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 #if defined(ELFOBJ) 31 #pragma weak tgammal = __tgammal 32 #endif 33 34 #include "libm.h" 35 #include <sys/isa_defs.h> 36 37 #if defined(_BIG_ENDIAN) 38 #define H0_WORD(x) ((unsigned *) &x)[0] 39 #define H3_WORD(x) ((unsigned *) &x)[3] 40 #define CHOPPED(x) (long double) ((double) (x)) 41 #else 42 #define H0_WORD(x) ((((int *) &x)[2] << 16) | \ 43 (0x0000ffff & (((unsigned *) &x)[1] >> 15))) 44 #define H3_WORD(x) ((unsigned *) &x)[0] 45 #define CHOPPED(x) (long double) ((float) (x)) 46 #endif 47 48 struct LDouble { 49 long double h, l; 50 }; 51 52 /* INDENT OFF */ 53 /* Primary interval GTi() */ 54 static const long double P1[] = { 55 +0.709086836199777919037185741507610124611513720557L, 56 +4.45754781206489035827915969367354835667391606951e-0001L, 57 +3.21049298735832382311662273882632210062918153852e-0002L, 58 -5.71296796342106617651765245858289197369688864350e-0003L, 59 +6.04666892891998977081619174969855831606965352773e-0003L, 60 +8.99106186996888711939627812174765258822658645168e-0004L, 61 -6.96496846144407741431207008527018441810175568949e-0005L, 62 +1.52597046118984020814225409300131445070213882429e-0005L, 63 +5.68521076168495673844711465407432189190681541547e-0007L, 64 +3.30749673519634895220582062520286565610418952979e-0008L, 65 }; 66 static const long double Q1[] = { 67 +1.0+0000L, 68 +1.35806511721671070408570853537257079579490650668e+0000L, 69 +2.97567810153429553405327140096063086994072952961e-0001L, 70 -1.52956835982588571502954372821681851681118097870e-0001L, 71 -2.88248519561420109768781615289082053597954521218e-0002L, 72 +1.03475311719937405219789948456313936302378395955e-0002L, 73 +4.12310203243891222368965360124391297374822742313e-0004L, 74 -3.12653708152290867248931925120380729518332507388e-0004L, 75 +2.36672170850409745237358105667757760527014332458e-0005L, 76 }; 77 static const long double P2[] = { 78 +0.428486815855585429730209907810650135255270600668084114L, 79 +2.62768479103809762805691743305424077975230551176e-0001L, 80 +3.81187532685392297608310837995193946591425896150e-0002L, 81 +3.00063075891811043820666846129131255948527925381e-0003L, 82 +2.47315407812279164228398470797498649142513408654e-0003L, 83 +3.62838199917848372586173483147214880464782938664e-0004L, 84 +3.43991105975492623982725644046473030098172692423e-0006L, 85 +4.56902151569603272237014240794257659159045432895e-0006L, 86 +2.13734755837595695602045100675540011352948958453e-0007L, 87 +9.74123440547918230781670266967882492234877125358e-0009L, 88 }; 89 static const long double Q2[] = { 90 +1.0L, 91 +9.18284118632506842664645516830761489700556179701e-0001L, 92 -6.41430858837830766045202076965923776189154874947e-0003L, 93 -1.24400885809771073213345747437964149775410921376e-0001L, 94 +4.69803798146251757538856567522481979624746875964e-0003L, 95 +7.18309447069495315914284705109868696262662082731e-0003L, 96 -8.75812626987894695112722600697653425786166399105e-0004L, 97 -1.23539972377769277995959339188431498626674835169e-0004L, 98 +3.10019017590151598732360097849672925448587547746e-0005L, 99 -1.77260223349332617658921874288026777465782364070e-0006L, 100 }; 101 static const long double P3[] = { 102 +0.3824094797345675048502747661075355640070439388902L, 103 +3.42198093076618495415854906335908427159833377774e-0001L, 104 +9.63828189500585568303961406863153237440702754858e-0002L, 105 +8.76069421042696384852462044188520252156846768667e-0003L, 106 +1.86477890389161491224872014149309015261897537488e-0003L, 107 +8.16871354540309895879974742853701311541286944191e-0004L, 108 +6.83783483674600322518695090864659381650125625216e-0005L, 109 -1.10168269719261574708565935172719209272190828456e-0006L, 110 +9.66243228508380420159234853278906717065629721016e-0007L, 111 +2.31858885579177250541163820671121664974334728142e-0008L, 112 }; 113 static const long double Q3[] = { 114 +1.0L, 115 +8.25479821168813634632437430090376252512793067339e-0001L, 116 -1.62251363073937769739639623669295110346015576320e-0002L, 117 -1.10621286905916732758745130629426559691187579852e-0001L, 118 +3.48309693970985612644446415789230015515365291459e-0003L, 119 +6.73553737487488333032431261131289672347043401328e-0003L, 120 -7.63222008393372630162743587811004613050245128051e-0004L, 121 -1.35792670669190631476784768961953711773073251336e-0004L, 122 +3.19610150954223587006220730065608156460205690618e-0005L, 123 -1.82096553862822346610109522015129585693354348322e-0006L, 124 }; 125 126 static const long double 127 #if defined(__x86) 128 GZ1_h = 0.938204627909682449364570100414084663498215377L, 129 GZ1_l = 4.518346116624229420055327632718530617227944106e-20L, 130 GZ2_h = 0.885603194410888700264725126309883762587560340L, 131 GZ2_l = 1.409077427270497062039119290776508217077297169e-20L, 132 GZ3_h = 0.936781411463652321613537060640553022494714241L, 133 GZ3_l = 5.309836440284827247897772963887219035221996813e-21L, 134 #else 135 GZ1_h = 0.938204627909682449409753561580326910854647031L, 136 GZ1_l = 4.684412162199460089642452580902345976446297037e-35L, 137 GZ2_h = 0.885603194410888700278815900582588658192658794L, 138 GZ2_l = 7.501529273890253789219935569758713534641074860e-35L, 139 GZ3_h = 0.936781411463652321618846897080837818855399840L, 140 GZ3_l = 