1 /* $OpenBSD: e_lgammal.c,v 1.3 2011/07/09 05:29:06 martynas 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_r
20 *
21 * Natural logarithm of gamma function
22 *
23 *
24 *
25 * SYNOPSIS:
26 *
27 * long double x, y, lgammal_r();
28 * int signgam;
29 *
30 * y = lgammal_r(x, &signgam);
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 through signgamp.
39 *
40 * The positive domain is partitioned into numerous segments for approximation.
41 * For x > 10,
42 * log gamma(x) = (x - 0.5) log(x) - x + log sqrt(2 pi) + 1/x R(1/x^2)
43 * Near the minimum at x = x0 = 1.46... the approximation is
44 * log gamma(x0 + z) = log gamma(x0) + z^2 P(z)/Q(z)
45 * for small z.
46 * Elsewhere between 0 and 10,
47 * log gamma(n + z) = log gamma(n) + z P(z)/Q(z)
48 * for various selected n and small z.
49 *
50 * The cosecant reflection formula is employed for negative arguments.
51 *
52 *
53 *
54 * ACCURACY:
55 *
56 *
57 * arithmetic domain # trials peak rms
58 * Relative error:
59 * IEEE 10, 30 100000 3.9e-34 9.8e-35
60 * IEEE 0, 10 100000 3.8e-34 5.3e-35
61 * Absolute error:
62 * IEEE -10, 0 100000 8.0e-34 8.0e-35
63 * IEEE -30, -10 100000 4.4e-34 1.0e-34
64 * IEEE -100, 100 100000 1.0e-34
65 *
66 * The absolute error criterion is the same as relative error
67 * when the function magnitude is greater than one but it is absolute
68 * when the magnitude is less than one.
69 *
70 */
71
72
73
74 static const long double PIL = 3.1415926535897932384626433832795028841972E0L;
75 static const long double MAXLGM = 1.0485738685148938358098967157129705071571E4928L;
76 static const long double one = 1.0L;
77 static const long double huge = 1.0e4000L;
78
79 /* log gamma(x) = ( x - 0.5 ) * log(x) - x + LS2PI + 1/x P(1/x^2)
80 1/x <= 0.0741 (x >= 13.495...)
81 Peak relative error 1.5e-36 */
82 static const long double ls2pi = 9.1893853320467274178032973640561763986140E-1L;
83 #define NRASY 12
84 static const long double RASY[NRASY + 1] =
85 {
86 8.333333333333333333333333333310437112111E-2L,
87 -2.777777777777777777777774789556228296902E-3L,
88 7.936507936507936507795933938448586499183E-4L,
89 -5.952380952380952041799269756378148574045E-4L,
90 8.417508417507928904209891117498524452523E-4L,
91 -1.917526917481263997778542329739806086290E-3L,
92 6.410256381217852504446848671499409919280E-3L,
93 -2.955064066900961649768101034477363301626E-2L,
94 1.796402955865634243663453415388336954675E-1L,
95 -1.391522089007758553455753477688592767741E0L,
96 1.326130089598399157988112385013829305510E1L,
97 -1.420412699593782497803472576479997819149E2L,
98 1.218058922427762808938869872528846787020E3L
99 };
100
101
102 /* log gamma(x+13) = log gamma(13) + x P(x)/Q(x)
103 -0.5 <= x <= 0.5
104 12.5 <= x+13 <= 13.5
105 Peak relative error 1.1e-36 */
106 static const long double lgam13a = 1.9987213134765625E1L;
107 static const long double lgam13b = 1.3608962611495173623870550785125024484248E-6L;
108 #define NRN13 7
109 static const long double RN13[NRN13 + 1] =
110 {
111 8.591478354823578150238226576156275285700E11L,
112 2.347931159756482741018258864137297157668E11L,
113 2.555408396679352028680662433943000804616E10L,
114 1.408581709264464345480765758902967123937E9L,
115 4.126759849752613822953004114044451046321E7L,
116 6.133298899622688505854211579222889943778E5L,
117 3.929248056293651597987893340755876578072E3L,
118 6.850783280018706668924952057996075215223E0L
119 };
120 #define NRD13 6
121 static const long double RD13[NRD13 + 1] =
122 {
123 3.401225382297342302296607039352935541669E11L,
124 8.756765276918037910363513243563234551784E10L,
125 8.873913342866613213078554180987647243903E9L,
126 4.483797255342763263361893016049310017973E8L,
127 1.178186288833066430952276702931512870676E7L,
128 1.519928623743264797939103740132278337476E5L,
129 7.989298844938119228411117593338850892311E2L
130 /* 1.0E0L */
131 };
132
133
134 /* log gamma(x+12) = log gamma(12) + x P(x)/Q(x)
135 -0.5 <= x <= 0.5
136 11.5 <= x+12 <= 12.5
137 Peak relative error 4.1e-36 */
138 static const long double lgam12a = 1.75023040771484375E1L;
139 static const long double lgam12b = 3.7687254483392876529072161996717039575982E-6L;
140 #define NRN12 7
141 static const long double RN12[NRN12 + 1] =
142 {
143 4.709859662695606986110997348630997559137E11L,
144 1.398713878079497115037857470168777995230E11L,
