From 9477dbe667f250ecd23f8fc0d56b942191526421 Mon Sep 17 00:00:00 2001
From: Franciszek Malinka <franciszek.malinka@gmail.com>
Date: Thu, 25 Feb 2021 14:42:55 +0100
Subject: Stare semestry, niepoukladane

---
 Semestr 3/anm/pracownia1/prog/program.ipynb | 2042 +++++++++++++++++++++++++++
 1 file changed, 2042 insertions(+)
 create mode 100644 Semestr 3/anm/pracownia1/prog/program.ipynb

(limited to 'Semestr 3/anm/pracownia1/prog/program.ipynb')

diff --git a/Semestr 3/anm/pracownia1/prog/program.ipynb b/Semestr 3/anm/pracownia1/prog/program.ipynb
new file mode 100644
index 0000000..e5d7c6a
--- /dev/null
+++ b/Semestr 3/anm/pracownia1/prog/program.ipynb	
@@ -0,0 +1,2042 @@
+{
+ "cells": [
+  {
+   "cell_type": "code",
+   "execution_count": 89,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "set_cordic_iterations (generic function with 1 method)"
+      ]
+     },
+     "execution_count": 89,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "include(\"program.jl\")"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 90,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "Plots.PlotlyBackend()"
+      ]
+     },
+     "execution_count": 90,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "using Plots\n",
+    "using Random\n",
+    "using Distributions\n",
+    "\n",
+    "plotly()"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 91,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# Zadanie 10, Franiszek malinka, Kacper Solecki\n",
+    "\n",
+    "# instrukcja:\n",
+    "# Nasz program udostępnia funkcje \n",
+    "\n",
+    "# -> taylor_sin(a, b) - sinus liczby a+bi liczony za pomocą szeregu Taylora\n",
+    "# -> taylor_cos(a, b) - cosinus liczby a+bi liczony za pomocą szeregu Taylora\n",
+    "# -> taylor_sinh(x) - sinus hiperboliczny liczby x liczony za pomocą szeregu Taylora\n",
+    "# -> taylor_cosh(x) - cosinus hiperboliczny liczby x liczony za pomocą szeregu Taylora\n",
+    "# -> cordic_sin(x) - sinus (rzeczywistej) liczby x liczony za pomocą algorytmu Cordic\n",
+    "# -> cordic_cos(x) - cosinus (rzeczywistej) liczby x liczony za pomocą algorytmu Cordic"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 92,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "0.9092974268256817"
+      ]
+     },
+     "execution_count": 92,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "# porównianie na sin(2), cos(2)\n",
+    "sin(2.0)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 93,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "(0.9092974268256817, -0.0)"
+      ]
+     },
+     "execution_count": 93,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "taylor_sin(2.0, 0.0)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 94,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "0.9092974280938506"
+      ]
+     },
+     "execution_count": 94,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "cordic_sin(2.0)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 95,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "-0.4161468365471424"
+      ]
+     },
+     "execution_count": 95,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "cos(2)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 96,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "(-0.41614683654714246, -0.0)"
+      ]
+     },
+     "execution_count": 96,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "taylor_cos(2.0, 0.0)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 97,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "-0.4161468353122473"
+      ]
+     },
+     "execution_count": 97,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "cordic_cos(2.0)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 98,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "-5991.431207677988 - 9240.89014825243im"
+      ]
+     },
+     "execution_count": 98,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "# porównianie na sin(10 + 10i)\n",
+    "sin(10 + 10im)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 99,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "(-5991.431207678, -9240.890148252452)"
