1 | /* Natural logarithm of gamma function. IBM Extended Precision version. |
2 | Copyright 2001 by Stephen L. Moshier <moshier@na-net.ornl.gov> |
3 | |
4 | This library is free software; you can redistribute it and/or |
5 | modify it under the terms of the GNU Lesser General Public |
6 | License as published by the Free Software Foundation; either |
7 | version 2.1 of the License, or (at your option) any later version. |
8 | |
9 | This library is distributed in the hope that it will be useful, |
10 | but WITHOUT ANY WARRANTY; without even the implied warranty of |
11 | MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU |
12 | Lesser General Public License for more details. |
13 | |
14 | You should have received a copy of the GNU Lesser General Public |
15 | License along with this library; if not, see |
16 | <https://www.gnu.org/licenses/>. */ |
17 | |
18 | /* This file was copied from sysdeps/ieee754/ldbl-128/e_lgammal_r.c. */ |
19 | |
20 | |
21 | #include <math.h> |
22 | #include <math_private.h> |
23 | #include <float.h> |
24 | #include <libm-alias-finite.h> |
25 | |
26 | static const long double PIL = 3.1415926535897932384626433832795028841972E0L; |
27 | static const long double MAXLGM = 0x5.d53649e2d469dbc1f01e99fd66p+1012L; |
28 | static const long double one = 1; |
29 | static const long double huge = LDBL_MAX; |
30 | |
31 | /* log gamma(x) = ( x - 0.5 ) * log(x) - x + LS2PI + 1/x P(1/x^2) |
32 | 1/x <= 0.0741 (x >= 13.495...) |
33 | Peak relative error 1.5e-36 */ |
34 | static const long double ls2pi = 9.1893853320467274178032973640561763986140E-1L; |
35 | #define NRASY 12 |
36 | static const long double RASY[NRASY + 1] = |
37 | { |
38 | 8.333333333333333333333333333310437112111E-2L, |
39 | -2.777777777777777777777774789556228296902E-3L, |
40 | 7.936507936507936507795933938448586499183E-4L, |
41 | -5.952380952380952041799269756378148574045E-4L, |
42 | 8.417508417507928904209891117498524452523E-4L, |
43 | -1.917526917481263997778542329739806086290E-3L, |
44 | 6.410256381217852504446848671499409919280E-3L, |
45 | -2.955064066900961649768101034477363301626E-2L, |
46 | 1.796402955865634243663453415388336954675E-1L, |
47 | -1.391522089007758553455753477688592767741E0L, |
48 | 1.326130089598399157988112385013829305510E1L, |
49 | -1.420412699593782497803472576479997819149E2L, |
50 | 1.218058922427762808938869872528846787020E3L |
51 | }; |
52 | |
53 | |
54 | /* log gamma(x+13) = log gamma(13) + x P(x)/Q(x) |
55 | -0.5 <= x <= 0.5 |
56 | 12.5 <= x+13 <= 13.5 |
57 | Peak relative error 1.1e-36 */ |
58 | static const long double lgam13a = 1.9987213134765625E1L; |
59 | static const long double lgam13b = 1.3608962611495173623870550785125024484248E-6L; |
60 | #define NRN13 7 |
61 | static const long double RN13[NRN13 + 1] = |
62 | { |
63 | 8.591478354823578150238226576156275285700E11L, |
64 | 2.347931159756482741018258864137297157668E11L, |
65 | 2.555408396679352028680662433943000804616E10L, |
66 | 1.408581709264464345480765758902967123937E9L, |
67 | 4.126759849752613822953004114044451046321E7L, |
68 | 6.133298899622688505854211579222889943778E5L, |
69 | 3.929248056293651597987893340755876578072E3L, |
70 | 6.850783280018706668924952057996075215223E0L |
71 | }; |
72 | #define NRD13 6 |
73 | static const long double RD13[NRD13 + 1] = |
74 | { |
75 | 3.401225382297342302296607039352935541669E11L, |
76 | 8.756765276918037910363513243563234551784E10L, |
77 | 8.873913342866613213078554180987647243903E9L, |
78 | 4.483797255342763263361893016049310017973E8L, |
79 | 1.178186288833066430952276702931512870676E7L, |
80 | 1.519928623743264797939103740132278337476E5L, |
81 | 7.989298844938119228411117593338850892311E2L |
82 | /* 1.0E0L */ |
83 | }; |
84 | |
85 | |
86 | /* log gamma(x+12) = log gamma(12) + x P(x)/Q(x) |
87 | -0.5 <= x <= 0.5 |
88 | 11.5 <= x+12 <= 12.5 |
89 | Peak relative error 4.1e-36 */ |
90 | static const long double lgam12a = 1.75023040771484375E1L; |
91 | static const long double lgam12b = 3.7687254483392876529072161996717039575982E-6L; |
92 | #define NRN12 7 |
93 | static const long double RN12[NRN12 + 1] = |
94 | { |
95 | 4.709859662695606986110997348630997559137E11L, |
96 | 1.398713878079497115037857470168777995230E11L, |
97 | 1.654654931821564315970930093932954900867E10L, |
98 | 9.916279414876676861193649489207282144036E8L, |
99 | 3.159604070526036074112008954113411389879E7L, |
100 | 5.109099197547205212294747623977502492861E5L, |
101 | 3.563054878276102790183396740969279826988E3L, |
102 | 6.769610657004672719224614163196946862747E0L |
103 | }; |
104 | #define NRD12 6 |
