usr/src/lib/libm/common/LD/__lgammal.c

   1 /*
   2  * CDDL HEADER START
   3  *
   4  * The contents of this file are subject to the terms of the
   5  * Common Development and Distribution License (the "License").
   6  * You may not use this file except in compliance with the License.
   7  *
   8  * You can obtain a copy of the license at usr/src/OPENSOLARIS.LICENSE
   9  * or http://www.opensolaris.org/os/licensing.
  10  * See the License for the specific language governing permissions
  11  * and limitations under the License.
  12  *
  13  * When distributing Covered Code, include this CDDL HEADER in each
  14  * file and include the License file at usr/src/OPENSOLARIS.LICENSE.
  15  * If applicable, add the following below this CDDL HEADER, with the
  16  * fields enclosed by brackets "[]" replaced with your own identifying
  17  * information: Portions Copyright [yyyy] [name of copyright owner]
  18  *
  19  * CDDL HEADER END
  20  */
  21
  22 /*
  23  * Copyright 2011 Nexenta Systems, Inc.  All rights reserved.
  24  */
  25 /*
  26  * Copyright 2006 Sun Microsystems, Inc.  All rights reserved.
  27  * Use is subject to license terms.
  28  */
  29
  30 /*
  31  * long double __k_lgammal(long double x, int *signgamlp);
  32  * K.C. Ng, August, 1989.
  33  *
  34  * We choose [1.5,2.5] to be the primary interval. Our algorithms
  35  * are mainly derived from
  36  *
  37  *
  38  *                             zeta(2)-1    2    zeta(3)-1    3
  39  * lgamma(2+s) = s*(1-euler) + --------- * s  -  --------- * s  + ...
  40  *                                 2                 3
  41  *
  42  *
  43  * Note 1. Since gamma(1+s)=s*gamma(s), hence
  44  *              lgamma(1+s) = log(s) + lgamma(s), or
  45  *              lgamma(s) = lgamma(1+s) - log(s).
  46  *         When s is really tiny (like roundoff), lgamma(1+s) ~ s(1-enler)
  47  *         Hence lgamma(s) ~ -log(s) for tiny s
  48  *
  49  */
  50
  51 #include "libm.h"
  52 #include "longdouble.h"
  53
  54 static long double neg(long double, int *);
  55 static long double poly(long double, const long double *, int);
  56 static long double polytail(long double);
  57 static long double primary(long double);
  58
  59 static const long double
  60 c0 =     0.0L,
  61 ch =     0.5L,
  62 c1 =     1.0L,
  63 c2 =     2.0L,
  64 c3 =     3.0L,
  65 c4 =     4.0L,
  66 c5 =     5.0L,
  67 c6 =     6.0L,
  68 pi =     3.1415926535897932384626433832795028841971L,
  69 tiny =   1.0e-40L;
  70
  71 long double
  72 __k_lgammal(long double x, int *signgamlp) {
  73         long double t, y;
  74         int i;
  75
  76         /* purge off +-inf, NaN and negative arguments */
  77         if (!finitel(x))
  78                 return (x*x);
  79         *signgamlp = 1;
  80         if (signbitl(x))
  81                 return (neg(x, signgamlp));
  82
  83         /* for x < 8.0 */
  84         if (x < 8.0L) {
  85             y = anintl(x);
  86             i = (int) y;
  87             switch (i) {
  88             case 0:
  89                 if (x < 1.0e-40L)
  90                         return (-logl(x));
  91                 else
  92                         return (primary(x)-log1pl(x))-logl(x);
  93             case 1:
  94                 return (primary(x-y)-logl(x));
  95             case 2:
  96                 return (primary(x-y));
  97             case 3:
  98                 return (primary(x-y)+logl(x-c1));
  99             case 4:
 100                 return (primary(x-y)+logl((x-c1)*(x-c2)));
 101             case 5:
 102                 return (primary(x-y)+logl((x-c1)*(x-c2)*(x-c3)));
 103             case 6:
 104                 return (primary(x-y)+logl((x-c1)*(x-c2)*(x-c3)*(x-c4)));
 105             case 7:
 106                 return (primary(x-y)+logl((x-c1)*(x-c2)*(x-c3)*(x-c4)*(x-c5)));
 107             case 8:
 108                 return primary(x-y)+
 109                         logl((x-c1)*(x-c2)*(x-c3)*(x-c4)*(x-c5)*(x-c6));
 110             }
 111         }
 112
 113         /* 8.0 <= x < 1.0e40 */
 114         if (x < 1.0e40L) {
 115             t = logl(x);
 116             return (x*(t-c1)-(ch*t-polytail(c1/x)));
 117         }
 118
 119         /* 1.0e40 <= x <= inf */
 120         return (x*(logl(x)-c1));
 121 }
 122
 123 static const long double an1[] = {              /* 20 terms */
 124         -0.0772156649015328606065120900824024309741L,
 125         3.224670334241132182362075833230130289059e-0001L,
 126         -6.735230105319809513324605383668929964120e-0002L,
 127         2.058080842778454787900092432928910226297e-0002L,
 128         -7.385551028673985266273054086081102125704e-0003L,
 129         2.890510330741523285758867304409628648727e-0003L,
 130         -1.192753911703260976581414338096267498555e-0003L,
 131         5.096695247430424562831956662855697824035e-0004L,
 132         -2.231547584535777978926798502084300123638e-0004L,
 133         9.945751278186384670278268034322157947635e-0005L,
 134         -4.492623673665547726647838474125147631082e-0005L,
 135         2.050721280617796810096993154281561168706e-0005L,
 136         -9.439487785617396552092393234044767313568e-0006L,
 137         4.374872903516051510689234173139793159340e-0006L,
 138         -2.039156676413643091040459825776029327487e-0006L,
 139         9.555777181318621470466563543806211523634e-0007L,
 140         -4.468344919709630637558538313482398989638e-0007L,
 141         2.216738086090045781773004477831059444178e-0007L,
 142         -7.472783403418388455860445842543843485916e-0008L,
 143         8.777317930927149922056782132706238921648e-0008L,
 144 };
 145
 146 static const long double an2[] = {              /* 20 terms */
 147   -.0772156649015328606062692723698127607018L,
 148    3.224670334241132182635552349060279118047e-0001L,
 149   -6.735230105319809367555642883133994818325e-0002L,
 150    2.058080842778459676880822202762143671813e-0002L,
 151   -7.385551028672828216011343150077846918930e-0003L,
 152    2.890510330762060607399561536905727853178e-0003L,
 153   -1.192753911419623262328187532759756368041e-0003L,
 154    5.096695278636456678258091134532258618614e-0004L,
 155   -2.231547306817535743052975194022893369135e-0004L,
 156    9.945771461633313282744264853986643877087e-0005L,
 157   -4.492503279458972037926876061257489481619e-0005L,
 158    2.051311416812082875492678651369394595613e-0005L,
 159   -9.415778282365955203915850761537462941165e-0006L,
 160    4.452428829045147098722932981088650055919e-0006L,
 161   -1.835024727987632579886951760650722695781e-0006L,
 162    1.379783080658545009579060714946381462565e-0006L,
 163    2.282637532109775156769736768748402175238e-0007L,
 164    1.002577375515900191362119718128149880168e-0006L,
 165    5.177028794262638311939991106423220002463e-0007L,
 166    3.127947245174847104122426445937830555755e-0007L,
 167 };
 168
 169 static const long double an3[] = {              /* 20 terms */
 170   -.0772156649015328227870646417729220690875L,
 171    3.224670334241156699881788955959915250365e-0001L,
 172   -6.735230105312273571375431059744975563170e-0002L,
 173    2.058080842924464587662846071337083809005e-0002L,
