usr/src/lib/libm/common/m9x/tgammaf.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 #pragma weak __tgammaf = tgammaf
  31
  32 /*
  33  * True gamma function
  34  *
  35  * float tgammaf(float x)
  36  *
  37  * Algorithm: see tgamma.c
  38  *
  39  * Maximum error observed: 0.87ulp (both positive and negative arguments)
  40  */
  41
  42 #include "libm.h"
  43 #include <math.h>
  44 #if defined(__SUNPRO_C)
  45 #include <sunmath.h>
  46 #endif
  47 #include <sys/isa_defs.h>
  48
  49 #if defined(_BIG_ENDIAN)
  50 #define HIWORD  0
  51 #define LOWORD  1
  52 #else
  53 #define HIWORD  1
  54 #define LOWORD  0
  55 #endif
  56 #define __HI(x) ((int *) &x)[HIWORD]
  57 #define __LO(x) ((unsigned *) &x)[LOWORD]
  58
  59 /* Coefficients for primary intervals GTi() */
  60 static const double cr[] = {
  61         /* p1 */
  62         +7.09087253435088360271451613398019280077561279443e-0001,
  63         -5.17229560788652108545141978238701790105241761089e-0001,
  64         +5.23403394528150789405825222323770647162337764327e-0001,
  65         -4.54586308717075010784041566069480411732634814899e-0001,
  66         +4.20596490915239085459964590559256913498190955233e-0001,
  67         -3.57307589712377520978332185838241458642142185789e-0001,
  68
  69         /* p2 */
  70         +4.28486983980295198166056119223984284434264344578e-0001,
  71         -1.30704539487709138528680121627899735386650103914e-0001,
  72         +1.60856285038051955072861219352655851542955430871e-0001,
  73         -9.22285161346010583774458802067371182158937943507e-0002,
  74         +7.19240511767225260740890292605070595560626179357e-0002,
  75         -4.88158265593355093703112238534484636193260459574e-0002,
  76
  77         /* p3 */
  78         +3.82409531118807759081121479786092134814808872880e-0001,
  79         +2.65309888180188647956400403013495759365167853426e-0002,
  80         +8.06815109775079171923561169415370309376296739835e-0002,
  81         -1.54821591666137613928840890835174351674007764799e-0002,
  82         +1.76308239242717268530498313416899188157165183405e-0002,
  83
  84         /* GZi and TZi */
  85         +0.9382046279096824494097535615803269576988,    /* GZ1 */
  86         +0.8856031944108887002788159005825887332080,    /* GZ2 */
  87         +0.9367814114636523216188468970808378497426,    /* GZ3 */
  88         -0.3517214357852935791015625,   /* TZ1 */
  89         +0.280530631542205810546875,    /* TZ3 */
  90 };
  91
  92 #define P10     cr[0]
  93 #define P11     cr[1]
  94 #define P12     cr[2]
  95 #define P13     cr[3]
  96 #define P14     cr[4]
  97 #define P15     cr[5]
  98 #define P20     cr[6]
  99 #define P21     cr[7]
 100 #define P22     cr[8]
 101 #define P23     cr[9]
 102 #define P24     cr[10]
 103 #define P25     cr[11]
 104 #define P30     cr[12]
 105 #define P31     cr[13]
 106 #define P32     cr[14]
 107 #define P33     cr[15]
 108 #define P34     cr[16]
 109 #define GZ1     cr[17]
 110 #define GZ2     cr[18]
 111 #define GZ3     cr[19]
 112 #define TZ1     cr[20]
 113 #define TZ3     cr[21]
 114
 115 /* compute gamma(y) for y in GT1 = [1.0000, 1.2845] */
 116 static double
 117 GT1(double y) {
 118         double z, r;
 119
 120         z = y * y;
