397 lines
10 KiB
C
397 lines
10 KiB
C
/**
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* This file has no copyright assigned and is placed in the Public Domain.
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* This file is part of the mingw-w64 runtime package.
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* No warranty is given; refer to the file DISCLAIMER.PD within this package.
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*/
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#include "cephes_mconf.h"
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/*
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gamma(x+2) = gamma(x+2) P(x)/Q(x)
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0 <= x <= 1
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Relative error
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n=7, d=8
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Peak error = 1.83e-20
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Relative error spread = 8.4e-23
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*/
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#if UNK
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static const uLD P[8] = {
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{ { 4.212760487471622013093E-5L } },
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{ { 4.542931960608009155600E-4L } },
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{ { 4.092666828394035500949E-3L } },
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{ { 2.385363243461108252554E-2L } },
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{ { 1.113062816019361559013E-1L } },
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{ { 3.629515436640239168939E-1L } },
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{ { 8.378004301573126728826E-1L } },
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{ { 1.000000000000000000009E0L } }
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};
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static const uLD Q[9] = {
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{ { -1.397148517476170440917E-5L } },
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{ { 2.346584059160635244282E-4L } },
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{ { -1.237799246653152231188E-3L } },
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{ { -7.955933682494738320586E-4L } },
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{ { 2.773706565840072979165E-2L } },
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{ { -4.633887671244534213831E-2L } },
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{ { -2.243510905670329164562E-1L } },
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{ { 4.150160950588455434583E-1L } },
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{ { 9.999999999999999999908E-1L } }
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};
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#endif
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#if IBMPC
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static const uLD P[8] = {
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{ { 0x434a,0x3f22,0x2bda,0xb0b2,0x3ff0, 0x0, 0x0, 0x0 } },
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{ { 0xf5aa,0xe82f,0x335b,0xee2e,0x3ff3, 0x0, 0x0, 0x0 } },
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{ { 0xbe6c,0x3757,0xc717,0x861b,0x3ff7, 0x0, 0x0, 0x0 } },
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{ { 0x7f43,0x5196,0xb166,0xc368,0x3ff9, 0x0, 0x0, 0x0 } },
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{ { 0x9549,0x8eb5,0x8c3a,0xe3f4,0x3ffb, 0x0, 0x0, 0x0 } },
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{ { 0x8d75,0x23af,0xc8e4,0xb9d4,0x3ffd, 0x0, 0x0, 0x0 } },
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{ { 0x29cf,0x19b3,0x16c8,0xd67a,0x3ffe, 0x0, 0x0, 0x0 } },
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{ { 0x0000,0x0000,0x0000,0x8000,0x3fff, 0x0, 0x0, 0x0 } }
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};
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static const uLD Q[9] = {
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{ { 0x5473,0x2de8,0x1268,0xea67,0xbfee, 0x0, 0x0, 0x0 } },
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{ { 0x334b,0xc2f0,0xa2dd,0xf60e,0x3ff2, 0x0, 0x0, 0x0 } },
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{ { 0xbeed,0x1853,0xa691,0xa23d,0xbff5, 0x0, 0x0, 0x0 } },
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{ { 0x296e,0x7cb1,0x5dfd,0xd08f,0xbff4, 0x0, 0x0, 0x0 } },
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{ { 0x0417,0x7989,0xd7bc,0xe338,0x3ff9, 0x0, 0x0, 0x0 } },
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{ { 0x3295,0x3698,0xd580,0xbdcd,0xbffa, 0x0, 0x0, 0x0 } },
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{ { 0x75ef,0x3ab7,0x4ad3,0xe5bc,0xbffc, 0x0, 0x0, 0x0 } },
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{ { 0xe458,0x2ec7,0xfd57,0xd47c,0x3ffd, 0x0, 0x0, 0x0 } },
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{ { 0x0000,0x0000,0x0000,0x8000,0x3fff, 0x0, 0x0, 0x0 } }
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};
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#endif
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#if MIEEE
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static const uLD P[8] = {
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{ { 0x3ff00000,0xb0b22bda,0x3f22434a, 0 } },
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{ { 0x3ff30000,0xee2e335b,0xe82ff5aa, 0 } },
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{ { 0x3ff70000,0x861bc717,0x3757be6c, 0 } },
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{ { 0x3ff90000,0xc368b166,0x51967f43, 0 } },
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{ { 0x3ffb0000,0xe3f48c3a,0x8eb59549, 0 } },
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{ { 0x3ffd0000,0xb9d4c8e4,0x23af8d75, 0 } },
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{ { 0x3ffe0000,0xd67a16c8,0x19b329cf, 0 } },
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{ { 0x3fff0000,0x80000000,0x00000000, 0 } }
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};
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static const uLD Q[9] = {
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{ { 0xbfee0000,0xea671268,0x2de85473, 0 } },
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{ { 0x3ff20000,0xf60ea2dd,0xc2f0334b, 0 } },
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{ { 0xbff50000,0xa23da691,0x1853beed, 0 } },
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{ { 0xbff40000,0xd08f5dfd,0x7cb1296e, 0 } },
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{ { 0x3ff90000,0xe338d7bc,0x79890417, 0 } },
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{ { 0xbffa0000,0xbdcdd580,0x36983295, 0 } },
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{ { 0xbffc0000,0xe5bc4ad3,0x3ab775ef, 0 } },
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{ { 0x3ffd0000,0xd47cfd57,0x2ec7e458, 0 } },
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{ { 0x3fff0000,0x80000000,0x00000000, 0 } }
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};