3.088721217404784363585591914529361687403776917e-35L, 141 #endif 142 TZ1 = -0.3517214357852935791015625L, 143 TZ3 = 0.280530631542205810546875L; 144 /* INDENT ON */ 145 146 /* INDENT OFF */ 147 /* 148 * compute gamma(y=yh+yl) for y in GT1 = [1.0000, 1.2845] 149 * ...assume yh got 53 or 24(i386) significant bits 150 */ 151 /* INDENT ON */ 152 static struct LDouble 153 GT1(long double yh, long double yl) { 154 long double t3, t4, y; 155 int i; 156 struct LDouble r; 157 158 y = yh + yl; 159 for (t4 = Q1[8], t3 = P1[8] + y * P1[9], i = 7; i >= 0; i--) { 160 t4 = t4 * y + Q1[i]; 161 t3 = t3 * y + P1[i]; 162 } 163 t3 = (y * y) * t3 / t4; 164 t3 += (TZ1 * yl + GZ1_l); 165 t4 = TZ1 * yh; 166 r.h = CHOPPED((t4 + GZ1_h + t3)); 167 t3 += (t4 - (r.h - GZ1_h)); 168 r.l = t3; 169 return (r); 170 } 171 172 /* INDENT OFF */ 173 /* 174 * compute gamma(y=yh+yl) for y in GT2 = [1.2844, 1.6374] 175 * ...assume yh got 53 significant bits 176 */ 177 /* INDENT ON */ 178 static struct LDouble 179 GT2(long double yh, long double yl) { 180 long double t3, t4, y; 181 int i; 182 struct LDouble r; 183 184 y = yh + yl; 185 for (t4 = Q2[9], t3 = P2[9], i = 8; i >= 0; i--) { 186 t4 = t4 * y + Q2[i]; 187 t3 = t3 * y + P2[i]; 188 } 189 t3 = GZ2_l + (y * y) * t3 / t4; 190 r.h = CHOPPED((GZ2_h + t3)); 191 r.l = t3 - (r.h - GZ2_h); 192 return (r); 193 } 194 195 /* INDENT OFF */ 196 /* 197 * compute gamma(y=yh+yl) for y in GT3 = [1.6373, 2.0000] 198 * ...assume yh got 53 significant bits 199 */ 200 /* INDENT ON */ 201 static struct LDouble 202 GT3(long double yh, long double yl) { 203 long double t3, t4, y; 204 int i; 205 struct LDouble r; 206 207 y = yh + yl; 208 for (t4 = Q3[9], t3 = P3[9], i = 8; i >= 0; i--) { 209 t4 = t4 * y + Q3[i]; 210 t3 = t3 * y + P3[i]; 211 } 212 t3 = (y * y) * t3 / t4; 213 t3 += (TZ3 * yl + GZ3_l); 214 t4 = TZ3 * yh; 215 r.h = CHOPPED((t4 + GZ3_h + t3)); 216 t3 += (t4 - (r.h - GZ3_h)); 217 r.l = t3; 218 return (r); 219 } 220 221 /* INDENT OFF */ 222 /* Hex value of GP[0] shoule be 3FB55555 55555555 */ 223 static const long double GP[] = { 224 +0.083333333333333333333333333333333172839171301L, 225 -2.77777777777777777777777777492501211999399424104e-0003L, 226 +7.93650793650793650793635650541638236350020883243e-0004L, 227 -5.95238095238095238057299772679324503339241961704e-0004L, 228 +8.41750841750841696138422987977683524926142600321e-0004L, 229 -1.91752691752686682825032547823699662178842123308e-0003L, 230 +6.41025641022403480921891559356473451161279359322e-0003L, 231 -2.95506535798414019189819587455577003732808185071e-0002L, 232 +1.79644367229970031486079180060923073476568732136e-0001L, 233 -1.39243086487274662174562872567057200255649290646e+0000L, 234 +1.34025874044417962188677816477842265259608269775e+0001L, 235 -1.56803713480127469414495545399982508700748274318e+0002L, 236 +2.18739841656201561694927630335099313968924493891e+0003L, 237 -3.55249848644100338419187038090925410976237921269e+0004L, 238 +6.43464880437835286216768959439484376449179576452e+0005L, 239 -1.20459154385577014992600342782821389605893904624e+0007L, 240 +2.09263249637351298563934942349749718491071093210e+0008L, 241 -2.96247483183169219343745316433899599834685703457e+0009L, 242 +2.88984933605896033154727626086506756972327292981e+0010L, 243 -1.40960434146030007732838382416230610302678063984e+0011L, /* 19 */ 244 }; 245 246 static const long double T3[] = { 247 +0.666666666666666666666666666666666634567834260213L, /* T3[0] */ 248 +0.400000000000000000000000000040853636176634934140L, /* T3[1] */ 249 +0.285714285714285714285696975252753987869020263448L, /* T3[2] */ 250 +0.222222222222222225593221101192317258554772129875L, /* T3[3] */ 251 +0.181818181817850192105847183461778186703779262916L, /* T3[4] */ 252 +0.153846169861348633757101285952333369222567014596L, /* T3[5] */ 253 +0.133033462889260193922261296772841229985047571265L, /* T3[6] */ 254 }; 255 256 static const long double c[] = { 257 0.0L, 258 1.0L, 259 2.0L, 260 0.5L, 261 1.0e-4930L, /* tiny */ 262 4.18937683105468750000e-01L, /* hln2pim1_h */ 263 8.50099203991780329736405617639861397473637783412817152e-07L, /* hln2pim1_l */ 264 0.418938533204672741780329736405617639861397473637783412817152L, /* hln2pim1 */ 265 2.16608493865351192653179168701171875e-02L, /* ln2_32hi */ 266 5.96317165397058692545083025235937919875797669127130e-12L, /* ln2_32lo */ 267 46.16624130844682903551758979206054839765267053289554989233L, /* invln2_32 */ 268 #if defined(__x86) 269 1.7555483429044629170023839037639845628291e+03L, /* overflow */ 270 #else 271 1.7555483429044629170038892160702032034177e+03L, /* overflow */ 272 #endif 273 }; 274 275 #define zero c[0] 276 #define one c[1] 277 #define two c[2] 278 #define half c[3] 279 #define tiny c[4] 280 #define hln2pim1_h c[5] 281 #define hln2pim1_l c[6] 282 #define hln2pim1 c[7] 283 #define ln2_32hi c[8] 284 #define ln2_32lo c[9] 285 #define invln2_32 c[10] 286 #define overflow c[11] 287 288 /* 289 * |exp(r) - (1+r+Et0*r^2+...