| 1 | /* origin: OpenBSD /usr/src/lib/libm/src/ld80/e_lgammal.c */
|
|---|
| 2 | /*
|
|---|
| 3 | * ====================================================
|
|---|
| 4 | * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
|
|---|
| 5 | *
|
|---|
| 6 | * Developed at SunPro, a Sun Microsystems, Inc. business.
|
|---|
| 7 | * Permission to use, copy, modify, and distribute this
|
|---|
| 8 | * software is freely granted, provided that this notice
|
|---|
| 9 | * is preserved.
|
|---|
| 10 | * ====================================================
|
|---|
| 11 | */
|
|---|
| 12 | /*
|
|---|
| 13 | * Copyright (c) 2008 Stephen L. Moshier <steve@moshier.net>
|
|---|
| 14 | *
|
|---|
| 15 | * Permission to use, copy, modify, and distribute this software for any
|
|---|
| 16 | * purpose with or without fee is hereby granted, provided that the above
|
|---|
| 17 | * copyright notice and this permission notice appear in all copies.
|
|---|
| 18 | *
|
|---|
| 19 | * THE SOFTWARE IS PROVIDED "AS IS" AND THE AUTHOR DISCLAIMS ALL WARRANTIES
|
|---|
| 20 | * WITH REGARD TO THIS SOFTWARE INCLUDING ALL IMPLIED WARRANTIES OF
|
|---|
| 21 | * MERCHANTABILITY AND FITNESS. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR
|
|---|
| 22 | * ANY SPECIAL, DIRECT, INDIRECT, OR CONSEQUENTIAL DAMAGES OR ANY DAMAGES
|
|---|
| 23 | * WHATSOEVER RESULTING FROM LOSS OF USE, DATA OR PROFITS, WHETHER IN AN
|
|---|
| 24 | * ACTION OF CONTRACT, NEGLIGENCE OR OTHER TORTIOUS ACTION, ARISING OUT OF
|
|---|
| 25 | * OR IN CONNECTION WITH THE USE OR PERFORMANCE OF THIS SOFTWARE.
|
|---|
| 26 | */
|
|---|
| 27 | /* lgammal(x)
|
|---|
| 28 | * Reentrant version of the logarithm of the Gamma function
|
|---|
| 29 | * with user provide pointer for the sign of Gamma(x).
|
|---|
| 30 | *
|
|---|
| 31 | * Method:
|
|---|
| 32 | * 1. Argument Reduction for 0 < x <= 8
|
|---|
| 33 | * Since gamma(1+s)=s*gamma(s), for x in [0,8], we may
|
|---|
| 34 | * reduce x to a number in [1.5,2.5] by
|
|---|
| 35 | * lgamma(1+s) = log(s) + lgamma(s)
|
|---|
| 36 | * for example,
|
|---|
| 37 | * lgamma(7.3) = log(6.3) + lgamma(6.3)
|
|---|
| 38 | * = log(6.3*5.3) + lgamma(5.3)
|
|---|
| 39 | * = log(6.3*5.3*4.3*3.3*2.3) + lgamma(2.3)
|
|---|
| 40 | * 2. Polynomial approximation of lgamma around its
|
|---|
| 41 | * minimun ymin=1.461632144968362245 to maintain monotonicity.
|
|---|
| 42 | * On [ymin-0.23, ymin+0.27] (i.e., [1.23164,1.73163]), use
|
|---|
| 43 | * Let z = x-ymin;
|
|---|
| 44 | * lgamma(x) = -1.214862905358496078218 + z^2*poly(z)
|
|---|
| 45 | * 2. Rational approximation in the primary interval [2,3]
|
|---|
| 46 | * We use the following approximation:
|
|---|
| 47 | * s = x-2.0;
|
|---|
| 48 | * lgamma(x) = 0.5*s + s*P(s)/Q(s)
|
|---|
| 49 | * Our algorithms are based on the following observation
|
|---|
| 50 | *
|
|---|
| 51 | * zeta(2)-1 2 zeta(3)-1 3
|
|---|
| 52 | * lgamma(2+s) = s*(1-Euler) + --------- * s - --------- * s + ...
|
|---|
| 53 | * 2 3
|
|---|
| 54 | *
|
|---|
| 55 | * where Euler = 0.5771... is the Euler constant, which is very
|
|---|
| 56 | * close to 0.5.