 121         r = TZ1 * y + z * ((P10 + y * P11 + z * P12) + (z * y) * (P13 + y *
 122                 P14 + z * P15));
 123         return (GZ1 + r);
 124 }
 125
 126 /* compute gamma(y) for y in GT2 = [1.2844, 1.6374] */
 127 static double
 128 GT2(double y) {
 129         double z;
 130
 131         z = y * y;
 132         return (GZ2 + z * ((P20 + y * P21 + z * P22) + (z * y) * (P23 + y *
 133                 P24 + z * P25)));
 134 }
 135
 136 /* compute gamma(y) for y in GT3 = [1.6373, 2.0000] */
 137 static double
 138 GT3(double y) {
 139 double z, r;
 140
 141         z = y * y;
 142         r = TZ3 * y + z * ((P30 + y * P31 + z * P32) + (z * y) * (P33 + y *
 143                 P34));
 144         return (GZ3 + r);
 145 }
 146
 147 /* INDENT OFF */
 148 static const double c[] = {
 149 +1.0,
 150 +2.0,
 151 +0.5,
 152 +1.0e-300,
 153 +6.666717231848518054693623697539230e-0001,                     /* A1=T3[0] */
 154 +8.33333330959694065245736888749042811909994573178e-0002,       /* GP[0] */
 155 -2.77765545601667179767706600890361535225507762168e-0003,       /* GP[1] */
 156 +7.77830853479775281781085278324621033523037489883e-0004,       /* GP[2] */
 157 +4.18938533204672741744150788368695779923320328369e-0001,       /* hln2pi   */
 158 +2.16608493924982901946e-02,                                    /* ln2_32 */
 159 +4.61662413084468283841e+01,                                    /* invln2_32 */
 160 +5.00004103388988968841156421415669985414073453720e-0001,       /* Et1 */
 161 +1.66667656752800761782778277828110208108687545908e-0001,       /* Et2 */
 162 };
 163
 164 #define one             c[0]
 165 #define two             c[1]
 166 #define half            c[2]
 167 #define tiny            c[3]
 168 #define A1              c[4]
 169 #define GP0             c[5]
 170 #define GP1             c[6]
 171 #define GP2             c[7]
 172 #define hln2pi          c[8]
 173 #define ln2_32          c[9]
 174 #define invln2_32       c[10]
 175 #define Et1             c[11]
 176 #define Et2             c[12]
 177
 178 /* S[j] = 2**(j/32.) for the final computation of exp(w) */
 179 static const double S[] = {
 180 +1.00000000000000000000e+00,    /* 3FF0000000000000 */
 181 +1.02189714865411662714e+00,    /* 3FF059B0D3158574 */
 182 +1.04427378242741375480e+00,    /* 3FF0B5586CF9890F */
 183 +1.06714040067682369717e+00,    /* 3FF11301D0125B51 */
 184 +1.09050773266525768967e+00,    /* 3FF172B83C7D517B */
 185 +1.11438674259589243221e+00,    /* 3FF1D4873168B9AA */
 186 +1.13878863475669156458e+00,    /* 3FF2387A6E756238 */
 187 +1.16372485877757747552e+00,    /* 3FF29E9DF51FDEE1 */
 188 +1.18920711500272102690e+00,    /* 3FF306FE0A31B715 */
 189 +1.21524735998046895524e+00,    /* 3FF371A7373AA9CB */
 190 +1.24185781207348400201e+00,    /* 3FF3DEA64C123422 */
 191 +1.26905095719173321989e+00,    /* 3FF44E086061892D */
 192 +1.29683955465100964055e+00,    /* 3FF4BFDAD5362A27 */
 193 +1.32523664315974132322e+00,    /* 3FF5342B569D4F82 */
 194 +1.35425554693689265129e+00,    /* 3FF5AB07DD485429 */
 195 +1.38390988196383202258e+00,    /* 3FF6247EB03A5585 */
 196 +1.41421356237309514547e+00,    /* 3FF6A09E667F3BCD */
 197 +1.44518080697704665027e+00,    /* 3FF71F75E8EC5F74 */