+Et10*r^12)| <= 2^(-128.88) for |r|<=ln2/64 290 */ 291 static const long double Et[] = { 292 +5.0000000000000000000e-1L, 293 +1.66666666666666666666666666666828835166292152466e-0001L, 294 +4.16666666666666666666666666666693398646592712189e-0002L, 295 +8.33333333333333333333331748774512601775591115951e-0003L, 296 +1.38888888888888888888888845356011511394764753997e-0003L, 297 +1.98412698412698413237140350092993252684198882102e-0004L, 298 +2.48015873015873016080222025357442659895814371694e-0005L, 299 +2.75573192239028921114572986441972140933432317798e-0006L, 300 +2.75573192239448470555548102895526369739856219317e-0007L, 301 +2.50521677867683935940853997995937600214167232477e-0008L, 302 +2.08767928899010367374984448513685566514152147362e-0009L, 303 }; 304 305 /* 306 * long double precision coefficients for computing log(x)-1 in tgamma. 307 * See "algorithm" for details 308 * 309 * log(x) - 1 = T1(n) + T2(j) + T3(s), where x = 2**n * y, 1<=y<2, 310 * j=[64*y], z[j]=1+j/64+1/128, s = (y-z[j])/(y+z[j]), and 311 * T1(n) = T1[2n,2n+1] = n*log(2)-1, 312 * T2(j) = T2[2j,2j+1] = log(z[j]), 313 * T3(s) = 2s + T3[0]s^3 + T3[1]s^5 + T3[2]s^7 + ... + T3[6]s^15 314 * Note 315 * (1) the leading entries are truncated to 24 binary point. 316 * (2) Remez error for T3(s) is bounded by 2**(-136.54) 317 */ 318 static const long double T1[] = { 319 -1.000000000000000000000000000000000000000000e+00L, 320 +0.000000000000000000000000000000000000000000e+00L, 321 -3.068528175354003906250000000000000000000000e-01L, 322 -1.904654299957767878541823431924500011926579e-09L, 323 +3.862943053245544433593750000000000000000000e-01L, 324 +5.579533617547508924291635313615100141107647e-08L, 325 +1.079441487789154052734375000000000000000000e+00L, 326 +5.389068187551732136437452970422650211661470e-08L, 327 +1.772588670253753662109375000000000000000000e+00L, 328 +5.198602757555955348583270627230200282215294e-08L, 329 +2.465735852718353271484375000000000000000000e+00L, 330 +5.008137327560178560729088284037750352769117e-08L, 331 +3.158883035182952880859375000000000000000000e+00L, 332 +4.817671897564401772874905940845299849351090e-08L, 333 +3.852030217647552490234375000000000000000000e+00L, 334 +4.627206467568624985020723597652849919904913e-08L, 335 +4.545177400112152099609375000000000000000000e+00L, 336 +4.436741037572848197166541254460399990458737e-08L, 337 +5.238324582576751708984375000000000000000000e+00L, 338 +4.246275607577071409312358911267950061012560e-08L, 339 +5.931471765041351318359375000000000000000000e+00L, 340 +4.055810177581294621458176568075500131566384e-08L, 341 }; 342 343 /* 344 * T2[2i,2i+1] = log(1+i/64+1/128) 345 */ 346 static const long double T2[] = { 347 +7.7821016311645507812500000000000000000000e-03L, 348 +3.8810890398166212900061136763678127453570e-08L, 349 +2.3167014122009277343750000000000000000000e-02L, 350 +4.5159525100885049160962289916579411752759e-08L, 351 +3.8318812847137451171875000000000000000000e-02L, 352 +5.1454999148021880325123797290345960518164e-08L, 353 +5.3244471549987792968750000000000000000000e-02L, 354 +4.2968824489897120193786528776939573415076e-08L, 355 +6.7950606346130371093750000000000000000000e-02L, 356 +5.5562377378300815277772629414034632394030e-08L, 357 +8.2443654537200927734375000000000000000000e-02L, 358 +1.4673873663533785068668307805914095366600e-08L, 359 +9.6729576587677001953125000000000000000000e-02L, 360 +4.9870874110342446056487463437015041543346e-08L, 361 +1.1081433296203613281250000000000000000000e-01L, 362 +3.3378253981382306169323211928098474801099e-08L, 363 +1.2470346689224243164062500000000000000000e-01L, 364 +1.1608714804222781515380863268491613205318e-08L, 365 +1.3840228319168090820312500000000000000000e-01L, 366 +3.9667438227482200873601649187393160823607e-08L, 367 +1.5191602706909179687500000000000000000000e-01L, 368 +1.4956750178196803424896884511327584958252e-08L, 369 +1.6524952650070190429687500000000000000000e-01L, 370 +4.6394605258578736449277240313729237989366e-08L, 371 +1.7840760946273803710937500000000000000000e-01L, 372 +4.8010080260010025241510941968354682199540e-08L, 373 +1.9139480590820312500000000000000000000000e-01L, 374 +4.7091426329609298807561308873447039132856e-08L, 375 +2.0421552658081054687500000000000000000000e-01L, 376 +1.4847880344628820386196239272213742113867e-08L, 377 +2.1687388420104980468750000000000000000000e-01L, 378 +5.4099564554931589525744347498478964801484e-08L, 379 +2.2937405109405517578125000000000000000000e-01L, 380 +4.9970790654210230725046139871550961365282e-08L, 381 +2.4171990156173706054687500000000000000000e-01L, 382 +3.5325408107597432515913513900103385655073e-08L, 383 +2.5391519069671630859375000000000000000000e-01L, 384 +1.9284247135543573297906606667466299224747e-08L, 385 +2.6596349477767944335937500000000000000000e-01L, 386 +5.3719458497979750926537543389268821141517e-08L, 387 +2.7786844968795776367187500000000000000000e-01L, 388 +1.3154985425144750329234012330820349974537e-09L, 389 +2.8963327407836914062500000000000000000000e-01L, 390 +1.8504673536253893055525668970003860369760e-08L, 391 +3.0126130580902099609375000000000000000000e-01L, 392 +2.4769140784919125538233755492657352680723e-08L, 393 +3.1275570392608642578125000000000000000000e-01L, 394 +6.0778104626049965596883190321597861455475e-09L, 395 +3.2411944866180419921875000000000000000000e-01L, 396 +1.9992407776871920760434987352182336158873e-08L, 397 +3.3535552024841308593750000000000000000000e-01L, 398 +2.1672724744319679579814166199074433006807e-08L, 399 +3.4646672010421752929687500000000000000000e-01L, 400 +4.7241991051621587188425772950711830538414e-08L, 