|
|---|
| 57 | *
|
|---|
| 58 | * 3. For x>=8, we have
|
|---|
| 59 | * lgamma(x)~(x-0.5)log(x)-x+0.5*log(2pi)+1/(12x)-1/(360x**3)+....
|
|---|
| 60 | * (better formula:
|
|---|
| 61 | * lgamma(x)~(x-0.5)*(log(x)-1)-.5*(log(2pi)-1) + ...)
|
|---|
| 62 | * Let z = 1/x, then we approximation
|
|---|
| 63 | * f(z) = lgamma(x) - (x-0.5)(log(x)-1)
|
|---|
| 64 | * by
|
|---|
| 65 | * 3 5 11
|
|---|
| 66 | * w = w0 + w1*z + w2*z + w3*z + ... + w6*z
|
|---|
| 67 | *
|
|---|
| 68 | * 4. For negative x, since (G is gamma function)
|
|---|
| 69 | * -x*G(-x)*G(x) = pi/sin(pi*x),
|
|---|
| 70 | * we have
|
|---|
| 71 | * G(x) = pi/(sin(pi*x)*(-x)*G(-x))
|
|---|
| 72 | * since G(-x) is positive, sign(G(x)) = sign(sin(pi*x)) for x<0
|
|---|
| 73 | * Hence, for x<0, signgam = sign(sin(pi*x)) and
|
|---|
| 74 | * lgamma(x) = log(|Gamma(x)|)
|
|---|
| 75 | * = log(pi/(|x*sin(pi*x)|)) - lgamma(-x);
|
|---|
| 76 | * Note: one should avoid compute pi*(-x) directly in the
|
|---|
| 77 | * computation of sin(pi*(-x)).
|
|---|
| 78 | *
|
|---|
| 79 | * 5. Special Cases
|
|---|
| 80 | * lgamma(2+s) ~ s*(1-Euler) for tiny s
|
|---|
| 81 | * lgamma(1)=lgamma(2)=0
|
|---|
| 82 | * lgamma(x) ~ -log(x) for tiny x
|
|---|
| 83 | * lgamma(0) = lgamma(inf) = inf
|
|---|
| 84 | * lgamma(-integer) = +-inf
|
|---|
| 85 | *
|
|---|
| 86 | */
|
|---|
| 87 |
|
|---|
| 88 | #define _GNU_SOURCE
|
|---|
| 89 | #include "libm.h"
|
|---|
| 90 | #include "libc.h"
|
|---|
| 91 |
|
|---|
| 92 | #if LDBL_MANT_DIG == 53 && LDBL_MAX_EXP == 1024
|
|---|
| 93 | double __lgamma_r(double x, int *sg);
|
|---|
| 94 |
|
|---|
| 95 | long double __lgammal_r(long double x, int *sg)
|
|---|
| 96 | {
|
|---|
| 97 | return __lgamma_r(x, sg);
|
|---|
| 98 | }
|
|---|
| 99 | #elif LDBL_MANT_DIG == 64 && LDBL_MAX_EXP == 16384
|
|---|
| 100 | static const long double
|
|---|
| 101 | pi = 3.14159265358979323846264L,
|
|---|
| 102 |
|
|---|
| 103 | /* lgam(1+x) = 0.5 x + x a(x)/b(x)
|
|---|
| 104 | -0.268402099609375 <= x <= 0
|
|---|
| 105 | peak relative error 6.6e-22 */
|
|---|
| 106 | a0 = -6.343246574721079391729402781192128239938E2L,
|
|---|
| 107 | a1 = 1.856560238672465796768677717168371401378E3L,
|
|---|
| 108 | a2 = 2.404733102163746263689288466865843408429E3L,
|
|---|
| 109 | a3 = 8.804188795790383497379532868917517596322E2L,
|
|---|
| 110 | a4 = 1.135361354097447729740103745999661157426E2L,
|
|---|
| 111 | a5 = 3.766956539107615557608581581190400021285E0L,
|
|---|
| 112 |
|
|---|
| 113 | b0 = 8.214973713960928795704317259806842490498E3L,
|
|---|
| 114 | b1 = 1.026343508841367384879065363925870888012E4L,
|
|---|
| 115 | b2 = 4.553337477045763320522762343132210919277E3L,
|
|---|
| 116 | b3 = 8.506975785032585797446253359230031874803E2L,
|
|---|
