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[/] [openrisc/] [trunk/] [gnu-dev/] [or1k-gcc/] [libquadmath/] [math/] [lgammaq.c] - Blame information for rev 742

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1 740 jeremybenn 
/*                                                      lgammal
2 
 *
3 
 *      Natural logarithm of gamma function
4 
 *
5 
 *
6 
 *
7 
 * SYNOPSIS:
8 
 *
9 
 * __float128 x, y, lgammal();
10 
 * extern int sgngam;
11 
 *
12 
 * y = lgammal(x);
13 
 *
14 
 *
15 
 *
16 
 * DESCRIPTION:
17 
 *
18 
 * Returns the base e (2.718...) logarithm of the absolute
19 
 * value of the gamma function of the argument.
20 
 * The sign (+1 or -1) of the gamma function is returned in a
21 
 * global (extern) variable named sgngam.
22 
 *
23 
 * The positive domain is partitioned into numerous segments for approximation.
24 
 * For x > 10,
25 
 *   log gamma(x) = (x - 0.5) log(x) - x + log sqrt(2 pi) + 1/x R(1/x^2)
26 
 * Near the minimum at x = x0 = 1.46... the approximation is
27 
 *   log gamma(x0 + z) = log gamma(x0) + z^2 P(z)/Q(z)
28 
 * for small z.
29 
 * Elsewhere between 0 and 10,
30 
 *   log gamma(n + z) = log gamma(n) + z P(z)/Q(z)
31 
 * for various selected n and small z.
32 
 *
33 
 * The cosecant reflection formula is employed for negative arguments.
34 
 *
35 
 *
36 
 *
37 
 * ACCURACY:
38 
 *
39 
 *
40 
 * arithmetic      domain        # trials     peak         rms
41 
 *                                            Relative error:
42 
 *    IEEE         10, 30         100000     3.9e-34     9.8e-35
43 
 *    IEEE          0, 10         100000     3.8e-34     5.3e-35
44 
 *                                            Absolute error:
45 
 *    IEEE         -10, 0         100000     8.0e-34     8.0e-35
46 
 *    IEEE         -30, -10       100000     4.4e-34     1.0e-34
47 
 *    IEEE        -100, 100       100000                 1.0e-34
48 
 *
49 
 * The absolute error criterion is the same as relative error
50 
 * when the function magnitude is greater than one but it is absolute
51 
 * when the magnitude is less than one.
52 
 *
53 
 */
54 
 
55 
/* Copyright 2001 by Stephen L. Moshier <moshier@na-net.ornl.gov>
56 
 
57 
    This library is free software; you can redistribute it and/or
58 
    modify it under the terms of the GNU Lesser General Public
59 
    License as published by the Free Software Foundation; either
60 
    version 2.1 of the License, or (at your option) any later version.
61 
 
62 
    This library is distributed in the hope that it will be useful,
63 
    but WITHOUT ANY WARRANTY; without even the implied warranty of
64 
    MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
65 
    Lesser General Public License for more details.
66 
 
67 
    You should have received a copy of the GNU Lesser General Public
68 
    License along with this library; if not, write to the Free Software
69 
    Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307  USA */
70 
 
71 
#include "quadmath-imp.h"
72 
 
73 
static const __float128 PIQ = 3.1415926535897932384626433832795028841972E0Q;
74 
static const __float128 MAXLGM = 1.0485738685148938358098967157129705071571E4928Q;
75 
static const __float128 one = 1.0Q;
76 
static const __float128 zero = 0.0Q;
77 
static const __float128 huge = 1.0e4000Q;
78 
 
79 
/* log gamma(x) = ( x - 0.5 ) * log(x) - x + LS2PI + 1/x P(1/x^2)
80 
   1/x <= 0.0741 (x >= 13.495...)
81 
   Peak relative error 1.5e-36  */
82 
static const __float128 ls2pi = 9.1893853320467274178032973640561763986140E-1Q;
83 
#define NRASY 12
84 
static const __float128 RASY[NRASY + 1] =
85 
{
86 
  8.333333333333333333333333333310437112111E-2Q,
87 
 -2.777777777777777777777774789556228296902E-3Q,
88 
  7.936507936507936507795933938448586499183E-4Q,
89 
 -5.952380952380952041799269756378148574045E-4Q,
90 
  8.417508417507928904209891117498524452523E-4Q,
91 
 -1.917526917481263997778542329739806086290E-3Q,
92 
  6.410256381217852504446848671499409919280E-3Q,
93 
 -2.955064066900961649768101034477363301626E-2Q,
94 
  1.796402955865634243663453415388336954675E-1Q,
95 
 -1.391522089007758553455753477688592767741E0Q,
96 
  1.326130089598399157988112385013829305510E1Q,
97 
 -1.420412699593782497803472576479997819149E2Q,
98 
  1.218058922427762808938869872528846787020E3Q
99 
};
100 
 