 198 +1.47682614593949934623e+00,    /* 3FF7A11473EB0187 */
 199 +1.50916442759342284141e+00,    /* 3FF82589994CCE13 */
 200 +1.54221082540794074411e+00,    /* 3FF8ACE5422AA0DB */
 201 +1.57598084510788649659e+00,    /* 3FF93737B0CDC5E5 */
 202 +1.61049033194925428347e+00,    /* 3FF9C49182A3F090 */
 203 +1.64575547815396494578e+00,    /* 3FFA5503B23E255D */
 204 +1.68179283050742900407e+00,    /* 3FFAE89F995AD3AD */
 205 +1.71861929812247793414e+00,    /* 3FFB7F76F2FB5E47 */
 206 +1.75625216037329945351e+00,    /* 3FFC199BDD85529C */
 207 +1.79470907500310716820e+00,    /* 3FFCB720DCEF9069 */
 208 +1.83400808640934243066e+00,    /* 3FFD5818DCFBA487 */
 209 +1.87416763411029996256e+00,    /* 3FFDFC97337B9B5F */
 210 +1.91520656139714740007e+00,    /* 3FFEA4AFA2A490DA */
 211 +1.95714412417540017941e+00,    /* 3FFF50765B6E4540 */
 212 };
 213 /* INDENT ON */
 214
 215 /* INDENT OFF */
 216 /*
 217  * return tgammaf(x) in double for 8<x<=35.040096283... using Stirling's formula
 218  *     log(G(x)) ~= (x-.5)*(log(x)-1) + .5(log(2*pi)-1) + (1/x)*P(1/(x*x))
 219  */
 220 /*
 221  * compute ss = log(x)-1
 222  *
 223  *  log(x) - 1 = T1(n) + T2(j) + T3(s), where x = 2**n * y,  1<=y<2,
 224  *  j=[64*y], z[j]=1+j/64+1/128, s = (y-z[j])/(y+z[j]), and
 225  *       T1(n-3) = n*log(2)-1,  n=3,4,5
 226  *       T2(j) = log(z[j]),
 227  *       T3(s) = 2s + A1*s^3
 228  *  Note
 229  *  (1) Remez error for T3(s) is bounded by 2**(-35.8)
 230  *      (see mpremez/work/Log/tgamma_log_2_outr1)
 231  */
 232
 233 static const double T1[] = { /* T1[j]=(j+3)*log(2)-1 */
 234 +1.079441541679835928251696364375e+00,
 235 +1.772588722239781237668928485833e+00,
 236 +2.465735902799726547086160607291e+00,
 237 };
 238
 239 static const double T2[] = {   /* T2[j]=log(1+j/64+1/128) */
 240 +7.782140442054948947462900061137e-03,
 241 +2.316705928153437822879916096229e-02,
 242 +3.831886430213659919375532512380e-02,
 243 +5.324451451881228286587019378653e-02,
 244 +6.795066190850774939456527777263e-02,
 245 +8.244366921107459126816006866831e-02,
 246 +9.672962645855111229557105648746e-02,
 247 +1.108143663402901141948061693232e-01,
 248 +1.247034785009572358634065153809e-01,
 249 +1.384023228591191356853258736016e-01,
 250 +1.519160420258419750718034248969e-01,
 251 +1.652495728953071628756114492772e-01,
 252 +1.784076574728182971194002415109e-01,
 253 +1.913948529996294546092988075613e-01,
 254 +2.042155414286908915038203861962e-01,
 255 +2.168739383006143596190895257443e-01,
 256 +2.293741010648458299914807250461e-01,
 257 +2.417199368871451681443075159135e-01,
 258 +2.539152099809634441373232979066e-01,
 259 +2.659635484971379413391259265375e-01,
 260 +2.778684510034563061863500329234e-01,
 261 +2.896332925830426768788930555257e-01,
 262 +3.012613305781617810128755382338e-01,
 263 +3.127557100038968883862465596883e-01,
 264 +3.241194686542119760906707604350e-01,
 265 +3.353555419211378302571795798142e-01,
 266 +3.464667673462085809184621884258e-01,
 267 +3.574558889218037742260094901409e-01,
 268 +3.683255611587076530482301540504e-01,
 269 +3.790783529349694583908533456310e-01,
 270 +3.897167511400252133704636040035e-01,
 271 +4.002431641270127069293251019951e-01,
 272 +4.106599249852683859343062031758e-01,