145 1.654654931821564315970930093932954900867E10L,
146 9.916279414876676861193649489207282144036E8L,
147 3.159604070526036074112008954113411389879E7L,
148 5.109099197547205212294747623977502492861E5L,
149 3.563054878276102790183396740969279826988E3L,
150 6.769610657004672719224614163196946862747E0L
151 };
152 #define NRD12 6
153 static const long double RD12[NRD12 + 1] =
154 {
155 1.928167007860968063912467318985802726613E11L,
156 5.383198282277806237247492369072266389233E10L,
157 5.915693215338294477444809323037871058363E9L,
158 3.241438287570196713148310560147925781342E8L,
159 9.236680081763754597872713592701048455890E6L,
160 1.292246897881650919242713651166596478850E5L,
161 7.366532445427159272584194816076600211171E2L
162 /* 1.0E0L */
163 };
164
165
166 /* log gamma(x+11) = log gamma(11) + x P(x)/Q(x)
167 -0.5 <= x <= 0.5
168 10.5 <= x+11 <= 11.5
169 Peak relative error 1.8e-35 */
170 static const long double lgam11a = 1.5104400634765625E1L;
171 static const long double lgam11b = 1.1938309890295225709329251070371882250744E-5L;
172 #define NRN11 7
173 static const long double RN11[NRN11 + 1] =
174 {
175 2.446960438029415837384622675816736622795E11L,
176 7.955444974446413315803799763901729640350E10L,
177 1.030555327949159293591618473447420338444E10L,
178 6.765022131195302709153994345470493334946E8L,
179 2.361892792609204855279723576041468347494E7L,
180 4.186623629779479136428005806072176490125E5L,
181 3.202506022088912768601325534149383594049E3L,
182 6.681356101133728289358838690666225691363E0L
183 };
184 #define NRD11 6
185 static const long double RD11[NRD11 + 1] =
186 {
187 1.040483786179428590683912396379079477432E11L,
188 3.172251138489229497223696648369823779729E10L,
189 3.806961885984850433709295832245848084614E9L,
190 2.278070344022934913730015420611609620171E8L,
191 7.089478198662651683977290023829391596481E6L,
192 1.083246385105903533237139380509590158658E5L,
193 6.744420991491385145885727942219463243597E2L
194 /* 1.0E0L */
195 };
196
197
198 /* log gamma(x+10) = log gamma(10) + x P(x)/Q(x)
199 -0.5 <= x <= 0.5
200 9.5 <= x+10 <= 10.5
201 Peak relative error 5.4e-37 */
202 static const long double lgam10a = 1.280181884765625E1L;
203 static const long double lgam10b = 8.6324252196112077178745667061642811492557E-6L;
204 #define NRN10 7
205 static const long double RN10[NRN10 + 1] =
206 {
207 -1.239059737177249934158597996648808363783E14L,
208 -4.725899566371458992365624673357356908719E13L,
209 -7.283906268647083312042059082837754850808E12L,
210 -5.802855515464011422171165179767478794637E11L,
211 -2.532349691157548788382820303182745897298E10L,
212 -5.884260178023777312587193693477072061820E8L,
213 -6.437774864512125749845840472131829114906E6L,
214 -2.350975266781548931856017239843273049384E4L
215 };
216 #define NRD10 7
217 static const long double RD10[NRD10 + 1] =
218 {
219 -5.502645997581822567468347817182347679552E13L,
220 -1.970266640239849804162284805400136473801E13L,
221 -2.819677689615038489384974042561531409392E12L,
222 -2.056105863694742752589691183194061265094E11L,
223 -8.053670086493258693186307810815819662078E9L,
224 -1.632090155573373286153427982504851867131E8L,
225 -1.483575879240631280658077826889223634921E6L,
226 -4.002806669713232271615885826373550502510E3L
227 /* 1.0E0L */
228 };
229
230
231 /* log gamma(x+9) = log gamma(9) + x P(x)/Q(x)
232 -0.5 <= x <= 0.5
233 8.5 <= x+9 <= 9.5
234 Peak relative error 3.6e-36 */
235 static const long double lgam9a = 1.06045989990234375E1L;
236 static const long double lgam9b = 3.9037218127284172274007216547549861681400E-6L;
237 #define NRN9 7
238 static const long double RN9[NRN9 + 1] =
239 {
240 -4.936332264202687973364500998984608306189E13L,
241 -2.101372682623700967335206138517766274855E13L,
242 -3.615893404644823888655732817505129444195E12L,
243 -3.217104993800878891194322691860075472926E11L,
244 -1.568465330337375725685439173603032921399E10L,
245 -4.073317518162025744377629219101510217761E8L,
246 -4.983232096406156139324846656819246974500E6L,
247 -2.036280038903695980912289722995505277253E4L
248 };
249 #define NRD9 7
250 static const long double RD9[NRD9 + 1] =
251 {
252 -2.306006080437656357167128541231915480393E13L,
253 -9.183606842453274924895648863832233799950E12L,
254 -1.461857965935942962087907301194381010380E12L,