105 | static const long double RD12[NRD12 + 1] = |
106 | { |
107 | 1.928167007860968063912467318985802726613E11L, |
108 | 5.383198282277806237247492369072266389233E10L, |
109 | 5.915693215338294477444809323037871058363E9L, |
110 | 3.241438287570196713148310560147925781342E8L, |
111 | 9.236680081763754597872713592701048455890E6L, |
112 | 1.292246897881650919242713651166596478850E5L, |
113 | 7.366532445427159272584194816076600211171E2L |
114 | /* 1.0E0L */ |
115 | }; |
116 | |
117 | |
118 | /* log gamma(x+11) = log gamma(11) + x P(x)/Q(x) |
119 | -0.5 <= x <= 0.5 |
120 | 10.5 <= x+11 <= 11.5 |
121 | Peak relative error 1.8e-35 */ |
122 | static const long double lgam11a = 1.5104400634765625E1L; |
123 | static const long double lgam11b = 1.1938309890295225709329251070371882250744E-5L; |
124 | #define NRN11 7 |
125 | static const long double RN11[NRN11 + 1] = |
126 | { |
127 | 2.446960438029415837384622675816736622795E11L, |
128 | 7.955444974446413315803799763901729640350E10L, |
129 | 1.030555327949159293591618473447420338444E10L, |
130 | 6.765022131195302709153994345470493334946E8L, |
131 | 2.361892792609204855279723576041468347494E7L, |
132 | 4.186623629779479136428005806072176490125E5L, |
133 | 3.202506022088912768601325534149383594049E3L, |
134 | 6.681356101133728289358838690666225691363E0L |
135 | }; |
136 | #define NRD11 6 |
137 | static const long double RD11[NRD11 + 1] = |
138 | { |
139 | 1.040483786179428590683912396379079477432E11L, |
140 | 3.172251138489229497223696648369823779729E10L, |
141 | 3.806961885984850433709295832245848084614E9L, |
142 | 2.278070344022934913730015420611609620171E8L, |
143 | 7.089478198662651683977290023829391596481E6L, |
144 | 1.083246385105903533237139380509590158658E5L, |
145 | 6.744420991491385145885727942219463243597E2L |
146 | /* 1.0E0L */ |
147 | }; |
148 | |
149 | |
150 | /* log gamma(x+10) = log gamma(10) + x P(x)/Q(x) |
151 | -0.5 <= x <= 0.5 |
152 | 9.5 <= x+10 <= 10.5 |
153 | Peak relative error 5.4e-37 */ |
154 | static const long double lgam10a = 1.280181884765625E1L; |
155 | static const long double lgam10b = 8.6324252196112077178745667061642811492557E-6L; |
156 | #define NRN10 7 |
157 | static const long double RN10[NRN10 + 1] = |
158 | { |
159 | -1.239059737177249934158597996648808363783E14L, |
160 | -4.725899566371458992365624673357356908719E13L, |
161 | -7.283906268647083312042059082837754850808E12L, |
162 | -5.802855515464011422171165179767478794637E11L, |
163 | -2.532349691157548788382820303182745897298E10L, |
164 | -5.884260178023777312587193693477072061820E8L, |
165 | -6.437774864512125749845840472131829114906E6L, |
166 | -2.350975266781548931856017239843273049384E4L |
167 | }; |
168 | #define NRD10 7 |
169 | static const long double RD10[NRD10 + 1] = |
170 | { |
171 | -5.502645997581822567468347817182347679552E13L, |
172 | -1.970266640239849804162284805400136473801E13L, |
173 | -2.819677689615038489384974042561531409392E12L, |
174 | -2.056105863694742752589691183194061265094E11L, |
175 | -8.053670086493258693186307810815819662078E9L, |
176 | -1.632090155573373286153427982504851867131E8L, |
177 | -1.483575879240631280658077826889223634921E6L, |
178 | -4.002806669713232271615885826373550502510E3L |
179 | /* 1.0E0L */ |
180 | }; |
181 | |
182 | |
183 | /* log gamma(x+9) = log gamma(9) + x P(x)/Q(x) |
184 | -0.5 <= x <= 0.5 |
185 | 8.5 <= x+9 <= 9.5 |
186 | Peak relative error 3.6e-36 */ |
187 | static const long double lgam9a = 1.06045989990234375E1L; |
188 | static const long double lgam9b = 3.9037218127284172274007216547549861681400E-6L; |
189 | #define NRN9 7 |
190 | static const long double RN9[NRN9 + 1] = |
191 | { |
192 | -4.936332264202687973364500998984608306189E13L, |
193 | -2.101372682623700967335206138517766274855E13L, |
194 | -3.615893404644823888655732817505129444195E12L, |
195 | -3.217104993800878891194322691860075472926E11L, |
196 | -1.568465330337375725685439173603032921399E10L, |
197 | -4.073317518162025744377629219101510217761E8L, |
198 | -4.983232096406156139324846656819246974500E6L, |
199 | -2.036280038903695980912289722995505277253E4L |
200 | }; |
201 | #define NRD9 7 |
202 | static const long double RD9[NRD9 + 1] = |
203 | { |
204 | -2.306006080437656357167128541231915480393E13L, |
205 | -9.183606842453274924895648863832233799950E12L, |
206 | -1.461857965935942962087907301194381010380E12L, |
207 | -1.185728254682789754150068652663124298303E11L, |
208 | -5.166285094703468567389566085480783070037E9L, |
209 | -1.164573656694603024184768200787835094317E8L, |
210 | -1.177343939483908678474886454113163527909E6L, |
211 | -3.529391059783109732159524500029157638736E3L |