101 
 
102 
/* log gamma(x+13) = log gamma(13) +  x P(x)/Q(x)
103 
   -0.5 <= x <= 0.5
104 
   12.5 <= x+13 <= 13.5
105 
   Peak relative error 1.1e-36  */
106 
static const __float128 lgam13a = 1.9987213134765625E1Q;
107 
static const __float128 lgam13b = 1.3608962611495173623870550785125024484248E-6Q;
108 
#define NRN13 7
109 
static const __float128 RN13[NRN13 + 1] =
110 
{
111 
  8.591478354823578150238226576156275285700E11Q,
112 
  2.347931159756482741018258864137297157668E11Q,
113 
  2.555408396679352028680662433943000804616E10Q,
114 
  1.408581709264464345480765758902967123937E9Q,
115 
  4.126759849752613822953004114044451046321E7Q,
116 
  6.133298899622688505854211579222889943778E5Q,
117 
  3.929248056293651597987893340755876578072E3Q,
118 
  6.850783280018706668924952057996075215223E0Q
119 
};
120 
#define NRD13 6
121 
static const __float128 RD13[NRD13 + 1] =
122 
{
123 
  3.401225382297342302296607039352935541669E11Q,
124 
  8.756765276918037910363513243563234551784E10Q,
125 
  8.873913342866613213078554180987647243903E9Q,
126 
  4.483797255342763263361893016049310017973E8Q,
127 
  1.178186288833066430952276702931512870676E7Q,
128 
  1.519928623743264797939103740132278337476E5Q,
129 
  7.989298844938119228411117593338850892311E2Q
130 
 /* 1.0E0Q */
131 
};
132 
 
133 
 
134 
/* log gamma(x+12) = log gamma(12) +  x P(x)/Q(x)
135 
   -0.5 <= x <= 0.5
136 
   11.5 <= x+12 <= 12.5
137 
   Peak relative error 4.1e-36  */
138 
static const __float128 lgam12a = 1.75023040771484375E1Q;
139 
static const __float128 lgam12b = 3.7687254483392876529072161996717039575982E-6Q;
140 
#define NRN12 7
141 
static const __float128 RN12[NRN12 + 1] =
142 
{
143 
  4.709859662695606986110997348630997559137E11Q,
144 
  1.398713878079497115037857470168777995230E11Q,
145 
  1.654654931821564315970930093932954900867E10Q,
146 
  9.916279414876676861193649489207282144036E8Q,
147 
  3.159604070526036074112008954113411389879E7Q,
148 
  5.109099197547205212294747623977502492861E5Q,
149 
  3.563054878276102790183396740969279826988E3Q,
150 
  6.769610657004672719224614163196946862747E0Q
151 
};
152 
#define NRD12 6
153 
static const __float128 RD12[NRD12 + 1] =
154 
{
155 
  1.928167007860968063912467318985802726613E11Q,
156 
  5.383198282277806237247492369072266389233E10Q,
157 
  5.915693215338294477444809323037871058363E9Q,
158 
  3.241438287570196713148310560147925781342E8Q,
159 
  9.236680081763754597872713592701048455890E6Q,
160 
  1.292246897881650919242713651166596478850E5Q,
161 
  7.366532445427159272584194816076600211171E2Q
162 
 /* 1.0E0Q */
163 
};
164 
 
165 
 
166 
/* log gamma(x+11) = log gamma(11) +  x P(x)/Q(x)
167 
   -0.5 <= x <= 0.5
168 
   10.5 <= x+11 <= 11.5
169 
   Peak relative error 1.8e-35  */
170 
static const __float128 lgam11a = 1.5104400634765625E1Q;
171 
static const __float128 lgam11b = 1.1938309890295225709329251070371882250744E-5Q;
172 
#define NRN11 7
173 
static const __float128 RN11[NRN11 + 1] =
174 
{
175 
  2.446960438029415837384622675816736622795E11Q,
176 
  7.955444974446413315803799763901729640350E10Q,
177 
  1.030555327949159293591618473447420338444E10Q,
178 
  6.765022131195302709153994345470493334946E8Q,
179 
  2.361892792609204855279723576041468347494E7Q,
180 
  4.186623629779479136428005806072176490125E5Q,
181 
  3.202506022088912768601325534149383594049E3Q,
182 
  6.681356101133728289358838690666225691363E0Q
183 
};
184 
#define NRD11 6
185 
static const __float128 RD11[NRD11 + 1] =
186 
{
187 
  1.040483786179428590683912396379079477432E11Q,
188 
  3.172251138489229497223696648369823779729E10Q,
189 
  3.806961885984850433709295832245848084614E9Q,
190 
  2.278070344022934913730015420611609620171E8Q,
191 
  7.089478198662651683977290023829391596481E6Q,
192 
  1.083246385105903533237139380509590158658E5Q,
193 
  6.744420991491385145885727942219463243597E2Q
194 
 /* 1.0E0Q */
195 
};
196 
 
197 
 
198 
/* log gamma(x+10) = log gamma(10) +  x P(x)/Q(x)
199 
   -0.5 <= x <= 0.5
200 
   9.5 <= x+10 <= 10.5
201 
   Peak relative error 5.4e-37  */
202 
static const __float128 lgam10a = 1.280181884765625E1Q;
203 
static const __float128 lgam10b = 8.6324252196112077178745667061642811492557E-6Q;
204 
#define NRN10 7
205 
static const __float128 RN10[NRN10 + 1] =
206 
{
207 
  -1.239059737177249934158597996648808363783E14Q,
208 
  -4.725899566371458992365624673357356908719E13Q,
209 
  -7.283906268647083312042059082837754850808E12Q,
210 
  -5.802855515464011422171165179767478794637E11Q,
211 
  -2.532349691157548788382820303182745897298E10Q,
212 
  -5.884260178023777312587193693477072061820E8Q,
213 
  -6.437774864512125749845840472131829114906E6Q,
214 
  -2.350975266781548931856017239843273049384E4Q
215 
};
216 
#define NRD10 7
217 
static const __float128 RD10[NRD10 + 1] =
218 
{
219 
  -5.502645997581822567468347817182347679552E13Q,
220 
  -1.970266640239849804162284805400136473801E13Q,
221 
  -2.819677689615038489384974042561531409392E12Q,
222 
  -2.056105863694742752589691183194061265094E11Q,
223 
  -8.053670086493258693186307810815819662078E9Q,
224 
  -1.632090155573373286153427982504851867131E8Q,
225 
  -1.483575879240631280658077826889223634921E6Q,
226 
  -4.002806669713232271615885826373550502510E3Q
227 
 /* 1.0E0Q */
228 
};
229 
 