| 117 | b4 = 6.042447899703295436820744186992189445813E1L,
|
|---|
| 118 | /* b5 = 1.000000000000000000000000000000000000000E0 */
|
|---|
| 119 |
|
|---|
| 120 |
|
|---|
| 121 | tc = 1.4616321449683623412626595423257213284682E0L,
|
|---|
| 122 | tf = -1.2148629053584961146050602565082954242826E-1, /* double precision */
|
|---|
| 123 | /* tt = (tail of tf), i.e. tf + tt has extended precision. */
|
|---|
| 124 | tt = 3.3649914684731379602768989080467587736363E-18L,
|
|---|
| 125 | /* lgam ( 1.4616321449683623412626595423257213284682E0 ) =
|
|---|
| 126 | -1.2148629053584960809551455717769158215135617312999903886372437313313530E-1 */
|
|---|
| 127 |
|
|---|
| 128 | /* lgam (x + tc) = tf + tt + x g(x)/h(x)
|
|---|
| 129 | -0.230003726999612341262659542325721328468 <= x
|
|---|
| 130 | <= 0.2699962730003876587373404576742786715318
|
|---|
| 131 | peak relative error 2.1e-21 */
|
|---|
| 132 | g0 = 3.645529916721223331888305293534095553827E-18L,
|
|---|
| 133 | g1 = 5.126654642791082497002594216163574795690E3L,
|
|---|
| 134 | g2 = 8.828603575854624811911631336122070070327E3L,
|
|---|
| 135 | g3 = 5.464186426932117031234820886525701595203E3L,
|
|---|
| 136 | g4 = 1.455427403530884193180776558102868592293E3L,
|
|---|
| 137 | g5 = 1.541735456969245924860307497029155838446E2L,
|
|---|
| 138 | g6 = 4.335498275274822298341872707453445815118E0L,
|
|---|
| 139 |
|
|---|
| 140 | h0 = 1.059584930106085509696730443974495979641E4L,
|
|---|
| 141 | h1 = 2.147921653490043010629481226937850618860E4L,
|
|---|
| 142 | h2 = 1.643014770044524804175197151958100656728E4L,
|
|---|
| 143 | h3 = 5.869021995186925517228323497501767586078E3L,
|
|---|
| 144 | h4 = 9.764244777714344488787381271643502742293E2L,
|
|---|
| 145 | h5 = 6.442485441570592541741092969581997002349E1L,
|
|---|
| 146 | /* h6 = 1.000000000000000000000000000000000000000E0 */
|
|---|
| 147 |
|
|---|
| 148 |
|
|---|
| 149 | /* lgam (x+1) = -0.5 x + x u(x)/v(x)
|
|---|
| 150 | -0.100006103515625 <= x <= 0.231639862060546875
|
|---|
| 151 | peak relative error 1.3e-21 */
|
|---|
| 152 | u0 = -8.886217500092090678492242071879342025627E1L,
|
|---|
| 153 | u1 = 6.840109978129177639438792958320783599310E2L,
|
|---|
| 154 | u2 = 2.042626104514127267855588786511809932433E3L,
|
|---|
| 155 | u3 = 1.911723903442667422201651063009856064275E3L,
|
|---|
| 156 | u4 = 7.447065275665887457628865263491667767695E2L,
|
|---|
| 157 | u5 = 1.132256494121790736268471016493103952637E2L,
|
|---|
| 158 | u6 = 4.484398885516614191003094714505960972894E0L,
|
|---|
| 159 |
|
|---|
| 160 | v0 = 1.150830924194461522996462401210374632929E3L,
|
|---|
| 161 | v1 = 3.399692260848747447377972081399737098610E3L,
|
|---|
| 162 | v2 = 3.786631705644460255229513563657226008015E3L,
|
|---|
