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 "libm_synonyms.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
61 c0 = 0.0L,
62 ch = 0.5L,
63 c1 = 1.0L,
64 c2 = 2.0L,
65 c3 = 3.0L,
66 c4 = 4.0L,
67 c5 = 5.0L,
68 c6 = 6.0L,
69 pi = 3.1415926535897932384626433832795028841971L,
70 tiny = 1.0e-40L;
71
72 long double
73 __k_lgammal(long double x, int *signgamlp) {
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 *signgamlp = 1;
81 if (signbitl(x))
82 return (neg(x, signgamlp));
83
84 /* for x < 8.0 */
85 if (x < 8.0L) {
86 y = anintl(x);
87 i = (int) y;
88 switch (i) {
89 case 0:
90 if (x < 1.0e-40L)
91 return (-logl(x));
92 else
93 return (primary(x)-log1pl(x))-logl(x);
94 case 1:
95 return (primary(x-y)-logl(x));
96 case 2:
97 return (primary(x-y));
98 case 3:
99 return (primary(x-y)+logl(x-c1));
100 case 4:
101 return (primary(x-y)+logl((x-c1)*(x-c2)));
102 case 5:
103 return (primary(x-y)+logl((x-c1)*(x-c2)*(x-c3)));
104 case 6:
105 return (primary(x-y)+logl((x-c1)*(x-c2)*(x-c3)*(x-c4)));
106 case 7:
107 return (primary(x-y)+logl((x-c1)*(x-c2)*(x-c3)*(x-c4)*(x-c5)));
108 case 8:
109 return primary(x-y)+
110 logl((x-c1)*(x-c2)*(x-c3)*(x-c4)*(x-c5)*(x-c6));
111 }
112 }
113
114 /* 8.0 <= x < 1.0e40 */
115 if (x < 1.0e40L) {
116 t = logl(x);
117 return (x*(t-c1)-(ch*t-polytail(c1/x)));
118 }
119
120 /* 1.0e40 <= x <= inf */
121 return (x*(logl(x)-c1));
122 }
123
124 static const long double an1[] = { /* 20 terms */
125 -0.0772156649015328606065120900824024309741L,
126 3.224670334241132182362075833230130289059e-0001L,
127 -6.735230105319809513324605383668929964120e-0002L,
128 2.058080842778454787900092432928910226297e-0002L,
129 -7.385551028673985266273054086081102125704e-0003L,
130 2.890510330741523285758867304409628648727e-0003L,
131 -1.192753911703260976581414338096267498555e-0003L,
132 5.096695247430424562831956662855697824035e-0004L,
133 -2.231547584535777978926798502084300123638e-0004L,
134 9.945751278186384670278268034322157947635e-0005L,
135 -4.492623673665547726647838474125147631082e-0005L,
136 2.050721280617796810096993154281561168706e-0005L,
137 -9.439487785617396552092393234044767313568e-0006L,
138 4.374872903516051510689234173139793159340e-0006L,
139 -2.039156676413643091040459825776029327487e-0006L,
140 9.555777181318621470466563543806211523634e-0007L,
141 -4.468344919709630637558538313482398989638e-0007L,
142 2.216738086090045781773004477831059444178e-0007L,
143 -7.472783403418388455860445842543843485916e-0008L,
144 8.777317930927149922056782132706238921648e-0008L,
145 };
146
147 static const long double an2[] = { /* 20 terms */
148 -.0772156649015328606062692723698127607018L,
149 3.224670334241132182635552349060279118047e-0001L,
150 -6.735230105319809367555642883133994818325e-0002L,
151 2.058080842778459676880822202762143671813e-0002L,
152 -7.385551028672828216011343150077846918930e-0003L,
153 2.890510330762060607399561536905727853178e-0003L,
154 -1.192753911419623262328187532759756368041e-0003L,
155 5.096695278636456678258091134532258618614e-0004L,
156 -2.231547306817535743052975194022893369135e-0004L,
157 9.945771461633313282744264853986643877087e-0005L,
158 -4.492503279458972037926876061257489481619e-0005L,
159 2.051311416812082875492678651369394595613e-0005L,
160 -9.415778282365955203915850761537462941165e-0006L,