212 | /* 1.0E0L */ |
213 | }; |
214 | |
215 | |
216 | /* log gamma(x+8) = log gamma(8) + x P(x)/Q(x) |
217 | -0.5 <= x <= 0.5 |
218 | 7.5 <= x+8 <= 8.5 |
219 | Peak relative error 2.4e-37 */ |
220 | static const long double lgam8a = 8.525146484375E0L; |
221 | static const long double lgam8b = 1.4876690414300165531036347125050759667737E-5L; |
222 | #define NRN8 8 |
223 | static const long double RN8[NRN8 + 1] = |
224 | { |
225 | 6.600775438203423546565361176829139703289E11L, |
226 | 3.406361267593790705240802723914281025800E11L, |
227 | 7.222460928505293914746983300555538432830E10L, |
228 | 8.102984106025088123058747466840656458342E9L, |
229 | 5.157620015986282905232150979772409345927E8L, |
230 | 1.851445288272645829028129389609068641517E7L, |
231 | 3.489261702223124354745894067468953756656E5L, |
232 | 2.892095396706665774434217489775617756014E3L, |
233 | 6.596977510622195827183948478627058738034E0L |
234 | }; |
235 | #define NRD8 7 |
236 | static const long double RD8[NRD8 + 1] = |
237 | { |
238 | 3.274776546520735414638114828622673016920E11L, |
239 | 1.581811207929065544043963828487733970107E11L, |
240 | 3.108725655667825188135393076860104546416E10L, |
241 | 3.193055010502912617128480163681842165730E9L, |
242 | 1.830871482669835106357529710116211541839E8L, |
243 | 5.790862854275238129848491555068073485086E6L, |
244 | 9.305213264307921522842678835618803553589E4L, |
245 | 6.216974105861848386918949336819572333622E2L |
246 | /* 1.0E0L */ |
247 | }; |
248 | |
249 | |
250 | /* log gamma(x+7) = log gamma(7) + x P(x)/Q(x) |
251 | -0.5 <= x <= 0.5 |
252 | 6.5 <= x+7 <= 7.5 |
253 | Peak relative error 3.2e-36 */ |
254 | static const long double lgam7a = 6.5792388916015625E0L; |
255 | static const long double lgam7b = 1.2320408538495060178292903945321122583007E-5L; |
256 | #define NRN7 8 |
257 | static const long double RN7[NRN7 + 1] = |
258 | { |
259 | 2.065019306969459407636744543358209942213E11L, |
260 | 1.226919919023736909889724951708796532847E11L, |
261 | 2.996157990374348596472241776917953749106E10L, |
262 | 3.873001919306801037344727168434909521030E9L, |
263 | 2.841575255593761593270885753992732145094E8L, |
264 | 1.176342515359431913664715324652399565551E7L, |
265 | 2.558097039684188723597519300356028511547E5L, |
266 | 2.448525238332609439023786244782810774702E3L, |
267 | 6.460280377802030953041566617300902020435E0L |
268 | }; |
269 | #define NRD7 7 |
270 | static const long double RD7[NRD7 + 1] = |
271 | { |
272 | 1.102646614598516998880874785339049304483E11L, |
273 | 6.099297512712715445879759589407189290040E10L, |
274 | 1.372898136289611312713283201112060238351E10L, |
275 | 1.615306270420293159907951633566635172343E9L, |
276 | 1.061114435798489135996614242842561967459E8L, |
277 | 3.845638971184305248268608902030718674691E6L, |
278 | 7.081730675423444975703917836972720495507E4L, |
279 | 5.423122582741398226693137276201344096370E2L |
280 | /* 1.0E0L */ |
281 | }; |
282 | |
283 | |
284 | /* log gamma(x+6) = log gamma(6) + x P(x)/Q(x) |
285 | -0.5 <= x <= 0.5 |
286 | 5.5 <= x+6 <= 6.5 |
287 | Peak relative error 6.2e-37 */ |
288 | static const long double lgam6a = 4.7874908447265625E0L; |
289 | static const long double lgam6b = 8.9805548349424770093452324304839959231517E-7L; |
290 | #define NRN6 8 |
291 | static const long double RN6[NRN6 + 1] = |
292 | { |
293 | -3.538412754670746879119162116819571823643E13L, |
294 | -2.613432593406849155765698121483394257148E13L, |
295 | -8.020670732770461579558867891923784753062E12L, |
296 | -1.322227822931250045347591780332435433420E12L, |
297 | -1.262809382777272476572558806855377129513E11L, |
298 | -7.015006277027660872284922325741197022467E9L, |
299 | -2.149320689089020841076532186783055727299E8L, |
300 | -3.167210585700002703820077565539658995316E6L, |
301 | -1.576834867378554185210279285358586385266E4L |
302 | }; |
303 | #define NRD6 8 |
304 | static const long double RD6[NRD6 + 1] = |
305 | { |
306 | -2.073955870771283609792355579558899389085E13L, |
307 | -1.421592856111673959642750863283919318175E13L, |
308 | -4.012134994918353924219048850264207074949E12L, |
309 | -6.013361045800992316498238470888523722431E11L, |
310 | -5.145382510136622274784240527039643430628E10L, |
311 | -2.510575820013409711678540476918249524123E9L, |
312 | -6.564058379709759600836745035871373240904E7L, |
313 | -7.861511116647120540275354855221373571536E5L, |
314 | -2.821943442729620524365661338459579270561E3L |
315 | /* 1.0E0L */ |
316 | }; |
317 | |
318 | |
319 | /* log gamma(x+5) = log gamma(5) + x P(x)/Q(x) |
320 | -0.5 <= x <= 0.5 |
321 | 4.5 <= x+5 <= 5.5 |
