502 lines
11 KiB
C
502 lines
11 KiB
C
/* gammal.c
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*
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* Gamma function
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*
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*
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*
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* SYNOPSIS:
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*
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* long double x, y, __tgammal_r();
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* int* sgngaml;
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* y = __tgammal_r( x, sgngaml );
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*
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* long double x, y, tgammal();
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* y = tgammal( x); *
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*
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*
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* DESCRIPTION:
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*
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* Returns gamma function of the argument. The result is
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* correctly signed. In the reentrant version the sign (+1 or -1)
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* is returned in the variable referenced by sgngamf.
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*
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* Arguments |x| <= 13 are reduced by recurrence and the function
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* approximated by a rational function of degree 7/8 in the
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* interval (2,3). Large arguments are handled by Stirling's
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* formula. Large negative arguments are made positive using
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* a reflection formula.
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*
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*
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* ACCURACY:
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*
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* Relative error:
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* arithmetic domain # trials peak rms
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* IEEE -40,+40 10000 3.6e-19 7.9e-20
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* IEEE -1755,+1755 10000 4.8e-18 6.5e-19
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*
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* Accuracy for large arguments is dominated by error in powl().
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*
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*/
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/*
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Copyright 1994 by Stephen L. Moshier
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*/
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/*
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* 26-11-2002 Modified for mingw.
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* Danny Smith <dannysmith@users.sourceforge.net>
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*/
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#ifndef __MINGW32__
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#include "mconf.h"
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#else
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#include "cephes_mconf.h"
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#endif
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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 long double 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 long double 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 unsigned short P[] = {
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0x434a,0x3f22,0x2bda,0xb0b2,0x3ff0, XPD
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0xf5aa,0xe82f,0x335b,0xee2e,0x3ff3, XPD
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0xbe6c,0x3757,0xc717,0x861b,0x3ff7, XPD
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0x7f43,0x5196,0xb166,0xc368,0x3ff9, XPD
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0x9549,0x8eb5,0x8c3a,0xe3f4,0x3ffb, XPD
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0x8d75,0x23af,0xc8e4,0xb9d4,0x3ffd, XPD
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0x29cf,0x19b3,0x16c8,0xd67a,0x3ffe, XPD
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0x0000,0x0000,0x0000,0x8000,0x3fff, XPD
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};
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static const unsigned short Q[] = {
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0x5473,0x2de8,0x1268,0xea67,0xbfee, XPD
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0x334b,0xc2f0,0xa2dd,0xf60e,0x3ff2, XPD
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0xbeed,0x1853,0xa691,0xa23d,0xbff5, XPD
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0x296e,0x7cb1,0x5dfd,0xd08f,0xbff4, XPD
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0x0417,0x7989,0xd7bc,0xe338,0x3ff9, XPD
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0x3295,0x3698,0xd580,0xbdcd,0xbffa, XPD
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0x75ef,0x3ab7,0x4ad3,0xe5bc,0xbffc, XPD
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0xe458,0x2ec7,0xfd57,0xd47c,0x3ffd, XPD
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0x0000,0x0000,0x0000,0x8000,0x3fff, XPD
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};
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#endif
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#if MIEEE
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static const long P[24] = {
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0x3ff00000,0xb0b22bda,0x3f22434a,
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0x3ff30000,0xee2e335b,0xe82ff5aa,
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0x3ff70000,0x861bc717,0x3757be6c,
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0x3ff90000,0xc368b166,0x51967f43,
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0x3ffb0000,0xe3f48c3a,0x8eb59549,
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0x3ffd0000,0xb9d4c8e4,0x23af8d75,
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0x3ffe0000,0xd67a16c8,0x19b329cf,
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0x3fff0000,0x80000000,0x00000000,
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};
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static const long Q[27] = {
