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 /*
  27  * Copyright 2006 Sun Microsystems, Inc.  All rights reserved.
  28  * Use is subject to license terms.
  29  */
  30 
  31 /*
  32  * long double __k_lgammal(long double x, int *signgamlp);
  33  * K.C. Ng, August, 1989.
  34  *
  35  * We choose [1.5,2.5] to be the primary interval. Our algorithms
  36  * are mainly derived from
  37  *
  38  *
  39  *                             zeta(2)-1    2    zeta(3)-1    3
  40  * lgamma(2+s) = s*(1-euler) + --------- * s  -  --------- * s  + ...
  41  *                                 2                 3
  42  *
  43  *
  44  * Note 1. Since gamma(1+s)=s*gamma(s), hence
  45  *              lgamma(1+s) = log(s) + lgamma(s), or
  46  *              lgamma(s) = lgamma(1+s) - log(s).
  47  *         When s is really tiny (like roundoff), lgamma(1+s) ~ s(1-enler)
  48  *         Hence lgamma(s) ~ -log(s) for tiny s
  49  *
  50  */
  51 
  52 #include "libm.h"
  53 #include "longdouble.h"
  54 
  55 static long double neg(long double, int *);
  56 static long double poly(long double, const long double *, int);
  57 static long double polytail(long double);
  58 static long double primary(long double);
  59 
  60 static const long double 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 {
  74         long double t, y;
  75         int i;
  76 
  77         /* purge off +-inf, NaN and negative arguments */
  78         if (!finitel(x))
  79                 return (x * x);
  80 
  81         *signgamlp = 1;
  82 
  83         if (signbitl(x))
  84                 return (neg(x, signgamlp));
  85 
  86         /* for x < 8.0 */
  87         if (x < 8.0L) {
  88                 y = anintl(x);
  89                 i = (int)y;
  90 
  91                 switch (i) {
  92                 case 0:
  93 
  94                         if (x < 1.0e-40L)
  95                                 return (-logl(x));
  96                         else
  97                                 return ((primary(x) - log1pl(x)) - logl(x));
  98 
  99                 case 1:
 100                         return (primary(x - y) - logl(x));
 101                 case 2:
 102                         return (primary(x - y));
 103                 case 3:
 104                         return (primary(x - y) + logl(x - c1));
 105                 case 4:
 106                         return (primary(x - y) + logl((x - c1) * (x - c2)));
 107                 case 5:
 108                         return (primary(x - y) + logl((x - c1) * (x - c2) * (x -
 109                             c3)));
 110                 case 6:
 111                         return (primary(x - y) + logl((x - c1) * (x - c2) * (x -
 112                             c3) * (x - c4)));
 113                 case 7:
 114                         return (primary(x - y) + logl((x - c1) * (x - c2) * (x -
 115                             c3) * (x - c4) * (x - c5)));
 116                 case 8:
 117                         return (primary(x - y) + logl((x - c1) * (x - c2) * (x -
 118                             c3) * (x - c4) * (x - c5) * (x - c6)));
 119                 }
 120         }
 121 
 122         /* 8.0 <= x < 1.0e40 */
 123         if (x < 1.0e40L) {
 124                 t = logl(x);
 125                 return (x * (t - c1) - (ch * t - polytail(c1 / x)));
 126         }
 127 
 128         /* 1.0e40 <= x <= inf */
 129         return (x * (logl(x) - c1));
 130 }
 131 
 132 static const long double an1[] = {      /* 20 terms */
 133         -0.0772156649015328606065120900824024309741L,
 134         3.224670334241132182362075833230130289059e-0001L,
 135         -6.735230105319809513324605383668929964120e-0002L,
 136         2.058080842778454787900092432928910226297e-0002L,
 137         -7.385551028673985266273054086081102125704e-0003L,
 138         2.890510330741523285758867304409628648727e-0003L,
 139         -1.192753911703260976581414338096267498555e-0003L,
 140         5.096695247430424562831956662855697824035e-0004L,
 141         -2.231547584535777978926798502084300123638e-0004L,
 142         9.945751278186384670278268034322157947635e-0005L,
 143         -4.492623673665547726647838474125147631082e-0005L,
 144         2.050721280617796810096993154281561168706e-0005L,
 145         -9.439487785617396552092393234044767313568e-0006L,
 146         4.374872903516051510689234173139793159340e-0006L,
 147         -2.039156676413643091040459825776029327487e-0006L,
 148         9.555777181318621470466563543806211523634e-0007L,
 149         -4.468344919709630637558538313482398989638e-0007L,