255 -1.185728254682789754150068652663124298303E11L,
256 -5.166285094703468567389566085480783070037E9L,
257 -1.164573656694603024184768200787835094317E8L,
258 -1.177343939483908678474886454113163527909E6L,
259 -3.529391059783109732159524500029157638736E3L
260 /* 1.0E0L */
261 };
262
263
264 /* log gamma(x+8) = log gamma(8) + x P(x)/Q(x)
265 -0.5 <= x <= 0.5
266 7.5 <= x+8 <= 8.5
267 Peak relative error 2.4e-37 */
268 static const long double lgam8a = 8.525146484375E0L;
269 static const long double lgam8b = 1.4876690414300165531036347125050759667737E-5L;
270 #define NRN8 8
271 static const long double RN8[NRN8 + 1] =
272 {
273 6.600775438203423546565361176829139703289E11L,
274 3.406361267593790705240802723914281025800E11L,
275 7.222460928505293914746983300555538432830E10L,
276 8.102984106025088123058747466840656458342E9L,
277 5.157620015986282905232150979772409345927E8L,
278 1.851445288272645829028129389609068641517E7L,
279 3.489261702223124354745894067468953756656E5L,
280 2.892095396706665774434217489775617756014E3L,
281 6.596977510622195827183948478627058738034E0L
282 };
283 #define NRD8 7
284 static const long double RD8[NRD8 + 1] =
285 {
286 3.274776546520735414638114828622673016920E11L,
287 1.581811207929065544043963828487733970107E11L,
288 3.108725655667825188135393076860104546416E10L,
289 3.193055010502912617128480163681842165730E9L,
290 1.830871482669835106357529710116211541839E8L,
291 5.790862854275238129848491555068073485086E6L,
292 9.305213264307921522842678835618803553589E4L,
293 6.216974105861848386918949336819572333622E2L
294 /* 1.0E0L */
295 };
296
297
298 /* log gamma(x+7) = log gamma(7) + x P(x)/Q(x)
299 -0.5 <= x <= 0.5
300 6.5 <= x+7 <= 7.5
301 Peak relative error 3.2e-36 */
302 static const long double lgam7a = 6.5792388916015625E0L;
303 static const long double lgam7b = 1.2320408538495060178292903945321122583007E-5L;
304 #define NRN7 8
305 static const long double RN7[NRN7 + 1] =
306 {
307 2.065019306969459407636744543358209942213E11L,
308 1.226919919023736909889724951708796532847E11L,
309 2.996157990374348596472241776917953749106E10L,
310 3.873001919306801037344727168434909521030E9L,
311 2.841575255593761593270885753992732145094E8L,
312 1.176342515359431913664715324652399565551E7L,
313 2.558097039684188723597519300356028511547E5L,
314 2.448525238332609439023786244782810774702E3L,
315 6.460280377802030953041566617300902020435E0L
316 };
317 #define NRD7 7
318 static const long double RD7[NRD7 + 1] =
319 {
320 1.102646614598516998880874785339049304483E11L,
321 6.099297512712715445879759589407189290040E10L,
322 1.372898136289611312713283201112060238351E10L,
323 1.615306270420293159907951633566635172343E9L,
324 1.061114435798489135996614242842561967459E8L,
325 3.845638971184305248268608902030718674691E6L,
326 7.081730675423444975703917836972720495507E4L,
327 5.423122582741398226693137276201344096370E2L
328 /* 1.0E0L */
329 };
330
331
332 /* log gamma(x+6) = log gamma(6) + x P(x)/Q(x)
333 -0.5 <= x <= 0.5
334 5.5 <= x+6 <= 6.5
335 Peak relative error 6.2e-37 */
336 static const long double lgam6a = 4.7874908447265625E0L;
337 static const long double lgam6b = 8.9805548349424770093452324304839959231517E-7L;
338 #define NRN6 8
339 static const long double RN6[NRN6 + 1] =
340 {
341 -3.538412754670746879119162116819571823643E13L,
342 -2.613432593406849155765698121483394257148E13L,
343 -8.020670732770461579558867891923784753062E12L,
344 -1.322227822931250045347591780332435433420E12L,
345 -1.262809382777272476572558806855377129513E11L,
346 -7.015006277027660872284922325741197022467E9L,
347 -2.149320689089020841076532186783055727299E8L,
348 -3.167210585700002703820077565539658995316E6L,
349 -1.576834867378554185210279285358586385266E4L
350 };
351 #define NRD6 8
352 static const long double RD6[NRD6 + 1] =
353 {
354 -2.073955870771283609792355579558899389085E13L,
355 -1.421592856111673959642750863283919318175E13L,
356 -4.012134994918353924219048850264207074949E12L,
357 -6.013361045800992316498238470888523722431E11L,
358 -5.145382510136622274784240527039643430628E10L,
359 -2.510575820013409711678540476918249524123E9L,
360 -6.564058379709759600836745035871373240904E7L,
361 -7.861511116647120540275354855221373571536E5L,
362 -2.821943442729620524365661338459579270561E3L
363 /* 1.0E0L */
364 };
365
366
