/* origin: OpenBSD /usr/src/lib/libm/src/ld80/e_lgammal.c */ /* * ==================================================== * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. * * Developed at SunPro, a Sun Microsystems, Inc. business. * Permission to use, copy, modify, and distribute this * software is freely granted, provided that this notice * is preserved. * ==================================================== */ /* * Copyright (c) 2008 Stephen L. Moshier * * Permission to use, copy, modify, and distribute this software for any * purpose with or without fee is hereby granted, provided that the above * copyright notice and this permission notice appear in all copies. * * THE SOFTWARE IS PROVIDED "AS IS" AND THE AUTHOR DISCLAIMS ALL WARRANTIES * WITH REGARD TO THIS SOFTWARE INCLUDING ALL IMPLIED WARRANTIES OF * MERCHANTABILITY AND FITNESS. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR * ANY SPECIAL, DIRECT, INDIRECT, OR CONSEQUENTIAL DAMAGES OR ANY DAMAGES * WHATSOEVER RESULTING FROM LOSS OF USE, DATA OR PROFITS, WHETHER IN AN * ACTION OF CONTRACT, NEGLIGENCE OR OTHER TORTIOUS ACTION, ARISING OUT OF * OR IN CONNECTION WITH THE USE OR PERFORMANCE OF THIS SOFTWARE. */ /* lgammal(x) * Reentrant version of the logarithm of the Gamma function * with user provide pointer for the sign of Gamma(x). * * Method: * 1. Argument Reduction for 0 < x <= 8 * Since gamma(1+s)=s*gamma(s), for x in [0,8], we may * reduce x to a number in [1.5,2.5] by * lgamma(1+s) = log(s) + lgamma(s) * for example, * lgamma(7.3) = log(6.3) + lgamma(6.3) * = log(6.3*5.3) + lgamma(5.3) * = log(6.3*5.3*4.3*3.3*2.3) + lgamma(2.3) * 2. Polynomial approximation of lgamma around its * minimun ymin=1.461632144968362245 to maintain monotonicity. * On [ymin-0.23, ymin+0.27] (i.e., [1.23164,1.73163]), use * Let z = x-ymin; * lgamma(x) = -1.214862905358496078218 + z^2*poly(z) * 2. Rational approximation in the primary interval [2,3] * We use the following approximation: * s = x-2.0; * lgamma(x) = 0.5*s + s*P(s)/Q(s) * Our algorithms are based on the following observation * * zeta(2)-1 2 zeta(3)-1 3 * lgamma(2+s) = s*(1-Euler) + --------- * s - --------- * s + ... * 2 3 * * where Euler = 0.5771... is the Euler constant, which is very * close to 0.5. * * 3. For x>=8, we have * lgamma(x)~(x-0.5)log(x)-x+0.5*log(2pi)+1/(12x)-1/(360x**3)+.... * (better formula: * lgamma(x)~(x-0.5)*(log(x)-1)-.5*(log(2pi)-1) + ...) * Let z = 1/x, then we approximation * f(z) = lgamma(x) - (x-0.5)(log(x)-1) * by * 3 5 11 * w = w0 + w1*z + w2*z + w3*z + ... + w6*z * * 4. For negative x, since (G is gamma function) * -x*G(-x)*G(x) = pi/sin(pi*x), * we have * G(x) = pi/(sin(pi*x)*(-x)*G(-x)) * since G(-x) is positive, sign(G(x)) = sign(sin(pi*x)) for x<0 * Hence, for x<0, signgam = sign(sin(pi*x)) and * lgamma(x) = log(|Gamma(x)|) * = log(pi/(|x*sin(pi*x)|)) - lgamma(-x); * Note: one should avoid compute pi*(-x) directly in the * computation of sin(pi*(-x)). * * 5. Special Cases * lgamma(2+s) ~ s*(1-Euler) for tiny s * lgamma(1)=lgamma(2)=0 * lgamma(x) ~ -log(x) for tiny x * lgamma(0) = lgamma(inf) = inf * lgamma(-integer) = +-inf * */ #define _GNU_SOURCE #include "libm.h" #if LDBL_MANT_DIG == 53 && LDBL_MAX_EXP == 1024 long double __lgammal_r(long double x, int *sg) { return __lgamma_r(x, sg); } #elif LDBL_MANT_DIG == 64 && LDBL_MAX_EXP == 16384 static const long double pi = 3.14159265358979323846264L, /* lgam(1+x) = 0.5 x + x a(x)/b(x) -0.268402099609375 <= x <= 0 peak relative error 6.6e-22 */ a0 = -6.343246574721079391729402781192128239938E2L, a1 = 1.856560238672465796768677717168371401378E3L, a2 = 2.404733102163746263689288466865843408429E3L, a3 = 8.804188795790383497379532868917517596322E2L, a4 = 1.135361354097447729740103745999661157426E2L, a5 = 3.766956539107615557608581581190400021285E0L, b0 = 8.214973713960928795704317259806842490498E3L, b1 = 1.026343508841367384879065363925870888012E4L, b2 = 4.553337477045763320522762343132210919277E3L, b3 = 8.506975785032585797446253359230031874803E2L, b4 = 6.042447899703295436820744186992189445813E1L, /* b5 = 1.000000000000000000000000000000000000000E0 */ tc = 1.4616321449683623412626595423257213284682E0L, tf = -1.2148629053584961146050602565082954242826E-1, /* double precision */ /* tt = (tail of tf), i.e. tf + tt has extended precision. */ tt = 3.3649914684731379602768989080467587736363E-18L, /* lgam ( 1.4616321449683623412626595423257213284682E0 ) = -1.2148629053584960809551455717769158215135617312999903886372437313313530E-1 */ /* lgam (x + tc) = tf + tt + x g(x)/h(x) -0.230003726999612341262659542325721328468 <= x <= 0.2699962730003876587373404576742786715318 peak relative error 2.1e-21 */ g0 = 3.645529916721223331888305293534095553827E-18L, g1 = 5.126654642791082497002594216163574795690E3L, g2 = 8.828603575854624811911631336122070070327E3L, g3 = 5.464186426932117031234820886525701595203E3L, g4 = 1.455427403530884193180776558102868592293E3L, g5 = 1.541735456969245924860307497029155838446E2L, g6 = 4.335498275274822298341872707453445815118E0L, h0 = 1.059584930106085509696730443974495979641E4L, h1 = 2.147921653490043010629481226937850618860E4L, h2 = 1.643014770044524804175197151958100656728E4L, h3 = 5.869021995186925517228323497501767586078E3L, h4 = 9.764244777714344488787381271643502742293E2L, h5 = 6.442485441570592541741092969581997002349E1L, /* h6 = 1.000000000000000000000000000000000000000E0 */ /* lgam (x+1) = -0.5 x + x u(x)/v(x) -0.100006103515625 <= x <= 0.231639862060546875 peak relative error 1.3e-21 */ u0 = -8.886217500092090678492242071879342025627E1L, u1 = 6.840109978129177639438792958320783599310E2L, u2 = 2.042626104514127267855588786511809932433E3L, u3 = 1.911723903442667422201651063009856064275E3L, u4 = 7.447065275665887457628865263491667767695E2L, u5 = 1.132256494121790736268471016493103952637E2L, u6 = 4.484398885516614191003094714505960972894E0L, v0 = 1.150830924194461522996462401210374632929E3L, v1 = 3.399692260848747447377972081399737098610E3L, v2 = 3.786631705644460255229513563657226008015E3L, v3 = 1.966450123004478374557778781564114347876E3L, v4 = 4.741359068914069299837355438370682773122E2L, v5 = 4.508989649747184050907206782117647852364E1L, /* v6 = 1.000000000000000000000000000000000000000E0 */ /* lgam (x+2) = .5 x + x s(x)/r(x) 0 <= x <= 1 peak relative error 7.2e-22 */ s0 = 1.454726263410661942989109455292824853344E6L, s1 = -3.901428390086348447890408306153378922752E6L, s2 = -6.573568698209374121847873064292963089438E6L, s3 = -3.319055881485044417245964508099095984643E6L, s4 = -7.094891568758439227560184618114707107977E5L, s5 = -6.263426646464505837422314539808112478303E4L, s6 = -1.684926520999477529949915657519454051529E3L, r0 = -1.883978160734303518163008696712983134698E7L, r1 = -2.815206082812062064902202753264922306830E7L, r2 = -1.600245495251915899081846093343626358398E7L, r3 = -4.310526301881305003489257052083370058799E6L, r4 = -5.563807682263923279438235987186184968542E5L, r5 = -3.027734654434169996032905158145259713083E4L, r6 = -4.501995652861105629217250715790764371267E2L, /* r6 = 1.000000000000000000000000000000000000000E0 */ /* lgam(x) = ( x - 0.5 ) * log(x) - x + LS2PI + 1/x w(1/x^2) x >= 8 Peak relative error 1.51e-21 w0 = LS2PI - 0.5 */ w0 = 4.189385332046727417803e-1L, w1 = 8.333333333333331447505E-2L, w2 = -2.777777777750349603440E-3L, w3 = 7.936507795855070755671E-4L, w4 = -5.952345851765688514613E-4L, w5 = 8.412723297322498080632E-4L, w6 = -1.880801938119376907179E-3L, w7 = 4.885026142432270781165E-3L; /* sin(pi*x) assuming x > 2^-1000, if sin(pi*x)==0 the sign is arbitrary */ static long double sin_pi(long double x) { int n; /* spurious inexact if odd int */ x *= 0.5; x = 2.0*(x - floorl(x)); /* x mod 2.0 */ n = (int)(x*4.0); n = (n+1)/2; x -= n*0.5f; x *= pi; switch (n) { default: /* case 4: */ case 0: return __sinl(x, 0.0, 0); case 1: return __cosl(x, 0.0); case 2: return __sinl(-x, 0.0, 0); case 3: return -__cosl(x, 0.0); } } long double __lgammal_r(long double x, int *sg) { long double t, y, z, nadj, p, p1, p2, q, r, w; union ldshape u = {x}; uint32_t ix = (u.i.se & 0x7fffU)<<16 | u.i.m>>48; int sign = u.i.se >> 15; int i; *sg = 1; /* purge off +-inf, NaN, +-0, tiny and negative arguments */ if (ix >= 0x7fff0000) return x * x; if (ix < 0x3fc08000) { /* |x|<2**-63, return -log(|x|) */ if (sign) { *sg = -1; x = -x; } return -logl(x); } if (sign) { x = -x; t = sin_pi(x); if (t == 0.0) return 1.0 / (x-x); /* -integer */ if (t > 0.0) *sg = -1; else t = -t; nadj = logl(pi / (t * x)); } /* purge off 1 and 2 (so the sign is ok with downward rounding) */ if ((ix == 0x3fff8000 || ix == 0x40008000) && u.i.m == 0) { r = 0; } else if (ix < 0x40008000) { /* x < 2.0 */ if (ix <= 0x3ffee666) { /* 8.99993896484375e-1 */ /* lgamma(x) = lgamma(x+1) - log(x) */ r = -logl(x); if (ix >= 0x3ffebb4a) { /* 7.31597900390625e-1 */ y = x - 1.0; i = 0; } else if (ix >= 0x3ffced33) { /* 2.31639862060546875e-1 */ y = x - (tc - 1.0); i = 1; } else { /* x < 0.23 */ y = x; i = 2; } } else { r = 0.0; if (ix >= 0x3fffdda6) { /* 1.73162841796875 */ /* [1.7316,2] */ y = x - 2.0; i = 0; } else if (ix >= 0x3fff9da6) { /* 1.23162841796875 */ /* [1.23,1.73] */ y = x - tc; i = 1; } else { /* [0.9, 1.23] */ y = x - 1.0; i = 2; } } switch (i) { case 0: p1 = a0 + y * (a1 + y * (a2 + y * (a3 + y * (a4 + y * a5)))); p2 = b0 + y * (b1 + y * (b2 + y * (b3 + y * (b4 + y)))); r += 0.5 * y + y * p1/p2; break; case 1: p1 = g0 + y * (g1 + y * (g2 + y * (g3 + y * (g4 + y * (g5 + y * g6))))); p2 = h0 + y * (h1 + y * (h2 + y * (h3 + y * (h4 + y * (h5 + y))))); p = tt + y * p1/p2; r += (tf + p); break; case 2: p1 = y * (u0 + y * (u1 + y * (u2 + y * (u3 + y * (u4 + y * (u5 + y * u6)))))); p2 = v0 + y * (v1 + y * (v2 + y * (v3 + y * (v4 + y * (v5 + y))))); r += (-0.5 * y + p1 / p2); } } else if (ix < 0x40028000) { /* 8.0 */ /* x < 8.0 */ i = (int)x; y = x - (double)i; p = y * (s0 + y * (s1 + y * (s2 + y * (s3 + y * (s4 + y * (s5 + y * s6)))))); q = r0 + y * (r1 + y * (r2 + y * (r3 + y * (r4 + y * (r5 + y * (r6 + y)))))); r = 0.5 * y + p / q; z = 1.0; /* lgamma(1+s) = log(s) + lgamma(s) */ switch (i) { case 7: z *= (y + 6.0); /* FALLTHRU */ case 6: z *= (y + 5.0); /* FALLTHRU */ case 5: z *= (y + 4.0); /* FALLTHRU */ case 4: z *= (y + 3.0); /* FALLTHRU */ case 3: z *= (y + 2.0); /* FALLTHRU */ r += logl(z); break; } } else if (ix < 0x40418000) { /* 2^66 */ /* 8.0 <= x < 2**66 */ t = logl(x); z = 1.0 / x; y = z * z; w = w0 + z * (w1 + y * (w2 + y * (w3 + y * (w4 + y * (w5 + y * (w6 + y * w7)))))); r = (x - 0.5) * (t - 1.0) + w; } else /* 2**66 <= x <= inf */ r = x * (logl(x) - 1.0); if (sign) r = nadj - r; return r; } #elif LDBL_MANT_DIG == 113 && LDBL_MAX_EXP == 16384 // TODO: broken implementation to make things compile long double __lgammal_r(long double x, int *sg) { return __lgamma_r(x, sg); } #endif long double lgammal(long double x) { return __lgammal_r(x, &__signgam); } weak_alias(__lgammal_r, lgammal_r);