416 lines
9.7 KiB
C
416 lines
9.7 KiB
C
/* lgaml()
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*
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* Natural logarithm of 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, __lgammal_r();
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* int* sgngaml;
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* y = __lgammal_r( x, sgngaml );
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*
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* long double x, y, lgammal();
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* y = lgammal( x);
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*
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*
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*
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* DESCRIPTION:
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*
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* Returns the base e (2.718...) logarithm of the absolute
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* value of the gamma function of the argument. In the reentrant
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* version, the sign (+1 or -1) of the gamma function is returned
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* in the variable referenced by sgngaml.
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*
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* For arguments greater than 33, the logarithm of the gamma
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* function is approximated by the logarithmic version of
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* Stirling's formula using a polynomial approximation of
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* degree 4. Arguments between -33 and +33 are reduced by
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* recurrence to the interval [2,3] of a rational approximation.
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* The cosecant reflection formula is employed for arguments
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* less than -33.
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*
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* Arguments greater than MAXLGML (10^4928) return MAXNUML.
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*
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*
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*
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* ACCURACY:
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*
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*
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* arithmetic domain # trials peak rms
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* IEEE -40, 40 100000 2.2e-19 4.6e-20
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* IEEE 10^-2000,10^+2000 20000 1.6e-19 3.3e-20
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* The error criterion was relative when the function magnitude
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* was greater than one but absolute when it was less than one.
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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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#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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#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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#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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#include "cephes_mconf.h"
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#endif /* __MINGW32__ */
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#if UNK
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static 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 uLD 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 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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#if UNK
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static 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 uLD 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 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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/* A[]: Stirling's formula expansion of log gamma
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* B[], C[]: log gamma function between 2 and 3
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*/
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/* log gamma(x) = ( x - 0.5 ) * log(x) - x + LS2PI + 1/x A(1/x^2)
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* x >= 8
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* Peak relative error 1.51e-21
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* Relative spread of error peaks 5.67e-21
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*/
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#if UNK
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static long double A[7] = {
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4.885026142432270781165E-3L,
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-1.880801938119376907179E-3L,
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8.412723297322498080632E-4L,
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-5.952345851765688514613E-4L,
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7.936507795855070755671E-4L,
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-2.777777777750349603440E-3L,
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8.333333333333331447505E-2L,
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};
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#endif
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#if IBMPC
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static const uLD A[] = {
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{ { 0xd984,0xcc08,0x91c2,0xa012,0x3ff7, XPD } },
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{ { 0x3d91,0x0304,0x3da1,0xf685,0xbff5, XPD } },
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{ { 0x3bdc,0xaad1,0xd492,0xdc88,0x3ff4, XPD } },
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{ { 0x8b20,0x9fce,0x844e,0x9c09,0xbff4, XPD } },
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{ { 0xf8f2,0x30e5,0x0092,0xd00d,0x3ff4, XPD } },
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{ { 0x4d88,0x03a8,0x60b6,0xb60b,0xbff6, XPD } },
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{ { 0x9fcc,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 long A[21] = {
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0x3ff70000,0xa01291c2,0xcc08d984,
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0xbff50000,0xf6853da1,0x03043d91,
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0x3ff40000,0xdc88d492,0xaad13bdc,
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0xbff40000,0x9c09844e,0x9fce8b20,
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0x3ff40000,0xd00d0092,0x30e5f8f2,
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0xbff60000,0xb60b60b6,0x03a84d88,
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0x3ffb0000,0xaaaaaaaa,0xaaaa9fcc,
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};
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#endif
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/* log gamma(x+2) = x B(x)/C(x)
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* 0 <= x <= 1
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* Peak relative error 7.16e-22
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* Relative spread of error peaks 4.78e-20
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*/
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#if UNK
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static long double B[7] = {
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-2.163690827643812857640E3L,
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-8.723871522843511459790E4L,
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-1.104326814691464261197E6L,
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-6.111225012005214299996E6L,
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-1.625568062543700591014E7L,
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-2.003937418103815175475E7L,
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-8.875666783650703802159E6L,
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};
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static long double C[7] = {
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/* 1.000000000000000000000E0L,*/
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-5.139481484435370143617E2L,
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-3.403570840534304670537E4L,
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-6.227441164066219501697E5L,
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-4.814940379411882186630E6L,
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-1.785433287045078156959E7L,
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-3.138646407656182662088E7L,
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-2.099336717757895876142E7L,
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};
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#endif
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#if IBMPC
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static const uLD B[] = {
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{ { 0x9557,0x4995,0x0da1,0x873b,0xc00a, XPD } },
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{ { 0xfe44,0x9af8,0x5b8c,0xaa63,0xc00f, XPD } },
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{ { 0x5aa8,0x7cf5,0x3684,0x86ce,0xc013, XPD } },
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{ { 0x259a,0x258c,0xf206,0xba7f,0xc015, XPD } },