 273 +4.209692946441296361288671615068e-01,
 274 +4.311734648183713408591724789556e-01,
 275 +4.412745608048752294894964416613e-01,
 276 +4.512746441394585851446923830790e-01,
 277 +4.611757151221701663679999255979e-01,
 278 +4.709797152187910125468978560564e-01,
 279 +4.806885293457519076766184554480e-01,
 280 +4.903039880451938381503461596457e-01,
 281 +4.998278695564493298213314152470e-01,
 282 +5.092619017898079468040749192283e-01,
 283 +5.186077642080456321529769963648e-01,
 284 +5.278670896208423851138922177783e-01,
 285 +5.370414658968836545667292441538e-01,
 286 +5.461324375981356503823972092312e-01,
 287 +5.551415075405015927154803595159e-01,
 288 +5.640701382848029660713842900902e-01,
 289 +5.729197535617855090927567266263e-01,
 290 +5.816917396346224825206107537254e-01,
 291 +5.903874466021763746419167081236e-01,
 292 +5.990081896460833993816000244617e-01,
 293 +6.075552502245417955010851527911e-01,
 294 +6.160298772155140196475659281967e-01,
 295 +6.244332880118935010425387440547e-01,
 296 +6.327666695710378295457864685036e-01,
 297 +6.410311794209312910556013344054e-01,
 298 +6.492279466251098188908399699053e-01,
 299 +6.573580727083600301418900232459e-01,
 300 +6.654226325450904489500926100067e-01,
 301 +6.734226752121667202979603888010e-01,
 302 +6.813592248079030689480715595681e-01,
 303 +6.892332812388089803249143378146e-01,
 304 };
 305 /* INDENT ON */
 306
 307 static double
 308 large_gam(double x) {
 309         double ss, zz, z, t1, t2, w, y, u;
 310         unsigned lx;
 311         int k, ix, j, m;
 312
 313         ix = __HI(x);
 314         lx = __LO(x);
 315         m = (ix >> 20) - 0x3ff;                 /* exponent of x, range:3-5 */
 316         ix = (ix & 0x000fffff) | 0x3ff00000;    /* y = scale x to [1,2] */
 317         __HI(y) = ix;
 318         __LO(y) = lx;
 319         __HI(z) = (ix & 0xffffc000) | 0x2000;   /* z[j]=1+j/64+1/128 */
 320         __LO(z) = 0;
 321         j = (ix >> 14) & 0x3f;
 322         t1 = y + z;
 323         t2 = y - z;
 324         u = t2 / t1;
 325         ss = T1[m - 3] + T2[j] + u * (two + A1 * (u * u));
 326                                                         /* ss = log(x)-1 */
 327         /*
 328          * compute ww = (x-.5)*(log(x)-1) + .5*(log(2pi)-1) + 1/x*(P(1/x^2)))
 329          * where ss = log(x) - 1
 330          */
 331         z = one / x;
 332         zz = z * z;
 333         w = ((x - half) * ss + hln2pi) + z * (GP0 + zz * GP1 + (zz * zz) * GP2);
 334         k = (int) (w * invln2_32 + half);
 335
 336         /* compute the exponential of w */
 337         j = k & 0x1f;
 338         m = k >> 5;
 339         z = w - (double) k *ln2_32;
 340         zz = S[j] * (one + z + (z * z) * (Et1 + z * Et2));
 341         __HI(zz) += m << 20;
 342         return (zz);
 343 }
 344 /* INDENT OFF */
 345 /*
 346  * kpsin(x)= sin(pi*x)/pi
 347  *                 3        5        7        9
 348  *      = x+ks[0]*x +ks[1]*x +ks[2]*x +ks[3]*x
 349  */
 350 static const double ks[] = {
 351 -1.64493404985645811354476665052005342839447790544e+0000,
 352 +8.11740794458351064092797249069438269367389272270e-0001,
 353 -1.90703144603551216933075809162889536878854055202e-0001,
 354 +2.55742333994264563281155312271481108635575331201e-0002,