401 +3.5745584964752197265625000000000000000000e-01L, 402 +3.9274281801569759490140904474434669956562e-08L, 403 +3.6832553148269653320312500000000000000000e-01L, 404 +2.9676011119845105154050398826897178765758e-08L, 405 +3.7907832860946655273437500000000000000000e-01L, 406 +2.4325502905656478345631019858881408009210e-08L, 407 +3.8971674442291259765625000000000000000000e-01L, 408 +6.7171126157142136040035208670510556529487e-09L, 409 +4.0024316310882568359375000000000000000000e-01L, 410 +1.0181870233355751019951311700799406124957e-09L, 411 +4.1065990924835205078125000000000000000000e-01L, 412 +1.5736916335153056203175822787661567534220e-08L, 413 +4.2096924781799316406250000000000000000000e-01L, 414 +4.6826136472066367161506795972449857268707e-08L, 415 +4.3117344379425048828125000000000000000000e-01L, 416 +2.1024120852577922478955594998480144051225e-08L, 417 +4.4127452373504638671875000000000000000000e-01L, 418 +3.7069828842770746441661301225362605528786e-08L, 419 +4.5127463340759277343750000000000000000000e-01L, 420 +1.0731865811707192383079012478685922879010e-08L, 421 +4.6117568016052246093750000000000000000000e-01L, 422 +3.4961647705430499925597855358603099030515e-08L, 423 +4.7097969055175781250000000000000000000000e-01L, 424 +2.4667033200046897856056359251373510964634e-08L, 425 +4.8068851232528686523437500000000000000000e-01L, 426 +1.7020465042442243455448011551208861216878e-08L, 427 +4.9030393362045288085937500000000000000000e-01L, 428 +5.4424740957290971159645746860530583309571e-08L, 429 +4.9982786178588867187500000000000000000000e-01L, 430 +7.7705606579463314152470441415126573566105e-09L, 431 +5.0926184654235839843750000000000000000000e-01L, 432 +5.5247449548366574919228323824878565745713e-08L, 433 +5.1860773563385009765625000000000000000000e-01L, 434 +2.8574195534496726996364798698556235730848e-08L, 435 +5.2786707878112792968750000000000000000000e-01L, 436 +1.0839714455426392217778300963558522088193e-08L, 437 +5.3704142570495605468750000000000000000000e-01L, 438 +4.0191927599879229244153832299023744345999e-08L, 439 +5.4613238573074340820312500000000000000000e-01L, 440 +5.1867392242179272209231209163864971792889e-08L, 441 +5.5514144897460937500000000000000000000000e-01L, 442 +5.8565892217715480359515904050170125743178e-08L, 443 +5.6407010555267333984375000000000000000000e-01L, 444 +3.2732129626227634290090190711817681692354e-08L, 445 +5.7291972637176513671875000000000000000000e-01L, 446 +2.7190020372374006726626261068626400393936e-08L, 447 +5.8169168233871459960937500000000000000000e-01L, 448 +5.7295907882911235753725372340709967597394e-08L, 449 +5.9038740396499633789062500000000000000000e-01L, 450 +4.2637180036751291708123598757577783615014e-08L, 451 +5.9900814294815063476562500000000000000000e-01L, 452 +4.6697932764615975024461651502060474048774e-08L, 453 +6.0755521059036254882812500000000000000000e-01L, 454 +3.9634179246672960152791125371893149820625e-08L, 455 +6.1602985858917236328125000000000000000000e-01L, 456 +1.8626341656366315928196700650292529688219e-08L, 457 +6.2443327903747558593750000000000000000000e-01L, 458 +8.9744179151050387440546731199093039879228e-09L, 459 +6.3276666402816772460937500000000000000000e-01L, 460 +5.5428701049364114685035797584887586099726e-09L, 461 +6.4103114604949951171875000000000000000000e-01L, 462 +3.3371431779336851334405392546708949047361e-08L, 463 +6.4922791719436645507812500000000000000000e-01L, 464 +2.9430743363812714969905311122271269100885e-08L, 465 +6.5735805034637451171875000000000000000000e-01L, 466 +2.2361985518423140023245936165514147093250e-08L, 467 +6.6542261838912963867187500000000000000000e-01L, 468 +1.4155960810278217610006660181148303091649e-08L, 469 +6.7342263460159301757812500000000000000000e-01L, 470 +4.0610573702719835388801017264750843477878e-08L, 471 +6.8135917186737060546875000000000000000000e-01L, 472 +5.2940532463479321559568089441735584156689e-08L, 473 +6.8923324346542358398437500000000000000000e-01L, 474 +3.7773385396340539337814603903232796216537e-08L, 475 }; 476 477 /* 478 * S[j],S_trail[j] = 2**(j/32.) for the final computation of exp(t+w) 479 */ 480 static const long double S[] = { 481 #if defined(__x86) 482 +1.0000000000000000000000000e+00L, 483 +1.0218971486541166782081522e+00L, 484 +1.0442737824274138402382006e+00L, 485 +1.0671404006768236181297224e+00L, 486 +1.0905077326652576591003302e+00L, 487 +1.1143867425958925362894369e+00L, 488 +1.1387886347566916536971221e+00L, 489 +1.1637248587775775137938619e+00L, 490 +1.1892071150027210666875674e+00L, 491 +1.2152473599804688780476325e+00L, 492 +1.2418578120734840485256747e+00L, 493 +1.2690509571917332224885722e+00L, 494 +1.2968395546510096659215822e+00L, 495 +1.3252366431597412945939118e+00L, 496 +1.3542555469368927282668852e+00L, 497 +1.3839098819638319548151403e+00L, 498 +1.4142135623730950487637881e+00L, 499 +1.4451808069770466200253470e+00L, 500 +1.4768261459394993113155431e+00L, 501 +1.5091644275934227397133885e+00L, 502 +1.5422108254079408235859630e+00L, 503 +1.5759808451078864864006862e+00L, 504 +1.6104903319492543080837174e+00L, 505 +1.6457554781539648445110730e+00L, 506 +1.6817928305074290860378350e+00L, 507 +1.7186192981224779156032914e+00L, 508 +1.7562521603732994831094730e+00L, 509 +1.7947090750031071864148413e+00L, 510 +1.8340080864093424633989166e+00L, 511 +1.8741676341102999013002103e+00L, 512 +1.9152065613971472938202589e+00L, 513 +1.9571441241754002689657438e+00L, 514 #else 515 +1.00000000000000000000000000000000000e+00L, 516 +1.02189714865411667823448013478329942e+00L, 517 +1.04427378242741384032196647873992910e+00L, 518 +1.06714040067682361816952112099280918e+00L, 519 +1.09050773266525765920701065576070789e+00L, 520 +1.11438674259589253630881295691960313e+00L, 521 +1.13878863475669165370383028384151134e+00L, 522 +1.16372485877757751381357359909218536e+00L, 523 +1.18920711500272106671749997056047593e+00L, 524 +1.21524735998046887811652025133879836e+00L, 525 +1.24185781207348404859367746872659561e+00L, 526 +1.26905095719173322255441908103233805e+00L, 527 +1.29683955465100966593375411779245118e+00L, 528 +1.32523664315974129462953709549872168e+00L, 529 +1.35425554693689272829801474014070273e+00L, 530 +1.38390988196383195487265952726519287e+00L, 531 +1.41421356237309504880168872420969798e+00L, 532 +1.44518080697704662003700624147167095e+00L, 533 +1.47682614593949931138690748037404985e+00L, 534 +1.50916442759342273976601955103319352e+00L, 535 +1.54221082540794082361229186209073479e+00L, 536 +1.57598084510788648645527016018190504e+00L, 537 +1.61049033194925430817952066735740067e+00L, 538 +1.64575547815396484451875672472582254e+00L, 539 +1.68179283050742908606225095246642969e+00L, 540 +1.71861929812247791562934437645631244e+00L, 541 +1.75625216037329948311216061937531314e+00L, 542 +1.79470907500310718642770324212778174e+00L, 543 +1.83400808640934246348708318958828892e+00L, 544 +1.87416763411029990132999894995444645e+00L, 545 +1.91520656139714729387261127029583086e+00L, 546 +1.95714412417540026901832225162687149e+00L, 547 #endif 548 }; 549 static const long double S_trail[] = { 550 #if defined(__x86) 551 +0.0000000000000000000000000e+00L, 552 +2.6327965667180882569382524e-20L, 553 +8.3765863521895191129661899e-20L, 554 +3.9798705777454504249209575e-20L, 555 +1.0668046596651558640993042e-19L, 556 +1.9376009847285360448117114e-20L, 557 +6.7081819456112953751277576e-21L, 558 +1.9711680502629186462729727e-20L, 559 +2.9932584438449523689104569e-20L, 560 +6.8887754153039109411061914e-20L, 561 +6.8002718741225378942847820e-20L, 562 +6.5846917376975403439742349e-20L, 563 +1.2171958727511372194876001e-20L, 564 +3.5625253228704087115438260e-20L, 565 +3.1129551559077560956309179e-20L, 566 +5.7519192396164779846216492e-20L, 567 +3.7900651177865141593101239e-20L, 568 +1.1659262405698741798080115e-20L, 569 +7.1364385105284695967172478e-20L, 570 +5.2631003710812203588788949e-20L, 571 +2.6328853788732632868460580e-20L, 572 +5.4583950085438242788190141e-20L, 573 +9.5803254376938269960718656e-20L, 574 +7.6837733983874245823512279e-21L, 575 +2.4415965910835093824202087e-20L, 576 +2.6052966871016580981769728e-20L, 577 +2.6876456344632553875309579e-21L, 578 +1.2861930155613700201703279e-20L, 579 +8.8166633394037485606572294e-20L, 580 +2.9788615389580190940837037e-20L, 581 +5.2352341619805098677422139e-20L, 582 +5.2578463064010463732242363e-20L, 583 #else 584 +0.00000000000000000000000000000000000e+00L, 585 +1.80506787420330954745573333054573786e-35L, 586 -9.37452029228042742195756741973083214e-35L, 587 -1.59696844729275877071290963023149997e-35L, 588 +9.11249341012502297851168610167248666e-35L, 589 -6.50422820697854828723037477525938871e-35L, 590 -8.14846884452585113732569176748815532e-35L, 591 -5.06621457672180031337233074514290335e-35L, 592 -1.35983097468881697374987563824591912e-35L, 593 +9.49742763556319647030771056643324660e-35L, 594 -3.28317052317699860161506596533391526e-36L, 595 -5.01723570938719041029018653045842895e-35L, 596 -2.39147479768910917162283430160264014e-35L, 597 -8.35057135763390881529889073794408385e-36L, 598 +7.03675688907326504242173719067187644e-35L, 599 -5.18248485306464645753689301856695619e-35L, 600 +9.42224254862183206569211673639406488e-35L, 601 -3.96750082539886230916730613021641828e-35L, 602 +7.14352899156330061452327361509276724e-35L, 603 +1.15987125286798512424651783410044433e-35L, 604 +4.69693347835811549530973921320187447e-35L, 605 -3.38651317599500471079924198499981917e-35L, 606 -8.58731877429824706886865593510387445e-35L, 607 -9.60595154874935050318549936224606909e-35L, 608 +9.60973393212801278450755869714178581e-35L, 609 +6.37839792144002843924476144978084855e-35L, 610 +7.79243078569586424945646112516927770e-35L, 611 +7.36133776758845652413193083663393220e-35L, 612 -6.47299514791334723003521457561217053e-35L, 613 +8.58747441795369869427879806229522962e-35L, 614 +2.37181542282517483569165122830269098e-35L, 615 -3.02689168209611877300459737342190031e-37L, 616 #endif 617 }; 618 /* INDENT ON */ 619 620 /* INDENT OFF */ 621 /* 622 * return tgamma(x) scaled by 2**-m for 8<x<=171.62... using Stirling's formula 623 * log(G(x)) ~= (x-.5)*(log(x)-1) + .5(log(2*pi)-1) + (1/x)*P(1/(x*x)) 624 * = L1 + L2 + L3, 625 */ 626 /* INDENT ON */ 627 static struct LDouble 628 large_gam(long double x, int *m) { 629 long double z, t1, t2, t3, z2, t5, w, y, u, r, v; 630 long double t24 = 16777216.0L, p24 = 1.0L / 16777216.0L; 631 int n2, j2, k, ix, j, i; 632 struct LDouble zz; 633 long double u2, ss_h, ss_l, r_h, w_h, w_l, t4; 634 635 /* INDENT OFF */ 636 /* 637 * compute ss = ss.h+ss.l = log(x)-1 (see tgamma_log.h for details) 638 * 639 * log(x) - 1 = T1(n) + T2(j) + T3(s), where x = 2**n * y, 1<=y<2, 640 * j=[64*y], z[j]=1+j/64+1/128, s = (y-z[j])/(y+z[j]), and 641 * T1(n) = T1[2n,2n+1] = n*log(2)-1, 642 * T2(j) = T2[2j,2j+1] = log(z[j]), 643 * T3(s) = 2s + T3[0]s^3 + T3[1]s^5 + ... + T3[6]s^15 644 * Note 645 * (1) the leading entries are truncated to 24 binary point. 