1 /* $OpenBSD: e_lgammal.c,v 1.5 2016/09/12 19:47:02 guenther Exp $ */ 2 3 /* 4 * Copyright (c) 2008 Stephen L. Moshier <steve@moshier.net> 5 * 6 * Permission to use, copy, modify, and distribute this software for any 7 * purpose with or without fee is hereby granted, provided that the above 8 * copyright notice and this permission notice appear in all copies. 9 * 10 * THE SOFTWARE IS PROVIDED "AS IS" AND THE AUTHOR DISCLAIMS ALL WARRANTIES 11 * WITH REGARD TO THIS SOFTWARE INCLUDING ALL IMPLIED WARRANTIES OF 12 * MERCHANTABILITY AND FITNESS. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR 13 * ANY SPECIAL, DIRECT, INDIRECT, OR CONSEQUENTIAL DAMAGES OR ANY DAMAGES 14 * WHATSOEVER RESULTING FROM LOSS OF USE, DATA OR PROFITS, WHETHER IN AN 15 * ACTION OF CONTRACT, NEGLIGENCE OR OTHER TORTIOUS ACTION, ARISING OUT OF 16 * OR IN CONNECTION WITH THE USE OR PERFORMANCE OF THIS SOFTWARE. 17 */ 18 19 /* lgammal 20 * 21 * Natural logarithm of gamma function 22 * 23 * 24 * 25 * SYNOPSIS: 26 * 27 * long double x, y, lgammal(); 28 * extern int signgam; 29 * 30 * y = lgammal(x); 31 * 32 * 33 * 34 * DESCRIPTION: 35 * 36 * Returns the base e (2.718...) logarithm of the absolute 37 * value of the gamma function of the argument. 38 * The sign (+1 or -1) of the gamma function is returned in a 39 * global (extern) variable named signgam. 40 * 41 * The positive domain is partitioned into numerous segments for approximation. 42 * For x > 10, 43 * log gamma(x) = (x - 0.5) log(x) - x + log sqrt(2 pi) + 1/x R(1/x^2) 44 * Near the minimum at x = x0 = 1.46... the approximation is 45 * log gamma(x0 + z) = log gamma(x0) + z^2 P(z)/Q(z) 46 * for small z. 47 * Elsewhere between 0 and 10, 48 * log gamma(n + z) = log gamma(n) + z P(z)/Q(z) 49 * for various selected n and small z. 50 * 51 * The cosecant reflection formula is employed for negative arguments. 52 * 53 * 54 * 55 * ACCURACY: 56 * 57 * 58 * arithmetic domain # trials peak rms 59 * Relative error: 60 * IEEE 10, 30 100000 3.9e-34 9.8e-35 61 * IEEE 0, 10 100000 3.8e-34 5.3e-35 62 * Absolute error: 63 * IEEE -10, 0 100000 8.0e-34 8.0e-35 64 * IEEE -30, -10 100000 4.4e-34 1.0e-34 65 * IEEE -100, 100 100000 1.0e-34 66 * 67 * The absolute error criterion is the same as relative error 68 * when the function magnitude is greater than one but it is absolute 69 * when the magnitude is less than one. 70 * 71 */ 72 73 #include <math.h> 74 75 #include "math_private.h" 76 77 static const long double PIL = 3.1415926535897932384626433832795028841972E0L; 78 static const long double MAXLGM = 1.0485738685148938358098967157129705071571E4928L; 79 static const long double one = 1.0L; 80 static const long double huge = 1.0e4000L; 81 82 /* log gamma(x) = ( x - 0.5 ) * log(x) - x + LS2PI + 1/x P(1/x^2) 83 1/x <= 0.0741 (x >= 13.495...) 84 Peak relative error 1.5e-36 */ 85 static const long double ls2pi = 9.1893853320467274178032973640561763986140E-1L; 86 #define NRASY 12 87 static const long double RASY[NRASY + 1] = 88 { 89 8.333333333333333333333333333310437112111E-2L, 90 -2.777777777777777777777774789556228296902E-3L, 91 7.936507936507936507795933938448586499183E-4L, 92 -5.952380952380952041799269756378148574045E-4L, 93 8.417508417507928904209891117498524452523E-4L, 94 -1.917526917481263997778542329739806086290E-3L, 95 6.410256381217852504446848671499409919280E-3L, 96 -2.955064066900961649768101034477363301626E-2L, 97 1.796402955865634243663453415388336954675E-1L, 98 -1.391522089007758553455753477688592767741E0L, 99 1.326130089598399157988112385013829305510E1L, 100 -1.420412699593782497803472576479997819149E2L, 101 1.218058922427762808938869872528846787020E3L 102 }; 103 104 105 /* log gamma(x+13) = log gamma(13) + x P(x)/Q(x) 106 -0.5 <= x <= 0.5 107 12.5 <= x+13 <= 13.5 108 Peak relative error 1.1e-36 */ 109 static const long double lgam13a = 1.9987213134765625E1L; 110 static const long double lgam13b = 1.3608962611495173623870550785125024484248E-6L; 111 #define NRN13 7 112 static const long double RN13[NRN13 + 1] = 113 { 114 8.591478354823578150238226576156275285700E11L, 115 2.347931159756482741018258864137297157668E11L, 116 2.555408396679352028680662433943000804616E10L, 117 1.408581709264464345480765758902967123937E9L, 118 