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#endif
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#define MAXGAML 1755.455L
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/*static const long double LOGPI = 1.14472988584940017414L;*/
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/* Stirling's formula for the gamma function
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gamma(x) = sqrt(2 pi) x^(x-.5) exp(-x) (1 + 1/x P(1/x))
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z(x) = x
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13 <= x <= 1024
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Relative error
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n=8, d=0
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Peak error = 9.44e-21
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Relative error spread = 8.8e-4
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*/
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#if UNK
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static const uLD STIR[9] = {
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{ { 7.147391378143610789273E-4L } },
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{ { -2.363848809501759061727E-5L } },
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{ { -5.950237554056330156018E-4L } },
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{ { 6.989332260623193171870E-5L } },
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{ { 7.840334842744753003862E-4L } },
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{ { -2.294719747873185405699E-4L } },
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{ { -2.681327161876304418288E-3L } },
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{ { 3.472222222230075327854E-3L } },
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{ { 8.333333333333331800504E-2L } }
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};
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#endif
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#if IBMPC
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static const uLD STIR[9] = {
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{ { 0x6ede,0x69f7,0x54e3,0xbb5d,0x3ff4, 0, 0, 0 } },
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{ { 0xc395,0x0295,0x4443,0xc64b,0xbfef, 0, 0, 0 } },
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{ { 0xba6f,0x7c59,0x5e47,0x9bfb,0xbff4, 0, 0, 0 } },
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{ { 0x5704,0x1a39,0xb11d,0x9293,0x3ff1, 0, 0, 0 } },
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{ { 0x30b7,0x1a21,0x98b2,0xcd87,0x3ff4, 0, 0, 0 } },
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{ { 0xbef3,0x7023,0x6a08,0xf09e,0xbff2, 0, 0, 0 } },
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{ { 0x3a1c,0x5ac8,0x3478,0xafb9,0xbff6, 0, 0, 0 } },
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{ { 0xc3c9,0x906e,0x38e3,0xe38e,0x3ff6, 0, 0, 0 } },
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{ { 0xa1d5,0xaaaa,0xaaaa,0xaaaa,0x3ffb, 0, 0, 0 } }
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};
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#endif
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#if MIEEE
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static const uLD STIR[9] = {
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{ { 0x3ff40000,0xbb5d54e3,0x69f76ede, 0 } },
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{ { 0xbfef0000,0xc64b4443,0x0295c395, 0 } },
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{ { 0xbff40000,0x9bfb5e47,0x7c59ba6f, 0 } },
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{ { 0x3ff10000,0x9293b11d,0x1a395704, 0 } },
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{ { 0x3ff40000,0xcd8798b2,0x1a2130b7, 0 } },
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{ { 0xbff20000,0xf09e6a08,0x7023bef3, 0 } },
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{ { 0xbff60000,0xafb93478,0x5ac83a1c, 0 } },
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{ { 0x3ff60000,0xe38e38e3,0x906ec3c9, 0 } },
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{ { 0x3ffb0000,0xaaaaaaaa,0xaaaaa1d5, 0 } }
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};
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#endif
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#define MAXSTIR 1024.0L
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static const long double SQTPI = 2.50662827463100050242E0L;
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/* 1/gamma(x) = z P(z)
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* z(x) = 1/x
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* 0 < x < 0.03125
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* Peak relative error 4.2e-23
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*/
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#if UNK
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static const uLD S[9] = {
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{ { -1.193945051381510095614E-3L } },
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{ { 7.220599478036909672331E-3L } },
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{ { -9.622023360406271645744E-3L } },
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{ { -4.219773360705915470089E-2L } },
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{ { 1.665386113720805206758E-1L } },
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{ { -4.200263503403344054473E-2L } },
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{ { -6.558780715202540684668E-1L } },
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{ { 5.772156649015328608253E-1L } },
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{ { 1.000000000000000000000E0L } }
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};
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#endif
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#if IBMPC
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static const uLD S[9] = {
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{ { 0xbaeb,0xd6d3,0x25e5,0x9c7e,0xbff5, 0, 0, 0 } },
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{ { 0xfe9a,0xceb4,0xc74e,0xec9a,0x3ff7, 0, 0, 0 } },
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{ { 0x9225,0xdfef,0xb0e9,0x9da5,0xbff8, 0, 0, 0 } },
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{ { 0x10b0,0xec17,0x87dc,0xacd7,0xbffa, 0, 0, 0 } },
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{ { 0x6b8d,0x7515,0x1905,0xaa89,0x3ffc, 0, 0, 0 } },
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{ { 0xf183,0x126b,0xf47d,0xac0a,0xbffa, 0, 0, 0 } },
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{ { 0x7bf6,0x57d1,0xa013,0xa7e7,0xbffe, 0, 0, 0 } },
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{ { 0xc7a9,0x7db0,0x67e3,0x93c4,0x3ffe, 0, 0, 0 } },
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{ { 0x0000,0x0000,0x0000,0x8000,0x3fff, 0, 0, 0 } }
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};
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#endif
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#if MIEEE
