.00089586291474760584255*t**14 + 0.0089336931214819976007*t**13 + 0.042095909802441250189*t**12 + 0.12482069131842848916*t**11 + 0.26604814866002116766*t**10 + 0.42347046743527135243*t**9 + 0.57833691404471361527*t**8 + 0.54769996073689544827*t**7 + 0.68765070665076017993*t**6 + 0.2278059784921081907*t**5 + 0.6501516075867275978*t**4 - 0.14644874880736715755*t**3 + 0.42333407306668857903*t**2 - 0.11968772580151045925*t + 0.096339574212779644172)\").subs(t, solve(10*(t + 1)/(t - 1) - y, t)[0].subs(y, t))\n",
    "n, d = together(r).as_numer_denom() # simplify/cancel here will add degree to the numerator and denominator\n",
    "rat_func = (Poly(n)/Poly(d).TC())/(Poly(d)/Poly(d).TC())\n",
    "rat_func"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "D = S(\"(10/(t - 1) - (10*t + 10)/(t - 1)**2)*exp((10*t + 10)/(t - 1)) - (-0.011109949473020816773*t**13 - 0.10498117392496721903*t**12 - 0.4662693207692269047*t**11 - 1.2944463271987387573*t**10 - 2.560791476062548363*t**9 - 3.7445986907826045347*t**8 - 4.6175581668981299287*t**7 - 3.9202376055011740667*t**6 - 4.1853692405031723697*t**5 - 1.2730163279308121306*t**4 - 2.6093036740226881464*t**3 + 0.37572825889837415047*t**2 - 0.84260381473372911342*t + 0.11395040244406088625)*(0.0011012754893812472661*t**14 - 0.010681747521049671358*t**13 + 0.047852308718456039572*t**12 - 0.13125412233037760452*t**11 + 0.24631294936212648611*t**10 - 0.33462134384239723802*t**9 + 0.3394635962011490975*t**8 - 0.2613214567551016616*t**7 + 0.15344584617039959246*t**6 - 0.068417469219647235265*t**5 + 0.022810045726230519721*t**4 - 0.0055160290938826628221*t**3 + 0.00091495897212439435512*t**2 - 0.00009321390565315082022*t + 4.402028268659469097e-6)/(0.00079356781950148691235*t**14 + 0.0080754749173051706947*t**13 + 0.038855776730768908725*t**12 + 0.11767693883624897794*t**11 + 0.2560791476062548363*t**10 + 0.41606652119806717052*t**9 + 0.57719477086226624108*t**8 + 0.56003394364302486668*t**7 + 0.69756154008386206162*t**6 + 0.25460326558616242613*t**5 + 0.65232591850567203661*t**4 - 0.12524275296612471682*t**3 + 0.42130190736686455671*t**2 - 0.11395040244406088625*t + 0.096961125131689425859)**2 - (0.015417856851337461725*t**13 - 0.13886271777364572765*t**12 + 0.57422770462147247487*t**11 - 1.4437953456341536497*t**10 + 2.4631294936212648611*t**9 - 3.0115920945815751422*t**8 + 2.71570876960919278*t**7 - 1.8292501972857116312*t**6 + 0.92067507702239755476*t**5 - 0.34208734609823617633*t**4 + 0.091240182904922078882*t**3 - 0.016548087281647988466*t**2 + 0.0018299179442487887102*t - 0.00009321390565315082022)/(0.00079356781950148691235*t**14 + 0.0080754749173051706947*t**13 + 0.038855776730768908725*t**12 + 0.11767693883624897794*t**11 + 0.2560791476062548363*t**10 + 0.41606652119806717052*t**9 + 0.57719477086226624108*t**8 + 0.56003394364302486668*t**7 + 0.69756154008386206162*t**6 + 0.25460326558616242613*t**5 + 0.65232591850567203661*t**4 - 0.12524275296612471682*t**3 + 0.42130190736686455671*t**2 - 0.11395040244406088625*t + 0.096961125131689425859)\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "D.subs(t, Float(\"-5.0000000006988898296e-7\"))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "interval = [-5.00000000069889e-07, 0.019999489999999787]\n",
    "sol = nsolve(D, interval, solver='bisect', prec=20)\n",
    "sol"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "D.subs(t, sol)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "interval[0] < sol < interval[1]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "interval[0]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "import mpmath\n",
    "mpmath.mp.dps = 20\n",
    "f = lambdify(t, D, 'mpmath')\n",
    "sol = mpmath.findroot(f, [0.57999921, 0.5999991999999998] , solver='bisect')\n",
