lib/libm/src/ld80/e_lgammal.c

   1 /*
   2  * ====================================================
   3  * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
   4  *
   5  * Developed at SunPro, a Sun Microsystems, Inc. business.
   6  * Permission to use, copy, modify, and distribute this
   7  * software is freely granted, provided that this notice
   8  * is preserved.
   9  * ====================================================
  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
  28 /* lgammal(x)
  29  * Reentrant version of the logarithm of the Gamma function
  30  * with user provide pointer for the sign of Gamma(x).
  31  *
  32  * Method:
  33  *   1. Argument Reduction for 0 < x <= 8
  34  *      Since gamma(1+s)=s*gamma(s), for x in [0,8], we may
  35  *      reduce x to a number in [1.5,2.5] by
  36  *              lgamma(1+s) = log(s) + lgamma(s)
  37  *      for example,
  38  *              lgamma(7.3) = log(6.3) + lgamma(6.3)
  39  *                          = log(6.3*5.3) + lgamma(5.3)
  40  *                          = log(6.3*5.3*4.3*3.3*2.3) + lgamma(2.3)
  41  *   2. Polynomial approximation of lgamma around its
  42  *      minimun ymin=1.461632144968362245 to maintain monotonicity.
  43  *      On [ymin-0.23, ymin+0.27] (i.e., [1.23164,1.73163]), use
  44  *              Let z = x-ymin;
  45  *              lgamma(x) = -1.214862905358496078218 + z^2*poly(z)
  46  *   2. Rational approximation in the primary interval [2,3]
  47  *      We use the following approximation:
  48  *              s = x-2.0;
  49  *              lgamma(x) = 0.5*s + s*P(s)/Q(s)
  50  *      Our algorithms are based on the following observation
  51  *
  52  *                             zeta(2)-1    2    zeta(3)-1    3
  53  * lgamma(2+s) = s*(1-Euler) + --------- * s  -  --------- * s  + ...
  54  *                                 2                 3
  55  *
  56  *      where Euler = 0.5771... is the Euler constant, which is very
  57  *      close to 0.5.
  58  *
  59  *   3. For x>=8, we have
  60  *      lgamma(x)~(x-0.5)log(x)-x+0.5*log(2pi)+1/(12x)-1/(360x**3)+....
  61  *      (better formula:
  62  *         lgamma(x)~(x-0.5)*(log(x)-1)-.5*(log(2pi)-1) + ...)
  63  *      Let z = 1/x, then we approximation
  64  *              f(z) = lgamma(x) - (x-0.5)(log(x)-1)
  65  *      by
  66  *                                  3       5             11
  67  *              w = w0 + w1*z + w2*z  + w3*z  + ... + w6*z
  68  *
  69  *   4. For negative x, since (G is gamma function)
  70  *              -x*G(-x)*G(x) = pi/sin(pi*x),
  71  *      we have
  72  *              G(x) = pi/(sin(pi*x)*(-x)*G(-x))
  73  *      since G(-x) is positive, sign(G(x)) = sign(sin(pi*x)) for x<0
  74  *      Hence, for x<0, signgam = sign(sin(pi*x)) and
  75  *              lgamma(x) = log(|Gamma(x)|)
  76  *                        = log(pi/(|x*sin(pi*x)|)) - lgamma(-x);
  77  *      Note: one should avoid compute pi*(-x) directly in the
  78  *            computation of sin(pi*(-x)).