 355 };
 356 /* INDENT ON */
 357
 358 static double
 359 kpsin(double x) {
 360         double z;
 361
 362         z = x * x;
 363         return (x + (x * z) * ((ks[0] + z * ks[1]) + (z * z) * (ks[2] + z *
 364                 ks[3])));
 365 }
 366
 367 /* INDENT OFF */
 368 /*
 369  * kpcos(x)= cos(pi*x)/pi
 370  *                     2        4        6
 371  *      = kc[0]+kc[1]*x +kc[2]*x +kc[3]*x
 372  */
 373 static const double kc[] = {
 374 +3.18309886183790671537767526745028724068919291480e-0001,
 375 -1.57079581447762568199467875065854538626594937791e+0000,
 376 +1.29183528092558692844073004029568674027807393862e+0000,
 377 -4.20232949771307685981015914425195471602739075537e-0001,
 378 };
 379 /* INDENT ON */
 380
 381 static double
 382 kpcos(double x) {
 383         double z;
 384
 385         z = x * x;
 386         return (kc[0] + z * (kc[1] + z * kc[2] + (z * z) * kc[3]));
 387 }
 388
 389 /* INDENT OFF */
 390 static const double
 391 t0z1 = 0.134861805732790769689793935774652917006,
 392 t0z2 = 0.461632144968362341262659542325721328468,
 393 t0z3 = 0.819773101100500601787868704921606996312;
 394         /* 1.134861805732790769689793935774652917006 */
 395 /* INDENT ON */
 396
 397 /*
 398  * gamma(x+i) for 0 <= x < 1
 399  */
 400 static double
 401 gam_n(int i, double x) {
 402         double rr = 0.0L, yy;
 403         double z1, z2;
 404
 405         /* compute yy = gamma(x+1) */
 406         if (x > 0.2845) {
 407                 if (x > 0.6374)
 408                         yy = GT3(x - t0z3);
 409                 else
 410                         yy = GT2(x - t0z2);
 411         } else
 412                 yy = GT1(x - t0z1);
 413
 414         /* compute gamma(x+i) = (x+i-1)*...*(x+1)*yy, 0<i<8 */
 415         switch (i) {
 416         case 0:         /* yy/x */
 417                 rr = yy / x;
 418                 break;
 419         case 1:         /* yy */
 420                 rr = yy;
 421                 break;
 422         case 2:         /* (x+1)*yy */
 423                 rr = (x + one) * yy;
 424                 break;
 425         case 3:         /* (x+2)*(x+1)*yy */
 426                 rr = (x + one) * (x + two) * yy;
 427                 break;
 428
 429         case 4:         /* (x+1)*(x+3)*(x+2)*yy */
 430                 rr = (x + one) * (x + two) * ((x + 3.0) * yy);
 431                 break;
 432         case 5:         /* ((x+1)*(x+4)*(x+2)*(x+3))*yy */
 433                 z1 = (x + two) * (x + 3.0) * yy;
 434                 z2 = (x + one) * (x + 4.0);
 435                 rr = z1 * z2;
 436                 break;
 437         case 6:         /* ((x+1)*(x+2)*(x+3)*(x+4)*(x+5))*yy */
 438                 z1 = (x + two) * (x + 3.0);
 439                 z2 = (x + 5.0) * yy;
 440                 rr = z1 * (z1 - two) * z2;
 441                 break;
 442         case 7:         /* ((x+1)*(x+2)*(x+3)*(x+4)*(x+5)*(x+6))*yy */
 443                 z1 = (x + two) * (x + 3.0);
 444                 z2 = (x + 5.0) * (x + 6.0) * yy;
 445                 rr = z1 * (z1 - two) * z2;
 446                 break;
 447         }
 448         return (rr);
 449 }
 450
 451 float
 452 tgammaf(float xf) {
 453         float zf;