230 
 
231 
/* log gamma(x+9) = log gamma(9) +  x P(x)/Q(x)
232 
   -0.5 <= x <= 0.5
233 
   8.5 <= x+9 <= 9.5
234 
   Peak relative error 3.6e-36  */
235 
static const __float128 lgam9a = 1.06045989990234375E1Q;
236 
static const __float128 lgam9b = 3.9037218127284172274007216547549861681400E-6Q;
237 
#define NRN9 7
238 
static const __float128 RN9[NRN9 + 1] =
239 
{
240 
  -4.936332264202687973364500998984608306189E13Q,
241 
  -2.101372682623700967335206138517766274855E13Q,
242 
  -3.615893404644823888655732817505129444195E12Q,
243 
  -3.217104993800878891194322691860075472926E11Q,
244 
  -1.568465330337375725685439173603032921399E10Q,
245 
  -4.073317518162025744377629219101510217761E8Q,
246 
  -4.983232096406156139324846656819246974500E6Q,
247 
  -2.036280038903695980912289722995505277253E4Q
248 
};
249 
#define NRD9 7
250 
static const __float128 RD9[NRD9 + 1] =
251 
{
252 
  -2.306006080437656357167128541231915480393E13Q,
253 
  -9.183606842453274924895648863832233799950E12Q,
254 
  -1.461857965935942962087907301194381010380E12Q,
255 
  -1.185728254682789754150068652663124298303E11Q,
256 
  -5.166285094703468567389566085480783070037E9Q,
257 
  -1.164573656694603024184768200787835094317E8Q,
258 
  -1.177343939483908678474886454113163527909E6Q,
259 
  -3.529391059783109732159524500029157638736E3Q
260 
  /* 1.0E0Q */
261 
};
262 
 
263 
 
264 
/* log gamma(x+8) = log gamma(8) +  x P(x)/Q(x)
265 
   -0.5 <= x <= 0.5
266 
   7.5 <= x+8 <= 8.5
267 
   Peak relative error 2.4e-37  */
268 
static const __float128 lgam8a = 8.525146484375E0Q;
269 
static const __float128 lgam8b = 1.4876690414300165531036347125050759667737E-5Q;
270 
#define NRN8 8
271 
static const __float128 RN8[NRN8 + 1] =
272 
{
273 
  6.600775438203423546565361176829139703289E11Q,
274 
  3.406361267593790705240802723914281025800E11Q,
275 
  7.222460928505293914746983300555538432830E10Q,
276 
  8.102984106025088123058747466840656458342E9Q,
277 
  5.157620015986282905232150979772409345927E8Q,
278 
  1.851445288272645829028129389609068641517E7Q,
279 
  3.489261702223124354745894067468953756656E5Q,
280 
  2.892095396706665774434217489775617756014E3Q,
281 
  6.596977510622195827183948478627058738034E0Q
282 
};
283 
#define NRD8 7
284 
static const __float128 RD8[NRD8 + 1] =
285 
{
286 
  3.274776546520735414638114828622673016920E11Q,
287 
  1.581811207929065544043963828487733970107E11Q,
288 
  3.108725655667825188135393076860104546416E10Q,
289 
  3.193055010502912617128480163681842165730E9Q,
290 
  1.830871482669835106357529710116211541839E8Q,
291 
  5.790862854275238129848491555068073485086E6Q,
292 
  9.305213264307921522842678835618803553589E4Q,
293 
  6.216974105861848386918949336819572333622E2Q
294 
  /* 1.0E0Q */
295 
};
296 
 
297 
 
298 
/* log gamma(x+7) = log gamma(7) +  x P(x)/Q(x)
299 
   -0.5 <= x <= 0.5
300 
   6.5 <= x+7 <= 7.5
301 
   Peak relative error 3.2e-36  */
302 
static const __float128 lgam7a = 6.5792388916015625E0Q;
303 
static const __float128 lgam7b = 1.2320408538495060178292903945321122583007E-5Q;
304 
#define NRN7 8
305 
static const __float128 RN7[NRN7 + 1] =
306 
{
307 
  2.065019306969459407636744543358209942213E11Q,
308 
  1.226919919023736909889724951708796532847E11Q,
309 
  2.996157990374348596472241776917953749106E10Q,
310 
  3.873001919306801037344727168434909521030E9Q,
311 
  2.841575255593761593270885753992732145094E8Q,
312 
  1.176342515359431913664715324652399565551E7Q,
313 
  2.558097039684188723597519300356028511547E5Q,
314 
  2.448525238332609439023786244782810774702E3Q,
315 
  6.460280377802030953041566617300902020435E0Q
316 
};
317 
#define NRD7 7
318 
static const __float128 RD7[NRD7 + 1] =
319 
{
320 
  1.102646614598516998880874785339049304483E11Q,
321 
  6.099297512712715445879759589407189290040E10Q,
322 
  1.372898136289611312713283201112060238351E10Q,
323 
  1.615306270420293159907951633566635172343E9Q,
324 
  1.061114435798489135996614242842561967459E8Q,
325 
  3.845638971184305248268608902030718674691E6Q,
326 
  7.081730675423444975703917836972720495507E4Q,
327 
  5.423122582741398226693137276201344096370E2Q
328 
  /* 1.0E0Q */
329 
};
330 
 