322 | Peak relative error 3.4e-37 */ |
323 | static const long double lgam5a = 3.17803955078125E0L; |
324 | static const long double lgam5b = 1.4279566695619646941601297055408873990961E-5L; |
325 | #define NRN5 9 |
326 | static const long double RN5[NRN5 + 1] = |
327 | { |
328 | 2.010952885441805899580403215533972172098E11L, |
329 | 1.916132681242540921354921906708215338584E11L, |
330 | 7.679102403710581712903937970163206882492E10L, |
331 | 1.680514903671382470108010973615268125169E10L, |
332 | 2.181011222911537259440775283277711588410E9L, |
333 | 1.705361119398837808244780667539728356096E8L, |
334 | 7.792391565652481864976147945997033946360E6L, |
335 | 1.910741381027985291688667214472560023819E5L, |
336 | 2.088138241893612679762260077783794329559E3L, |
337 | 6.330318119566998299106803922739066556550E0L |
338 | }; |
339 | #define NRD5 8 |
340 | static const long double RD5[NRD5 + 1] = |
341 | { |
342 | 1.335189758138651840605141370223112376176E11L, |
343 | 1.174130445739492885895466097516530211283E11L, |
344 | 4.308006619274572338118732154886328519910E10L, |
345 | 8.547402888692578655814445003283720677468E9L, |
346 | 9.934628078575618309542580800421370730906E8L, |
347 | 6.847107420092173812998096295422311820672E7L, |
348 | 2.698552646016599923609773122139463150403E6L, |
349 | 5.526516251532464176412113632726150253215E4L, |
350 | 4.772343321713697385780533022595450486932E2L |
351 | /* 1.0E0L */ |
352 | }; |
353 | |
354 | |
355 | /* log gamma(x+4) = log gamma(4) + x P(x)/Q(x) |
356 | -0.5 <= x <= 0.5 |
357 | 3.5 <= x+4 <= 4.5 |
358 | Peak relative error 6.7e-37 */ |
359 | static const long double lgam4a = 1.791748046875E0L; |
360 | static const long double lgam4b = 1.1422353055000812477358380702272722990692E-5L; |
361 | #define NRN4 9 |
362 | static const long double RN4[NRN4 + 1] = |
363 | { |
364 | -1.026583408246155508572442242188887829208E13L, |
365 | -1.306476685384622809290193031208776258809E13L, |
366 | -7.051088602207062164232806511992978915508E12L, |
367 | -2.100849457735620004967624442027793656108E12L, |
368 | -3.767473790774546963588549871673843260569E11L, |
369 | -4.156387497364909963498394522336575984206E10L, |
370 | -2.764021460668011732047778992419118757746E9L, |
371 | -1.036617204107109779944986471142938641399E8L, |
372 | -1.895730886640349026257780896972598305443E6L, |
373 | -1.180509051468390914200720003907727988201E4L |
374 | }; |
375 | #define NRD4 9 |
376 | static const long double RD4[NRD4 + 1] = |
377 | { |
378 | -8.172669122056002077809119378047536240889E12L, |
379 | -9.477592426087986751343695251801814226960E12L, |
380 | -4.629448850139318158743900253637212801682E12L, |
381 | -1.237965465892012573255370078308035272942E12L, |
382 | -1.971624313506929845158062177061297598956E11L, |
383 | -1.905434843346570533229942397763361493610E10L, |
384 | -1.089409357680461419743730978512856675984E9L, |
385 | -3.416703082301143192939774401370222822430E7L, |
386 | -4.981791914177103793218433195857635265295E5L, |
387 | -2.192507743896742751483055798411231453733E3L |
388 | /* 1.0E0L */ |
389 | }; |
390 | |
391 | |
392 | /* log gamma(x+3) = log gamma(3) + x P(x)/Q(x) |
393 | -0.25 <= x <= 0.5 |
394 | 2.75 <= x+3 <= 3.5 |
395 | Peak relative error 6.0e-37 */ |
396 | static const long double lgam3a = 6.93145751953125E-1L; |
397 | static const long double lgam3b = 1.4286068203094172321214581765680755001344E-6L; |
398 | |
399 | #define NRN3 9 |
400 | static const long double RN3[NRN3 + 1] = |
401 | { |
402 | -4.813901815114776281494823863935820876670E11L, |
403 | -8.425592975288250400493910291066881992620E11L, |
404 | -6.228685507402467503655405482985516909157E11L, |
405 | -2.531972054436786351403749276956707260499E11L, |
406 | -6.170200796658926701311867484296426831687E10L, |
407 | -9.211477458528156048231908798456365081135E9L, |
408 | -8.251806236175037114064561038908691305583E8L, |
409 | -4.147886355917831049939930101151160447495E7L, |
410 | -1.010851868928346082547075956946476932162E6L, |
411 | -8.333374463411801009783402800801201603736E3L |
412 | }; |
413 | #define NRD3 9 |
414 | static const long double RD3[NRD3 + 1] = |
415 | { |
416 | -5.216713843111675050627304523368029262450E11L, |
417 | -8.014292925418308759369583419234079164391E11L, |
418 | -5.180106858220030014546267824392678611990E11L, |
419 | -1.830406975497439003897734969120997840011E11L, |
420 | -3.845274631904879621945745960119924118925E10L, |
421 | -4.891033385370523863288908070309417710903E9L, |
422 | -3.670172254411328640353855768698287474282E8L, |
423 | -1.505316381525727713026364396635522516989E7L, |
424 | -2.856327162923716881454613540575964890347E5L, |