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static const long S[9] = {
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{ { 0xbff50000,0x9c7e25e5,0xd6d3baeb, 0 } },
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{ { 0x3ff70000,0xec9ac74e,0xceb4fe9a, 0 } },
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{ { 0xbff80000,0x9da5b0e9,0xdfef9225, 0 } },
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{ { 0xbffa0000,0xacd787dc,0xec1710b0, 0 } },
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{ { 0x3ffc0000,0xaa891905,0x75156b8d, 0 } },
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{ { 0xbffa0000,0xac0af47d,0x126bf183, 0 } },
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{ { 0xbffe0000,0xa7e7a013,0x57d17bf6, 0 } },
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{ { 0x3ffe0000,0x93c467e3,0x7db0c7a9, 0 } },
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{ { 0x3fff0000,0x80000000,0x00000000, 0 } }
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};
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#endif
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/* 1/gamma(-x) = z P(z)
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* z(x) = 1/x
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* 0 < x < 0.03125
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* Peak relative error 5.16e-23
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* Relative error spread = 2.5e-24
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*/
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#if UNK
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static const uLD SN[9] = {
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{ { 1.133374167243894382010E-3L } },
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{ { 7.220837261893170325704E-3L } },
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{ { 9.621911155035976733706E-3L } },
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{ { -4.219773343731191721664E-2L } },
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{ { -1.665386113944413519335E-1L } },
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{ { -4.200263503402112910504E-2L } },
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{ { 6.558780715202536547116E-1L } },
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{ { 5.772156649015328608727E-1L } },
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{ { -1.000000000000000000000E0L } }
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};
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#endif
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#if IBMPC
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static const uLD SN[9] = {
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{ { 0x5dd1,0x02de,0xb9f7,0x948d,0x3ff5, 0, 0, 0 } },
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{ { 0x989b,0xdd68,0xc5f1,0xec9c,0x3ff7, 0, 0, 0 } },
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{ { 0x2ca1,0x18f0,0x386f,0x9da5,0x3ff8, 0, 0, 0 } },
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{ { 0x783f,0x41dd,0x87d1,0xacd7,0xbffa, 0, 0, 0 } },
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{ { 0x7a5b,0xd76d,0x1905,0xaa89,0xbffc, 0, 0, 0 } },
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{ { 0x7f64,0x1234,0xf47d,0xac0a,0xbffa, 0, 0, 0 } },
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{ { 0x5e26,0x57d1,0xa013,0xa7e7,0x3ffe, 0, 0, 0 } },
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{ { 0xc7aa,0x7db0,0x67e3,0x93c4,0x3ffe, 0, 0, 0 } },
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{ { 0x0000,0x0000,0x0000,0x8000,0xbfff, 0, 0, 0 } }
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};
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#endif
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#if MIEEE
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static const uLD SN[9] = {
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{ { 0x3ff50000,0x948db9f7,0x02de5dd1, 0 } },
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{ { 0x3ff70000,0xec9cc5f1,0xdd68989b, 0 } },
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{ { 0x3ff80000,0x9da5386f,0x18f02ca1, 0 } },
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{ { 0xbffa0000,0xacd787d1,0x41dd783f, 0 } },
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{ { 0xbffc0000,0xaa891905,0xd76d7a5b, 0 } },
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{ { 0xbffa0000,0xac0af47d,0x12347f64, 0 } },
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{ { 0x3ffe0000,0xa7e7a013,0x57d15e26, 0 } },
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{ { 0x3ffe0000,0x93c467e3,0x7db0c7aa, 0 } },
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{ { 0xbfff0000,0x80000000,0x00000000, 0 } }
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};
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#endif
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static long double stirf (long double);
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/* Gamma function computed by Stirling's formula. */
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static long double stirf(long double x)
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{
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long double y, w, v;
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w = 1.0L/x;
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/* For large x, use rational coefficients from the analytical expansion. */
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if (x > 1024.0L)
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w = (((((6.97281375836585777429E-5L * w
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+ 7.84039221720066627474E-4L) * w
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- 2.29472093621399176955E-4L) * w
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- 2.68132716049382716049E-3L) * w
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+ 3.47222222222222222222E-3L) * w
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+ 8.33333333333333333333E-2L) * w
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+ 1.0L;
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else
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w = 1.0L + w * polevll( w, STIR, 8 );
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y = expl(x);
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if (x > MAXSTIR)
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{ /* Avoid overflow in pow() */
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v = powl(x, 0.5L * x - 0.25L);
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y = v * (v / y);
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}
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else
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{
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y = powl(x, x - 0.5L) / y;
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}
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y = SQTPI * y * w;
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return (y);
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}
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long double __tgammal_r(long double, int *);