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0xbfee0000,0xea671268,0x2de85473,
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0x3ff20000,0xf60ea2dd,0xc2f0334b,
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0xbff50000,0xa23da691,0x1853beed,
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0xbff40000,0xd08f5dfd,0x7cb1296e,
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0x3ff90000,0xe338d7bc,0x79890417,
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0xbffa0000,0xbdcdd580,0x36983295,
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0xbffc0000,0xe5bc4ad3,0x3ab775ef,
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0x3ffd0000,0xd47cfd57,0x2ec7e458,
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0x3fff0000,0x80000000,0x00000000,
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};
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#endif
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/*
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static const long double P[] = {
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-3.01525602666895735709e0L,
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-3.25157411956062339893e1L,
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-2.92929976820724030353e2L,
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-1.70730828800510297666e3L,
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-7.96667499622741999770e3L,
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-2.59780216007146401957e4L,
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-5.99650230220855581642e4L,
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-7.15743521530849602425e4L
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};
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static const long double Q[] = {
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1.00000000000000000000e0L,
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-1.67955233807178858919e1L,
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8.85946791747759881659e1L,
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5.69440799097468430177e1L,
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-1.98526250512761318471e3L,
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3.31667508019495079814e3L,
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1.60577839621734713377e4L,
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-2.97045081369399940529e4L,
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-7.15743521530849602412e4L
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};
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*/
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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 long double 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 unsigned short STIR[] = {
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0x6ede,0x69f7,0x54e3,0xbb5d,0x3ff4, XPD
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0xc395,0x0295,0x4443,0xc64b,0xbfef, XPD
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0xba6f,0x7c59,0x5e47,0x9bfb,0xbff4, XPD
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0x5704,0x1a39,0xb11d,0x9293,0x3ff1, XPD
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0x30b7,0x1a21,0x98b2,0xcd87,0x3ff4, XPD
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0xbef3,0x7023,0x6a08,0xf09e,0xbff2, XPD
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0x3a1c,0x5ac8,0x3478,0xafb9,0xbff6, XPD
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0xc3c9,0x906e,0x38e3,0xe38e,0x3ff6, XPD
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0xa1d5,0xaaaa,0xaaaa,0xaaaa,0x3ffb, XPD
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};
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#endif
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#if MIEEE
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static const long STIR[27] = {
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0x3ff40000,0xbb5d54e3,0x69f76ede,
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0xbfef0000,0xc64b4443,0x0295c395,
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0xbff40000,0x9bfb5e47,0x7c59ba6f,
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0x3ff10000,0x9293b11d,0x1a395704,
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0x3ff40000,0xcd8798b2,0x1a2130b7,
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0xbff20000,0xf09e6a08,0x7023bef3,
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0xbff60000,0xafb93478,0x5ac83a1c,
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0x3ff60000,0xe38e38e3,0x906ec3c9,
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0x3ffb0000,0xaaaaaaaa,0xaaaaa1d5,
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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 long double 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 unsigned short S[] = {
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0xbaeb,0xd6d3,0x25e5,0x9c7e,0xbff5, XPD
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0xfe9a,0xceb4,0xc74e,0xec9a,0x3ff7, XPD
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0x9225,0xdfef,0xb0e9,0x9da5,0xbff8, XPD
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0x10b0,0xec17,0x87dc,0xacd7,0xbffa, XPD
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0x6b8d,0x7515,0x1905,0xaa89,0x3ffc, XPD
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0xf183,0x126b,0xf47d,0xac0a,0xbffa, XPD
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0x7bf6,0x57d1,0xa013,0xa7e7,0xbffe, XPD
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0xc7a9,0x7db0,0x67e3,0x93c4,0x3ffe, XPD
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0x0000,0x0000,0x0000,0x8000,0x3fff, XPD
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};
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#endif
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#if MIEEE
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static const long S[27] = {
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0xbff50000,0x9c7e25e5,0xd6d3baeb,
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0x3ff70000,0xec9ac74e,0xceb4fe9a,
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0xbff80000,0x9da5b0e9,0xdfef9225,
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0xbffa0000,0xacd787dc,0xec1710b0,
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0x3ffc0000,0xaa891905,0x75156b8d,
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0xbffa0000,0xac0af47d,0x126bf183,
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0xbffe0000,0xa7e7a013,0x57d17bf6,
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0x3ffe0000,0x93c467e3,0x7db0c7a9,
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0x3fff0000,0x80000000,0x00000000,
