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 #pragma weak __tgammal = tgammal
32
33 #include "libm.h"
34 #include <sys/isa_defs.h>
35
36 #if defined(_BIG_ENDIAN)
37 #define H0_WORD(x) ((unsigned *)&x)[0]
38 #define H3_WORD(x) ((unsigned *)&x)[3]
39 #define CHOPPED(x) (long double)((double)(x))
40 #else
41 #define H0_WORD(x) ((((int *)&x)[2] << 16) | (0x0000ffff & \
42 (((unsigned *)&x)[1] >> 15)))
43 #define H3_WORD(x) ((unsigned *)&x)[0]
44 #define CHOPPED(x) (long double)((float)(x))
45 #endif
46
47 struct LDouble {
48 long double h, l;
49 };
50
51 /*
52 * Primary interval GTi()
53 */
54 static const long double P1[] = {
55 +0.709086836199777919037185741507610124611513720557L,
56 +4.45754781206489035827915969367354835667391606951e-0001L,
57 +3.21049298735832382311662273882632210062918153852e-0002L,
58 -5.71296796342106617651765245858289197369688864350e-0003L,
59 +6.04666892891998977081619174969855831606965352773e-0003L,
60 +8.99106186996888711939627812174765258822658645168e-0004L,
61 -6.96496846144407741431207008527018441810175568949e-0005L,
62 +1.52597046118984020814225409300131445070213882429e-0005L,
63 +5.68521076168495673844711465407432189190681541547e-0007L,
64 +3.30749673519634895220582062520286565610418952979e-0008L,
65 };
66
67 static const long double Q1[] = {
68 +1.0 + 0000L,
69 +1.35806511721671070408570853537257079579490650668e+0000L,
70 +2.97567810153429553405327140096063086994072952961e-0001L,
71 -1.52956835982588571502954372821681851681118097870e-0001L,
72 -2.88248519561420109768781615289082053597954521218e-0002L,
73 +1.03475311719937405219789948456313936302378395955e-0002L,
74 +4.12310203243891222368965360124391297374822742313e-0004L,
75 -3.12653708152290867248931925120380729518332507388e-0004L,
76 +2.36672170850409745237358105667757760527014332458e-0005L,
77 };
78
79 static const long double P2[] = {
80 +0.428486815855585429730209907810650135255270600668084114L,
81 +2.62768479103809762805691743305424077975230551176e-0001L,
82 +3.81187532685392297608310837995193946591425896150e-0002L,
83 +3.00063075891811043820666846129131255948527925381e-0003L,
84 +2.47315407812279164228398470797498649142513408654e-0003L,
85 +3.62838199917848372586173483147214880464782938664e-0004L,
86 +3.43991105975492623982725644046473030098172692423e-0006L,
87 +4.56902151569603272237014240794257659159045432895e-0006L,
88 +2.13734755837595695602045100675540011352948958453e-0007L,
89 +9.74123440547918230781670266967882492234877125358e-0009L,
90 };
91
92 static const long double Q2[] = {
93 +1.0L,
94 +9.18284118632506842664645516830761489700556179701e-0001L,
95 -6.41430858837830766045202076965923776189154874947e-0003L,
96 -1.24400885809771073213345747437964149775410921376e-0001L,
97 +4.69803798146251757538856567522481979624746875964e-0003L,
98 +7.18309447069495315914284705109868696262662082731e-0003L,
99 -8.75812626987894695112722600697653425786166399105e-0004L,
100 -1.23539972377769277995959339188431498626674835169e-0004L,
101 +3.10019017590151598732360097849672925448587547746e-0005L,
102 -1.77260223349332617658921874288026777465782364070e-0006L,
103 };
104
105 static const long double P3[] = {
106 +0.3824094797345675048502747661075355640070439388902L,
107 +3.42198093076618495415854906335908427159833377774e-0001L,
108 +9.63828189500585568303961406863153237440702754858e-0002L,
109 +8.76069421042696384852462044188520252156846768667e-0003L,
110 +1.86477890389161491224872014149309015261897537488e-0003L,
111 +8.16871354540309895879974742853701311541286944191e-0004L,
112 +6.83783483674600322518695090864659381650125625216e-0005L,
113 -1.10168269719261574708565935172719209272190828456e-0006L,
114 +9.66243228508380420159234853278906717065629721016e-0007L,
115 +2.31858885579177250541163820671121664974334728142e-0008L,
116 };
117
118 static const long double Q3[] = {
119 +1.0L, +8.25479821168813634632437430090376252512793067339e-0001L,
120 -1.62251363073937769739639623669295110346015576320e-0002L,
121 -1.10621286905916732758745130629426559691187579852e-0001L,
122 +3.48309693970985612644446415789230015515365291459e-0003L,
123 +6.73553737487488333032431261131289672347043401328e-0003L,
124 -7.63222008393372630162743587811004613050245128051e-0004L,
125 -1.35792670669190631476784768961953711773073251336e-0004L,
126 +3.19610150954223587006220730065608156460205690618e-0005L,
127 -1.82096553862822346610109522015129585693354348322e-0006L,
128 };
129
130 static const long double
131 #if defined(__x86)
132 GZ1_h = 0.938204627909682449364570100414084663498215377L,
133 GZ1_l = 4.518346116624229420055327632718530617227944106e-20L,
134 GZ2_h = 0.885603194410888700264725126309883762587560340L,
135 GZ2_l = 1.409077427270497062039119290776508217077297169e-20L,
136 GZ3_h = 0.936781411463652321613537060640553022494714241L,
137 GZ3_l = 5.309836440284827247897772963887219035221996813e-21L,
138 #else
139 GZ1_h = 0.938204627909682449409753561580326910854647031L,
140 GZ1_l = 4.684412162199460089642452580902345976446297037e-35L,
141 GZ2_h = 0.885603194410888700278815900582588658192658794L,
142 GZ2_l = 7.501529273890253789219935569758713534641074860e-35L,
143 GZ3_h = 0.936781411463652321618846897080837818855399840L,
144 GZ3_l = 3.088721217404784363585591914529361687403776917e-35L,
145 #endif
146 TZ1 = -0.3517214357852935791015625L,
147 TZ3 = 0.280530631542205810546875L;
148
149
150 /*
151 * compute gamma(y=yh+yl) for y in GT1 = [1.0000, 1.2845]
152 * ...assume yh got 53 or 24(i386) significant bits
153 */
154 static struct LDouble
155 GT1(long double yh, long double yl)
156 {
157 long double t3, t4, y;
158 int i;
159 struct LDouble r;
160
161 y = yh + yl;
162