331 
 
332 
/* log gamma(x+6) = log gamma(6) +  x P(x)/Q(x)
333 
   -0.5 <= x <= 0.5
334 
   5.5 <= x+6 <= 6.5
335 
   Peak relative error 6.2e-37  */
336 
static const __float128 lgam6a = 4.7874908447265625E0Q;
337 
static const __float128 lgam6b = 8.9805548349424770093452324304839959231517E-7Q;
338 
#define NRN6 8
339 
static const __float128 RN6[NRN6 + 1] =
340 
{
341 
  -3.538412754670746879119162116819571823643E13Q,
342 
  -2.613432593406849155765698121483394257148E13Q,
343 
  -8.020670732770461579558867891923784753062E12Q,
344 
  -1.322227822931250045347591780332435433420E12Q,
345 
  -1.262809382777272476572558806855377129513E11Q,
346 
  -7.015006277027660872284922325741197022467E9Q,
347 
  -2.149320689089020841076532186783055727299E8Q,
348 
  -3.167210585700002703820077565539658995316E6Q,
349 
  -1.576834867378554185210279285358586385266E4Q
350 
};
351 
#define NRD6 8
352 
static const __float128 RD6[NRD6 + 1] =
353 
{
354 
  -2.073955870771283609792355579558899389085E13Q,
355 
  -1.421592856111673959642750863283919318175E13Q,
356 
  -4.012134994918353924219048850264207074949E12Q,
357 
  -6.013361045800992316498238470888523722431E11Q,
358 
  -5.145382510136622274784240527039643430628E10Q,
359 
  -2.510575820013409711678540476918249524123E9Q,
360 
  -6.564058379709759600836745035871373240904E7Q,
361 
  -7.861511116647120540275354855221373571536E5Q,
362 
  -2.821943442729620524365661338459579270561E3Q
363 
  /* 1.0E0Q */
364 
};
365 
 
366 
 
367 
/* log gamma(x+5) = log gamma(5) +  x P(x)/Q(x)
368 
   -0.5 <= x <= 0.5
369 
   4.5 <= x+5 <= 5.5
370 
   Peak relative error 3.4e-37  */
371 
static const __float128 lgam5a = 3.17803955078125E0Q;
372 
static const __float128 lgam5b = 1.4279566695619646941601297055408873990961E-5Q;
373 
#define NRN5 9
374 
static const __float128 RN5[NRN5 + 1] =
375 
{
376 
  2.010952885441805899580403215533972172098E11Q,
377 
  1.916132681242540921354921906708215338584E11Q,
378 
  7.679102403710581712903937970163206882492E10Q,
379 
  1.680514903671382470108010973615268125169E10Q,
380 
  2.181011222911537259440775283277711588410E9Q,
381 
  1.705361119398837808244780667539728356096E8Q,
382 
  7.792391565652481864976147945997033946360E6Q,
383 
  1.910741381027985291688667214472560023819E5Q,
384 
  2.088138241893612679762260077783794329559E3Q,
385 
  6.330318119566998299106803922739066556550E0Q
386 
};
387 
#define NRD5 8
388 
static const __float128 RD5[NRD5 + 1] =
389 
{
390 
  1.335189758138651840605141370223112376176E11Q,
391 
  1.174130445739492885895466097516530211283E11Q,
392 
  4.308006619274572338118732154886328519910E10Q,
393 
  8.547402888692578655814445003283720677468E9Q,
394 
  9.934628078575618309542580800421370730906E8Q,
395 
  6.847107420092173812998096295422311820672E7Q,
396 
  2.698552646016599923609773122139463150403E6Q,
397 
  5.526516251532464176412113632726150253215E4Q,
398 
  4.772343321713697385780533022595450486932E2Q
399 
  /* 1.0E0Q */
400 
};
401 
 
402 
 
403 
/* log gamma(x+4) = log gamma(4) +  x P(x)/Q(x)
404 
   -0.5 <= x <= 0.5
405 
   3.5 <= x+4 <= 4.5
406 
   Peak relative error 6.7e-37  */
407 
static const __float128 lgam4a = 1.791748046875E0Q;
408 
static const __float128 lgam4b = 1.1422353055000812477358380702272722990692E-5Q;
409 
#define NRN4 9
410 
static const __float128 RN4[NRN4 + 1] =
411 
{
412 
  -1.026583408246155508572442242188887829208E13Q,
413 
  -1.306476685384622809290193031208776258809E13Q,
414 
  -7.051088602207062164232806511992978915508E12Q,
415 
  -2.100849457735620004967624442027793656108E12Q,
416 
  -3.767473790774546963588549871673843260569E11Q,
417 
  -4.156387497364909963498394522336575984206E10Q,
418 
  -2.764021460668011732047778992419118757746E9Q,
419 
  -1.036617204107109779944986471142938641399E8Q,
420 
  -1.895730886640349026257780896972598305443E6Q,
421 
  -1.180509051468390914200720003907727988201E4Q
422 
};
423 
#define NRD4 9
424 
static const __float128 RD4[NRD4 + 1] =
425 
{
426 
  -8.172669122056002077809119378047536240889E12Q,
427 
  -9.477592426087986751343695251801814226960E12Q,
428 
  -4.629448850139318158743900253637212801682E12Q,
429 
  -1.237965465892012573255370078308035272942E12Q,
430 
  -1.971624313506929845158062177061297598956E11Q,
431 
  -1.905434843346570533229942397763361493610E10Q,
432 
  -1.089409357680461419743730978512856675984E9Q,
433 
  -3.416703082301143192939774401370222822430E7Q,
434 
  -4.981791914177103793218433195857635265295E5Q,
435 
  -2.192507743896742751483055798411231453733E3Q
436 
  /* 1.0E0Q */
437 
};
438 
 