    "sol"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "f(sol)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "plot(-exp(-t) + S(\"(0.00421099590689821779984639140604*t**2 + 0.188333501989274148910007547496*t + 0.992641329830419396033230547887)/(0.572249579048364881552280260776*t**2 - 0.669301542710871251180032333275*t + 1.0)\"), (t, 0, 100))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "rat_func"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "plot(rat_func - exp(-t), (t, 0, 100))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "plot(rat_func - S(\"(0.00421099590689821779984639140604*t**2 + 0.188333501989274148910007547496*t + 0.992641329830419396033230547887)/(0.572249579048364881552280260776*t**2 - 0.669301542710871251180032333275*t + 1.0)\"), (t, 0, 100))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "rat_func"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "S(\"(0.00421099590689821779984639140604*t**2 + 0.188333501989274148910007547496*t + 0.992641329830419396033230547887)/(0.572249579048364881552280260776*t**2 - 0.669301542710871251180032333275*t + 1.0)\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "rat_func == S(\"(0.00421099590689821779984639140604*t**2 + 0.188333501989274148910007547496*t + 0.992641329830419396033230547887)/(0.572249579048364881552280260776*t**2 - 0.669301542710871251180032333275*t + 1.0)\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "str(rat_func) == \"(0.00421099590689821779984639140604*t**2 + 0.188333501989274148910007547496*t + 0.992641329830419396033230547887)/(0.572249579048364881552280260776*t**2 - 0.669301542710871251180032333275*t + 1.0)\""
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "plot(rat_func)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "plot(S(\"(0.00421099590689821779984639140604*t**2 + 0.188333501989274148910007547496*t + 0.992641329830419396033230547887)/(0.572249579048364881552280260776*t**2 - 0.669301542710871251180032333275*t + 1.0)\"))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "good_rat_func = S(\"(0.00421099590689821779984639140604*t**2 + 0.188333501989274148910007547496*t + 0.992641329830419396033230547887)/(0.572249579048364881552280260776*t**2 - 0.669301542710871251180032333275*t + 1.0)\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "plot(rat_func - exp(-t), (t, 0, 100))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "plot(good_rat_func - exp(-t), (t, 0, 100))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "plot(-exp(-t) + good_rat_func, (t, 0, 100))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "plot(-exp(-t) - S(\"(0.00421099590689821779984639140604*t**2 + 0.188333501989274148910007547496*t + 0.992641329830419396033230547887)/(0.572249579048364881552280260776*t**2 - 0.669301542710871251180032333275*t + 1.0)\"), (t, 0, 100))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "plot(-exp(-t) + S(srepr(rat_func)), (t, 0, 100))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "plot(-exp(-t) + S(str(rat_func)), (t, 0, 100))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "plot(rat_func - exp(-t), (t, 0, 100), adaptive=False, nb_of_points=1000)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "import numpy\n",
    "import matplotlib.pyplot as plt\n",
    "vals = numpy.linspace(0, 100, 1000)\n",
    "plt.plot(vals, lambdify(t, rat_func.subs(t, -t) - exp(-t), 'numpy')(vals))\n",
    "                "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "degree = 14\n",
    "points = [chebyshevt_root(2*(degree + 1) + 1, 2*(degree + 1) - j) for j in range(1, 2*(degree + 1) + 1)]\n",
    "[i.evalf() for i in points]\n",
    "c = symbols('c', positive=True)\n",
    "len(points)\n",
    "solve(sqrt(c*(1 + points[degree])/(1 - points[degree])*c*(1 + points[degree+1])/(1 - points[degree+1])) - c, c)\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "sqrt(c*(1 + points[degree])/(1 - points[degree])*c*(1 + points[degree+1])/(1 - points[degree+1])) - c"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "E = S(\"exp((10.8*t + 10.8)/(t - 1)) - (0.000292923440172061376307712067378*t**18 - 0.00382369712669093978523377528192*t**17 + 0.0235117968300681682735465482247*t**16 - 0.0904961683900329437749507666096*t**15 + 0.244347315405413563336773282433*t**14 - 0.491533156343580993021894164578*t**13 + 0.763512364001241488427664101087*t**12 - 0.936481460181764203042245859202*t**11 + 0.919553635779126092586969716435*t**10 - 0.728338033009921941415427470583*t**9 + 0.466494241057100272722298949598*t**8 - 0.241066390548087834899533957827*t**7 + 0.0997744318509568132657709953788*t**6 - 0.0326299433259599107384290896079*t**5 + 0.00824748059197329933751330243673*t**4 - 0.00155480878299504771864337728231*t**3 + 0.000205891550849325966892424491264*t**2 - 0.0000170923433737289258328111955236*t + 