163 for (t4 = Q1[8], t3 = P1[8] + y * P1[9], i = 7; i >= 0; i--) {
164 t4 = t4 * y + Q1[i];
165 t3 = t3 * y + P1[i];
166 }
167
168 t3 = (y * y) * t3 / t4;
169 t3 += (TZ1 * yl + GZ1_l);
170 t4 = TZ1 * yh;
171 r.h = CHOPPED((t4 + GZ1_h + t3));
172 t3 += (t4 - (r.h - GZ1_h));
173 r.l = t3;
174 return (r);
175 }
176
177
178 /*
179 * compute gamma(y=yh+yl) for y in GT2 = [1.2844, 1.6374]
180 * ...assume yh got 53 significant bits
181 */
182 static struct LDouble
183 GT2(long double yh, long double yl)
184 {
185 long double t3, t4, y;
186 int i;
187 struct LDouble r;
188
189 y = yh + yl;
190
191 for (t4 = Q2[9], t3 = P2[9], i = 8; i >= 0; i--) {
192 t4 = t4 * y + Q2[i];
193 t3 = t3 * y + P2[i];
194 }
195
196 t3 = GZ2_l + (y * y) * t3 / t4;
197 r.h = CHOPPED((GZ2_h + t3));
198 r.l = t3 - (r.h - GZ2_h);
199 return (r);
200 }
201
202
203 /*
204 * compute gamma(y=yh+yl) for y in GT3 = [1.6373, 2.0000]
205 * ...assume yh got 53 significant bits
206 */
207 static struct LDouble
208 GT3(long double yh, long double yl)
209 {
210 long double t3, t4, y;
211 int i;
212 struct LDouble r;
213
214 y = yh + yl;
215
216 for (t4 = Q3[9], t3 = P3[9], i = 8; i >= 0; i--) {
217 t4 = t4 * y + Q3[i];
218 t3 = t3 * y + P3[i];
219 }
220
221 t3 = (y * y) * t3 / t4;
222 t3 += (TZ3 * yl + GZ3_l);
223 t4 = TZ3 * yh;
224 r.h = CHOPPED((t4 + GZ3_h + t3));
225 t3 += (t4 - (r.h - GZ3_h));
226 r.l = t3;
227 return (r);
228 }
229
230 /*
231 * Hex value of GP[0] shoule be 3FB55555 55555555
232 */
233 static const long double GP[] = {
234 +0.083333333333333333333333333333333172839171301L,
235 -2.77777777777777777777777777492501211999399424104e-0003L,
236 +7.93650793650793650793635650541638236350020883243e-0004L,
237 -5.95238095238095238057299772679324503339241961704e-0004L,
238 +8.41750841750841696138422987977683524926142600321e-0004L,
239 -1.91752691752686682825032547823699662178842123308e-0003L,
240 +6.41025641022403480921891559356473451161279359322e-0003L,
241 -2.95506535798414019189819587455577003732808185071e-0002L,
242 +1.79644367229970031486079180060923073476568732136e-0001L,
243 -1.39243086487274662174562872567057200255649290646e+0000L,
244 +1.34025874044417962188677816477842265259608269775e+0001L,
245 -1.56803713480127469414495545399982508700748274318e+0002L,
246 +2.18739841656201561694927630335099313968924493891e+0003L,
247 -3.55249848644100338419187038090925410976237921269e+0004L,
248 +6.43464880437835286216768959439484376449179576452e+0005L,
249 -1.20459154385577014992600342782821389605893904624e+0007L,
250 +2.09263249637351298563934942349749718491071093210e+0008L,
251 -2.96247483183169219343745316433899599834685703457e+0009L,
252 +2.88984933605896033154727626086506756972327292981e+0010L,
253 -1.40960434146030007732838382416230610302678063984e+0011L, /* 19 */
254 };
255
256 static const long double T3[] = {
257 +0.666666666666666666666666666666666634567834260213L, /* T3[0] */
258 +0.400000000000000000000000000040853636176634934140L, /* T3[1] */
259 +0.285714285714285714285696975252753987869020263448L, /* T3[2] */
260 +0.222222222222222225593221101192317258554772129875L, /* T3[3] */
261 +0.181818181817850192105847183461778186703779262916L, /* T3[4] */
262 +0.153846169861348633757101285952333369222567014596L, /* T3[5] */
263 +0.133033462889260193922261296772841229985047571265L, /* T3[6] */
264 };
265 /* BEGIN CSTYLED */
266 static const long double c[] = {
267 0.0L,
268 1.0L,
269 2.0L,
270 0.5L,
271 1.0e-4930L, /* tiny */
272 4.18937683105468750000e-01L, /* hln2pim1_h */
273 8.50099203991780329736405617639861397473637783412817152e-07L, /* hln2pim1_l */
274 0.418938533204672741780329736405617639861397473637783412817152L, /* hln2pim1 */
275 2.16608493865351192653179168701171875e-02L, /* ln2_32hi */
276 5.96317165397058692545083025235937919875797669127130e-12L, /* ln2_32lo */
277 46.16624130844682903551758979206054839765267053289554989233L, /* invln2_32 */
278 #if defined(__x86)
279 1.7555483429044629170023839037639845628291e+03L, /* overflow */
280 #else
281 1.7555483429044629170038892160702032034177e+03L, /* overflow */
282 #endif
283 };
284 /* END CSTYLED */
285
286 #define zero c[0]
287 #define one c[1]
288 #define two c[2]
289 #define half c[3]
290 #define tiny c[4]
291 #define hln2pim1_h c[5]
292 #define hln2pim1_l c[6]
293 #define hln2pim1 c[7]
294 #define ln2_32hi c[8]
295 #define ln2_32lo c[9]
296 #define invln2_32 c[10]
297 #define overflow c[11]
298
299 /*
300 * |exp(r) - (1+r+Et0*r^2+...+Et10*r^12)| <= 2^(-128.88) for |r|<=ln2/64
301 */
302 static const long double Et[] = {
303 +5.0000000000000000000e-1L,
304 +1.66666666666666666666666666666828835166292152466e-0001L,
305 +4.16666666666666666666666666666693398646592712189e-0002L,
306 +8.33333333333333333333331748774512601775591115951e-0003L,
307 +1.38888888888888888888888845356011511394764753997e-0003L,
308 +1.98412698412698413237140350092993252684198882102e-0004L,
309 +2.48015873015873016080222025357442659895814371694e-0005L,
310 +2.75573192239028921114572986441972140933432317798e-0006L,
311 +2.75573192239448470555548102895526369739856219317e-0007L,
312 +2.50521677867683935940853997995937600214167232477e-0008L,
313 +2.08767928899010367374984448513685566514152147362e-0009L,
314 };
315
316 /*
317 * long double precision coefficients for computing log(x)-1 in tgamma.
318 * See "algorithm" for details
319 *
320 * log(x) - 1 = T1(n) + T2(j) + T3(s), where x = 2**n * y, 1<=y<2,
321 * j=[64*y], z[j]=1+j/64+1/128, s = (y-z[j])/(y+z[j]), and
322 * T1(n) = T1[2n,2n+1] = n*log(2)-1,
323 * T2(j) = T2[2j,2j+1] = log(z[j]),
324 * T3(s) = 2s + T3[0]s^3 + T3[1]s^5 + T3[2]s^7 + ... + T3[6]s^15
325 * Note
326 * (1) the leading entries are truncated to 24 binary point.