367 /* log gamma(x+5) = log gamma(5) + x P(x)/Q(x)
368 -0.5 <= x <= 0.5
369 4.5 <= x+5 <= 5.5
370 Peak relative error 3.4e-37 */
371 static const long double lgam5a = 3.17803955078125E0L;
372 static const long double lgam5b = 1.4279566695619646941601297055408873990961E-5L;
373 #define NRN5 9
374 static const long double RN5[NRN5 + 1] =
375 {
376 2.010952885441805899580403215533972172098E11L,
377 1.916132681242540921354921906708215338584E11L,
378 7.679102403710581712903937970163206882492E10L,
379 1.680514903671382470108010973615268125169E10L,
380 2.181011222911537259440775283277711588410E9L,
381 1.705361119398837808244780667539728356096E8L,
382 7.792391565652481864976147945997033946360E6L,
383 1.910741381027985291688667214472560023819E5L,
384 2.088138241893612679762260077783794329559E3L,
385 6.330318119566998299106803922739066556550E0L
386 };
387 #define NRD5 8
388 static const long double RD5[NRD5 + 1] =
389 {
390 1.335189758138651840605141370223112376176E11L,
391 1.174130445739492885895466097516530211283E11L,
392 4.308006619274572338118732154886328519910E10L,
393 8.547402888692578655814445003283720677468E9L,
394 9.934628078575618309542580800421370730906E8L,
395 6.847107420092173812998096295422311820672E7L,
396 2.698552646016599923609773122139463150403E6L,
397 5.526516251532464176412113632726150253215E4L,
398 4.772343321713697385780533022595450486932E2L
399 /* 1.0E0L */
400 };
401
402
403 /* log gamma(x+4) = log gamma(4) + x P(x)/Q(x)
404 -0.5 <= x <= 0.5
405 3.5 <= x+4 <= 4.5
406 Peak relative error 6.7e-37 */
407 static const long double lgam4a = 1.791748046875E0L;
408 static const long double lgam4b = 1.1422353055000812477358380702272722990692E-5L;
409 #define NRN4 9
410 static const long double RN4[NRN4 + 1] =
411 {
412 -1.026583408246155508572442242188887829208E13L,
413 -1.306476685384622809290193031208776258809E13L,
414 -7.051088602207062164232806511992978915508E12L,
415 -2.100849457735620004967624442027793656108E12L,
416 -3.767473790774546963588549871673843260569E11L,
417 -4.156387497364909963498394522336575984206E10L,
418 -2.764021460668011732047778992419118757746E9L,
419 -1.036617204107109779944986471142938641399E8L,
420 -1.895730886640349026257780896972598305443E6L,
421 -1.180509051468390914200720003907727988201E4L
422 };
423 #define NRD4 9
424 static const long double RD4[NRD4 + 1] =
425 {
426 -8.172669122056002077809119378047536240889E12L,
427 -9.477592426087986751343695251801814226960E12L,
428 -4.629448850139318158743900253637212801682E12L,
429 -1.237965465892012573255370078308035272942E12L,
430 -1.971624313506929845158062177061297598956E11L,
431 -1.905434843346570533229942397763361493610E10L,
432 -1.089409357680461419743730978512856675984E9L,
433 -3.416703082301143192939774401370222822430E7L,
434 -4.981791914177103793218433195857635265295E5L,
435 -2.192507743896742751483055798411231453733E3L
436 /* 1.0E0L */
437 };
438
439
440 /* log gamma(x+3) = log gamma(3) + x P(x)/Q(x)
441 -0.25 <= x <= 0.5
442 2.75 <= x+3 <= 3.5
443 Peak relative error 6.0e-37 */
444 static const long double lgam3a = 6.93145751953125E-1L;
445 static const long double lgam3b = 1.4286068203094172321214581765680755001344E-6L;
446
447 #define NRN3 9
448 static const long double RN3[NRN3 + 1] =
449 {
450 -4.813901815114776281494823863935820876670E11L,
451 -8.425592975288250400493910291066881992620E11L,
452 -6.228685507402467503655405482985516909157E11L,
453 -2.531972054436786351403749276956707260499E11L,
454 -6.170200796658926701311867484296426831687E10L,
455 -9.211477458528156048231908798456365081135E9L,
456 -8.251806236175037114064561038908691305583E8L,
457 -4.147886355917831049939930101151160447495E7L,
458 -1.010851868928346082547075956946476932162E6L,
459 -8.333374463411801009783402800801201603736E3L
460 };
461 #define NRD3 9
462 static const long double RD3[NRD3 + 1] =
463 {
464 -5.216713843111675050627304523368029262450E11L,
465 -8.014292925418308759369583419234079164391E11L,
466 -5.180106858220030014546267824392678611990E11L,
467 -1.830406975497439003897734969120997840011E11L,
468 -3.845274631904879621945745960119924118925E10L,
469 -4.891033385370523863288908070309417710903E9L,
470 -3.670172254411328640353855768698287474282E8L,
471 -1.505316381525727713026364396635522516989E7L,
472 -2.856327162923716881454613540575964890347E5L,
