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degree-13 polynomial approximation of sine
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| { | |
| "cells": [ | |
| { | |
| "metadata": { | |
| "trusted": true | |
| }, | |
| "cell_type": "code", | |
| "source": "using Polynomials\nusing Plots\n\nfunction chebyshev_approx(d, F::Polynomial{CT}) where CT\n P = F\n for k in degree(F):-1:d+1\n degree(P) < k && continue\n a = coeffs(P)[end]\n T = ChebyshevT([zeros(CT, k); a/CT(2)^(k-1)])\n P = P - T\n end\n P\nend\n\nfunction maclaurin_sinpio4(d, CT=BigFloat)\n s = [isodd(k) ? (-1)^((k-1)÷2)*(big(π)/4)^k/factorial(CT(k)) : zero(CT) for k in 0:d]\n Polynomial(s)\nend\n\nfunction maclaurin_sin(d, CT=BigFloat)\n s = [isodd(k) ? (-1)^((k-1)÷2)/factorial(CT(k)) : zero(CT) for k in 0:d]\n Polynomial(s)\nend\n\nfunction ordinaryscale(F::Polynomial{CT}) where CT\n c = coeffs(F) ./ [(CT(π)/4)^k for k in 0:degree(F)]\n Polynomial(c)\nend", | |
| "execution_count": 1, | |
| "outputs": [ | |
| { | |
| "output_type": "execute_result", | |
| "execution_count": 1, | |
| "data": { | |
| "text/plain": "ordinaryscale (generic function with 1 method)" | |
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| "metadata": {} | |
| } | |
| ] | |
| }, | |
| { | |
| "metadata": { | |
| "trusted": true | |
| }, | |
| "cell_type": "code", | |
| "source": "@show precision(BigFloat)\n\nS1 = chebyshev_approx(13, maclaurin_sinpio4(61)) |> ordinaryscale\nS2 = chebyshev_approx(13, maclaurin_sinpio4(63)) |> ordinaryscale\nS3 = chebyshev_approx(13, maclaurin_sinpio4(65)) |> ordinaryscale\n[\n S1 - S2\n S2 - S3\n]", | |
| "execution_count": 2, | |
| "outputs": [ | |
| { | |
| "output_type": "stream", | |
| "text": "precision(BigFloat) = 256\n", | |
| "name": "stdout" | |
| }, | |
| { | |
| "output_type": "execute_result", | |
| "execution_count": 2, | |
| "data": { | |
| "text/plain": "2-element Vector{Polynomial{BigFloat}}:\n Polynomial(-2.010764683385948796148028192762378503362648273919774541869690518399838312334015e-87*x^13)\n Polynomial(0.0)" | |
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| ] | |
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| "metadata": { | |
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| }, | |
| "cell_type": "code", | |
| "source": "# @show old_precition = precision(BigFloat)\n# setprecision(1024)\n\n# S1 = chebyshev_approx(13, maclaurin_sinpio4(169)) |> ordinaryscale\n# S2 = chebyshev_approx(13, maclaurin_sinpio4(171)) |> ordinaryscale\n# S3 = chebyshev_approx(13, maclaurin_sinpio4(173)) |> ordinaryscale\n# [\n# S1 - S2\n# S2 - S3\n# ]", | |
| "execution_count": 3, | |
| "outputs": [] | |
| }, | |
| { | |
| "metadata": { | |
| "trusted": true | |
| }, | |
| "cell_type": "code", | |
| "source": "S = S2\ns = coeffs(S2)\nS", | |
| "execution_count": 4, | |
| "outputs": [ | |
| { | |
| "output_type": "execute_result", | |
| "execution_count": 4, | |
| "data": { | |
| "text/plain": "Polynomial(0.9999999999999999764211612009725588855111463008826924315423464769054326509759891*x - 0.1666666666666652393535348747726998605240410270492158545928182832987126305327973*x^3 + 0.008333333333308337379310370484936396053881316821516016008049291759834554025541592*x^5 - 0.0001984126982196706774095496759095037380395398697503813794740059976612433177454158*x^7 + 2.75573115707744123859880350508182954104581095022903878633952359659471623882716e-06*x^9 - 2.505048281275841971620868327213456634147834603978717581559607950713258016414098e-08*x^11 + 1.588305691336997706018324915122180570003704510781773494445823588813343496455814e-10*x^13)", | |
| "text/html": "0.9999999999999999764211612009725588855111463008826924315423464769054326509759891∙x - 0.1666666666666652393535348747726998605240410270492158545928182832987126305327973∙x<sup>3</sup> + 0.008333333333308337379310370484936396053881316821516016008049291759834554025541592∙x<sup>5</sup> - 0.0001984126982196706774095496759095037380395398697503813794740059976612433177454158∙x<sup>7</sup> + 2.75573115707744123859880350508182954104581095022903878633952359659471623882716e-06∙x<sup>9</sup> - 2.505048281275841971620868327213456634147834603978717581559607950713258016414098e-08∙x<sup>11</sup> + 1.588305691336997706018324915122180570003704510781773494445823588813343496455814e-10∙x<sup>13</sup>", | |
| "text/latex": "$0.9999999999999999764211612009725588855111463008826924315423464769054326509759891\\cdot x - 0.1666666666666652393535348747726998605240410270492158545928182832987126305327973\\cdot x^{3} + 0.008333333333308337379310370484936396053881316821516016008049291759834554025541592\\cdot x^{5} - 0.0001984126982196706774095496759095037380395398697503813794740059976612433177454158\\cdot x^{7} + 2.75573115707744123859880350508182954104581095022903878633952359659471623882716e-06\\cdot x^{9} - 2.505048281275841971620868327213456634147834603978717581559607950713258016414098e-08\\cdot x^{11} + 1.588305691336997706018324915122180570003704510781773494445823588813343496455814e-10\\cdot x^{13}$" | |
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| "metadata": {} | |
| } | |
| ] | |
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| { | |
| "metadata": {}, | |
| "cell_type": "markdown", | |
| "source": "https://github.com/JuliaLang/julia/blob/master/base/special/trig.jl#L54-L62\n\n```julia\n# Coefficients in 13th order polynomial approximation on [0; π/4]\n# sin(x) ≈ x + S1*x³ + S2*x⁵ + S3*x⁷ + S4*x⁹ + S5*x¹¹ + S6*x¹³\n# D for double, S for sin, number is the order of x-1\nconst DS1 = -1.66666666666666324348e-01\nconst DS2 = 8.33333333332248946124e-03\nconst DS3 = -1.98412698298579493134e-04\nconst DS4 = 2.75573137070700676789e-06\nconst DS5 = -2.50507602534068634195e-08\nconst DS6 = 1.58969099521155010221e-10\n```" | |
| }, | |
| { | |
| "metadata": { | |
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| "cell_type": "code", | |
| "source": "s64 = Float64.(s)\ndisplay(s64[begin+1:2:end])\nS64 = Polynomial(s64)", | |
| "execution_count": 5, | |
| "outputs": [ | |
| { | |
| "output_type": "display_data", | |
| "data": { | |
| "text/plain": "7-element Vector{Float64}:\n 1.0\n -0.16666666666666524\n 0.008333333333308338\n -0.0001984126982196707\n 2.7557311570774412e-6\n -2.5050482812758418e-8\n 1.5883056913369978e-10" | |
| }, | |
| "metadata": {} | |
| }, | |
| { | |
| "output_type": "execute_result", | |
| "execution_count": 5, | |
| "data": { | |
| "text/plain": "Polynomial(1.0*x - 0.16666666666666524*x^3 + 0.008333333333308338*x^5 - 0.0001984126982196707*x^7 + 2.7557311570774412e-6*x^9 - 2.5050482812758418e-8*x^11 + 1.5883056913369978e-10*x^13)", | |
| "text/html": "1.0∙x - 0.16666666666666524∙x<sup>3</sup> + 0.008333333333308338∙x<sup>5</sup> - 0.0001984126982196707∙x<sup>7</sup> + 2.7557311570774412e-6∙x<sup>9</sup> - 2.5050482812758418e-8∙x<sup>11</sup> + 1.5883056913369978e-10∙x<sup>13</sup>", | |
| "text/latex": "$1.0\\cdot x - 0.16666666666666524\\cdot x^{3} + 0.008333333333308338\\cdot x^{5} - 0.0001984126982196707\\cdot x^{7} + 2.7557311570774412e-6\\cdot x^{9} - 2.5050482812758418e-8\\cdot x^{11} + 1.5883056913369978e-10\\cdot x^{13}$" | |
| }, | |
| "metadata": {} | |
| } | |
| ] | |
| }, | |
| { | |
| "metadata": { | |
| "trusted": true | |
| }, | |
| "cell_type": "code", | |
| "source": "s64m = Float64.(coeffs(maclaurin_sin(13)))\ndisplay(s64m[begin+1:2:end])\nS64m = Polynomial(s64m)", | |
| "execution_count": 6, | |
| "outputs": [ | |
| { | |
| "output_type": "display_data", | |
| "data": { | |
| "text/plain": "7-element Vector{Float64}:\n 1.0\n -0.16666666666666666\n 0.008333333333333333\n -0.0001984126984126984\n 2.7557319223985893e-6\n -2.505210838544172e-8\n 1.6059043836821613e-10" | |
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| "metadata": {} | |
| }, | |
| { | |
| "output_type": "execute_result", | |
| "execution_count": 6, | |
| "data": { | |
| "text/plain": "Polynomial(1.0*x - 0.16666666666666666*x^3 + 0.008333333333333333*x^5 - 0.0001984126984126984*x^7 + 2.7557319223985893e-6*x^9 - 2.505210838544172e-8*x^11 + 1.6059043836821613e-10*x^13)", | |
| "text/html": "1.0∙x - 0.16666666666666666∙x<sup>3</sup> + 0.008333333333333333∙x<sup>5</sup> - 0.0001984126984126984∙x<sup>7</sup> + 2.7557319223985893e-6∙x<sup>9</sup> - 2.505210838544172e-8∙x<sup>11</sup> + 1.6059043836821613e-10∙x<sup>13</sup>", | |
| "text/latex": "$1.0\\cdot x - 0.16666666666666666\\cdot x^{3} + 0.008333333333333333\\cdot x^{5} - 0.0001984126984126984\\cdot x^{7} + 2.7557319223985893e-6\\cdot x^{9} - 2.505210838544172e-8\\cdot x^{11} + 1.6059043836821613e-10\\cdot x^{13}$" | |
| }, | |
| "metadata": {} | |
| } | |
| ] | |
| }, | |
| { | |
| "metadata": { | |
| "trusted": true | |
| }, | |
| "cell_type": "code", | |
| "source": "π/4", | |
| "execution_count": 7, | |
| "outputs": [ | |
| { | |
| "output_type": "execute_result", | |
| "execution_count": 7, | |
| "data": { | |
| "text/plain": "0.7853981633974483" | |
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| "metadata": {} | |
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| }, | |
| { | |
| "metadata": { | |
| "trusted": true | |
| }, | |
| "cell_type": "code", | |
| "source": "S64(0.78)", | |
| "execution_count": 8, | |
| "outputs": [ | |
| { | |
| "output_type": "execute_result", | |
| "execution_count": 8, | |
| "data": { | |
| "text/plain": "0.7032794192004101" | |
| }, | |
| "metadata": {} | |
| } | |
| ] | |
| }, | |
| { | |
| "metadata": { | |
| "trusted": true | |
| }, | |
| "cell_type": "code", | |
| "source": "S64m(0.78)", | |
| "execution_count": 9, | |
| "outputs": [ | |
| { | |
| "output_type": "execute_result", | |
| "execution_count": 9, | |
| "data": { | |
| "text/plain": "0.7032794192004286" | |
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| "metadata": {} | |
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| ] | |
| }, | |
| { | |
| "metadata": { | |
| "trusted": true | |
| }, | |
| "cell_type": "code", | |
| "source": "sin(0.78)", | |
| "execution_count": 10, | |
| "outputs": [ | |
| { | |
| "output_type": "execute_result", | |
| "execution_count": 10, | |
| "data": { | |
| "text/plain": "0.7032794192004101" | |
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| "metadata": {} | |
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| ] | |
| }, | |
| { | |
| "metadata": { | |
| "trusted": true | |
| }, | |
| "cell_type": "code", | |
| "source": "x = range(eps(), π/4; length=1001)\n@show extrema(S64.(x) .- sin.(x))\n@show extrema(S64m.(x) .- sin.(x));", | |
| "execution_count": 11, | |
| "outputs": [ | |
| { | |
| "output_type": "stream", | |
| "text": "extrema(S64.(x) .- sin.(x)) = (-1.1102230246251565e-16, 1.1102230246251565e-16)\nextrema(S64m.(x) .- sin.(x)) = (-5.551115123125783e-17, 2.042810365310288e-14)\n", | |
| "name": "stdout" | |
| } | |
| ] | |
| }, | |
| { | |
| "metadata": { | |
| "trusted": true | |
| }, | |
| "cell_type": "code", | |
| "source": "x = range(eps(), π/4; length=1001)\ny = 10^16 .* (S64.(x) .- sin.(x))\nplot(x, y; label=\"\", title=\"10¹⁶ (S64(x) - sin(x)) for Float64 x in (0, π/4]\")", | |
| "execution_count": 12, | |
| "outputs": [ | |
| { | |
| "output_type": "execute_result", | |
| "execution_count": 12, | |
| "data": { | |
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612.224,804.96 614.274,644.232 616.324,804.96 618.374,804.96 620.424,804.96 622.475,644.232 624.525,965.688 626.575,804.96 628.625,804.96 \n 630.675,804.96 632.725,804.96 634.775,965.688 636.825,804.96 638.875,804.96 640.925,804.96 642.975,804.96 645.026,804.96 647.076,965.688 649.126,804.96 \n 651.176,804.96 653.226,804.96 655.276,804.96 657.326,804.96 659.376,804.96 661.426,804.96 663.476,644.232 665.527,804.96 667.577,644.232 669.627,804.96 \n 671.677,965.688 673.727,804.96 675.777,804.96 677.827,644.232 679.877,804.96 681.927,804.96 683.977,804.96 686.027,965.688 688.078,804.96 690.128,804.96 \n 692.178,804.96 694.228,804.96 696.278,804.96 698.328,804.96 700.378,804.96 702.428,804.96 704.478,804.96 706.528,804.96 708.579,644.232 710.629,804.96 \n 712.679,804.96 714.729,804.96 716.779,804.96 718.829,804.96 720.879,804.96 722.929,804.96 724.979,804.96 727.029,804.96 729.08,804.96 731.13,804.96 \n 733.18,644.232 735.23,804.96 737.28,804.96 739.33,644.232 741.38,804.96 743.43,644.232 745.48,804.96 747.53,804.96 749.58,804.96 751.631,804.96 \n 753.681,804.96 755.731,804.96 757.781,804.96 759.831,804.96 761.881,804.96 763.931,804.96 765.981,804.96 768.031,804.96 770.081,644.232 772.132,804.96 \n 774.182,804.96 776.232,804.96 778.282,804.96 780.332,644.232 782.382,804.96 784.432,804.96 786.482,804.96 788.532,804.96 790.582,804.96 792.633,804.96 \n 794.683,804.96 796.733,804.96 798.783,644.232 800.833,644.232 802.883,804.96 804.933,804.96 806.983,644.232 809.033,804.96 811.083,644.232 813.133,804.96 \n 815.184,644.232 817.234,804.96 819.284,804.96 821.334,804.96 823.384,804.96 825.434,804.96 827.484,804.96 829.534,644.232 831.584,804.96 833.634,804.96 \n 835.685,804.96 837.735,804.96 839.785,804.96 841.835,644.232 843.885,804.96 845.935,804.96 847.985,644.232 850.035,804.96 852.085,804.96 854.135,644.232 \n 856.185,804.96 858.236,644.232 860.286,804.96 862.336,644.232 864.386,804.96 866.436,644.232 868.486,804.96 870.536,804.96 872.586,804.96 874.636,804.96 \n 876.686,804.96 878.737,804.96 880.787,644.232 882.837,804.96 884.887,804.96 886.937,644.232 888.987,804.96 891.037,804.96 893.087,804.96 895.137,804.96 \n 897.187,644.232 899.238,804.96 901.288,804.96 903.338,804.96 905.388,804.96 907.438,804.96 909.488,804.96 911.538,804.96 913.588,483.503 915.638,804.96 \n 917.688,804.96 919.738,804.96 921.789,804.96 923.839,804.96 925.889,804.96 927.939,804.96 929.989,804.96 932.039,804.96 934.089,804.96 936.139,804.96 \n 938.189,804.96 940.239,483.503 942.29,804.96 944.34,804.96 946.39,804.96 948.44,483.503 950.49,804.96 952.54,483.503 954.59,804.96 956.64,804.96 \n 958.69,804.96 960.74,804.96 962.791,483.503 964.841,804.96 966.891,804.96 968.941,804.96 970.991,804.96 973.041,804.96 975.091,804.96 977.141,483.503 \n 979.191,483.503 981.241,804.96 983.291,804.96 985.342,804.96 987.392,804.96 989.442,804.96 991.492,804.96 993.542,804.96 995.592,483.503 997.642,804.96 \n 999.692,804.96 1001.74,483.503 1003.79,804.96 1005.84,804.96 1007.89,804.96 1009.94,804.96 1011.99,804.96 1014.04,804.96 1016.09,804.96 1018.14,804.96 \n 1020.19,483.503 1022.24,804.96 1024.29,804.96 1026.34,804.96 1028.39,804.96 1030.44,804.96 1032.49,804.96 1034.54,804.96 1036.59,804.96 1038.64,483.503 \n 1040.69,804.96 1042.74,804.96 1044.79,483.503 1046.84,483.503 1048.89,804.96 1050.94,804.96 1052.99,483.503 1055.04,804.96 1057.09,804.96 1059.15,483.503 \n 1061.2,804.96 1063.25,804.96 1065.3,804.96 1067.35,804.96 1069.4,804.96 1071.45,804.96 1073.5,483.503 1075.55,483.503 1077.6,804.96 1079.65,804.96 \n 1081.7,804.96 1083.75,804.96 1085.8,804.96 1087.85,804.96 1089.9,804.96 1091.95,804.96 1094,804.96 1096.05,804.96 1098.1,804.96 1100.15,804.96 \n 1102.2,804.96 1104.25,804.96 1106.3,483.503 1108.35,804.96 1110.4,804.96 1112.45,804.96 1114.5,804.96 1116.55,804.96 1118.6,804.96 1120.65,804.96 \n 1122.7,804.96 1124.75,804.96 1126.8,804.96 1128.85,804.96 1130.9,483.503 1132.95,804.96 1135,804.96 1137.05,804.96 1139.1,804.96 1141.15,804.96 \n 1143.2,804.96 1145.25,804.96 1147.3,804.96 1149.35,804.96 1151.4,804.96 1153.45,804.96 1155.5,1126.42 1157.55,804.96 1159.6,483.503 1161.65,804.96 \n 1163.7,804.96 1165.75,804.96 1167.8,804.96 1169.85,804.96 1171.9,483.503 1173.95,483.503 1176,804.96 1178.05,804.96 1180.1,1126.42 1182.15,804.96 \n 1184.2,483.503 1186.25,804.96 