327 * (2) Remez error for T3(s) is bounded by 2**(-136.54)
328 */
329 static const long double T1[] = {
330 -1.000000000000000000000000000000000000000000e+00L,
331 +0.000000000000000000000000000000000000000000e+00L,
332 -3.068528175354003906250000000000000000000000e-01L,
333 -1.904654299957767878541823431924500011926579e-09L,
334 +3.862943053245544433593750000000000000000000e-01L,
335 +5.579533617547508924291635313615100141107647e-08L,
336 +1.079441487789154052734375000000000000000000e+00L,
337 +5.389068187551732136437452970422650211661470e-08L,
338 +1.772588670253753662109375000000000000000000e+00L,
339 +5.198602757555955348583270627230200282215294e-08L,
340 +2.465735852718353271484375000000000000000000e+00L,
341 +5.008137327560178560729088284037750352769117e-08L,
342 +3.158883035182952880859375000000000000000000e+00L,
343 +4.817671897564401772874905940845299849351090e-08L,
344 +3.852030217647552490234375000000000000000000e+00L,
345 +4.627206467568624985020723597652849919904913e-08L,
346 +4.545177400112152099609375000000000000000000e+00L,
347 +4.436741037572848197166541254460399990458737e-08L,
348 +5.238324582576751708984375000000000000000000e+00L,
349 +4.246275607577071409312358911267950061012560e-08L,
350 +5.931471765041351318359375000000000000000000e+00L,
351 +4.055810177581294621458176568075500131566384e-08L,
352 };
353
354 /*
355 * T2[2i,2i+1] = log(1+i/64+1/128)
356 */
357 static const long double T2[] = {
358 +7.7821016311645507812500000000000000000000e-03L,
359 +3.8810890398166212900061136763678127453570e-08L,
360 +2.3167014122009277343750000000000000000000e-02L,
361 +4.5159525100885049160962289916579411752759e-08L,
362 +3.8318812847137451171875000000000000000000e-02L,
363 +5.1454999148021880325123797290345960518164e-08L,
364 +5.3244471549987792968750000000000000000000e-02L,
365 +4.2968824489897120193786528776939573415076e-08L,
366 +6.7950606346130371093750000000000000000000e-02L,
367 +5.5562377378300815277772629414034632394030e-08L,
368 +8.2443654537200927734375000000000000000000e-02L,
369 +1.4673873663533785068668307805914095366600e-08L,
370 +9.6729576587677001953125000000000000000000e-02L,
371 +4.9870874110342446056487463437015041543346e-08L,
372 +1.1081433296203613281250000000000000000000e-01L,
373 +3.3378253981382306169323211928098474801099e-08L,
374 +1.2470346689224243164062500000000000000000e-01L,
375 +1.1608714804222781515380863268491613205318e-08L,
376 +1.3840228319168090820312500000000000000000e-01L,
377 +3.9667438227482200873601649187393160823607e-08L,
378 +1.5191602706909179687500000000000000000000e-01L,
379 +1.4956750178196803424896884511327584958252e-08L,
380 +1.6524952650070190429687500000000000000000e-01L,
381 +4.6394605258578736449277240313729237989366e-08L,
382 +1.7840760946273803710937500000000000000000e-01L,
383 +4.8010080260010025241510941968354682199540e-08L,
384 +1.9139480590820312500000000000000000000000e-01L,
385 +4.7091426329609298807561308873447039132856e-08L,
386 +2.0421552658081054687500000000000000000000e-01L,
387 +1.4847880344628820386196239272213742113867e-08L,
388 +2.1687388420104980468750000000000000000000e-01L,
389 +5.4099564554931589525744347498478964801484e-08L,
390 +2.2937405109405517578125000000000000000000e-01L,
391 +4.9970790654210230725046139871550961365282e-08L,
392 +2.4171990156173706054687500000000000000000e-01L,
393 +3.5325408107597432515913513900103385655073e-08L,
394 +2.5391519069671630859375000000000000000000e-01L,
395 +1.9284247135543573297906606667466299224747e-08L,
396 +2.6596349477767944335937500000000000000000e-01L,
397 +5.3719458497979750926537543389268821141517e-08L,
398 +2.7786844968795776367187500000000000000000e-01L,
399 +1.3154985425144750329234012330820349974537e-09L,
400 +2.8963327407836914062500000000000000000000e-01L,
401 +1.8504673536253893055525668970003860369760e-08L,
402 +3.0126130580902099609375000000000000000000e-01L,
403 +2.4769140784919125538233755492657352680723e-08L,
404 +3.1275570392608642578125000000000000000000e-01L,
405 +6.0778104626049965596883190321597861455475e-09L,
406 +3.2411944866180419921875000000000000000000e-01L,
407 +1.9992407776871920760434987352182336158873e-08L,
408 +3.3535552024841308593750000000000000000000e-01L,
409 +2.1672724744319679579814166199074433006807e-08L,
410 +3.4646672010421752929687500000000000000000e-01L,
411 +4.7241991051621587188425772950711830538414e-08L,
412 +3.5745584964752197265625000000000000000000e-01L,
413 +3.9274281801569759490140904474434669956562e-08L,
414 +3.6832553148269653320312500000000000000000e-01L,
415 +2.9676011119845105154050398826897178765758e-08L,
416 +3.7907832860946655273437500000000000000000e-01L,
417 +2.4325502905656478345631019858881408009210e-08L,
418 +3.8971674442291259765625000000000000000000e-01L,
419 +6.7171126157142136040035208670510556529487e-09L,
420 +4.0024316310882568359375000000000000000000e-01L,
421 +1.0181870233355751019951311700799406124957e-09L,
422 +4.1065990924835205078125000000000000000000e-01L,
423 +1.5736916335153056203175822787661567534220e-08L,
424 +4.2096924781799316406250000000000000000000e-01L,
