[388] | 1 | /* origin: OpenBSD /usr/src/lib/libm/src/ld80/e_tgammal.c */
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| 2 | /*
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| 3 | * Copyright (c) 2008 Stephen L. Moshier <steve@moshier.net>
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| 4 | *
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| 5 | * Permission to use, copy, modify, and distribute this software for any
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| 6 | * purpose with or without fee is hereby granted, provided that the above
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| 7 | * copyright notice and this permission notice appear in all copies.
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| 8 | *
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| 9 | * THE SOFTWARE IS PROVIDED "AS IS" AND THE AUTHOR DISCLAIMS ALL WARRANTIES
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| 10 | * WITH REGARD TO THIS SOFTWARE INCLUDING ALL IMPLIED WARRANTIES OF
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| 11 | * MERCHANTABILITY AND FITNESS. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR
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| 12 | * ANY SPECIAL, DIRECT, INDIRECT, OR CONSEQUENTIAL DAMAGES OR ANY DAMAGES
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| 13 | * WHATSOEVER RESULTING FROM LOSS OF USE, DATA OR PROFITS, WHETHER IN AN
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| 14 | * ACTION OF CONTRACT, NEGLIGENCE OR OTHER TORTIOUS ACTION, ARISING OUT OF
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| 15 | * OR IN CONNECTION WITH THE USE OR PERFORMANCE OF THIS SOFTWARE.
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| 16 | */
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| 17 | /*
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| 18 | * Gamma function
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| 19 | *
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| 20 | *
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| 21 | * SYNOPSIS:
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| 22 | *
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| 23 | * long double x, y, tgammal();
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| 24 | *
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| 25 | * y = tgammal( x );
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| 26 | *
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| 27 | *
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| 28 | * DESCRIPTION:
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| 29 | *
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| 30 | * Returns gamma function of the argument. The result is
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| 31 | * correctly signed.
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| 32 | *
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| 33 | * Arguments |x| <= 13 are reduced by recurrence and the function
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| 34 | * approximated by a rational function of degree 7/8 in the
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| 35 | * interval (2,3). Large arguments are handled by Stirling's
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| 36 | * formula. Large negative arguments are made positive using
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| 37 | * a reflection formula.
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| 38 | *
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| 39 | *
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| 40 | * ACCURACY:
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| 41 | *
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| 42 | * Relative error:
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| 43 | * arithmetic domain # trials peak rms
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| 44 | * IEEE -40,+40 10000 3.6e-19 7.9e-20
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| 45 | * IEEE -1755,+1755 10000 4.8e-18 6.5e-19
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| 46 | *
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| 47 | * Accuracy for large arguments is dominated by error in powl().
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| 48 | *
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| 49 | */
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| 50 |
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| 51 | #include "libm.h"
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| 52 |
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| 53 | #if LDBL_MANT_DIG == 53 && LDBL_MAX_EXP == 1024
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| 54 | long double tgammal(long double x)
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| 55 | {
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| 56 | return tgamma(x);
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| 57 | }
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| 58 | #elif LDBL_MANT_DIG == 64 && LDBL_MAX_EXP == 16384
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| 59 | /*
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| 60 | tgamma(x+2) = tgamma(x+2) P(x)/Q(x)
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| 61 | 0 <= x <= 1
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| 62 | Relative error
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| 63 | n=7, d=8
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| 64 | Peak error = 1.83e-20
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| 65 | Relative error spread = 8.4e-23
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| 66 | */
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| 67 | static const long double P[8] = {
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| 68 | 4.212760487471622013093E-5L,
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| 69 | 4.542931960608009155600E-4L,
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| 70 | 4.092666828394035500949E-3L,
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| 71 | 2.385363243461108252554E-2L,
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| 72 | 1.113062816019361559013E-1L,
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| 73 | 3.629515436640239168939E-1L,
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| 74 | 8.378004301573126728826E-1L,
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| 75 | 1.000000000000000000009E0L,
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| 76 | };
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| 77 | static const long double Q[9] = {
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| 78 | -1.397148517476170440917E-5L,
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| 79 | 2.346584059160635244282E-4L,
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| 80 | -1.237799246653152231188E-3L,
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| 81 | -7.955933682494738320586E-4L,
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| 82 | 2.773706565840072979165E-2L,
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| 83 | -4.633887671244534213831E-2L,
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| 84 | -2.243510905670329164562E-1L,
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| 85 | 4.150160950588455434583E-1L,
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| 86 | 9.999999999999999999908E-1L,
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| 87 | };