1188.3,804.96 1190.35,483.503 1192.4,804.96 1194.45,483.503 1196.5,804.96 1198.55,483.503 1200.6,483.503 1202.65,804.96 \n 1204.7,804.96 1206.75,804.96 1208.8,483.503 1210.85,804.96 1212.9,804.96 1214.95,804.96 1217,483.503 1219.05,804.96 1221.1,1126.42 1223.15,804.96 \n 1225.2,804.96 1227.25,804.96 1229.3,804.96 1231.35,804.96 1233.4,804.96 1235.45,804.96 1237.5,483.503 1239.55,804.96 1241.6,804.96 1243.65,804.96 \n 1245.7,804.96 1247.75,483.503 1249.8,483.503 1251.85,483.503 1253.9,804.96 1255.95,1126.42 1258,804.96 1260.05,483.503 1262.1,483.503 1264.15,804.96 \n 1266.2,483.503 1268.25,804.96 1270.3,804.96 1272.36,804.96 1274.41,483.503 1276.46,804.96 1278.51,804.96 1280.56,804.96 1282.61,804.96 1284.66,804.96 \n 1286.71,804.96 1288.76,804.96 1290.81,804.96 1292.86,804.96 1294.91,804.96 1296.96,804.96 1299.01,804.96 1301.06,804.96 1303.11,804.96 1305.16,804.96 \n 1307.21,804.96 1309.26,1126.42 1311.31,804.96 1313.36,804.96 1315.41,804.96 1317.46,483.503 1319.51,804.96 1321.56,804.96 1323.61,804.96 1325.66,804.96 \n 1327.71,483.503 1329.76,804.96 1331.81,804.96 1333.86,804.96 1335.91,1126.42 1337.96,804.96 1340.01,483.503 1342.06,804.96 1344.11,804.96 1346.16,804.96 \n 1348.21,1126.42 1350.26,804.96 1352.31,804.96 1354.36,804.96 1356.41,804.96 1358.46,483.503 1360.51,483.503 1362.56,804.96 1364.61,804.96 1366.66,483.503 \n 1368.71,804.96 1370.76,804.96 1372.81,804.96 1374.86,804.96 1376.91,804.96 1378.96,1126.42 1381.01,804.96 1383.06,804.96 1385.11,804.96 1387.16,804.96 \n 1389.21,804.96 1391.26,804.96 1393.31,804.96 1395.36,483.503 1397.41,1126.42 1399.46,804.96 1401.51,804.96 1403.56,804.96 1405.61,804.96 1407.66,804.96 \n 1409.71,804.96 1411.76,804.96 1413.81,1126.42 1415.86,483.503 1417.91,1126.42 1419.96,483.503 1422.01,804.96 1424.06,1126.42 1426.11,804.96 1428.16,483.503 \n 1430.21,483.503 1432.26,804.96 1434.31,804.96 1436.36,483.503 1438.41,804.96 1440.46,804.96 1442.51,804.96 1444.56,804.96 1446.61,804.96 1448.66,804.96 \n 1450.71,1126.42 1452.76,804.96 1454.81,804.96 1456.86,483.503 1458.91,804.96 1460.96,483.503 1463.01,483.503 1465.06,804.96 1467.11,804.96 1469.16,1126.42 \n 1471.21,483.503 1473.26,483.503 1475.31,804.96 1477.36,804.96 1479.41,804.96 1481.46,483.503 1483.52,804.96 1485.57,804.96 1487.62,483.503 1489.67,804.96 \n 1491.72,804.96 1493.77,1126.42 1495.82,483.503 1497.87,804.96 1499.92,804.96 1501.97,804.96 1504.02,804.96 1506.07,483.503 1508.12,804.96 1510.17,804.96 \n 1512.22,804.96 1514.27,1126.42 1516.32,483.503 1518.37,483.503 1520.42,483.503 1522.47,804.96 1524.52,804.96 1526.57,804.96 1528.62,483.503 1530.67,804.96 \n 1532.72,483.503 1534.77,804.96 1536.82,483.503 1538.87,804.96 1540.92,804.96 1542.97,804.96 1545.02,804.96 1547.07,804.96 1549.12,804.96 1551.17,483.503 \n 1553.22,804.96 1555.27,804.96 1557.32,804.96 1559.37,804.96 1561.42,483.503 1563.47,804.96 1565.52,804.96 1567.57,804.96 1569.62,804.96 1571.67,483.503 \n 1573.72,804.96 1575.77,804.96 1577.82,483.503 1579.87,804.96 1581.92,804.96 1583.97,804.96 1586.02,483.503 1588.07,804.96 1590.12,804.96 1592.17,483.503 \n 1594.22,804.96 1596.27,483.503 1598.32,804.96 1600.37,1126.42 1602.42,804.96 1604.47,483.503 1606.52,804.96 1608.57,804.96 1610.62,804.96 1612.67,804.96 \n 1614.72,804.96 1616.77,804.96 1618.82,162.047 1620.87,804.96 1622.92,804.96 1624.97,804.96 1627.02,804.96 1629.07,804.96 1631.12,804.96 1633.17,804.96 \n 1635.22,162.047 1637.27,804.96 1639.32,804.96 1641.37,804.96 1643.42,804.96 1645.47,804.96 1647.52,162.047 1649.57,804.96 1651.62,162.047 1653.67,804.96 \n 1655.72,804.96 1657.77,804.96 1659.82,804.96 1661.87,804.96 1663.92,804.96 1665.97,804.96 1668.02,804.96 1670.07,162.047 1672.12,804.96 1674.17,804.96 \n 1676.22,804.96 1678.27,162.047 1680.32,162.047 1682.37,804.96 1684.42,804.96 1686.47,162.047 1688.52,804.96 1690.57,804.96 1692.62,804.96 1694.67,804.96 \n 1696.73,804.96 1698.78,162.047 1700.83,162.047 1702.88,804.96 1704.93,804.96 1706.98,804.96 1709.03,804.96 1711.08,804.96 1713.13,162.047 1715.18,162.047 \n 1717.23,804.96 1719.28,804.96 1721.33,804.96 1723.38,162.047 1725.43,804.96 1727.48,804.96 1729.53,1447.87 1731.58,804.96 1733.63,804.96 1735.68,804.96 \n 1737.73,804.96 1739.78,1447.87 1741.83,804.96 1743.88,804.96 1745.93,804.96 1747.98,804.96 1750.03,804.96 1752.08,804.96 1754.13,804.96 1756.18,162.047 \n 1758.23,804.96 1760.28,804.96 1762.33,804.96 1764.38,162.047 1766.43,804.96 1768.48,162.047 1770.53,804.96 1772.58,804.96 1774.63,804.96 1776.68,804.96 \n 1778.73,804.96 1780.78,804.96 1782.83,162.047 1784.88,804.96 1786.93,804.96 1788.98,804.96 1791.03,804.96 1793.08,804.96 1795.13,804.96 1797.18,162.047 \n 1799.23,162.047 1801.28,804.96 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| }, | |
| "metadata": {} | |
| } | |
| ] | |
| }, | |
| { | |
| "metadata": { | |
| "trusted": true | |
| }, | |
| "cell_type": "code", | |
| "source": "x = range(eps(), π/4; length=1001)\ny = 10^16 .* (S64m.(x) .- sin.(x))\nplot(x, y; label=\"\", title=\"10¹⁶ (S64m(x) - sin(x)) for Float64 x in (0, π/4]\")", | |
| "execution_count": 13, | |
| "outputs": [ | |
| { | |
| "output_type": "execute_result", | |
| "execution_count": 13, | |
| "data": { | |
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338.56,1444.39 340.626,1444.39 342.692,1444.39 344.758,1444.39 346.824,1444.39 \n 348.89,1444.39 350.956,1444.39 353.021,1444.39 355.087,1444.39 357.153,1444.39 359.219,1444.82 361.285,1444.39 363.351,1444.39 365.416,1444.39 367.482,1444.82 \n 369.548,1444.39 371.614,1444.39 373.68,1444.82 375.746,1444.39 377.811,1444.39 379.877,1444.39 381.943,1444.82 384.009,1444.39 386.075,1444.39 388.141,1444.39 \n 390.207,1443.52 392.272,1443.52 394.338,1444.39 396.404,1444.39 398.47,1443.52 400.536,1444.39 402.602,1444.39 404.667,1444.39 406.733,1444.39 408.799,1444.39 \n 410.865,1443.52 412.931,1444.39 414.997,1444.39 417.062,1443.52 419.128,1444.39 421.194,1444.39 423.26,1444.39 425.326,1444.39 427.392,1444.39 429.457,1443.52 \n 431.523,1444.39 433.589,1444.39 435.655,1444.39 437.721,1445.26 439.787,1444.39 441.853,1444.39 443.918,1443.52 445.984,1444.39 448.05,1444.39 450.116,1444.39 \n 452.182,1444.39 454.248,1444.39 456.313,1444.39 458.379,1444.39 460.445,1444.39 462.511,1444.39 464.577,1444.39 466.643,1444.39 468.708,1443.52 470.774,1444.39 \n 472.84,1444.39 474.906,1444.39 476.972,1444.39 479.038,1444.39 481.104,1444.39 483.169,1444.39 485.235,1444.39 487.301,1444.39 489.367,1445.26 491.433,1444.39 \n 493.499,1443.52 495.564,1444.39 497.63,1444.39 499.696,1444.39 501.762,1444.39 503.828,1444.39 505.894,1444.39 507.959,1444.39 510.025,1443.52 512.091,1444.39 \n 514.157,1445.26 516.223,1444.39 518.289,1444.39 520.355,1444.39 522.42,1444.39 524.486,1444.39 526.552,1443.52 528.618,1444.39 530.684,1444.39 532.75,1443.52 \n 534.815,1445.26 536.881,1444.39 538.947,1444.39 541.013,1445.26 543.079,1444.39 545.145,1445.26 547.21,1444.39 549.276,1444.39 551.342,1444.39 553.408,1445.26 \n 555.474,1444.39 557.54,1444.39 559.606,1444.39 561.671,1444.39 563.737,1444.39 565.803,1444.39 567.869,1446.13 569.935,1444.39 572.001,1444.39 574.066,1442.65 \n 576.132,1444.39 578.198,1444.39 580.264,1444.39 582.33,1444.39 584.396,1444.39 586.461,1444.39 588.527,1444.39 590.593,1446.13 592.659,1444.39 594.725,1444.39 \n 596.791,1444.39 598.856,1444.39 600.922,1444.39 602.988,1444.39 605.054,1444.39 607.12,1444.39 609.186,1444.39 611.252,1446.13 613.317,1444.39 615.383,1444.39 \n 617.449,1444.39 619.515,1444.39 621.581,1446.13 623.647,1444.39 625.712,1446.13 627.778,1444.39 629.844,1444.39 631.91,1444.39 633.976,1446.13 636.042,1444.39 \n 638.107,1444.39 640.173,1444.39 642.239,1444.39 644.305,1444.39 646.371,1444.39 648.437,1444.39 650.503,1442.65 652.568,1444.39 654.634,1444.39 656.7,1444.39 \n 658.766,1446.13 660.832,1444.39 662.898,1444.39 664.963,1444.39 667.029,1444.39 669.095,1444.39 671.161,1444.39 673.227,1446.13 675.293,1444.39 677.358,1446.13 \n 679.424,1444.39 681.49,1444.39 683.556,1444.39 685.622,1444.39 687.688,1444.39 689.754,1446.13 691.819,1444.39 693.885,1444.39 695.951,1444.39 698.017,1444.39 \n 700.083,1444.39 702.149,1444.39 704.214,1444.39 706.28,1444.39 708.346,1444.39 710.412,1444.39 712.478,1444.39 714.544,1444.39 716.609,1444.39 718.675,1446.13 \n 720.741,1442.65 722.807,1444.39 724.873,1444.39 726.939,1442.65 729.005,1444.39 731.07,1442.65 733.136,1444.39 735.202,1444.39 737.268,1444.39 739.334,1444.39 \n 741.4,1444.39 743.465,1446.13 745.531,1446.13 747.597,1446.13 749.663,1444.39 751.729,1444.39 753.795,1444.39 755.86,1444.39 757.926,1444.39 759.992,1446.13 \n 762.058,1444.39 764.124,1446.13 766.19,1444.39 768.255,1442.65 770.321,1444.39 772.387,1444.39 774.453,1444.39 776.519,1444.39 778.585,1444.39 780.651,1444.39 \n 782.716,1444.39 784.782,1446.13 786.848,1444.39 788.914,1444.39 790.98,1446.13 793.046,1446.13 795.111,1442.65 797.177,1444.39 799.243,1442.65 801.309,1444.39 \n 803.375,1444.39 805.441,1444.39 807.506,1444.39 809.572,1444.39 811.638,1444.39 813.704,1444.39 815.77,1444.39 817.836,1442.65 819.902,1444.39 821.967,1444.39 \n 824.033,1444.39 826.099,1444.39 828.165,1444.39 830.231,1444.39 832.297,1444.39 834.362,1444.39 836.428,1442.65 838.494,1446.13 840.56,1444.39 842.626,1444.39 \n 844.692,1444.39 846.757,1442.65 848.823,1444.39 850.889,1442.65 852.955,1446.13 855.021,1444.39 857.087,1444.39 859.153,1444.39 861.218,1444.39 863.284,1444.39 \n 865.35,1444.39 867.416,1444.39 869.482,1444.39 871.548,1444.39 873.613,1444.39 875.679,1442.65 877.745,1444.39 879.811,1444.39 881.877,1444.39 883.943,1444.39 \n 886.008,1444.39 888.074,1446.13 890.14,1444.39 892.206,1444.39 894.272,1444.39 896.338,1444.39 898.404,1444.39 900.469,1444.39 902.535,1444.39 904.601,1444.39 \n 906.667,1444.39 908.733,1444.39 910.799,1444.39 912.864,1444.39 914.93,1444.39 916.996,1444.39 919.062,1444.39 921.128,1444.39 923.194,1444.39 925.259,1444.39 \n 927.325,1444.39 929.391,1440.9 931.457,1444.39 933.523,1444.39 935.589,1444.39 937.654,1440.9 939.72,1444.39 941.786,1444.39 943.852,1447.87 945.918,1444.39 \n 947.984,1447.87 950.05,1444.39 952.115,1444.39 954.181,1444.39 956.247,1444.39 958.313,1444.39 960.379,1444.39 962.445,1444.39 964.51,1444.39 966.576,1444.39 \n 968.642,1444.39 970.708,1447.87 972.774,1444.39 974.84,1444.39 976.905,1444.39 978.971,1444.39 981.037,1444.39 983.103,1444.39 985.169,1440.9 987.235,1444.39 \n 989.301,1444.39 991.366,1444.39 993.432,1444.39 995.498,1444.39 997.564,1444.39 999.63,1444.39 1001.7,1444.39 1003.76,1444.39 1005.83,1444.39 1007.89,1444.39 \n 1009.96,1440.9 1012.02,1444.39 1014.09,1444.39 1016.16,1444.39 1018.22,1444.39 1020.29,1444.39 1022.35,1444.39 1024.42,1444.39 1026.49,1444.39 1028.55,1444.39 \n 1030.62,1444.39 1032.68,1444.39 1034.75,1444.39 1036.81,1440.9 1038.88,1444.39 1040.95,1444.39 1043.01,1440.9 1045.08,1444.39 1047.14,1444.39 1049.21,1440.9 \n 1051.28,1444.39 1053.34,1444.39 1055.41,1444.39 1057.47,1444.39 1059.54,1444.39 1061.6,1444.39 1063.67,1440.9 1065.74,1440.9 1067.8,1444.39 1069.87,1444.39 \n 1071.93,1444.39 1074,1444.39 1076.07,1444.39 1078.13,1444.39 1080.2,1447.87 1082.26,1444.39 1084.33,1444.39 1086.4,1444.39 1088.46,1444.39 1090.53,1444.39 \n 1092.59,1444.39 1094.66,1444.39 1096.72,1444.39 1098.79,1447.87 1100.86,1444.39 1102.92,1444.39 1104.99,1444.39 1107.05,1444.39 1109.12,1444.39 1111.19,1444.39 \n 1113.25,1444.39 1115.32,1444.39 1117.38,1444.39 1119.45,1444.39 1121.51,1440.9 1123.58,1444.39 1125.65,1444.39 1127.71,1444.39 1129.78,1444.39 1131.84,1444.39 \n 1133.91,1444.39 1135.98,1447.87 1138.04,1444.39 1140.11,1444.39 1142.17,1444.39 1144.24,1444.39 1146.3,1447.87 1148.37,1444.39 1150.44,1440.9 1152.5,1444.39 \n 1154.57,1444.39 1156.63,1447.87 1158.7,1444.39 1160.77,1444.39 1162.83,1444.39 1164.9,1440.9 1166.96,1444.39 1169.03,1444.39 1171.09,1447.87 1173.16,1444.39 \n 1175.23,1440.9 1177.29,1444.39 1179.36,1444.39 1181.42,1440.9 1183.49,1444.39 1185.56,1444.39 1187.62,1444.39 1189.69,1440.9 1191.75,1444.39 1193.82,1444.39 \n 1195.88,1444.39 1197.95,1444.39 1200.02,1444.39 1202.08,1447.87 1204.15,1444.39 1206.21,1444.39 1208.28,1440.9 1210.35,1444.39 1212.41,1447.87 1214.48,1444.39 \n 1216.54,1444.39 1218.61,1444.39 1220.67,1444.39 1222.74,1444.39 1224.81,1444.39 1226.87,1444.39 1228.94,1444.39 1231,1444.39 1233.07,1444.39 1235.14,1444.39 \n 1237.2,1444.39 1239.27,1440.9 1241.33,1444.39 1243.4,1440.9 1245.46,1444.39 1247.53,1447.87 1249.6,1444.39 1251.66,1440.9 1253.73,1440.9 1255.79,1444.39 \n 1257.86,1440.9 1259.93,1444.39 1261.99,1444.39 1264.06,1444.39 1266.12,1440.9 1268.19,1444.39 1270.25,1444.39 1272.32,1447.87 1274.39,1444.39 1276.45,1444.39 \n 1278.52,1444.39 1280.58,1444.39 1282.65,1444.39 1284.72,1444.39 1286.78,1444.39 1288.85,1447.87 1290.91,1444.39 1292.98,1444.39 1295.05,1444.39 1297.11,1444.39 \n 1299.18,1444.39 1301.24,1447.87 1303.31,1444.39 1305.37,1444.39 1307.44,1444.39 1309.51,1440.9 1311.57,1444.39 1313.64,1444.39 1315.7,1444.39 1317.77,1444.39 \n 1319.84,1440.9 1321.9,1444.39 1323.97,1444.39 1326.03,1444.39 1328.1,1447.87 1330.16,1444.39 1332.23,1444.39 1334.3,1444.39 1336.36,1444.39 1338.43,1444.39 \n 1340.49,1447.87 1342.56,1444.39 1344.63,1444.39 1346.69,1444.39 1348.76,1444.39 1350.82,1444.39 1352.89,1440.9 1354.95,1444.39 1357.02,1444.39 1359.09,1440.9 \n 1361.15,1444.39 1363.22,1444.39 1365.28,1444.39 1367.35,1444.39 1369.42,1444.39 1371.48,1447.87 1373.55,1444.39 1375.61,1444.39 1377.68,1444.39 1379.74,1444.39 \n 1381.81,1444.39 1383.88,1444.39 1385.94,1444.39 1388.01,1444.39 1390.07,1447.87 1392.14,1444.39 1394.21,1444.39 1396.27,1444.39 1398.34,1444.39 1400.4,1444.39 \n 1402.47,1444.39 1404.53,1444.39 1406.6,1447.87 1408.67,1440.9 1410.73,1447.87 1412.8,1440.9 1414.86,1444.39 1416.93,1447.87 1419,1444.39 1421.06,1444.39 \n 1423.13,1440.9 1425.19,1444.39 1427.26,1444.39 1429.32,1444.39 1431.39,1444.39 1433.46,1444.39 1435.52,1444.39 1437.59,1444.39 1439.65,1444.39 1441.72,1444.39 \n 1443.79,1447.87 1445.85,1444.39 1447.92,1444.39 1449.98,1440.9 1452.05,1444.39 1454.11,1440.9 1456.18,1440.9 1458.25,1444.39 1460.31,1444.39 1462.38,1447.87 \n 1464.44,1440.9 1466.51,1440.9 1468.58,1444.39 1470.64,1444.39 1472.71,1444.39 1474.77,1440.9 1476.84,1444.39 1478.9,1444.39 1480.97,1440.9 1483.04,1444.39 \n 1485.1,1444.39 1487.17,1447.87 1489.23,1440.9 1491.3,1444.39 1493.37,1444.39 1495.43,1444.39 1497.5,1444.39 1499.56,1440.9 1501.63,1444.39 1503.7,1444.39 \n 1505.76,1444.39 1507.83,1444.39 1509.89,1440.9 1511.96,1440.9 1514.02,1440.9 1516.09,1444.39 1518.16,1440.9 1520.22,1444.39 1522.29,1440.9 1524.35,1440.9 \n 1526.42,1440.9 1528.49,1444.39 1530.55,1440.9 1532.62,1440.9 1534.68,1444.39 1536.75,1444.39 1538.81,1440.9 1540.88,1440.9 1542.95,1444.39 1545.01,1440.9 \n 1547.08,1444.39 1549.14,1444.39 1551.21,1440.9 1553.28,1444.39 1555.34,1440.9 1557.41,1444.39 1559.47,1444.39 1561.54,1444.39 1563.6,1440.9 1565.67,1440.9 \n 1567.74,1440.9 1569.8,1444.39 1571.87,1440.9 1573.93,1440.9 1576,1440.9 1578.07,1444.39 1580.13,1437.42 1582.2,1440.9 1584.26,1440.9 1586.33,1440.9 \n 1588.39,1440.9 1590.46,1440.9 1592.53,1444.39 1594.59,1444.39 1596.66,1444.39 1598.72,1440.9 1600.79,1440.9 1602.86,1444.39 1604.92,1437.42 1606.99,1444.39 \n 1609.05,1444.39 1611.12,1444.39 1613.18,1437.42 1615.25,1444.39 1617.32,1444.39 1619.38,1444.39 1621.45,1444.39 1623.51,1444.39 1625.58,1444.39 1627.65,1437.42 \n 1629.71,1437.42 1631.78,1437.42 1633.84,1437.42 1635.91,1444.39 1637.97,1444.39 1640.04,1444.39 1642.11,1437.42 1644.17,1444.39 1646.24,1437.42 1648.3,1444.39 \n 1650.37,1444.39 1652.44,1444.39 1654.5,1444.39 1656.57,1437.42 1658.63,1437.42 1660.7,1444.39 1662.76,1437.42 1664.83,1437.42 