4.126759849752613822953004114044451046321E7L, 119 6.133298899622688505854211579222889943778E5L, 120 3.929248056293651597987893340755876578072E3L, 121 6.850783280018706668924952057996075215223E0L 122 }; 123 #define NRD13 6 124 static const long double RD13[NRD13 + 1] = 125 { 126 3.401225382297342302296607039352935541669E11L, 127 8.756765276918037910363513243563234551784E10L, 128 8.873913342866613213078554180987647243903E9L, 129 4.483797255342763263361893016049310017973E8L, 130 1.178186288833066430952276702931512870676E7L, 131 1.519928623743264797939103740132278337476E5L, 132 7.989298844938119228411117593338850892311E2L 133 /* 1.0E0L */ 134 }; 135 136 137 /* log gamma(x+12) = log gamma(12) + x P(x)/Q(x) 138 -0.5 <= x <= 0.5 139 11.5 <= x+12 <= 12.5 140 Peak relative error 4.1e-36 */ 141 static const long double lgam12a = 1.75023040771484375E1L; 142 static const long double lgam12b = 3.7687254483392876529072161996717039575982E-6L; 143 #define NRN12 7 144 static const long double RN12[NRN12 + 1] = 145 { 146 4.709859662695606986110997348630997559137E11L, 147 1.398713878079497115037857470168777995230E11L, 148 1.654654931821564315970930093932954900867E10L, 149 9.916279414876676861193649489207282144036E8L, 150 3.159604070526036074112008954113411389879E7L, 151 5.109099197547205212294747623977502492861E5L, 152 3.563054878276102790183396740969279826988E3L, 153 6.769610657004672719224614163196946862747E0L 154 }; 155 #define NRD12 6 156 static const long double RD12[NRD12 + 1] = 157 { 158 1.928167007860968063912467318985802726613E11L, 159 5.383198282277806237247492369072266389233E10L, 160 5.915693215338294477444809323037871058363E9L, 161 3.241438287570196713148310560147925781342E8L, 162 9.236680081763754597872713592701048455890E6L, 163 1.292246897881650919242713651166596478850E5L, 164 7.366532445427159272584194816076600211171E2L 165 /* 1.0E0L */ 166 }; 167 168 169 /* log gamma(x+11) = log gamma(11) + x P(x)/Q(x) 170 -0.5 <= x <= 0.5 171 10.5 <= x+11 <= 11.5 172 Peak relative error 1.8e-35 */ 173 static const long double lgam11a = 1.5104400634765625E1L; 174 static const long double lgam11b = 1.1938309890295225709329251070371882250744E-5L; 175 #define NRN11 7 176 static const long double RN11[NRN11 + 1] = 177 { 178 2.446960438029415837384622675816736622795E11L, 179 7.955444974446413315803799763901729640350E10L, 180 1.030555327949159293591618473447420338444E10L, 181 6.765022131195302709153994345470493334946E8L, 182 2.361892792609204855279723576041468347494E7L, 183 4.186623629779479136428005806072176490125E5L, 184 3.202506022088912768601325534149383594049E3L, 185 6.681356101133728289358838690666225691363E0L 186 }; 187 #define NRD11 6 188 static const long double RD11[NRD11 + 1] = 189 { 190 1.040483786179428590683912396379079477432E11L, 191 3.172251138489229497223696648369823779729E10L, 192 3.806961885984850433709295832245848084614E9L, 193 2.278070344022934913730015420611609620171E8L, 194 7.089478198662651683977290023829391596481E6L, 195 1.083246385105903533237139380509590158658E5L, 196 6.744420991491385145885727942219463243597E2L 197 /* 1.0E0L */ 198 }; 199 200 201 /* log gamma(x+10) = log gamma(10) + x P(x)/Q(x) 202 -0.5 <= x <= 0.5 203 9.5 <= x+10 <= 10.5 204 Peak relative error 5.4e-37 */ 205 static const long double lgam10a = 1.280181884765625E1L; 206 static const long double lgam10b = 8.6324252196112077178745667061642811492557E-6L; 207 #define NRN10 7 208 static const long double RN10[NRN10 + 1] = 209 { 210 -1.239059737177249934158597996648808363783E14L, 211 -4.725899566371458992365624673357356908719E13L, 212 -7.283906268647083312042059082837754850808E12L, 213 -5.802855515464011422171165179767478794637E11L, 214 -2.532349691157548788382820303182745897298E10L, 215 -5.884260178023777312587193693477072061820E8L, 216 -6.437774864512125749845840472131829114906E6L, 217 -2.350975266781548931856017239843273049384E4L 218 }; 219 #define NRD10 7 220 static const long double RD10[NRD10 + 1] = 221 { 222 -5.502645997581822567468347817182347679552E13L, 223 -1.970266640239849804162284805400136473801E13L, 224 -2.819677689615038489384974042561531409392E12L, 225 -2.056105863694742752589691183194061265094E11L, 226 -8.053670086493258693186307810815819662078E9L, 227 -1.632090155573373286153427982504851867131E8L, 228 -1.483575879240631280658077826889223634921E6L, 229 -4.002806669713232271615885826373550502510E3L 230 /* 1.0E0L */ 231 }; 232 233 234 /* log gamma(x+9) = log gamma(9) + x P(x)/Q(x) 235 -0.5 <= x <= 0.5 236 8.5 <= x+9 <= 9.5 237 Peak relative error 3.6e-36 */ 238 static const long double lgam9a = 1.06045989990234375E1L; 239 static const long double lgam9b = 3.9037218127284172274007216547549861681400E-6L; 240 #define NRN9 7 241 static const long double RN9[NRN9 + 1] = 242 { 243 -4.936332264202687973364500998984608306189E13L, 244 -2.101372682623700967335206138517766274855E13L, 245 -3.615893404644823888655732817505129444195E12L, 246 -3.217104993800878891194322691860075472926E11L, 247 -1.568465330337375725685439173603032921399E10L, 248 -4.073317518162025744377629219101510217761E8L, 249 -4.983232096406156139324846656819246974500E6L, 250 -2.036280038903695980912289722995505277253E4L 251 }; 252 #define NRD9 7 253 static const long double RD9[NRD9 + 1] = 254 { 255 -2.306006080437656357167128541231915480393E13L, 256 -9.183606842453274924895648863832233799950E12L, 257 -1.461857965935942962087907301194381010380E12L, 258 -1.185728254682789754150068652663124298303E11L, 259 -5.166285094703468567389566085480783070037E9L, 260 -1.164573656694603024184768200787835094317E8L, 261 -1.177343939483908678474886454113163527909E6L, 262 -3.529391059783109732159524500029157638736E3L 263 /* 1.0E0L */ 264 }; 265 266 267 /* log gamma(x+8) = log gamma(8) + x P(x)/Q(x) 268 -0.5 <= x <= 0.5 269 7.5 <= x+8 <= 8.5 270 Peak relative error 2.4e-37 */ 271 static const long double lgam8a = 8.525146484375E0L; 272 static const long double lgam8b = 1.4876690414300165531036347125050759667737E-5L; 273 #define NRN8 8 274 static const long double RN8[NRN8 + 1] = 275 { 276 6.600775438203423546565361176829139703289E11L, 277 3.406361267593790705240802723914281025800E11L, 278 7.222460928505293914746983300555538432830E10L, 279 8.102984106025088123058747466840656458342E9L, 280 5.157620015986282905232150979772409345927E8L, 281 1.851445288272645829028129389609068641517E7L, 282 3.489261702223124354745894067468953756656E5L, 283 2.892095396706665774434217489775617756014E3L, 284 6.596977510622195827183948478627058738034E0L 285 }; 286 #define NRD8 7 287 static const long double RD8[NRD8 + 1] = 288 { 289 3.274776546520735414638114828622673016920E11L, 290 1.581811207929065544043963828487733970107E11L, 291 3.108725655667825188135393076860104546416E10L, 292 3.193055010502912617128480163681842165730E9L, 293 1.830871482669835106357529710116211541839E8L, 294 5.790862854275238129848491555068073485086E6L, 295 9.305213264307921522842678835618803553589E4L, 296 6.216974105861848386918949336819572333622E2L 297 /* 1.0E0L */ 298 }; 299 300 301 /* log gamma(x+7) = log gamma(7) + x P(x)/Q(x) 302 -0.5 <= x <= 0.5 303 6.5 <= x+7 <= 7.5 304 Peak relative error 3.2e-36 */ 305 static const long double lgam7a = 6.5792388916015625E0L; 306 static const long double lgam7b = 1.2320408538495060178292903945321122583007E-5L; 307 #define NRN7 8 308 static const long double RN7[NRN7 + 1] = 309 { 310 2.065019306969459407636744543358209942213E11L, 311 1.226919919023736909889724951708796532847E11L, 312 2.996157990374348596472241776917953749106E10L, 313 3.873001919306801037344727168434909521030E9L, 314 2.841575255593761593270885753992732145094E8L, 315 1.176342515359431913664715324652399565551E7L, 316 2.558097039684188723597519300356028511547E5L, 317 2.448525238332609439023786244782810774702E3L, 318 6.460280377802030953041566617300902020435E0L 319 }; 320 #define NRD7 7 321 static const long double RD7[NRD7 + 1] = 322 { 323 1.102646614598516998880874785339049304483E11L, 324 6.099297512712715445879759589407189290040E10L, 325 1.372898136289611312713283201112060238351E10L, 326 1.615306270420293159907951633566635172343E9L, 327 1.061114435798489135996614242842561967459E8L, 328 3.845638971184305248268608902030718674691E6L, 329 7.081730675423444975703917836972720495507E4L, 330 5.423122582741398226693137276201344096370E2L 331 /* 1.0E0L */ 332 }; 333 334 335 /* log gamma(x+6) = log gamma(6) + x P(x)/Q(x) 336 -0.5 <= x <= 0.5 337 5.5 <= x+6 <= 6.5 338 Peak relative error 6.2e-37 */ 339 static const