0.000000669545506459572399276839032838)/(0.0000256271908755749476501555060643*t**18 + 0.00029976157640698698493412804827*t**17 + 0.0016792592359715410041296380456*t**16 + 0.00593759917905712411027230635685*t**15 + 0.0156202804404542365771258126305*t**14 + 0.0289577506063902781990988819345*t**13 + 0.0560418346810373552008650596658*t**12 + 0.0418316309497562848580964550569*t**11 + 0.151260111256952231777215316531*t**10 - 0.0963027347150387933031292538397*t**9 + 0.434868911821397452965639691116*t**8 - 0.535112011457860838676364996918*t**7 + 0.885545060524724480506124658926*t**6 - 0.910515237881500396140647664512*t**5 + 0.901052413999683103211771128713*t**4 - 0.618793687227890880788897623539*t**3 + 0.360336843257148540488631327245*t**2 - 0.128932571601980421778511686416*t + 0.0328216571239099274214356723826)\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "plot(E, (t, -1, 1))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "E.replace(exp, lambda x: 0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "c = 10.8\n",
    "prec = 30\n",
    "r = -E.replace(exp, lambda x: 0)\n",
    "inv = solve(-c*(t + 1)/(t - 1) - y, t)[0].subs(y, t)\n",
    "n, d = together(r.subs(t, inv)).as_numer_denom() # simplify/cancel here will add degree to the numerator and denominator\n",
    "rat_func = (Poly(n)/Poly(d).TC())/(Poly(d)/Poly(d).TC())\n",
    "rat_func"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "inv = solve(-c*(t + 1)/(t - 1) - y, t)[0].subs(y, t)\n",
    "n, d = together(r.subs(t, inv.evalf(prec))).as_numer_denom() # simplify/cancel here will add degree to the numerator and denominator\n",
    "rat_func = expand(n/d.subs(t, 0))/expand(d/d.subs(t, 0))\n",
    "rat_func"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "r.subs(t, inv)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "master_rat_func = S(\"(0.000000000000000000000000000000000000076480564016753411391758926810678059166246681879403496152487645239211280144217888827074664937622241433057155411729238059005620362*t**18 - 0.00000000000000000000000000000000144349696189967720280773581225860550659908207558078615762681841828055896712976164458203824215880891842340327527192908591610821*t**17 + 0.0000000000000000000000000000044910880325416024060232183011837489287970058235258370172770028208341880031097425649152413600814540715147042236183237457793362906*t**16 - 0.0000000000000000000000000055904194675876842224749371778659158365295791811734393172655086431051636828145602210004215923885798770655051649836756001927074635*t**15 + 0.0000000000000000000000036534617094118107737307644008627750110509773097939005142805165889454907972711014257711631906543393485730765556180421132890852392*t**14 - 0.000000000000000000001452005752480375025477783595297687855428318333130316532815795944869337169252966327504293176967865458909941895728922083075100905*t**13 + 0.00000000000000000038258486902246032408006621010963716586950687991236057815854665336890002346614805976419029288917692940037303895895123632103288712*t**12 - 0.000000000000000070617682439427615900771877450592882868606546484110524383009238796767626116906296729855457707191810570035340443853799274578659569*t**11 + 0.0000000000000094758123346723567170908152172735939652157451575834910841031916583217101183394528014444063348758109776739928297123459805390104553*t**10 - 0.00000000000094773868840452502508269303119154845985934338599523295887294825053761377691930897288612488861758600642472630646115958140782655741*t**9 + 0.000000000071794537029977599630049677051557682902685713470170819790111653928229512768189817446726011375388897055644123749103346310180957703*t**8 - 0.0000000041548085790436606814151882639426540923873459207739075135022708664862920165014122761594444148444999444676937908166192319430948739*t**7 + 0.00000018396482187290156607582394998664929895412165041697075842037042924223322317897465358352879794796218422276896933375527141853756368*t**6 - 0.0000061940471597289213538311760411627015047318480396138859471763151560132556207793110081461448458843840327989497213331241784911809188*t**5 + 0.00015609017337127813834188030193516319436098147682156485822323423924211374867515035340573616843723433765654338888756861271039524759*t**4 - 0.0028562580858878094323585065185801663096872149540312530171997790387127408262872701820963659684205263711498876442924046259043818741*t**3 + 