646 * (2) Remez error for T3(s) is bounded by 2**(-72.4) 647 * 2**(-24) 648 * _________V___________________ 649 * T1(n): |_________|___________________| 650 * _______ ______________________ 651 * T2(j): |_______|______________________| 652 * ____ _______________________ 653 * 2s: |____|_______________________| 654 * __________________________ 655 * + T3(s)-2s: |__________________________| 656 * ------------------------------------------- 657 * [leading] + [Trailing] 658 */ 659 /* INDENT ON */ 660 ix = H0_WORD(x); 661 n2 = (ix >> 16) - 0x3fff; /* exponent of x, range:3-10 */ 662 y = scalbnl(x, -n2); /* y = scale x to [1,2] */ 663 n2 += n2; /* 2n */ 664 j = (ix >> 10) & 0x3f; /* j */ 665 z = 1.0078125L + (long double) j * 0.015625L; /* z[j]=1+j/64+1/128 */ 666 j2 = j + j; 667 t1 = y + z; 668 t2 = y - z; 669 r = one / t1; 670 u = r * t2; /* u = (y-z)/(y+z) */ 671 t1 = CHOPPED(t1); 672 t4 = T2[j2 + 1] + T1[n2 + 1]; 673 z2 = u * u; 674 k = H0_WORD(u) & 0x7fffffff; 675 t3 = T2[j2] + T1[n2]; 676 for (t5 = T3[6], i = 5; i >= 0; i--) 677 t5 = z2 * t5 + T3[i]; 678 if ((k >> 16) < 0x3fec) { /* |u|<2**-19 */ 679 t2 = t4 + u * (two + z2 * t5); 680 } else { 681 t5 = t4 + (u * z2) * t5; 682 u2 = u + u; 683 v = (long double) ((int) (u2 * t24)) * p24; 684 t2 = t5 + r * ((two * t2 - v * t1) - v * (y - (t1 - z))); 685 t3 += v; 686 } 687 ss_h = CHOPPED((t2 + t3)); 688 ss_l = t2 - (ss_h - t3); 689 /* INDENT OFF */ 690 /* 691 * compute ww = (x-.5)*(log(x)-1) + .5*(log(2pi)-1) + 1/x*(P(1/x^2))) 692 * where ss = log(x) - 1 in already in extra precision 693 */ 694 /* INDENT ON */ 695 z = one / x; 696 r = x - half; 697 r_h = CHOPPED((r)); 698 w_h = r_h * ss_h + hln2pim1_h; 699 z2 = z * z; 700 w = (r - r_h) * ss_h + r * ss_l; 701 t1 = GP[19]; 702 for (i = 18; i > 0; i--) 703 t1 = z2 * t1 + GP[i]; 704 w += hln2pim1_l; 705 w_l = z * (GP[0] + z2 * t1) + w; 706 k = (int) ((w_h + w_l) * invln2_32 + half); 707 708 /* compute the exponential of w_h+w_l */ 709 710 j = k & 0x1f; 711 *m = k >> 5; 712 t3 = (long double) k; 713 714 /* perform w - k*ln2_32 (represent as w_h - w_l) */ 715 t1 = w_h - t3 * ln2_32hi; 716 t2 = t3 * ln2_32lo; 717 w = t2 - w_l; 718 w_h = t1 - w; 719 w_l = w - (t1 - w_h); 720 721 /* compute exp(w_h-w_l) */ 722 z = w_h - w_l; 723 for (t1 = Et[10], i = 9; i >= 0; i--) 724 t1 = z * t1 + Et[i]; 725 t3 = w_h - (w_l - (z * z) * t1); /* t3 = expm1(z) */ 726 zz.l = S_trail[j] * (one + t3) + S[j] * t3; 727 zz.h = S[j]; 728 return (zz); 729 } 730 731 /* INDENT OFF */ 732 /* 733 * kpsin(x)= sin(pi*x)/pi 734 * 3 5 7 9 11 27 735 * = x+ks[0]*x +ks[1]*x +ks[2]*x +ks[3]*x +ks[4]*x + ... + ks[12]*x 736 */ 737 static const long double ks[] = { 738 -1.64493406684822643647241516664602518705158902870e+0000L, 739 +8.11742425283353643637002772405874238094995726160e-0001L, 740 -1.90751824122084213696472111835337366232282723933e-0001L, 741 +2.61478478176548005046532613563241288115395517084e-0002L, 742 -2.34608103545582363750893072647117829448016479971e-0003L, 743 +1.48428793031071003684606647212534027556262040158e-0004L, 744 -6.97587366165638046518462722252768122615952898698e-0006L, 745 +2.53121740413702536928659271747187500934840057929e-0007L, 746 -7.30471182221385990397683641695766121301933621956e-0009L, 747 +1.71653847451163495739958249695549313987973589884e-0010L, 748 -3.34813314714560776122245796929054813458341420565e-0012L, 749 +5.50724992262622033449487808306969135431411753047e-0014L, 750 -7.67678132753577998601234393215802221104236979928e-0016L, 751 }; 752 /* INDENT ON */ 753 754 /* 755 * assume x is not tiny and positive 756 */ 757 static struct LDouble 758 kpsin(long double x) { 759 long double z, t1, t2; 760 struct LDouble xx; 761 int i; 762 763 z = x * x; 764 xx.h = x; 765 for (t2 = ks[12], i = 11; i > 0; i--) 766 t2 = z * t2 + ks[i]; 767 t1 = z * x; 768 t2 *= z * t1; 769 xx.l = t1 * ks[0] + t2; 770 return (xx); 771 } 772 773 /* INDENT OFF */ 774 /* 775 * kpcos(x)= cos(pi*x)/pi 776 * 2 4 6 8 10 12 777 * = 1/pi +kc[0]*x +kc[1]*x +kc[2]*x +kc[3]*x +kc[4]*x +kc[5]*x 778 * 779 * 2 4 6 8 10 22 780 * = 1/pi - pi/2*x +kc[0]*x +kc[1]*x +kc[2]*x +kc[3]*x +...