<?php
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namespace PhpOffice\PhpSpreadsheet\Calculation\Statistical\Distributions;
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use PhpOffice\PhpSpreadsheet\Calculation\Functions;
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use PhpOffice\PhpSpreadsheet\Calculation\Information\ExcelError;
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abstract class GammaBase
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{
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private const LOG_GAMMA_X_MAX_VALUE = 2.55e305;
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private const EPS = 2.22e-16;
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private const MAX_VALUE = 1.2e308;
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private const SQRT2PI = 2.5066282746310005024157652848110452530069867406099;
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private const MAX_ITERATIONS = 256;
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protected static function calculateDistribution(float $value, float $a, float $b, bool $cumulative): float
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{
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if ($cumulative) {
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return self::incompleteGamma($a, $value / $b) / self::gammaValue($a);
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}
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return (1 / ($b ** $a * self::gammaValue($a))) * $value ** ($a - 1) * exp(0 - ($value / $b));
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}
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/** @return float|string */
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protected static function calculateInverse(float $probability, float $alpha, float $beta)
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{
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$xLo = 0;
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$xHi = $alpha * $beta * 5;
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$dx = 1024;
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$x = $xNew = 1;
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$i = 0;
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while ((abs($dx) > Functions::PRECISION) && (++$i <= self::MAX_ITERATIONS)) {
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// Apply Newton-Raphson step
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$result = self::calculateDistribution($x, $alpha, $beta, true);
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if (!is_float($result)) {
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return ExcelError::NA();
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}
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$error = $result - $probability;
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if ($error == 0.0) {
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$dx = 0;
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} elseif ($error < 0.0) {
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$xLo = $x;
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} else {
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$xHi = $x;
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}
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$pdf = self::calculateDistribution($x, $alpha, $beta, false);
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// Avoid division by zero
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if (!is_float($pdf)) {
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return ExcelError::NA();
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}
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if ($pdf !== 0.0) {
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$dx = $error / $pdf;
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$xNew = $x - $dx;
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}
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// If the NR fails to converge (which for example may be the
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// case if the initial guess is too rough) we apply a bisection
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// step to determine a more narrow interval around the root.
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if (($xNew < $xLo) || ($xNew > $xHi) || ($pdf == 0.0)) {
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$xNew = ($xLo + $xHi) / 2;
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$dx = $xNew - $x;
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}
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$x = $xNew;
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}
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if ($i === self::MAX_ITERATIONS) {
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return ExcelError::NA();
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}
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return $x;
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}
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//
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// Implementation of the incomplete Gamma function
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//
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public static function incompleteGamma(float $a, float $x): float
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{
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static $max = 32;
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$summer = 0;
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for ($n = 0; $n <= $max; ++$n) {
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$divisor = $a;
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for ($i = 1; $i <= $n; ++$i) {
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$divisor *= ($a + $i);
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}
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$summer += ($x ** $n / $divisor);
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}
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return $x ** $a * exp(0 - $x) * $summer;
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}
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//
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// Implementation of the Gamma function
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//
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public static function gammaValue(float $value): float
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{
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if ($value == 0.0) {
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return 0;
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}
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static $p0 = 1.000000000190015;
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static $p = [
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1 => 76.18009172947146,
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2 => -86.50532032941677,
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3 => 24.01409824083091,
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4 => -1.231739572450155,
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5 => 1.208650973866179e-3,
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6 => -5.395239384953e-6,
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];
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$y = $x = $value;
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$tmp = $x + 5.5;
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$tmp -= ($x + 0.5) * log($tmp);
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$summer = $p0;
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for ($j = 1; $j <= 6; ++$j) {
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$summer += ($p[$j] / ++$y);
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}
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return exp(0 - $tmp + log(self::SQRT2PI * $summer / $x));
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}
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private const LG_D1 = -0.5772156649015328605195174;
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private const LG_D2 = 0.4227843350984671393993777;
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private const LG_D4 = 1.791759469228055000094023;
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private const LG_P1 = [
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4.945235359296727046734888,
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201.8112620856775083915565,
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2290.838373831346393026739,