425 +4.6826136472066367161506795972449857268707e-08L,
426 +4.3117344379425048828125000000000000000000e-01L,
427 +2.1024120852577922478955594998480144051225e-08L,
428 +4.4127452373504638671875000000000000000000e-01L,
429 +3.7069828842770746441661301225362605528786e-08L,
430 +4.5127463340759277343750000000000000000000e-01L,
431 +1.0731865811707192383079012478685922879010e-08L,
432 +4.6117568016052246093750000000000000000000e-01L,
433 +3.4961647705430499925597855358603099030515e-08L,
434 +4.7097969055175781250000000000000000000000e-01L,
435 +2.4667033200046897856056359251373510964634e-08L,
436 +4.8068851232528686523437500000000000000000e-01L,
437 +1.7020465042442243455448011551208861216878e-08L,
438 +4.9030393362045288085937500000000000000000e-01L,
439 +5.4424740957290971159645746860530583309571e-08L,
440 +4.9982786178588867187500000000000000000000e-01L,
441 +7.7705606579463314152470441415126573566105e-09L,
442 +5.0926184654235839843750000000000000000000e-01L,
443 +5.5247449548366574919228323824878565745713e-08L,
444 +5.1860773563385009765625000000000000000000e-01L,
445 +2.8574195534496726996364798698556235730848e-08L,
446 +5.2786707878112792968750000000000000000000e-01L,
447 +1.0839714455426392217778300963558522088193e-08L,
448 +5.3704142570495605468750000000000000000000e-01L,
449 +4.0191927599879229244153832299023744345999e-08L,
450 +5.4613238573074340820312500000000000000000e-01L,
451 +5.1867392242179272209231209163864971792889e-08L,
452 +5.5514144897460937500000000000000000000000e-01L,
453 +5.8565892217715480359515904050170125743178e-08L,
454 +5.6407010555267333984375000000000000000000e-01L,
455 +3.2732129626227634290090190711817681692354e-08L,
456 +5.7291972637176513671875000000000000000000e-01L,
457 +2.7190020372374006726626261068626400393936e-08L,
458 +5.8169168233871459960937500000000000000000e-01L,
459 +5.7295907882911235753725372340709967597394e-08L,
460 +5.9038740396499633789062500000000000000000e-01L,
461 +4.2637180036751291708123598757577783615014e-08L,
462 +5.9900814294815063476562500000000000000000e-01L,
463 +4.6697932764615975024461651502060474048774e-08L,
464 +6.0755521059036254882812500000000000000000e-01L,
465 +3.9634179246672960152791125371893149820625e-08L,
466 +6.1602985858917236328125000000000000000000e-01L,
467 +1.8626341656366315928196700650292529688219e-08L,
468 +6.2443327903747558593750000000000000000000e-01L,
469 +8.9744179151050387440546731199093039879228e-09L,
470 +6.3276666402816772460937500000000000000000e-01L,
471 +5.5428701049364114685035797584887586099726e-09L,
472 +6.4103114604949951171875000000000000000000e-01L,
473 +3.3371431779336851334405392546708949047361e-08L,
474 +6.4922791719436645507812500000000000000000e-01L,
475 +2.9430743363812714969905311122271269100885e-08L,
476 +6.5735805034637451171875000000000000000000e-01L,
477 +2.2361985518423140023245936165514147093250e-08L,
478 +6.6542261838912963867187500000000000000000e-01L,
479 +1.4155960810278217610006660181148303091649e-08L,
480 +6.7342263460159301757812500000000000000000e-01L,
481 +4.0610573702719835388801017264750843477878e-08L,
482 +6.8135917186737060546875000000000000000000e-01L,
483 +5.2940532463479321559568089441735584156689e-08L,
484 +6.8923324346542358398437500000000000000000e-01L,
485 +3.7773385396340539337814603903232796216537e-08L,
486 };
487
488 /*
489 * S[j],S_trail[j] = 2**(j/32.) for the final computation of exp(t+w)
490 */
491 static const long double S[] = {
492 #if defined(__x86)
493 +1.0000000000000000000000000e+00L,
494 +1.0218971486541166782081522e+00L,
495 +1.0442737824274138402382006e+00L,
496 +1.0671404006768236181297224e+00L,
497 +1.0905077326652576591003302e+00L,
498 +1.1143867425958925362894369e+00L,
499 +1.1387886347566916536971221e+00L,
500 +1.1637248587775775137938619e+00L,
501 +1.1892071150027210666875674e+00L,
502 +1.2152473599804688780476325e+00L,
503 +1.2418578120734840485256747e+00L,
504 +1.2690509571917332224885722e+00L,
505 +1.2968395546510096659215822e+00L,
506 +1.3252366431597412945939118e+00L,
507 +1.3542555469368927282668852e+00L,
508 +1.3839098819638319548151403e+00L,
509 +1.4142135623730950487637881e+00L,
510 +1.4451808069770466200253470e+00L,
511 +1.4768261459394993113155431e+00L,
512 +1.5091644275934227397133885e+00L,
513 +1.5422108254079408235859630e+00L,
514 +1.5759808451078864864006862e+00L,
515 +1.6104903319492543080837174e+00L,
516 +1.6457554781539648445110730e+00L,
517 +1.6817928305074290860378350e+00L,
518 +1.7186192981224779156032914e+00L,
519 +1.7562521603732994831094730e+00L,
520 +1.7947090750031071864148413e+00L,
521 +1.8340080864093424633989166e+00L,
522 +1.8741676341102999013002103e+00L,
523 +1.9152065613971472938202589e+00L,
524 +1.9571441241754002689657438e+00L,
525 #else
526 +1.00000000000000000000000000000000000e+00L,
527 +1.02189714865411667823448013478329942e+00L,
528 +1.04427378242741384032196647873992910e+00L,
529 +1.06714040067682361816952112099280918e+00L,
530 +1.09050773266525765920701065576070789e+00L,
531 +1.11438674259589253630881295691960313e+00L,
532 +1.13878863475669165370383028384151134e+00L,
533 +1.16372485877757751381357359909218536e+00L,