 150         2.216738086090045781773004477831059444178e-0007L,
 151         -7.472783403418388455860445842543843485916e-0008L,
 152         8.777317930927149922056782132706238921648e-0008L,
 153 };
 154 
 155 static const long double an2[] = {      /* 20 terms */
 156         -.0772156649015328606062692723698127607018L,
 157         3.224670334241132182635552349060279118047e-0001L,
 158         -6.735230105319809367555642883133994818325e-0002L,
 159         2.058080842778459676880822202762143671813e-0002L,
 160         -7.385551028672828216011343150077846918930e-0003L,
 161         2.890510330762060607399561536905727853178e-0003L,
 162         -1.192753911419623262328187532759756368041e-0003L,
 163         5.096695278636456678258091134532258618614e-0004L,
 164         -2.231547306817535743052975194022893369135e-0004L,
 165         9.945771461633313282744264853986643877087e-0005L,
 166         -4.492503279458972037926876061257489481619e-0005L,
 167         2.051311416812082875492678651369394595613e-0005L,
 168         -9.415778282365955203915850761537462941165e-0006L,
 169         4.452428829045147098722932981088650055919e-0006L,
 170         -1.835024727987632579886951760650722695781e-0006L,
 171         1.379783080658545009579060714946381462565e-0006L,
 172         2.282637532109775156769736768748402175238e-0007L,
 173         1.002577375515900191362119718128149880168e-0006L,
 174         5.177028794262638311939991106423220002463e-0007L,
 175         3.127947245174847104122426445937830555755e-0007L,
 176 };
 177 
 178 static const long double an3[] = {      /* 20 terms */
 179         -.0772156649015328227870646417729220690875L,
 180         3.224670334241156699881788955959915250365e-0001L,
 181         -6.735230105312273571375431059744975563170e-0002L,
 182         2.058080842924464587662846071337083809005e-0002L,
 183         -7.385551008677271654723604653956131791619e-0003L,
 184         2.890510536479782086197110272583833176602e-0003L,
 185         -1.192752262076857692740571567808259138697e-0003L,
 186         5.096800771149805289371135155128380707889e-0004L,
 187         -2.231000836682831335505058492409860123647e-0004L,
 188         9.968912171073936803871803966360595275047e-0005L,
 189         -4.412020779327746243544387946167256187258e-0005L,
 190         2.281374113541454151067016632998630209049e-0005L,
 191         -4.028361291428629491824694655287954266830e-0006L,
 192         1.470694920619518924598956849226530750139e-0005L,
 193         1.381686137617987197975289545582377713772e-0005L,
 194         2.012493539265777728944759982054970441601e-0005L,
 195         1.723917864208965490251560644681933675799e-0005L,
 196         1.202954035243788300138608765425123713395e-0005L,
 197         5.079851887558623092776296577030850938146e-0006L,
 198         1.220657945824153751555138592006604026282e-0006L,
 199 };
 200 
 201 static const long double an4[] = {      /* 21 terms */
 202         -.0772156649015732285350261816697540392371L,
 203         3.224670334221752060691751340365212226097e-0001L,
 204         -6.735230109744009693977755991488196368279e-0002L,
 205         2.058080778913037626909954141611580783216e-0002L,
 206         -7.385557567931505621170483708950557506819e-0003L,
 207         2.890459838416254326340844289785254883436e-0003L,
 208         -1.193059036207136762877351596966718455737e-0003L,
 209         5.081914708100372836613371356529568937869e-0004L,
 210         -2.289855016133600313131553005982542045338e-0004L,
 211         8.053454537980585879620331053833498511491e-0005L,
 212         -9.574620532104845821243493405855672438998e-0005L,
 213         -9.269085628207107155601445001196317715686e-0005L,
 214         -2.183276779859490461716196344776208220180e-0004L,
 215         -3.134834305597571096452454999737269668868e-0004L,
 216         -3.973878894951937437018305986901392888619e-0004L,
 217         -3.953352414899222799161275564386488057119e-0004L,
 218         -3.136740932204038779362660900621212816511e-0004L,
 219         -1.884502253819634073946130825196078627664e-0004L,
 220         -8.192655799958926853585332542123631379301e-0005L,
 221         -2.292183750010571062891605074281744854436e-0005L,
 222         -3.223980628729716864927724265781406614294e-0006L,
 223 };
 224 
 225 static const long double ap1[] = {      /* 19 terms */
 226         -0.0772156649015328606065120900824024296961L,
 227         3.224670334241132182362075833230047956465e-0001L,
 228         -6.735230105319809513324605382963943777301e-0002L,
 229         2.058080842778454787900092126606252375465e-0002L,
