/* * ==================================================== * 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. * ==================================================== */ /* Long double expansions are Copyright (C) 2001 Stephen L. Moshier and are incorporated herein by permission of the author. The author reserves the right to distribute this material elsewhere under different copying permissions. These modifications are distributed here under the following terms: This library is free software; you can redistribute it and/or modify it under the terms of the GNU Lesser General Public License as published by the Free Software Foundation; either version 2.1 of the License, or (at your option) any later version. This library is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public License for more details. You should have received a copy of the GNU Lesser General Public License along with this library; if not, write to the Free Software Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA */ /* __ieee754_lgammal_r(x, signgamp) * 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 * */ #include "math.h" #include "math_private.h" #ifdef __STDC__ static const long double #else static long double #endif half = 0.5L, one = 1.0L, pi = 3.14159265358979323846264L, two63 = 9.223372036854775808e18L, /* 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; #ifdef __STDC__ static const long double zero = 0.0L; #else static long double zero = 0.0L; #endif #ifdef __STDC__ static long double sin_pi (long double x) #else static long double sin_pi (x) long double x; #endif { long double y, z; int n, ix; u_int32_t se, i0, i1; GET_LDOUBLE_WORDS (se, i0, i1, x); ix = se & 0x7fff; ix = (ix << 16) | (i0 >> 16); if (ix < 0x3ffd8000) /* 0.25 */ return __sinl (pi * x); y = -x; /* x is assume negative */ /* * argument reduction, make sure inexact flag not raised if input * is an integer */ z = __floorl (y); if (z != y) { /* inexact anyway */ y *= 0.5; y = 2.0*(y - __floorl(y)); /* y = |x| mod 2.0 */ n = (int) (y*4.0); } else { if (ix >= 0x403f8000) /* 2^64 */ { y = zero; n = 0; /* y must be even */ } else { if (ix < 0x403e8000) /* 2^63 */ z = y + two63; /* exact */ GET_LDOUBLE_WORDS (se, i0, i1, z); n = i1 & 1; y = n; n <<= 2; } } switch (n) { case 0: y = __sinl (pi * y); break; case 1: case 2: y = __cosl (pi * (half - y)); break; case 3: case 4: y = __sinl (pi * (one - y)); break; case 5: case 6: y = -__cosl (pi * (y - 1.5)); break; default: y = __sinl (pi * (y - 2.0)); break; } return -y; } #ifdef __STDC__ long double __ieee754_lgammal_r (long double x, int *signgamp) #else long double __ieee754_lgammal_r (x, signgamp) long double x; int *signgamp; #endif { long double t, y, z, nadj, p, p1, p2, q, r, w; int i, ix; u_int32_t se, i0, i1; *signgamp = 1; GET_LDOUBLE_WORDS (se, i0, i1, x); ix = se & 0x7fff; if ((ix | i0 | i1) == 0) { if (se & 0x8000) *signgamp = -1; return one / fabsl (x); } ix = (ix << 16) | (i0 >> 16); /* purge off +-inf, NaN, +-0, and negative arguments */ if (ix >= 0x7fff0000) return x * x; if (ix < 0x3fc08000) /* 2^-63 */ { /* |x|<2**-63, return -log(|x|) */ if (se & 0x8000) { *signgamp = -1; return -__ieee754_logl (-x); } else return -__ieee754_logl (x); } if (se & 0x8000) { t = sin_pi (x); if (t == zero) return one / fabsl (t); /* -integer */ nadj = __ieee754_logl (pi / fabsl (t * x)); if (t < zero) *signgamp = -1; x = -x; } /* purge off 1 and 2 */ if ((((ix - 0x3fff8000) | i0 | i1) == 0) || (((ix - 0x40008000) | i0 | i1) == 0)) r = 0; else if (ix < 0x40008000) /* 2.0 */ { /* x < 2.0 */ if (ix <= 0x3ffee666) /* 8.99993896484375e-1 */ { /* lgamma(x) = lgamma(x+1) - log(x) */ r = -__ieee754_logl (x); if (ix >= 0x3ffebb4a) /* 7.31597900390625e-1 */ { y = x - one; i = 0; } else if (ix >= 0x3ffced33)/* 2.31639862060546875e-1 */ { y = x - (tc - one); i = 1; } else { /* x < 0.23 */ y = x; i = 2; } } else { r = zero; 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 - one; 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 += half * 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 += (-half * y + p1 / p2); } } else if (ix < 0x40028000) /* 8.0 */ { /* x < 8.0 */ i = (int) x; t = zero; 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 = half * y + p / q; z = one; /* 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 += __ieee754_logl (z); break; } } else if (ix < 0x40418000) /* 2^66 */ { /* 8.0 <= x < 2**66 */ t = __ieee754_logl (x); z = one / x; y = z * z; w = w0 + z * (w1 + y * (w2 + y * (w3 + y * (w4 + y * (w5 + y * (w6 + y * w7)))))); r = (x - half) * (t - one) + w; } else /* 2**66 <= x <= inf */ r = x * (__ieee754_logl (x) - one); if (se & 0x8000) r = nadj - r; return r; }