439 
 
440 
/* log gamma(x+3) = log gamma(3) +  x P(x)/Q(x)
441 
   -0.25 <= x <= 0.5
442 
   2.75 <= x+3 <= 3.5
443 
   Peak relative error 6.0e-37  */
444 
static const __float128 lgam3a = 6.93145751953125E-1Q;
445 
static const __float128 lgam3b = 1.4286068203094172321214581765680755001344E-6Q;
446 
 
447 
#define NRN3 9
448 
static const __float128 RN3[NRN3 + 1] =
449 
{
450 
  -4.813901815114776281494823863935820876670E11Q,
451 
  -8.425592975288250400493910291066881992620E11Q,
452 
  -6.228685507402467503655405482985516909157E11Q,
453 
  -2.531972054436786351403749276956707260499E11Q,
454 
  -6.170200796658926701311867484296426831687E10Q,
455 
  -9.211477458528156048231908798456365081135E9Q,
456 
  -8.251806236175037114064561038908691305583E8Q,
457 
  -4.147886355917831049939930101151160447495E7Q,
458 
  -1.010851868928346082547075956946476932162E6Q,
459 
  -8.333374463411801009783402800801201603736E3Q
460 
};
461 
#define NRD3 9
462 
static const __float128 RD3[NRD3 + 1] =
463 
{
464 
  -5.216713843111675050627304523368029262450E11Q,
465 
  -8.014292925418308759369583419234079164391E11Q,
466 
  -5.180106858220030014546267824392678611990E11Q,
467 
  -1.830406975497439003897734969120997840011E11Q,
468 
  -3.845274631904879621945745960119924118925E10Q,
469 
  -4.891033385370523863288908070309417710903E9Q,
470 
  -3.670172254411328640353855768698287474282E8Q,
471 
  -1.505316381525727713026364396635522516989E7Q,
472 
  -2.856327162923716881454613540575964890347E5Q,
473 
  -1.622140448015769906847567212766206894547E3Q
474 
  /* 1.0E0Q */
475 
};
476 
 
477 
 
478 
/* log gamma(x+2.5) = log gamma(2.5) +  x P(x)/Q(x)
479 
   -0.125 <= x <= 0.25
480 
   2.375 <= x+2.5 <= 2.75  */
481 
static const __float128 lgam2r5a = 2.8466796875E-1Q;
482 
static const __float128 lgam2r5b = 1.4901722919159632494669682701924320137696E-5Q;
483 
#define NRN2r5 8
484 
static const __float128 RN2r5[NRN2r5 + 1] =
485 
{
486 
  -4.676454313888335499356699817678862233205E9Q,
487 
  -9.361888347911187924389905984624216340639E9Q,
488 
  -7.695353600835685037920815799526540237703E9Q,
489 
  -3.364370100981509060441853085968900734521E9Q,
490 
  -8.449902011848163568670361316804900559863E8Q,
491 
  -1.225249050950801905108001246436783022179E8Q,
492 
  -9.732972931077110161639900388121650470926E6Q,
493 
  -3.695711763932153505623248207576425983573E5Q,
494 
  -4.717341584067827676530426007495274711306E3Q
495 
};
496 
#define NRD2r5 8
497 
static const __float128 RD2r5[NRD2r5 + 1] =
498 
{
499 
  -6.650657966618993679456019224416926875619E9Q,
500 
  -1.099511409330635807899718829033488771623E10Q,
501 
  -7.482546968307837168164311101447116903148E9Q,
502 
  -2.702967190056506495988922973755870557217E9Q,
503 
  -5.570008176482922704972943389590409280950E8Q,
504 
  -6.536934032192792470926310043166993233231E7Q,
505 
  -4.101991193844953082400035444146067511725E6Q,
506 
  -1.174082735875715802334430481065526664020E5Q,
507 
  -9.932840389994157592102947657277692978511E2Q
508 
  /* 1.0E0Q */
509 
};
510 
 
511 
 
512 
/* log gamma(x+2) = x P(x)/Q(x)
513 
   -0.125 <= x <= +0.375
514 
   1.875 <= x+2 <= 2.375
515 
   Peak relative error 4.6e-36  */
516 
#define NRN2 9
517 
static const __float128 RN2[NRN2 + 1] =
518 
{
519 
  -3.716661929737318153526921358113793421524E9Q,
520 
  -1.138816715030710406922819131397532331321E10Q,
521 
  -1.421017419363526524544402598734013569950E10Q,
522 
  -9.510432842542519665483662502132010331451E9Q,
523 
  -3.747528562099410197957514973274474767329E9Q,
524 
  -8.923565763363912474488712255317033616626E8Q,
525 
  -1.261396653700237624185350402781338231697E8Q,
526 
  -9.918402520255661797735331317081425749014E6Q,
527 
  -3.753996255897143855113273724233104768831E5Q,
528 
  -4.778761333044147141559311805999540765612E3Q
529 
};
530 
#define NRD2 9
531 
static const __float128 RD2[NRD2 + 1] =
532 
{
533 
  -8.790916836764308497770359421351673950111E9Q,
534 
  -2.023108608053212516399197678553737477486E10Q,
535 
  -1.958067901852022239294231785363504458367E10Q,
536 
  -1.035515043621003101254252481625188704529E10Q,
537 
  -3.253884432621336737640841276619272224476E9Q,
538 
  -6.186383531162456814954947669274235815544E8Q,
539 
  -6.932557847749518463038934953605969951466E7Q,
540 
  -4.240731768287359608773351626528479703758E6Q,
541 
  -1.197343995089189188078944689846348116630E5Q,
542 
  -1.004622911670588064824904487064114090920E3Q
543 
/* 1.0E0 */
544 
};
545 
 