long double lgam6a = 4.7874908447265625E0L; 340 static const long double lgam6b = 8.9805548349424770093452324304839959231517E-7L; 341 #define NRN6 8 342 static const long double RN6[NRN6 + 1] = 343 { 344 -3.538412754670746879119162116819571823643E13L, 345 -2.613432593406849155765698121483394257148E13L, 346 -8.020670732770461579558867891923784753062E12L, 347 -1.322227822931250045347591780332435433420E12L, 348 -1.262809382777272476572558806855377129513E11L, 349 -7.015006277027660872284922325741197022467E9L, 350 -2.149320689089020841076532186783055727299E8L, 351 -3.167210585700002703820077565539658995316E6L, 352 -1.576834867378554185210279285358586385266E4L 353 }; 354 #define NRD6 8 355 static const long double RD6[NRD6 + 1] = 356 { 357 -2.073955870771283609792355579558899389085E13L, 358 -1.421592856111673959642750863283919318175E13L, 359 -4.012134994918353924219048850264207074949E12L, 360 -6.013361045800992316498238470888523722431E11L, 361 -5.145382510136622274784240527039643430628E10L, 362 -2.510575820013409711678540476918249524123E9L, 363 -6.564058379709759600836745035871373240904E7L, 364 -7.861511116647120540275354855221373571536E5L, 365 -2.821943442729620524365661338459579270561E3L 366 /* 1.0E0L */ 367 }; 368 369 370 /* log gamma(x+5) = log gamma(5) + x P(x)/Q(x) 371 -0.5 <= x <= 0.5 372 4.5 <= x+5 <= 5.5 373 Peak relative error 3.4e-37 */ 374 static const long double lgam5a = 3.17803955078125E0L; 375 static const long double lgam5b = 1.4279566695619646941601297055408873990961E-5L; 376 #define NRN5 9 377 static const long double RN5[NRN5 + 1] = 378 { 379 2.010952885441805899580403215533972172098E11L, 380 1.916132681242540921354921906708215338584E11L, 381 7.679102403710581712903937970163206882492E10L, 382 1.680514903671382470108010973615268125169E10L, 383 2.181011222911537259440775283277711588410E9L, 384 1.705361119398837808244780667539728356096E8L, 385 7.792391565652481864976147945997033946360E6L, 386 1.910741381027985291688667214472560023819E5L, 387 2.088138241893612679762260077783794329559E3L, 388 6.330318119566998299106803922739066556550E0L 389 }; 390 #define NRD5 8 391 static const long double RD5[NRD5 + 1] = 392 { 393 1.335189758138651840605141370223112376176E11L, 394 1.174130445739492885895466097516530211283E11L, 395 4.308006619274572338118732154886328519910E10L, 396 8.547402888692578655814445003283720677468E9L, 397 9.934628078575618309542580800421370730906E8L, 398 6.847107420092173812998096295422311820672E7L, 399 2.698552646016599923609773122139463150403E6L, 400 5.526516251532464176412113632726150253215E4L, 401 4.772343321713697385780533022595450486932E2L 402 /* 1.0E0L */ 403 }; 404 405 406 /* log gamma(x+4) = log gamma(4) + x P(x)/Q(x) 407 -0.5 <= x <= 0.5 408 3.5 <= x+4 <= 4.5 409 Peak relative error 6.7e-37 */ 410 static const long double lgam4a = 1.791748046875E0L; 411 static const long double lgam4b = 1.1422353055000812477358380702272722990692E-5L; 412 #define NRN4 9 413 static const long double RN4[NRN4 + 1] = 414 { 415 -1.026583408246155508572442242188887829208E13L, 416 -1.306476685384622809290193031208776258809E13L, 417 -7.051088602207062164232806511992978915508E12L, 418 -2.100849457735620004967624442027793656108E12L, 419 -3.767473790774546963588549871673843260569E11L, 420 -4.156387497364909963498394522336575984206E10L, 421 -2.764021460668011732047778992419118757746E9L, 422 -1.036617204107109779944986471142938641399E8L, 423 -1.895730886640349026257780896972598305443E6L, 424 -1.180509051468390914200720003907727988201E4L 425 }; 426 #define NRD4 9 427 static const long double RD4[NRD4 + 1] = 428 { 429 -8.172669122056002077809119378047536240889E12L, 430 -9.477592426087986751343695251801814226960E12L, 431 -4.629448850139318158743900253637212801682E12L, 432 -1.237965465892012573255370078308035272942E12L, 433 -1.971624313506929845158062177061297598956E11L, 434 -1.905434843346570533229942397763361493610E10L, 435 -1.089409357680461419743730978512856675984E9L, 436 -3.416703082301143192939774401370222822430E7L, 437 -4.981791914177103793218433195857635265295E5L, 438 -2.192507743896742751483055798411231453733E3L 439 /* 1.0E0L */ 440 }; 441 442 443 /* log gamma(x+3) = log gamma(3) + x P(x)/Q(x) 444 -0.25 <= x <= 0.5 445 2.75 <= x+3 <= 