 230         -7.385551028673985266272518231365020063941e-0003L,
 231         2.890510330741523285681704570797770736423e-0003L,
 232         -1.192753911703260971285304221165990244515e-0003L,
 233         5.096695247430420878696018188830886972245e-0004L,
 234         -2.231547584535654004647639737841526025095e-0004L,
 235         9.945751278137201960636098805852315982919e-0005L,
 236         -4.492623672777606053587919463929044226280e-0005L,
 237         2.050721258703289487603702670753053765201e-0005L,
 238         -9.439485626565616989352750672499008021041e-0006L,
 239         4.374838162403994645138200419356844574219e-0006L,
 240         -2.038979492862555348577006944451002161496e-0006L,
 241         9.536763152382263548086981191378885102802e-0007L,
 242         -4.426111214332434049863595231916564014913e-0007L,
 243         1.911148847512947464234633846270287546882e-0007L,
 244         -5.788673944861923038157839080272303519671e-0008L,
 245 };
 246 
 247 static const long double ap2[] = {      /* 19 terms */
 248         -0.077215664901532860606428624449354836087L,
 249         3.224670334241132182271948744265855440139e-0001L,
 250         -6.735230105319809467356126599005051676203e-0002L,
 251         2.058080842778453315716389815213496002588e-0002L,
 252         -7.385551028673653323064118422580096222959e-0003L,
 253         2.890510330735923572088003424849289006039e-0003L,
 254         -1.192753911629952368606185543945790688144e-0003L,
 255         5.096695239806718875364547587043220998766e-0004L,
 256         -2.231547520600616108991867127392089144886e-0004L,
 257         9.945746913898151120612322833059416008973e-0005L,
 258         -4.492599307461977003570224943054585729684e-0005L,
 259         2.050609891889165453592046505651759999090e-0005L,
 260         -9.435329866734193796540515247917165988579e-0006L,
 261         4.362267138522223236241016136585565144581e-0006L,
 262         -2.008556356653246579300491601497510230557e-0006L,
 263         8.961498103387207161105347118042844354395e-0007L,
 264         -3.614187228330216282235692806488341157741e-0007L,
 265         1.136978988247816860500420915014777753153e-0007L,
 266         -2.000532786387196664019286514899782691776e-0008L,
 267 };
 268 
 269 static const long double ap3[] = {      /* 19 terms */
 270         -0.077215664901532859888521470795348856446L,
 271         3.224670334241131733364048614484228443077e-0001L,
 272         -6.735230105319676541660495145259038151576e-0002L,
 273         2.058080842775975461837768839015444273830e-0002L,
 274         -7.385551028347615729728618066663566606906e-0003L,
 275         2.890510327517954083379032008643080256676e-0003L,
 276         -1.192753886919470728001821137439430882603e-0003L,
 277         5.096693728898932234814903769146577482912e-0004L,
 278         -2.231540055048827662528594010961874258037e-0004L,
 279         9.945446210018649311491619999438833843723e-0005L,
 280         -4.491608206598064519190236245753867697750e-0005L,
 281         2.047939071322271016498065052853746466669e-0005L,
 282         -9.376824046522786006677541036631536790762e-0006L,
 283         4.259329829498149111582277209189150127347e-0006L,
 284         -1.866064770421594266702176289764212873428e-0006L,
 285         7.462066721137579592928128104534957135669e-0007L,
 286         -2.483546217529077735074007138457678727371e-0007L,
 287         5.915166576378161473299324673649144297574e-0008L,
 288         -7.334139641706988966966252333759604701905e-0009L,
 289 };
 290 
 291 static const long double ap4[] = {      /* 19 terms */
 292         -0.0772156649015326785569313252637238673675L,
 293         3.224670334241051435008842685722468344822e-0001L,
 294         -6.735230105302832007479431772160948499254e-0002L,
 295         2.058080842553481183648529360967441889912e-0002L,
 296         -7.385551007602909242024706804659879199244e-0003L,
 297         2.890510182473907253939821312248303471206e-0003L,
 298         -1.192753098427856770847894497586825614450e-0003L,
 299         5.096659636418811568063339214203693550804e-0004L,
 300         -2.231421144004355691166194259675004483639e-0004L,
 301         9.942073842343832132754332881883387625136e-0005L,
 302         -4.483809261973204531263252655050701205397e-0005L,
 303         2.033260142610284888319116654931994447173e-0005L,
 304         -9.153539544026646699870528191410440585796e-0006L,
 305         3.988460469925482725894144688699584997971e-0006L,
 306         -1.609692980087029172567957221850825977621e-0006L,
 307         5.634916377249975825399706694496688803488e-0007L,