 174   -7.385551008677271654723604653956131791619e-0003L,
 175    2.890510536479782086197110272583833176602e-0003L,
 176   -1.192752262076857692740571567808259138697e-0003L,
 177    5.096800771149805289371135155128380707889e-0004L,
 178   -2.231000836682831335505058492409860123647e-0004L,
 179    9.968912171073936803871803966360595275047e-0005L,
 180   -4.412020779327746243544387946167256187258e-0005L,
 181    2.281374113541454151067016632998630209049e-0005L,
 182   -4.028361291428629491824694655287954266830e-0006L,
 183    1.470694920619518924598956849226530750139e-0005L,
 184    1.381686137617987197975289545582377713772e-0005L,
 185    2.012493539265777728944759982054970441601e-0005L,
 186    1.723917864208965490251560644681933675799e-0005L,
 187    1.202954035243788300138608765425123713395e-0005L,
 188    5.079851887558623092776296577030850938146e-0006L,
 189    1.220657945824153751555138592006604026282e-0006L,
 190 };
 191
 192 static const long double an4[] = {              /* 21 terms */
 193   -.0772156649015732285350261816697540392371L,
 194    3.224670334221752060691751340365212226097e-0001L,
 195   -6.735230109744009693977755991488196368279e-0002L,
 196    2.058080778913037626909954141611580783216e-0002L,
 197   -7.385557567931505621170483708950557506819e-0003L,
 198    2.890459838416254326340844289785254883436e-0003L,
 199   -1.193059036207136762877351596966718455737e-0003L,
 200    5.081914708100372836613371356529568937869e-0004L,
 201   -2.289855016133600313131553005982542045338e-0004L,
 202    8.053454537980585879620331053833498511491e-0005L,
 203   -9.574620532104845821243493405855672438998e-0005L,
 204   -9.269085628207107155601445001196317715686e-0005L,
 205   -2.183276779859490461716196344776208220180e-0004L,
 206   -3.134834305597571096452454999737269668868e-0004L,
 207   -3.973878894951937437018305986901392888619e-0004L,
 208   -3.953352414899222799161275564386488057119e-0004L,
 209   -3.136740932204038779362660900621212816511e-0004L,
 210   -1.884502253819634073946130825196078627664e-0004L,
 211   -8.192655799958926853585332542123631379301e-0005L,
 212   -2.292183750010571062891605074281744854436e-0005L,
 213   -3.223980628729716864927724265781406614294e-0006L,
 214 };
 215
 216 static const long double ap1[] = {                      /* 19 terms */
 217   -0.0772156649015328606065120900824024296961L,
 218    3.224670334241132182362075833230047956465e-0001L,
 219   -6.735230105319809513324605382963943777301e-0002L,
 220    2.058080842778454787900092126606252375465e-0002L,
 221   -7.385551028673985266272518231365020063941e-0003L,
 222    2.890510330741523285681704570797770736423e-0003L,
 223   -1.192753911703260971285304221165990244515e-0003L,
 224    5.096695247430420878696018188830886972245e-0004L,
 225   -2.231547584535654004647639737841526025095e-0004L,
 226    9.945751278137201960636098805852315982919e-0005L,
 227   -4.492623672777606053587919463929044226280e-0005L,
 228    2.050721258703289487603702670753053765201e-0005L,
 229   -9.439485626565616989352750672499008021041e-0006L,
 230    4.374838162403994645138200419356844574219e-0006L,
 231   -2.038979492862555348577006944451002161496e-0006L,
 232    9.536763152382263548086981191378885102802e-0007L,
 233   -4.426111214332434049863595231916564014913e-0007L,
 234    1.911148847512947464234633846270287546882e-0007L,
 235   -5.788673944861923038157839080272303519671e-0008L,
 236 };
 237
 238 static const long double ap2[] = {                      /* 19 terms */
 239   -0.077215664901532860606428624449354836087L,