3.5 446 Peak relative error 6.0e-37 */ 447 static const long double lgam3a = 6.93145751953125E-1L; 448 static const long double lgam3b = 1.4286068203094172321214581765680755001344E-6L; 449 450 #define NRN3 9 451 static const long double RN3[NRN3 + 1] = 452 { 453 -4.813901815114776281494823863935820876670E11L, 454 -8.425592975288250400493910291066881992620E11L, 455 -6.228685507402467503655405482985516909157E11L, 456 -2.531972054436786351403749276956707260499E11L, 457 -6.170200796658926701311867484296426831687E10L, 458 -9.211477458528156048231908798456365081135E9L, 459 -8.251806236175037114064561038908691305583E8L, 460 -4.147886355917831049939930101151160447495E7L, 461 -1.010851868928346082547075956946476932162E6L, 462 -8.333374463411801009783402800801201603736E3L 463 }; 464 #define NRD3 9 465 static const long double RD3[NRD3 + 1] = 466 { 467 -5.216713843111675050627304523368029262450E11L, 468 -8.014292925418308759369583419234079164391E11L, 469 -5.180106858220030014546267824392678611990E11L, 470 -1.830406975497439003897734969120997840011E11L, 471 -3.845274631904879621945745960119924118925E10L, 472 -4.891033385370523863288908070309417710903E9L, 473 -3.670172254411328640353855768698287474282E8L, 474 -1.505316381525727713026364396635522516989E7L, 475 -2.856327162923716881454613540575964890347E5L, 476 -1.622140448015769906847567212766206894547E3L 477 /* 1.0E0L */ 478 }; 479 480 481 /* log gamma(x+2.5) = log gamma(2.5) + x P(x)/Q(x) 482 -0.125 <= x <= 0.25 483 2.375 <= x+2.5 <= 2.75 */ 484 static const long double lgam2r5a = 2.8466796875E-1L; 485 static const long double lgam2r5b = 1.4901722919159632494669682701924320137696E-5L; 486 #define NRN2r5 8 487 static const long double RN2r5[NRN2r5 + 1] = 488 { 489 -4.676454313888335499356699817678862233205E9L, 490 -9.361888347911187924389905984624216340639E9L, 491 -7.695353600835685037920815799526540237703E9L, 492 -3.364370100981509060441853085968900734521E9L, 493 -8.449902011848163568670361316804900559863E8L, 494 -1.225249050950801905108001246436783022179E8L, 495 -9.732972931077110161639900388121650470926E6L, 496 -3.695711763932153505623248207576425983573E5L, 497 -4.717341584067827676530426007495274711306E3L 498 }; 499 #define NRD2r5 8 500 static const long double RD2r5[NRD2r5 + 1] = 501 { 502 -6.650657966618993679456019224416926875619E9L, 503 -1.099511409330635807899718829033488771623E10L, 504 -7.482546968307837168164311101447116903148E9L, 505 -2.702967190056506495988922973755870557217E9L, 506 -5.570008176482922704972943389590409280950E8L, 507 -6.536934032192792470926310043166993233231E7L, 508 -4.101991193844953082400035444146067511725E6L, 509 -1.174082735875715802334430481065526664020E5L, 510 -9.932840389994157592102947657277692978511E2L 511 /* 1.0E0L */ 512 }; 513 514 515 /* log gamma(x+2) = x P(x)/Q(x) 516 -0.125 <= x <= +0.375 517 1.875 <= x+2 <= 2.375 518 Peak relative error 4.6e-36 */ 519 #define NRN2 9 520 static const long double RN2[NRN2 + 1] = 521 { 522 -3.716661929737318153526921358113793421524E9L, 523 -1.138816715030710406922819131397532331321E10L, 524 -1.421017419363526524544402598734013569950E10L, 525 -9.510432842542519665483662502132010331451E9L, 526 -3.747528562099410197957514973274474767329E9L, 527 -8.923565763363912474488712255317033616626E8L, 528 -1.261396653700237624185350402781338231697E8L, 529 -9.918402520255661797735331317081425749014E6L, 530 -3.753996255897143855113273724233104768831E5L, 531 -4.778761333044147141559311805999540765612E3L 532 }; 533 #define NRD2 9 534 static const long double RD2[NRD2 + 1] = 535 { 536 -8.790916836764308497770359421351673950111E9L, 537 -2.023108608053212516399197678553737477486E10L, 538 -1.958067901852022239294231785363504458367E10L, 539 -1.035515043621003101254252481625188704529E10L, 540 -3.253884432621336737640841276619272224476E9L, 541 -6.186383531162456814954947669274235815544E8L, 542 -6.932557847749518463038934953605969951466E7L, 543 -4.240731768287359608773351626528479703758E6L, 544 -1.197343995089189188078944689846348116630E5L, 545 -1.004622911670588064824904487064114090920E3L 546 /* 1.0E0 */ 547 }; 548 549 550 /* log gamma(x+1.75) = log gamma(1.75) + x P(x)/Q(x) 551 -0.125 <= x <= +0.125 552 1.625 <= x+1.75 <= 1.875 553 Peak relative