+      ]
+     },
+     "execution_count": 99,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "taylor_sin(10, 10)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 73,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "rel_error (generic function with 1 method)"
+      ]
+     },
+     "execution_count": 73,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "# w ten sposób liczony jest błąd względny zarówno dla liczb rzeczywistych jak i zespolonych\n",
+    "function rel_error(x, y)\n",
+    "    if x == 0\n",
+    "        return 0\n",
+    "    end\n",
+    "    return abs((x-y)/x)\n",
+    "end"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 100,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "taylor_error_of_iterations (generic function with 1 method)"
+      ]
+     },
+     "execution_count": 100,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "function taylor_error_of_iterations(x)\n",
+    "    set_taylor_iterations(x)\n",
+    "    return rel_error(sin(100+100im), taylor_sin(100, 100)[1] + taylor_sin(100, 100)[2]*im)\n",
+    "end"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 101,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "cordic_error_of_iterations (generic function with 1 method)"
+      ]
+     },
+     "execution_count": 101,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "function cordic_error_of_iterations(x)\n",
+    "    set_cordic_iterations(x)\n",
+    "    return rel_error(sin(100), cordic_sin(100.0))\n",
+    "end"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 111,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "taylor_error_of_iterations2 (generic function with 1 method)"
+      ]
+     },
+     "execution_count": 111,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "function taylor_error_of_iterations2(x)\n",
+    "    set_taylor_iterations(x)\n",
+    "    return rel_error(sin(100), taylor_sin(100.0, 0.0)[1])\n",
+    "end"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 112,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "1:20"
+      ]
+     },
+     "execution_count": 112,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "X = 1:20"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# Przykładowe błędy w zależności od liczby iteracji:"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 113,
+   "metadata": {},
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+       "<!DOCTYPE html>\n",
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+       "    <head>\n",
+       "        <title>Plots.jl</title>\n",
+       "        <meta http-equiv=\"content-type\" content=\"text/html; charset=UTF-8\">\n",
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+       "        \"ticktext\": [\n",
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+       "    \"plot_bgcolor\": \"rgba(255, 255, 255, 1.000)\",\n",
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+       "        \"title\": \"relative error\",\n",
+       "        \"mirror\": false,\n",
+       "        \"tickangle\": 0,\n",
+       "        \"showline\": true,\n",
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+       "        \"tickcolor\": \"rgb(0, 0, 0)\",\n",
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+       "            \"0.2\",\n",
+       "            \"0.3\",\n",
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+       "        \"zeroline\": false,\n",
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+       "    \"legend\": {\n",
+       "        \"yanchor\": \"auto\",\n",
+       "        \"xanchor\": \"auto\",\n",
+       "        \"bordercolor\": \"rgba(0, 0, 0, 1.000)\",\n",
+       "        \"bgcolor\": \"rgba(255, 255, 255, 1.000)\",\n",
+       "        \"font\": {\n",
+       "            \"color\": \"rgba(0, 0, 0, 1.000)\",\n",
+       "            \"family\": \"sans-serif\",\n",
+       "            \"size\": 11\n",
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+       "        \"tracegroupgap\": 0,\n",
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+       "        \"borderwidth\": 1,\n",
