1 /*
2 * CDDL HEADER START
3 *
4 * The contents of this file are subject to the terms of the
5 * Common Development and Distribution License (the "License").
6 * You may not use this file except in compliance with the License.
7 *
8 * You can obtain a copy of the license at usr/src/OPENSOLARIS.LICENSE
9 * or http://www.opensolaris.org/os/licensing.
10 * See the License for the specific language governing permissions
11 * and limitations under the License.
12 *
13 * When distributing Covered Code, include this CDDL HEADER in each
14 * file and include the License file at usr/src/OPENSOLARIS.LICENSE.
15 * If applicable, add the following below this CDDL HEADER, with the
16 * fields enclosed by brackets "[]" replaced with your own identifying
17 * information: Portions Copyright [yyyy] [name of copyright owner]
18 *
19 * CDDL HEADER END
20 */
21
22 /*
23 * Copyright 2011 Nexenta Systems, Inc. All rights reserved.
24 */
25
26 /*
27 * Copyright 2006 Sun Microsystems, Inc. All rights reserved.
28 * Use is subject to license terms.
29 */
30
31 /*
32 * long double __k_lgammal(long double x, int *signgamlp);
33 * K.C. Ng, August, 1989.
34 *
35 * We choose [1.5,2.5] to be the primary interval. Our algorithms
36 * are mainly derived from
37 *
38 *
39 * zeta(2)-1 2 zeta(3)-1 3
40 * lgamma(2+s) = s*(1-euler) + --------- * s - --------- * s + ...
41 * 2 3
42 *
43 *
44 * Note 1. Since gamma(1+s)=s*gamma(s), hence
45 * lgamma(1+s) = log(s) + lgamma(s), or
46 * lgamma(s) = lgamma(1+s) - log(s).
47 * When s is really tiny (like roundoff), lgamma(1+s) ~ s(1-enler)
48 * Hence lgamma(s) ~ -log(s) for tiny s
49 *
50 */
51
52 #include "libm.h"
53 #include "longdouble.h"
54
55 static long double neg(long double, int *);
56 static long double poly(long double, const long double *, int);
57 static long double polytail(long double);
58 static long double primary(long double);
59
60 static const long double c0 = 0.0L,
61 ch = 0.5L,
62 c1 = 1.0L,
63 c2 = 2.0L,
64 c3 = 3.0L,
65 c4 = 4.0L,
66 c5 = 5.0L,
67 c6 = 6.0L,
68 pi = 3.1415926535897932384626433832795028841971L,
69 tiny = 1.0e-40L;
70
71 long double
72 __k_lgammal(long double x, int *signgamlp)
73 {
74 long double t, y;
75 int i;
76
77 /* purge off +-inf, NaN and negative arguments */
78 if (!finitel(x))
79 return (x * x);
80
81 *signgamlp = 1;
82
83 if (signbitl(x))
84 return (neg(x, signgamlp));
85
86 /* for x < 8.0 */
87 if (x < 8.0L) {
88 y = anintl(x);
89 i = (int)y;
90
91 switch (i) {
92 case 0:
93
94 if (x < 1.0e-40L)
95 return (-logl(x));
96 else
97 return ((primary(x) - log1pl(x)) - logl(x));
98
99 case 1:
100 return (primary(x - y) - logl(x));
101 case 2:
102 return (primary(x - y));
103 case 3:
104 return (primary(x - y) + logl(x - c1));
105 case 4:
106 return (primary(x - y) + logl((x - c1) * (x - c2)));
107 case 5:
108 return (primary(x - y) + logl((x - c1) * (x - c2) * (x -
109 c3)));
110 case 6:
111 return (primary(x - y) + logl((x - c1) * (x - c2) * (x -
112 c3) * (x - c4)));
113 case 7:
114 return (primary(x - y) + logl((x - c1) * (x - c2) * (x -
115 c3) * (x - c4) * (x - c5)));
116 case 8:
117 return (primary(x - y) + logl((x - c1) * (x - c2) * (x -
118 c3) * (x - c4) * (x - c5) * (x - c6)));
119 }
120 }
121
122 /* 8.0 <= x < 1.0e40 */
123 if (x < 1.0e40L) {
124 t = logl(x);
125 return (x * (t - c1) - (ch * t - polytail(c1 / x)));
126 }
127
128 /* 1.0e40 <= x <= inf */
129 return (x * (logl(x) - c1));
130 }
131
132 static const long double an1[] = { /* 20 terms */
133 -0.0772156649015328606065120900824024309741L,
134 3.224670334241132182362075833230130289059e-0001L,
135 -6.735230105319809513324605383668929964120e-0002L,
136 2.058080842778454787900092432928910226297e-0002L,
137 -7.385551028673985266273054086081102125704e-0003L,
138 2.890510330741523285758867304409628648727e-0003L,
139 -1.192753911703260976581414338096267498555e-0003L,
140 5.096695247430424562831956662855697824035e-0004L,