425 | -1.622140448015769906847567212766206894547E3L |
426 | /* 1.0E0L */ |
427 | }; |
428 | |
429 | |
430 | /* log gamma(x+2.5) = log gamma(2.5) + x P(x)/Q(x) |
431 | -0.125 <= x <= 0.25 |
432 | 2.375 <= x+2.5 <= 2.75 */ |
433 | static const long double lgam2r5a = 2.8466796875E-1L; |
434 | static const long double lgam2r5b = 1.4901722919159632494669682701924320137696E-5L; |
435 | #define NRN2r5 8 |
436 | static const long double RN2r5[NRN2r5 + 1] = |
437 | { |
438 | -4.676454313888335499356699817678862233205E9L, |
439 | -9.361888347911187924389905984624216340639E9L, |
440 | -7.695353600835685037920815799526540237703E9L, |
441 | -3.364370100981509060441853085968900734521E9L, |
442 | -8.449902011848163568670361316804900559863E8L, |
443 | -1.225249050950801905108001246436783022179E8L, |
444 | -9.732972931077110161639900388121650470926E6L, |
445 | -3.695711763932153505623248207576425983573E5L, |
446 | -4.717341584067827676530426007495274711306E3L |
447 | }; |
448 | #define NRD2r5 8 |
449 | static const long double RD2r5[NRD2r5 + 1] = |
450 | { |
451 | -6.650657966618993679456019224416926875619E9L, |
452 | -1.099511409330635807899718829033488771623E10L, |
453 | -7.482546968307837168164311101447116903148E9L, |
454 | -2.702967190056506495988922973755870557217E9L, |
455 | -5.570008176482922704972943389590409280950E8L, |
456 | -6.536934032192792470926310043166993233231E7L, |
457 | -4.101991193844953082400035444146067511725E6L, |
458 | -1.174082735875715802334430481065526664020E5L, |
459 | -9.932840389994157592102947657277692978511E2L |
460 | /* 1.0E0L */ |
461 | }; |
462 | |
463 | |
464 | /* log gamma(x+2) = x P(x)/Q(x) |
465 | -0.125 <= x <= +0.375 |
466 | 1.875 <= x+2 <= 2.375 |
467 | Peak relative error 4.6e-36 */ |
468 | #define NRN2 9 |
469 | static const long double RN2[NRN2 + 1] = |
470 | { |
471 | -3.716661929737318153526921358113793421524E9L, |
472 | -1.138816715030710406922819131397532331321E10L, |
473 | -1.421017419363526524544402598734013569950E10L, |
474 | -9.510432842542519665483662502132010331451E9L, |
475 | -3.747528562099410197957514973274474767329E9L, |
476 | -8.923565763363912474488712255317033616626E8L, |
477 | -1.261396653700237624185350402781338231697E8L, |
478 | -9.918402520255661797735331317081425749014E6L, |
479 | -3.753996255897143855113273724233104768831E5L, |
480 | -4.778761333044147141559311805999540765612E3L |
481 | }; |
482 | #define NRD2 9 |
483 | static const long double RD2[NRD2 + 1] = |
484 | { |
485 | -8.790916836764308497770359421351673950111E9L, |
486 | -2.023108608053212516399197678553737477486E10L, |
487 | -1.958067901852022239294231785363504458367E10L, |
488 | -1.035515043621003101254252481625188704529E10L, |
489 | -3.253884432621336737640841276619272224476E9L, |
490 | -6.186383531162456814954947669274235815544E8L, |
491 | -6.932557847749518463038934953605969951466E7L, |
492 | -4.240731768287359608773351626528479703758E6L, |
493 | -1.197343995089189188078944689846348116630E5L, |
494 | -1.004622911670588064824904487064114090920E3L |
495 | /* 1.0E0 */ |
496 | }; |
497 | |
498 | |
499 | /* log gamma(x+1.75) = log gamma(1.75) + x P(x)/Q(x) |
500 | -0.125 <= x <= +0.125 |
501 | 1.625 <= x+1.75 <= 1.875 |
502 | Peak relative error 9.2e-37 */ |
503 | static const long double lgam1r75a = -8.441162109375E-2L; |
504 | static const long double lgam1r75b = 1.0500073264444042213965868602268256157604E-5L; |
505 | #define NRN1r75 8 |
506 | static const long double RN1r75[NRN1r75 + 1] = |
507 | { |
508 | -5.221061693929833937710891646275798251513E7L, |
509 | -2.052466337474314812817883030472496436993E8L, |
510 | -2.952718275974940270675670705084125640069E8L, |
511 | -2.132294039648116684922965964126389017840E8L, |
512 | -8.554103077186505960591321962207519908489E7L, |
513 | -1.940250901348870867323943119132071960050E7L, |
514 | -2.379394147112756860769336400290402208435E6L, |
515 | -1.384060879999526222029386539622255797389E5L, |
516 | -2.698453601378319296159355612094598695530E3L |
517 | }; |
518 | #define NRD1r75 8 |
519 | static const long double RD1r75[NRD1r75 + 1] = |
520 | { |
521 | -2.109754689501705828789976311354395393605E8L, |
522 | -5.036651829232895725959911504899241062286E8L, |
523 | -4.954234699418689764943486770327295098084E8L, |
524 | -2.589558042412676610775157783898195339410E8L, |
525 | -7.731476117252958268044969614034776883031E7L, |
526 | -1.316721702252481296030801191240867486965E7L, |
527 | -1.201296501404876774861190604303728810836E6L, |
528 | -5.007966406976106636109459072523610273928E4L, |
529 | -6.155817990560743422008969155276229018209E2L |
530 | /* 1.0E0L */ |