| 163 | v3 = 1.966450123004478374557778781564114347876E3L,
|
|---|
| 164 | v4 = 4.741359068914069299837355438370682773122E2L,
|
|---|
| 165 | v5 = 4.508989649747184050907206782117647852364E1L,
|
|---|
| 166 | /* v6 = 1.000000000000000000000000000000000000000E0 */
|
|---|
| 167 |
|
|---|
| 168 |
|
|---|
| 169 | /* lgam (x+2) = .5 x + x s(x)/r(x)
|
|---|
| 170 | 0 <= x <= 1
|
|---|
| 171 | peak relative error 7.2e-22 */
|
|---|
| 172 | s0 = 1.454726263410661942989109455292824853344E6L,
|
|---|
| 173 | s1 = -3.901428390086348447890408306153378922752E6L,
|
|---|
| 174 | s2 = -6.573568698209374121847873064292963089438E6L,
|
|---|
| 175 | s3 = -3.319055881485044417245964508099095984643E6L,
|
|---|
| 176 | s4 = -7.094891568758439227560184618114707107977E5L,
|
|---|
| 177 | s5 = -6.263426646464505837422314539808112478303E4L,
|
|---|
| 178 | s6 = -1.684926520999477529949915657519454051529E3L,
|
|---|
| 179 |
|
|---|
| 180 | r0 = -1.883978160734303518163008696712983134698E7L,
|
|---|
| 181 | r1 = -2.815206082812062064902202753264922306830E7L,
|
|---|
| 182 | r2 = -1.600245495251915899081846093343626358398E7L,
|
|---|
| 183 | r3 = -4.310526301881305003489257052083370058799E6L,
|
|---|
| 184 | r4 = -5.563807682263923279438235987186184968542E5L,
|
|---|
| 185 | r5 = -3.027734654434169996032905158145259713083E4L,
|
|---|
| 186 | r6 = -4.501995652861105629217250715790764371267E2L,
|
|---|
| 187 | /* r6 = 1.000000000000000000000000000000000000000E0 */
|
|---|
| 188 |
|
|---|
| 189 |
|
|---|
| 190 | /* lgam(x) = ( x - 0.5 ) * log(x) - x + LS2PI + 1/x w(1/x^2)
|
|---|
| 191 | x >= 8
|
|---|
| 192 | Peak relative error 1.51e-21
|
|---|
| 193 | w0 = LS2PI - 0.5 */
|
|---|
| 194 | w0 = 4.189385332046727417803e-1L,
|
|---|
| 195 | w1 = 8.333333333333331447505E-2L,
|
|---|
| 196 | w2 = -2.777777777750349603440E-3L,
|
|---|
| 197 | w3 = 7.936507795855070755671E-4L,
|
|---|
| 198 | w4 = -5.952345851765688514613E-4L,
|
|---|
| 199 | w5 = 8.412723297322498080632E-4L,
|
|---|
| 200 | w6 = -1.880801938119376907179E-3L,
|
|---|
| 201 | w7 = 4.885026142432270781165E-3L;
|
|---|
| 202 |
|
|---|
| 203 | /* sin(pi*x) assuming x > 2^-1000, if sin(pi*x)==0 the sign is arbitrary */
|
|---|
| 204 | static long double sin_pi(long double x)
|
|---|
| 205 | {
|
|---|
| 206 | int n;
|
|---|
| 207 |
|
|---|
| 208 | /* spurious inexact if odd int */
|
|---|
| 209 | x *= 0.5;
|
|---|
| 210 | x = 2.0*(x - floorl(x)); /* x mod 2.0 */
|
|---|
| 211 |
|
|---|
| 212 | n = (int)(x*4.0);
|
|---|
| 213 | n = (n+1)/2;
|
|---|
| 214 | x -= n*0.5f;
|
|---|
| 215 | x *= pi;
|
|---|
| 216 |
|
|---|
| 217 | switch (n) {
|
|---|
| 218 | default: /* case 4: */
|
|---|
| 219 | case 0: return __sinl(x, 0.0, 0);
|
|---|
| 220 | case 1: return __cosl(x, 0.0);
|