141 -2.231547584535777978926798502084300123638e-0004L,
142 9.945751278186384670278268034322157947635e-0005L,
143 -4.492623673665547726647838474125147631082e-0005L,
144 2.050721280617796810096993154281561168706e-0005L,
145 -9.439487785617396552092393234044767313568e-0006L,
146 4.374872903516051510689234173139793159340e-0006L,
147 -2.039156676413643091040459825776029327487e-0006L,
148 9.555777181318621470466563543806211523634e-0007L,
149 -4.468344919709630637558538313482398989638e-0007L,
150 2.216738086090045781773004477831059444178e-0007L,
151 -7.472783403418388455860445842543843485916e-0008L,
152 8.777317930927149922056782132706238921648e-0008L,
153 };
154
155 static const long double an2[] = { /* 20 terms */
156 -.0772156649015328606062692723698127607018L,
157 3.224670334241132182635552349060279118047e-0001L,
158 -6.735230105319809367555642883133994818325e-0002L,
159 2.058080842778459676880822202762143671813e-0002L,
160 -7.385551028672828216011343150077846918930e-0003L,
161 2.890510330762060607399561536905727853178e-0003L,
162 -1.192753911419623262328187532759756368041e-0003L,
163 5.096695278636456678258091134532258618614e-0004L,
164 -2.231547306817535743052975194022893369135e-0004L,
165 9.945771461633313282744264853986643877087e-0005L,
166 -4.492503279458972037926876061257489481619e-0005L,
167 2.051311416812082875492678651369394595613e-0005L,
168 -9.415778282365955203915850761537462941165e-0006L,
169 4.452428829045147098722932981088650055919e-0006L,
170 -1.835024727987632579886951760650722695781e-0006L,
171 1.379783080658545009579060714946381462565e-0006L,
172 2.282637532109775156769736768748402175238e-0007L,
173 1.002577375515900191362119718128149880168e-0006L,
174 5.177028794262638311939991106423220002463e-0007L,
175 3.127947245174847104122426445937830555755e-0007L,
176 };
177
178 static const long double an3[] = { /* 20 terms */
179 -.0772156649015328227870646417729220690875L,
180 3.224670334241156699881788955959915250365e-0001L,
181 -6.735230105312273571375431059744975563170e-0002L,
182 2.058080842924464587662846071337083809005e-0002L,
183 -7.385551008677271654723604653956131791619e-0003L,
184 2.890510536479782086197110272583833176602e-0003L,
185 -1.192752262076857692740571567808259138697e-0003L,
186 5.096800771149805289371135155128380707889e-0004L,
187 -2.231000836682831335505058492409860123647e-0004L,
188 9.968912171073936803871803966360595275047e-0005L,
189 -4.412020779327746243544387946167256187258e-0005L,
190 2.281374113541454151067016632998630209049e-0005L,
191 -4.028361291428629491824694655287954266830e-0006L,
192 1.470694920619518924598956849226530750139e-0005L,
193 1.381686137617987197975289545582377713772e-0005L,
194 2.012493539265777728944759982054970441601e-0005L,
195 1.723917864208965490251560644681933675799e-0005L,
196 1.202954035243788300138608765425123713395e-0005L,
197 5.079851887558623092776296577030850938146e-0006L,
198 1.220657945824153751555138592006604026282e-0006L,
199 };
200
201 static const long double an4[] = { /* 21 terms */
202 -.0772156649015732285350261816697540392371L,
203 3.224670334221752060691751340365212226097e-0001L,
204 -6.735230109744009693977755991488196368279e-0002L,
205 2.058080778913037626909954141611580783216e-0002L,
206 -7.385557567931505621170483708950557506819e-0003L,
207 2.890459838416254326340844289785254883436e-0003L,
208 -1.193059036207136762877351596966718455737e-0003L,
209 5.081914708100372836613371356529568937869e-0004L,
210 -2.289855016133600313131553005982542045338e-0004L,
211 8.053454537980585879620331053833498511491e-0005L,
212 -9.574620532104845821243493405855672438998e-0005L,
213 -9.269085628207107155601445001196317715686e-0005L,
214 -2.183276779859490461716196344776208220180e-0004L,
215 -3.134834305597571096452454999737269668868e-0004L,
216 -3.973878894951937437018305986901392888619e-0004L,
217 -3.953352414899222799161275564386488057119e-0004L,