 240    3.224670334241132182271948744265855440139e-0001L,
 241   -6.735230105319809467356126599005051676203e-0002L,
 242    2.058080842778453315716389815213496002588e-0002L,
 243   -7.385551028673653323064118422580096222959e-0003L,
 244    2.890510330735923572088003424849289006039e-0003L,
 245   -1.192753911629952368606185543945790688144e-0003L,
 246    5.096695239806718875364547587043220998766e-0004L,
 247   -2.231547520600616108991867127392089144886e-0004L,
 248    9.945746913898151120612322833059416008973e-0005L,
 249   -4.492599307461977003570224943054585729684e-0005L,
 250    2.050609891889165453592046505651759999090e-0005L,
 251   -9.435329866734193796540515247917165988579e-0006L,
 252    4.362267138522223236241016136585565144581e-0006L,
 253   -2.008556356653246579300491601497510230557e-0006L,
 254    8.961498103387207161105347118042844354395e-0007L,
 255   -3.614187228330216282235692806488341157741e-0007L,
 256    1.136978988247816860500420915014777753153e-0007L,
 257   -2.000532786387196664019286514899782691776e-0008L,
 258 };
 259
 260 static const long double ap3[] = {                      /* 19 terms */
 261   -0.077215664901532859888521470795348856446L,
 262    3.224670334241131733364048614484228443077e-0001L,
 263   -6.735230105319676541660495145259038151576e-0002L,
 264    2.058080842775975461837768839015444273830e-0002L,
 265   -7.385551028347615729728618066663566606906e-0003L,
 266    2.890510327517954083379032008643080256676e-0003L,
 267   -1.192753886919470728001821137439430882603e-0003L,
 268    5.096693728898932234814903769146577482912e-0004L,
 269   -2.231540055048827662528594010961874258037e-0004L,
 270    9.945446210018649311491619999438833843723e-0005L,
 271   -4.491608206598064519190236245753867697750e-0005L,
 272    2.047939071322271016498065052853746466669e-0005L,
 273   -9.376824046522786006677541036631536790762e-0006L,
 274    4.259329829498149111582277209189150127347e-0006L,
 275   -1.866064770421594266702176289764212873428e-0006L,
 276    7.462066721137579592928128104534957135669e-0007L,
 277   -2.483546217529077735074007138457678727371e-0007L,
 278    5.915166576378161473299324673649144297574e-0008L,
 279   -7.334139641706988966966252333759604701905e-0009L,
 280 };
 281
 282 static const long double ap4[] = {                      /* 19 terms */
 283   -0.0772156649015326785569313252637238673675L,
 284    3.224670334241051435008842685722468344822e-0001L,
 285   -6.735230105302832007479431772160948499254e-0002L,
 286    2.058080842553481183648529360967441889912e-0002L,
 287   -7.385551007602909242024706804659879199244e-0003L,
 288    2.890510182473907253939821312248303471206e-0003L,
 289   -1.192753098427856770847894497586825614450e-0003L,
 290    5.096659636418811568063339214203693550804e-0004L,
 291   -2.231421144004355691166194259675004483639e-0004L,
 292    9.942073842343832132754332881883387625136e-0005L,
 293   -4.483809261973204531263252655050701205397e-0005L,
 294    2.033260142610284888319116654931994447173e-0005L,
 295   -9.153539544026646699870528191410440585796e-0006L,
 296    3.988460469925482725894144688699584997971e-0006L,
 297   -1.609692980087029172567957221850825977621e-0006L,
 298    5.634916377249975825399706694496688803488e-0007L,
 299   -1.560065465929518563549083208482591437696e-0007L,
 300    2.961350193868935325526962209019387821584e-0008L,
 301   -2.834602215195368130104649234505033159842e-0009L,
 302 };
 303
 304 static long double
 305 primary(long double s) {        /* assume |s|<=0.5 */
 306         int i;
 307
 308         i = (int) (8.0L * (s + 0.5L));