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| 88 |
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| 89 | /*
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| 90 | static const long double P[] = {
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| 91 | -3.01525602666895735709e0L,
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| 92 | -3.25157411956062339893e1L,
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| 93 | -2.92929976820724030353e2L,
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| 94 | -1.70730828800510297666e3L,
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| 95 | -7.96667499622741999770e3L,
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| 96 | -2.59780216007146401957e4L,
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| 97 | -5.99650230220855581642e4L,
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| 98 | -7.15743521530849602425e4L
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| 99 | };
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| 100 | static const long double Q[] = {
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| 101 | 1.00000000000000000000e0L,
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| 102 | -1.67955233807178858919e1L,
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| 103 | 8.85946791747759881659e1L,
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| 104 | 5.69440799097468430177e1L,
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| 105 | -1.98526250512761318471e3L,
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| 106 | 3.31667508019495079814e3L,
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| 107 | 1.60577839621734713377e4L,
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| 108 | -2.97045081369399940529e4L,
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| 109 | -7.15743521530849602412e4L
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| 110 | };
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| 111 | */
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| 112 | #define MAXGAML 1755.455L
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| 113 | /*static const long double LOGPI = 1.14472988584940017414L;*/
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| 114 |
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| 115 | /* Stirling's formula for the gamma function
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| 116 | tgamma(x) = sqrt(2 pi) x^(x-.5) exp(-x) (1 + 1/x P(1/x))
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| 117 | z(x) = x
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| 118 | 13 <= x <= 1024
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| 119 | Relative error
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| 120 | n=8, d=0
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| 121 | Peak error = 9.44e-21
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| 122 | Relative error spread = 8.8e-4
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| 123 | */
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| 124 | static const long double STIR[9] = {
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| 125 | 7.147391378143610789273E-4L,
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| 126 | -2.363848809501759061727E-5L,
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| 127 | -5.950237554056330156018E-4L,
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| 128 | 6.989332260623193171870E-5L,
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| 129 | 7.840334842744753003862E-4L,
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| 130 | -2.294719747873185405699E-4L,
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| 131 | -2.681327161876304418288E-3L,
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| 132 | 3.472222222230075327854E-3L,
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| 133 | 8.333333333333331800504E-2L,
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| 134 | };
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| 135 |
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| 136 | #define MAXSTIR 1024.0L
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| 137 | static const long double SQTPI = 2.50662827463100050242E0L;
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| 138 |
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| 139 | /* 1/tgamma(x) = z P(z)
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| 140 | * z(x) = 1/x
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| 141 | * 0 < x < 0.03125
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| 142 | * Peak relative error 4.2e-23
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| 143 | */
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| 144 | static const long double S[9] = {
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| 145 | -1.193945051381510095614E-3L,
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| 146 | 7.220599478036909672331E-3L,
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| 147 | -9.622023360406271645744E-3L,
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| 148 | -4.219773360705915470089E-2L,
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| 149 | 1.665386113720805206758E-1L,
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| 150 | -4.200263503403344054473E-2L,
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| 151 | -6.558780715202540684668E-1L,
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| 152 | 5.772156649015328608253E-1L,
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| 153 | 1.000000000000000000000E0L,
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| 154 | };
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| 155 |
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| 156 | /* 1/tgamma(-x) = z P(z)
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| 157 | * z(x) = 1/x
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| 158 | * 0 < x < 0.03125
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| 159 | * Peak relative error 5.16e-23
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| 160 | * Relative error spread = 2.5e-24
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| 161 | */
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| 162 | static const long double SN[9] = {
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| 163 | 1.133374167243894382010E-3L,
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| 164 | 7.220837261893170325704E-3L,
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| 165 | 9.621911155035976733706E-3L,
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| 166 | -4.219773343731191721664E-2L,
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| 167 | -1.665386113944413519335E-1L,
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| 168 | -4.200263503402112910504E-2L,
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| 169 | 6.558780715202536547116E-1L,
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| 170 | 5.772156649015328608727E-1L,
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| 171 | -1.000000000000000000000E0L,
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| 172 | };
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| 173 |
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| 174 | static const long double PIL = 3.1415926535897932384626L;
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| 175 |
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| 176 | /* Gamma function computed by Stirling's formula.