218 -3.136740932204038779362660900621212816511e-0004L,
219 -1.884502253819634073946130825196078627664e-0004L,
220 -8.192655799958926853585332542123631379301e-0005L,
221 -2.292183750010571062891605074281744854436e-0005L,
222 -3.223980628729716864927724265781406614294e-0006L,
223 };
224
225 static const long double ap1[] = { /* 19 terms */
226 -0.0772156649015328606065120900824024296961L,
227 3.224670334241132182362075833230047956465e-0001L,
228 -6.735230105319809513324605382963943777301e-0002L,
229 2.058080842778454787900092126606252375465e-0002L,
230 -7.385551028673985266272518231365020063941e-0003L,
231 2.890510330741523285681704570797770736423e-0003L,
232 -1.192753911703260971285304221165990244515e-0003L,
233 5.096695247430420878696018188830886972245e-0004L,
234 -2.231547584535654004647639737841526025095e-0004L,
235 9.945751278137201960636098805852315982919e-0005L,
236 -4.492623672777606053587919463929044226280e-0005L,
237 2.050721258703289487603702670753053765201e-0005L,
238 -9.439485626565616989352750672499008021041e-0006L,
239 4.374838162403994645138200419356844574219e-0006L,
240 -2.038979492862555348577006944451002161496e-0006L,
241 9.536763152382263548086981191378885102802e-0007L,
242 -4.426111214332434049863595231916564014913e-0007L,
243 1.911148847512947464234633846270287546882e-0007L,
244 -5.788673944861923038157839080272303519671e-0008L,
245 };
246
247 static const long double ap2[] = { /* 19 terms */
248 -0.077215664901532860606428624449354836087L,
249 3.224670334241132182271948744265855440139e-0001L,
250 -6.735230105319809467356126599005051676203e-0002L,
251 2.058080842778453315716389815213496002588e-0002L,
252 -7.385551028673653323064118422580096222959e-0003L,
253 2.890510330735923572088003424849289006039e-0003L,
254 -1.192753911629952368606185543945790688144e-0003L,
255 5.096695239806718875364547587043220998766e-0004L,
256 -2.231547520600616108991867127392089144886e-0004L,
257 9.945746913898151120612322833059416008973e-0005L,
258 -4.492599307461977003570224943054585729684e-0005L,
259 2.050609891889165453592046505651759999090e-0005L,
260 -9.435329866734193796540515247917165988579e-0006L,
261 4.362267138522223236241016136585565144581e-0006L,
262 -2.008556356653246579300491601497510230557e-0006L,
263 8.961498103387207161105347118042844354395e-0007L,
264 -3.614187228330216282235692806488341157741e-0007L,
265 1.136978988247816860500420915014777753153e-0007L,
266 -2.000532786387196664019286514899782691776e-0008L,
267 };
268
269 static const long double ap3[] = { /* 19 terms */
270 -0.077215664901532859888521470795348856446L,
271 3.224670334241131733364048614484228443077e-0001L,
272 -6.735230105319676541660495145259038151576e-0002L,
273 2.058080842775975461837768839015444273830e-0002L,
274 -7.385551028347615729728618066663566606906e-0003L,
275 2.890510327517954083379032008643080256676e-0003L,
276 -1.192753886919470728001821137439430882603e-0003L,
277 5.096693728898932234814903769146577482912e-0004L,
278 -2.231540055048827662528594010961874258037e-0004L,
279 9.945446210018649311491619999438833843723e-0005L,
280 -4.491608206598064519190236245753867697750e-0005L,
281 2.047939071322271016498065052853746466669e-0005L,
282 -9.376824046522786006677541036631536790762e-0006L,
283 4.259329829498149111582277209189150127347e-0006L,
284 -1.866064770421594266702176289764212873428e-0006L,
285 7.462066721137579592928128104534957135669e-0007L,
286 -2.483546217529077735074007138457678727371e-0007L,
287 5.915166576378161473299324673649144297574e-0008L,
288 -7.334139641706988966966252333759604701905e-0009L,
289 };
290
291 static const long double ap4[] = { /* 19 terms */
292 -0.0772156649015326785569313252637238673675L,
293 3.224670334241051435008842685722468344822e-0001L,
294 -6.735230105302832007479431772160948499254e-0002L,
295 2.058080842553481183648529360967441889912e-0002L,
296 -7.385551007602909242024706804659879199244e-0003L,