 308         -1.560065465929518563549083208482591437696e-0007L,
 309         2.961350193868935325526962209019387821584e-0008L,
 310         -2.834602215195368130104649234505033159842e-0009L,
 311 };
 312 
 313 static long double
 314 primary(long double s)                  /* assume |s|<=0.5 */
 315 {
 316         int i;
 317 
 318         i = (int)(8.0L * (s + 0.5L));
 319 
 320         switch (i) {
 321         case 0:
 322                 return (ch * s + s * poly(s, an4, 21));
 323         case 1:
 324                 return (ch * s + s * poly(s, an3, 20));
 325         case 2:
 326                 return (ch * s + s * poly(s, an2, 20));
 327         case 3:
 328                 return (ch * s + s * poly(s, an1, 20));
 329         case 4:
 330                 return (ch * s + s * poly(s, ap1, 19));
 331         case 5:
 332                 return (ch * s + s * poly(s, ap2, 19));
 333         case 6:
 334                 return (ch * s + s * poly(s, ap3, 19));
 335         case 7:
 336                 return (ch * s + s * poly(s, ap4, 19));
 337         }
 338 
 339         /* NOTREACHED */
 340         return (0.0L);
 341 }
 342 
 343 static long double
 344 poly(long double s, const long double *p, int n)
 345 {
 346         long double y;
 347         int i;
 348 
 349         y = p[n - 1];
 350 
 351         for (i = n - 2; i >= 0; i--)
 352                 y = p[i] + s * y;
 353 
 354         return (y);
 355 }
 356 
 357 static const long double pt[] = {
 358         9.189385332046727417803297364056176804663e-0001L,
 359         8.333333333333333333333333333331286969123e-0002L,
 360         -2.777777777777777777777777553194796036402e-0003L,
 361         7.936507936507936507927283071433584248176e-0004L,
 362         -5.952380952380952362351042163192634108297e-0004L,
 363         8.417508417508395661774286645578379460131e-0004L,
 364         -1.917526917525263651186066417934685675649e-0003L,
 365         6.410256409395203164659292973142293199083e-0003L,
 366         -2.955065327248303301763594514012418438188e-0002L,
 367         1.796442830099067542945998615411893822886e-0001L,
 368         -1.392413465829723742489974310411118662919e+0000L,
 369         1.339984238037267658352656597960492029261e+0001L,
 370         -1.564707657605373662425785904278645727813e+0002L,
 371         2.156323807499211356127813962223067079300e+0003L,
 372         -3.330486427626223184647299834137041307569e+0004L,
 373         5.235535072011889213611369254140123518699e+0005L,
 374         -7.258160984602220710491988573430212593080e+0006L,
 375         7.316526934569686459641438882340322673357e+0007L,
 376         -3.806450279064900548836571789284896711473e+0008L,
 377 };
 378 
 379 static long double
 380 polytail(long double s)
 381 {
 382         long double t, z;
 383         int i;
 384 
 385         z = s * s;
 386         t = pt[18];
 387 
 388         for (i = 17; i >= 1; i--)
 389                 t = pt[i] + z * t;
 390 
 391         return (pt[0] + s * t);
 392 }
 393 
 394 static long double
 395 neg(long double z, int *signgamlp)
 396 {
 397         long double t, p;
 398 
 399         /* BEGIN CSTYLED */
 400         /*
 401          * written by K.C. Ng,  Feb 2, 1989.
 402          *
 403          * Since
 404          *              -z*G(-z)*G(z) = pi/sin(pi*z),
 405          * we have
 406          *      G(-z) = -pi/(sin(pi*z)*G(z)*z)
 407          *                =  pi/(sin(pi*(-z))*G(z)*z)
 408          * Algorithm
 409          *              z = |z|
 410          *              t = sinpi(z); ...note that when z>2**112, z is an int
 411          *              and hence t=0.
 412          *
 413          *              if (t == 0.0) return 1.0/0.0;
 414          *              if (t< 0.0) *signgamlp = -1; else t= -t;
 415          *              if (z<1.0e-40)       ...tiny z
 416          *                  return -log(z);
 417          *              else
 418          *                  return log(pi/(t*z))-lgamma(z);
 419          *
 420          */
 421         /* END CSTYLED */
 422 
 423         t = sinpil(z);                  /* t := sin(pi*z) */
 424 
 425         if (t == c0)                    /* return   1.0/0.0 =  +INF */
 426                 return (c1 / c0);
 427 
 428         z = -z;
 429 
 430         if (z <= tiny)
 431                 p = -logl(z);
 432         else
 433                 p = logl(pi / (fabsl(t) * z)) - __k_lgammal(z, signgamlp);
 434 
 435         if (t < c0)
 436                 *signgamlp = -1;
 437 
 438         return (p);
 439 }