161 4.452428829045147098722932981088650055919e-0006L,
162 -1.835024727987632579886951760650722695781e-0006L,
163 1.379783080658545009579060714946381462565e-0006L,
164 2.282637532109775156769736768748402175238e-0007L,
165 1.002577375515900191362119718128149880168e-0006L,
166 5.177028794262638311939991106423220002463e-0007L,
167 3.127947245174847104122426445937830555755e-0007L,
168 };
169
170 static const long double an3[] = { /* 20 terms */
171 -.0772156649015328227870646417729220690875L,
172 3.224670334241156699881788955959915250365e-0001L,
173 -6.735230105312273571375431059744975563170e-0002L,
174 2.058080842924464587662846071337083809005e-0002L,
175 -7.385551008677271654723604653956131791619e-0003L,
176 2.890510536479782086197110272583833176602e-0003L,
177 -1.192752262076857692740571567808259138697e-0003L,
178 5.096800771149805289371135155128380707889e-0004L,
179 -2.231000836682831335505058492409860123647e-0004L,
180 9.968912171073936803871803966360595275047e-0005L,
181 -4.412020779327746243544387946167256187258e-0005L,
182 2.281374113541454151067016632998630209049e-0005L,
183 -4.028361291428629491824694655287954266830e-0006L,
184 1.470694920619518924598956849226530750139e-0005L,
185 1.381686137617987197975289545582377713772e-0005L,
186 2.012493539265777728944759982054970441601e-0005L,
187 1.723917864208965490251560644681933675799e-0005L,
188 1.202954035243788300138608765425123713395e-0005L,
189 5.079851887558623092776296577030850938146e-0006L,
190 1.220657945824153751555138592006604026282e-0006L,
191 };
192
193 static const long double an4[] = { /* 21 terms */
194 -.0772156649015732285350261816697540392371L,
195 3.224670334221752060691751340365212226097e-0001L,
196 -6.735230109744009693977755991488196368279e-0002L,
197 2.058080778913037626909954141611580783216e-0002L,
198 -7.385557567931505621170483708950557506819e-0003L,
199 2.890459838416254326340844289785254883436e-0003L,
200 -1.193059036207136762877351596966718455737e-0003L,
201 5.081914708100372836613371356529568937869e-0004L,
202 -2.289855016133600313131553005982542045338e-0004L,
203 8.053454537980585879620331053833498511491e-0005L,
204 -9.574620532104845821243493405855672438998e-0005L,
205 -9.269085628207107155601445001196317715686e-0005L,
206 -2.183276779859490461716196344776208220180e-0004L,
207 -3.134834305597571096452454999737269668868e-0004L,
208 -3.973878894951937437018305986901392888619e-0004L,
209 -3.953352414899222799161275564386488057119e-0004L,
210 -3.136740932204038779362660900621212816511e-0004L,
211 -1.884502253819634073946130825196078627664e-0004L,
212 -8.192655799958926853585332542123631379301e-0005L,
213 -2.292183750010571062891605074281744854436e-0005L,
214 -3.223980628729716864927724265781406614294e-0006L,
215 };
216
217 static const long double ap1[] = { /* 19 terms */
218 -0.0772156649015328606065120900824024296961L,
219 3.224670334241132182362075833230047956465e-0001L,
220 -6.735230105319809513324605382963943777301e-0002L,
221 2.058080842778454787900092126606252375465e-0002L,
222 -7.385551028673985266272518231365020063941e-0003L,
223 2.890510330741523285681704570797770736423e-0003L,
224 -1.192753911703260971285304221165990244515e-0003L,
225 5.096695247430420878696018188830886972245e-0004L,
226 -2.231547584535654004647639737841526025095e-0004L,
227 9.945751278137201960636098805852315982919e-0005L,