 454         double ss, ww;
 455         double x, y, z;
 456         int i, j, k, ix, hx, xk;
 457
 458         hx = *(int *) &xf;
 459         ix = hx & 0x7fffffff;
 460
 461         x = (double) xf;
 462         if (ix < 0x33800000)
 463                 return (1.0F / xf);     /* |x| < 2**-24 */
 464
 465         if (ix >= 0x7f800000)
 466                 return (xf * ((hx < 0)? 0.0F : xf)); /* +-Inf or NaN */
 467
 468         if (hx > 0x420C290F)    /* x > 35.040096283... overflow */
 469                 return (float)(x / tiny);
 470
 471         if (hx >= 0x41000000)   /* x >= 8 */
 472                 return ((float) large_gam(x));
 473
 474         if (hx > 0) {           /* 0 < x < 8 */
 475                 i = (int) xf;
 476                 return ((float) gam_n(i, x - (double) i));
 477         }
 478
 479         /* negative x */
 480         /* INDENT OFF */
 481         /*
 482          * compute xk =
 483          *      -2 ... x is an even int (-inf is considered even)
 484          *      -1 ... x is an odd int
 485          *      +0 ... x is not an int but chopped to an even int
 486          *      +1 ... x is not an int but chopped to an odd int
 487          */
 488         /* INDENT ON */
 489         xk = 0;
 490         if (ix >= 0x4b000000) {
 491                 if (ix > 0x4b000000)
 492                         xk = -2;
 493                 else
 494                         xk = -2 + (ix & 1);
 495         } else if (ix >= 0x3f800000) {
 496                 k = (ix >> 23) - 0x7f;
 497                 j = ix >> (23 - k);
 498                 if ((j << (23 - k)) == ix)
 499                         xk = -2 + (j & 1);
 500                 else
 501                         xk = j & 1;
 502         }
 503         if (xk < 0) {
 504                 /* 0/0 invalid NaN, ideally gamma(-n)= (-1)**(n+1) * inf */
 505                 zf = xf - xf;
 506                 return (zf / zf);
 507         }
 508
 509         /* negative underflow thresold */
 510         if (ix > 0x4224000B) {  /* x < -(41+11ulp) */
 511                 if (xk == 0)
 512                         z = -tiny;
 513                 else
 514                         z = tiny;
 515                 return ((float)z);
 516         }
 517
 518         /* INDENT OFF */
 519         /* now compute gamma(x) by  -1/((sin(pi*y)/pi)*gamma(1+y)), y = -x */
 520         /*
 521          * First compute ss = -sin(pi*y)/pi , so that
 522          * gamma(x) = 1/(ss*gamma(1+y))
 523          */
 524         /* INDENT ON */
 525         y = -x;
 526         j = (int) y;
 527         z = y - (double) j;
 528         if (z > 0.3183098861837906715377675)
 529                 if (z > 0.6816901138162093284622325)
 530                         ss = kpsin(one - z);
 531                 else
 532                         ss = kpcos(0.5 - z);
 533         else
 534                 ss = kpsin(z);
 535         if (xk == 0)
 536                 ss = -ss;
 537
 538         /* Then compute ww = gamma(1+y)  */
 539         if (j < 7)
 540                 ww = gam_n(j + 1, z);
 541         else
 542                 ww = large_gam(y + one);
 543
 544         /* return 1/(ss*ww) */
 545         return ((float) (one / (ww * ss)));
 546 }