473 -1.622140448015769906847567212766206894547E3L
474 /* 1.0E0L */
475 };
476
477
478 /* log gamma(x+2.5) = log gamma(2.5) + x P(x)/Q(x)
479 -0.125 <= x <= 0.25
480 2.375 <= x+2.5 <= 2.75 */
481 static const long double lgam2r5a = 2.8466796875E-1L;
482 static const long double lgam2r5b = 1.4901722919159632494669682701924320137696E-5L;
483 #define NRN2r5 8
484 static const long double RN2r5[NRN2r5 + 1] =
485 {
486 -4.676454313888335499356699817678862233205E9L,
487 -9.361888347911187924389905984624216340639E9L,
488 -7.695353600835685037920815799526540237703E9L,
489 -3.364370100981509060441853085968900734521E9L,
490 -8.449902011848163568670361316804900559863E8L,
491 -1.225249050950801905108001246436783022179E8L,
492 -9.732972931077110161639900388121650470926E6L,
493 -3.695711763932153505623248207576425983573E5L,
494 -4.717341584067827676530426007495274711306E3L
495 };
496 #define NRD2r5 8
497 static const long double RD2r5[NRD2r5 + 1] =
498 {
499 -6.650657966618993679456019224416926875619E9L,
500 -1.099511409330635807899718829033488771623E10L,
501 -7.482546968307837168164311101447116903148E9L,
502 -2.702967190056506495988922973755870557217E9L,
503 -5.570008176482922704972943389590409280950E8L,
504 -6.536934032192792470926310043166993233231E7L,
505 -4.101991193844953082400035444146067511725E6L,
506 -1.174082735875715802334430481065526664020E5L,
507 -9.932840389994157592102947657277692978511E2L
508 /* 1.0E0L */
509 };
510
511
512 /* log gamma(x+2) = x P(x)/Q(x)
513 -0.125 <= x <= +0.375
514 1.875 <= x+2 <= 2.375
515 Peak relative error 4.6e-36 */
516 #define NRN2 9
517 static const long double RN2[NRN2 + 1] =
518 {
519 -3.716661929737318153526921358113793421524E9L,
520 -1.138816715030710406922819131397532331321E10L,
521 -1.421017419363526524544402598734013569950E10L,
522 -9.510432842542519665483662502132010331451E9L,
523 -3.747528562099410197957514973274474767329E9L,
524 -8.923565763363912474488712255317033616626E8L,
525 -1.261396653700237624185350402781338231697E8L,
526 -9.918402520255661797735331317081425749014E6L,
527 -3.753996255897143855113273724233104768831E5L,
528 -4.778761333044147141559311805999540765612E3L
529 };
530 #define NRD2 9
531 static const long double RD2[NRD2 + 1] =
532 {
533 -8.790916836764308497770359421351673950111E9L,
534 -2.023108608053212516399197678553737477486E10L,
535 -1.958067901852022239294231785363504458367E10L,
536 -1.035515043621003101254252481625188704529E10L,
537 -3.253884432621336737640841276619272224476E9L,
538 -6.186383531162456814954947669274235815544E8L,
539 -6.932557847749518463038934953605969951466E7L,
540 -4.240731768287359608773351626528479703758E6L,
541 -1.197343995089189188078944689846348116630E5L,
542 -1.004622911670588064824904487064114090920E3L
543 /* 1.0E0 */
544 };
545
546
547 /* log gamma(x+1.75) = log gamma(1.75) + x P(x)/Q(x)
548 -0.125 <= x <= +0.125
549 1.625 <= x+1.75 <= 1.875
550 Peak relative error 9.2e-37 */
551 static const long double lgam1r75a = -8.441162109375E-2L;
552 static const long double lgam1r75b = 1.0500073264444042213965868602268256157604E-5L;
553 #define NRN1r75 8
554 static const long double RN1r75[NRN1r75 + 1] =
555 {
556 -5.221061693929833937710891646275798251513E7L,
557 -2.052466337474314812817883030472496436993E8L,
558 -2.952718275974940270675670705084125640069E8L,
559 -2.132294039648116684922965964126389017840E8L,
560 -8.554103077186505960591321962207519908489E7L,
561 -1.940250901348870867323943119132071960050E7L,
562 -2.379394147112756860769336400290402208435E6L,
563 -1.384060879999526222029386539622255797389E5L,
564 -2.698453601378319296159355612094598695530E3L
565 };
566 #define NRD1r75 8
567 static const long double RD1r75[NRD1r75 + 1] =
568 {
569 -2.109754689501705828789976311354395393605E8L,
570 -5.036651829232895725959911504899241062286E8L,
571 -4.954234699418689764943486770327295098084E8L,
572 -2.589558042412676610775157783898195339410E8L,
573 -7.731476117252958268044969614034776883031E7L,
574 -1.316721702252481296030801191240867486965E7L,
575 -1.201296501404876774861190604303728810836E6L,
576 -5.007966406976106636109459072523610273928E4L,
577 -6.155817990560743422008969155276229018209E2L
578 /* 1.0E0L */
579 };
580
581
582 /* log gamma(x+x0) = y0 + x^2 P(x)/Q(x)
583 -0.0867 <= x <= +0.1634
584 1.374932... <= x+x0 <= 1.625032...