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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 long double 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 unsigned short SN[] = {
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0x5dd1,0x02de,0xb9f7,0x948d,0x3ff5, XPD
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0x989b,0xdd68,0xc5f1,0xec9c,0x3ff7, XPD
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0x2ca1,0x18f0,0x386f,0x9da5,0x3ff8, XPD
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0x783f,0x41dd,0x87d1,0xacd7,0xbffa, XPD
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0x7a5b,0xd76d,0x1905,0xaa89,0xbffc, XPD
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0x7f64,0x1234,0xf47d,0xac0a,0xbffa, XPD
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0x5e26,0x57d1,0xa013,0xa7e7,0x3ffe, XPD
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0xc7aa,0x7db0,0x67e3,0x93c4,0x3ffe, XPD
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0x0000,0x0000,0x0000,0x8000,0xbfff, XPD
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};
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#endif
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#if MIEEE
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static const long SN[27] = {
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0x3ff50000,0x948db9f7,0x02de5dd1,
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0x3ff70000,0xec9cc5f1,0xdd68989b,
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0x3ff80000,0x9da5386f,0x18f02ca1,
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0xbffa0000,0xacd787d1,0x41dd783f,
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0xbffc0000,0xaa891905,0xd76d7a5b,
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0xbffa0000,0xac0af47d,0x12347f64,
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0x3ffe0000,0xa7e7a013,0x57d15e26,
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0x3ffe0000,0x93c467e3,0x7db0c7aa,
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0xbfff0000,0x80000000,0x00000000,
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};
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#endif
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#ifndef __MINGW32__
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extern long double MAXLOGL, MAXNUML, PIL;
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/* #define PIL 3.14159265358979323846L */
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/* #define MAXNUML 1.189731495357231765021263853E4932L */
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#ifdef ANSIPROT
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extern long double fabsl ( long double );
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extern long double lgaml ( long double );
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extern long double logl ( long double );
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extern long double expl ( long double );
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extern long double gammal ( long double );
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extern long double sinl ( long double );
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extern long double floorl ( long double );
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extern long double powl ( long double, long double );
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extern long double polevll ( long double, void *, int );
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extern long double p1evll ( long double, void *, int );
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extern int isnanl ( long double );
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extern int isfinitel ( long double );
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static long double stirf ( long double );
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#else
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long double fabsl(), lgaml(), logl(), expl(), gammal(), sinl();
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long double floorl(), powl(), polevll(), p1evll(), isnanl(), isfinitel();
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static long double stirf();
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#endif
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#ifdef INFINITIES
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extern long double INFINITYL;
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#endif
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#ifdef NANS
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extern long double NANL;
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#endif
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#else /* __MINGW32__ */
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static long double stirf ( long double );
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#endif
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/* Gamma function computed by Stirling's formula. */
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static long double stirf(x)
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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 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(NANL);
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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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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 INFINITIES
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return (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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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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|
{
|
|
p += 1.0L;
|
|
z = q - p;
|
|
}
|
|
z = q * sinl( PIL * z );
|
|
z = fabsl(z) * stirf(q);
|
|
if( z <= PIL/MAXNUML )
|
|
{
|
|
goverf:
|
|
_SET_ERRNO(ERANGE);
|
|
mtherr( "tgammal", OVERFLOW );
|
|
#ifdef INFINITIES
|
|
return( *sgngaml * INFINITYL);
|
|
#else
|
|
return( *sgngaml * MAXNUML);
|
|
#endif
|
|
}
|
|
z = PIL/z;
|
|
}
|
|
else
|
|
{
|
|
z = stirf(x);
|
|
}
|
|
return( *sgngaml * z );
|
|
}
|
|
|
|
z = 1.0L;
|
|
while( x >= 3.0L )
|
|
{
|
|
x -= 1.0L;
|
|
z *= x;
|
|
}
|
|
|
|
while( x < -0.03125L )
|
|
{
|
|
z /= x;
|
|
x += 1.0L;
|
|
}
|
|
|
|
if( x <= 0.03125L )
|
|
goto Small;
|
|
|
|
while( x < 2.0L )
|
|
{
|
|
z /= x;
|
|
x += 1.0L;
|
|
}
|
|
|
|
if( x == 2.0L )
|
|
return(z);
|
|
|
|
x -= 2.0L;
|
|
p = polevll( x, P, 7 );
|
|
q = polevll( x, Q, 8 );
|
|
return( z * p / q );
|
|
|
|
Small:
|
|
if( x == 0.0L )
|
|
{
|
|
goto gsing;
|
|
}
|
|
else
|
|
{
|
|
if( x < 0.0L )
|
|
{
|
|
x = -x;
|
|
q = z / (x * polevll( x, SN, 8 ));
|
|
}
|
|
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));
|
|
}
|
|
|