228 -4.492623672777606053587919463929044226280e-0005L,
229 2.050721258703289487603702670753053765201e-0005L,
230 -9.439485626565616989352750672499008021041e-0006L,
231 4.374838162403994645138200419356844574219e-0006L,
232 -2.038979492862555348577006944451002161496e-0006L,
233 9.536763152382263548086981191378885102802e-0007L,
234 -4.426111214332434049863595231916564014913e-0007L,
235 1.911148847512947464234633846270287546882e-0007L,
236 -5.788673944861923038157839080272303519671e-0008L,
237 };
238
239 static const long double ap2[] = { /* 19 terms */
240 -0.077215664901532860606428624449354836087L,
241 3.224670334241132182271948744265855440139e-0001L,
242 -6.735230105319809467356126599005051676203e-0002L,
243 2.058080842778453315716389815213496002588e-0002L,
244 -7.385551028673653323064118422580096222959e-0003L,
245 2.890510330735923572088003424849289006039e-0003L,
246 -1.192753911629952368606185543945790688144e-0003L,
247 5.096695239806718875364547587043220998766e-0004L,
248 -2.231547520600616108991867127392089144886e-0004L,
249 9.945746913898151120612322833059416008973e-0005L,
250 -4.492599307461977003570224943054585729684e-0005L,
251 2.050609891889165453592046505651759999090e-0005L,
252 -9.435329866734193796540515247917165988579e-0006L,
253 4.362267138522223236241016136585565144581e-0006L,
254 -2.008556356653246579300491601497510230557e-0006L,
255 8.961498103387207161105347118042844354395e-0007L,
256 -3.614187228330216282235692806488341157741e-0007L,
257 1.136978988247816860500420915014777753153e-0007L,
258 -2.000532786387196664019286514899782691776e-0008L,
259 };
260
261 static const long double ap3[] = { /* 19 terms */
262 -0.077215664901532859888521470795348856446L,
263 3.224670334241131733364048614484228443077e-0001L,
264 -6.735230105319676541660495145259038151576e-0002L,
265 2.058080842775975461837768839015444273830e-0002L,
266 -7.385551028347615729728618066663566606906e-0003L,
267 2.890510327517954083379032008643080256676e-0003L,
268 -1.192753886919470728001821137439430882603e-0003L,
269 5.096693728898932234814903769146577482912e-0004L,
270 -2.231540055048827662528594010961874258037e-0004L,
271 9.945446210018649311491619999438833843723e-0005L,
272 -4.491608206598064519190236245753867697750e-0005L,
273 2.047939071322271016498065052853746466669e-0005L,
274 -9.376824046522786006677541036631536790762e-0006L,
275 4.259329829498149111582277209189150127347e-0006L,
276 -1.866064770421594266702176289764212873428e-0006L,
277 7.462066721137579592928128104534957135669e-0007L,
278 -2.483546217529077735074007138457678727371e-0007L,
279 5.915166576378161473299324673649144297574e-0008L,
280 -7.334139641706988966966252333759604701905e-0009L,
281 };
282
283 static const long double ap4[] = { /* 19 terms */
284 -0.0772156649015326785569313252637238673675L,
285 3.224670334241051435008842685722468344822e-0001L,
286 -6.735230105302832007479431772160948499254e-0002L,
287 2.058080842553481183648529360967441889912e-0002L,
288 -7.385551007602909242024706804659879199244e-0003L,
289 2.890510182473907253939821312248303471206e-0003L,
290 -1.192753098427856770847894497586825614450e-0003L,
291 5.096659636418811568063339214203693550804e-0004L,
292 -2.231421144004355691166194259675004483639e-0004L,
293 9.942073842343832132754332881883387625136e-0005L,
294 -4.483809261973204531263252655050701205397e-0005L,