297 2.890510182473907253939821312248303471206e-0003L,
298 -1.192753098427856770847894497586825614450e-0003L,
299 5.096659636418811568063339214203693550804e-0004L,
300 -2.231421144004355691166194259675004483639e-0004L,
301 9.942073842343832132754332881883387625136e-0005L,
302 -4.483809261973204531263252655050701205397e-0005L,
303 2.033260142610284888319116654931994447173e-0005L,
304 -9.153539544026646699870528191410440585796e-0006L,
305 3.988460469925482725894144688699584997971e-0006L,
306 -1.609692980087029172567957221850825977621e-0006L,
307 5.634916377249975825399706694496688803488e-0007L,
308 -1.560065465929518563549083208482591437696e-0007L,
309 2.961350193868935325526962209019387821584e-0008L,
310 -2.834602215195368130104649234505033159842e-0009L,
311 };
312
313 static long double
314 primary(long double s) /* assume |s|<=0.5 */
315 {
316 int i;
317
318 i = (int)(8.0L * (s + 0.5L));
319
320 switch (i) {
321 case 0:
322 return (ch * s + s * poly(s, an4, 21));
323 case 1:
324 return (ch * s + s * poly(s, an3, 20));
325 case 2:
326 return (ch * s + s * poly(s, an2, 20));
327 case 3:
328 return (ch * s + s * poly(s, an1, 20));
329 case 4:
330 return (ch * s + s * poly(s, ap1, 19));
331 case 5:
332 return (ch * s + s * poly(s, ap2, 19));
333 case 6:
334 return (ch * s + s * poly(s, ap3, 19));
335 case 7:
336 return (ch * s + s * poly(s, ap4, 19));
337 }
338
339 /* NOTREACHED */
340 return (0.0L);
341 }
342
343 static long double
344 poly(long double s, const long double *p, int n)
345 {
346 long double y;
347 int i;
348
349 y = p[n - 1];
350
351 for (i = n - 2; i >= 0; i--)
352 y = p[i] + s * y;
353
354 return (y);
355 }
356
357 static const long double pt[] = {
358 9.189385332046727417803297364056176804663e-0001L,
359 8.333333333333333333333333333331286969123e-0002L,
360 -2.777777777777777777777777553194796036402e-0003L,
361 7.936507936507936507927283071433584248176e-0004L,
362 -5.952380952380952362351042163192634108297e-0004L,
363 8.417508417508395661774286645578379460131e-0004L,
364 -1.917526917525263651186066417934685675649e-0003L,
365 6.410256409395203164659292973142293199083e-0003L,
366 -2.955065327248303301763594514012418438188e-0002L,
367 1.796442830099067542945998615411893822886e-0001L,
368 -1.392413465829723742489974310411118662919e+0000L,
369 1.339984238037267658352656597960492029261e+0001L,
370 -1.564707657605373662425785904278645727813e+0002L,
371 2.156323807499211356127813962223067079300e+0003L,
372 -3.330486427626223184647299834137041307569e+0004L,
373 5.235535072011889213611369254140123518699e+0005L,
374 -7.258160984602220710491988573430212593080e+0006L,
375 7.316526934569686459641438882340322673357e+0007L,
376 -3.806450279064900548836571789284896711473e+0008L,
377 };
378
379 static long double
380 polytail(long double s)
381 {
382 long double t, z;
383 int i;
384
385 z = s * s;
386 t = pt[18];
387
388 for (i = 17; i >= 1; i--)
389 t = pt[i] + z * t;
390
391 return (pt[0] + s * t);
392 }
393
394 static long double
395 neg(long double z, int *signgamlp)
396 {
397 long double t, p;
398
399 /* BEGIN CSTYLED */
400 /*
401 * written by K.C. Ng, Feb 2, 1989.
402 *
403 * Since
404 * -z*G(-z)*G(z) = pi/sin(pi*z),
405 * we have
406 * G(-z) = -pi/(sin(pi*z)*G(z)*z)
407 * = pi/(sin(pi*(-z))*G(z)*z)
408 * Algorithm
409 * z = |z|
410 * t = sinpi(z); ...note that when z>2**112, z is an int
411 * and hence t=0.
412 *
413 * if (t == 0.0) return 1.0/0.0;
414 * if (t< 0.0) *signgamlp = -1; else t= -t;
415 * if (z<1.0e-40) ...tiny z
416 * return -log(z);
417 * else
418 * return log(pi/(t*z))-lgamma(z);
419 *
420 */
421 /* END CSTYLED */
422
423 t = sinpil(z); /* t := sin(pi*z) */
424
425 if (t == c0) /* return 1.0/0.0 = +INF */
426 return (c1 / c0);
427
428 z = -z;
429
430 if (z <= tiny)
431 p = -logl(z);
432 else
433 p = logl(pi / (fabsl(t) * z)) - __k_lgammal(z, signgamlp);
434
435 if (t < c0)
436 *signgamlp = -1;
437
438 return (p);
439 }