1666.9,1437.42 1668.96,1437.42 \n 1671.03,1437.42 1673.09,1437.42 1675.16,1437.42 1677.23,1444.39 1679.29,1437.42 1681.36,1437.42 1683.42,1437.42 1685.49,1437.42 1687.55,1437.42 1689.62,1437.42 \n 1691.69,1437.42 1693.75,1437.42 1695.82,1437.42 1697.88,1437.42 1699.95,1430.45 1702.02,1430.45 1704.08,1437.42 1706.15,1430.45 1708.21,1430.45 1710.28,1430.45 \n 1712.35,1437.42 1714.41,1437.42 1716.48,1430.45 1718.54,1430.45 1720.61,1430.45 1722.67,1430.45 1724.74,1437.42 1726.81,1437.42 1728.87,1430.45 1730.94,1430.45 \n 1733,1430.45 1735.07,1437.42 1737.14,1430.45 1739.2,1437.42 1741.27,1430.45 1743.33,1430.45 1745.4,1437.42 1747.46,1430.45 1749.53,1430.45 1751.6,1430.45 \n 1753.66,1437.42 1755.73,1423.48 1757.79,1423.48 1759.86,1423.48 1761.93,1430.45 1763.99,1423.48 1766.06,1423.48 1768.12,1430.45 1770.19,1430.45 1772.25,1430.45 \n 1774.32,1423.48 1776.39,1430.45 1778.45,1423.48 1780.52,1423.48 1782.58,1423.48 1784.65,1423.48 1786.72,1423.48 1788.78,1423.48 1790.85,1430.45 1792.91,1423.48 \n 1794.98,1416.51 1797.04,1423.48 1799.11,1423.48 1801.18,1423.48 1803.24,1423.48 1805.31,1416.51 1807.37,1416.51 1809.44,1416.51 1811.51,1423.48 1813.57,1416.51 \n 1815.64,1416.51 1817.7,1416.51 1819.77,1416.51 1821.83,1416.51 1823.9,1416.51 1825.97,1416.51 1828.03,1416.51 1830.1,1416.51 1832.16,1416.51 1834.23,1409.54 \n 1836.3,1409.54 1838.36,1416.51 1840.43,1416.51 1842.49,1402.57 1844.56,1402.57 1846.62,1409.54 1848.69,1409.54 1850.76,1409.54 1852.82,1409.54 1854.89,1409.54 \n 1856.95,1409.54 1859.02,1409.54 1861.09,1402.57 1863.15,1402.57 1865.22,1402.57 1867.28,1402.57 1869.35,1402.57 1871.41,1402.57 1873.48,1402.57 1875.55,1402.57 \n 1877.61,1402.57 1879.68,1395.6 1881.74,1395.6 1883.81,1395.6 1885.88,1388.63 1887.94,1395.6 1890.01,1395.6 1892.07,1388.63 1894.14,1388.63 1896.2,1388.63 \n 1898.27,1395.6 1900.34,1388.63 1902.4,1388.63 1904.47,1388.63 1906.53,1388.63 1908.6,1388.63 1910.67,1381.67 1912.73,1374.7 1914.8,1381.67 1916.86,1381.67 \n 1918.93,1374.7 1921,1374.7 1923.06,1374.7 1925.13,1374.7 1927.19,1374.7 1929.26,1374.7 1931.32,1367.73 1933.39,1367.73 1935.46,1367.73 1937.52,1367.73 \n 1939.59,1367.73 1941.65,1367.73 1943.72,1367.73 1945.79,1360.76 1947.85,1360.76 1949.92,1353.79 1951.98,1360.76 1954.05,1353.79 1956.11,1353.79 1958.18,1346.82 \n 1960.25,1353.79 1962.31,1353.79 1964.38,1346.82 1966.44,1339.85 1968.51,1346.82 1970.58,1339.85 1972.64,1339.85 1974.71,1339.85 1976.77,1339.85 1978.84,1332.88 \n 1980.9,1332.88 1982.97,1332.88 1985.04,1332.88 1987.1,1325.91 1989.17,1325.91 1991.23,1325.91 1993.3,1318.94 1995.37,1318.94 1997.43,1311.97 1999.5,1311.97 \n 2001.56,1311.97 2003.63,1305 2005.69,1305 2007.76,1305 2009.83,1305 2011.89,1298.03 2013.96,1298.03 2016.02,1291.07 2018.09,1298.03 2020.16,1291.07 \n 2022.22,1291.07 2024.29,1284.1 2026.35,1277.13 2028.42,1277.13 2030.48,1277.13 2032.55,1270.16 2034.62,1270.16 2036.68,1263.19 2038.75,1263.19 2040.81,1256.22 \n 2042.88,1256.22 2044.95,1249.25 2047.01,1249.25 2049.08,1249.25 2051.14,1242.28 2053.21,1242.28 2055.27,1235.31 2057.34,1235.31 2059.41,1228.34 2061.47,1221.37 \n 2063.54,1221.37 2065.6,1214.4 2067.67,1207.43 2069.74,1207.43 2071.8,1207.43 2073.87,1207.43 2075.93,1193.5 2078,1193.5 2080.06,1193.5 2082.13,1193.5 \n 2084.2,1186.53 2086.26,1179.56 2088.33,1172.59 2090.39,1172.59 2092.46,1165.62 2094.53,1158.65 2096.59,1151.68 2098.66,1144.71 2100.72,1144.71 2102.79,1137.74 \n 2104.85,1130.77 2106.92,1130.77 2108.99,1123.8 2111.05,1116.83 2113.12,1109.86 2115.18,1102.9 2117.25,1095.93 2119.32,1095.93 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| }, | |
| "metadata": {} | |
| } | |
| ] | |
| }, | |
| { | |
| "metadata": { | |
| "trusted": true | |
| }, | |
| "cell_type": "code", | |
| "source": "x = range(eps(), π/4; length=1001)\n@show extrema((S64.(x) .- sin.(x)) ./ sin.(x))\n@show extrema((S64m.(x) .- sin.(x)) ./ sin.(x));", | |
| "execution_count": 14, | |
| "outputs": [ | |
| { | |
| "output_type": "stream", | |
| "text": "extrema((S64.(x) .- sin.(x)) ./ sin.(x)) = (-2.0567317166037288e-16, 2.2101724769669492e-16)\nextrema((S64m.(x) .- sin.(x)) ./ sin.(x)) = (-2.134928416610595e-16, 2.888970123978146e-14)\n", | |
| "name": "stdout" | |
| } | |
| ] | |
| }, | |
| { | |
| "metadata": { | |
| "trusted": true | |
| }, | |
| "cell_type": "code", | |
| "source": "x = range(eps(), π/4; length=1001)\ny = 10^16 .* (S64.(x) .- sin.(x)) ./ sin.(x)\nplot(x, y; label=\"\", title=\"10¹⁶ (S64(x) - sin(x))/sin(x) for Float64 x in (0, π/4]\")", | |
| "execution_count": 15, | |
| "outputs": [ | |
| { | |
| "output_type": "execute_result", | |
| "execution_count": 15, | |
| "data": { | |
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332.207,828.08 334.292,828.08 336.377,828.08 338.463,828.08 340.548,1237.85 342.633,828.08 344.718,828.08 346.803,828.08 348.889,1214.12 \n 350.974,828.08 353.059,828.08 355.144,828.08 357.229,828.08 359.315,828.08 361.4,828.08 363.485,1178.6 365.57,828.08 367.655,828.08 369.74,828.08 \n 371.826,162.047 373.911,170.259 375.996,828.08 378.081,828.08 380.166,193.72 382.252,828.08 384.337,828.08 386.422,828.08 388.507,828.08 390.592,828.08 \n 392.678,235.947 394.763,828.08 396.848,828.08 398.933,255.016 401.018,828.08 403.104,267.059 405.189,828.08 407.274,828.08 409.359,828.08 411.444,289.683 \n 413.529,828.08 415.615,828.08 417.7,828.08 419.785,828.08 421.87,828.08 423.955,828.08 426.041,325.163 428.126,828.08 430.211,828.08 432.296,828.08 \n 434.381,828.08 436.467,828.08 438.552,828.08 440.637,828.08 442.722,828.08 444.807,828.08 446.892,828.08 448.978,828.08 451.063,376.182 453.148,828.08 \n 455.233,828.08 457.318,828.08 459.404,828.08 461.489,828.08 463.574,397.984 465.659,828.08 467.744,828.08 469.83,828.08 471.915,828.08 474,828.08 \n 476.085,417.77 478.17,828.08 480.256,828.08 482.341,828.08 484.426,828.08 486.511,828.08 488.596,828.08 490.681,828.08 492.767,441.471 494.852,828.08 \n 496.937,1209.19 499.022,828.08 501.107,828.08 503.193,828.08 505.278,828.08 507.363,828.08 509.448,462.57 511.533,828.08 513.619,828.08 515.704,469.896 \n 517.789,1183.89 519.874,828.08 521.959,828.08 524.045,1176.94 526.13,828.08 528.215,1172.46 530.3,828.08 532.385,828.08 534.47,828.08 536.556,1163.84 \n 538.641,828.08 540.726,828.08 542.811,828.08 544.896,828.08 546.982,828.08 549.067,828.08 551.152,828.08 553.237,828.08 555.322,828.08 557.408,196.077 \n 559.493,828.08 561.578,828.08 563.663,828.08 565.748,828.08 567.833,828.08 569.919,217.616 572.004,828.08 574.089,1431.69 576.174,227.839 578.259,828.08 \n 580.345,828.08 582.43,828.08 584.515,240.944 586.6,828.08 588.685,828.08 590.771,828.08 592.856,253.484 594.941,1399.62 597.026,828.08 599.111,828.08 \n 601.197,828.08 603.282,828.08 605.367,1384.85 607.452,828.08 609.537,828.08 611.622,828.08 613.708,828.08 615.793,828.08 617.878,1368.11 619.963,828.08 \n 622.048,828.08 624.134,828.08 626.219,828.08 628.304,828.08 630.389,828.08 632.474,828.08 634.56,308.851 636.645,828.08 638.73,313.799 640.815,828.08 \n 642.9,1337.51 644.986,828.08 647.071,828.08 649.156,325.763 651.241,828.08 653.326,828.08 655.411,828.08 657.497,1321.22 659.582,828.08 661.667,828.08 \n 663.752,828.08 665.837,828.08 667.923,828.08 670.008,828.08 672.093,828.08 674.178,828.08 676.263,828.08 678.349,828.08 680.434,358.49 682.519,828.08 \n 684.604,828.08 686.689,828.08 688.775,828.08 690.86,828.08 692.945,828.08 695.03,828.08 697.115,828.08 699.2,828.08 701.286,828.08 703.371,828.08 \n 705.456,381.711 707.541,828.08 709.626,828.08 711.712,387.156 713.797,828.08 715.882,390.712 717.967,828.08 720.052,828.08 722.138,828.08 724.223,828.08 \n 726.308,828.08 728.393,828.08 730.478,828.08 732.563,828.08 734.649,828.08 736.734,828.08 738.819,828.08 740.904,828.08 742.989,412.469 745.075,828.08 \n 747.16,828.08 749.245,828.08 751.33,828.08 753.415,420.261 755.501,828.08 757.586,828.08 759.671,828.08 761.756,828.08 763.841,828.08 765.927,828.08 \n 768.012,828.08 770.097,828.08 772.182,433.559 774.267,434.982 776.352,828.08 778.438,828.08 780.523,439.189 782.608,828.08 784.693,441.943 786.778,828.08 \n 788.864,444.657 790.949,828.08 793.034,828.08 795.119,828.08 797.204,828.08 799.29,828.08 801.375,828.08 803.46,453.855 805.545,828.08 807.63,828.08 \n 809.716,828.08 811.801,828.08 813.886,828.08 815.971,461.387 818.056,828.08 820.141,828.08 822.227,465.037 824.312,828.08 826.397,828.08 828.482,468.614 \n 830.567,828.08 832.653,470.958 834.738,828.08 836.823,473.271 838.908,828.08 840.993,475.553 843.079,828.08 845.164,828.08 847.249,828.08 849.334,828.08 \n 851.419,828.08 853.504,828.08 855.59,483.309 857.675,828.08 859.76,828.08 861.845,486.526 863.93,828.08 866.016,828.08 868.101,828.08 870.186,828.08 \n 872.271,491.753 874.356,828.08 876.442,828.08 878.527,828.08 880.612,828.08 882.697,828.08 884.782,828.08 886.868,828.08 888.953,171.481 891.038,828.08 \n 893.123,828.08 895.208,828.08 897.293,828.08 899.379,828.08 901.464,828.08 903.549,828.08 905.634,828.08 907.719,828.08 909.805,828.08 911.89,828.08 \n 913.975,828.08 916.06,195.945 918.145,828.08 920.231,828.08 922.316,828.08 924.401,203.097 926.486,828.08 928.571,206.61 930.657,828.08 932.742,828.08 \n 934.827,828.08 936.912,828.08 938.997,215.217 941.082,828.08 943.168,828.08 945.253,828.08 947.338,828.08 949.423,828.08 951.508,828.08 953.594,226.857 \n 955.679,228.483 957.764,828.08 959.849,828.08 961.934,828.08 964.02,828.08 966.105,828.08 968.19,828.08 970.275,828.08 972.36,241.164 974.445,828.08 \n 976.531,828.08 978.616,245.777 980.701,828.08 982.786,828.08 984.871,828.08 986.957,828.08 989.042,828.08 991.127,828.08 993.212,828.08 995.297,828.08 \n 997.383,259.171 999.468,828.08 1001.55,828.08 1003.64,828.08 1005.72,828.08 1007.81,828.08 1009.89,828.08 1011.98,828.08 1014.06,828.08 1016.15,271.935 \n 1018.23,828.08 1020.32,828.08 1022.4,276.058 1024.49,277.417 1026.58,828.08 1028.66,828.08 1030.75,281.455 1032.83,828.08 1034.92,828.08 1037,285.431 \n 1039.09,828.08 1041.17,828.08 1043.26,828.08 1045.34,828.08 1047.43,828.08 1049.51,828.08 1051.6,294.476 1053.68,295.742 1055.77,828.08 1057.85,828.08 \n 1059.94,828.08 1062.02,828.08 1064.11,828.08 1066.19,828.08 1068.28,828.08 1070.36,828.08 1072.45,828.08 1074.53,828.08 1076.62,828.08 1078.7,828.08 \n 1080.79,828.08 1082.88,828.08 1084.96,314.003 1087.05,828.08 1089.13,828.08 1091.22,828.08 1093.3,828.08 1095.39,828.08 1097.47,828.08 1099.56,828.08 \n 1101.64,828.08 1103.73,828.08 1105.81,828.08 1107.9,828.08 1109.98,327.685 1112.07,828.08 1114.15,828.08 1116.24,828.08 1118.32,828.08 1120.41,828.08 \n 1122.49,828.08 1124.58,828.08 1126.66,828.08 1128.75,828.08 1130.83,828.08 1132.92,828.08 1135.01,1315.54 1137.09,828.08 1139.18,342.701 1141.26,828.08 \n 1143.35,828.08 1145.43,828.08 1147.52,828.08 1149.6,828.08 1151.69,348.847 1153.77,349.855 1155.86,828.08 1157.94,828.08 1160.03,1303.31 1162.11,828.08 \n 1164.2,354.829 1166.28,828.08 1168.37,828.08 1170.45,357.76 1172.54,828.08 1174.62,359.693 1176.71,828.08 1178.79,361.609 1180.88,362.56 1182.96,828.08 \n 1185.05,828.08 1187.13,828.08 1189.22,366.325 1191.31,828.08 1193.39,828.08 1195.48,828.08 1197.56,370.026 1199.65,828.08 1201.73,1284.31 1203.82,828.08 \n 1205.9,828.08 1207.99,828.08 1210.07,828.08 1212.16,828.08 1214.24,828.08 1216.33,828.08 1218.41,379.002 1220.5,828.08 1222.58,828.08 1224.67,828.08 \n 1226.75,828.08 1228.84,383.35 1230.92,384.208 1233.01,385.064 1235.09,828.08 1237.18,1269.4 1239.26,828.08 1241.35,388.448 1243.43,389.286 1245.52,828.08 \n 1247.61,390.95 1249.69,828.08 1251.78,828.08 1253.86,828.08 1255.95,394.239 1258.03,828.08 1260.12,828.08 1262.2,828.08 1264.29,828.08 1266.37,828.08 \n 1268.46,828.08 1270.54,828.08 1272.63,828.08 1274.71,828.08 1276.8,828.08 1278.88,828.08 1280.97,828.08 1283.05,828.08 1285.14,828.08 1287.22,828.08 \n 1289.31,828.08 1291.39,1248.53 1293.48,828.08 1295.56,828.08 1297.65,828.08 1299.74,410.656 1301.82,828.08 1303.91,828.08 1305.99,828.08 1308.08,828.08 \n 1310.16,414.367 1312.25,828.08 1314.33,828.08 1316.42,828.08 1318.5,1238.88 1320.59,828.08 1322.67,418.726 1324.76,828.08 1326.84,828.08 1328.93,828.08 \n 1331.01,1234.58 1333.1,828.08 1335.18,828.08 1337.27,828.08 1339.35,828.08 1341.44,425.078 1343.52,425.77 1345.61,828.08 1347.69,828.08 1349.78,427.832 \n 1351.86,828.08 1353.95,828.08 1356.04,828.08 1358.12,828.08 1360.21,828.08 1362.29,1224.27 1364.38,828.08 1366.46,828.08 1368.55,828.08 1370.63,828.08 \n 1372.72,828.08 1374.8,828.08 1376.89,828.08 1378.97,437.151 1381.06,1218.36 1383.14,828.08 1385.23,828.08 1387.31,828.08 1389.4,828.08 1391.48,828.08 \n 1393.57,828.08 1395.65,828.08 1397.74,1213.27 1399.82,443.519 1401.91,1212.02 1403.99,444.765 1406.08,828.08 1408.16,1210.16 1410.25,828.08 1412.34,447.23 \n 1414.42,447.84 1416.51,828.08 1418.59,828.08 1420.68,449.659 1422.76,828.08 1424.85,828.08 1426.93,828.08 1429.02,828.08 1431.1,828.08 1433.19,828.08 \n 1435.27,1202.33 1437.36,828.08 1439.44,828.08 1441.53,455.584 1443.61,828.08 1445.7,456.744 1447.78,457.321 1449.87,828.08 1451.95,828.08 1454.04,1197.12 \n 1456.12,459.609 1458.21,460.176 1460.29,828.08 1462.38,828.08 1464.47,828.08 1466.55,462.424 1468.64,828.08 1470.72,828.08 1472.81,464.09 1474.89,828.08 \n 1476.98,828.08 1479.06,1190.42 1481.15,466.285 1483.23,828.08 1485.32,828.08 1487.4,828.08 1489.49,828.08 1491.57,468.986 1493.66,828.08 1495.74,828.08 \n 1497.83,828.08 1499.91,1185.04 1502,471.642 1504.08,472.168 1506.17,472.692 1508.25,828.08 1510.34,828.08 1512.42,828.08 1514.51,474.771 1516.59,828.08 \n 1518.68,475.8 1520.77,828.08 1522.85,476.822 1524.94,828.08 1527.02,828.08 1529.11,828.08 1531.19,828.08 1533.28,828.08 1535.36,828.08 1537.45,480.347 \n 1539.53,828.08 1541.62,828.08 1543.7,828.08 1545.79,828.08 1547.87,482.815 1549.96,828.08 1552.04,828.08 1554.13,828.08 1556.21,828.08 1558.3,485.243 \n 1560.38,828.08 1562.47,828.08 1564.55,486.681 1566.64,828.08 1568.72,828.08 1570.81,828.08 1572.89,488.577 1574.98,828.08 1577.07,828.08 1579.15,489.983 \n 1581.24,828.08 1583.32,490.913 1585.41,828.08 1587.49,1164.32 1589.58,828.08 1591.66,492.755 1593.75,828.08 1595.83,828.08 1597.92,828.08 1600,828.08 \n 1602.09,828.08 1604.17,828.08 1606.26,163.765 1608.34,828.08 1610.43,828.08 1612.51,828.08 1614.6,828.08 1616.68,828.08 1618.77,828.08 1620.85,828.08 \n 1622.94,170.836 1625.02,828.08 1627.11,828.08 1629.2,828.08 1631.28,828.08 1633.37,828.08 1635.45,176.024 1637.54,828.08 1639.62,177.732 1641.71,828.08 \n 1643.79,828.08 1645.88,828.08 1647.96,828.08 1650.05,828.08 1652.13,828.08 1654.22,828.08 1656.3,828.08 1658.39,185.29 1660.47,828.08 1662.56,828.08 \n 1664.64,828.08 1666.73,188.583 1668.81,189.4 1670.9,828.08 1672.98,828.08 1675.07,191.836 1677.15,828.08 1679.24,828.08 1681.32,828.08 1683.41,828.08 \n 1685.5,828.08 1687.58,196.642 1689.67,197.434 1691.75,828.08 1693.84,828.08 1695.92,828.08 1698.01,828.08 1700.09,828.08 1702.18,202.141 1704.26,202.917 \n 1706.35,828.08 1708.43,828.08 1710.52,828.08 1712.6,205.999 1714.69,828.08 1716.77,828.08 1718.86,1447.87 1720.94,828.08 1723.03,828.08 1725.11,828.08 \n 1727.2,828.08 1729.28,1444.11 1731.37,828.08 1733.45,828.08 1735.54,828.08 1737.62,828.08 1739.71,828.08 1741.8,828.08 1743.88,828.08 1745.97,217.969 \n 1748.05,828.08 1750.14,828.08 1752.22,828.08 1754.31,220.875 1756.39,828.08 1758.48,222.315 1760.56,828.08 1762.65,828.08 1764.73,828.08 1766.82,828.08 \n 1768.9,828.08 1770.99,828.08 1773.07,227.291 1775.16,828.08 1777.24,828.08 1779.33,828.08 1781.41,828.08 1783.5,828.08 1785.58,828.08 1787.67,232.169 \n 1789.75,232.858 1791.84,828.08 1793.93,828.08 1796.01,828.08 1798.1,828.08 1800.18,236.273 1802.27,828.08 