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{ { 0xbe18,0x1ca3,0xc0a0,0xf80a,0xc016, XPD } },
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{ { 0x168f,0x2c42,0x6717,0x98e3,0xc017, XPD } },
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{ { 0x2051,0x9d55,0x92c8,0x876e,0xc016, XPD } }
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};
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static const uLD C[] = {
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{ { 0xaa77,0xcf2f,0xae76,0x807c,0xc008, XPD } },
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{ { 0xb280,0x0d74,0xb55a,0x84f3,0xc00e, XPD } },
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{ { 0xa505,0xcd30,0x81dc,0x9809,0xc012, XPD } },
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{ { 0x3369,0x4246,0xb8c2,0x92f0,0xc015, XPD } },
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{ { 0x63cf,0x6aee,0xbe6f,0x8837,0xc017, XPD } },
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{ { 0x26bb,0xccc7,0xb009,0xef75,0xc017, XPD } },
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{ { 0x462b,0xbae8,0xab96,0xa02a,0xc017, XPD } }
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};
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#endif
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#if MIEEE
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static long B[21] = {
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0xc00a0000,0x873b0da1,0x49959557,
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0xc00f0000,0xaa635b8c,0x9af8fe44,
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0xc0130000,0x86ce3684,0x7cf55aa8,
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0xc0150000,0xba7ff206,0x258c259a,
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0xc0160000,0xf80ac0a0,0x1ca3be18,
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0xc0170000,0x98e36717,0x2c42168f,
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0xc0160000,0x876e92c8,0x9d552051,
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};
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static long C[21] = {
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/*0x3fff0000,0x80000000,0x00000000,*/
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0xc0080000,0x807cae76,0xcf2faa77,
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0xc00e0000,0x84f3b55a,0x0d74b280,
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0xc0120000,0x980981dc,0xcd30a505,
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0xc0150000,0x92f0b8c2,0x42463369,
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0xc0170000,0x8837be6f,0x6aee63cf,
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0xc0170000,0xef75b009,0xccc726bb,
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0xc0170000,0xa02aab96,0xbae8462b,
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};
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#endif
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/* log( sqrt( 2*pi ) ) */
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static const long double LS2PI = 0.91893853320467274178L;
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#define MAXLGM 1.04848146839019521116e+4928L
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/* Logarithm of gamma function */
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/* Reentrant version */
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long double __lgammal_r(long double x, int* sgngaml)
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{
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long double p, q, w, z, f, nx;
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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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if( !isfinitel(x) )
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return(INFINITYL);
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#endif
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if( x < -34.0L )
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{
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q = -x;
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w = __lgammal_r(q, sgngaml); /* note this modifies sgngam! */
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p = floorl(q);
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if( p == q )
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{
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lgsing:
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_SET_ERRNO(EDOM);
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mtherr( "lgammal", SING );
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#ifdef INFINITIES
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return (INFINITYL);
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#else
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return (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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else
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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 = p - q;
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}
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z = q * sinl( PIL * z );
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if( z == 0.0L )
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goto lgsing;
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/* z = LOGPI - logl( z ) - w; */
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z = logl( PIL/z ) - w;
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return( z );
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}
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if( x < 13.0L )
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{
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z = 1.0L;
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nx = floorl( x + 0.5L );
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f = x - nx;
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while( x >= 3.0L )
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{
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nx -= 1.0L;
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x = nx + f;
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z *= x;
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}
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while( x < 2.0L )
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{
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if( fabsl(x) <= 0.03125 )
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goto lsmall;
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z /= nx + f;
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nx += 1.0L;
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x = nx + f;
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}
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if( z < 0.0L )
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{
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*sgngaml = -1;
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z = -z;
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}
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else
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*sgngaml = 1;
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if( x == 2.0L )
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return( logl(z) );
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x = (nx - 2.0L) + f;
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p = x * polevll( x, B, 6 ) / p1evll( x, C, 7);
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return( logl(z) + p );
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}
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if( x > MAXLGM )
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{
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_SET_ERRNO(ERANGE);
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mtherr( "lgammal", 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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q = ( x - 0.5L ) * logl(x) - x + LS2PI;
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if( x > 1.0e10L )
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return(q);
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p = 1.0L/(x*x);
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q += polevll( p, A, 6 ) / x;
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return( q );
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lsmall:
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if( x == 0.0L )
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goto lgsing;
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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
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q = z / (x * polevll( x, S, 8 ));
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if( q < 0.0L )
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{
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*sgngaml = -1;
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q = -q;
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}
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else
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*sgngaml = 1;
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q = logl( q );
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return(q);
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}
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/* This is the C99 version */
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long double lgammal(long double x)
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{
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int local_sgngaml=0;
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return (__lgammal_r(x, &local_sgngaml));
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}
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