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