1804.35,237.625 1806.44,828.08 1808.52,828.08 \n 1810.61,828.08 1812.69,828.08 1814.78,828.08 1816.86,828.08 1818.95,242.3 1821.03,828.08 1823.12,828.08 1825.2,828.08 1827.29,828.08 1829.37,245.583 \n 1831.46,246.234 1833.54,828.08 1835.63,828.08 1837.71,248.176 1839.8,248.82 1841.88,828.08 1843.97,828.08 1846.05,828.08 1848.14,828.08 1850.23,828.08 \n 1852.31,828.08 1854.4,828.08 1856.48,828.08 1858.57,828.08 1860.65,255.162 1862.74,828.08 1864.82,828.08 1866.91,828.08 1868.99,257.651 1871.08,828.08 \n 1873.16,828.08 1875.25,259.499 1877.33,828.08 1879.42,260.723 1881.5,261.333 1883.59,828.08 1885.67,828.08 1887.76,263.151 1889.84,263.754 1891.93,828.08 \n 1894.01,828.08 1896.1,828.08 1898.18,828.08 1900.27,266.744 1902.35,828.08 1904.44,828.08 1906.53,828.08 1908.61,269.107 1910.7,828.08 1912.78,828.08 \n 1914.87,270.863 1916.95,271.445 1919.04,828.08 1921.12,828.08 1923.21,273.182 1925.29,828.08 1927.38,828.08 1929.46,274.905 1931.55,828.08 1933.63,828.08 \n 1935.72,828.08 1937.8,828.08 1939.89,828.08 1941.97,828.08 1944.06,828.08 1946.14,828.08 1948.23,828.08 1950.31,828.08 1952.4,828.08 1954.48,281.661 \n 1956.57,828.08 1958.66,828.08 1960.74,828.08 1962.83,828.08 1964.91,828.08 1967,828.08 1969.08,828.08 1971.17,828.08 1973.25,828.08 1975.34,287.131 \n 1977.42,828.08 1979.51,828.08 1981.59,828.08 1983.68,828.08 1985.76,828.08 1987.85,828.08 1989.93,1365.28 1992.02,291.405 1994.1,291.933 1996.19,292.459 \n 1998.27,828.08 2000.36,828.08 2002.44,294.031 2004.53,1361.61 2006.61,828.08 2008.7,295.59 2010.78,828.08 2012.87,828.08 2014.96,828.08 2017.04,828.08 \n 2019.13,828.08 2021.21,828.08 2023.3,299.183 2025.38,299.691 2027.47,828.08 2029.55,828.08 2031.64,828.08 2033.72,301.71 2035.81,828.08 2037.89,828.08 \n 2039.98,828.08 2042.06,303.708 2044.15,828.08 2046.23,828.08 2048.32,305.193 2050.4,828.08 2052.49,828.08 2054.57,828.08 2056.66,828.08 2058.74,828.08 \n 2060.83,308.13 2062.91,828.08 2065,309.1 2067.08,309.582 2069.17,310.064 2071.26,828.08 2073.34,828.08 2075.43,828.08 2077.51,828.08 2079.6,828.08 \n 2081.68,828.08 2083.77,828.08 2085.85,828.08 2087.94,1341.82 2090.02,828.08 2092.11,828.08 2094.19,828.08 2096.28,828.08 2098.36,828.08 2100.45,828.08 \n 2102.53,828.08 2104.62,828.08 2106.7,828.08 2108.79,318.987 2110.87,319.445 2112.96,319.902 2115.04,320.358 2117.13,828.08 2119.21,828.08 2121.3,828.08 \n 2123.39,828.08 2125.47,322.62 2127.56,323.069 2129.64,828.08 2131.73,323.963 2133.81,828.08 2135.9,324.854 2137.98,828.08 2140.07,828.08 2142.15,828.08 \n 2144.24,828.08 2146.32,828.08 2148.41,828.08 2150.49,828.08 2152.58,328.372 2154.66,828.08 2156.75,828.08 2158.83,828.08 2160.92,330.105 2163,828.08 \n 2165.09,828.08 2167.17,828.08 2169.26,828.08 2171.34,828.08 2173.43,332.673 2175.51,828.08 2177.6,828.08 2179.69,333.943 2181.77,828.08 2183.86,828.08 \n 2185.94,828.08 2188.03,828.08 2190.11,828.08 2192.2,828.08 2194.28,828.08 2196.37,828.08 2198.45,828.08 2200.54,828.08 2202.62,828.08 2204.71,828.08 \n 2206.79,828.08 2208.88,339.75 2210.96,828.08 2213.05,828.08 2215.13,340.969 2217.22,828.08 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| }, | |
| "metadata": {} | |
| } | |
| ] | |
| }, | |
| { | |
| "metadata": { | |
| "trusted": true | |
| }, | |
| "cell_type": "code", | |
| "source": "x = range(eps(), π/4; length=1001)\ny = 10^16 .* (S64m.(x) .- sin.(x)) ./ sin.(x)\nplot(x, y; label=\"\", title=\"10¹⁶ (S64m(x) - sin(x))/sin(x) for Float64 x in (0, π/4]\")", | |
| "execution_count": 16, | |
| "outputs": [ | |
| { | |
| "output_type": "execute_result", | |
| "execution_count": 16, | |
| "data": { | |
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266.256,1438.44 268.322,1438.44 270.388,1438.44 272.454,1438.44 274.519,1438.44 276.585,1438.44 278.651,1445.95 280.717,1438.44 282.783,1431.47 284.849,1438.44 \n 286.914,1431.93 288.98,1438.44 291.046,1438.44 293.112,1438.44 295.178,1444.18 297.244,1438.44 299.309,1438.44 301.375,1438.44 303.441,1438.44 305.507,1438.44 \n 307.573,1438.44 309.639,1428.92 311.705,1438.44 313.77,1438.44 315.836,1447.31 317.902,1438.44 319.968,1438.44 322.034,1446.75 324.1,1438.44 326.165,1438.44 \n 328.231,1438.44 330.297,1438.44 332.363,1438.44 334.429,1438.44 336.495,1438.44 338.56,1438.44 340.626,1438.44 342.692,1438.44 344.758,1438.44 346.824,1438.44 \n 348.89,1438.44 350.956,1438.44 353.021,1438.44 355.087,1438.44 357.153,1438.44 359.219,1444.45 361.285,1438.44 363.351,1438.44 365.416,1438.44 367.482,1444.1 \n 369.548,1438.44 371.614,1438.44 373.68,1443.86 375.746,1438.44 377.811,1438.44 379.877,1438.44 381.943,1443.58 384.009,1438.44 386.075,1438.44 388.141,1438.44 \n 390.207,1428.68 392.272,1428.8 394.338,1438.44 396.404,1438.44 398.47,1429.14 400.536,1438.44 402.602,1438.44 404.667,1438.44 406.733,1438.44 408.799,1438.44 \n 410.865,1429.76 412.931,1438.44 414.997,1438.44 417.062,1430.04 419.128,1438.44 421.194,1438.44 423.26,1438.44 425.326,1438.44 427.392,1438.44 429.457,1430.55 \n 431.523,1438.44 433.589,1438.44 435.655,1438.44 437.721,1446.03 439.787,1438.44 441.853,1438.44 443.918,1431.07 445.984,1438.44 448.05,1438.44 450.116,1438.44 \n 452.182,1438.44 454.248,1438.44 456.313,1438.44 458.379,1438.44 460.445,1438.44 462.511,1438.44 464.577,1438.44 466.643,1438.44 468.708,1431.82 470.774,1438.44 \n 472.84,1438.44 474.906,1438.44 476.972,1438.44 479.038,1438.44 481.104,1438.44 483.169,1438.44 485.235,1438.44 487.301,1438.44 489.367,1444.55 491.433,1438.44 \n 493.499,1432.42 495.564,1438.44 497.63,1438.44 499.696,1438.44 501.762,1438.44 503.828,1438.44 505.894,1438.44 507.959,1438.44 510.025,1432.77 512.091,1438.44 \n 514.157,1444.03 516.223,1438.44 518.289,1438.44 520.355,1438.44 522.42,1438.44 524.486,1438.44 526.552,1433.08 528.618,1438.44 530.684,1438.44 532.75,1433.19 \n 534.815,1443.66 536.881,1438.44 538.947,1438.44 541.013,1443.56 543.079,1438.44 545.145,1443.49 547.21,1438.44 549.276,1438.44 551.342,1438.44 553.408,1443.36 \n 555.474,1438.44 557.54,1438.44 559.606,1438.44 561.671,1438.44 563.737,1438.44 565.803,1438.44 567.869,1447.87 569.935,1438.44 572.001,1438.44 574.066,1429.17 \n 576.132,1438.44 578.198,1438.44 580.264,1438.44 582.33,1438.44 584.396,1438.44 586.461,1438.44 588.527,1438.44 590.593,1447.29 592.659,1438.44 594.725,1438.44 \n 596.791,1438.44 598.856,1438.44 600.922,1438.44 602.988,1438.44 605.054,1438.44 607.12,1438.44 609.186,1438.44 611.252,1446.82 613.317,1438.44 615.383,1438.44 \n 617.449,1438.44 619.515,1438.44 621.581,1446.6 623.647,1438.44 625.712,1446.52 627.778,1438.44 629.844,1438.44 631.91,1438.44 633.976,1446.36 636.042,1438.44 \n 638.107,1438.44 640.173,1438.44 642.239,1438.44 644.305,1438.44 646.371,1438.44 648.437,1438.44 650.503,1430.83 652.568,1438.44 654.634,1438.44 656.7,1438.44 \n 658.766,1445.91 660.832,1438.44 662.898,1438.44 664.963,1438.44 667.029,1438.44 669.095,1438.44 671.161,1438.44 673.227,1445.67 675.293,1438.44 677.358,1445.61 \n 679.424,1438.44 681.49,1438.44 683.556,1438.44 685.622,1438.44 687.688,1438.44 689.754,1445.42 691.819,1438.44 693.885,1438.44 695.951,1438.44 698.017,1438.44 \n 700.083,1438.44 702.149,1438.44 704.214,1438.44 706.28,1438.44 708.346,1438.44 710.412,1438.44 712.478,1438.44 714.544,1438.44 716.609,1438.44 718.675,1445.01 \n 720.741,1431.9 722.807,1438.44 724.873,1438.44 726.939,1431.98 729.005,1438.44 731.07,1432.03 733.136,1438.44 735.202,1438.44 737.268,1438.44 739.334,1438.44 \n 741.4,1438.44 743.465,1444.7 745.531,1444.68 747.597,1444.65 749.663,1438.44 751.729,1438.44 753.795,1438.44 755.86,1438.44 757.926,1438.44 759.992,1444.51 \n 762.058,1438.44 764.124,1444.46 766.19,1438.44 768.255,1432.46 770.321,1438.44 772.387,1438.44 774.453,1438.44 776.519,1438.44 778.585,1438.44 780.651,1438.44 \n 782.716,1438.44 784.782,1444.25 786.848,1438.44 788.914,1438.44 790.98,1444.18 793.046,1444.16 795.111,1432.74 797.177,1438.44 799.243,1432.78 801.309,1438.44 \n 803.375,1438.44 805.441,1438.44 807.506,1438.44 809.572,1438.44 811.638,1438.44 813.704,1438.44 815.77,1438.44 817.836,1432.95 819.902,1438.44 821.967,1438.44 \n 824.033,1438.44 826.099,1438.44 828.165,1438.44 830.231,1438.44 832.297,1438.44 834.362,1438.44 836.428,1433.12 838.494,1443.75 840.56,1438.44 842.626,1438.44 \n 844.692,1438.44 846.757,1433.2 848.823,1438.44 850.889,1433.24 852.955,1443.63 855.021,1438.44 857.087,1438.44 859.153,1438.44 861.218,1438.44 863.284,1438.44 \n 865.35,1438.44 867.416,1438.44 869.482,1438.44 871.548,1438.44 873.613,1438.44 875.679,1433.43 877.745,1438.44 879.811,1438.44 881.877,1438.44 883.943,1438.44 \n 886.008,1438.44 888.074,1443.36 890.14,1438.44 892.206,1438.44 894.272,1438.44 896.338,1438.44 898.404,1438.44 900.469,1438.44 902.535,1438.44 904.601,1438.44 \n 906.667,1438.44 908.733,1438.44 910.799,1438.44 912.864,1438.44 914.93,1438.44 916.996,1438.44 919.062,1438.44 921.128,1438.44 923.194,1438.44 925.259,1438.44 \n 927.325,1438.44 929.391,1429.17 931.457,1438.44 933.523,1438.44 935.589,1438.44 937.654,1429.28 939.72,1438.44 941.786,1438.44 943.852,1447.53 945.918,1438.44 \n 947.984,1447.48 950.05,1438.44 952.115,1438.44 954.181,1438.44 956.247,1438.44 958.313,1438.44 960.379,1438.44 962.445,1438.44 964.51,1438.44 966.576,1438.44 \n 968.642,1438.44 970.708,1447.21 972.774,1438.44 974.84,1438.44 976.905,1438.44 978.971,1438.44 981.037,1438.44 983.103,1438.44 985.169,1429.84 987.235,1438.44 \n 989.301,1438.44 991.366,1438.44 993.432,1438.44 995.498,1438.44 997.564,1438.44 999.63,1438.44 1001.7,1438.44 1003.76,1438.44 1005.83,1438.44 1007.89,1438.44 \n 1009.96,1430.1 1012.02,1438.44 1014.09,1438.44 1016.16,1438.44 1018.22,1438.44 1020.29,1438.44 1022.35,1438.44 1024.42,1438.44 1026.49,1438.44 1028.55,1438.44 \n 1030.62,1438.44 1032.68,1438.44 1034.75,1438.44 1036.81,1430.37 1038.88,1438.44 1040.95,1438.44 1043.01,1430.43 1045.08,1438.44 1047.14,1438.44 1049.21,1430.48 \n 1051.28,1438.44 1053.34,1438.44 1055.41,1438.44 1057.47,1438.44 1059.54,1438.44 1061.6,1438.44 1063.67,1430.62 1065.74,1430.64 1067.8,1438.44 1069.87,1438.44 \n 1071.93,1438.44 1074,1438.44 1076.07,1438.44 1078.13,1438.44 1080.2,1446.12 1082.26,1438.44 1084.33,1438.44 1086.4,1438.44 1088.46,1438.44 1090.53,1438.44 \n 1092.59,1438.44 1094.66,1438.44 1096.72,1438.44 1098.79,1445.96 1100.86,1438.44 1102.92,1438.44 1104.99,1438.44 1107.05,1438.44 1109.12,1438.44 1111.19,1438.44 \n 1113.25,1438.44 1115.32,1438.44 1117.38,1438.44 1119.45,1438.44 1121.51,1431.1 1123.58,1438.44 1125.65,1438.44 1127.71,1438.44 1129.78,1438.44 1131.84,1438.44 \n 1133.91,1438.44 1135.98,1445.67 1138.04,1438.44 1140.11,1438.44 1142.17,1438.44 1144.24,1438.44 1146.3,1445.59 1148.37,1438.44 1150.44,1431.32 1152.5,1438.44 \n 1154.57,1438.44 1156.63,1445.51 1158.7,1438.44 1160.77,1438.44 1162.83,1438.44 1164.9,1431.43 1166.96,1438.44 1169.03,1438.44 1171.09,1445.41 1173.16,1438.44 \n 1175.23,1431.5 1177.29,1438.44 1179.36,1438.44 1181.42,1431.55 1183.49,1438.44 1185.56,1438.44 1187.62,1438.44 1189.69,1431.6 1191.75,1438.44 1193.82,1438.44 \n 1195.88,1438.44 1197.95,1438.44 1200.02,1438.44 1202.08,1445.2 1204.15,1438.44 1206.21,1438.44 1208.28,1431.72 1210.35,1438.44 1212.41,1445.13 1214.48,1438.44 \n 1216.54,1438.44 1218.61,1438.44 1220.67,1438.44 1222.74,1438.44 1224.81,1438.44 1226.87,1438.44 1228.94,1438.44 1231,1438.44 1233.07,1438.44 1235.14,1438.44 \n 1237.2,1438.44 1239.27,1431.92 1241.33,1438.44 1243.4,1431.95 1245.46,1438.44 1247.53,1444.91 1249.6,1438.44 1251.66,1432 1253.73,1432.01 1255.79,1438.44 \n 1257.86,1432.03 1259.93,1438.44 1261.99,1438.44 1264.06,1438.44 1266.12,1432.08 1268.19,1438.44 1270.25,1438.44 1272.32,1444.77 1274.39,1438.44 1276.45,1438.44 \n 1278.52,1438.44 1280.58,1438.44 1282.65,1438.44 1284.72,1438.44 1286.78,1438.44 1288.85,1444.67 1290.91,1438.44 1292.98,1438.44 1295.05,1438.44 1297.11,1438.44 \n 1299.18,1438.44 1301.24,1444.6 1303.31,1438.44 1305.37,1438.44 1307.44,1438.44 1309.51,1432.32 1311.57,1438.44 1313.64,1438.44 1315.7,1438.44 1317.77,1438.44 \n 1319.84,1432.38 1321.9,1438.44 1323.97,1438.44 1326.03,1438.44 1328.1,1444.46 1330.16,1438.44 1332.23,1438.44 1334.3,1438.44 1336.36,1438.44 1338.43,1438.44 \n 1340.49,1444.4 1342.56,1438.44 1344.63,1438.44 1346.69,1438.44 1348.76,1438.44 1350.82,1438.44 1352.89,1432.54 1354.95,1438.44 1357.02,1438.44 1359.09,1432.57 \n 1361.15,1438.44 1363.22,1438.44 1365.28,1438.44 1367.35,1438.44 1369.42,1438.44 1371.48,1444.25 1373.55,1438.44 1375.61,1438.44 1377.68,1438.44 1379.74,1438.44 \n 1381.81,1438.44 1383.88,1438.44 1385.94,1438.44 1388.01,1438.44 1390.07,1444.16 1392.14,1438.44 1394.21,1438.44 1396.27,1438.44 1398.34,1438.44 1400.4,1438.44 \n 1402.47,1438.44 1404.53,1438.44 1406.6,1444.09 1408.67,1432.8 1410.73,1444.07 1412.8,1432.82 1414.86,1438.44 1416.93,1444.04 1419,1438.44 1421.06,1438.44 \n 1423.13,1432.87 1425.19,1438.44 1427.26,1438.44 1429.32,1438.44 1431.39,1438.44 1433.46,1438.44 1435.52,1438.44 1437.59,1438.44 1439.65,1438.44 1441.72,1438.44 \n 1443.79,1443.93 1445.85,1438.44 1447.92,1438.44 1449.98,1432.98 1452.05,1438.44 1454.11,1433 1456.18,1433 1458.25,1438.44 1460.31,1438.44 1462.38,1443.85 \n 1464.44,1433.04 1466.51,1433.05 1468.58,1438.44 1470.64,1438.44 1472.71,1438.44 1474.77,1433.08 1476.84,1438.44 1478.9,1438.44 1480.97,1433.1 1483.04,1438.44 \n 1485.1,1438.44 1487.17,1443.75 1489.23,1433.14 1491.3,1438.44 1493.37,1438.44 1495.43,1438.44 1497.5,1438.44 1499.56,1433.18 1501.63,1438.44 1503.7,1438.44 \n 1505.76,1438.44 1507.83,1438.44 1509.89,1433.21 1511.96,1433.22 1514.02,1433.23 1516.09,1438.44 1518.16,1433.25 1520.22,1438.44 1522.29,1433.26 1524.35,1433.27 \n 1526.42,1433.28 1528.49,1438.44 1530.55,1433.29 1532.62,1433.3 1534.68,1438.44 1536.75,1438.44 1538.81,1433.32 1540.88,1433.33 1542.95,1438.44 1545.01,1433.34 \n 1547.08,1438.44 1549.14,1438.44 1551.21,1433.36 1553.28,1438.44 1555.34,1433.38 1557.41,1438.44 1559.47,1438.44 1561.54,1438.44 1563.6,1433.41 1565.67,1433.41 \n 1567.74,1433.42 1569.8,1438.44 1571.87,1433.44 1573.93,1433.44 1576,1433.45 1578.07,1438.44 1580.13,1428.49 1582.2,1433.47 1584.26,1433.48 1586.33,1433.48 \n 1588.39,1433.49 1590.46,1433.5 1592.53,1438.44 1594.59,1438.44 1596.66,1438.44 1598.72,1433.52 1600.79,1433.53 1602.86,1438.44 1604.92,1428.65 1606.99,1438.44 \n 1609.05,1438.44 1611.12,1438.44 1613.18,1428.7 1615.25,1438.44 1617.32,1438.44 1619.38,1438.44 1621.45,1438.44 1623.51,1438.44 1625.58,1438.44 1627.65,1428.79 \n 1629.71,1428.8 1631.78,1428.82 1633.84,1428.83 1635.91,1438.44 1637.97,1438.44 1640.04,1438.44 1642.11,1428.88 1644.17,1438.44 1646.24,1428.91 1648.3,1438.44 \n 1650.37,1438.44 1652.44,1438.44 1654.5,1438.44 1656.57,1428.97 1658.63,1428.98 1660.7,1438.44 1662.76,1429 1664.83,1429.02 1666.9,1429.03 1668.96,1429.04 \n 1671.03,1429.05 1673.09,1429.06 1675.16,1429.08 1677.23,1438.44 1679.29,1429.1 1681.36,1429.11 1683.42,1429.12 1685.49,1429.14 1687.55,1429.15 1689.62,1429.16 \n 1691.69,1429.17 1693.75,1429.18 1695.82,1429.19 1697.88,1429.21 1699.95,1419.99 1702.02,1420.02 1704.08,1429.24 1706.15,1420.06 1708.21,1420.09 1710.28,1420.11 \n 1712.35,1429.29 1714.41,1429.3 1716.48,1420.18 1718.54,1420.2 1720.61,1420.22 1722.67,1420.24 1724.74,1429.35 1726.81,1429.36 1728.87,1420.31 1730.94,1420.33 \n 1733,1420.36 1735.07,1429.41 1737.14,1420.4 1739.2,1429.43 1741.27,1420.44 1743.33,1420.46 1745.4,1429.46 1747.46,1420.51 1749.53,1420.53 1751.6,1420.55 \n 1753.66,1429.51 1755.73,1411.67 1757.79,1411.7 1759.86,1411.73 1761.93,1420.66 1763.99,1411.8 1766.06,1411.83 1768.12,1420.72 1770.19,1420.74 1772.25,1420.76 \n 1774.32,1411.95 1776.39,1420.8 1778.45,1412.02 1780.52,1412.05 1782.58,1412.08 1784.65,1412.11 1786.72,1412.14 1788.78,1412.17 1790.85,1420.95 