  79  *
  80  *   5. Special Cases
  81  *              lgamma(2+s) ~ s*(1-Euler) for tiny s
  82  *              lgamma(1)=lgamma(2)=0
  83  *              lgamma(x) ~ -log(x) for tiny x
  84  *              lgamma(0) = lgamma(inf) = inf
  85  *              lgamma(-integer) = +-inf
  86  *
  87  */
  88
  89 #include <math.h>
  90
  91 #include "math_private.h"
  92
  93 static const long double
  94   half = 0.5L,
  95   one = 1.0L,
  96   pi = 3.14159265358979323846264L,
  97   two63 = 9.223372036854775808e18L,
  98
  99   /* lgam(1+x) = 0.5 x + x a(x)/b(x)
 100      -0.268402099609375 <= x <= 0
 101      peak relative error 6.6e-22 */
 102   a0 = -6.343246574721079391729402781192128239938E2L,
 103   a1 =  1.856560238672465796768677717168371401378E3L,
 104   a2 =  2.404733102163746263689288466865843408429E3L,
 105   a3 =  8.804188795790383497379532868917517596322E2L,
 106   a4 =  1.135361354097447729740103745999661157426E2L,
 107   a5 =  3.766956539107615557608581581190400021285E0L,
 108
 109   b0 =  8.214973713960928795704317259806842490498E3L,
 110   b1 =  1.026343508841367384879065363925870888012E4L,
 111   b2 =  4.553337477045763320522762343132210919277E3L,
 112   b3 =  8.506975785032585797446253359230031874803E2L,
 113   b4 =  6.042447899703295436820744186992189445813E1L,
 114   /* b5 =  1.000000000000000000000000000000000000000E0 */
 115
 116
 117   tc =  1.4616321449683623412626595423257213284682E0L,
 118   tf = -1.2148629053584961146050602565082954242826E-1,/* double precision */
 119 /* tt = (tail of tf), i.e. tf + tt has extended precision. */
 120   tt = 3.3649914684731379602768989080467587736363E-18L,
 121   /* lgam ( 1.4616321449683623412626595423257213284682E0 ) =
 122 -1.2148629053584960809551455717769158215135617312999903886372437313313530E-1 */
 123
 124   /* lgam (x + tc) = tf + tt + x g(x)/h(x)
 125      - 0.230003726999612341262659542325721328468 <= x
 126      <= 0.2699962730003876587373404576742786715318
 127      peak relative error 2.1e-21 */
 128   g0 = 3.645529916721223331888305293534095553827E-18L,
 129   g1 = 5.126654642791082497002594216163574795690E3L,
 130   g2 = 8.828603575854624811911631336122070070327E3L,
 131   g3 = 5.464186426932117031234820886525701595203E3L,
 132   g4 = 1.455427403530884193180776558102868592293E3L,
 133   g5 = 1.541735456969245924860307497029155838446E2L,
 134   g6 = 4.335498275274822298341872707453445815118E0L,
 135
 136   h0 = 1.059584930106085509696730443974495979641E4L,
 137   h1 =  2.147921653490043010629481226937850618860E4L,
 138   h2 = 1.643014770044524804175197151958100656728E4L,
 139   h3 =  5.869021995186925517228323497501767586078E3L,
 140   h4 =  9.764244777714344488787381271643502742293E2L,
 141   h5 =  6.442485441570592541741092969581997002349E1L,
 142   /* h6 = 1.000000000000000000000000000000000000000E0 */
 143
 144
 145   /* lgam (x+1) = -0.5 x + x u(x)/v(x)
 146      -0.100006103515625 <= x <= 0.231639862060546875
 147      peak relative error 1.3e-21 */
 148   u0 = -8.886217500092090678492242071879342025627E1L,
 149   u1 =  6.840109978129177639438792958320783599310E2L,
 150   u2 =  2.042626104514127267855588786511809932433E3L,
 151   u3 =  1.911723903442667422201651063009856064275E3L,
 152   u4 =  7.447065275665887457628865263491667767695E2L,
 153   u5 =  1.132256494121790736268471016493103952637E2L,
 154   u6 =  4.484398885516614191003094714505960972894E0L,
 155
 156   v0 =  1.150830924194461522996462401210374632929E3L,
 157   v1 =  3.399692260848747447377972081399737098610E3L,
 158   v2 =  3.786631705644460255229513563657226008015E3L,
 159   v3 =  1.966450123004478374557778781564114347876E3L,
 160   v4 =  4.741359068914069299837355438370682773122E2L,
 161   v5 =  4.508989649747184050907206782117647852364E1L,
 162   /* v6 =  1.000000000000000000000000000000000000000E0 */
 163
 164
 165   /* lgam (x+2) = .5 x + x s(x)/r(x)
 166      0 <= x <= 1
 167      peak relative error 7.2e-22 */
