src/math/lgammal.c

   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);