1792.91,1412.23 \n 1794.98,1403.53 1797.04,1412.29 1799.11,1412.32 1801.18,1412.35 1803.24,1412.38 1805.31,1403.73 1807.37,1403.77 1809.44,1403.81 1811.51,1412.5 1813.57,1403.89 \n 1815.64,1403.93 1817.7,1403.97 1819.77,1404.01 1821.83,1404.05 1823.9,1404.09 1825.97,1404.13 1828.03,1404.16 1830.1,1404.2 1832.16,1404.24 1834.23,1395.74 \n 1836.3,1395.79 1838.36,1404.36 1840.43,1404.39 1842.49,1387.43 1844.56,1387.48 1846.62,1396.02 1848.69,1396.07 1850.76,1396.12 1852.82,1396.16 1854.89,1396.21 \n 1856.95,1396.26 1859.02,1396.3 1861.09,1387.93 1863.15,1387.99 1865.22,1388.04 1867.28,1388.1 1869.35,1388.15 1871.41,1388.21 1873.48,1388.26 1875.55,1388.32 \n 1877.61,1388.37 1879.68,1380.09 1881.74,1380.15 1883.81,1380.21 1885.88,1371.97 1887.94,1380.34 1890.01,1380.4 1892.07,1372.18 1894.14,1372.25 1896.2,1372.32 \n 1898.27,1380.65 1900.34,1372.46 1902.4,1372.53 1904.47,1372.6 1906.53,1372.67 1908.6,1372.74 1910.67,1364.61 1912.73,1356.49 1914.8,1364.76 1916.86,1364.84 \n 1918.93,1356.75 1921,1356.83 1923.06,1356.92 1925.13,1357 1927.19,1357.09 1929.26,1357.17 1931.32,1349.14 1933.39,1349.23 1935.46,1349.32 1937.52,1349.41 \n 1939.59,1349.5 1941.65,1349.59 1943.72,1349.69 1945.79,1341.72 1947.85,1341.82 1949.92,1333.87 1951.98,1342.01 1954.05,1334.08 1956.11,1334.19 1958.18,1326.28 \n 1960.25,1334.4 1962.31,1334.51 1964.38,1326.62 1966.44,1318.76 1968.51,1326.85 1970.58,1319 1972.64,1319.12 1974.71,1319.24 1976.77,1319.36 1978.84,1311.54 \n 1980.9,1311.67 1982.97,1311.8 1985.04,1311.92 1987.1,1304.15 1989.17,1304.28 1991.23,1304.41 1993.3,1296.67 1995.37,1296.81 1997.43,1289.09 1999.5,1289.24 \n 2001.56,1289.38 2003.63,1281.69 2005.69,1281.84 2007.76,1282 2009.83,1282.15 2011.89,1274.49 2013.96,1274.65 2016.02,1267.02 2018.09,1274.97 2020.16,1267.35 \n 2022.22,1267.52 2024.29,1259.92 2026.35,1252.34 2028.42,1252.52 2030.48,1252.69 2032.55,1245.14 2034.62,1245.32 2036.68,1237.79 2038.75,1237.98 2040.81,1230.47 \n 2042.88,1230.67 2044.95,1223.18 2047.01,1223.38 2049.08,1223.58 2051.14,1216.12 2053.21,1216.33 2055.27,1208.89 2057.34,1209.1 2059.41,1201.68 2061.47,1194.27 \n 2063.54,1194.5 2065.6,1187.11 2067.67,1179.74 2069.74,1179.98 2071.8,1180.22 2073.87,1180.46 2075.93,1165.53 2078,1165.79 2080.06,1166.04 2082.13,1166.29 \n 2084.2,1158.99 2086.26,1151.7 2088.33,1144.42 2090.39,1144.69 2092.46,1137.43 2094.53,1130.19 2096.59,1122.96 2098.66,1115.74 2100.72,1116.04 2102.79,1108.84 \n 2104.85,1101.65 2106.92,1101.96 2108.99,1094.79 2111.05,1087.63 2113.12,1080.49 2115.18,1073.36 2117.25,1066.25 2119.32,1066.58 2121.38,1059.48 2123.45,1052.4 \n 2125.51,1045.32 2127.58,1038.26 2129.65,1031.21 2131.71,1031.58 2133.78,1017.15 2135.84,1024.91 2137.91,1010.52 2139.97,1003.53 2142.04,996.544 2144.11,989.573 \n 2146.17,982.615 2148.24,975.669 2150.3,976.073 2152.37,961.811 2154.44,954.901 2156.5,948.001 2158.57,941.114 2160.63,934.238 2162.7,920.073 2164.76,920.521 \n 2166.83,906.392 2168.9,906.85 2170.96,892.756 2173.03,885.956 2175.09,871.903 2177.16,872.389 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| "metadata": {} | |
| } | |
| ] | |
| }, | |
| { | |
| "metadata": { | |
| "trusted": true | |
| }, | |
| "cell_type": "code", | |
| "source": "DS1 = -1.66666666666666324348e-01\nDS2 = 8.33333333332248946124e-03\nDS3 = -1.98412698298579493134e-04\nDS4 = 2.75573137070700676789e-06\nDS5 = -2.50507602534068634195e-08\nDS6 = 1.58969099521155010221e-10\n\nsin_Float64 = Polynomial(BigFloat[0.0, 1.0, 0.0, DS1, 0.0, DS2, 0.0, DS3, 0.0, DS4, 0.0, DS5, 0.0, DS6])", | |
| "execution_count": 17, | |
| "outputs": [ | |
| { | |
| "output_type": "execute_result", | |
| "execution_count": 17, | |
| "data": { | |
| "text/plain": "Polynomial(1.0*x - 0.1666666666666663243479007405767333693802356719970703125*x^3 + 0.0083333333333224894612367705803990247659385204315185546875*x^5 - 0.00019841269829857949313391130008454865674139000475406646728515625*x^7 + 2.75573137070700676789039053338203899556901887990534305572509765625e-06*x^9 - 2.5050760253406863419538120707312500723418224879424087703227996826171875e-08*x^11 + 1.5896909952115501022112634498906040547705487142593483440577983856201171875e-10*x^13)", | |
| "text/html": "1.0∙x - 0.1666666666666663243479007405767333693802356719970703125∙x<sup>3</sup> + 0.0083333333333224894612367705803990247659385204315185546875∙x<sup>5</sup> - 0.00019841269829857949313391130008454865674139000475406646728515625∙x<sup>7</sup> + 2.75573137070700676789039053338203899556901887990534305572509765625e-06∙x<sup>9</sup> - 2.5050760253406863419538120707312500723418224879424087703227996826171875e-08∙x<sup>11</sup> + 1.5896909952115501022112634498906040547705487142593483440577983856201171875e-10∙x<sup>13</sup>", | |
| "text/latex": "$1.0\\cdot x - 0.1666666666666663243479007405767333693802356719970703125\\cdot x^{3} + 0.0083333333333224894612367705803990247659385204315185546875\\cdot x^{5} - 0.00019841269829857949313391130008454865674139000475406646728515625\\cdot x^{7} + 2.75573137070700676789039053338203899556901887990534305572509765625e-06\\cdot x^{9} - 2.5050760253406863419538120707312500723418224879424087703227996826171875e-08\\cdot x^{11} + 1.5896909952115501022112634498906040547705487142593483440577983856201171875e-10\\cdot x^{13}$" | |
| }, | |
| "metadata": {} | |
| } | |
| ] | |
| }, | |
| { | |
| "metadata": { | |
| "trusted": true | |
| }, | |
| "cell_type": "code", | |
| "source": "x = range(eps(BigFloat), big(π)/4; length=1001)\n@show extrema((S64.(x) .- sin.(x)))\n@show extrema((S.(x) .- sin.(x)))\n@show extrema((sin_Float64.(x) .- sin.(x)))\n;", | |
| "execution_count": 18, | |
| "outputs": [ | |
| { | |
| "output_type": "stream", | |
| "text": "extrema(S64.(x) .- sin.(x)) = (0.0, 1.858746143958339058632659372872706811762238366251681555149420183435388355009553e-17)\nextrema(S.(x) .- sin.(x)) = (-1.234443929454079361788252684700880340775434640802344016276874337727306069801397e-18, 1.234357785430595994202668714467051814976695856920202366479957759989256279954844e-18)\nextrema(sin_Float64.(x) .- sin.(x)) = (-2.666890902692519776786628397121481900078450964560409641356415489047233954435966e-18, 2.674485140569285607847344891524957213176247872326852756133208513286352090792979e-18)\n", | |
| "name": "stdout" | |
| } | |
| ] | |
| }, | |
| { | |
| "metadata": { | |
| "trusted": true | |
| }, | |
| "cell_type": "code", | |
| "source": "x = range(eps(BigFloat), big(π)/4; length=1001)\ny = 10^16 .* (S64.(x) .- sin.(x))\nplot(x, y; label=\"\", title=\"10¹⁶ (S64(x) - sin(x)) for BigFloat x in (0, π/4]\")", | |
| "execution_count": 19, | |
| "outputs": [ | |
| { | |
| "output_type": "execute_result", | |
| "execution_count": 19, | |
| "data": { | |
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427.58,1412.91 429.635,1411.85 431.689,1410.76 433.744,1409.66 435.798,1408.54 437.853,1407.39 439.908,1406.23 \n 441.962,1405.05 444.017,1403.85 446.071,1402.63 448.126,1401.39 450.181,1400.13 452.235,1398.85 454.29,1397.56 456.345,1396.24 458.399,1394.9 460.454,1393.55 \n 462.508,1392.17 464.563,1390.78 466.618,1389.37 468.672,1387.93 470.727,1386.48 472.781,1385.01 474.836,1383.52 476.891,1382.01 478.945,1380.48 481,1378.93 \n 483.055,1377.37 485.109,1375.78 487.164,1374.18 489.218,1372.55 491.273,1370.91 493.328,1369.25 495.382,1367.57 497.437,1365.87 499.491,1364.16 501.546,1362.42 \n 503.601,1360.67 505.655,1358.9 507.71,1357.11 509.765,1355.3 511.819,1353.48 513.874,1351.63 515.928,1349.78 517.983,1347.9 520.038,1346 522.092,1344.09 \n 524.147,1342.16 526.202,1340.22 528.256,1338.26 530.311,1336.28 532.365,1334.28 534.42,1332.27 536.475,1330.24 538.529,1328.2 540.584,1326.14 542.638,1324.06 \n 544.693,1321.97 546.748,1319.87 548.802,1317.75 550.857,1315.61 552.912,1313.46 554.966,1311.3 557.021,1309.12 559.075,1306.92 561.13,1304.72 563.185,1302.5 \n 565.239,1300.26 567.294,1298.02 569.348,1295.75 571.403,1293.48 573.458,1291.2 575.512,1288.9 577.567,1286.59 579.622,1284.27 581.676,1281.93 583.731,1279.59 \n 585.785,1277.23 587.84,1274.86 589.895,1272.48 591.949,1270.1 594.004,1267.7 596.058,1265.29 598.113,1262.87 600.168,1260.44 602.222,1258.01 604.277,1255.56 \n 606.332,1253.11 608.386,1250.64 610.441,1248.17 612.495,1245.7 614.55,1243.21 616.605,1240.72 618.659,1238.22 620.714,1235.71 622.769,1233.2 624.823,1230.68 \n 626.878,1228.15 628.932,1225.62 630.987,1223.08 633.042,1220.54 635.096,1218 637.151,1215.45 639.205,1212.89 641.26,1210.34 643.315,1207.77 645.369,1205.21 \n 647.424,1202.64 649.479,1200.07 651.533,1197.5 653.588,1194.93 655.642,1192.35 657.697,1189.77 659.752,1187.19 661.806,1184.61 663.861,1182.03 665.915,1179.45 \n 667.97,1176.88 670.025,1174.3 672.079,1171.72 674.134,1169.14 676.189,1166.56 678.243,1163.99 680.298,1161.42 682.352,1158.85 684.407,1156.28 686.462,1153.72 \n 688.516,1151.16 690.571,1148.6 692.625,1146.04 694.68,1143.49 696.735,1140.95 698.789,1138.41 700.844,1135.87 702.899,1133.34 704.953,1130.82 707.008,1128.3 \n 709.062,1125.79 711.117,1123.28 713.172,1120.79 715.226,1118.29 717.281,1115.81 719.335,1113.33 721.39,1110.87 723.445,1108.41 725.499,1105.96 727.554,1103.51 \n 729.609,1101.08 731.663,1098.66 733.718,1096.24 735.772,1093.84 737.827,1091.45 739.882,1089.06 741.936,1086.69 743.991,1084.33 746.046,1081.98 748.1,1079.65 \n 750.155,1077.32 752.209,1075.01 754.264,1072.71 756.319,1070.42 758.373,1068.14 760.428,1065.88 762.482,1063.63 764.537,1061.4 766.592,1059.18 768.646,1056.97 \n 770.701,1054.78 772.756,1052.6 774.81,1050.44 776.865,1048.3 778.919,1046.16 780.974,1044.05 783.029,1041.95 785.083,1039.86 787.138,1037.8 789.192,1035.75 \n 791.247,1033.71 793.302,1031.69 795.356,1029.69 797.411,1027.71 799.466,1025.74 801.52,1023.8 803.575,1021.87 805.629,1019.95 807.684,1018.06 809.739,1016.18 \n 811.793,1014.33 813.848,1012.49 815.902,1010.67 817.957,1008.87 820.012,1007.09 822.066,1005.32 824.121,1003.58 826.176,1001.86 828.23,1000.16 830.285,998.472 \n 832.339,996.808 834.394,995.165 836.449,993.541 838.503,991.938 840.558,990.355 842.613,988.793 844.667,987.251 846.722,985.73 848.776,984.23 850.831,982.751 \n 852.886,981.293 854.94,979.855 856.995,978.439 859.049,977.044 861.104,975.669 863.159,974.317 865.213,972.985 867.268,971.675 869.323,970.386 871.377,969.118 \n 873.432,967.872 875.486,966.647 877.541,965.444 879.596,964.262 881.65,963.102 883.705,961.963 885.759,960.845 887.814,959.749 889.869,958.674 891.923,957.621 \n 893.978,956.589 896.033,955.578 898.087,954.589 900.142,953.621 902.196,952.674 904.251,951.748 906.306,950.844 908.36,949.96 910.415,949.097 912.469,948.255 \n 914.524,947.434 916.579,946.634 918.633,945.854 920.688,945.095 922.743,944.356 924.797,943.638 926.852,942.94 928.906,942.262 930.961,941.603 933.016,940.965 \n 935.07,940.346 937.125,939.747 939.18,939.168 941.234,938.607 943.289,938.066 945.343,937.544 947.398,937.04 949.453,936.555 951.507,936.089 953.562,935.641 \n 955.616,935.211 957.671,934.799 959.726,934.404 961.78,934.027 963.835,933.668 965.89,933.325 967.944,933 969.999,932.691 972.053,932.399 974.108,932.123 \n 976.163,931.863 978.217,931.618 980.272,931.389 982.326,931.176 984.381,930.977 986.436,930.794 988.49,930.625 990.545,930.47 992.6,930.329 994.654,930.202 \n 996.709,930.088 998.763,929.988 1000.82,929.901 1002.87,929.826 1004.93,929.763 1006.98,929.713 1009.04,929.674 1011.09,929.647 1013.15,929.631 1015.2,929.625 \n 1017.25,929.631 1019.31,929.646 1021.36,929.672 1023.42,929.707 1025.47,929.751 1027.53,929.804 1029.58,929.866 1031.64,929.936 1033.69,930.014 1035.75,930.1 \n 1037.8,930.193 1039.86,930.293 1041.91,930.399 1043.96,930.512 1046.02,930.63 1048.07,930.754 1050.13,930.883 1052.18,931.017 1054.24,931.156 1056.29,931.299 \n 1058.35,931.445 1060.4,931.595 1062.46,931.747 1064.51,931.903 1066.57,932.06 1068.62,932.22 1070.67,932.381 1072.73,932.543 1074.78,932.706 1076.84,932.869 \n 1078.89,933.032 1080.95,933.195 1083,933.358 1085.06,933.519 1087.11,933.678 1089.17,933.836 1091.22,933.992 1093.28,934.145 1095.33,934.294 1097.38,934.441 \n 1099.44,934.584 1101.49,934.722 1103.55,934.856 1105.6,934.985 1107.66,935.109 1109.71,935.227 1111.77,935.339 1113.82,935.445 1115.88,935.543 1117.93,935.635 \n 1119.99,935.719 1122.04,935.795 1124.1,935.863 1126.15,935.922 1128.2,935.972 1130.26,936.013 1132.31,936.043 1134.37,936.064 1136.42,936.074 1138.48,936.073 \n 1140.53,936.061 1142.59,936.038 1144.64,936.002 1146.7,935.954 1148.75,935.894 1150.81,935.82 1152.86,935.733 1154.91,935.633 1156.97,935.518 1159.02,935.389 \n 1161.08,935.245 1163.13,935.087 1165.19,934.912 1167.24,934.723 1169.3,934.517 1171.35,934.295 1173.41,934.056 1175.46,933.8 1177.52,933.527 1179.57,933.237 \n 1181.62,932.929 1183.68,932.602 1185.73,932.257 1187.79,931.894 1189.84,931.511 1191.9,931.109 1193.95,930.688 1196.01,930.247 1198.06,929.785 1200.12,929.304 \n 1202.17,928.802 1204.23,928.279 1206.28,927.735 1208.33,927.169 1210.39,926.583 1212.44,925.974 1214.5,925.344 1216.55,924.691 1218.61,924.016 1220.66,923.318 \n 1222.72,922.598 1224.77,921.855 1226.83,921.088 1228.88,920.298 1230.94,919.485 1232.99,918.648 1235.04,917.787 1237.1,916.902 1239.15,915.994 1241.21,915.06 \n 1243.26,914.103 1245.32,913.121 1247.37,912.114 1249.43,911.083 1251.48,910.027 1253.54,908.946 1255.59,907.839 1257.65,906.708 1259.7,905.552 1261.75,904.37 \n 1263.81,903.163 1265.86,901.931 1267.92,900.673 1269.97,899.389 1272.03,898.081 1274.08,896.746 1276.14,895.387 1278.19,894.001 1280.25,892.59 1282.3,891.154 \n 1284.36,889.692 1286.41,888.205 1288.46,886.692 1290.52,885.153 1292.57,883.59 1294.63,882.001 1296.68,880.386 1298.74,878.747 1300.79,877.082 1302.85,875.392 \n 1304.9,873.678 1306.96,871.938 1309.01,870.174 1311.07,868.385 1313.12,866.572 1315.17,864.734 1317.23,862.872 1319.28,860.986 1321.34,859.076 1323.39,857.142 \n 1325.45,855.185 1327.5,853.204 1329.56,851.2 1331.61,849.173 1333.67,847.123 1335.72,845.051 1337.78,842.956 1339.83,840.839 1341.88,838.7 1343.94,836.539 \n 1345.99,834.357 1348.05,832.154 1350.1,829.93 1352.16,827.685 1354.21,825.42 1356.27,823.135 1358.32,820.83 1360.38,818.506 1362.43,816.162 1364.49,813.8 \n 1366.54,811.419 1368.59,809.02 1370.65,806.603 1372.7,804.169 1374.76,801.718 1376.81,799.25 1378.87,796.765 1380.92,794.264 1382.98,791.748 1385.03,789.217 \n 1387.09,786.67 1389.14,784.109 1391.2,781.535 1393.25,778.946 1395.3,776.344 1397.36,773.73 1399.41,771.103 1401.47,768.464 1403.52,765.814 1405.58,763.153 \n 1407.63,760.481 1409.69,757.799 1411.74,755.108 1413.8,752.407 1415.85,749.698 1417.91,746.98 1419.96,744.255 1422.01,741.523 1424.07,738.784 1426.12,736.038 \n 1428.18,733.287 1430.23,730.531 1432.29,727.77 1434.34,725.004 1436.4,722.235 1438.45,719.463 1440.51,716.689 1442.56,713.912 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| }, | |
| "metadata": {} | |
| } | |
| ] | |
| }, | |
| { | |
| "metadata": { | |
| "trusted": true | |
| }, | |
| "cell_type": "code", | |
| "source": "x = range(eps(BigFloat), big(π)/4; length=1001)\ny = 10^16 .* (S.(x) .- sin.(x))\nplot(x, y; label=\"\", title=\"10¹⁶ (S(x) - sin(x)) for BigFloat x in (0, π/4]\")", | |
| "execution_count": 20, | |
| "outputs": [ | |
| { | |
| "output_type": "execute_result", | |
| "execution_count": 20, | |
| "data": { | |