546 
 
547 
/* log gamma(x+1.75) = log gamma(1.75) +  x P(x)/Q(x)
548 
   -0.125 <= x <= +0.125
549 
   1.625 <= x+1.75 <= 1.875
550 
   Peak relative error 9.2e-37 */
551 
static const __float128 lgam1r75a = -8.441162109375E-2Q;
552 
static const __float128 lgam1r75b = 1.0500073264444042213965868602268256157604E-5Q;
553 
#define NRN1r75 8
554 
static const __float128 RN1r75[NRN1r75 + 1] =
555 
{
556 
  -5.221061693929833937710891646275798251513E7Q,
557 
  -2.052466337474314812817883030472496436993E8Q,
558 
  -2.952718275974940270675670705084125640069E8Q,
559 
  -2.132294039648116684922965964126389017840E8Q,
560 
  -8.554103077186505960591321962207519908489E7Q,
561 
  -1.940250901348870867323943119132071960050E7Q,
562 
  -2.379394147112756860769336400290402208435E6Q,
563 
  -1.384060879999526222029386539622255797389E5Q,
564 
  -2.698453601378319296159355612094598695530E3Q
565 
};
566 
#define NRD1r75 8
567 
static const __float128 RD1r75[NRD1r75 + 1] =
568 
{
569 
  -2.109754689501705828789976311354395393605E8Q,
570 
  -5.036651829232895725959911504899241062286E8Q,
571 
  -4.954234699418689764943486770327295098084E8Q,
572 
  -2.589558042412676610775157783898195339410E8Q,
573 
  -7.731476117252958268044969614034776883031E7Q,
574 
  -1.316721702252481296030801191240867486965E7Q,
575 
  -1.201296501404876774861190604303728810836E6Q,
576 
  -5.007966406976106636109459072523610273928E4Q,
577 
  -6.155817990560743422008969155276229018209E2Q
578 
  /* 1.0E0Q */
579 
};
580 
 
581 
 
582 
/* log gamma(x+x0) = y0 +  x^2 P(x)/Q(x)
583 
   -0.0867 <= x <= +0.1634
584 
   1.374932... <= x+x0 <= 1.625032...
585 
   Peak relative error 4.0e-36  */
586 
static const __float128 x0a = 1.4616241455078125Q;
587 
static const __float128 x0b = 7.9994605498412626595423257213002588621246E-6Q;
588 
static const __float128 y0a = -1.21490478515625E-1Q;
589 
static const __float128 y0b = 4.1879797753919044854428223084178486438269E-6Q;
590 
#define NRN1r5 8
591 
static const __float128 RN1r5[NRN1r5 + 1] =
592 
{
593 
  6.827103657233705798067415468881313128066E5Q,
594 
  1.910041815932269464714909706705242148108E6Q,
595 
  2.194344176925978377083808566251427771951E6Q,
596 
  1.332921400100891472195055269688876427962E6Q,
597 
  4.589080973377307211815655093824787123508E5Q,
598 
  8.900334161263456942727083580232613796141E4Q,
599 
  9.053840838306019753209127312097612455236E3Q,
600 
  4.053367147553353374151852319743594873771E2Q,
601 
  5.040631576303952022968949605613514584950E0Q
602 
};
603 
#define NRD1r5 8
604 
static const __float128 RD1r5[NRD1r5 + 1] =
605 
{
606 
  1.411036368843183477558773688484699813355E6Q,
607 
  4.378121767236251950226362443134306184849E6Q,
608 
  5.682322855631723455425929877581697918168E6Q,
609 
  3.999065731556977782435009349967042222375E6Q,
610 
  1.653651390456781293163585493620758410333E6Q,
611 
  4.067774359067489605179546964969435858311E5Q,
612 
  5.741463295366557346748361781768833633256E4Q,
613 
  4.226404539738182992856094681115746692030E3Q,
614 
  1.316980975410327975566999780608618774469E2Q,
615 
  /* 1.0E0Q */
616 
};
617 
 
618 
 
619 
/* log gamma(x+1.25) = log gamma(1.25) +  x P(x)/Q(x)
620 
   -.125 <= x <= +.125
621 
   1.125 <= x+1.25 <= 1.375
622 
   Peak relative error = 4.9e-36 */
623 
static const __float128 lgam1r25a = -9.82818603515625E-2Q;
624 
static const __float128 lgam1r25b = 1.0023929749338536146197303364159774377296E-5Q;
625 
#define NRN1r25 9
626 
static const __float128 RN1r25[NRN1r25 + 1] =
627 
{
628 
  -9.054787275312026472896002240379580536760E4Q,
629 
  -8.685076892989927640126560802094680794471E4Q,
630 
  2.797898965448019916967849727279076547109E5Q,
631 
  6.175520827134342734546868356396008898299E5Q,
632 
  5.179626599589134831538516906517372619641E5Q,
633 
  2.253076616239043944538380039205558242161E5Q,
634 
  5.312653119599957228630544772499197307195E4Q,
635 
  6.434329437514083776052669599834938898255E3Q,
636 
  3.385414416983114598582554037612347549220E2Q,
637 
  4.907821957946273805080625052510832015792E0Q
638 
};
639 
#define NRD1r25 8
640 
static const __float128 RD1r25[NRD1r25 + 1] =
641 
{
642 
  3.980939377333448005389084785896660309000E5Q,
643 
  1.429634893085231519692365775184490465542E6Q,
644 
  2.145438946455476062850151428438668234336E6Q,
645 
  1.743786661358280837020848127465970357893E6Q,
646 
  8.316364251289743923178092656080441655273E5Q,
647 
  2.355732939106812496699621491135458324294E5Q,
648 
  3.822267399625696880571810137601310855419E4Q,
649 
  3.228463206479133236028576845538387620856E3Q,
650 
  1.152133170470059555646301189220117965514E2Q
651 
  /* 1.0E0Q */
652 
};
653 
 