531 | }; |
532 | |
533 | |
534 | /* log gamma(x+x0) = y0 + x^2 P(x)/Q(x) |
535 | -0.0867 <= x <= +0.1634 |
536 | 1.374932... <= x+x0 <= 1.625032... |
537 | Peak relative error 4.0e-36 */ |
538 | static const long double x0a = 1.4616241455078125L; |
539 | static const long double x0b = 7.9994605498412626595423257213002588621246E-6L; |
540 | static const long double y0a = -1.21490478515625E-1L; |
541 | static const long double y0b = 4.1879797753919044854428223084178486438269E-6L; |
542 | #define NRN1r5 8 |
543 | static const long double RN1r5[NRN1r5 + 1] = |
544 | { |
545 | 6.827103657233705798067415468881313128066E5L, |
546 | 1.910041815932269464714909706705242148108E6L, |
547 | 2.194344176925978377083808566251427771951E6L, |
548 | 1.332921400100891472195055269688876427962E6L, |
549 | 4.589080973377307211815655093824787123508E5L, |
550 | 8.900334161263456942727083580232613796141E4L, |
551 | 9.053840838306019753209127312097612455236E3L, |
552 | 4.053367147553353374151852319743594873771E2L, |
553 | 5.040631576303952022968949605613514584950E0L |
554 | }; |
555 | #define NRD1r5 8 |
556 | static const long double RD1r5[NRD1r5 + 1] = |
557 | { |
558 | 1.411036368843183477558773688484699813355E6L, |
559 | 4.378121767236251950226362443134306184849E6L, |
560 | 5.682322855631723455425929877581697918168E6L, |
561 | 3.999065731556977782435009349967042222375E6L, |
562 | 1.653651390456781293163585493620758410333E6L, |
563 | 4.067774359067489605179546964969435858311E5L, |
564 | 5.741463295366557346748361781768833633256E4L, |
565 | 4.226404539738182992856094681115746692030E3L, |
566 | 1.316980975410327975566999780608618774469E2L, |
567 | /* 1.0E0L */ |
568 | }; |
569 | |
570 | |
571 | /* log gamma(x+1.25) = log gamma(1.25) + x P(x)/Q(x) |
572 | -.125 <= x <= +.125 |
573 | 1.125 <= x+1.25 <= 1.375 |
574 | Peak relative error = 4.9e-36 */ |
575 | static const long double lgam1r25a = -9.82818603515625E-2L; |
576 | static const long double lgam1r25b = 1.0023929749338536146197303364159774377296E-5L; |
577 | #define NRN1r25 9 |
578 | static const long double RN1r25[NRN1r25 + 1] = |
579 | { |
580 | -9.054787275312026472896002240379580536760E4L, |
581 | -8.685076892989927640126560802094680794471E4L, |
582 | 2.797898965448019916967849727279076547109E5L, |
583 | 6.175520827134342734546868356396008898299E5L, |
584 | 5.179626599589134831538516906517372619641E5L, |
585 | 2.253076616239043944538380039205558242161E5L, |
586 | 5.312653119599957228630544772499197307195E4L, |
587 | 6.434329437514083776052669599834938898255E3L, |
588 | 3.385414416983114598582554037612347549220E2L, |
589 | 4.907821957946273805080625052510832015792E0L |
590 | }; |
591 | #define NRD1r25 8 |
592 | static const long double RD1r25[NRD1r25 + 1] = |
593 | { |
594 | 3.980939377333448005389084785896660309000E5L, |
595 | 1.429634893085231519692365775184490465542E6L, |
596 | 2.145438946455476062850151428438668234336E6L, |
597 | 1.743786661358280837020848127465970357893E6L, |
598 | 8.316364251289743923178092656080441655273E5L, |
599 | 2.355732939106812496699621491135458324294E5L, |
600 | 3.822267399625696880571810137601310855419E4L, |
601 | 3.228463206479133236028576845538387620856E3L, |
602 | 1.152133170470059555646301189220117965514E2L |
603 | /* 1.0E0L */ |
604 | }; |
605 | |
606 | |
607 | /* log gamma(x + 1) = x P(x)/Q(x) |
608 | 0.0 <= x <= +0.125 |
609 | 1.0 <= x+1 <= 1.125 |
610 | Peak relative error 1.1e-35 */ |
611 | #define NRN1 8 |
612 | static const long double RN1[NRN1 + 1] = |
613 | { |
614 | -9.987560186094800756471055681088744738818E3L, |
615 | -2.506039379419574361949680225279376329742E4L, |
616 | -1.386770737662176516403363873617457652991E4L, |
617 | 1.439445846078103202928677244188837130744E4L, |
618 | 2.159612048879650471489449668295139990693E4L, |
619 | 1.047439813638144485276023138173676047079E4L, |
620 | 2.250316398054332592560412486630769139961E3L, |
621 | 1.958510425467720733041971651126443864041E2L, |
622 | 4.516830313569454663374271993200291219855E0L |
623 | }; |
624 | #define NRD1 7 |
625 | static const long double RD1[NRD1 + 1] = |
626 | { |
627 | 1.730299573175751778863269333703788214547E4L, |
628 | 6.807080914851328611903744668028014678148E4L, |
629 | 1.090071629101496938655806063184092302439E5L, |
630 | 9.124354356415154289343303999616003884080E4L, |
631 | 4.262071638655772404431164427024003253954E4L, |
632 | 1.096981664067373953673982635805821283581E4L, |
633 | 1.431229503796575892151252708527595787588E3L, |
634 | 7.734110684303689320830401788262295992921E1L |
635 | /* 1.0E0 */ |
636 | }; |
637 | |
638 | |
639 | /* log gamma(x + 1) = x P(x)/Q(x) |