534 +1.18920711500272106671749997056047593e+00L,
535 +1.21524735998046887811652025133879836e+00L,
536 +1.24185781207348404859367746872659561e+00L,
537 +1.26905095719173322255441908103233805e+00L,
538 +1.29683955465100966593375411779245118e+00L,
539 +1.32523664315974129462953709549872168e+00L,
540 +1.35425554693689272829801474014070273e+00L,
541 +1.38390988196383195487265952726519287e+00L,
542 +1.41421356237309504880168872420969798e+00L,
543 +1.44518080697704662003700624147167095e+00L,
544 +1.47682614593949931138690748037404985e+00L,
545 +1.50916442759342273976601955103319352e+00L,
546 +1.54221082540794082361229186209073479e+00L,
547 +1.57598084510788648645527016018190504e+00L,
548 +1.61049033194925430817952066735740067e+00L,
549 +1.64575547815396484451875672472582254e+00L,
550 +1.68179283050742908606225095246642969e+00L,
551 +1.71861929812247791562934437645631244e+00L,
552 +1.75625216037329948311216061937531314e+00L,
553 +1.79470907500310718642770324212778174e+00L,
554 +1.83400808640934246348708318958828892e+00L,
555 +1.87416763411029990132999894995444645e+00L,
556 +1.91520656139714729387261127029583086e+00L,
557 +1.95714412417540026901832225162687149e+00L,
558 #endif
559 };
560
561 static const long double S_trail[] = {
562 #if defined(__x86)
563 +0.0000000000000000000000000e+00L,
564 +2.6327965667180882569382524e-20L,
565 +8.3765863521895191129661899e-20L,
566 +3.9798705777454504249209575e-20L,
567 +1.0668046596651558640993042e-19L,
568 +1.9376009847285360448117114e-20L,
569 +6.7081819456112953751277576e-21L,
570 +1.9711680502629186462729727e-20L,
571 +2.9932584438449523689104569e-20L,
572 +6.8887754153039109411061914e-20L,
573 +6.8002718741225378942847820e-20L,
574 +6.5846917376975403439742349e-20L,
575 +1.2171958727511372194876001e-20L,
576 +3.5625253228704087115438260e-20L,
577 +3.1129551559077560956309179e-20L,
578 +5.7519192396164779846216492e-20L,
579 +3.7900651177865141593101239e-20L,
580 +1.1659262405698741798080115e-20L,
581 +7.1364385105284695967172478e-20L,
582 +5.2631003710812203588788949e-20L,
583 +2.6328853788732632868460580e-20L,
584 +5.4583950085438242788190141e-20L,
585 +9.5803254376938269960718656e-20L,
586 +7.6837733983874245823512279e-21L,
587 +2.4415965910835093824202087e-20L,
588 +2.6052966871016580981769728e-20L,
589 +2.6876456344632553875309579e-21L,
590 +1.2861930155613700201703279e-20L,
591 +8.8166633394037485606572294e-20L,
592 +2.9788615389580190940837037e-20L,
593 +5.2352341619805098677422139e-20L,
594 +5.2578463064010463732242363e-20L,
595 #else
596 +0.00000000000000000000000000000000000e+00L,
597 +1.80506787420330954745573333054573786e-35L,
598 -9.37452029228042742195756741973083214e-35L,
599 -1.59696844729275877071290963023149997e-35L,
600 +9.11249341012502297851168610167248666e-35L,
601 -6.50422820697854828723037477525938871e-35L,
602 -8.14846884452585113732569176748815532e-35L,
603 -5.06621457672180031337233074514290335e-35L,
604 -1.35983097468881697374987563824591912e-35L,
605 +9.49742763556319647030771056643324660e-35L,
606 -3.28317052317699860161506596533391526e-36L,
607 -5.01723570938719041029018653045842895e-35L,
608 -2.39147479768910917162283430160264014e-35L,
609 -8.35057135763390881529889073794408385e-36L,
610 +7.03675688907326504242173719067187644e-35L,
611 -5.18248485306464645753689301856695619e-35L,
612 +9.42224254862183206569211673639406488e-35L,
613 -3.96750082539886230916730613021641828e-35L,
614 +7.14352899156330061452327361509276724e-35L,
615 +1.15987125286798512424651783410044433e-35L,
616 +4.69693347835811549530973921320187447e-35L,
617 -3.38651317599500471079924198499981917e-35L,
618 -8.58731877429824706886865593510387445e-35L,
619 -9.60595154874935050318549936224606909e-35L,
620 +9.60973393212801278450755869714178581e-35L,
621 +6.37839792144002843924476144978084855e-35L,
622 +7.79243078569586424945646112516927770e-35L,
623 +7.36133776758845652413193083663393220e-35L,
624 -6.47299514791334723003521457561217053e-35L,
625 +8.58747441795369869427879806229522962e-35L,
626 +2.37181542282517483569165122830269098e-35L,
627 -3.02689168209611877300459737342190031e-37L,
628 #endif
629 };
630
631
632 /*
633 * return tgamma(x) scaled by 2**-m for 8<x<=171.62... using Stirling's formula
634 * log(G(x)) ~= (x-.5)*(log(x)-1) + .5(log(2*pi)-1) + (1/x)*P(1/(x*x))
635 * = L1 + L2 + L3,
636 */
637 static struct LDouble
638 large_gam(long double x, int *m)
639 {
640 long double z, t1, t2, t3, z2, t5, w, y, u, r, v;
641 long double t24 = 16777216.0L, p24 = 1.0L / 16777216.0L;
642 int n2, j2, k, ix, j, i;
643 struct LDouble zz;
644 long double u2, ss_h, ss_l, r_h, w_h, w_l, t4;
645
646 /* BEGIN CSTYLED */
647 /*
648 * compute ss = ss.h+ss.l = log(x)-1 (see tgamma_log.h for details)
649 *
650 * log(x) - 1 = T1(n) + T2(j) + T3(s), where x = 2**n * y, 1<=y<2,
651 * j=[64*y], z[j]=1+j/64+1/128, s = (y-z[j])/(y+z[j]), and
652 * T1(n) = T1[2n,2n+1] = n*log(2)-1,
653 * T2(j) = T2[2j,2j+1] = log(z[j]),
654 * T3(s) = 2s + T3[0]s^3 + T3[1]s^5 + ... + T3[6]s^15
655 * Note
656 * (1) the leading entries are truncated to 24 binary point.
657 * (2) Remez error for T3(s) is bounded by 2**(-72.4)