654 
 
655 
/* log gamma(x + 1) = x P(x)/Q(x)
656 
   0.0 <= x <= +0.125
657 
   1.0 <= x+1 <= 1.125
658 
   Peak relative error 1.1e-35  */
659 
#define NRN1 8
660 
static const __float128 RN1[NRN1 + 1] =
661 
{
662 
  -9.987560186094800756471055681088744738818E3Q,
663 
  -2.506039379419574361949680225279376329742E4Q,
664 
  -1.386770737662176516403363873617457652991E4Q,
665 
  1.439445846078103202928677244188837130744E4Q,
666 
  2.159612048879650471489449668295139990693E4Q,
667 
  1.047439813638144485276023138173676047079E4Q,
668 
  2.250316398054332592560412486630769139961E3Q,
669 
  1.958510425467720733041971651126443864041E2Q,
670 
  4.516830313569454663374271993200291219855E0Q
671 
};
672 
#define NRD1 7
673 
static const __float128 RD1[NRD1 + 1] =
674 
{
675 
  1.730299573175751778863269333703788214547E4Q,
676 
  6.807080914851328611903744668028014678148E4Q,
677 
  1.090071629101496938655806063184092302439E5Q,
678 
  9.124354356415154289343303999616003884080E4Q,
679 
  4.262071638655772404431164427024003253954E4Q,
680 
  1.096981664067373953673982635805821283581E4Q,
681 
  1.431229503796575892151252708527595787588E3Q,
682 
  7.734110684303689320830401788262295992921E1Q
683 
 /* 1.0E0 */
684 
};
685 
 
686 
 
687 
/* log gamma(x + 1) = x P(x)/Q(x)
688 
   -0.125 <= x <= 0
689 
   0.875 <= x+1 <= 1.0
690 
   Peak relative error 7.0e-37  */
691 
#define NRNr9 8
692 
static const __float128 RNr9[NRNr9 + 1] =
693 
{
694 
  4.441379198241760069548832023257571176884E5Q,
695 
  1.273072988367176540909122090089580368732E6Q,
696 
  9.732422305818501557502584486510048387724E5Q,
697 
  -5.040539994443998275271644292272870348684E5Q,
698 
  -1.208719055525609446357448132109723786736E6Q,
699 
  -7.434275365370936547146540554419058907156E5Q,
700 
  -2.075642969983377738209203358199008185741E5Q,
701 
  -2.565534860781128618589288075109372218042E4Q,
702 
  -1.032901669542994124131223797515913955938E3Q,
703 
};
704 
#define NRDr9 8
705 
static const __float128 RDr9[NRDr9 + 1] =
706 
{
707 
  -7.694488331323118759486182246005193998007E5Q,
708 
  -3.301918855321234414232308938454112213751E6Q,
709 
  -5.856830900232338906742924836032279404702E6Q,
710 
  -5.540672519616151584486240871424021377540E6Q,
711 
  -3.006530901041386626148342989181721176919E6Q,
712 
  -9.350378280513062139466966374330795935163E5Q,
713 
  -1.566179100031063346901755685375732739511E5Q,
714 
  -1.205016539620260779274902967231510804992E4Q,
715 
  -2.724583156305709733221564484006088794284E2Q
716 
/* 1.0E0 */
717 
};
718 
 
719 
 
720 
/* Evaluate P[n] x^n  +  P[n-1] x^(n-1)  +  ...  +  P[0] */
721 
 
722 
static __float128
723 
neval (__float128 x, const __float128 *p, int n)
724 
{
725 
  __float128 y;
726 
 
727 
  p += n;
728 
  y = *p--;
729 
  do
730 
    {
731 
      y = y * x + *p--;
732 
    }
733 
  while (--n > 0);
734 
  return y;
735 
}
736 
 
737 
 
738 
/* Evaluate x^n+1  +  P[n] x^(n)  +  P[n-1] x^(n-1)  +  ...  +  P[0] */
739 
 
740 
static __float128
741 
deval (__float128 x, const __float128 *p, int n)
742 
{
743 
  __float128 y;
744 
 
745 
  p += n;
746 
  y = x + *p--;
747 
  do
748 
    {
749 
      y = y * x + *p--;
750 
    }
751 
  while (--n > 0);
752 
  return y;
753 
}
754 
 
755 
 
756 
__float128
757 
lgammaq (__float128 x)
758 
{
759 
  __float128 p, q, w, z, nx;
760 
  int i, nn, sign;
761 
 
762 
  sign = 1;
763 
 
764 
  if (! finiteq (x))
765 
    return x * x;
766 
 
767 
  if (x == 0.0Q)
768 
    {
769 
      if (signbitq (x))
770 
        sign = -1;
771 
    }
772 
 