+kc[9]*x 781 * 782 * -pi/2*x*x = (npi_2_h + npi_2_l) * (x_f+x_l)*(x_f+x_l) 783 * = npi_2_h*(x_f+x_l)*(x_f+x_l) + npi_2_l*x*x 784 * = npi_2_h*x_f*x_f + npi_2_h*(x*x-x_f*x_f) + npi_2_l*x*x 785 * = npi_2_h*x_f*x_f + npi_2_h*(x+x_f)*(x-x_f) + npi_2_l*x*x 786 * Here x_f = (long double) (float)x 787 * Note that pi/2(in hex) = 788 * 1.921FB54442D18469898CC51701B839A252049C1114CF98E804177D4C76273644A29 789 * npi_2_h = -pi/2 chopped to 25 bits = -1.921FB50000000000000000000000000 = 790 * -1.570796310901641845703125000000000 and 791 * npi_2_l = 792 * -0.0000004442D18469898CC51701B839A252049C1114CF98E804177D4C76273644A29 = 793 * -.0000000158932547735281966916397514420985846996875529104874722961539 = 794 * -1.5893254773528196691639751442098584699687552910487472296153e-8 795 * 1/pi(in hex) = 796 * .517CC1B727220A94FE13ABE8FA9A6EE06DB14ACC9E21C820FF28B1D5EF5DE2B 797 * will be splitted into: 798 * one_pi_h = 1/pi chopped to 48 bits = .517CC1B727220000000000... and 799 * one_pi_l = .0000000000000A94FE13ABE8FA9A6EE06DB14ACC9E21C820FF28B1D5EF5DE2B 800 */ 801 802 static const long double 803 #if defined(__x86) 804 one_pi_h = 0.3183098861481994390487670898437500L, /* 31 bits */ 805 one_pi_l = 3.559123248900043690127872406891929148e-11L, 806 #else 807 one_pi_h = 0.31830988618379052468299050815403461456298828125L, 808 one_pi_l = 1.46854777018590994109505931010230912897495334688117e-16L, 809 #endif 810 npi_2_h = -1.570796310901641845703125000000000L, 811 npi_2_l = -1.5893254773528196691639751442098584699687552910e-8L; 812 813 static const long double kc[] = { 814 +1.29192819501249250731151312779548918765320728489e+0000L, 815 -4.25027339979557573976029596929319207009444090366e-0001L, 816 +7.49080661650990096109672954618317623888421628613e-0002L, 817 -8.21458866111282287985539464173976555436050215120e-0003L, 818 +6.14202578809529228503205255165761204750211603402e-0004L, 819 -3.33073432691149607007217330302595267179545908740e-0005L, 820 +1.36970959047832085796809745461530865597993680204e-0006L, 821 -4.41780774262583514450246512727201806217271097336e-0008L, 822 +1.14741409212381858820016567664488123478660705759e-0009L, 823 -2.44261236114707374558437500654381006300502749632e-0011L, 824 }; 825 /* INDENT ON */ 826 827 /* 828 * assume x is not tiny and positive 829 */ 830 static struct LDouble 831 kpcos(long double x) { 832 long double z, t1, t2, t3, t4, x4, x8; 833 int i; 834 struct LDouble xx; 835 836 z = x * x; 837 xx.h = one_pi_h; 838 t1 = (long double) ((float) x); 839 x4 = z * z; 840 t2 = npi_2_l * z + npi_2_h * (x + t1) * (x - t1); 841 for (i = 8, t3 = kc[9]; i >= 0; i--) 842 t3 = z * t3 + kc[i]; 843 t3 = one_pi_l + x4 * t3; 844 t4 = t1 * t1 * npi_2_h; 845 x8 = t2 + t3; 846 xx.l = x8 + t4; 847 return (xx); 848 } 849 850 /* INDENT OFF */ 851 static const long double 852 /* 0.13486180573279076968979393577465291700642511139552429398233 */ 853 #if defined(__x86) 854 t0z1 = 0.1348618057327907696779385054997035808810L, 855 t0z1_l = 1.1855430274949336125392717150257379614654e-20L, 856 #else 857 t0z1 = 0.1348618057327907696897939357746529168654L, 858 t0z1_l = 1.4102088588676879418739164486159514674310e-37L, 859 #endif 860 /* 0.46163214496836234126265954232572132846819620400644635129599 */ 861 #if defined(__x86) 862 t0z2 = 0.4616321449683623412538115843295472018326L, 863 t0z2_l = 8.84795799617412663558532305039261747030640e-21L, 864 #else 865 t0z2 = 0.46163214496836234126265954232572132343318L, 866 t0z2_l = 5.03501162329616380465302666480916271611101e-36L, 867 #endif 868 /* 0.81977310110050060178786870492160699631174407846245179119586 */ 869 #if defined(__x86) 870 t0z3 = 0.81977310110050060178773362329351925836817L, 871 t0z3_l = 1.350816280877379435658077052534574556256230e-22L 872 #else 873 t0z3 = 0.8197731011005006017878687049216069516957449L, 874 t0z3_l = 4.461599916947014419045492615933551648857380e-35L 875 #endif 876 ; 877 /* INDENT ON */ 878 879 /* 880 * gamma(x+i) for 0 <= x < 1 881 */ 882 static struct LDouble 883 gam_n(int i, long double x) { 884 struct LDouble rr = {0.0L, 0.0L}, yy; 885 long double r1, r2, t2, z, xh, xl, yh, yl, zh, z1, z2, zl, x5, wh, wl; 886 887 /* compute yy = gamma(x+1) */ 888 if (x > 0.2845L) { 889 if (x > 0.6374L) { 890 r1 = x - t0z3; 891 r2 = CHOPPED((r1 - t0z3_l)); 892 t2 = r1 - r2; 893 yy = GT3(r2, t2 - t0z3_l); 894 } else { 895 r1 = x - t0z2; 896 r2 = CHOPPED((r1 - t0z2_l)); 897 t2 = r1 - r2; 898 yy = GT2(r2, t2 - t0z2_l); 899 } 900 } else { 901 r1 = x - t0z1; 902 r2 = CHOPPED((r1 - t0z1_l)); 903 t2 = r1 - r2; 904 yy = GT1(r2, t2 - t0z1_l); 905 } 906 /* compute gamma(x+i) = (x+i-1)*...*(x+1)*yy, 0<i<8 */ 907 switch (i) { 908 case 0: /* yy/x */ 909 r1 = one / x; 910 xh = CHOPPED((x)); /* x is not tiny */ 911 rr.h = CHOPPED(((yy.h + yy.l) * r1)); 912 rr.l = r1 * (yy.h - rr.h * xh) - ((r1 * rr.h) * (x - xh) - 913 r1 * yy.l); 914 break; 915 case 1: /* yy */ 916 rr.h = yy.h; 917 rr.l = yy.l; 918 break; 919 case 2: /* (x+1)*yy */ 920 z = x + one; /* may not be exact */ 921 zh = CHOPPED((z)); 922 rr.h = zh * yy.h; 923 rr.l = z * yy.l + (x - (zh - one)) * yy.h; 924 break; 925 case 3: /* (x+2)*(x+1)*yy */ 926 z1 = x + one; 927 z2 = x + 2.0L; 928 z = z1 * z2; 929 xh = CHOPPED((z)); 930 zh = CHOPPED((z1)); 931 xl = (x - (zh - one)) * (z2 + zh) - (xh - zh * (zh + one)); 932 933 rr.h = xh * yy.h; 934 rr.l = z * yy.l + xl * yy.h; 935 break; 936 937 case 4: /* (x+1)*(x+3)*(x+2)*yy */ 938 z1 = x + 2.0L; 939 z2 = (x + one) * (x + 3.0L); 940 zh = CHOPPED(z1); 941 zl = x - (zh - 2.0L); 942 xh = CHOPPED(z2); 943 xl = zl * (zh + z1) - (xh - (zh * zh - one)); 944 945 /* wh+wl=(x+2)*yy */ 946 wh = CHOPPED((z1 * (yy.h + yy.l))); 947 wl = (zl * yy.h + z1 * yy.l) - (wh - zh * yy.h); 948 949 rr.h = xh * wh; 950 rr.l = z2 * wl + xl * wh; 951 952 break; 953 case 5: /* ((x+1)*(x+4)*(x+2)*(x+3))*yy */ 954 z1 = x + 2.0L; 955 z2 = x + 3.0L; 956 z = z1 * z2; 957 zh = CHOPPED((z1)); 958 yh = CHOPPED((z)); 959 yl = (x - (zh - 2.0L)) * (z2 + zh) - (yh - zh * (zh + one)); 960 z2 = z - 2.0L; 961 z *= z2; 962 xh = CHOPPED((z)); 963 xl = yl * (z2 + yh) - (xh - yh * (yh - 2.0L)); 964 rr.h = xh * yy.h; 965 rr.l = z * yy.l + xl * yy.h; 966 break; 967 case 6: /* ((x+1)*(x+2)*(x+3)*(x+4)*(x+5))*yy */ 968 z1 = x + 2.0L; 969 z2 = x + 3.0L; 970 z = z1 * z2; 971 zh = CHOPPED((z1)); 972 yh = CHOPPED((z)); 973 z1 = x - (zh - 2.0L); 974 yl = z1 * (z2 + zh) - (yh - zh * (zh + one)); 975 z2 = z - 2.0L; 976 x5 = x + 5.0L; 977 z *= z2; 978 xh = CHOPPED(z); 979 zh += 3.0; 980 xl = yl * (z2 + yh) - (xh - yh * (yh - 2.0L)); 981 /* xh+xl=(x+1)*...