error 9.2e-37 */ 554 static const long double lgam1r75a = -8.441162109375E-2L; 555 static const long double lgam1r75b = 1.0500073264444042213965868602268256157604E-5L; 556 #define NRN1r75 8 557 static const long double RN1r75[NRN1r75 + 1] = 558 { 559 -5.221061693929833937710891646275798251513E7L, 560 -2.052466337474314812817883030472496436993E8L, 561 -2.952718275974940270675670705084125640069E8L, 562 -2.132294039648116684922965964126389017840E8L, 563 -8.554103077186505960591321962207519908489E7L, 564 -1.940250901348870867323943119132071960050E7L, 565 -2.379394147112756860769336400290402208435E6L, 566 -1.384060879999526222029386539622255797389E5L, 567 -2.698453601378319296159355612094598695530E3L 568 }; 569 #define NRD1r75 8 570 static const long double RD1r75[NRD1r75 + 1] = 571 { 572 -2.109754689501705828789976311354395393605E8L, 573 -5.036651829232895725959911504899241062286E8L, 574 -4.954234699418689764943486770327295098084E8L, 575 -2.589558042412676610775157783898195339410E8L, 576 -7.731476117252958268044969614034776883031E7L, 577 -1.316721702252481296030801191240867486965E7L, 578 -1.201296501404876774861190604303728810836E6L, 579 -5.007966406976106636109459072523610273928E4L, 580 -6.155817990560743422008969155276229018209E2L 581 /* 1.0E0L */ 582 }; 583 584 585 /* log gamma(x+x0) = y0 + x^2 P(x)/Q(x) 586 -0.0867 <= x <= +0.1634 587 1.374932... <= x+x0 <= 1.625032... 588 Peak relative error 4.0e-36 */ 589 static const long double x0a = 1.4616241455078125L; 590 static const long double x0b = 7.9994605498412626595423257213002588621246E-6L; 591 static const long double y0a = -1.21490478515625E-1L; 592 static const long double y0b = 4.1879797753919044854428223084178486438269E-6L; 593 #define NRN1r5 8 594 static const long double RN1r5[NRN1r5 + 1] = 595 { 596 6.827103657233705798067415468881313128066E5L, 597 1.910041815932269464714909706705242148108E6L, 598 2.194344176925978377083808566251427771951E6L, 599 1.332921400100891472195055269688876427962E6L, 600 4.589080973377307211815655093824787123508E5L, 601 8.900334161263456942727083580232613796141E4L, 602 9.053840838306019753209127312097612455236E3L, 603 4.053367147553353374151852319743594873771E2L, 604 5.040631576303952022968949605613514584950E0L 605 }; 606 #define NRD1r5 8 607 static const long double RD1r5[NRD1r5 + 1] = 608 { 609 1.411036368843183477558773688484699813355E6L, 610 4.378121767236251950226362443134306184849E6L, 611 5.682322855631723455425929877581697918168E6L, 612 3.999065731556977782435009349967042222375E6L, 613 1.653651390456781293163585493620758410333E6L, 614 4.067774359067489605179546964969435858311E5L, 615 5.741463295366557346748361781768833633256E4L, 616 4.226404539738182992856094681115746692030E3L, 617 1.316980975410327975566999780608618774469E2L, 618 /* 1.0E0L */ 619 }; 620 621 622 /* log gamma(x+1.25) = log gamma(1.25) + x P(x)/Q(x) 623 -.125 <= x <= +.125 624 1.125 <= x+1.25 <= 1.375 625 Peak relative error = 4.9e-36 */ 626 static const long double lgam1r25a = -9.82818603515625E-2L; 627 static const long double lgam1r25b = 1.0023929749338536146197303364159774377296E-5L; 628 #define NRN1r25 9 629 static const long double RN1r25[NRN1r25 + 1] = 630 { 631 -9.054787275312026472896002240379580536760E4L, 632 -8.685076892989927640126560802094680794471E4L, 633 2.797898965448019916967849727279076547109E5L, 634 6.175520827134342734546868356396008898299E5L, 635 5.179626599589134831538516906517372619641E5L, 636 2.253076616239043944538380039205558242161E5L, 637 5.312653119599957228630544772499197307195E4L, 638 6.434329437514083776052669599834938898255E3L, 639 3.385414416983114598582554037612347549220E2L, 640 4.907821957946273805080625052510832015792E0L 641 }; 642 #define NRD1r25 8 643 static const long double RD1r25[NRD1r25 + 1] = 644 { 645 3.980939377333448005389084785896660309000E5L, 646 1.429634893085231519692365775184490465542E6L, 647 2.145438946455476062850151428438668234336E6L, 648 1.743786661358280837020848127465970357893E6L, 649 8.316364251289743923178092656080441655273E5L, 650 2.355732939106812496699621491135458324294E5L, 651 3.822267399625696880571810137601310855419E4L, 652 3.228463206479133236028576845538387620856E3L, 653 1.152133170470059555646301189220117965514E2L 654 /* 1.0E0L */ 655 }; 656 657 658 /* log gamma(x + 1) = x P(x)/Q(x) 659 0.0 <= x <= +0.125 660 1.0 <= x+1 <= 1.125 661 Peak relative error 1.1e-35 */ 662 #define NRN1 8 663 static const long double RN1[NRN1 + 1] = 664 { 665 -9.987560186094800756471055681088744738818E3L, 666 -2.506039379419574361949680225279376329742E4L, 667 -1.386770737662176516403363873617457652991E4L, 668 1.439445846078103202928677244188837130744E4L, 669 2.159612048879650471489449668295139990693E4L, 670 1.047439813638144485276023138173676047079E4L, 671 2.250316398054332592560412486630769139961E3L, 672 1.958510425467720733041971651126443864041E2L, 673 4.516830313569454663374271993200291219855E0L 674 }; 675 #define NRD1 7 676 static const long double RD1[NRD1 + 1] = 677 { 678 1.730299573175751778863269333703788214547E4L, 679 6.807080914851328611903744668028014678148E4L, 680 1.090071629101496938655806063184092302439E5L, 681 9.124354356415154289343303999616003884080E4L, 682 4.262071638655772404431164427024003253954E4L, 683 1.096981664067373953673982635805821283581E4L, 684 1.431229503796575892151252708527595787588E3L, 685 7.734110684303689320830401788262295992921E1L 686 /* 1.0E0 */ 687 }; 688 689 690 /* log gamma(x + 1) = x P(x)/Q(x) 691 -0.125 <= x <= 0 692 0.875 <= x+1 <= 1.0 693 Peak relative error 7.0e-37 */ 694 #define NRNr9 8 695 static const long double RNr9[NRNr9 + 1] = 696 { 697 4.441379198241760069548832023257571176884E5L, 698 1.273072988367176540909122090089580368732E6L, 699 9.732422305818501557502584486510048387724E5L, 700 -5.040539994443998275271644292272870348684E5L, 701 -1.208719055525609446357448132109723786736E6L, 702 -7.434275365370936547146540554419058907156E5L, 703 -2.075642969983377738209203358199008185741E5L, 704 -2.565534860781128618589288075109372218042E4L, 705 -1.032901669542994124131223797515913955938E3L, 706 }; 707 #define NRDr9 8 708 static const long double RDr9[NRDr9 + 1] = 709 { 710 -7.694488331323118759486182246005193998007E5L, 711 -3.301918855321234414232308938454112213751E6L, 712 -5.856830900232338906742924836032279404702E6L, 713 -5.540672519616151584486240871424021377540E6L, 714 -3.006530901041386626148342989181721176919E6L, 715 -9.350378280513062139466966374330795935163E5L, 716 -1.566179100031063346901755685375732739511E5L, 717 -1.205016539620260779274902967231510804992E4L, 718 -2.724583156305709733221564484006088794284E2L 719 /* 1.0E0 */ 720 }; 721 722 723 /* Evaluate P[n] x^n + P[n-1] x^(n-1) + ... + P[0] */ 724 725 static long double 726 neval (long double x, const long double *p, int n) 727 { 728 long double y; 729 730 p += n; 731 y = *p--; 732 do 733 { 734 y = y * x + *p--; 735 } 736 while (--n > 0); 737 return y; 738 } 739 740 741 /* Evaluate x^n+1 + P[n] x^(n) + P[n-1] x^(n-1) + ... + P[0] */ 742 743 static long double 744 deval (long double x, const long double *p, int n) 745 { 746 long double y; 747 748 p += n; 749 y = x + *p--; 750 do 751 { 752 y = y * x + *p--; 753 } 754 while (--n > 0); 755 return y; 756 } 757 758 759 long double 760 lgammal(long double x) 761 { 762 long double p, q, w, z, nx; 763 int i, nn; 764 765 signgam = 1; 766 767 if (! isfinite (x)) 768 return x * x; 769 770 if (x == 0.0L) 771 { 772 if (signbit (x)) 773 signgam = -1; 774 return one / fabsl (x); 775 } 776 777 if (x < 0.0L) 778 { 779 q = -x; 780 p = floorl (q); 781 if (p == q) 782 return (one / (p - p)); 783 i = p; 784 if ((i & 1) == 0) 785 signgam = -1; 786 else 787 signgam = 1; 788 z = q - p; 789 if (z > 0.5L) 790 { 791 p += 1.0L; 792 z = p - q; 793 } 794 z = q * sinl (PIL * z); 795 if (z == 0.0L) 796 return (signgam * huge * huge); 797 w = lgammal (q); 798 z = logl (PIL / z) - w; 799 return (z); 800 } 801 802 if (x < 13.5L) 803 { 804 p = 0.0L; 805 nx = floorl (x + 0.5L); 806 nn = nx; 807 switch (nn) 808 { 809 case 0: 810 /* log gamma (x + 1) = log(x) + log gamma(x) */ 811 if (x <= 0.125) 812 { 813 p = x * neval (x, RN1, NRN1) / deval (x, RD1, NRD1); 814 } 815 else if (x <= 0.375) 816 { 817 z = x - 0.25L; 818 p = z * neval (z, RN1r25, NRN1r25) / deval (z, RD1r25, NRD1r25); 819 p += lgam1r25b; 820 p += lgam1r25a; 821 } 822 else if (x <= 0.625) 823 { 824 z = x + (1.0L - x0a); 825 z = z - x0b; 826 p = neval (z, RN1r5, NRN1r5) / deval (z, RD1r5, NRD1r5); 827 