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 "libm_synonyms.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 61 c0 = 0.0L, 62 ch = 0.5L, 63 c1 = 1.0L, 64 c2 = 2.0L, 65 c3 = 3.0L, 66 c4 = 4.0L, 67 c5 = 5.0L, 68 c6 = 6.0L, 69 pi = 3.1415926535897932384626433832795028841971L, 70 tiny = 1.0e-40L; 71 72 long double 73 __k_lgammal(long double x, int *signgamlp) { 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 *signgamlp = 1; 81 if (signbitl(x)) 82 return (neg(x, signgamlp)); 83 84 /* for x < 8.0 */ 85 if (x < 8.0L) { 86 y = anintl(x); 87 i = (int) y; 88 switch (i) { 89 case 0: 90 if (x < 1.0e-40L) 91 return (-logl(x)); 92 else 93 return (primary(x)-log1pl(x))-logl(x); 94 case 1: 95 return (primary(x-y)-logl(x)); 96 case 2: 97 return (primary(x-y)); 98 case 3: 99 return (primary(x-y)+logl(x-c1)); 100 case 4: 101 return (primary(x-y)+logl((x-c1)*(x-c2))); 102 case 5: 103 return (primary(x-y)+logl((x-c1)*(x-c2)*(x-c3))); 104 case 6: 105 return (primary(x-y)+logl((x-c1)*(x-c2)*(x-c3)*(x-c4))); 106 case 7: 107 return (primary(x-y)+logl((x-c1)*(x-c2)*(x-c3)*(x-c4)*(x-c5))); 108 case 8: 109 return primary(x-y)+ 110 logl((x-c1)*(x-c2)*(x-c3)*(x-c4)*(x-c5)*(x-c6)); 111 } 112 } 113 114 /* 8.0 <= x < 1.0e40 */ 115 if (x < 1.0e40L) { 116 t = logl(x); 117 return (x*(t-c1)-(ch*t-polytail(c1/x))); 118 } 119 120 /* 1.0e40 <= x <= inf */ 121 return (x*(logl(x)-c1)); 122 } 123 124 static const long double an1[] = { /* 20 terms */ 125 -0.0772156649015328606065120900824024309741L, 126 3.224670334241132182362075833230130289059e-0001L, 127 -6.735230105319809513324605383668929964120e-0002L, 128 2.058080842778454787900092432928910226297e-0002L, 129 -7.385551028673985266273054086081102125704e-0003L, 130 2.890510330741523285758867304409628648727e-0003L, 131 -1.192753911703260976581414338096267498555e-0003L, 132 5.096695247430424562831956662855697824035e-0004L, 133 -2.231547584535777978926798502084300123638e-0004L, 134 9.945751278186384670278268034322157947635e-0005L, 135 -4.492623673665547726647838474125147631082e-0005L, 136 2.050721280617796810096993154281561168706e-0005L, 137 -9.439487785617396552092393234044767313568e-0006L, 138 4.374872903516051510689234173139793159340e-0006L, 139 -2.039156676413643091040459825776029327487e-0006L, 140 9.555777181318621470466563543806211523634e-0007L, 141 -4.468344919709630637558538313482398989638e-0007L, 142 2.216738086090045781773004477831059444178e-0007L, 143 -7.472783403418388455860445842543843485916e-0008L, 144 8.777317930927149922056782132706238921648e-0008L, 145 }; 146 147 static const long double an2[] = { /* 20 terms */ 148 -.0772156649015328606062692723698127607018L, 149 3.224670334241132182635552349060279118047e-0001L, 150 -6.735230105319809367555642883133994818325e-0002L, 151 2.058080842778459676880822202762143671813e-0002L, 152 -7.385551028672828216011343150077846918930e-0003L, 153 2.890510330762060607399561536905727853178e-0003L, 154 -1.192753911419623262328187532759756368041e-0003L, 155 5.096695278636456678258091134532258618614e-0004L, 156 -2.231547306817535743052975194022893369135e-0004L, 157 9.945771461633313282744264853986643877087e-0005L, 158 -4.492503279458972037926876061257489481619e-0005L, 159 2.051311416812082875492678651369394595613e-0005L, 160 -9.415778282365955203915850761537462941165e-0006L, 161 