 168   s0 =  1.454726263410661942989109455292824853344E6L,
 169   s1 = -3.901428390086348447890408306153378922752E6L,
 170   s2 = -6.573568698209374121847873064292963089438E6L,
 171   s3 = -3.319055881485044417245964508099095984643E6L,
 172   s4 = -7.094891568758439227560184618114707107977E5L,
 173   s5 = -6.263426646464505837422314539808112478303E4L,
 174   s6 = -1.684926520999477529949915657519454051529E3L,
 175
 176   r0 = -1.883978160734303518163008696712983134698E7L,
 177   r1 = -2.815206082812062064902202753264922306830E7L,
 178   r2 = -1.600245495251915899081846093343626358398E7L,
 179   r3 = -4.310526301881305003489257052083370058799E6L,
 180   r4 = -5.563807682263923279438235987186184968542E5L,
 181   r5 = -3.027734654434169996032905158145259713083E4L,
 182   r6 = -4.501995652861105629217250715790764371267E2L,
 183   /* r6 =  1.000000000000000000000000000000000000000E0 */
 184
 185
 186 /* lgam(x) = ( x - 0.5 ) * log(x) - x + LS2PI + 1/x w(1/x^2)
 187    x >= 8
 188    Peak relative error 1.51e-21
 189    w0 = LS2PI - 0.5 */
 190   w0 =  4.189385332046727417803e-1L,
 191   w1 =  8.333333333333331447505E-2L,
 192   w2 = -2.777777777750349603440E-3L,
 193   w3 =  7.936507795855070755671E-4L,
 194   w4 = -5.952345851765688514613E-4L,
 195   w5 =  8.412723297322498080632E-4L,
 196   w6 = -1.880801938119376907179E-3L,
 197   w7 =  4.885026142432270781165E-3L;
 198
 199 static const long double zero = 0.0L;
 200
 201 static long double
 202 sin_pi(long double x)
 203 {
 204   long double y, z;
 205   int n, ix;
 206   u_int32_t se, i0, i1;
 207
 208   GET_LDOUBLE_WORDS (se, i0, i1, x);
 209   ix = se & 0x7fff;
 210   ix = (ix << 16) | (i0 >> 16);
 211   if (ix < 0x3ffd8000) /* 0.25 */
 212     return sinl (pi * x);
 213   y = -x;                       /* x is assume negative */
 214
 215   /*
 216    * argument reduction, make sure inexact flag not raised if input
 217    * is an integer
 218    */
 219   z = floorl (y);
 220   if (z != y)
 221     {                           /* inexact anyway */
 222       y  *= 0.5;
 223       y = 2.0*(y - floorl(y));          /* y = |x| mod 2.0 */
 224       n = (int) (y*4.0);
 225     }
 226   else
 227     {
 228       if (ix >= 0x403f8000)  /* 2^64 */
 229         {
 230           y = zero; n = 0;              /* y must be even */
 231         }
 232       else
 233         {
 234         if (ix < 0x403e8000)  /* 2^63 */
 235           z = y + two63;        /* exact */
 236         GET_LDOUBLE_WORDS (se, i0, i1, z);
 237         n = i1 & 1;
 238         y  = n;
 239         n <<= 2;
 240       }
 241     }
 242
 243   switch (n)
 244     {
 245     case 0:
 246       y = sinl (pi * y);
 247       break;
 248     case 1:
 249     case 2:
 250       y = cosl (pi * (half - y));
 251       break;
 252     case 3:
 253     case 4:
 254       y = sinl (pi * (one - y));
 255       break;
 256     case 5:
 257     case 6:
 258       y = -cosl (pi * (y - 1.5));
 259       break;
 260     default:
 261       y = sinl (pi * (y - 2.0));
 262       break;
 263     }
 264   return -y;
 265 }
 266
 267
 268 long double
 269 lgammal(long double x)
 270 {
 271   long double t, y, z, nadj, p, p1, p2, q, r, w;
 272   int i, ix;
 273   u_int32_t se, i0, i1;
 274
 275   signgam = 1;
 276   GET_LDOUBLE_WORDS (se, i0, i1, x);
 277   ix = se & 0x7fff;
 278
 279   if ((ix | i0 | i1) == 0)
 280     {
 281       if (se & 0x8000)
 282         signgam = -1;
 283       return one / fabsl (x);
 284     }
 285
 286   ix = (ix << 16) | (i0 >> 16);
 287
 288   /* purge off +-inf, NaN, +-0, and negative arguments */
 289   if (ix >= 0x7fff0000)
 290     return x * x;
 291
 292   if (ix < 0x3fc08000) /* 2^-63 */
 293     {                           /* |x|<2**-63, return -log(|x|) */
 294       if (se & 0x8000)
 295         {
 296           signgam = -1;
 297           return -logl (-x);
 298         }
 299       else
 300         return -logl (x);
 301     }