658 * 2**(-24)
659 * _________V___________________
660 * T1(n): |_________|___________________|
661 * _______ ______________________
662 * T2(j): |_______|______________________|
663 * ____ _______________________
664 * 2s: |____|_______________________|
665 * __________________________
666 * + T3(s)-2s: |__________________________|
667 * -------------------------------------------
668 * [leading] + [Trailing]
669 */
670 /* END CSTYLED */
671 ix = H0_WORD(x);
672 n2 = (ix >> 16) - 0x3fff; /* exponent of x, range:3-10 */
673 y = scalbnl(x, -n2); /* y = scale x to [1,2] */
674 n2 += n2; /* 2n */
675 j = (ix >> 10) & 0x3f; /* j */
676 z = 1.0078125L + (long double)j * 0.015625L; /* z[j]=1+j/64+1/128 */
677 j2 = j + j;
678 t1 = y + z;
679 t2 = y - z;
680 r = one / t1;
681 u = r * t2; /* u = (y-z)/(y+z) */
682 t1 = CHOPPED(t1);
683 t4 = T2[j2 + 1] + T1[n2 + 1];
684 z2 = u * u;
685 k = H0_WORD(u) & 0x7fffffff;
686 t3 = T2[j2] + T1[n2];
687
688 for (t5 = T3[6], i = 5; i >= 0; i--)
689 t5 = z2 * t5 + T3[i];
690
691 if ((k >> 16) < 0x3fec) { /* |u|<2**-19 */
692 t2 = t4 + u * (two + z2 * t5);
693 } else {
694 t5 = t4 + (u * z2) * t5;
695 u2 = u + u;
696 v = (long double)((int)(u2 * t24)) * p24;
697 t2 = t5 + r * ((two * t2 - v * t1) - v * (y - (t1 - z)));
698 t3 += v;
699 }
700
701 ss_h = CHOPPED((t2 + t3));
702 ss_l = t2 - (ss_h - t3);
703
704 /*
705 * compute ww = (x-.5)*(log(x)-1) + .5*(log(2pi)-1) + 1/x*(P(1/x^2)))
706 * where ss = log(x) - 1 in already in extra precision
707 */
708 z = one / x;
709 r = x - half;
710 r_h = CHOPPED((r));
711 w_h = r_h * ss_h + hln2pim1_h;
712 z2 = z * z;
713 w = (r - r_h) * ss_h + r * ss_l;
714 t1 = GP[19];
715
716 for (i = 18; i > 0; i--)
717 t1 = z2 * t1 + GP[i];
718
719 w += hln2pim1_l;
720 w_l = z * (GP[0] + z2 * t1) + w;
721 k = (int)((w_h + w_l) * invln2_32 + half);
722
723 /* compute the exponential of w_h+w_l */
724
725 j = k & 0x1f;
726 *m = k >> 5;
727 t3 = (long double)k;
728
729 /* perform w - k*ln2_32 (represent as w_h - w_l) */
730 t1 = w_h - t3 * ln2_32hi;
731 t2 = t3 * ln2_32lo;
732 w = t2 - w_l;
733 w_h = t1 - w;
734 w_l = w - (t1 - w_h);
735
736 /* compute exp(w_h-w_l) */
737 z = w_h - w_l;
738
739 for (t1 = Et[10], i = 9; i >= 0; i--)
740 t1 = z * t1 + Et[i];
741
742 t3 = w_h - (w_l - (z * z) * t1); /* t3 = expm1(z) */
743 zz.l = S_trail[j] * (one + t3) + S[j] * t3;
744 zz.h = S[j];
745 return (zz);
746 }
747
748
749 /*
750 * kpsin(x)= sin(pi*x)/pi
751 * 3 5 7 9 11 27
752 * = x+ks[0]*x +ks[1]*x +ks[2]*x +ks[3]*x +ks[4]*x + ... + ks[12]*x
753 */
754 static const long double ks[] = {
755 -1.64493406684822643647241516664602518705158902870e+0000L,
756 +8.11742425283353643637002772405874238094995726160e-0001L,
757 -1.90751824122084213696472111835337366232282723933e-0001L,
758 +2.61478478176548005046532613563241288115395517084e-0002L,
759 -2.34608103545582363750893072647117829448016479971e-0003L,
760 +1.48428793031071003684606647212534027556262040158e-0004L,
761 -6.97587366165638046518462722252768122615952898698e-0006L,
762 +2.53121740413702536928659271747187500934840057929e-0007L,
763 -7.30471182221385990397683641695766121301933621956e-0009L,
764 +1.71653847451163495739958249695549313987973589884e-0010L,
765 -3.34813314714560776122245796929054813458341420565e-0012L,
766 +5.50724992262622033449487808306969135431411753047e-0014L,
767 -7.67678132753577998601234393215802221104236979928e-0016L,
768 };
769
770 /*
771 * assume x is not tiny and positive
772 */
773 static struct LDouble
774 kpsin(long double x)
775 {
776 long double z, t1, t2;
777 struct LDouble xx;
778 int i;
779
780 z = x * x;
781 xx.h = x;
782
783 for (t2 = ks[12], i = 11; i > 0; i--)
784 t2 = z * t2 + ks[i];
785
786 t1 = z * x;
787 t2 *= z * t1;
788 xx.l = t1 * ks[0] + t2;
789 return (xx);
790 }
791
792
793 /*
794 * kpcos(x)= cos(pi*x)/pi
795 * 2 4 6 8 10 12
796 * = 1/pi +kc[0]*x +kc[1]*x +kc[2]*x +kc[3]*x +kc[4]*x +kc[5]*x
797 *
798 * 2 4 6 8 10 22
799 * = 1/pi - pi/2*x +kc[0]*x +kc[1]*x +kc[2]*x +kc[3]*x +...+kc[9]*x
800 *
801 * -pi/2*x*x = (npi_2_h + npi_2_l) * (x_f+x_l)*(x_f+x_l)
802 * = npi_2_h*(x_f+x_l)*(x_f+x_l) + npi_2_l*x*x
803 * = npi_2_h*x_f*x_f + npi_2_h*(x*x-x_f*x_f) + npi_2_l*x*x
804 * = npi_2_h*x_f*x_f + npi_2_h*(x+x_f)*(x-x_f) + npi_2_l*x*x
805 * Here x_f = (long double) (float)x
806 * Note that pi/2(in hex) =
807 * 1.921FB54442D18469898CC51701B839A252049C1114CF98E804177D4C76273644A29
808 * npi_2_h = -pi/2 chopped to 25 bits = -1.921FB50000000000000000000000000 =
809 * -1.570796310901641845703125000000000 and
810 * npi_2_l =
811 * -0.0000004442D18469898CC51701B839A252049C1114CF98E804177D4C76273644A29 =
812 * -.0000000158932547735281966916397514420985846996875529104874722961539 =
813 * -1.5893254773528196691639751442098584699687552910487472296153e-8
814 * 1/pi(in hex) =
815 * .517CC1B727220A94FE13ABE8FA9A6EE06DB14ACC9E21C820FF28B1D5EF5DE2B
816 * will be splitted into:
817 * one_pi_h = 1/pi chopped to 48 bits = .517CC1B727220000000000... and
818 * one_pi_l = .0000000000000A94FE13ABE8FA9A6EE06DB14ACC9E21C820FF28B1D5EF5DE2B
819 */
820