+       "        \"traceorder\": \"normal\",\n",
+       "        \"x\": 1.0\n",
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+       "    \"width\": 600\n",
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+       ");\n",
+       "    </script>\n",
+       "\n",
+       "    </body>\n",
+       "</html>\n"
+      ]
+     },
+     "execution_count": 109,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "plot(cordic_error_of_iterations, X, title=\"Cordic relative error calculating sin(100)\", xguide = \"iterations\", yguide = \"relative error\")"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 106,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# Poniżej znajdują się funkcje testujące, na podstawie których powstała tabelka z błędami w sprawozdaniu"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 18,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "MersenneTwister(UInt32[0x00003039], Random.DSFMT.DSFMT_state(Int32[-870096391, 1072918504, -1812426662, 1073255081, -733866021, 1073404543, 807620846, 1073368448, 1919433844, 1072852359  …  -362113007, 1073100625, -166402106, 1073460158, -1907020342, 721295190, -750225566, -1300227565, 382, 0]), [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0  …  0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], UInt128[0x00000000000000000000000000000000, 0x00000000000000000000000000000000, 0x00000000000000000000000000000000, 0x00000000000000000000000000000000, 0x00000000000000000000000000000000, 0x00000000000000000000000000000000, 0x00000000000000000000000000000000, 0x00000000000000000000000000000000, 0x00000000000000000000000000000000, 0x00000000000000000000000000000000  …  0x00000000000000000000000000000000, 0x00000000000000000000000000000000, 0x00000000000000000000000000000000, 0x00000000000000000000000000000000, 0x00000000000000000000000000000000, 0x00000000000000000000000000000000, 0x00000000000000000000000000000000, 0x00000000000000000000000000000000, 0x00000000000000000000000000000000, 0x00000000000000000000000000000000], 1002, 0)"
+      ]
+     },
+     "execution_count": 18,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "TESTS = 100000000\n",
+    "\n",
+    "Random.seed!(12345)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 19,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "taylor_test_error_real (generic function with 3 methods)"
+      ]
+     },
+     "execution_count": 19,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "function taylor_test_error_real(l::Float64=floatmin(), r::Float64=floatmax())\n",
+    "    res = BigFloat(0)\n",
+    "    abs_res = BigFloat(0)\n",
+    "    maksi_rel = BigFloat(0)\n",
+    "    maksi_abs = BigFloat(0)\n",
+    "    for i = 1:TESTS\n",
+    "        if i % 100000 == 0\n",
+    "            println(i)\n",
+    "        end\n",
+    "        x = rand(Uniform(l, r))\n",
+    "        lib_sin = sin(x)\n",
+    "        if lib_sin == 0\n",
+    "            continue\n",
+    "        end\n",
+    "        my_sin = taylor_sin(x, 0)\n",
+    "        error = rel_error(lib_sin, my_sin[1])\n",
+    "        abs_error = abs(my_sin[1] - lib_sin)\n",
+    "        res += error\n",
+    "        abs_res += abs_error\n",
+    "        maksi_rel = max(maksi_rel, error)\n",
+    "        maksi_abs = max(maksi_abs, abs_error)\n",
+    "    end\n",
+    "    return (res/TESTS, maksi_rel, abs_res/TESTS, maksi_abs)\n",
+    "end\n",
+    "\n",
+    "# (floatmin(), floatmax()):\n",
+    "# (1.887844299668514797145972383393008309519973773948872165524132116232181033410611e-15, \n",
+    "# 3.16719187748669057932019506480803006098767582443542778491973876953125e-08,\n",
+    "# 1.1794041986528804301572959036155385792454808324691839516162872314453125e-16,\n",
+    "# 8.8817841970012523233890533447265625e-16)\n",
+    "\n",
+    "# (-pi/2, pi/2):\n",
+    "# (1.471587646915289673578957365178574707202863924359834292944840261618821841693717e-15, \n",
+    "# 1.1848604479598457485905096801294400510329296594136394560337066650390625e-08, \n",
+    "# 9.765754183892570637182101557852154094518937199609354138374328613281249999999994e-17, \n",