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851.411,231.977 \n 853.411,227.491 855.41,223.145 857.409,218.94 859.408,214.877 861.407,210.957 863.406,207.181 865.405,203.551 867.404,200.066 869.404,196.729 871.403,193.54 \n 873.402,190.501 875.401,187.611 877.4,184.872 879.399,182.285 881.398,179.851 883.397,177.569 885.397,175.442 887.396,173.469 889.395,171.651 891.394,169.99 \n 893.393,168.484 895.392,167.136 897.391,165.944 899.39,164.911 901.39,164.036 903.389,163.32 905.388,162.762 907.387,162.364 909.386,162.126 911.385,162.047 \n 913.384,162.128 915.384,162.369 917.383,162.771 919.382,163.332 921.381,164.054 923.38,164.936 925.379,165.979 927.378,167.181 929.377,168.544 931.377,170.066 \n 933.376,171.748 935.375,173.589 937.374,175.59 939.373,177.749 941.372,180.066 943.371,182.542 945.37,185.175 947.37,187.965 949.369,190.911 951.368,194.014 \n 953.367,197.272 955.366,200.684 957.365,204.251 959.364,207.97 961.363,211.842 963.363,215.866 965.362,220.041 967.361,224.366 969.36,228.84 971.359,233.462 \n 973.358,238.23 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| }, | |
| "metadata": {} | |
| } | |
| ] | |
| }, | |
| { | |
| "metadata": { | |
| "trusted": true | |
| }, | |
| "cell_type": "code", | |
| "source": "x = range(eps(BigFloat), big(π)/4; length=1001)\ny = 10^16 .* (sin_Float64.(x) .- sin.(x))\nplot(x, y; label=\"\", title=\"10¹⁶ (sin_Float64(x) - sin(x)) for BigFloat x in (0, π/4]\")", | |
| "execution_count": 21, | |
| "outputs": [ | |
| { | |
| "output_type": "execute_result", | |
| "execution_count": 21, | |
| "data": { | |
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657.973,682.557 659.998,681.929 662.023,681.321 664.048,680.732 666.072,680.163 668.097,679.615 670.122,679.088 \n 672.147,678.582 674.172,678.097 676.197,677.635 678.221,677.194 680.246,676.776 682.271,676.381 684.296,676.01 686.321,675.662 688.346,675.338 690.37,675.038 \n 692.395,674.764 694.42,674.514 696.445,674.289 698.47,674.09 700.494,673.917 702.519,673.771 704.544,673.651 706.569,673.558 708.594,673.492 710.619,673.453 \n 712.643,673.443 714.668,673.46 716.693,673.506 718.718,673.58 720.743,673.684 722.768,673.816 724.792,673.978 726.817,674.169 728.842,674.391 730.867,674.642 \n 732.892,674.924 734.917,675.237 736.941,675.58 738.966,675.954 740.991,676.36 743.016,676.797 745.041,677.266 747.066,677.766 749.09,678.299 751.115,678.863 \n 753.14,679.46 755.165,680.09 757.19,680.752 759.215,681.447 761.239,682.174 763.264,682.935 765.289,683.729 767.314,684.556 769.339,685.417 771.364,686.311 \n 773.388,687.238 775.413,688.199 777.438,689.194 779.463,690.223 781.488,691.285 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909.052,821.879 911.077,824.78 913.102,827.698 \n 915.126,830.632 917.151,833.582 919.176,836.546 921.201,839.525 923.226,842.518 925.251,845.523 927.275,848.541 929.3,851.571 931.325,854.612 933.35,857.664 \n 935.375,860.725 937.4,863.796 939.424,866.875 941.449,869.962 943.474,873.056 945.499,876.156 947.524,879.263 949.549,882.374 951.573,885.489 953.598,888.609 \n 955.623,891.731 957.648,894.855 959.673,897.98 961.698,901.106 963.722,904.232 965.747,907.358 967.772,910.481 969.797,913.602 971.822,916.72 973.847,919.834 \n 975.871,922.943 977.896,926.046 979.921,929.143 981.946,932.233 983.971,935.315 985.996,938.388 988.02,941.452 990.045,944.505 992.07,947.547 994.095,950.577 \n 996.12,953.594 998.145,956.597 1000.17,959.586 1002.19,962.559 1004.22,965.516 1006.24,968.456 1008.27,971.379 1010.29,974.282 1012.32,977.166 1014.34,980.03 \n 1016.37,982.872 1018.39,985.692 1020.42,988.49 1022.44,991.263 1024.47,994.012 1026.49,996.736 1028.52,999.433 1030.54,1002.1 1032.57,1004.74 1034.59,1007.36 \n 1036.62,1009.94 1038.64,1012.49 1040.67,1015.02 1042.69,1017.5 1044.72,1019.96 1046.74,1022.38 1048.77,1024.77 1050.79,1027.12 1052.81,1029.44 1054.84,1031.72 \n 1056.86,1033.96 1058.89,1036.16 1060.91,1038.32 1062.94,1040.44 1064.96,1042.53 1066.99,1044.56 1069.01,1046.56 1071.04,1048.51 1073.06,1050.42 1075.09,1052.28 \n 1077.11,1054.1 1079.14,1055.87 1081.16,1057.59 1083.19,1059.27 1085.21,1060.89 1087.24,1062.47 1089.26,1064 1091.29,1065.47 1093.31,1066.89 1095.34,1068.27 \n 1097.36,1069.58 1099.39,1070.85 1101.41,1072.06 1103.44,1073.21 1105.46,1074.31 1107.49,1075.35 1109.51,1076.34 1111.54,1077.26 1113.56,1078.13 1115.58,1078.94 \n 1117.61,1079.69 1119.63,1080.38 1121.66,1081.01 1123.68,1081.58 1125.71,1082.09 1127.73,1082.53 1129.76,1082.92 1131.78,1083.23 1133.81,1083.49 1135.83,1083.68 \n 1137.86,1083.81 1139.88,1083.87 1141.91,1083.86 1143.93,1083.79 1145.96,1083.65 1147.98,1083.45 1150.01,1083.18 1152.03,1082.84 1154.06,1082.43 1156.08,1081.96 \n 1158.11,1081.41 1160.13,1080.8 1162.16,1080.12 1164.18,1079.37 1166.21,1078.55 1168.23,1077.66 1170.26,1076.7 1172.28,1075.67 1174.3,1074.56 1176.33,1073.39 \n 1178.35,1072.15 1180.38,1070.83 1182.4,1069.45 1184.43,1067.99 1186.45,1066.47 1188.48,1064.87 1190.5,1063.2 1192.53,1061.46 1194.55,1059.65 1196.58,1057.76 \n 1198.6,1055.81 1200.63,1053.78 1202.65,1051.68 1204.68,1049.52 1206.7,1047.28 1208.73,1044.97 1210.75,1042.59 1212.78,1040.14 1214.8,1037.61 1216.83,1035.02 \n 1218.85,1032.36 1220.88,1029.63 1222.9,1026.83 1224.93,1023.96 1226.95,1021.03 1228.98,1018.02 1231,1014.95 1233.02,1011.81 1235.05,1008.6 1237.07,1005.33 \n 1239.1,1001.99 1241.12,998.582 1243.15,995.111 1245.17,991.575 1247.2,987.975 1249.22,984.312 1251.25,980.585 1253.27,976.796 1255.3,972.946 1257.32,969.034 \n 1259.35,965.062 1261.37,961.03 1263.4,956.939 1265.42,952.79 1267.45,948.583 1269.47,944.32 1271.5,940 1273.52,935.626 1275.55,931.197 1277.57,926.715 \n 1279.6,922.18 1281.62,917.593 1283.65,912.956 1285.67,908.269 1287.7,903.533 1289.72,898.75 1291.74,893.919 1293.77,889.043 1295.79,884.122 1297.82,879.158 \n 1299.84,874.151 1301.87,869.102 1303.89,864.014 1305.92,858.886 1307.94,853.72 1309.97,848.517 1311.99,843.279 1314.02,838.007 1316.04,832.701 1318.07,827.364 \n 1320.09,821.996 1322.12,816.599 1324.14,811.174 1326.17,805.722 1328.19,800.245 1330.22,794.745 1332.24,789.221 1334.27,783.677 1336.29,778.114 1338.32,772.532 \n 1340.34,766.933 1342.37,761.32 1344.39,755.692 1346.42,750.053 1348.44,744.402 1350.46,738.743 1352.49,733.076 1354.51,727.402 1356.54,721.725 1358.56,716.044 \n 1360.59,710.362 1362.61,704.681 1364.64,699.001 1366.66,693.325 1368.69,687.655 1370.71,681.991 1372.74,676.336 1374.76,670.691 1376.79,665.058 1378.81,659.438 \n 1380.84,653.834 1382.86,648.247 1384.89,642.679 1386.91,637.131 1388.94,631.606 1390.96,626.105 1392.99,620.629 1395.01,615.181 1397.04,609.763 1399.06,604.375 \n 1401.09,599.02 1403.11,593.7 1405.14,588.416 1407.16,583.171 1409.19,577.965 1411.21,572.801 1413.23,567.681 1415.26,562.605 1417.28,557.577 1419.31,552.598 \n 1421.33,547.67 1423.36,542.794 1425.38,537.972 1427.41,533.207 1429.43,528.499 1431.46,523.85 1433.48,519.264 1435.51,514.74 1437.53,510.281 1439.56,505.888 \n 1441.58,501.564 1443.61,497.31 1445.63,493.128 1447.66,489.019 1449.68,484.986 1451.71,481.029 1453.73,477.151 1455.76,473.354 1457.78,469.638 1459.81,466.006 \n 1461.83,462.459 1463.86,459 1465.88,455.628 1467.91,452.347 1469.93,449.157 1471.95,446.061 1473.98,443.059 1476,440.154 1478.03,437.346 1480.05,434.638 \n 1482.08,432.03 1484.1,429.525 1486.13,427.123 1488.15,424.827 1490.18,422.637 1492.2,420.554 1494.23,418.581 1496.25,416.718 1498.28,414.967 1500.3,413.329 \n 1502.33,411.805 1504.35,410.396 1506.38,409.105 1508.4,407.93 1510.43,406.875 1512.45,405.94 1514.48,405.126 1516.5,404.434 1518.53,403.864 1520.55,403.419 \n 1522.58,403.099 1524.6,402.905 1526.63,402.837 1528.65,402.897 1530.67,403.085 1532.7,403.403 1534.72,403.85 1536.75,404.428 1538.77,405.137 1540.8,405.977 \n 1542.82,406.95 1544.85,408.056 1546.87,409.295 1548.9,410.667 1550.92,412.174 1552.95,413.816 1554.97,415.592 1557,417.503 1559.02,419.55 1561.05,421.732 \n 1563.07,424.05 1565.1,426.504 1567.12,429.093 1569.15,431.818 1571.17,434.679 1573.2,437.676 1575.22,440.808 1577.25,444.075 1579.27,447.478 1581.3,451.015 \n 1583.32,454.687 1585.35,458.492 1587.37,462.431 1589.39,466.504 1591.42,470.709 1593.44,475.046 1595.47,479.514 1597.49,484.113 1599.52,488.841 1601.54,493.699 \n 1603.57,498.685 1605.59,503.798 1607.62,509.038 1609.64,514.402 1611.67,519.891 1613.69,525.503 1615.72,531.237 1617.74,537.091 1619.77,543.065 1621.79,549.157 \n 1623.82,555.365 1625.84,561.689 1627.87,568.125 1629.89,574.674 1631.92,581.334 1633.94,588.101 1635.97,594.976 1637.99,601.956 1640.02,609.039 1642.04,616.224 \n 1644.07,623.507 1646.09,630.889 1648.12,638.365 1650.14,645.934 1652.16,653.595 1654.19,661.344 1656.21,669.179 1658.24,677.099 1660.26,685.1 1662.29,693.18 \n 1664.31,701.336 1666.34,709.567 1668.36,717.869 1670.39,726.24 1672.41,734.677 1674.44,743.177 1676.46,751.737 1678.49,760.355 1680.51,769.027 1682.54,777.751 \n 1684.56,786.523 1686.59,795.341 1688.61,804.2 1690.64,813.099 1692.66,822.034 1694.69,831.002 1696.71,839.998 1698.74,849.021 1700.76,858.066 1702.79,867.13 \n 1704.81,876.21 1706.84,885.301 1708.86,894.401 1710.88,903.506 1712.91,912.612 1714.93,921.715 1716.96,930.812 1718.98,939.899 1721.01,948.972 1723.03,958.027 \n 1725.06,967.061 1727.08,976.07 1729.11,985.049 1731.13,993.995 1733.16,1002.9 1735.18,1011.77 1737.21,1020.59 1739.23,1029.36 1741.26,1038.08 1743.28,1046.75 \n 1745.31,1055.34 1747.33,1063.88 1749.36,1072.34 1751.38,1080.73 1753.41,1089.04 1755.43,1097.27 1757.46,1105.42 1759.48,1113.47 1761.51,1121.43 1763.53,1129.28 \n 1765.56,1137.04 1767.58,1144.68 1769.6,1152.22 1771.63,1159.64 1773.65,1166.94 1775.68,1174.11 1777.7,1181.16 1779.73,1188.07 1781.75,1194.85 1783.78,1201.48 \n 1785.8,1207.97 1787.83,1214.31 1789.85,1220.5 1791.88,1226.53 1793.9,1232.4 1795.93,1238.1 1797.95,1243.64 1799.98,1249 1802,1254.18 1804.03,1259.19 \n 1806.05,1264.01 1808.08,1268.64 1810.1,1273.07 1812.13,1277.32 1814.15,1281.36 1816.18,1285.2 1818.2,1288.83 1820.23,1292.25 1822.25,1295.46 1824.28,1298.46 \n 1826.3,1301.23 1828.32,1303.78 1830.35,1306.1 1832.37,1308.2 1834.4,1310.06 1836.42,1311.69 1838.45,1313.08 1840.47,1314.23 1842.5,1315.13 1844.52,1315.79 \n 1846.55,1316.2 1848.57,1316.36 1850.6,1316.27 1852.62,1315.93 1854.65,1315.32 1856.67,1314.46 1858.7,1313.34 1860.72,1311.95 1862.75,1310.31 1864.77,1308.39 \n 1866.8,1306.21 1868.82,1303.76 1870.85,1301.04 1872.87,1298.06 1874.9,1294.8 1876.92,1291.27 1878.95,1287.46 1880.97,1283.39 1883,1279.04 1885.02,1274.42 \n 1887.04,1269.53 1889.07,1264.37 1891.09,1258.93 1893.12,1253.22 1895.14,1247.24 1897.17,1240.99 1899.19,1234.47 1901.22,1227.68 1903.24,1220.63 1905.27,1213.31 \n 1907.29,1205.73 1909.32,1197.89 1911.34,1189.78 1913.37,1181.42 1915.39,1172.81 1917.42,1163.94 1919.44,1154.82 1921.47,1145.46 1923.49,1135.85 1925.52,1126 \n 1927.54,1115.92 1929.57,1105.6 1931.59,1095.06 1933.62,1084.29 1935.64,1073.3 1937.67,1062.09 1939.69,1050.67 1941.72,1039.05 1943.74,1027.23 1945.77,1015.21 \n 1947.79,1003 1949.81,990.61 1951.84,978.04 1953.86,965.299 1955.89,952.393 1957.91,939.328 1959.94,926.111 1961.96,912.75 1963.99,899.252 1966.01,885.624 \n 1968.04,871.874 1970.06,858.011 1972.09,844.041 1974.11,829.974 1976.14,815.818 1978.16,801.583 1980.19,787.276 1982.21,772.907 1984.24,758.485 1986.26,744.021 \n 1988.29,729.524 1990.31,715.004 1992.34,700.471 1994.36,685.935 1996.39,671.407 1998.41,656.899 2000.44,642.42 2002.46,627.982 2004.49,613.597 2006.51,599.275 \n 2008.53,585.028 2010.56,570.869 2012.58,556.809 2014.61,542.86 2016.63,529.035 2018.66,515.345 2020.68,501.804 2022.71,488.424 2024.73,475.218 2026.76,462.199 \n 2028.78,449.38 2030.81,436.773 2032.83,424.393 2034.86,412.252 2036.88,400.364 2038.91,388.742 2040.93,377.399 2042.96,366.35 2044.98,355.607 2047.01,345.184 \n 2049.03,335.096 2051.06,325.354 2053.08,315.974 2055.11,306.968 2057.13,298.35 2059.16,290.134 2061.18,282.333 2063.21,274.959 2065.23,268.028 2067.25,261.551 \n 2069.28,255.541 2071.3,250.012 2073.33,244.975 2075.35,240.445 2077.38,236.432 2079.4,232.95 2081.43,230.009 2083.45,227.622 2085.48,225.799 2087.5,224.553 \n 2089.53,223.893 2091.55,223.831 2093.58,224.375 2095.6,225.536 2097.63,227.324 2099.65,229.746 2101.68,232.811 2103.7,236.528 2105.73,240.902 2107.75,245.942 \n 2109.78,251.654 2111.8,258.042 2113.83,265.113 2115.85,272.87 2117.88,281.317 2119.9,290.457 2121.93,300.292 2123.95,310.824 2125.97,322.053 2128,333.978 \n 2130.02,346.599 2132.05,359.913 2134.07,373.917 2136.1,388.607 2138.12,403.978 2140.15,420.023 2142.17,436.734 2144.2,454.103 2146.22,472.121 2148.25,490.775 \n 2150.27,510.053 2152.3,529.941 2154.32,550.424 2156.35,571.485 2158.37,593.105 2160.4,615.264 2162.42,637.942 2164.45,661.113 2166.47,684.753 2168.5,708.836 \n 2170.52,733.331 2172.55,758.21 2174.57,783.438 2176.6,808.98 2178.62,834.801 2180.65,860.86 2182.67,887.116 2184.7,913.525 2186.72,940.042 2188.74,966.617 \n 2190.77,993.199 2192.79,1019.73 2194.82,1046.17 2196.84,1072.43 2198.87,1098.48 2200.89,1124.23 2202.92,1149.63 2204.94,1174.59 2206.97,1199.05 2208.99,1222.93 \n 2211.02,1246.14 2213.04,1268.61 2215.07,1290.24 2217.09,1310.94 2219.12,1330.61 2221.14,1349.16 2223.17,1366.48 2225.19,1382.47 2227.22,1397 2229.24,1409.98 \n 2231.27,1421.26 2233.29,1430.74 2235.32,1438.28 2237.34,1443.74 2239.37,1446.99 2241.39,1447.87 2243.42,1446.25 2245.44,1441.97 2247.46,1434.86 2249.49,1424.76 \n 2251.51,1411.5 2253.54,1394.9 2255.56,1374.78 2257.59,1350.95 2259.61,1323.21 2261.64,1291.37 2263.66,1255.2 2265.69,1214.51 2267.71,1169.06 2269.74,1118.62 \n 2271.76,1062.97 2273.79,1001.86 2275.81,935.03 2277.84,862.232 2279.86,783.196 2281.89,697.648 2283.91,605.306 2285.94,505.878 2287.96,399.066 2289.99,284.561 \n 2292.01,162.047 \n \"/>\n</svg>\n" | |
| }, | |
| "metadata": {} | |
| } | |
| ] | |
| }, | |
| { | |
| "metadata": { | |
| "trusted": true | |
| }, | |
| "cell_type": "code", | |
| "source": "x = range(eps(BigFloat), big(π)/4; length=1001)\n@show extrema((S64.(x) .- sin.(x)) ./ sin.(x))\n@show extrema((S.(x) .- sin.(x)) ./ sin.(x))\n@show extrema((sin_Float64.(x) .- sin.(x)) ./ sin.(x))\n;", | |
| "execution_count": 22, | |
| "outputs": [ | |
| { | |
| "output_type": "stream", | |
| "text": "extrema((S64.(x) .- sin.(x)) ./ sin.(x)) = (0.0, 2.890144982484280529488206685108906698594729179793535372587255855640231647975821e-17)\nextrema((S.(x) .- sin.(x)) ./ sin.(x)) = (-2.357883879902744111448885369911730756845765352309456734902401092423818885492969e-17, 5.246071091669231850361295500455225787555322581784755759323665643566342456694278e-18)\nextrema((sin_Float64.(x) .- sin.(x)) ./ sin.(x)) = (-3.847842345440139437818398333331234612529064011915260797872510610933335926744214e-18, 3.78229315815839727130921771273231327900685383833531492053827666419786601539418e-18)\n", | |
| "name": "stdout" | |
| } | |
| ] | |
| }, | |
| { | |
| "metadata": { | |
| "trusted": true | |
| }, | |
| "cell_type": "code", | |
| "source": "x = range(eps(BigFloat), big(π)/4; length=1001)\ny = 10^16 .* (S64.(x) .- sin.(x)) ./ sin.(x)\nplot(x, y; label=\"\", title=\"10¹⁶ (S64(x) - sin(x))/sin(x) for BigFloat x in (0, π/4]\")", | |
| "execution_count": 23, | |
| "outputs": [ | |
| { | |
| "output_type": "execute_result", | |
| "execution_count": 23, | |