640 | -0.125 <= x <= 0 |
641 | 0.875 <= x+1 <= 1.0 |
642 | Peak relative error 7.0e-37 */ |
643 | #define NRNr9 8 |
644 | static const long double RNr9[NRNr9 + 1] = |
645 | { |
646 | 4.441379198241760069548832023257571176884E5L, |
647 | 1.273072988367176540909122090089580368732E6L, |
648 | 9.732422305818501557502584486510048387724E5L, |
649 | -5.040539994443998275271644292272870348684E5L, |
650 | -1.208719055525609446357448132109723786736E6L, |
651 | -7.434275365370936547146540554419058907156E5L, |
652 | -2.075642969983377738209203358199008185741E5L, |
653 | -2.565534860781128618589288075109372218042E4L, |
654 | -1.032901669542994124131223797515913955938E3L, |
655 | }; |
656 | #define NRDr9 8 |
657 | static const long double RDr9[NRDr9 + 1] = |
658 | { |
659 | -7.694488331323118759486182246005193998007E5L, |
660 | -3.301918855321234414232308938454112213751E6L, |
661 | -5.856830900232338906742924836032279404702E6L, |
662 | -5.540672519616151584486240871424021377540E6L, |
663 | -3.006530901041386626148342989181721176919E6L, |
664 | -9.350378280513062139466966374330795935163E5L, |
665 | -1.566179100031063346901755685375732739511E5L, |
666 | -1.205016539620260779274902967231510804992E4L, |
667 | -2.724583156305709733221564484006088794284E2L |
668 | /* 1.0E0 */ |
669 | }; |
670 | |
671 | |
672 | /* Evaluate P[n] x^n + P[n-1] x^(n-1) + ... + P[0] */ |
673 | |
674 | static long double |
675 | neval (long double x, const long double *p, int n) |
676 | { |
677 | long double y; |
678 | |
679 | p += n; |
680 | y = *p--; |
681 | do |
682 | { |
683 | y = y * x + *p--; |
684 | } |
685 | while (--n > 0); |
686 | return y; |
687 | } |
688 | |
689 | |
690 | /* Evaluate x^n+1 + P[n] x^(n) + P[n-1] x^(n-1) + ... + P[0] */ |
691 | |
692 | static long double |
693 | deval (long double x, const long double *p, int n) |
694 | { |
695 | long double y; |
696 | |
697 | p += n; |
698 | y = x + *p--; |
699 | do |
700 | { |
701 | y = y * x + *p--; |
702 | } |
703 | while (--n > 0); |
704 | return y; |
705 | } |
706 | |
707 | |
708 | long double |
709 | __ieee754_lgammal_r (long double x, int *signgamp) |
710 | { |
711 | long double p, q, w, z, nx; |
712 | int i, nn; |
713 | |
714 | *signgamp = 1; |
715 | |
716 | if (! isfinite (x)) |
717 | return x * x; |
718 | |
719 | if (x == 0) |
720 | { |
721 | if (signbit (x)) |
722 | *signgamp = -1; |
723 | } |
724 | |
725 | if (x < 0) |
726 | { |
727 | if (x < -2 && x > -48) |
728 | return __lgamma_negl (x, signgamp); |
729 | q = -x; |
730 | p = floorl (q); |
731 | if (p == q) |
732 | return (one / fabsl (x: p - p)); |
733 | long double halfp = p * 0.5L; |
734 | if (halfp == floorl (halfp)) |
735 | *signgamp = -1; |
736 | else |
737 | *signgamp = 1; |
738 | if (q < 0x1p-120L) |
739 | return -__logl (x: q); |
740 | z = q - p; |
741 | if (z > 0.5L) |
742 | { |
743 | p += 1; |
744 | z = p - q; |
745 | } |
746 | z = q * __sinl (x: PIL * z); |
747 | w = __ieee754_lgammal_r (x: q, signgamp: &i); |
748 | z = __logl (x: PIL / z) - w; |
749 | return (z); |
750 | } |
751 | |
752 | if (x < 13.5L) |
753 | { |
754 | p = 0; |
755 | nx = floorl (x + 0.5L); |
756 | nn = nx; |
757 | switch (nn) |
758 | { |
759 | case 0: |
760 | /* log gamma (x + 1) = log(x) + log gamma(x) */ |
761 | if (x < 0x1p-120L) |
762 | return -__logl (x: x); |
763 | else if (x <= 0.125) |
764 | { |
765 | p = x * neval (x, p: RN1, NRN1) / deval (x, p: RD1, NRD1); |
766 | } |
767 | else if (x <= 0.375) |
768 | { |
769 | z = x - 0.25L; |
770 | p = z * neval (x: z, p: RN1r25, NRN1r25) / deval (x: z, p: RD1r25, NRD1r25); |
771 | p += lgam1r25b; |
772 | p += lgam1r25a; |
773 | } |
774 | else if (x <= 0.625) |
775 | { |
776 | z = x + (1 - x0a); |
777 | z = z - x0b; |
778 | p = neval (x: z, p: RN1r5, NRN1r5) / deval (x: z, p: RD1r5, NRD1r5); |
779 | p = p * z * z; |
780 | p = p + y0b; |
781 | p = p + y0a; |
782 | } |
783 | else if (x <= 0.875) |
784 | { |
785 | z = x - 0.75L; |
786 | p = z * neval (x: z, p: RN1r75, NRN1r75) / deval (x: z, p: RD1r75, NRD1r75); |
787 | p += lgam1r75b; |
788 | p += lgam1r75a; |
789 | } |
790 | else |
791 | { |
792 | z = x - 1; |
793 | p = z * neval (x: z, p: RN2, NRN2) / deval (x: z, p: RD2, NRD2); |
794 | } |
795 | p = p - __logl (x: x); |
796 | break; |
797 | |
798 | case 1: |
799 | if (x < 0.875L) |
800 | { |
801 | if (x <= 0.625) |
802 | { |
803 | z = x + (1 - x0a); |
804 | z = z - x0b; |
805 | p = neval (x: z, p: RN1r5, NRN1r5) / deval (x: z, p: RD1r5, NRD1r5); |
806 | p = p * z * z; |
807 | p = p + y0b; |
808 | p = p + y0a; |
809 | } |
810 | else if (x <= 0.875) |
811 | { |
812 | z = x - 0.75L; |