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11319.67205903380828685045,
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28557.24635671635335736389,
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38484.96228443793359990269,
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26377.48787624195437963534,
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7225.813979700288197698961,
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];
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private const LG_P2 = [
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4.974607845568932035012064,
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542.4138599891070494101986,
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15506.93864978364947665077,
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184793.2904445632425417223,
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1088204.76946882876749847,
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3338152.967987029735917223,
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5106661.678927352456275255,
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3074109.054850539556250927,
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];
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private const LG_P4 = [
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14745.02166059939948905062,
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2426813.369486704502836312,
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121475557.4045093227939592,
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2663432449.630976949898078,
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29403789566.34553899906876,
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170266573776.5398868392998,
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492612579337.743088758812,
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560625185622.3951465078242,
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];
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private const LG_Q1 = [
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67.48212550303777196073036,
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1113.332393857199323513008,
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7738.757056935398733233834,
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27639.87074403340708898585,
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54993.10206226157329794414,
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61611.22180066002127833352,
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36351.27591501940507276287,
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8785.536302431013170870835,
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];
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private const LG_Q2 = [
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183.0328399370592604055942,
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7765.049321445005871323047,
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133190.3827966074194402448,
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1136705.821321969608938755,
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5267964.117437946917577538,
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13467014.54311101692290052,
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17827365.30353274213975932,
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9533095.591844353613395747,
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];
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private const LG_Q4 = [
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2690.530175870899333379843,
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639388.5654300092398984238,
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41355999.30241388052042842,
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1120872109.61614794137657,
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14886137286.78813811542398,
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101680358627.2438228077304,
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341747634550.7377132798597,
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446315818741.9713286462081,
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];
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private const LG_C = [
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-0.001910444077728,
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8.4171387781295e-4,
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-5.952379913043012e-4,
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7.93650793500350248e-4,
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-0.002777777777777681622553,
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0.08333333333333333331554247,
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0.0057083835261,
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];
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// Rough estimate of the fourth root of logGamma_xBig
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private const LG_FRTBIG = 2.25e76;
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private const PNT68 = 0.6796875;
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// Function cache for logGamma
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private static float $logGammaCacheResult = 0.0;
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private static float $logGammaCacheX = 0.0;
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/**
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* logGamma function.
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*
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* Original author was Jaco van Kooten. Ported to PHP by Paul Meagher.
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*
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* The natural logarithm of the gamma function. <br />
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* Based on public domain NETLIB (Fortran) code by W. J. Cody and L. Stoltz <br />
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* Applied Mathematics Division <br />
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* Argonne National Laboratory <br />
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* Argonne, IL 60439 <br />
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* <p>
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* References:
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* <ol>
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* <li>W. J. Cody and K. E. Hillstrom, 'Chebyshev Approximations for the Natural
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* Logarithm of the Gamma Function,' Math. Comp. 21, 1967, pp. 198-203.</li>
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* <li>K. E. Hillstrom, ANL/AMD Program ANLC366S, DGAMMA/DLGAMA, May, 1969.</li>
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* <li>Hart, Et. Al., Computer Approximations, Wiley and sons, New York, 1968.</li>
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* </ol>
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* </p>
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* <p>
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* From the original documentation:
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* </p>
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* <p>
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* This routine calculates the LOG(GAMMA) function for a positive real argument X.
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* Computation is based on an algorithm outlined in references 1 and 2.
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* The program uses rational functions that theoretically approximate LOG(GAMMA)
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* to at least 18 significant decimal digits. The approximation for X > 12 is from
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* reference 3, while approximations for X < 12.0 are similar to those in reference
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* 1, but are unpublished. The accuracy achieved depends on the arithmetic system,
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* the compiler, the intrinsic functions, and proper selection of the
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* machine-dependent constants.
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* </p>
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* <p>
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* Error returns: <br />
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* The program returns the value XINF for X .LE. 0.0 or when overflow would occur.