+    "# 5.5511151231257827021181583404541015625e-16)\n",
+    "\n",
+    "# (0, 1):\n",
+    "# (8.693695902799099432701533207691913249153884601349429181102457242502623557811573e-17,\n",
+    "# 6.661260307992334044328275268948192015174572739102942797728701407322660088539124e-16,\n",
+    "# 4.293257315426284893844499634951716871000826358795166015624999999999999999999994e-17,\n",
+    "# 4.44089209850062616169452667236328125e-16)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 20,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "taylor_test_error_complex (generic function with 3 methods)"
+      ]
+     },
+     "execution_count": 20,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "function taylor_test_error_complex(l::Float64=-100.0, r::Float64=100.0)\n",
+    "    res = BigFloat(0)\n",
+    "    abs_res = BigFloat(0)\n",
+    "    maksi_rel = BigFloat(0)\n",
+    "    maksi_abs = BigFloat(0)\n",
+    "    for i = 1:TESTS\n",
+    "        if i % 100000 == 0\n",
+    "            println(i)\n",
+    "        end\n",
+    "        x = rand(Uniform(l, r))\n",
+    "        y = rand(Uniform(max(l, -Float64(√(BigFloat(r)*r - BigFloat(x)*x))), \n",
+    "                        Float64(√(BigFloat(r)*r - BigFloat(x)*x))))\n",
+    "        lib_sin = sin(x + y*im)\n",
+    "        my_sin = taylor_sin(x, y)\n",
+    "        error = rel_error(lib_sin, my_sin[1] + my_sin[2]*im)\n",
+    "        abs_error = abs(lib_sin - (my_sin[1] + my_sin[2]*im))\n",
+    "        res += error\n",
+    "        abs_res += abs_error\n",
+    "        maksi_rel = max(maksi_rel, error)\n",
+    "        maksi_abs = max(maksi_abs, abs_error)\n",
+    "    end\n",
+    "    return (res/TESTS, maksi_rel, abs_res/TESTS, maksi_abs)\n",
+    "end\n",
+    "\n",
+    "# (-100, 100):\n",
+    "# (4.932205036590292360305897845543684560590114030155004375572792447173773555907229e-15, \n",
+    "# 1.3111008357751143737471652583705182364137709072338111582212150096893310546875e-13, \n",
+    "# 1.688623533003329462861070079404255492323042928202526655997186385923664654746476e+26, \n",
+    "# 5.89784569029861503624382775296e+29)\n",
+    "\n",
+    "# (-2pi, 2pi):\n",
+    "# (4.338436856498561167962902801400526155223569336855327458414068651872587652334067e-16, \n",
+    "# 1.48720543982594402760972427363260419015678071019692652043886482715606689453125e-11, \n",
+    "# 1.364745868545483273874507699553481910023596725366415789061836204439613629002538e-14, \n",
+    "# 8.7095846425677781478128738959826782468909289747216462274082005023956298828125e-13)\n",
+    "\n",
+    "# (0, 1):\n",
+    "# (1.596935223079780368874812440778376297707878344605454825588075017177200118204992e-16, \n",
+    "# 1.098997011961567777204023105931451003520679665648174250236479565501213073730469e-15, \n",
+    "# 1.124298405324025732059699593805301650508046127888472394113736655893442950571177e-16, \n",
+    "# 1.110569915127177230816030746289393434073728902933275719533412484452128410339355e-15)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 21,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "cordic_test_error (generic function with 3 methods)"
+      ]
+     },
+     "execution_count": 21,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "function cordic_test_error(l::Float64=floatmin(), r::Float64=floatmax())\n",
+    "    res = BigFloat(0)\n",
+    "    abs_res = BigFloat(0)\n",
+    "    maksi_rel = BigFloat(0)\n",
+    "    maksi_abs = BigFloat(0)\n",
+    "    for i = 1:TESTS\n",
+    "        if i % 100000 == 0\n",
+    "            println(i)\n",
+    "        end\n",
+    "        x = rand(Uniform(l, r))\n",
+    "        lib_sin = sin(x)\n",
+    "        my_sin = cordic_sin(x)\n",
+    "        error = rel_error(lib_sin, my_sin)\n",
+    "        abs_error = abs(lib_sin - my_sin)\n",
+    "        res += error\n",
+    "        abs_res += abs_error\n",
+    "        if error > maksi_rel\n",
+    "            worst_rel = x\n",
+    "        end\n",
+    "        maksi_rel = max(maksi_rel, error)\n",
+    "        maksi_abs = max(maksi_abs, abs_error)\n",