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252.937,1445.37 254.992,1444.71 \n 257.047,1443.97 259.101,1443.15 261.156,1442.25 263.211,1441.28 265.265,1440.23 267.32,1439.1 269.374,1437.89 271.429,1436.61 273.484,1435.25 275.538,1433.81 \n 277.593,1432.3 279.647,1430.71 281.702,1429.05 283.757,1427.31 285.811,1425.49 287.866,1423.6 289.921,1421.63 291.975,1419.59 294.03,1417.47 296.084,1415.28 \n 298.139,1413.02 300.194,1410.68 302.248,1408.27 304.303,1405.79 306.358,1403.23 308.412,1400.6 310.467,1397.9 312.521,1395.12 314.576,1392.28 316.631,1389.36 \n 318.685,1386.38 320.74,1383.32 322.794,1380.19 324.849,1377 326.904,1373.73 328.958,1370.4 331.013,1366.99 333.068,1363.52 335.122,1359.98 337.177,1356.38 \n 339.231,1352.71 341.286,1348.97 343.341,1345.17 345.395,1341.3 347.45,1337.37 349.504,1333.37 351.559,1329.31 353.614,1325.19 355.668,1321 357.723,1316.76 \n 359.778,1312.45 361.832,1308.08 363.887,1303.65 365.941,1299.16 367.996,1294.61 370.051,1290 372.105,1285.34 374.16,1280.62 376.214,1275.84 378.269,1271 \n 380.324,1266.11 382.378,1261.16 384.433,1256.16 386.488,1251.11 388.542,1246 390.597,1240.84 392.651,1235.63 394.706,1230.37 396.761,1225.06 398.815,1219.69 \n 400.87,1214.28 402.924,1208.82 404.979,1203.31 407.034,1197.76 409.088,1192.16 411.143,1186.51 413.198,1180.82 415.252,1175.09 417.307,1169.31 419.361,1163.49 \n 421.416,1157.62 423.471,1151.72 425.525,1145.77 427.58,1139.79 429.635,1133.76 431.689,1127.7 433.744,1121.6 435.798,1115.47 437.853,1109.29 439.908,1103.09 \n 441.962,1096.85 444.017,1090.57 446.071,1084.26 448.126,1077.92 450.181,1071.55 452.235,1065.15 454.29,1058.72 456.345,1052.26 458.399,1045.77 460.454,1039.26 \n 462.508,1032.72 464.563,1026.15 466.618,1019.56 468.672,1012.94 470.727,1006.3 472.781,999.642 474.836,992.96 476.891,986.257 478.945,979.534 481,972.792 \n 483.055,966.033 485.109,959.255 487.164,952.462 489.218,945.653 491.273,938.829 493.328,931.991 495.382,925.14 497.437,918.277 499.491,911.402 501.546,904.517 \n 503.601,897.622 505.655,890.719 507.71,883.807 509.765,876.888 511.819,869.963 513.874,863.032 515.928,856.097 517.983,849.157 520.038,842.215 522.092,835.271 \n 524.147,828.325 526.202,821.379 528.256,814.434 530.311,807.489 532.365,800.547 534.42,793.608 536.475,786.672 538.529,779.741 540.584,772.815 542.638,765.896 \n 544.693,758.983 546.748,752.079 548.802,745.183 550.857,738.297 552.912,731.421 554.966,724.556 557.021,717.703 559.075,710.863 561.13,704.036 563.185,697.224 \n 565.239,690.427 567.294,683.647 569.348,676.882 571.403,670.136 573.458,663.408 575.512,656.699 577.567,650.01 579.622,643.341 581.676,636.694 583.731,630.069 \n 585.785,623.468 587.84,616.89 589.895,610.336 591.949,603.808 594.004,597.306 596.058,590.83 598.113,584.382 600.168,577.962 602.222,571.571 604.277,565.21 \n 606.332,558.879 608.386,552.579 610.441,546.311 612.495,540.075 614.55,533.873 616.605,527.704 618.659,521.57 620.714,515.471 622.769,509.408 624.823,503.381 \n 626.878,497.392 628.932,491.44 630.987,485.527 633.042,479.653 635.096,473.818 637.151,468.024 639.205,462.271 641.26,456.56 643.315,450.89 645.369,445.263 \n 647.424,439.68 649.479,434.141 651.533,428.646 653.588,423.196 655.642,417.792 657.697,412.433 659.752,407.122 661.806,401.858 663.861,396.641 665.915,391.473 \n 667.97,386.353 670.025,381.283 672.079,376.263 674.134,371.292 676.189,366.373 678.243,361.505 680.298,356.688 682.352,351.923 684.407,347.211 686.462,342.552 \n 688.516,337.946 690.571,333.394 692.625,328.896 694.68,324.453 696.735,320.065 698.789,315.732 700.844,311.455 702.899,307.234 704.953,303.069 707.008,298.961 \n 709.062,294.91 711.117,290.917 713.172,286.981 715.226,283.104 717.281,279.285 719.335,275.524 721.39,271.822 723.445,268.18 725.499,264.597 727.554,261.073 \n 729.609,257.61 731.663,254.207 733.718,250.864 735.772,247.581 737.827,244.359 739.882,241.199 741.936,238.099 743.991,235.061 746.046,232.084 748.1,229.169 \n 750.155,226.315 752.209,223.523 754.264,220.794 756.319,218.126 758.373,215.52 760.428,212.977 762.482,210.496 764.537,208.078 766.592,205.722 768.646,203.429 \n 770.701,201.198 772.756,199.03 774.81,196.924 776.865,194.881 778.919,192.9 780.974,190.983 783.029,189.127 785.083,187.335 787.138,185.604 789.192,183.937 \n 791.247,182.331 793.302,180.788 795.356,179.307 797.411,177.889 799.466,176.532 801.52,175.237 803.575,174.004 805.629,172.833 807.684,171.723 809.739,170.675 \n 811.793,169.687 813.848,168.761 815.902,167.896 817.957,167.091 820.012,166.347 822.066,165.663 824.121,165.039 826.176,164.475 828.23,163.971 830.285,163.525 \n 832.339,163.139 834.394,162.812 836.449,162.543 838.503,162.332 840.558,162.18 842.613,162.085 844.667,162.047 846.722,162.066 848.776,162.142 850.831,162.274 \n 852.886,162.463 854.94,162.707 856.995,163.006 859.049,163.36 861.104,163.768 863.159,164.231 865.213,164.747 867.268,165.317 869.323,165.94 871.377,166.615 \n 873.432,167.342 875.486,168.12 877.541,168.95 879.596,169.83 881.65,170.761 883.705,171.741 885.759,172.771 887.814,173.849 889.869,174.976 891.923,176.151 \n 893.978,177.372 896.033,178.641 898.087,179.956 900.142,181.316 902.196,182.722 904.251,184.172 906.306,185.666 908.36,187.204 910.415,188.784 912.469,190.408 \n 914.524,192.072 916.579,193.778 918.633,195.525 920.688,197.312 922.743,199.138 924.797,201.003 926.852,202.906 928.906,204.847 930.961,206.825 933.016,208.839 \n 935.07,210.888 937.125,212.973 939.18,215.092 941.234,217.245 943.289,219.431 945.343,221.65 947.398,223.9 949.453,226.182 951.507,228.493 953.562,230.835 \n 955.616,233.206 957.671,235.605 959.726,238.032 961.78,240.486 963.835,242.966 965.89,245.472 967.944,248.003 969.999,250.558 972.053,253.136 974.108,255.738 \n 976.163,258.361 978.217,261.006 980.272,263.672 982.326,266.358 984.381,269.062 986.436,271.786 988.49,274.527 990.545,277.285 992.6,280.059 994.654,282.85 \n 996.709,285.655 998.763,288.474 1000.82,291.306 1002.87,294.151 1004.93,297.008 1006.98,299.877 1009.04,302.755 1011.09,305.644 1013.15,308.541 1015.2,311.447 \n 1017.25,314.36 1019.31,317.28 1021.36,320.206 1023.42,323.137 1025.47,326.073 1027.53,329.013 1029.58,331.956 1031.64,334.901 1033.69,337.848 1035.75,340.795 \n 1037.8,343.743 1039.86,346.691 1041.91,349.637 1043.96,352.581 1046.02,355.522 1048.07,358.46 1050.13,361.393 1052.18,364.322 1054.24,367.245 1056.29,370.162 \n 1058.35,373.072 1060.4,375.974 1062.46,378.868 1064.51,381.753 1066.57,384.628 1068.62,387.492 1070.67,390.346 1072.73,393.187 1074.78,396.016 1076.84,398.832 \n 1078.89,401.633 1080.95,404.421 1083,407.193 1085.06,409.949 1087.11,412.689 1089.17,415.412 1091.22,418.117 1093.28,420.803 1095.33,423.47 1097.38,426.118 \n 1099.44,428.746 1101.49,431.352 1103.55,433.937 1105.6,436.5 1107.66,439.041 1109.71,441.558 1111.77,444.051 1113.82,446.52 1115.88,448.963 1117.93,451.381 \n 1119.99,453.773 1122.04,456.139 1124.1,458.477 1126.15,460.787 1128.2,463.069 1130.26,465.323 1132.31,467.547 1134.37,469.741 1136.42,471.905 1138.48,474.038 \n 1140.53,476.14 1142.59,478.21 1144.64,480.248 1146.7,482.254 1148.75,484.226 1150.81,486.165 1152.86,488.07 1154.91,489.94 1156.97,491.776 1159.02,493.576 \n 1161.08,495.341 1163.13,497.07 1165.19,498.763 1167.24,500.418 1169.3,502.037 1171.35,503.619 1173.41,505.162 1175.46,506.668 1177.52,508.135 1179.57,509.564 \n 1181.62,510.953 1183.68,512.304 1185.73,513.614 1187.79,514.885 1189.84,516.116 1191.9,517.307 1193.95,518.457 1196.01,519.566 1198.06,520.634 1200.12,521.661 \n 1202.17,522.647 1204.23,523.591 1206.28,524.493 1208.33,525.353 1210.39,526.171 1212.44,526.948 1214.5,527.681 1216.55,528.372 1218.61,529.021 1220.66,529.627 \n 1222.72,530.19 1224.77,530.711 1226.83,531.188 1228.88,531.623 1230.94,532.014 1232.99,532.363 1235.04,532.668 1237.1,532.931 1239.15,533.15 1241.21,533.327 \n 1243.26,533.46 1245.32,533.55 1247.37,533.598 1249.43,533.602 1251.48,533.564 1253.54,533.483 1255.59,533.359 1257.65,533.193 1259.7,532.985 1261.75,532.734 \n 1263.81,532.441 1265.86,532.106 1267.92,531.729 1269.97,531.31 1272.03,530.85 1274.08,530.349 1276.14,529.806 1278.19,529.223 1280.25,528.599 1282.3,527.934 \n 1284.36,527.23 1286.41,526.485 1288.46,525.701 1290.52,524.877 1292.57,524.014 1294.63,523.113 1296.68,522.172 1298.74,521.194 1300.79,520.178 1302.85,519.124 \n 1304.9,518.034 1306.96,516.906 1309.01,515.742 1311.07,514.542 1313.12,513.306 1315.17,512.035 1317.23,510.73 1319.28,509.39 1321.34,508.015 1323.39,506.608 \n 1325.45,505.167 1327.5,503.693 1329.56,502.188 1331.61,500.651 1333.67,499.082 1335.72,497.483 1337.78,495.853 1339.83,494.194 1341.88,492.506 1343.94,490.789 \n 1345.99,489.044 1348.05,487.272 1350.1,485.472 1352.16,483.646 1354.21,481.794 1356.27,479.917 1358.32,478.015 1360.38,476.089 1362.43,474.139 1364.49,472.167 \n 1366.54,470.172 1368.59,468.155 1370.65,466.118 1372.7,464.06 1374.76,461.982 1376.81,459.885 1378.87,457.77 1380.92,455.637 1382.98,453.487 1385.03,451.32 \n 1387.09,449.137 1389.14,446.94 1391.2,444.727 1393.25,442.501 1395.3,440.262 1397.36,438.011 1399.41,435.748 1401.47,433.474 1403.52,431.19 1405.58,428.896 \n 1407.63,426.593 1409.69,424.283 1411.74,421.965 1413.8,419.64 1415.85,417.309 1417.91,414.974 1419.96,412.634 1422.01,410.29 1424.07,407.943 1426.12,405.594 \n 1428.18,403.243 1430.23,400.892 1432.29,398.541 1434.34,396.19 1436.4,393.842 1438.45,391.495 1440.51,389.151 1442.56,386.811 1444.62,384.476 1446.67,382.146 \n 1448.72,379.822 1450.78,377.504 1452.83,375.195 1454.89,372.893 1456.94,370.6 1459,368.318 1461.05,366.045 1463.11,363.784 1465.16,361.535 1467.22,359.298 \n 1469.27,357.075 1471.33,354.866 1473.38,352.671 1475.43,350.493 1477.49,348.33 1479.54,346.185 1481.6,344.057 1483.65,341.948 1485.71,339.857 1487.76,337.787 \n 1489.82,335.737 1491.87,333.708 1493.93,331.701 1495.98,329.717 1498.04,327.756 1500.09,325.818 1502.14,323.905 1504.2,322.017 1506.25,320.155 1508.31,318.32 \n 1510.36,316.511 1512.42,314.73 1514.47,312.977 1516.53,311.254 1518.58,309.559 1520.64,307.895 1522.69,306.261 1524.75,304.658 1526.8,303.087 1528.85,301.549 \n 1530.91,300.043 1532.96,298.57 1535.02,297.131 1537.07,295.727 1539.13,294.357 1541.18,293.023 1543.24,291.725 1545.29,290.463 1547.35,289.237 1549.4,288.049 \n 1551.46,286.898 1553.51,285.785 1555.56,284.711 1557.62,283.675 1559.67,282.679 1561.73,281.722 1563.78,280.805 1565.84,279.929 1567.89,279.092 1569.95,278.297 \n 1572,277.543 1574.06,276.83 1576.11,276.159 1578.17,275.53 1580.22,274.943 1582.27,274.399 1584.33,273.897 1586.38,273.438 1588.44,273.022 1590.49,272.65 \n 1592.55,272.32 1594.6,272.035 1596.66,271.792 1598.71,271.594 1600.77,271.439 1602.82,271.328 1604.88,271.262 1606.93,271.239 1608.98,271.26 1611.04,271.325 \n 1613.09,271.434 1615.15,271.587 1617.2,271.784 1619.26,272.025 1621.31,272.31 1623.37,272.639 1625.42,273.011 1627.48,273.427 1629.53,273.886 1631.59,274.388 \n 1633.64,274.933 1635.69,275.521 1637.75,276.152 1639.8,276.825 1641.86,277.54 1643.91,278.297 1645.97,279.095 1648.02,279.935 1650.08,280.816 1652.13,281.737 \n 1654.19,282.699 1656.24,283.701 1658.3,284.742 1660.35,285.822 1662.4,286.941 1664.46,288.099 1666.51,289.294 1668.57,290.526 1670.62,291.796 1672.68,293.101 \n 1674.73,294.443 1676.79,295.82 1678.84,297.231 1680.9,298.677 1682.95,300.156 1685.01,301.668 1687.06,303.213 1689.11,304.789 1691.17,306.396 1693.22,308.034 \n 1695.28,309.701 1697.33,311.398 1699.39,313.122 1701.44,314.874 1703.5,316.653 1705.55,318.458 1707.61,320.288 1709.66,322.143 1711.72,324.021 1713.77,325.922 \n 1715.82,327.845 1717.88,329.789 1719.93,331.754 1721.99,333.737 1724.04,335.74 1726.1,337.76 1728.15,339.796 1730.21,341.848 1732.26,343.915 1734.32,345.996 \n 1736.37,348.09 1738.43,350.196 1740.48,352.312 1742.53,354.439 1744.59,356.574 1746.64,358.718 1748.7,360.868 1750.75,363.024 1752.81,365.185 1754.86,367.35 \n 1756.92,369.517 1758.97,371.686 1761.03,373.856 1763.08,376.025 1765.14,378.193 1767.19,380.358 1769.24,382.519 1771.3,384.675 1773.35,386.825 1775.41,388.969 \n 1777.46,391.104 1779.52,393.229 1781.57,395.344 1783.63,397.448 1785.68,399.539 1787.74,401.616 1789.79,403.678 1791.85,405.724 1793.9,407.753 1795.95,409.764 \n 1798.01,411.755 1800.06,413.726 1802.12,415.675 1804.17,417.601 1806.23,419.504 1808.28,421.381 1810.34,423.232 1812.39,425.056 1814.45,426.852 1816.5,428.619 \n 1818.56,430.355 1820.61,432.059 1822.66,433.731 1824.72,435.369 1826.77,436.973 1828.83,438.541 1830.88,440.072 1832.94,441.566 1834.99,443.021 1837.05,444.436 \n 1839.1,445.811 1841.16,447.144 1843.21,448.434 1845.27,449.681 1847.32,450.883 1849.37,452.041 1851.43,453.152 1853.48,454.216 1855.54,455.232 1857.59,456.2 \n 1859.65,457.118 1861.7,457.986 1863.76,458.803 1865.81,459.569 1867.87,460.282 1869.92,460.941 1871.98,461.547 1874.03,462.099 1876.08,462.596 1878.14,463.037 \n 1880.19,463.422 1882.25,463.75 1884.3,464.021 1886.36,464.235 1888.41,464.39 1890.47,464.487 1892.52,464.525 1894.58,464.504 1896.63,464.423 1898.69,464.283 \n 1900.74,464.082 1902.79,463.821 1904.85,463.5 1906.9,463.118 1908.96,462.676 1911.01,462.172 1913.07,461.608 1915.12,460.983 1917.18,460.298 1919.23,459.551 \n 1921.29,458.744 1923.34,457.877 1925.4,456.95 1927.45,455.962 1929.5,454.915 1931.56,453.809 1933.61,452.643 1935.67,451.419 1937.72,450.137 1939.78,448.798 \n 1941.83,447.401 1943.89,445.948 1945.94,444.439 1948,442.875 1950.05,441.257 1952.11,439.585 1954.16,437.86 1956.21,436.083 1958.27,434.256 1960.32,432.378 \n 1962.38,430.451 1964.43,428.476 1966.49,426.454 1968.54,424.387 1970.6,422.275 1972.65,420.119 1974.71,417.922 1976.76,415.684 1978.82,413.407 1980.87,411.092 \n 1982.92,408.74 1984.98,406.354 1987.03,403.935 1989.09,401.484 1991.14,399.003 1993.2,396.493 1995.25,393.958 1997.31,391.397 1999.36,388.814 2001.42,386.211 \n 2003.47,383.588 2005.53,380.948 2007.58,378.293 2009.63,375.626 2011.69,372.948 2013.74,370.261 2015.8,367.568 2017.85,364.871 2019.91,362.172 2021.96,359.473 \n 2024.02,356.777 2026.07,354.087 2028.13,351.404 2030.18,348.731 2032.24,346.071 2034.29,343.426 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2291.12,278.38 \n \"/>\n</svg>\n" | |
| }, | |
| "metadata": {} | |
| } | |
| ] | |
| }, | |
| { | |
| "metadata": { | |
| "trusted": true | |
| }, | |
| "cell_type": "code", | |
| "source": "x = range(eps(BigFloat), big(π)/4; length=1001)\ny = 10^16 .* (S.(x) .- sin.(x)) ./ sin.(x)\nplot(x, y; label=\"\", title=\"10¹⁶ (S(x) - sin(x))/sin(x) for BigFloat x in (0, π/4]\")", | |
| "execution_count": 24, | |
| "outputs": [ | |
| { | |
| "output_type": "execute_result", | |
| "execution_count": 24, | |
| "data": { | |