p = p * z * z; 828 p = p + y0b; 829 p = p + y0a; 830 } 831 else if (x <= 0.875) 832 { 833 z = x - 0.75L; 834 p = z * neval (z, RN1r75, NRN1r75) / deval (z, RD1r75, NRD1r75); 835 p += lgam1r75b; 836 p += lgam1r75a; 837 } 838 else 839 { 840 z = x - 1.0L; 841 p = z * neval (z, RN2, NRN2) / deval (z, RD2, NRD2); 842 } 843 p = p - logl (x); 844 break; 845 846 case 1: 847 if (x < 0.875L) 848 { 849 if (x <= 0.625) 850 { 851 z = x + (1.0L - x0a); 852 z = z - x0b; 853 p = neval (z, RN1r5, NRN1r5) / deval (z, RD1r5, NRD1r5); 854 p = p * z * z; 855 p = p + y0b; 856 p = p + y0a; 857 } 858 else if (x <= 0.875) 859 { 860 z = x - 0.75L; 861 p = z * neval (z, RN1r75, NRN1r75) 862 / deval (z, RD1r75, NRD1r75); 863 p += lgam1r75b; 864 p += lgam1r75a; 865 } 866 else 867 { 868 z = x - 1.0L; 869 p = z * neval (z, RN2, NRN2) / deval (z, RD2, NRD2); 870 } 871 p = p - logl (x); 872 } 873 else if (x < 1.0L) 874 { 875 z = x - 1.0L; 876 p = z * neval (z, RNr9, NRNr9) / deval (z, RDr9, NRDr9); 877 } 878 else if (x == 1.0L) 879 p = 0.0L; 880 else if (x <= 1.125L) 881 { 882 z = x - 1.0L; 883 p = z * neval (z, RN1, NRN1) / deval (z, RD1, NRD1); 884 } 885 else if (x <= 1.375) 886 { 887 z = x - 1.25L; 888 p = z * neval (z, RN1r25, NRN1r25) / deval (z, RD1r25, NRD1r25); 889 p += lgam1r25b; 890 p += lgam1r25a; 891 } 892 else 893 { 894 /* 1.375 <= x+x0 <= 1.625 */ 895 z = x - x0a; 896 z = z - x0b; 897 p = neval (z, RN1r5, NRN1r5) / deval (z, RD1r5, NRD1r5); 898 p = p * z * z; 899 p = p + y0b; 900 p = p + y0a; 901 } 902 break; 903 904 case 2: 905 if (x < 1.625L) 906 { 907 z = x - x0a; 908 z = z - x0b; 909 p = neval (z, RN1r5, NRN1r5) / deval (z, RD1r5, NRD1r5); 910 p = p * z * z; 911 p = p + y0b; 912 p = p + y0a; 913 } 914 else if (x < 1.875L) 915 { 916 z = x - 1.75L; 917 p = z * neval (z, RN1r75, NRN1r75) / deval (z, RD1r75, NRD1r75); 918 p += lgam1r75b; 919 p += lgam1r75a; 920 } 921 else if (x == 2.0L) 922 p = 0.0L; 923 else if (x < 2.375L) 924 { 925 z = x - 2.0L; 926 p = z * neval (z, RN2, NRN2) / deval (z, RD2, NRD2); 927 } 928 else 929 { 930 z = x - 2.5L; 931 p = z * neval (z, RN2r5, NRN2r5) / deval (z, RD2r5, NRD2r5); 932 p += lgam2r5b; 933 p += lgam2r5a; 934 } 935 break; 936 937 case 3: 938 if (x < 2.75) 939 { 940 z = x - 2.5L; 941 p = z * neval (z, RN2r5, NRN2r5) / deval (z, RD2r5, NRD2r5); 942 p += lgam2r5b; 943 p += lgam2r5a; 944 } 945 else 946 { 947 z = x - 3.0L; 948 p = z * neval (z, RN3, NRN3) / deval (z, RD3, NRD3); 949 p += lgam3b; 950 p += lgam3a; 951 } 952 break; 953 954 case 4: 955 z = x - 4.0L; 956 p = z * neval (z, RN4, NRN4) / deval (z, RD4, NRD4); 957 p += lgam4b; 958 p += lgam4a; 959 break; 960 961 case 5: 962 z = x - 5.0L; 963 p = z * neval (z, RN5, NRN5) / deval (z, RD5, NRD5); 964 p += lgam5b; 965 p += lgam5a; 966 break; 967 968 case 6: 969 z = x - 6.0L; 970 p = z * neval (z, RN6, NRN6) / deval (z, RD6, NRD6); 971 p += lgam6b; 972 p += lgam6a; 973 break; 974 975 case 7: 976 z = x - 7.0L; 977 p = z * neval (z, RN7, NRN7) / deval (z, RD7, NRD7); 978 p += lgam7b; 979 p += lgam7a; 980 break; 981 982 case 8: 983 z = x - 8.0L; 984 p = z * neval (z, RN8, NRN8) / deval (z, RD8, NRD8); 985 p += lgam8b; 986 p += lgam8a; 987 break; 988 989 case 9: 990 z = x - 9.0L; 991 p = z * neval (z, RN9, NRN9) / deval (z, RD9, NRD9); 992 p += lgam9b; 993 p += lgam9a; 994 break; 995 996 case 10: 997 z = x - 10.0L; 998 p = z * neval (z, RN10, NRN10) / deval (z, RD10, NRD10); 999 p += lgam10b; 1000 p += lgam10a; 1001 break; 1002 1003 case 11: 1004 z = x - 11.0L; 1005 p = z * neval (z, RN11, NRN11) / deval (z, RD11, NRD11); 1006 p += lgam11b; 1007 p += lgam11a; 1008 break; 1009 1010 case 12: 1011 z = x - 12.0L; 1012 p = z * neval (z, RN12, NRN12) / deval (z, RD12, NRD12); 1013 p += lgam12b; 1014 p += lgam12a; 1015 break; 1016 1017 case 13: 1018 z = x - 13.0L; 1019 p = z * neval (z, RN13, NRN13) / deval (z, RD13, NRD13); 1020 p += lgam13b; 1021 p += lgam13a; 1022 break; 1023 } 1024 return p; 1025 } 1026 1027 if (x > MAXLGM) 1028 return (signgam * huge * huge); 1029 1030 q = ls2pi - x; 1031 q = (x - 0.5L) * logl (x) + q; 1032 if (x > 1.0e18L) 1033 return (q); 1034 1035 p = 1.0L / (x * x); 1036 q += neval (p, RASY, NRASY) / x; 1037 return (q); 1038 } 1039 DEF_STD(lgammal); 1040