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* The computation is believed to be free of underflow and overflow.
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* </p>
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*
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* @version 1.1
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*
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* @author Jaco van Kooten
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*
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* @return float MAX_VALUE for x < 0.0 or when overflow would occur, i.e. x > 2.55E305
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*/
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public static function logGamma(float $x): float
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{
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if ($x == self::$logGammaCacheX) {
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return self::$logGammaCacheResult;
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}
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$y = $x;
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if ($y > 0.0 && $y <= self::LOG_GAMMA_X_MAX_VALUE) {
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if ($y <= self::EPS) {
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$res = -log($y);
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} elseif ($y <= 1.5) {
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$res = self::logGamma1($y);
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} elseif ($y <= 4.0) {
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$res = self::logGamma2($y);
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} elseif ($y <= 12.0) {
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$res = self::logGamma3($y);
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} else {
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$res = self::logGamma4($y);
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}
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} else {
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// --------------------------
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// Return for bad arguments
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// --------------------------
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$res = self::MAX_VALUE;
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}
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// ------------------------------
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// Final adjustments and return
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// ------------------------------
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self::$logGammaCacheX = $x;
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self::$logGammaCacheResult = $res;
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return $res;
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}
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private static function logGamma1(float $y): float
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{
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// ---------------------
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// EPS .LT. X .LE. 1.5
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// ---------------------
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if ($y < self::PNT68) {
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$corr = -log($y);
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$xm1 = $y;
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} else {
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$corr = 0.0;
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$xm1 = $y - 1.0;
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}
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$xden = 1.0;
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$xnum = 0.0;
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if ($y <= 0.5 || $y >= self::PNT68) {
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for ($i = 0; $i < 8; ++$i) {
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$xnum = $xnum * $xm1 + self::LG_P1[$i];
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$xden = $xden * $xm1 + self::LG_Q1[$i];
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}
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return $corr + $xm1 * (self::LG_D1 + $xm1 * ($xnum / $xden));
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}
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$xm2 = $y - 1.0;
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for ($i = 0; $i < 8; ++$i) {
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$xnum = $xnum * $xm2 + self::LG_P2[$i];
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$xden = $xden * $xm2 + self::LG_Q2[$i];
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}
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return $corr + $xm2 * (self::LG_D2 + $xm2 * ($xnum / $xden));
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}
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private static function logGamma2(float $y): float
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{
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// ---------------------
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// 1.5 .LT. X .LE. 4.0
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// ---------------------
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$xm2 = $y - 2.0;
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$xden = 1.0;
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$xnum = 0.0;
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for ($i = 0; $i < 8; ++$i) {
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$xnum = $xnum * $xm2 + self::LG_P2[$i];
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$xden = $xden * $xm2 + self::LG_Q2[$i];
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}
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return $xm2 * (self::LG_D2 + $xm2 * ($xnum / $xden));
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}
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protected static function logGamma3(float $y): float
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{
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// ----------------------
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// 4.0 .LT. X .LE. 12.0
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// ----------------------
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$xm4 = $y - 4.0;
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$xden = -1.0;
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$xnum = 0.0;
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for ($i = 0; $i < 8; ++$i) {
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$xnum = $xnum * $xm4 + self::LG_P4[$i];
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$xden = $xden * $xm4 + self::LG_Q4[$i];
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}
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return self::LG_D4 + $xm4 * ($xnum / $xden);
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}
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protected static function logGamma4(float $y): float
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{
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// ---------------------------------
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// Evaluate for argument .GE. 12.0
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// ---------------------------------
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$res = 0.0;
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if ($y <= self::LG_FRTBIG) {
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$res = self::LG_C[6];
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$ysq = $y * $y;
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for ($i = 0; $i < 6; ++$i) {
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$res = $res / $ysq + self::LG_C[$i];
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}
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$res /= $y;
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$corr = log($y);
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$res = $res + log(self::SQRT2PI) - 0.5 * $corr;
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$res += $y * ($corr - 1.0);
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}
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return $res;
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}
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}
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