|---|
| 221 | case 2: return __sinl(-x, 0.0, 0);
|
|---|
| 222 | case 3: return -__cosl(x, 0.0);
|
|---|
| 223 | }
|
|---|
| 224 | }
|
|---|
| 225 |
|
|---|
| 226 | long double __lgammal_r(long double x, int *sg) {
|
|---|
| 227 | long double t, y, z, nadj, p, p1, p2, q, r, w;
|
|---|
| 228 | union ldshape u = {x};
|
|---|
| 229 | uint32_t ix = (u.i.se & 0x7fffU)<<16 | u.i.m>>48;
|
|---|
| 230 | int sign = u.i.se >> 15;
|
|---|
| 231 | int i;
|
|---|
| 232 |
|
|---|
| 233 | *sg = 1;
|
|---|
| 234 |
|
|---|
| 235 | /* purge off +-inf, NaN, +-0, tiny and negative arguments */
|
|---|
| 236 | if (ix >= 0x7fff0000)
|
|---|
| 237 | return x * x;
|
|---|
| 238 | if (ix < 0x3fc08000) { /* |x|<2**-63, return -log(|x|) */
|
|---|
| 239 | if (sign) {
|
|---|
| 240 | *sg = -1;
|
|---|
| 241 | x = -x;
|
|---|
| 242 | }
|
|---|
| 243 | return -logl(x);
|
|---|
| 244 | }
|
|---|
| 245 | if (sign) {
|
|---|
| 246 | x = -x;
|
|---|
| 247 | t = sin_pi(x);
|
|---|
| 248 | if (t == 0.0)
|
|---|
| 249 | return 1.0 / (x-x); /* -integer */
|
|---|
| 250 | if (t > 0.0)
|
|---|
| 251 | *sg = -1;
|
|---|
| 252 | else
|
|---|
| 253 | t = -t;
|
|---|
| 254 | nadj = logl(pi / (t * x));
|
|---|
| 255 | }
|
|---|
| 256 |
|
|---|
| 257 | /* purge off 1 and 2 (so the sign is ok with downward rounding) */
|
|---|
| 258 | if ((ix == 0x3fff8000 || ix == 0x40008000) && u.i.m == 0) {
|
|---|
| 259 | r = 0;
|
|---|
| 260 | } else if (ix < 0x40008000) { /* x < 2.0 */
|
|---|
| 261 | if (ix <= 0x3ffee666) { /* 8.99993896484375e-1 */
|
|---|
| 262 | /* lgamma(x) = lgamma(x+1) - log(x) */
|
|---|
| 263 | r = -logl(x);
|
|---|
| 264 | if (ix >= 0x3ffebb4a) { /* 7.31597900390625e-1 */
|
|---|
| 265 | y = x - 1.0;
|
|---|
| 266 | i = 0;
|
|---|
| 267 | } else if (ix >= 0x3ffced33) { /* 2.31639862060546875e-1 */
|
|---|
| 268 | y = x - (tc - 1.0);
|
|---|
| 269 | i = 1;
|
|---|
| 270 | } else { /* x < 0.23 */
|
|---|
| 271 | y = x;
|
|---|
| 272 | i = 2;
|
|---|
| 273 | }
|
|---|
| 274 | } else {
|
|---|
| 275 | r = 0.0;
|
|---|
| 276 | if (ix >= 0x3fffdda6) { /* 1.73162841796875 */
|
|---|
| 277 | /* [1.7316,2] */
|
|---|
| 278 | y = x - 2.0;
|
|---|
| 279 | i = 0;
|
|---|
| 280 | } else if (ix >= 0x3fff9da6) { /* 1.23162841796875 */
|
|---|
| 281 | /* [1.23,1.73] */
|
|---|
| 282 | y = x - tc;
|
|---|
| 283 | i = 1;
|
|---|
| 284 | } else {
|
|---|
| 285 | /* [0.9, 1.23] */
|
|---|
| 286 | y = x - 1.0;
|
|---|
| 287 | i = 2;
|
|---|
| 288 | }
|
|---|
| 289 | }
|
|---|
| 290 | switch (i) {
|
|---|
| 291 | case 0:
|
|---|
| 292 | p1 = a0 + y * (a1 + y * (a2 + y * (a3 + y * (a4 + y * a5))));
|
|---|
| 293 | p2 = b0 + y * (b1 + y * (b2 + y * (b3 + y * (b4 + y))));
|
|---|
| 294 | r += 0.5 * y + y * p1/p2;
|
|---|
| 295 | break;
|
|---|