4.452428829045147098722932981088650055919e-0006L, 162 -1.835024727987632579886951760650722695781e-0006L, 163 1.379783080658545009579060714946381462565e-0006L, 164 2.282637532109775156769736768748402175238e-0007L, 165 1.002577375515900191362119718128149880168e-0006L, 166 5.177028794262638311939991106423220002463e-0007L, 167 3.127947245174847104122426445937830555755e-0007L, 168 }; 169 170 static const long double an3[] = { /* 20 terms */ 171 -.0772156649015328227870646417729220690875L, 172 3.224670334241156699881788955959915250365e-0001L, 173 -6.735230105312273571375431059744975563170e-0002L, 174 2.058080842924464587662846071337083809005e-0002L, 175 -7.385551008677271654723604653956131791619e-0003L, 176 2.890510536479782086197110272583833176602e-0003L, 177 -1.192752262076857692740571567808259138697e-0003L, 178 5.096800771149805289371135155128380707889e-0004L, 179 -2.231000836682831335505058492409860123647e-0004L, 180 9.968912171073936803871803966360595275047e-0005L, 181 -4.412020779327746243544387946167256187258e-0005L, 182 2.281374113541454151067016632998630209049e-0005L, 183 -4.028361291428629491824694655287954266830e-0006L, 184 1.470694920619518924598956849226530750139e-0005L, 185 1.381686137617987197975289545582377713772e-0005L, 186 2.012493539265777728944759982054970441601e-0005L, 187 1.723917864208965490251560644681933675799e-0005L, 188 1.202954035243788300138608765425123713395e-0005L, 189 5.079851887558623092776296577030850938146e-0006L, 190 1.220657945824153751555138592006604026282e-0006L, 191 }; 192 193 static const long double an4[] = { /* 21 terms */ 194 -.0772156649015732285350261816697540392371L, 195 3.224670334221752060691751340365212226097e-0001L, 196 -6.735230109744009693977755991488196368279e-0002L, 197 2.058080778913037626909954141611580783216e-0002L, 198 -7.385557567931505621170483708950557506819e-0003L, 199 2.890459838416254326340844289785254883436e-0003L, 200 -1.193059036207136762877351596966718455737e-0003L, 201 5.081914708100372836613371356529568937869e-0004L, 202 -2.289855016133600313131553005982542045338e-0004L, 203 8.053454537980585879620331053833498511491e-0005L, 204 -9.574620532104845821243493405855672438998e-0005L, 205 -9.269085628207107155601445001196317715686e-0005L, 206 -2.183276779859490461716196344776208220180e-0004L, 207 -3.134834305597571096452454999737269668868e-0004L, 208 -3.973878894951937437018305986901392888619e-0004L, 209 -3.953352414899222799161275564386488057119e-0004L, 210 -3.136740932204038779362660900621212816511e-0004L, 211 -1.884502253819634073946130825196078627664e-0004L, 212 -8.192655799958926853585332542123631379301e-0005L, 213 -2.292183750010571062891605074281744854436e-0005L, 214 -3.223980628729716864927724265781406614294e-0006L, 215 }; 216 217 static const long double ap1[] = { /* 19 terms */ 218 -0.0772156649015328606065120900824024296961L, 219 3.224670334241132182362075833230047956465e-0001L, 220 -6.735230105319809513324605382963943777301e-0002L, 221 2.058080842778454787900092126606252375465e-0002L, 222 -7.385551028673985266272518231365020063941e-0003L, 223 2.890510330741523285681704570797770736423e-0003L, 224 -1.192753911703260971285304221165990244515e-0003L, 225 5.096695247430420878696018188830886972245e-0004L, 226 -2.231547584535654004647639737841526025095e-0004L, 227 9.945751278137201960636098805852315982919e-0005L, 228 -4.492623672777606053587919463929044226280e-0005L, 229 2.050721258703289487603702670753053765201e-0005L, 230 -9.439485626565616989352750672499008021041e-0006L, 231 4.374838162403994645138200419356844574219e-0006L, 232 -2.038979492862555348577006944451002161496e-0006L, 233 9.536763152382263548086981191378885102802e-0007L, 234 -4.426111214332434049863595231916564014913e-0007L, 235 1.911148847512947464234633846270287546882e-0007L, 236 -5.788673944861923038157839080272303519671e-0008L, 237 }; 238 239 static const long double ap2[] = { /* 19 terms */ 240 -0.077215664901532860606428624449354836087L, 241 3.224670334241132182271948744265855440139e-0001L, 242 -6.735230105319809467356126599005051676203e-0002L, 243 2.058080842778453315716389815213496002588e-0002L, 244 -7.385551028673653323064118422580096222959e-0003L, 245 2.890510330735923572088003424849289006039e-0003L, 246 -1.192753911629952368606185543945790688144e-0003L, 247 5.096695239806718875364547587043220998766e-0004L, 248 -2.231547520600616108991867127392089144886e-0004L, 249 9.945746913898151120612322833059416008973e-0005L, 250 -4.492599307461977003570224943054585729684e-0005L, 251 2.050609891889165453592046505651759999090e-0005L, 252 -9.435329866734193796540515247917165988579e-0006L, 253 4.362267138522223236241016136585565144581e-0006L, 254 -2.008556356653246579300491601497510230557e-0006L, 255 8.961498103387207161105347118042844354395e-0007L, 256 -3.614187228330216282235692806488341157741e-0007L, 257 1.136978988247816860500420915014777753153e-0007L, 258 -2.000532786387196664019286514899782691776e-0008L, 259 }; 260 261 static const long double ap3[] = { /* 19 terms */ 262 -0.077215664901532859888521470795348856446L, 263 3.224670334241131733364048614484228443077e-0001L, 264 -6.735230105319676541660495145259038151576e-0002L, 265 2.058080842775975461837768839015444273830e-0002L, 266 -7.385551028347615729728618066663566606906e-0003L, 267 2.890510327517954083379032008643080256676e-0003L, 268 -1.192753886919470728001821137439430882603e-0003L, 269 5.096693728898932234814903769146577482912e-0004L, 270 -2.231540055048827662528594010961874258037e-0004L, 271 9.945446210018649311491619999438833843723e-0005L, 272 -4.491608206598064519190236245753867697750e-0005L, 273 2.047939071322271016498065052853746466669e-0005L, 274 -9.376824046522786006677541036631536790762e-0006L, 275 4.259329829498149111582277209189150127347e-0006L, 276 -1.866064770421594266702176289764212873428e-0006L, 277 7.462066721137579592928128104534957135669e-0007L, 278 -2.483546217529077735074007138457678727371e-0007L, 279 5.915166576378161473299324673649144297574e-0008L, 280 -7.334139641706988966966252333759604701905e-0009L, 281 }; 282 283 static const long double ap4[] = { /* 19 terms */ 284 -0.0772156649015326785569313252637238673675L, 285 3.224670334241051435008842685722468344822e-0001L, 286 -6.735230105302832007479431772160948499254e-0002L, 287 2.058080842553481183648529360967441889912e-0002L, 288 -7.385551007602909242024706804659879199244e-0003L, 289 2.890510182473907253939821312248303471206e-0003L, 290 -1.192753098427856770847894497586825614450e-0003L, 291 5.096659636418811568063339214203693550804e-0004L, 292 -2.231421144004355691166194259675004483639e-0004L, 293 9.942073842343832132754332881883387625136e-0005L, 294 -4.483809261973204531263252655050701205397e-0005L, 295 