*(x+4) */ 982 /* wh+wl=(x+5)*yy */ 983 wh = CHOPPED((x5 * (yy.h + yy.l))); 984 wl = (z1 * yy.h + x5 * yy.l) - (wh - zh * yy.h); 985 rr.h = wh * xh; 986 rr.l = z * wl + xl * wh; 987 break; 988 case 7: /* ((x+1)*(x+2)*(x+3)*(x+4)*(x+5)*(x+6))*yy */ 989 z1 = x + 3.0L; 990 z2 = x + 4.0L; 991 z = z2 * z1; 992 zh = CHOPPED((z1)); 993 yh = CHOPPED((z)); /* yh+yl = (x+3)(x+4) */ 994 yl = (x - (zh - 3.0L)) * (z2 + zh) - (yh - (zh * (zh + one))); 995 z1 = x + 6.0L; 996 z2 = z - 2.0L; /* z2 = (x+2)*(x+5) */ 997 z *= z2; 998 xh = CHOPPED((z)); 999 xl = yl * (z2 + yh) - (xh - yh * (yh - 2.0L)); 1000 /* xh+xl=(x+2)*...*(x+5) */ 1001 /* wh+wl=(x+1)(x+6)*yy */ 1002 z2 -= 4.0L; /* z2 = (x+1)(x+6) */ 1003 wh = CHOPPED((z2 * (yy.h + yy.l))); 1004 wl = (z2 * yy.l + yl * yy.h) - (wh - (yh - 6.0L) * yy.h); 1005 rr.h = wh * xh; 1006 rr.l = z * wl + xl * wh; 1007 } 1008 return (rr); 1009 } 1010 1011 long double 1012 tgammal(long double x) { 1013 struct LDouble ss, ww; 1014 long double t, t1, t2, t3, t4, t5, w, y, z, z1, z2, z3, z5; 1015 int i, j, m, ix, hx, xk; 1016 unsigned lx; 1017 1018 hx = H0_WORD(x); 1019 lx = H3_WORD(x); 1020 ix = hx & 0x7fffffff; 1021 y = x; 1022 if (ix < 0x3f8e0000) { /* x < 2**-113 */ 1023 return (one / x); 1024 } 1025 if (ix >= 0x7fff0000) 1026 return (x * ((hx < 0)? zero : x)); /* Inf or NaN */ 1027 if (x > overflow) /* overflow threshold */ 1028 return (x * 1.0e4932L); 1029 if (hx >= 0x40020000) { /* x >= 8 */ 1030 ww = large_gam(x, &m); 1031 w = ww.h + ww.l; 1032 return (scalbnl(w, m)); 1033 } 1034 1035 if (hx > 0) { /* 0 < x < 8 */ 1036 i = (int) x; 1037 ww = gam_n(i, x - (long double) i); 1038 return (ww.h + ww.l); 1039 } 1040 /* INDENT OFF */ 1041 /* negative x */ 1042 /* 1043 * compute xk = 1044 * -2 ... x is an even int (-inf is considered an even #) 1045 * -1 ... x is an odd int 1046 * +0 ... x is not an int but chopped to an even int 1047 * +1 ... x is not an int but chopped to an odd int 1048 */ 1049 /* INDENT ON */ 1050 xk = 0; 1051 #if defined(__x86) 1052 if (ix >= 0x403e0000) { /* x >= 2**63 } */ 1053 if (ix >= 0x403f0000) 1054 xk = -2; 1055 else 1056 xk = -2 + (lx & 1); 1057 #else 1058 if (ix >= 0x406f0000) { /* x >= 2**112 */ 1059 if (ix >= 0x40700000) 1060 xk = -2; 1061 else 1062 xk = -2 + (lx & 1); 1063 #endif 1064 } else if (ix >= 0x3fff0000) { 1065 w = -x; 1066 t1 = floorl(w); 1067 t2 = t1 * half; 1068 t3 = floorl(t2); 1069 if (t1 == w) { 1070 if (t2 == t3) 1071 xk = -2; 1072 else 1073 xk = -1; 1074 } else { 1075 if (t2 == t3) 1076 xk = 0; 1077 else 1078 xk = 1; 1079 } 1080 } 1081 1082 if (xk < 0) { 1083 /* return NaN. Ideally gamma(-n)= (-1)**(n+1) * inf */ 1084 return (x - x) / (x - x); 1085 } 1086 1087 /* 1088 * negative underflow thresold -(1774+9ulp) 1089 */ 1090 if (x < -1774.0000000000000000000000000000017749370L) { 1091 z = tiny / x; 1092 if (xk == 1) 1093 z = -z; 1094 return (z * tiny); 1095 } 1096 1097 /* INDENT OFF */ 1098 /* 1099 * now compute gamma(x) by -1/((sin(pi*y)/pi)*gamma(1+y)), y = -x 1100 */ 1101 /* 1102 * First compute ss = -sin(pi*y)/pi so that 1103 * gamma(x) = 1/(ss*gamma(1+y)) 1104 */ 1105 /* INDENT ON */ 1106 y = -x; 1107 j = (int) y; 1108 z = y - (long double) j; 1109 if (z > 0.3183098861837906715377675L) 1110 if (z > 0.6816901138162093284622325L) 1111 ss = kpsin(one - z); 1112 else 1113 ss = kpcos(0.5L - z); 1114 else 1115 ss = kpsin(z); 1116 if (xk == 0) { 1117 ss.h = -ss.h; 1118 ss.l = -ss.l; 1119 } 1120 1121 /* Then compute ww = gamma(1+y), note that result scale to 2**m */ 1122 m = 0; 1123 if (j < 7) { 1124 ww = gam_n(j + 1, z); 1125 } else { 1126 w = y + one; 1127 if ((lx & 1) == 0) { /* y+1 exact (note that y<184) */ 1128 ww = large_gam(w, &m); 1129 } else { 1130 t = w - one; 1131 if (t == y) { /* y+one exact */ 1132 ww = large_gam(w, &m); 1133 } else { /* use y*gamma(y) */ 1134 if (j == 7) 1135 ww = gam_n(j, z); 1136 else 1137 ww = large_gam(y, &m); 1138 t4 = ww.h + ww.l; 1139 t1 = CHOPPED((y)); 1140 t2 = CHOPPED((t4)); 1141 /* t4 will not be too large */ 1142 ww.l = y * (ww.l - (t2 - ww.h)) + (y - t1) * t2; 1143 ww.h = t1 * t2; 1144 } 1145 } 1146 } 1147 1148 /* compute 1/(ss*ww) */ 1149 t3 = ss.h + ss.l; 1150 t4 = ww.h + ww.l; 1151 t1 = CHOPPED((t3)); 1152 t2 = CHOPPED((t4)); 1153 z1 = ss.l - (t1 - ss.h); /* (t1,z1) = ss */ 1154 z2 = ww.l - (t2 - ww.h); /* (t2,z2) = ww */ 1155 t3 = t3 * t4; /* t3 = ss*ww */ 1156 z3 = one / t3; /* z3 = 1/(ss*ww) */ 1157 t5 = t1 * t2; 1158 z5 = z1 * t4 + t1 * z2; /* (t5,z5) = ss*ww */ 1159 t1 = CHOPPED((t3)); /* (t1,z1) = ss*ww */ 1160 z1 = z5 - (t1 - t5); 1161 t2 = CHOPPED((z3)); /* leading 1/(ss*ww) */ 1162 z2 = z3 * (t2 * z1 - (one - t2 * t1)); 1163 z = t2 - z2; 1164 1165 return (scalbnl(z, -m)); 1166 } 1167