821 static const long double
822 #if defined(__x86)
823 one_pi_h = 0.3183098861481994390487670898437500L, /* 31 bits */
824 one_pi_l = 3.559123248900043690127872406891929148e-11L,
825 #else
826 one_pi_h = 0.31830988618379052468299050815403461456298828125L,
827 one_pi_l = 1.46854777018590994109505931010230912897495334688117e-16L,
828 #endif
829 npi_2_h = -1.570796310901641845703125000000000L,
830 npi_2_l = -1.5893254773528196691639751442098584699687552910e-8L;
831 static const long double kc[] = {
832 +1.29192819501249250731151312779548918765320728489e+0000L,
833 -4.25027339979557573976029596929319207009444090366e-0001L,
834 +7.49080661650990096109672954618317623888421628613e-0002L,
835 -8.21458866111282287985539464173976555436050215120e-0003L,
836 +6.14202578809529228503205255165761204750211603402e-0004L,
837 -3.33073432691149607007217330302595267179545908740e-0005L,
838 +1.36970959047832085796809745461530865597993680204e-0006L,
839 -4.41780774262583514450246512727201806217271097336e-0008L,
840 +1.14741409212381858820016567664488123478660705759e-0009L,
841 -2.44261236114707374558437500654381006300502749632e-0011L,
842 };
843
844 /*
845 * assume x is not tiny and positive
846 */
847 static struct LDouble
848 kpcos(long double x)
849 {
850 long double z, t1, t2, t3, t4, x4, x8;
851 int i;
852 struct LDouble xx;
853
854 z = x * x;
855 xx.h = one_pi_h;
856 t1 = (long double)((float)x);
857 x4 = z * z;
858 t2 = npi_2_l * z + npi_2_h * (x + t1) * (x - t1);
859
860 for (i = 8, t3 = kc[9]; i >= 0; i--)
861 t3 = z * t3 + kc[i];
862
863 t3 = one_pi_l + x4 * t3;
864 t4 = t1 * t1 * npi_2_h;
865 x8 = t2 + t3;
866 xx.l = x8 + t4;
867 return (xx);
868 }
869
870 static const long double
871 /* 0.13486180573279076968979393577465291700642511139552429398233 */
872 #if defined(__x86)
873 t0z1 = 0.1348618057327907696779385054997035808810L,
874 t0z1_l = 1.1855430274949336125392717150257379614654e-20L,
875 #else
876 t0z1 = 0.1348618057327907696897939357746529168654L,
877 t0z1_l = 1.4102088588676879418739164486159514674310e-37L,
878 #endif
879 /* 0.46163214496836234126265954232572132846819620400644635129599 */
880 #if defined(__x86)
881 t0z2 = 0.4616321449683623412538115843295472018326L,
882 t0z2_l = 8.84795799617412663558532305039261747030640e-21L,
883 #else
884 t0z2 = 0.46163214496836234126265954232572132343318L,
885 t0z2_l = 5.03501162329616380465302666480916271611101e-36L,
886 #endif
887 /* 0.81977310110050060178786870492160699631174407846245179119586 */
888 #if defined(__x86)
889 t0z3 = 0.81977310110050060178773362329351925836817L,
890 t0z3_l = 1.350816280877379435658077052534574556256230e-22L
891 #else
892 t0z3 = 0.8197731011005006017878687049216069516957449L,
893 t0z3_l = 4.461599916947014419045492615933551648857380e-35L
894 #endif
895 ;
896
897 /*
898 * gamma(x+i) for 0 <= x < 1
899 */
900 static struct LDouble
901 gam_n(int i, long double x)
902 {
903 struct LDouble rr = { 0.0L, 0.0L }, yy;
904 long double r1, r2, t2, z, xh, xl, yh, yl, zh, z1, z2, zl, x5, wh, wl;
905
906 /* compute yy = gamma(x+1) */
907 if (x > 0.2845L) {
908 if (x > 0.6374L) {
909 r1 = x - t0z3;
910 r2 = CHOPPED((r1 - t0z3_l));
911 t2 = r1 - r2;
912 yy = GT3(r2, t2 - t0z3_l);
913 } else {
914 r1 = x - t0z2;
915 r2 = CHOPPED((r1 - t0z2_l));
916 t2 = r1 - r2;
917 yy = GT2(r2, t2 - t0z2_l);
918 }
919 } else {
920 r1 = x - t0z1;
921 r2 = CHOPPED((r1 - t0z1_l));
922 t2 = r1 - r2;
923 yy = GT1(r2, t2 - t0z1_l);
924 }
925
926 /* compute gamma(x+i) = (x+i-1)*...*(x+1)*yy, 0<i<8 */
927 switch (i) {
928 case 0: /* yy/x */
929 r1 = one / x;
930 xh = CHOPPED((x)); /* x is not tiny */
931 rr.h = CHOPPED(((yy.h + yy.l) * r1));
932 rr.l = r1 * (yy.h - rr.h * xh) - ((r1 * rr.h) * (x - xh) - r1 *
933 yy.l);
934 break;
935 case 1: /* yy */
936 rr.h = yy.h;
937 rr.l = yy.l;
938 break;
939 case 2: /* (x+1)*yy */
940 z = x + one; /* may not be exact */
941 zh = CHOPPED((z));
942 rr.h = zh * yy.h;
943 rr.l = z * yy.l + (x - (zh - one)) * yy.h;
944 break;
945 case 3: /* (x+2)*(x+1)*yy */
946 z1 = x + one;
947 z2 = x + 2.0L;
948 z = z1 * z2;
949 xh = CHOPPED((z));
950 zh = CHOPPED((z1));
951 xl = (x - (zh - one)) * (z2 + zh) - (xh - zh * (zh + one));
952
953 rr.h = xh * yy.h;
954 rr.l = z * yy.l + xl * yy.h;
955 break;
956
957 case 4: /* (x+1)*(x+3)*(x+2)*yy */
958 z1 = x + 2.0L;
959 z2 = (x + one) * (x + 3.0L);
960 zh = CHOPPED(z1);
961 zl = x - (zh - 2.0L);
962 xh = CHOPPED(z2);
963 xl = zl * (zh + z1) - (xh - (zh * zh - one));
964
965 /* wh+wl=(x+2)*yy */
966 wh = CHOPPED((z1 * (yy.h + yy.l)));
967 wl = (zl * yy.h + z1 * yy.l) - (wh - zh * yy.h);
968
969 rr.h = xh * wh;
970 rr.l = z2 * wl + xl * wh;
971
972 break;
973 case 5: /* ((x+1)*(x+4)*(x+2)*(x+3))*yy */
974 z1 = x + 2.0L;
975 z2 = x + 3.0L;
976 z = z1 * z2;
977 zh = CHOPPED((z1));
978 yh = CHOPPED((z));
979 yl = (x - (zh - 2.0L)) * (z2 + zh) - (yh - zh * (zh + one));
980 z2 = z - 2.0L;
981 z *= z2;
982 xh = CHOPPED((z));
983 xl = yl * (z2 + yh) - (xh - yh * (yh - 2.0L));
984 rr.h = xh * yy.h;
985 rr.l = z * yy.l + xl * yy.h;
986 break;
987 case 6: /* ((x+1)*(x+2)*(x+3)*(x+4)*(x+5))*yy */
988 z1 = x + 2.0L;