585 Peak relative error 4.0e-36 */
586 static const long double x0a = 1.4616241455078125L;
587 static const long double x0b = 7.9994605498412626595423257213002588621246E-6L;
588 static const long double y0a = -1.21490478515625E-1L;
589 static const long double y0b = 4.1879797753919044854428223084178486438269E-6L;
590 #define NRN1r5 8
591 static const long double RN1r5[NRN1r5 + 1] =
592 {
593 6.827103657233705798067415468881313128066E5L,
594 1.910041815932269464714909706705242148108E6L,
595 2.194344176925978377083808566251427771951E6L,
596 1.332921400100891472195055269688876427962E6L,
597 4.589080973377307211815655093824787123508E5L,
598 8.900334161263456942727083580232613796141E4L,
599 9.053840838306019753209127312097612455236E3L,
600 4.053367147553353374151852319743594873771E2L,
601 5.040631576303952022968949605613514584950E0L
602 };
603 #define NRD1r5 8
604 static const long double RD1r5[NRD1r5 + 1] =
605 {
606 1.411036368843183477558773688484699813355E6L,
607 4.378121767236251950226362443134306184849E6L,
608 5.682322855631723455425929877581697918168E6L,
609 3.999065731556977782435009349967042222375E6L,
610 1.653651390456781293163585493620758410333E6L,
611 4.067774359067489605179546964969435858311E5L,
612 5.741463295366557346748361781768833633256E4L,
613 4.226404539738182992856094681115746692030E3L,
614 1.316980975410327975566999780608618774469E2L,
615 /* 1.0E0L */
616 };
617
618
619 /* log gamma(x+1.25) = log gamma(1.25) + x P(x)/Q(x)
620 -.125 <= x <= +.125
621 1.125 <= x+1.25 <= 1.375
622 Peak relative error = 4.9e-36 */
623 static const long double lgam1r25a = -9.82818603515625E-2L;
624 static const long double lgam1r25b = 1.0023929749338536146197303364159774377296E-5L;
625 #define NRN1r25 9
626 static const long double RN1r25[NRN1r25 + 1] =
627 {
628 -9.054787275312026472896002240379580536760E4L,
629 -8.685076892989927640126560802094680794471E4L,
630 2.797898965448019916967849727279076547109E5L,
631 6.175520827134342734546868356396008898299E5L,
632 5.179626599589134831538516906517372619641E5L,
633 2.253076616239043944538380039205558242161E5L,
634 5.312653119599957228630544772499197307195E4L,
635 6.434329437514083776052669599834938898255E3L,
636 3.385414416983114598582554037612347549220E2L,
637 4.907821957946273805080625052510832015792E0L
638 };
639 #define NRD1r25 8
640 static const long double RD1r25[NRD1r25 + 1] =
641 {
642 3.980939377333448005389084785896660309000E5L,
643 1.429634893085231519692365775184490465542E6L,
644 2.145438946455476062850151428438668234336E6L,
645 1.743786661358280837020848127465970357893E6L,
646 8.316364251289743923178092656080441655273E5L,
647 2.355732939106812496699621491135458324294E5L,
648 3.822267399625696880571810137601310855419E4L,
649 3.228463206479133236028576845538387620856E3L,
650 1.152133170470059555646301189220117965514E2L
651 /* 1.0E0L */
652 };
653
654
655 /* log gamma(x + 1) = x P(x)/Q(x)
656 0.0 <= x <= +0.125
657 1.0 <= x+1 <= 1.125
658 Peak relative error 1.1e-35 */
659 #define NRN1 8
660 static const long double RN1[NRN1 + 1] =
661 {
662 -9.987560186094800756471055681088744738818E3L,
663 -2.506039379419574361949680225279376329742E4L,
664 -1.386770737662176516403363873617457652991E4L,
665 1.439445846078103202928677244188837130744E4L,
666 2.159612048879650471489449668295139990693E4L,
667 1.047439813638144485276023138173676047079E4L,
668 2.250316398054332592560412486630769139961E3L,
669 1.958510425467720733041971651126443864041E2L,
670 4.516830313569454663374271993200291219855E0L
671 };
672 #define NRD1 7
673 static const long double RD1[NRD1 + 1] =
674 {
675 1.730299573175751778863269333703788214547E4L,
676 6.807080914851328611903744668028014678148E4L,
677 1.090071629101496938655806063184092302439E5L,
678 9.124354356415154289343303999616003884080E4L,
679 4.262071638655772404431164427024003253954E4L,