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long double __tgammal_r(long double x, int* sgngaml)
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{
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long double p, q, z;
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int i;
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*sgngaml = 1;
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#ifdef NANS
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if (isnanl(x))
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return x;
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#endif
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#ifdef INFINITIES
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#ifdef NANS
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if (x == INFINITYL)
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return (x);
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if (x == -INFINITYL)
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return (NANL);
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#else
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if (!isfinite(x))
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return (x);
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#endif
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#endif
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if (x == 0.0L)
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return copysignl(HUGE_VALL, x);
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q = fabsl(x);
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if (q > 13.0L)
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{
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if (q > MAXGAML)
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goto goverf;
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if (x < 0.0L)
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{
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p = floorl(q);
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if (p == q)
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{
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gsing:
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_SET_ERRNO(EDOM);
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mtherr("tgammal", SING);
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#ifdef NANS
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return (NAN);
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#else
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return (*sgngaml * MAXNUML);
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#endif
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}
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i = p;
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if ((i & 1) == 0)
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*sgngaml = -1;
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z = q - p;
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if (z > 0.5L)
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{
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p += 1.0L;
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z = q - p;
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}
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z = q * sinl(PIL * z);
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z = fabsl(z) * stirf(q);
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if (z <= PIL/MAXNUML)
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{
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goverf:
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_SET_ERRNO(ERANGE);
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mtherr("tgammal", OVERFLOW);
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#ifdef INFINITIES
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return(*sgngaml * INFINITYL);
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#else
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return(*sgngaml * MAXNUML);
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#endif
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}
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z = PIL/z;
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}
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else
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{
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z = stirf(x);
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}
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return (*sgngaml * z);
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}
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z = 1.0L;
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while (x >= 3.0L)
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{
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x -= 1.0L;
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z *= x;
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}
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while (x < -0.03125L)
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{
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z /= x;
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x += 1.0L;
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}
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if (x <= 0.03125L)
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goto Small;
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while (x < 2.0L)
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{
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z /= x;
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x += 1.0L;
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}
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if (x == 2.0L)
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return (z);
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x -= 2.0L;
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p = polevll( x, P, 7 );
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q = polevll( x, Q, 8 );
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return (z * p / q);
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Small:
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if (x == 0.0L)
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{
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goto gsing;
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}
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else
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{
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if (x < 0.0L)
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{
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x = -x;
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q = z / (x * polevll(x, SN, 8));
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}
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|
else
|
|
q = z / (x * polevll(x, S, 8));
|
|
}
|
|
return q;
|
|
}
|
|
|
|
/* This is the C99 version. */
|
|
long double tgammal(long double x)
|
|
{
|
|
int local_sgngaml = 0;
|
|
return (__tgammal_r(x, &local_sgngaml));
|
|
}
|
|
|