 302   if (se & 0x8000)
 303     {
 304       t = sin_pi (x);
 305       if (t == zero)
 306         return one / fabsl (t); /* -integer */
 307       nadj = logl (pi / fabsl (t * x));
 308       if (t < zero)
 309         signgam = -1;
 310       x = -x;
 311     }
 312
 313   /* purge off 1 and 2 */
 314   if ((((ix - 0x3fff8000) | i0 | i1) == 0)
 315       || (((ix - 0x40008000) | i0 | i1) == 0))
 316     r = 0;
 317   else if (ix < 0x40008000) /* 2.0 */
 318     {
 319       /* x < 2.0 */
 320       if (ix <= 0x3ffee666) /* 8.99993896484375e-1 */
 321         {
 322           /* lgamma(x) = lgamma(x+1) - log(x) */
 323           r = -logl (x);
 324           if (ix >= 0x3ffebb4a) /* 7.31597900390625e-1 */
 325             {
 326               y = x - one;
 327               i = 0;
 328             }
 329           else if (ix >= 0x3ffced33)/* 2.31639862060546875e-1 */
 330             {
 331               y = x - (tc - one);
 332               i = 1;
 333             }
 334           else
 335             {
 336               /* x < 0.23 */
 337               y = x;
 338               i = 2;
 339             }
 340         }
 341       else
 342         {
 343           r = zero;
 344           if (ix >= 0x3fffdda6) /* 1.73162841796875 */
 345             {
 346               /* [1.7316,2] */
 347               y = x - 2.0;
 348               i = 0;
 349             }
 350           else if (ix >= 0x3fff9da6)/* 1.23162841796875 */
 351             {
 352               /* [1.23,1.73] */
 353               y = x - tc;
 354               i = 1;
 355             }
 356           else
 357             {
 358               /* [0.9, 1.23] */
 359               y = x - one;
 360               i = 2;
 361             }
 362         }
 363       switch (i)
 364         {
 365         case 0:
 366           p1 = a0 + y * (a1 + y * (a2 + y * (a3 + y * (a4 + y * a5))));
 367           p2 = b0 + y * (b1 + y * (b2 + y * (b3 + y * (b4 + y))));
 368           r += half * y + y * p1/p2;
 369           break;
 370         case 1:
 371     p1 = g0 + y * (g1 + y * (g2 + y * (g3 + y * (g4 + y * (g5 + y * g6)))));
 372     p2 = h0 + y * (h1 + y * (h2 + y * (h3 + y * (h4 + y * (h5 + y)))));
 373     p = tt + y * p1/p2;
 374           r += (tf + p);
 375           break;
 376         case 2:
 377  p1 = y * (u0 + y * (u1 + y * (u2 + y * (u3 + y * (u4 + y * (u5 + y * u6))))));
 378       p2 = v0 + y * (v1 + y * (v2 + y * (v3 + y * (v4 + y * (v5 + y)))));
 379           r += (-half * y + p1 / p2);
 380         }
 381     }
 382   else if (ix < 0x40028000) /* 8.0 */
 383     {
 384       /* x < 8.0 */
 385       i = (int) x;
 386       t = zero;
 387       y = x - (double) i;
 388   p = y *
 389      (s0 + y * (s1 + y * (s2 + y * (s3 + y * (s4 + y * (s5 + y * s6))))));
 390   q = r0 + y * (r1 + y * (r2 + y * (r3 + y * (r4 + y * (r5 + y * (r6 + y))))));
 391       r = half * y + p / q;
 392       z = one;                  /* lgamma(1+s) = log(s) + lgamma(s) */
 393       switch (i)
 394         {
 395         case 7:
 396           z *= (y + 6.0);       /* FALLTHRU */
 397         case 6:
 398           z *= (y + 5.0);       /* FALLTHRU */
 399         case 5:
 400           z *= (y + 4.0);       /* FALLTHRU */
 401         case 4:
 402           z *= (y + 3.0);       /* FALLTHRU */
 403         case 3:
 404           z *= (y + 2.0);       /* FALLTHRU */
 405           r += logl (z);
 406           break;
 407         }
 408     }
 409   else if (ix < 0x40418000) /* 2^66 */
 410     {
 411       /* 8.0 <= x < 2**66 */
 412       t = logl (x);
 413       z = one / x;
 414       y = z * z;
 415       w = w0 + z * (w1
 416           + y * (w2 + y * (w3 + y * (w4 + y * (w5 + y * (w6 + y * w7))))));
 417       r = (x - half) * (t - one) + w;
 418     }
 419   else
 420     /* 2**66 <= x <= inf */
 421     r = x * (logl (x) - one);
 422   if (se & 0x8000)
 423     r = nadj - r;
 424   return r;
 425 }