680 1.096981664067373953673982635805821283581E4L,
681 1.431229503796575892151252708527595787588E3L,
682 7.734110684303689320830401788262295992921E1L
683 /* 1.0E0 */
684 };
685
686
687 /* log gamma(x + 1) = x P(x)/Q(x)
688 -0.125 <= x <= 0
689 0.875 <= x+1 <= 1.0
690 Peak relative error 7.0e-37 */
691 #define NRNr9 8
692 static const long double RNr9[NRNr9 + 1] =
693 {
694 4.441379198241760069548832023257571176884E5L,
695 1.273072988367176540909122090089580368732E6L,
696 9.732422305818501557502584486510048387724E5L,
697 -5.040539994443998275271644292272870348684E5L,
698 -1.208719055525609446357448132109723786736E6L,
699 -7.434275365370936547146540554419058907156E5L,
700 -2.075642969983377738209203358199008185741E5L,
701 -2.565534860781128618589288075109372218042E4L,
702 -1.032901669542994124131223797515913955938E3L,
703 };
704 #define NRDr9 8
705 static const long double RDr9[NRDr9 + 1] =
706 {
707 -7.694488331323118759486182246005193998007E5L,
708 -3.301918855321234414232308938454112213751E6L,
709 -5.856830900232338906742924836032279404702E6L,
710 -5.540672519616151584486240871424021377540E6L,
711 -3.006530901041386626148342989181721176919E6L,
712 -9.350378280513062139466966374330795935163E5L,
713 -1.566179100031063346901755685375732739511E5L,
714 -1.205016539620260779274902967231510804992E4L,
715 -2.724583156305709733221564484006088794284E2L
716 /* 1.0E0 */
717 };
718
719
720 /* Evaluate P[n] x^n + P[n-1] x^(n-1) + ... + P[0] */
721
722 static long double
neval(long double x,const long double * p,int n)723 neval (long double x, const long double *p, int n)
724 {
725 long double y;
726
727 p += n;
728 y = *p--;
729 do
730 {
731 y = y * x + *p--;
732 }
733 while (--n > 0);
734 return y;
735 }
736
737
738 /* Evaluate x^n+1 + P[n] x^(n) + P[n-1] x^(n-1) + ... + P[0] */
739
740 static long double
deval(long double x,const long double * p,int n)741 deval (long double x, const long double *p, int n)
742 {
743 long double y;
744
745 p += n;
746 y = x + *p--;
747 do
748 {
749 y = y * x + *p--;
750 }
751 while (--n > 0);
752 return y;
753 }
754
755
756 long double
lgammal_r(long double x,int * signgamp)757 lgammal_r(long double x, int *signgamp)
758 {
759 long double p, q, w, z, nx;
760 int i, nn;
761
762 *signgamp = 1;
763
764 if (!isfinite (x))
765 return x * x;
766
767 if (x == 0.0L)
768 {
769 if (signbit (x))
770 *signgamp = -1;
771 return __math_divzerol(0);
772 }
773
774 if (x < 0.0L)
775 {
776 int discard;
777 q = -x;
778 p = floorl (q);
779 if (p == q)
780 return __math_divzerol(0);
781 i = p;
782 if ((i & 1) == 0)
783 *signgamp = -1;
784 else
785 *signgamp = 1;
786 z = q - p;
787 if (z > 0.5L)
788 {
789 p += 1.0L;
790 z = p - q;
791 }
792 z = q * sinl (PIL * z);
793 if (z == 0.0L)
794 return __math_oflowl(*signgamp < 0);
795 w = lgammal_r (q, &discard);
796 z = logl (PIL / z) - w;
797 return (z);
798 }
799
800 if (x < 13.5L)
801 {
802 p = 0.0L;
803 nx = floorl (x + 0.5L);
804 nn = nx;
805 switch (nn)
806 {
807 case 0:
808 /* log gamma (x + 1) = log(x) + log gamma(x) */
809 if (x <= 0.125L)
810 {
811 p = x * neval (x, RN1, NRN1) / deval (x, RD1, NRD1);
812 }
813 else if (x <= 0.375L)
814 {
815 z = x - 0.25L;
816 p = z * neval (z, RN1r25, NRN1r25) / deval (z, RD1r25, NRD1r25);
817 p += lgam1r25b;
818 p += lgam1r25a;
819 }
820 else if (x <= 0.625L)
821 {
822 z = x + (1.0L - x0a);
823 z = z - x0b;
824 p = neval (z, RN1r5, NRN1r5) / deval (z, RD1r5, NRD1r5);
825 p = p * z * z;
826 p = p + y0b;
827 p = p + y0a;
828 }
829 else if (x <= 0.875L)
830 {
831 z = x - 0.75L;
832 p = z * neval (z, RN1r75, NRN1r75) / deval (z, RD1r75, NRD1r75);
833 p += lgam1r75b;
834 p += lgam1r75a;
835 }
836 else
837 {
838 z = x - 1.0L;
839 p = z * neval (z, RN2, NRN2) / deval (z, RD2, NRD2);
840 }
841 p = p - logl (x);
842 break;
843
844 case 1:
845 if (x < 0.875L)