989 z2 = x + 3.0L;
990 z = z1 * z2;
991 zh = CHOPPED((z1));
992 yh = CHOPPED((z));
993 z1 = x - (zh - 2.0L);
994 yl = z1 * (z2 + zh) - (yh - zh * (zh + one));
995 z2 = z - 2.0L;
996 x5 = x + 5.0L;
997 z *= z2;
998 xh = CHOPPED(z);
999 zh += 3.0;
1000 xl = yl * (z2 + yh) - (xh - yh * (yh - 2.0L));
1001
1002 /*
1003 * xh+xl=(x+1)*...*(x+4)
1004 * wh+wl=(x+5)*yy
1005 */
1006 wh = CHOPPED((x5 * (yy.h + yy.l)));
1007 wl = (z1 * yy.h + x5 * yy.l) - (wh - zh * yy.h);
1008 rr.h = wh * xh;
1009 rr.l = z * wl + xl * wh;
1010 break;
1011 case 7: /* ((x+1)*(x+2)*(x+3)*(x+4)*(x+5)*(x+6))*yy */
1012 z1 = x + 3.0L;
1013 z2 = x + 4.0L;
1014 z = z2 * z1;
1015 zh = CHOPPED((z1));
1016 yh = CHOPPED((z)); /* yh+yl = (x+3)(x+4) */
1017 yl = (x - (zh - 3.0L)) * (z2 + zh) - (yh - (zh * (zh + one)));
1018 z1 = x + 6.0L;
1019 z2 = z - 2.0L; /* z2 = (x+2)*(x+5) */
1020 z *= z2;
1021 xh = CHOPPED((z));
1022 xl = yl * (z2 + yh) - (xh - yh * (yh - 2.0L));
1023
1024 /*
1025 * xh+xl=(x+2)*...*(x+5)
1026 * wh+wl=(x+1)(x+6)*yy
1027 */
1028 z2 -= 4.0L; /* z2 = (x+1)(x+6) */
1029 wh = CHOPPED((z2 * (yy.h + yy.l)));
1030 wl = (z2 * yy.l + yl * yy.h) - (wh - (yh - 6.0L) * yy.h);
1031 rr.h = wh * xh;
1032 rr.l = z * wl + xl * wh;
1033 }
1034
1035 return (rr);
1036 }
1037
1038 long double
1039 tgammal(long double x)
1040 {
1041 struct LDouble ss, ww;
1042 long double t, t1, t2, t3, t4, t5, w, y, z, z1, z2, z3, z5;
1043 int i, j, m, ix, hx, xk;
1044 unsigned lx;
1045
1046 hx = H0_WORD(x);
1047 lx = H3_WORD(x);
1048 ix = hx & 0x7fffffff;
1049 y = x;
1050
1051 if (ix < 0x3f8e0000) /* x < 2**-113 */
1052 return (one / x);
1053
1054 if (ix >= 0x7fff0000)
1055 return (x * ((hx < 0) ? zero : x)); /* Inf or NaN */
1056
1057 if (x > overflow) /* overflow threshold */
1058 return (x * 1.0e4932L);
1059
1060 if (hx >= 0x40020000) { /* x >= 8 */
1061 ww = large_gam(x, &m);
1062 w = ww.h + ww.l;
1063 return (scalbnl(w, m));
1064 }
1065
1066 if (hx > 0) { /* 0 < x < 8 */
1067 i = (int)x;
1068 ww = gam_n(i, x - (long double)i);
1069 return (ww.h + ww.l);
1070 }
1071
1072 /*
1073 * negative x
1074 */
1075
1076 /*
1077 * compute xk =
1078 * -2 ... x is an even int (-inf is considered an even #)
1079 * -1 ... x is an odd int
1080 * +0 ... x is not an int but chopped to an even int
1081 * +1 ... x is not an int but chopped to an odd int
1082 */
1083 xk = 0;
1084 #if defined(__x86)
1085 if (ix >= 0x403e0000) { /* x >= 2**63 } */
1086 if (ix >= 0x403f0000)
1087 xk = -2;
1088 else
1089 xk = -2 + (lx & 1);
1090 #else
1091 if (ix >= 0x406f0000) { /* x >= 2**112 */
1092 if (ix >= 0x40700000)
1093 xk = -2;
1094 else
1095 xk = -2 + (lx & 1);
1096 #endif
1097 } else if (ix >= 0x3fff0000) {
1098 w = -x;
1099 t1 = floorl(w);
1100 t2 = t1 * half;
1101 t3 = floorl(t2);
1102
1103 if (t1 == w) {
1104 if (t2 == t3)
1105 xk = -2;
1106 else
1107 xk = -1;
1108 } else {
1109 if (t2 == t3)
1110 xk = 0;
1111 else
1112 xk = 1;
1113 }
1114 }
1115
1116 if (xk < 0) {
1117 /* return NaN. Ideally gamma(-n)= (-1)**(n+1) * inf */
1118 return ((x - x) / (x - x));
1119 }
1120
1121 /*
1122 * negative underflow thresold -(1774+9ulp)
1123 */
1124 if (x < -1774.0000000000000000000000000000017749370L) {
1125 z = tiny / x;
1126
1127 if (xk == 1)
1128 z = -z;
1129
1130 return (z * tiny);
1131 }
1132
1133
1134 /*
1135 * now compute gamma(x) by -1/((sin(pi*y)/pi)*gamma(1+y)), y = -x
1136 */
1137
1138 /*
1139 * First compute ss = -sin(pi*y)/pi so that
1140 * gamma(x) = 1/(ss*gamma(1+y))
1141 */
1142 y = -x;
1143 j = (int)y;
1144 z = y - (long double)j;
1145
1146 if (z > 0.3183098861837906715377675L) {
1147 if (z > 0.6816901138162093284622325L)
1148 ss = kpsin(one - z);
1149 else
1150 ss = kpcos(0.5L - z);
1151 } else {
1152 ss = kpsin(z);
1153 }
1154
1155 if (xk == 0) {
1156 ss.h = -ss.h;
1157 ss.l = -ss.l;
1158 }
1159
1160 /* Then compute ww = gamma(1+y), note that result scale to 2**m */
1161 m = 0;
1162
1163 if (j < 7) {
1164 ww = gam_n(j + 1, z);
1165 } else {
1166 w = y + one;
1167
1168 if ((lx & 1) == 0) { /* y+1 exact (note that y<184) */
1169 ww = large_gam(w, &m);
1170 } else {
1171 t = w - one;
1172
1173 if (t == y) { /* y+one exact */
1174 ww = large_gam(w, &m);
1175 } else { /* use y*gamma(y) */
1176 if (j == 7)
1177 ww = gam_n(j, z);
1178 else
1179 ww = large_gam(y, &m);
1180
1181 t4 = ww.h + ww.l;
1182 t1 = CHOPPED((y));
1183 t2 = CHOPPED((t4));
1184 /* t4 will not be too large */
1185 ww.l = y * (ww.l - (t2 - ww.h)) + (y - t1) * t2;
1186 ww.h = t1 * t2;
1187 }
1188 }
1189 }
1190
1191 /* compute 1/(ss*ww) */
1192 t3 = ss.h + ss.l;
1193 t4 = ww.h + ww.l;
1194 t1 = CHOPPED((t3));
1195 t2 = CHOPPED((t4));
1196 z1 = ss.l - (t1 - ss.h); /* (t1,z1) = ss */
1197 z2 = ww.l - (t2 - ww.h); /* (t2,z2) = ww */
1198 t3 = t3 * t4; /* t3 = ss*ww */
1199 z3 = one / t3; /* z3 = 1/(ss*ww) */
1200 t5 = t1 * t2;
1201 z5 = z1 * t4 + t1 * z2; /* (t5,z5) = ss*ww */
1202 t1 = CHOPPED((t3)); /* (t1,z1) = ss*ww */
1203 z1 = z5 - (t1 - t5);
1204 t2 = CHOPPED((z3)); /* leading 1/(ss*ww) */
1205 z2 = z3 * (t2 * z1 - (one - t2 * t1));
1206 z = t2 - z2;
1207
1208 return (scalbnl(z, -m));
1209 }