0.035908020274519704107157835421234360862435489836434974542107322036865342386568144753587976535805549361943459578492579177565909169*t**2 - 0.27789748837786198960284734868450774184090800132807427206186974905648882760573987089502539075626928473970338728172474858726571638*t + 1.0)/(0.000000000000000000031040251435758559612107016726409522805133890008185285868034602477549411378597769320246238650254246564453320307940208743427452132*t**18 + 0.00000000000000000033278408185195783331139523007623347013904881323415039608612228753896462298159222572181471639107826698580483779471762583452979632*t**17 + 0.000000000000000036787450097487433810641016956456458321996969855733085703659877884336499129663780877438684132168500286553243353222376638553979316*t**16 + 0.00000000000000086471483697461572289931466729671519305573126608639342123901368411795553895792792034205116888738244515155185481746831557263871812*t**15 + 0.000000000000028542945098191436824578047622161061818620141667875379414297168053758179023344405243390741611910581146082198383107320705069583298*t**14 + 0.00000000000069925714572325319966132548491379676104543251464498742295512527923847731544663020484057497681632084412599937152698907579711598455*t**13 + 0.000000000016023022340045153918478559722974792676358615762326800442501765171455792601242697969576723610509994457322049182740923202818976629*t**12 + 0.00000000032288400362818264475862911376930287544499616030122169108718711417899107361671375705975465433486011408164287643509933657504241192*t**11 + 0.0000000058031481469953025925067093526070113494249880473036061932865265507723402865048713484792246133690205685646178264769891316933338782*t**10 + 0.000000092334736432550055237788885523966859751018477031256929995376082333113413768260370819988748007438778907195512402675485670731415463*t**9 + 0.0000012926132501100766989333427587529904458908220243641974136275253050146198705426564792856481298765046890202380566214250561052101974*t**8 + 0.000015764526871377686388685138109739710158659275647616228857451800259851474387813943355076456887413075988332171386231137024453368698*t**7 + 0.00016523598870223734044973391186293737035539948649029834978809577442020543781821653294313415106131238369918759029115415865449394346*t**6 + 0.0014607084457312565069881359757159810508484208317079191023373636723638109071771404868435318582711990519471832277113738367820431404*t**5 + 0.010604260828765659024101011711311726965939177316289266086487560938209634069089784251192671810661317429692738097439733970455088636*t**4 + 0.060769684666293428066362095643979446983198257906551701462685569300225217899448253566777035915490895538454676999338813585125899174*t**3 + 0.25801053189666603350610238873105407999382491457750647353754793173734492220703368984743001028426676940884044469045936407185268167*t**2 + 0.72210251162213748406312012324555238365117862900849998495383644968647977930578316078230461402023282743548534135417525983920899434*t + 1.0)\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "smichr_nsolve_rat_func = S(\"(0.000000000000000000000000000000000000076480564016753411391758926810678059166246681879403496152487645239211280144217888827074664937622241433057155411729238059005620362*t**18 - 0.00000000000000000000000000000000144349696189967720280773581225860550659908207558078615762681841828055896712976164458203824215880891842340327527192908591610821*t**17 + 0.0000000000000000000000000000044910880325416024060232183011837489287970058235258370172770028208341880031097425649152413600814540715147042236183237457793362906*t**16 - 0.0000000000000000000000000055904194675876842224749371778659158365295791811734393172655086431051636828145602210004215923885798770655051649836756001927074635*t**15 + 0.0000000000000000000000036534617094118107737307644008627750110509773097939005142805165889454907972711014257711631906543393485730765556180421132890852392*t**14 - 0.000000000000000000001452005752480375025477783595297687855428318333130316532815795944869337169252966327504293176967865458909941895728922083075100905*t**13 + 0.00000000000000000038258486902246032408006621010963716586950687991236057815854665336890002346614805976419029288917692940037303895895123632103288712*t**12 - 0.000000000000000070617682439427615900771877450592882868606546484110524383009238796767626116906296729855457707191810570035340443853799274578659569*t**11 + 0.0000000000000094758123346723567170908152172735939652157451575834910841031916583217101183394528014444063348758109776739928297123459805390104553*t**10 - 0.00000000000094773868840452502508269303119154845985934338599523295887294825053761377691930897288612488861758600642472630646115958140782655741*t**9 + 0.000000000071794537029977599630049677051557682902685713470170819790111653928229512768189817446726011375388897055644123749103346310180957703*t**8 - 0.0000000041548085790436606814151882639426540923873459207739075135022708664862920165014122761594444148444999444676937908166192319430948739*t**7 + 0.00000018396482187290156607582394998664929895412165041697075842037042924223322317897465358352879794796218422276896933375527141853756368*t**6 - 0.0000061940471597289213538311760411627015047318480396138859471763151560132556207793110081461448458843840327989497213331241784911809188*t**5 + 0.00015609017337127813834188030193516319436098147682156485822323423924211374867515035340573616843723433765654338888756861271039524759*t**4 - 0.0028562580858878094323585065185801663096872149540312530171997790387127408262872701820963659684205263711498876442924046259043818741*t**3 + 0.035908020274519704107157835421234360862435489836434974542107322036865342386568144753587976535805549361943459578492579177565909169*t**2 - 0.27789748837786198960284734868450774184090800132807427206186974905648882760573987089502539075626928473970338728172474858726571638*t + 1.0)/(0.000000000000000000031040251435758559612107016726409522805133890008185285868034602477549411378597769320246238650254246564453320307940208743427452132*t**18 + 0.00000000000000000033278408185195783331139523007623347013904881323415039608612228753896462298159222572181471639107826698580483779471762583452979632*t**17 + 0.000000000000000036787450097487433810641016956456458321996969855733085703659877884336499129663780877438684132168500286553243353222376638553979316*t**16 + 0.00000000000000086471483697461572289931466729671519305573126608639342123901368411795553895792792034205116888738244515155185481746831557263871812*t**15 + 0.000000000000028542945098191436824578047622161061818620141667875379414297168053758179023344405243390741611910581146082198383107320705069583298*t**14 + 0.00000000000069925714572325319966132548491379676104543251464498742295512527923847731544663020484057497681632084412599937152698907579711598455*t**13 + 0.000000000016023022340045153918478559722974792676358615762326800442501765171455792601242697969576723610509994457322049182740923202818976629*t**12 + 0.00000000032288400362818264475862911376930287544499616030122169108718711417899107361671375705975465433486011408164287643509933657504241192*t**11 + 0.0000000058031481469953025925067093526070113494249880473036061932865265507723402865048713484792246133690205685646178264769891316933338782*t**10 + 0.000000092334736432550055237788885523966859751018477031256929995376082333113413768260370819988748007438778907195512402675485670731415463*t**9 + 0.0000012926132501100766989333427587529904458908220243641974136275253050146198705426564792856481298765046890202380566214250561052101974*t**8 + 0.000015764526871377686388685138109739710158659275647616228857451800259851474387813943355076456887413075988332171386231137024453368698*t**7 + 0.00016523598870223734044973391186293737035539948649029834978809577442020543781821653294313415106131238369918759029115415865449394346*t**6 + 0.0014607084457312565069881359757159810508484208317079191023373636723638109071771404868435318582711990519471832277113738367820431404*t**5 + 0.010604260828765659024101011711311726965939177316289266086487560938209634069089784251192671810661317429692738097439733970455088636*t**4 + 0.060769684666293428066362095643979446983198257906551701462685569300225217899448253566777035915490895538454676999338813585125899174*t**3 + 0.25801053189666603350610238873105407999382491457750647353754793173734492220703368984743001028426676940884044469045936407185268167*t**2 + 0.72210251162213748406312012324555238365117862900849998495383644968647977930578316078230461402023282743548534135417525983920899434*t + 1.0)\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "master_rat_func == smichr_nsolve_rat_func"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
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