2.033260142610284888319116654931994447173e-0005L, 296 -9.153539544026646699870528191410440585796e-0006L, 297 3.988460469925482725894144688699584997971e-0006L, 298 -1.609692980087029172567957221850825977621e-0006L, 299 5.634916377249975825399706694496688803488e-0007L, 300 -1.560065465929518563549083208482591437696e-0007L, 301 2.961350193868935325526962209019387821584e-0008L, 302 -2.834602215195368130104649234505033159842e-0009L, 303 }; 304 305 static long double 306 primary(long double s) { /* assume |s|<=0.5 */ 307 int i; 308 309 i = (int) (8.0L * (s + 0.5L)); 310 switch (i) { 311 case 0: return ch*s+s*poly(s, an4, 21); 312 case 1: return ch*s+s*poly(s, an3, 20); 313 case 2: return ch*s+s*poly(s, an2, 20); 314 case 3: return ch*s+s*poly(s, an1, 20); 315 case 4: return ch*s+s*poly(s, ap1, 19); 316 case 5: return ch*s+s*poly(s, ap2, 19); 317 case 6: return ch*s+s*poly(s, ap3, 19); 318 case 7: return ch*s+s*poly(s, ap4, 19); 319 } 320 /* NOTREACHED */ 321 return (0.0L); 322 } 323 324 static long double 325 poly(long double s, const long double *p, int n) { 326 long double y; 327 int i; 328 y = p[n-1]; 329 for (i = n-2; i >= 0; i--) y = p[i]+s*y; 330 return (y); 331 } 332 333 static const long double pt[] = { 334 9.189385332046727417803297364056176804663e-0001L, 335 8.333333333333333333333333333331286969123e-0002L, 336 -2.777777777777777777777777553194796036402e-0003L, 337 7.936507936507936507927283071433584248176e-0004L, 338 -5.952380952380952362351042163192634108297e-0004L, 339 8.417508417508395661774286645578379460131e-0004L, 340 -1.917526917525263651186066417934685675649e-0003L, 341 6.410256409395203164659292973142293199083e-0003L, 342 -2.955065327248303301763594514012418438188e-0002L, 343 1.796442830099067542945998615411893822886e-0001L, 344 -1.392413465829723742489974310411118662919e+0000L, 345 1.339984238037267658352656597960492029261e+0001L, 346 -1.564707657605373662425785904278645727813e+0002L, 347 2.156323807499211356127813962223067079300e+0003L, 348 -3.330486427626223184647299834137041307569e+0004L, 349 5.235535072011889213611369254140123518699e+0005L, 350 -7.258160984602220710491988573430212593080e+0006L, 351 7.316526934569686459641438882340322673357e+0007L, 352 -3.806450279064900548836571789284896711473e+0008L, 353 }; 354 355 static long double 356 polytail(long double s) { 357 long double t, z; 358 int i; 359 z = s*s; 360 t = pt[18]; 361 for (i = 17; i >= 1; i--) t = pt[i]+z*t; 362 return (pt[0]+s*t); 363 } 364 365 static long double 366 neg(long double z, int *signgamlp) { 367 long double t, p; 368 369 /* 370 * written by K.C. Ng, Feb 2, 1989. 371 * 372 * Since 373 * -z*G(-z)*G(z) = pi/sin(pi*z), 374 * we have 375 * G(-z) = -pi/(sin(pi*z)*G(z)*z) 376 * = pi/(sin(pi*(-z))*G(z)*z) 377 * Algorithm 378 * z = |z| 379 * t = sinpi(z); ...note that when z>2**112, z is an int 380 * and hence t=0. 381 * 382 * if (t == 0.0) return 1.0/0.0; 383 * if (t< 0.0) *signgamlp = -1; else t= -t; 384 * if (z<1.0e-40) ...tiny z 385 * return -log(z); 386 * else 387 * return log(pi/(t*z))-lgamma(z); 388 * 389 */ 390 391 t = sinpil(z); /* t := sin(pi*z) */ 392 if (t == c0) /* return 1.0/0.0 = +INF */ 393 return (c1/c0); 394 395 z = -z; 396 if (z <= tiny) 397 p = -logl(z); 398 else 399 p = logl(pi/(fabsl(t)*z)) - __k_lgammal(z, signgamlp); 400 if (t < c0) *signgamlp = -1; 401 return (p); 402 } 403