813 | p = z * neval (x: z, p: RN1r75, NRN1r75) |
814 | / deval (x: z, p: RD1r75, NRD1r75); |
815 | p += lgam1r75b; |
816 | p += lgam1r75a; |
817 | } |
818 | else |
819 | { |
820 | z = x - 1; |
821 | p = z * neval (x: z, p: RN2, NRN2) / deval (x: z, p: RD2, NRD2); |
822 | } |
823 | p = p - __logl (x: x); |
824 | } |
825 | else if (x < 1) |
826 | { |
827 | z = x - 1; |
828 | p = z * neval (x: z, p: RNr9, NRNr9) / deval (x: z, p: RDr9, NRDr9); |
829 | } |
830 | else if (x == 1) |
831 | p = 0; |
832 | else if (x <= 1.125L) |
833 | { |
834 | z = x - 1; |
835 | p = z * neval (x: z, p: RN1, NRN1) / deval (x: z, p: RD1, NRD1); |
836 | } |
837 | else if (x <= 1.375) |
838 | { |
839 | z = x - 1.25L; |
840 | p = z * neval (x: z, p: RN1r25, NRN1r25) / deval (x: z, p: RD1r25, NRD1r25); |
841 | p += lgam1r25b; |
842 | p += lgam1r25a; |
843 | } |
844 | else |
845 | { |
846 | /* 1.375 <= x+x0 <= 1.625 */ |
847 | z = x - x0a; |
848 | z = z - x0b; |
849 | p = neval (x: z, p: RN1r5, NRN1r5) / deval (x: z, p: RD1r5, NRD1r5); |
850 | p = p * z * z; |
851 | p = p + y0b; |
852 | p = p + y0a; |
853 | } |
854 | break; |
855 | |
856 | case 2: |
857 | if (x < 1.625L) |
858 | { |
859 | z = x - x0a; |
860 | z = z - x0b; |
861 | p = neval (x: z, p: RN1r5, NRN1r5) / deval (x: z, p: RD1r5, NRD1r5); |
862 | p = p * z * z; |
863 | p = p + y0b; |
864 | p = p + y0a; |
865 | } |
866 | else if (x < 1.875L) |
867 | { |
868 | z = x - 1.75L; |
869 | p = z * neval (x: z, p: RN1r75, NRN1r75) / deval (x: z, p: RD1r75, NRD1r75); |
870 | p += lgam1r75b; |
871 | p += lgam1r75a; |
872 | } |
873 | else if (x == 2) |
874 | p = 0; |
875 | else if (x < 2.375L) |
876 | { |
877 | z = x - 2; |
878 | p = z * neval (x: z, p: RN2, NRN2) / deval (x: z, p: RD2, NRD2); |
879 | } |
880 | else |
881 | { |
882 | z = x - 2.5L; |
883 | p = z * neval (x: z, p: RN2r5, NRN2r5) / deval (x: z, p: RD2r5, NRD2r5); |
884 | p += lgam2r5b; |
885 | p += lgam2r5a; |
886 | } |
887 | break; |
888 | |
889 | case 3: |
890 | if (x < 2.75) |
891 | { |
892 | z = x - 2.5L; |
893 | p = z * neval (x: z, p: RN2r5, NRN2r5) / deval (x: z, p: RD2r5, NRD2r5); |
894 | p += lgam2r5b; |
895 | p += lgam2r5a; |
896 | } |
897 | else |
898 | { |
899 | z = x - 3; |
900 | p = z * neval (x: z, p: RN3, NRN3) / deval (x: z, p: RD3, NRD3); |
901 | p += lgam3b; |
902 | p += lgam3a; |
903 | } |
904 | break; |
905 | |
906 | case 4: |
907 | z = x - 4; |
908 | p = z * neval (x: z, p: RN4, NRN4) / deval (x: z, p: RD4, NRD4); |
909 | p += lgam4b; |
910 | p += lgam4a; |
911 | break; |
912 | |
913 | case 5: |
914 | z = x - 5; |
915 | p = z * neval (x: z, p: RN5, NRN5) / deval (x: z, p: RD5, NRD5); |
916 | p += lgam5b; |
917 | p += lgam5a; |
918 | break; |
919 | |
920 | case 6: |
921 | z = x - 6; |
922 | p = z * neval (x: z, p: RN6, NRN6) / deval (x: z, p: RD6, NRD6); |
923 | p += lgam6b; |
924 | p += lgam6a; |
925 | break; |
926 | |
927 | case 7: |
928 | z = x - 7; |
929 | p = z * neval (x: z, p: RN7, NRN7) / deval (x: z, p: RD7, NRD7); |
930 | p += lgam7b; |
931 | p += lgam7a; |
932 | break; |
933 | |
934 | case 8: |
935 | z = x - 8; |
936 | p = z * neval (x: z, p: RN8, NRN8) / deval (x: z, p: RD8, NRD8); |
937 | p += lgam8b; |
938 | p += lgam8a; |
939 | break; |
940 | |
941 | case 9: |
942 | z = x - 9; |
943 | p = z * neval (x: z, p: RN9, NRN9) / deval (x: z, p: RD9, NRD9); |
944 | p += lgam9b; |
945 | p += lgam9a; |
946 | break; |
947 | |
948 | case 10: |
949 | z = x - 10; |
950 | p = z * neval (x: z, p: RN10, NRN10) / deval (x: z, p: RD10, NRD10); |
951 | p += lgam10b; |
952 | p += lgam10a; |
953 | break; |
954 | |
955 | case 11: |
956 | z = x - 11; |
957 | p = z * neval (x: z, p: RN11, NRN11) / deval (x: z, p: RD11, NRD11); |
958 | p += lgam11b; |
959 | p += lgam11a; |
960 | break; |
961 | |
962 | case 12: |
963 | z = x - 12; |
964 | p = z * neval (x: z, p: RN12, NRN12) / deval (x: z, p: RD12, NRD12); |
965 | p += lgam12b; |
966 | p += lgam12a; |
967 | break; |
968 | |
969 | case 13: |
970 | z = x - 13; |
971 | p = z * neval (x: z, p: RN13, NRN13) / deval (x: z, p: RD13, NRD13); |
972 | p += lgam13b; |
973 | p += lgam13a; |
974 | break; |
975 | } |
976 | return p; |
977 | } |
978 | |
979 | if (x > MAXLGM) |
980 | return (*signgamp * huge * huge); |
981 | |
982 | if (x > 0x1p120L) |
983 | return x * (__logl (x: x) - 1); |
984 | q = ls2pi - x; |
985 | q = (x - 0.5L) * __logl (x: x) + q; |
986 | if (x > 1.0e18L) |
987 | return (q); |
988 | |
989 | p = 1 / (x * x); |
990 | q += neval (x: p, p: RASY, NRASY) / x; |
991 | return (q); |
992 | } |
993 | libm_alias_finite (__ieee754_lgammal_r, __lgammal_r) |
994 | |