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733.066,293.902 735.091,289.917 737.116,285.989 739.14,282.119 741.165,278.308 743.19,274.556 745.214,270.862 747.239,267.229 749.263,263.655 751.288,260.141 \n 753.313,256.687 755.337,253.293 757.362,249.96 759.386,246.688 761.411,243.477 763.436,240.327 765.46,237.238 767.485,234.211 769.509,231.245 771.534,228.342 \n 773.559,225.5 775.583,222.721 777.608,220.003 779.633,217.349 781.657,214.756 783.682,212.226 785.706,209.759 787.731,207.354 789.756,205.012 791.78,202.733 \n 793.805,200.516 795.829,198.362 797.854,196.272 799.879,194.244 801.903,192.279 803.928,190.377 805.952,188.537 807.977,186.761 810.002,185.047 812.026,183.396 \n 814.051,181.807 816.076,180.281 818.1,178.818 820.125,177.416 822.149,176.078 824.174,174.801 826.199,173.586 828.223,172.434 830.248,171.343 832.272,170.313 \n 834.297,169.345 836.322,168.439 838.346,167.593 840.371,166.809 842.396,166.085 844.42,165.421 846.445,164.818 848.469,164.275 850.494,163.792 852.519,163.368 \n 854.543,163.004 856.568,162.698 858.592,162.452 860.617,162.264 862.642,162.134 864.666,162.061 866.691,162.047 868.715,162.09 870.74,162.189 872.765,162.345 \n 874.789,162.558 876.814,162.826 878.839,163.149 880.863,163.528 882.888,163.962 884.912,164.45 886.937,164.991 888.962,165.587 890.986,166.235 893.011,166.936 \n 895.035,167.69 897.06,168.495 899.085,169.351 901.109,170.259 903.134,171.217 905.158,172.224 907.183,173.282 909.208,174.388 911.232,175.543 913.257,176.745 \n 915.282,177.996 917.306,179.293 919.331,180.636 921.355,182.026 923.38,183.461 925.405,184.94 927.429,186.464 929.454,188.032 931.478,189.643 933.503,191.296 \n 935.528,192.991 937.552,194.728 939.577,196.506 941.601,198.323 943.626,200.181 945.651,202.077 947.675,204.012 949.7,205.984 951.725,207.994 953.749,210.04 \n 955.774,212.122 957.798,214.239 959.823,216.391 961.848,218.577 963.872,220.796 965.897,223.048 967.921,225.331 969.946,227.646 971.971,229.992 973.995,232.367 \n 976.02,234.772 978.045,237.205 980.069,239.667 982.094,242.155 984.118,244.67 986.143,247.211 988.168,249.776 990.192,252.367 992.217,254.98 994.241,257.617 \n 996.266,260.276 998.291,262.957 1000.32,265.658 1002.34,268.38 1004.36,271.121 1006.39,273.88 1008.41,276.657 1010.44,279.452 1012.46,282.263 1014.49,285.09 \n 1016.51,287.932 1018.54,290.788 1020.56,293.657 1022.59,296.54 1024.61,299.434 1026.64,302.34 1028.66,305.256 1030.68,308.182 1032.71,311.117 1034.73,314.061 \n 1036.76,317.012 1038.78,319.97 1040.81,322.934 1042.83,325.904 1044.86,328.878 1046.88,331.856 1048.91,334.837 1050.93,337.821 1052.96,340.807 1054.98,343.793 \n 1057,346.78 1059.03,349.766 1061.05,352.751 1063.08,355.734 1065.1,358.715 1067.13,361.692 1069.15,364.665 1071.18,367.633 1073.2,370.596 1075.23,373.553 \n 1077.25,376.502 1079.28,379.444 1081.3,382.378 1083.32,385.302 1085.35,388.217 1087.37,391.121 1089.4,394.015 1091.42,396.896 1093.45,399.765 1095.47,402.621 \n 1097.5,405.463 1099.52,408.291 1101.55,411.103 1103.57,413.9 1105.6,416.68 1107.62,419.443 1109.64,422.188 1111.67,424.915 1113.69,427.623 1115.72,430.312 \n 1117.74,432.98 1119.77,435.627 1121.79,438.253 1123.82,440.856 1125.84,443.437 1127.87,445.995 1129.89,448.529 1131.91,451.039 1133.94,453.523 1135.96,455.982 \n 1137.99,458.415 1140.01,460.821 1142.04,463.2 1144.06,465.552 1146.09,467.875 1148.11,470.169 1150.14,472.434 1152.16,474.669 1154.19,476.874 1156.21,479.048 \n 1158.23,481.191 1160.26,483.302 1162.28,485.381 1164.31,487.427 1166.33,489.441 1168.36,491.42 1170.38,493.366 1172.41,495.277 1174.43,497.154 1176.46,498.995 \n 1178.48,500.801 1180.51,502.571 1182.53,504.305 1184.55,506.001 1186.58,507.661 1188.6,509.283 1190.63,510.868 1192.65,512.414 1194.68,513.923 1196.7,515.392 \n 1198.73,516.822 1200.75,518.213 1202.78,519.565 1204.8,520.877 1206.83,522.148 1208.85,523.379 1210.87,524.57 1212.9,525.72 1214.92,526.829 1216.95,527.896 \n 1218.97,528.923 1221,529.907 1223.02,530.85 1225.05,531.751 1227.07,532.61 1229.1,533.426 1231.12,534.2 1233.15,534.932 1235.17,535.621 1237.19,536.268 \n 1239.22,536.871 1241.24,537.432 1243.27,537.95 1245.29,538.425 1247.32,538.857 1249.34,539.245 1251.37,539.591 1253.39,539.894 1255.42,540.153 1257.44,540.37 \n 1259.47,540.543 1261.49,540.674 1263.51,540.761 1265.54,540.806 1267.56,540.808 1269.59,540.767 1271.61,540.683 1273.64,540.557 1275.66,540.388 1277.69,540.177 \n 1279.71,539.924 1281.74,539.629 1283.76,539.292 1285.79,538.913 1287.81,538.493 1289.83,538.032 1291.86,537.529 1293.88,536.985 1295.91,536.401 1297.93,535.776 \n 1299.96,535.111 1301.98,534.406 1304.01,533.661 1306.03,532.877 1308.06,532.054 1310.08,531.192 1312.11,530.292 1314.13,529.353 1316.15,528.377 1318.18,527.363 \n 1320.2,526.311 1322.23,525.223 1324.25,524.099 1326.28,522.939 1328.3,521.743 1330.33,520.511 1332.35,519.245 1334.38,517.944 1336.4,516.61 1338.43,515.242 \n 1340.45,513.841 1342.47,512.407 1344.5,510.941 1346.52,509.443 1348.55,507.914 1350.57,506.355 1352.6,504.765 1354.62,503.145 1356.65,501.497 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| }, | |
| "metadata": {} | |
| } | |
| ] | |
| }, | |
| { | |
| "metadata": { | |
| "trusted": true | |
| }, | |
| "cell_type": "code", | |
| "source": "x = range(eps(BigFloat), big(π)/4; length=1001)\ny = 10^16 .* (sin_Float64.(x) .- sin.(x)) ./ sin.(x)\nplot(x, y; label=\"\", title=\"10¹⁶ (sin_Float64(x) - sin(x))/sin(x) for BigFloat x in (0, π/4]\")", | |
| "execution_count": 25, | |
| "outputs": [ | |
| { | |
| "output_type": "execute_result", | |
| "execution_count": 25, | |
| "data": { | |
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867.717,677.491 869.74,685.262 871.763,693.062 873.786,700.891 \n 875.81,708.746 877.833,716.626 879.856,724.53 881.879,732.456 883.902,740.402 885.926,748.368 887.949,756.352 889.972,764.352 891.995,772.367 894.018,780.394 \n 896.042,788.434 898.065,796.484 900.088,804.542 902.111,812.607 904.135,820.678 906.158,828.752 908.181,836.829 910.204,844.907 912.227,852.984 914.251,861.058 \n 916.274,869.129 918.297,877.194 920.32,885.251 922.343,893.3 924.367,901.338 926.39,909.365 928.413,917.377 930.436,925.374 932.46,933.355 934.483,941.317 \n 936.506,949.258 938.529,957.178 940.552,965.075 942.576,972.946 944.599,980.791 946.622,988.608 948.645,996.394 950.668,1004.15 952.692,1011.87 954.715,1019.56 \n 956.738,1027.21 958.761,1034.82 960.785,1042.39 962.808,1049.93 964.831,1057.41 966.854,1064.86 968.877,1072.26 970.901,1079.61 972.924,1086.91 974.947,1094.16 \n 976.97,1101.35 978.993,1108.5 981.017,1115.58 983.04,1122.61 985.063,1129.59 987.086,1136.5 989.11,1143.34 991.133,1150.13 993.156,1156.85 995.179,1163.5 \n 997.202,1170.09 999.226,1176.61 1001.25,1183.05 1003.27,1189.42 1005.3,1195.72 1007.32,1201.95 1009.34,1208.09 1011.36,1214.16 1013.39,1220.15 1015.41,1226.05 \n 1017.43,1231.88 1019.46,1237.62 1021.48,1243.27 1023.5,1248.84 1025.53,1254.32 1027.55,1259.71 1029.57,1265.01 1031.6,1270.22 1033.62,1275.33 1035.64,1280.35 \n 1037.67,1285.27 1039.69,1290.09 1041.71,1294.82 1043.74,1299.45 1045.76,1303.97 1047.78,1308.4 1049.81,1312.72 1051.83,1316.94 1053.85,1321.05 1055.88,1325.05 \n 1057.9,1328.95 1059.92,1332.73 1061.95,1336.41 1063.97,1339.98 1065.99,1343.43 1068.01,1346.78 1070.04,1350 1072.06,1353.12 1074.08,1356.11 1076.11,1359 \n 1078.13,1361.76 1080.15,1364.4 1082.18,1366.93 1084.2,1369.34 1086.22,1371.62 1088.25,1373.78 1090.27,1375.83 1092.29,1377.74 1094.32,1379.54 1096.34,1381.21 \n 1098.36,1382.75 1100.39,1384.17 1102.41,1385.47 1104.43,1386.63 1106.46,1387.67 1108.48,1388.58 1110.5,1389.36 1112.53,1390.02 1114.55,1390.54 1116.57,1390.94 \n 1118.6,1391.2 1120.62,1391.34 1122.64,1391.34 1124.66,1391.21 1126.69,1390.96 1128.71,1390.57 1130.73,1390.04 1132.76,1389.39 1134.78,1388.61 1136.8,1387.69 \n 1138.83,1386.64 1140.85,1385.46 1142.87,1384.14 1144.9,1382.7 1146.92,1381.12 1148.94,1379.4 1150.97,1377.56 1152.99,1375.58 1155.01,1373.48 1157.04,1371.24 \n 1159.06,1368.87 1161.08,1366.36 1163.11,1363.73 1165.13,1360.97 1167.15,1358.07 1169.18,1355.05 1171.2,1351.89 1173.22,1348.61 1175.25,1345.2 1177.27,1341.66 \n 1179.29,1337.99 1181.31,1334.19 1183.34,1330.27 1185.36,1326.22 1187.38,1322.05 1189.41,1317.75 1191.43,1313.33 1193.45,1308.78 1195.48,1304.12 1197.5,1299.33 \n 1199.52,1294.41 1201.55,1289.38 1203.57,1284.23 1205.59,1278.97 1207.62,1273.58 1209.64,1268.08 1211.66,1262.46 1213.69,1256.73 1215.71,1250.89 1217.73,1244.93 \n 1219.76,1238.86 1221.78,1232.69 1223.8,1226.4 1225.83,1220.01 1227.85,1213.51 1229.87,1206.91 1231.9,1200.2 1233.92,1193.4 1235.94,1186.49 1237.96,1179.48 \n 1239.99,1172.37 1242.01,1165.17 1244.03,1157.87 1246.06,1150.48 1248.08,1143 1250.1,1135.43 1252.13,1127.77 1254.15,1120.02 1256.17,1112.19 1258.2,1104.27 \n 1260.22,1096.27 1262.24,1088.2 1264.27,1080.04 1266.29,1071.81 1268.31,1063.5 1270.34,1055.12 1272.36,1046.67 1274.38,1038.15 1276.41,1029.56 1278.43,1020.91 \n 1280.45,1012.19 1282.48,1003.42 1284.5,994.58 1286.52,985.687 1288.55,976.739 1290.57,967.738 1292.59,958.687 1294.61,949.586 1296.64,940.439 1298.66,931.248 \n 1300.68,922.014 1302.71,912.741 1304.73,903.429 1306.75,894.082 1308.78,884.702 1310.8,875.291 1312.82,865.852 1314.85,856.386 1316.87,846.896 1318.89,837.385 \n 1320.92,827.854 1322.94,818.306 1324.96,808.744 1326.99,799.17 1329.01,789.586 1331.03,779.995 1333.06,770.4 1335.08,760.802 1337.1,751.204 1339.13,741.609 \n 1341.15,732.019 1343.17,722.437 1345.2,712.865 1347.22,703.306 1349.24,693.761 1351.26,684.235 1353.29,674.729 1355.31,665.246 1357.33,655.788 1359.36,646.359 \n 1361.38,636.959 1363.4,627.593 1365.43,618.263 1367.45,608.971 1369.47,599.719 1371.5,590.511 1373.52,581.349 1375.54,572.236 1377.57,563.174 1379.59,554.165 \n 1381.61,545.212 1383.64,536.319 1385.66,527.486 1387.68,518.718 1389.71,510.016 1391.73,501.383 1393.75,492.822 1395.78,484.334 1397.8,475.924 1399.82,467.592 \n 1401.85,459.342 1403.87,451.176 1405.89,443.097 1407.91,435.106 1409.94,427.208 1411.96,419.403 1413.98,411.694 1416.01,404.084 1418.03,396.576 1420.05,389.171 \n 1422.08,381.872 1424.1,374.681 1426.12,367.601 1428.15,360.634 1430.17,353.782 1432.19,347.047 1434.22,340.433 1436.24,333.94 1438.26,327.571 1440.29,321.329 \n 1442.31,315.215 1444.33,309.232 1446.36,303.381 1448.38,297.665 1450.4,292.086 1452.43,286.646 1454.45,281.347 1456.47,276.191 1458.5,271.179 1460.52,266.314 \n 1462.54,261.597 1464.56,257.031 1466.59,252.617 1468.61,248.356 1470.63,244.252 1472.66,240.304 1474.68,236.516 1476.7,232.888 1478.73,229.422 1480.75,226.12 \n 1482.77,222.983 1484.8,220.013 1486.82,217.211 1488.84,214.579 1490.87,212.117 1492.89,209.827 1494.91,207.711 1496.94,205.769 1498.96,204.003 1500.98,202.414 \n 1503.01,201.002 1505.03,199.77 1507.05,198.717 1509.08,197.846 1511.1,197.156 1513.12,196.648 1515.15,196.324 1517.17,196.184 1519.19,196.228 1521.21,196.458 \n 1523.24,196.874 1525.26,197.476 1527.28,198.265 1529.31,199.241 1531.33,200.405 1533.35,201.756 1535.38,203.296 1537.4,205.024 1539.42,206.941 1541.45,209.046 \n 1543.47,211.34 1545.49,213.822 1547.52,216.493 1549.54,219.352 1551.56,222.399 1553.59,225.633 1555.61,229.055 1557.63,232.664 1559.66,236.46 1561.68,240.441 \n 1563.7,244.608 1565.73,248.96 1567.75,253.495 1569.77,258.214 1571.8,263.115 1573.82,268.198 1575.84,273.461 1577.86,278.904 1579.89,284.525 1581.91,290.324 \n 1583.93,296.298 1585.96,302.447 1587.98,308.77 1590,315.264 1592.03,321.929 1594.05,328.763 1596.07,335.764 1598.1,342.931 1600.12,350.262 1602.14,357.754 \n 1604.17,365.407 1606.19,373.219 1608.21,381.186 1610.24,389.307 1612.26,397.581 1614.28,406.005 1616.31,414.576 1618.33,423.292 1620.35,432.151 1622.38,441.151 \n 1624.4,450.288 1626.42,459.561 1628.45,468.966 1630.47,478.501 1632.49,488.164 1634.51,497.95 1636.54,507.858 1638.56,517.885 1640.58,528.027 1642.61,538.282 \n 1644.63,548.645 1646.65,559.115 1648.68,569.687 1650.7,580.359 1652.72,591.127 1654.75,601.988 1656.77,612.937 1658.79,623.973 1660.82,635.09 1662.84,646.286 \n 1664.86,657.556 1666.89,668.897 1668.91,680.305 1670.93,691.776 1672.96,703.307 1674.98,714.893 1677,726.53 1679.03,738.215 1681.05,749.942 1683.07,761.709 \n 1685.1,773.51 1687.12,785.342 1689.14,797.2 1691.16,809.08 1693.19,820.978 1695.21,832.888 1697.23,844.808 1699.26,856.731 1701.28,868.654 1703.3,880.573 \n 1705.33,892.482 1707.35,904.377 1709.37,916.253 1711.4,928.105 1713.42,939.93 1715.44,951.722 1717.47,963.476 1719.49,975.188 1721.51,986.853 1723.54,998.466 \n 1725.56,1010.02 1727.58,1021.52 1729.61,1032.94 1731.63,1044.3 1733.65,1055.58 1735.68,1066.78 1737.7,1077.89 1739.72,1088.92 1741.75,1099.84 1743.77,1110.67 \n 1745.79,1121.39 1747.81,1132 1749.84,1142.49 1751.86,1152.86 1753.88,1163.11 1755.91,1173.23 1757.93,1183.21 1759.95,1193.05 1761.98,1202.75 1764,1212.3 \n 1766.02,1221.69 1768.05,1230.93 1770.07,1240 1772.09,1248.9 1774.12,1257.63 1776.14,1266.18 1778.16,1274.54 1780.19,1282.72 1782.21,1290.71 1784.23,1298.5 \n 1786.26,1306.09 1788.28,1313.48 1790.3,1320.66 1792.33,1327.62 1794.35,1334.36 1796.37,1340.88 1798.4,1347.18 1800.42,1353.24 1802.44,1359.07 1804.46,1364.66 \n 1806.49,1370.01 1808.51,1375.11 1810.53,1379.96 1812.56,1384.56 1814.58,1388.9 1816.6,1392.98 1818.63,1396.79 1820.65,1400.34 1822.67,1403.61 1824.7,1406.62 \n 1826.72,1409.34 1828.74,1411.78 1830.77,1413.94 1832.79,1415.82 1834.81,1417.41 1836.84,1418.7 1838.86,1419.7 1840.88,1420.41 1842.91,1420.82 1844.93,1420.93 \n 1846.95,1420.73 1848.98,1420.24 1851,1419.43 1853.02,1418.32 1855.05,1416.91 1857.07,1415.18 1859.09,1413.14 1861.11,1410.79 1863.14,1408.12 1865.16,1405.15 \n 1867.18,1401.85 1869.21,1398.25 1871.23,1394.33 1873.25,1390.09 1875.28,1385.54 1877.3,1380.67 1879.32,1375.49 1881.35,1370 1883.37,1364.19 1885.39,1358.07 \n 1887.42,1351.64 1889.44,1344.9 1891.46,1337.85 1893.49,1330.49 1895.51,1322.83 1897.53,1314.87 1899.56,1306.6 1901.58,1298.04 1903.6,1289.18 1905.63,1280.02 \n 1907.65,1270.58 1909.67,1260.84 1911.7,1250.83 1913.72,1240.53 1915.74,1229.95 1917.76,1219.11 1919.79,1207.99 1921.81,1196.61 1923.83,1184.97 1925.86,1173.07 \n 1927.88,1160.92 1929.9,1148.53 1931.93,1135.89 1933.95,1123.03 1935.97,1109.93 1938,1096.61 1940.02,1083.08 1942.04,1069.33 1944.07,1055.38 1946.09,1041.23 \n 1948.11,1026.9 1950.14,1012.38 1952.16,997.688 1954.18,982.828 1956.21,967.809 1958.23,952.639 1960.25,937.326 1962.28,921.878 1964.3,906.306 1966.32,890.616 \n 1968.35,874.82 1970.37,858.925 1972.39,842.942 1974.41,826.88 1976.44,810.749 1978.46,794.56 1980.48,778.322 1982.51,762.047 1984.53,745.745 1986.55,729.427 \n 1988.58,713.104 1990.6,696.788 1992.62,680.489 1994.65,664.22 1996.67,647.992 1998.69,631.818 2000.72,615.708 2002.74,599.677 2004.76,583.735 2006.79,567.896 \n 2008.81,552.172 2010.83,536.577 2012.86,521.123 2014.88,505.823 2016.9,490.69 2018.93,475.738 2020.95,460.98 2022.97,446.43 2025,432.1 2027.02,418.006 \n 2029.04,404.16 2031.06,390.576 2033.09,377.269 2035.11,364.251 2037.13,351.538 2039.16,339.142 2041.18,327.077 2043.2,315.359 2045.23,303.999 2047.25,293.013 \n 2049.27,282.415 2051.3,272.217 2053.32,262.434 2055.34,253.079 2057.37,244.166 2059.39,235.708 2061.41,227.719 2063.44,220.212 2065.46,213.2 2067.48,206.695 \n 2069.51,200.711 2071.53,195.26 2073.55,190.354 2075.58,186.006 2077.6,182.226 2079.62,179.027 2081.65,176.42 2083.67,174.416 2085.69,173.024 2087.71,172.257 \n 2089.74,172.123 2091.76,172.631 2093.78,173.792 2095.81,175.613 2097.83,178.103 2099.85,181.269 2101.88,185.118 2103.9,189.657 2105.92,194.892 2107.95,200.828 \n 2109.97,207.471 2111.99,214.823 2114.02,222.889 2116.04,231.672 2118.06,241.173 2120.09,251.393 2122.11,262.333 2124.13,273.992 2126.16,286.369 2128.18,299.462 \n 2130.2,313.268 2132.23,327.781 2134.25,342.998 2136.27,358.912 2138.3,375.515 2140.32,392.799 2142.34,410.754 2144.36,429.369 2146.39,448.633 2148.41,468.531 \n 2150.43,489.049 2152.46,510.17 2154.48,531.877 2156.5,554.15 2158.53,576.97 2160.55,600.313 2162.57,624.155 2164.6,648.471 2166.62,673.234 2168.64,698.415 \n 2170.67,723.981 2172.69,749.901 2174.71,776.14 2176.74,802.659 2178.76,829.422 2180.78,856.385 2182.81,883.507 2184.83,910.74 2186.85,938.038 2188.88,965.35 \n 2190.9,992.622 2192.92,1019.8 2194.95,1046.83 2196.97,1073.64 2198.99,1100.18 2201.01,1126.37 2203.04,1152.15 2205.06,1177.45 2207.08,1202.19 2209.11,1226.29 \n 2211.13,1249.67 2213.15,1272.25 2215.18,1293.93 2217.2,1314.64 2219.22,1334.26 2221.25,1352.71 2223.27,1369.87 2225.29,1385.65 2227.32,1399.94 2229.34,1412.61 \n 2231.36,1423.55 2233.39,1432.64 2235.41,1439.75 2237.43,1444.75 2239.46,1447.51 2241.48,1447.87 2243.5,1445.71 2245.53,1440.86 2247.55,1433.18 2249.57,1422.5 \n 2251.6,1408.65 2253.62,1391.48 2255.64,1370.79 2257.66,1346.41 2259.69,1318.16 2261.71,1285.84 2263.73,1249.24 2265.76,1208.18 2267.78,1162.42 2269.8,1111.77 \n 2271.83,1055.99 2273.85,994.847 2275.87,928.113 2277.9,855.541 2279.92,776.879 2281.94,691.87 2283.97,600.247 2285.99,501.738 2288.01,396.062 2290.04,282.931 \n 2292.06,162.047 \n \"/>\n</svg>\n" | |
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| "display_name": "Julia 1.6.0-rc1 depwarn -O3", | |
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| "description": "degree-13 polynomial approximation of sine", | |
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