+    "    end\n",
+    "    return (res/TESTS, maksi_rel, abs_res/TESTS, maksi_abs)\n",
+    "end\n",
+    "\n",
+    "# (floatmin(), floatmax()):\n",
+    "# (3.099880824631376815575307358441341907045753361742215192539218280437518515668677e-08, \n",
+    "# 0.457561153670805575988111968399607576429843902587890625, \n",
+    "# 2.459716652636021482355597144179802356154379561203882076370064169168472290039072e-09, \n",
+    "# 0.0006041780891818948617810747236944735050201416015625)\n",
+    "\n",
+    "# (-2pi, 2pi):\n",
+    "# (2.769658715752475495709394998775060901506630522496771093654899307916206208091117e-08, \n",
+    "# 0.11834204003306579566778822254491387866437435150146484375, \n",
+    "# 2.532059440779907667675144447194875727078638982803227008844260126352310180664052e-09,\n",
+    "# 0.00552917548107156875403234153054654598236083984375)\n",
+    "\n",
+    "# (0, 1):\n",
+    "# (4.176404604808155838824592152607760760141260709650975486490997166423577713345588e-08, \n",
+    "# 0.091828765031669201679420666550868190824985504150390625, \n",
+    "# 2.613683444981852927700279986835644064485650872597943816799670457839965820312493e-09, \n",
+    "# 0.00052619288922584050993691562325693666934967041015625)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 22,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "taylor_without_reduction_test_error (generic function with 3 methods)"
+      ]
+     },
+     "execution_count": 22,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "function taylor_without_reduction_test_error(l::Float64=-100.0, r::Float64=100.0)\n",
+    "    res = BigFloat(0)\n",
+    "    abs_res = BigFloat(0)\n",
+    "    maksi_rel = BigFloat(0)\n",
+    "    maksi_abs = BigFloat(0)\n",
+    "    for i = 1:TESTS\n",
+    "        if i % 1000000 == 0\n",
+    "            println(i)\n",
+    "        end\n",
+    "        x = rand(Uniform(l, r))\n",
+    "        y = rand(Uniform(max(l, -Float64(√(BigFloat(r)*r - BigFloat(x)*x))), \n",
+    "                        Float64(√(BigFloat(r)*r - BigFloat(x)*x))))\n",
+    "        lib_sin = sin(x + y*im)\n",
+    "        my_sin = taylor_sin_no_reduction(x, y)\n",
+    "        error = rel_error(lib_sin, my_sin[1] + my_sin[2]*im)\n",
+    "        abs_error = abs(lib_sin - (my_sin[1] + my_sin[2]*im))\n",
+    "        res += error\n",
+    "        abs_res += abs_error\n",
+    "        maksi_rel = max(maksi_rel, error)\n",
+    "        maksi_abs = max(maksi_abs, abs_error)\n",
+    "    end\n",
+    "    return (res/TESTS, maksi_rel, abs_res/TESTS, maksi_abs)\n",
+    "end\n",
+    "\n",
+    "# (-100, 100)\n",
+    "# (4.774091809397734982069398193189465079787514988283523440828527859306283137571149e+23, \n",
+    "# 4.48814142545670189837451264e+26, \n",
+    "# 7.758560481134976967771949796127369173267383351574525337904198599731007318070319e+40, \n",
+    "# 2.20832987186165589366506156220211970162294784e+44)\n",
+    "\n",
+    "# (-2pi, 2pi)\n",
+    "# (0.6332711088634405192103194531076134843075526902544601426097735760298574150340518, \n",
+    "# 1.0, \n",
+    "# 23.44057586605533515691829807979128873527513778367553433852055381911453864572971, \n",
+    "# 267.74654227273646256435313262045383453369140625)\n",
+    "\n",
+    "# (0, 1)\n",
+    "# (1.589482169544726703219739509256918022523030217883325972454504856003547167066932e-16, \n",
+    "# 1.291897416767691567199962520855285151964115327068161054313577551511116325855255e-15, \n",
+    "# 1.118367257755837281340217148887719929595000777959128862241583039814976641146415e-16, \n",
+    "# 1.115760330918745818020084658567032229219617364690542160587938269600272178649902e-15)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": []
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": "Julia 1.4.1",
+   "language": "julia",
+   "name": "julia-1.4"
+  },
+  "language_info": {
+   "file_extension": ".jl",
+   "mimetype": "application/julia",
+   "name": "julia",
+   "version": "1.4.1"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 4
+}
-- 
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