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Diffstat (limited to 'src/math/lgammal.c')
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diff --git a/src/math/lgammal.c b/src/math/lgammal.c new file mode 100644 index 00000000..603477c9 --- /dev/null +++ b/src/math/lgammal.c @@ -0,0 +1,393 @@ +/* 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 <steve@moshier.net> + * + * 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 + * + */ + +#include "libm.h" + +long double lgammal(long double x) +{ + return lgammal_r(x, &signgam); +} + +#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 +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; + +static const long double zero = 0.0L; + +static long double sin_pi(long double x) +{ + long double y, z; + int n, ix; + uint32_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; /* y must be even */ + n = 0; + } 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; +} + +long double lgammal_r(long double x, int *sg) { + long double t, y, z, nadj, p, p1, p2, q, r, w; + int i, ix; + uint32_t se, i0, i1; + + *sg = 1; + GET_LDOUBLE_WORDS(se, i0, i1, x); + ix = se & 0x7fff; + + if ((ix | i0 | i1) == 0) { + if (se & 0x8000) + *sg = -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) { /* |x|<2**-63, return -log(|x|) */ + if (se & 0x8000) { + *sg = -1; + return -logl(-x); + } + return -logl(x); + } + if (se & 0x8000) { + t = sin_pi (x); + if (t == zero) + return one / fabsl(t); /* -integer */ + nadj = logl(pi / fabsl(t * x)); + if (t < zero) + *sg = -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) { /* 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 - 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 += logl (z); + break; + } + } else if (ix < 0x40418000) { /* 2^66 */ + /* 8.0 <= x < 2**66 */ + t = 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 * (logl (x) - one); + if (se & 0x8000) + r = nadj - r; + return r; +} +#endif |