 309         switch (i) {
 310         case 0: return ch*s+s*poly(s, an4, 21);
 311         case 1: return ch*s+s*poly(s, an3, 20);
 312         case 2: return ch*s+s*poly(s, an2, 20);
 313         case 3: return ch*s+s*poly(s, an1, 20);
 314         case 4: return ch*s+s*poly(s, ap1, 19);
 315         case 5: return ch*s+s*poly(s, ap2, 19);
 316         case 6: return ch*s+s*poly(s, ap3, 19);
 317         case 7: return ch*s+s*poly(s, ap4, 19);
 318         }
 319         /* NOTREACHED */
 320     return (0.0L);
 321 }
 322
 323 static long double
 324 poly(long double s, const long double *p, int n) {
 325         long double y;
 326         int i;
 327         y = p[n-1];
 328         for (i = n-2; i >= 0; i--) y = p[i]+s*y;
 329         return (y);
 330 }
 331
 332 static const long double pt[] = {
 333    9.189385332046727417803297364056176804663e-0001L,
 334    8.333333333333333333333333333331286969123e-0002L,
 335   -2.777777777777777777777777553194796036402e-0003L,
 336    7.936507936507936507927283071433584248176e-0004L,
 337   -5.952380952380952362351042163192634108297e-0004L,
 338    8.417508417508395661774286645578379460131e-0004L,
 339   -1.917526917525263651186066417934685675649e-0003L,
 340    6.410256409395203164659292973142293199083e-0003L,
 341   -2.955065327248303301763594514012418438188e-0002L,
 342    1.796442830099067542945998615411893822886e-0001L,
 343   -1.392413465829723742489974310411118662919e+0000L,
 344    1.339984238037267658352656597960492029261e+0001L,
 345   -1.564707657605373662425785904278645727813e+0002L,
 346    2.156323807499211356127813962223067079300e+0003L,
 347   -3.330486427626223184647299834137041307569e+0004L,
 348    5.235535072011889213611369254140123518699e+0005L,
 349   -7.258160984602220710491988573430212593080e+0006L,
 350    7.316526934569686459641438882340322673357e+0007L,
 351   -3.806450279064900548836571789284896711473e+0008L,
 352 };
 353
 354 static long double
 355 polytail(long double s) {
 356         long double t, z;
 357         int i;
 358         z = s*s;
 359         t = pt[18];
 360         for (i = 17; i >= 1; i--) t = pt[i]+z*t;
 361         return (pt[0]+s*t);
 362 }
 363
 364 static long double
 365 neg(long double z, int *signgamlp) {
 366         long double t, p;
 367
 368         /*
 369          * written by K.C. Ng,  Feb 2, 1989.
 370          *
 371          * Since
 372          *              -z*G(-z)*G(z) = pi/sin(pi*z),
 373          * we have
 374          *      G(-z) = -pi/(sin(pi*z)*G(z)*z)
 375          *                =  pi/(sin(pi*(-z))*G(z)*z)
 376          * Algorithm
 377          *              z = |z|
 378          *              t = sinpi(z); ...note that when z>2**112, z is an int
 379          *              and hence t=0.
 380          *
 381          *              if (t == 0.0) return 1.0/0.0;
 382          *              if (t< 0.0) *signgamlp = -1; else t= -t;
 383          *              if (z<1.0e-40)  ...tiny z
 384          *                  return -log(z);
 385          *              else
 386          *                  return log(pi/(t*z))-lgamma(z);
 387          *
 388          */
 389
 390         t = sinpil(z);                  /* t := sin(pi*z) */
 391         if (t == c0)                    /* return   1.0/0.0 =  +INF */
 392             return (c1/c0);
 393
 394         z = -z;
 395         if (z <= tiny)
 396             p = -logl(z);
 397         else
 398                 p = logl(pi/(fabsl(t)*z)) - __k_lgammal(z, signgamlp);
 399         if (t < c0) *signgamlp = -1;
 400         return (p);
 401 }