| 296 | case 1:
|
|---|
| 297 | p1 = g0 + y * (g1 + y * (g2 + y * (g3 + y * (g4 + y * (g5 + y * g6)))));
|
|---|
| 298 | p2 = h0 + y * (h1 + y * (h2 + y * (h3 + y * (h4 + y * (h5 + y)))));
|
|---|
| 299 | p = tt + y * p1/p2;
|
|---|
| 300 | r += (tf + p);
|
|---|
| 301 | break;
|
|---|
| 302 | case 2:
|
|---|
| 303 | p1 = y * (u0 + y * (u1 + y * (u2 + y * (u3 + y * (u4 + y * (u5 + y * u6))))));
|
|---|
| 304 | p2 = v0 + y * (v1 + y * (v2 + y * (v3 + y * (v4 + y * (v5 + y)))));
|
|---|
| 305 | r += (-0.5 * y + p1 / p2);
|
|---|
| 306 | }
|
|---|
| 307 | } else if (ix < 0x40028000) { /* 8.0 */
|
|---|
| 308 | /* x < 8.0 */
|
|---|
| 309 | i = (int)x;
|
|---|
| 310 | y = x - (double)i;
|
|---|
| 311 | p = y * (s0 + y * (s1 + y * (s2 + y * (s3 + y * (s4 + y * (s5 + y * s6))))));
|
|---|
| 312 | q = r0 + y * (r1 + y * (r2 + y * (r3 + y * (r4 + y * (r5 + y * (r6 + y))))));
|
|---|
| 313 | r = 0.5 * y + p / q;
|
|---|
| 314 | z = 1.0;
|
|---|
| 315 | /* lgamma(1+s) = log(s) + lgamma(s) */
|
|---|
| 316 | switch (i) {
|
|---|
| 317 | case 7:
|
|---|
| 318 | z *= (y + 6.0); /* FALLTHRU */
|
|---|
| 319 | case 6:
|
|---|
| 320 | z *= (y + 5.0); /* FALLTHRU */
|
|---|
| 321 | case 5:
|
|---|
| 322 | z *= (y + 4.0); /* FALLTHRU */
|
|---|
| 323 | case 4:
|
|---|
| 324 | z *= (y + 3.0); /* FALLTHRU */
|
|---|
| 325 | case 3:
|
|---|
| 326 | z *= (y + 2.0); /* FALLTHRU */
|
|---|
| 327 | r += logl(z);
|
|---|
| 328 | break;
|
|---|
| 329 | }
|
|---|
| 330 | } else if (ix < 0x40418000) { /* 2^66 */
|
|---|
| 331 | /* 8.0 <= x < 2**66 */
|
|---|
| 332 | t = logl(x);
|
|---|
| 333 | z = 1.0 / x;
|
|---|
| 334 | y = z * z;
|
|---|
| 335 | w = w0 + z * (w1 + y * (w2 + y * (w3 + y * (w4 + y * (w5 + y * (w6 + y * w7))))));
|
|---|
| 336 | r = (x - 0.5) * (t - 1.0) + w;
|
|---|
| 337 | } else /* 2**66 <= x <= inf */
|
|---|
| 338 | r = x * (logl(x) - 1.0);
|
|---|
| 339 | if (sign)
|
|---|
| 340 | r = nadj - r;
|
|---|
| 341 | return r;
|
|---|
| 342 | }
|
|---|
| 343 | #elif LDBL_MANT_DIG == 113 && LDBL_MAX_EXP == 16384
|
|---|
| 344 | // TODO: broken implementation to make things compile
|
|---|
| 345 | double __lgamma_r(double x, int *sg);
|
|---|
| 346 |
|
|---|
| 347 | long double __lgammal_r(long double x, int *sg)
|
|---|
| 348 | {
|
|---|
| 349 | return __lgamma_r(x, sg);
|
|---|
| 350 | }
|
|---|
| 351 | #endif
|
|---|
| 352 |
|
|---|
| 353 | extern int __signgam;
|
|---|
| 354 |
|
|---|
| 355 | long double lgammal(long double x)
|
|---|
| 356 | {
|
|---|
| 357 | return __lgammal_r(x, &__signgam);
|
|---|
| 358 | }
|
|---|
| 359 |
|
|---|
| 360 | weak_alias(__lgammal_r, lgammal_r);
|
|---|
![(please configure the [header_logo] section in trac.ini)](/trac_user/contrib/chrome/site/your_project_logo.png){kind=logo}