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| 177 | */
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| 178 | static long double stirf(long double x)
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| 179 | {
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| 180 | long double y, w, v;
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| 181 |
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| 182 | w = 1.0/x;
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| 183 | /* For large x, use rational coefficients from the analytical expansion. */
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| 184 | if (x > 1024.0)
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| 185 | w = (((((6.97281375836585777429E-5L * w
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| 186 | + 7.84039221720066627474E-4L) * w
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| 187 | - 2.29472093621399176955E-4L) * w
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| 188 | - 2.68132716049382716049E-3L) * w
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| 189 | + 3.47222222222222222222E-3L) * w
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| 190 | + 8.33333333333333333333E-2L) * w
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| 191 | + 1.0;
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| 192 | else
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| 193 | w = 1.0 + w * __polevll(w, STIR, 8);
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| 194 | y = expl(x);
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| 195 | if (x > MAXSTIR) { /* Avoid overflow in pow() */
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| 196 | v = powl(x, 0.5L * x - 0.25L);
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| 197 | y = v * (v / y);
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| 198 | } else {
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| 199 | y = powl(x, x - 0.5L) / y;
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| 200 | }
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| 201 | y = SQTPI * y * w;
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| 202 | return y;
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| 203 | }
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| 204 |
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| 205 | long double tgammal(long double x)
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| 206 | {
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| 207 | long double p, q, z;
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| 208 |
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| 209 | if (!isfinite(x))
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| 210 | return x + INFINITY;
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| 211 |
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| 212 | q = fabsl(x);
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| 213 | if (q > 13.0) {
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| 214 | if (x < 0.0) {
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| 215 | p = floorl(q);
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| 216 | z = q - p;
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| 217 | if (z == 0)
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| 218 | return 0 / z;
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| 219 | if (q > MAXGAML) {
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| 220 | z = 0;
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| 221 | } else {
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| 222 | if (z > 0.5) {
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| 223 | p += 1.0;
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| 224 | z = q - p;
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| 225 | }
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| 226 | z = q * sinl(PIL * z);
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| 227 | z = fabsl(z) * stirf(q);
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| 228 | z = PIL/z;
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| 229 | }
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| 230 | if (0.5 * p == floorl(q * 0.5))
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| 231 | z = -z;
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| 232 | } else if (x > MAXGAML) {
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| 233 | z = x * 0x1p16383L;
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| 234 | } else {
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| 235 | z = stirf(x);
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| 236 | }
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| 237 | return z;
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| 238 | }
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| 239 |
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| 240 | z = 1.0;
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| 241 | while (x >= 3.0) {
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| 242 | x -= 1.0;
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| 243 | z *= x;
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| 244 | }
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| 245 | while (x < -0.03125L) {
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| 246 | z /= x;
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| 247 | x += 1.0;
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| 248 | }
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| 249 | if (x <= 0.03125L)
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| 250 | goto small;
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| 251 | while (x < 2.0) {
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| 252 | z /= x;
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| 253 | x += 1.0;
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| 254 | }
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| 255 | if (x == 2.0)
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| 256 | return z;
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| 257 |
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| 258 | x -= 2.0;
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| 259 | p = __polevll(x, P, 7);
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| 260 | q = __polevll(x, Q, 8);
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| 261 | z = z * p / q;
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| 262 | return z;
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| 263 |
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| 264 | small:
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| 265 | /* z==1 if x was originally +-0 */
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| 266 | if (x == 0 && z != 1)
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| 267 | return x / x;
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| 268 | if (x < 0.0) {
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| 269 | x = -x;
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| 270 | q = z / (x * __polevll(x, SN, 8));
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| 271 | } else
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| 272 | q = z / (x * __polevll(x, S, 8));
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| 273 | return q;
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| 274 | }
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| 275 | #elif LDBL_MANT_DIG == 113 && LDBL_MAX_EXP == 16384
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| 276 | // TODO: broken implementation to make things compile
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| 277 | long double tgammal(long double x)
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| 278 | {
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| 279 | return tgamma(x);
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| 280 | }
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| 281 | #endif
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