295 2.033260142610284888319116654931994447173e-0005L,
296 -9.153539544026646699870528191410440585796e-0006L,
297 3.988460469925482725894144688699584997971e-0006L,
298 -1.609692980087029172567957221850825977621e-0006L,
299 5.634916377249975825399706694496688803488e-0007L,
300 -1.560065465929518563549083208482591437696e-0007L,
301 2.961350193868935325526962209019387821584e-0008L,
302 -2.834602215195368130104649234505033159842e-0009L,
303 };
304
305 static long double
306 primary(long double s) { /* assume |s|<=0.5 */
307 int i;
308
309 i = (int) (8.0L * (s + 0.5L));
310 switch (i) {
311 case 0: return ch*s+s*poly(s, an4, 21);
312 case 1: return ch*s+s*poly(s, an3, 20);
313 case 2: return ch*s+s*poly(s, an2, 20);
314 case 3: return ch*s+s*poly(s, an1, 20);
315 case 4: return ch*s+s*poly(s, ap1, 19);
316 case 5: return ch*s+s*poly(s, ap2, 19);
317 case 6: return ch*s+s*poly(s, ap3, 19);
318 case 7: return ch*s+s*poly(s, ap4, 19);
319 }
320 /* NOTREACHED */
321 return (0.0L);
322 }
323
324 static long double
325 poly(long double s, const long double *p, int n) {
326 long double y;
327 int i;
328 y = p[n-1];
329 for (i = n-2; i >= 0; i--) y = p[i]+s*y;
330 return (y);
331 }
332
333 static const long double pt[] = {
334 9.189385332046727417803297364056176804663e-0001L,
335 8.333333333333333333333333333331286969123e-0002L,
336 -2.777777777777777777777777553194796036402e-0003L,
337 7.936507936507936507927283071433584248176e-0004L,
338 -5.952380952380952362351042163192634108297e-0004L,
339 8.417508417508395661774286645578379460131e-0004L,
340 -1.917526917525263651186066417934685675649e-0003L,
341 6.410256409395203164659292973142293199083e-0003L,
342 -2.955065327248303301763594514012418438188e-0002L,
343 1.796442830099067542945998615411893822886e-0001L,
344 -1.392413465829723742489974310411118662919e+0000L,
345 1.339984238037267658352656597960492029261e+0001L,
346 -1.564707657605373662425785904278645727813e+0002L,
347 2.156323807499211356127813962223067079300e+0003L,
348 -3.330486427626223184647299834137041307569e+0004L,
349 5.235535072011889213611369254140123518699e+0005L,
350 -7.258160984602220710491988573430212593080e+0006L,
351 7.316526934569686459641438882340322673357e+0007L,
352 -3.806450279064900548836571789284896711473e+0008L,
353 };
354
355 static long double
356 polytail(long double s) {
357 long double t, z;
358 int i;
359 z = s*s;
360 t = pt[18];
361 for (i = 17; i >= 1; i--) t = pt[i]+z*t;
362 return (pt[0]+s*t);
363 }
364
365 static long double
366 neg(long double z, int *signgamlp) {
367 long double t, p;
368
369 /*
370 * written by K.C. Ng, Feb 2, 1989.
371 *
372 * Since
373 * -z*G(-z)*G(z) = pi/sin(pi*z),
374 * we have
375 * G(-z) = -pi/(sin(pi*z)*G(z)*z)
376 * = pi/(sin(pi*(-z))*G(z)*z)
377 * Algorithm
378 * z = |z|
379 * t = sinpi(z); ...note that when z>2**112, z is an int
380 * and hence t=0.
381 *
382 * if(t==0.0) return 1.0/0.0;
383 * if(t< 0.0) *signgamlp = -1; else t= -t;
384 * if(z<1.0e-40) ...tiny z
385 * return -log(z);
386 * else
387 * return log(pi/(t*z))-lgamma(z);
388 *
389 */
390
391 t = sinpil(z); /* t := sin(pi*z) */
392 if (t == c0) /* return 1.0/0.0 = +INF */
393 return (c1/c0);
394
395 z = -z;
396 if (z <= tiny)
397 p = -logl(z);
398 else
399 p = logl(pi/(fabsl(t)*z)) - __k_lgammal(z, signgamlp);
400 if (t < c0) *signgamlp = -1;
401 return (p);
402 }