846 {
847 if (x <= 0.625L)
848 {
849 z = x + (1.0L - x0a);
850 z = z - x0b;
851 p = neval (z, RN1r5, NRN1r5) / deval (z, RD1r5, NRD1r5);
852 p = p * z * z;
853 p = p + y0b;
854 p = p + y0a;
855 }
856 else if (x <= 0.875L)
857 {
858 z = x - 0.75L;
859 p = z * neval (z, RN1r75, NRN1r75)
860 / deval (z, RD1r75, NRD1r75);
861 p += lgam1r75b;
862 p += lgam1r75a;
863 }
864 else
865 {
866 z = x - 1.0L;
867 p = z * neval (z, RN2, NRN2) / deval (z, RD2, NRD2);
868 }
869 p = p - logl (x);
870 }
871 else if (x < 1.0L)
872 {
873 z = x - 1.0L;
874 p = z * neval (z, RNr9, NRNr9) / deval (z, RDr9, NRDr9);
875 }
876 else if (x == 1.0L)
877 p = 0.0L;
878 else if (x <= 1.125L)
879 {
880 z = x - 1.0L;
881 p = z * neval (z, RN1, NRN1) / deval (z, RD1, NRD1);
882 }
883 else if (x <= 1.375L)
884 {
885 z = x - 1.25L;
886 p = z * neval (z, RN1r25, NRN1r25) / deval (z, RD1r25, NRD1r25);
887 p += lgam1r25b;
888 p += lgam1r25a;
889 }
890 else
891 {
892 /* 1.375 <= x+x0 <= 1.625 */
893 z = x - x0a;
894 z = z - x0b;
895 p = neval (z, RN1r5, NRN1r5) / deval (z, RD1r5, NRD1r5);
896 p = p * z * z;
897 p = p + y0b;
898 p = p + y0a;
899 }
900 break;
901
902 case 2:
903 if (x < 1.625L)
904 {
905 z = x - x0a;
906 z = z - x0b;
907 p = neval (z, RN1r5, NRN1r5) / deval (z, RD1r5, NRD1r5);
908 p = p * z * z;
909 p = p + y0b;
910 p = p + y0a;
911 }
912 else if (x < 1.875L)
913 {
914 z = x - 1.75L;
915 p = z * neval (z, RN1r75, NRN1r75) / deval (z, RD1r75, NRD1r75);
916 p += lgam1r75b;
917 p += lgam1r75a;
918 }
919 else if (x == 2.0L)
920 p = 0.0L;
921 else if (x < 2.375L)
922 {
923 z = x - 2.0L;
924 p = z * neval (z, RN2, NRN2) / deval (z, RD2, NRD2);
925 }
926 else
927 {
928 z = x - 2.5L;
929 p = z * neval (z, RN2r5, NRN2r5) / deval (z, RD2r5, NRD2r5);
930 p += lgam2r5b;
931 p += lgam2r5a;
932 }
933 break;
934
935 case 3:
936 if (x < 2.75L)
937 {
938 z = x - 2.5L;
939 p = z * neval (z, RN2r5, NRN2r5) / deval (z, RD2r5, NRD2r5);
940 p += lgam2r5b;
941 p += lgam2r5a;
942 }
943 else
944 {
945 z = x - 3.0L;
946 p = z * neval (z, RN3, NRN3) / deval (z, RD3, NRD3);
947 p += lgam3b;
948 p += lgam3a;
949 }
950 break;
951
952 case 4:
953 z = x - 4.0L;
954 p = z * neval (z, RN4, NRN4) / deval (z, RD4, NRD4);
955 p += lgam4b;
956 p += lgam4a;
957 break;
958
959 case 5:
960 z = x - 5.0L;
961 p = z * neval (z, RN5, NRN5) / deval (z, RD5, NRD5);
962 p += lgam5b;
963 p += lgam5a;
964 break;
965
966 case 6:
967 z = x - 6.0L;
968 p = z * neval (z, RN6, NRN6) / deval (z, RD6, NRD6);
969 p += lgam6b;
970 p += lgam6a;
971 break;
972
973 case 7:
974 z = x - 7.0L;
975 p = z * neval (z, RN7, NRN7) / deval (z, RD7, NRD7);
976 p += lgam7b;
977 p += lgam7a;
978 break;
979
980 case 8:
981 z = x - 8.0L;
982 p = z * neval (z, RN8, NRN8) / deval (z, RD8, NRD8);
983 p += lgam8b;
984 p += lgam8a;
985 break;
986
987 case 9:
988 z = x - 9.0L;
989 p = z * neval (z, RN9, NRN9) / deval (z, RD9, NRD9);
990 p += lgam9b;
991 p += lgam9a;
992 break;
993
994 case 10:
995 z = x - 10.0L;
996 p = z * neval (z, RN10, NRN10) / deval (z, RD10, NRD10);
997 p += lgam10b;
998 p += lgam10a;
999 break;
1000
1001 case 11:
1002 z = x - 11.0L;
1003 p = z * neval (z, RN11, NRN11) / deval (z, RD11, NRD11);
1004 p += lgam11b;
1005 p += lgam11a;
1006 break;
1007
1008 case 12:
1009 z = x - 12.0L;
1010 p = z * neval (z, RN12, NRN12) / deval (z, RD12, NRD12);
1011 p += lgam12b;
1012 p += lgam12a;
1013 break;
1014
1015 case 13:
1016 z = x - 13.0L;
1017 p = z * neval (z, RN13, NRN13) / deval (z, RD13, NRD13);
1018 p += lgam13b;
1019 p += lgam13a;
1020 break;
1021 }
1022 return p;
1023 }
1024
1025 if (x > MAXLGM)
1026 return __math_oflowl(*signgamp < 0);
1027
1028 q = ls2pi - x;
1029 q = (x - 0.5L) * logl (x) + q;
1030 if (x > 1.0e18L)
1031 return (q);
1032
1033 p = 1.0L / (x * x);
1034 q += neval (p, RASY, NRASY) / x;
1035 return (q);
1036 }
1037