773 
  if (x < 0.0Q)
774 
    {
775 
      q = -x;
776 
      p = floorq (q);
777 
      if (p == q)
778 
        return (one / (p - p));
779 
      i = p;
780 
      if ((i & 1) == 0)
781 
        sign = -1;
782 
      else
783 
        sign = 1;
784 
      z = q - p;
785 
      if (z > 0.5Q)
786 
        {
787 
          p += 1.0Q;
788 
          z = p - q;
789 
        }
790 
      z = q * sinq (PIQ * z);
791 
      if (z == 0.0Q)
792 
        return (sign * huge * huge);
793 
      w = lgammaq (q);
794 
      z = logq (PIQ / z) - w;
795 
      return (z);
796 
    }
797 
 
798 
  if (x < 13.5Q)
799 
    {
800 
      p = 0.0Q;
801 
      nx = floorq (x + 0.5Q);
802 
      nn = nx;
803 
      switch (nn)
804 
        {
805 
        case 0:
806 
          /* log gamma (x + 1) = log(x) + log gamma(x) */
807 
          if (x <= 0.125)
808 
            {
809 
              p = x * neval (x, RN1, NRN1) / deval (x, RD1, NRD1);
810 
            }
811 
          else if (x <= 0.375)
812 
            {
813 
              z = x - 0.25Q;
814 
              p = z * neval (z, RN1r25, NRN1r25) / deval (z, RD1r25, NRD1r25);
815 
              p += lgam1r25b;
816 
              p += lgam1r25a;
817 
            }
818 
          else if (x <= 0.625)
819 
            {
820 
              z = x + (1.0Q - x0a);
821 
              z = z - x0b;
822 
              p = neval (z, RN1r5, NRN1r5) / deval (z, RD1r5, NRD1r5);
823 
              p = p * z * z;
824 
              p = p + y0b;
825 
              p = p + y0a;
826 
            }
827 
          else if (x <= 0.875)
828 
            {
829 
              z = x - 0.75Q;
830 
              p = z * neval (z, RN1r75, NRN1r75) / deval (z, RD1r75, NRD1r75);
831 
              p += lgam1r75b;
832 
              p += lgam1r75a;
833 
            }
834 
          else
835 
            {
836 
              z = x - 1.0Q;
837 
              p = z * neval (z, RN2, NRN2) / deval (z, RD2, NRD2);
838 
            }
839 
          p = p - logq (x);
840 
          break;
841 
 
842 
        case 1:
843 
          if (x < 0.875Q)
844 
            {
845 
              if (x <= 0.625)
846 
                {
847 
                  z = x + (1.0Q - x0a);
848 
                  z = z - x0b;
849 
                  p = neval (z, RN1r5, NRN1r5) / deval (z, RD1r5, NRD1r5);
850 
                  p = p * z * z;
851 
                  p = p + y0b;
852 
                  p = p + y0a;
853 
                }
854 
              else if (x <= 0.875)
855 
                {
856 
                  z = x - 0.75Q;
857 
                  p = z * neval (z, RN1r75, NRN1r75)
858 
                        / deval (z, RD1r75, NRD1r75);
859 
                  p += lgam1r75b;
860 
                  p += lgam1r75a;
861 
                }
862 
              else
863 
                {
864 
                  z = x - 1.0Q;
865 
                  p = z * neval (z, RN2, NRN2) / deval (z, RD2, NRD2);
866 
                }
867 
              p = p - logq (x);
868 
            }
869 
          else if (x < 1.0Q)
870 
            {
871 
              z = x - 1.0Q;
872 
              p = z * neval (z, RNr9, NRNr9) / deval (z, RDr9, NRDr9);
873 
            }
874 
          else if (x == 1.0Q)
875 
            p = 0.0Q;
876 
          else if (x <= 1.125Q)
877 
            {
878 
              z = x - 1.0Q;
879 
              p = z * neval (z, RN1, NRN1) / deval (z, RD1, NRD1);
880 
            }
881 
          else if (x <= 1.375)
882 
            {
883 
              z = x - 1.25Q;
884 
              p = z * neval (z, RN1r25, NRN1r25) / deval (z, RD1r25, NRD1r25);
885 
              p += lgam1r25b;
886 
              p += lgam1r25a;
887 
            }
888 
          else
889 
            {
890 
              /* 1.375 <= x+x0 <= 1.625 */
891 
              z = x - x0a;
892 
              z = z - x0b;
893 
              p = neval (z, RN1r5, NRN1r5) / deval (z, RD1r5, NRD1r5);
894 
              p = p * z * z;
895 
              p = p + y0b;
896 
              p = p + y0a;
897 
            }
898 
          break;
899 
 
900 
        case 2:
901 
          if (x < 1.625Q)
902 
            {
903 
              z = x - x0a;
904 
              z = z - x0b;
905 
              p = neval (z, RN1r5, NRN1r5) / deval (z, RD1r5, NRD1r5);
906 
              p = p * z * z;
907 
              p = p + y0b;
908 
              p = p + y0a;
909 
            }
910 
          else if (x < 1.875Q)
911 
            {
912 
              z = x - 1.75Q;
913 
              p = z * neval (z, RN1r75, NRN1r75) / deval (z, RD1r75, NRD1r75);
914 
              p += lgam1r75b;
915 
              p += lgam1r75a;
916 
            }
917 
          else if (x == 2.0Q)
918 
            p = 0.0Q;
919 
          else if (x < 2.375Q)
920 
            {
921 
              z = x - 2.0Q;
922 
              p = z * neval (z, RN2, NRN2) / deval (z, RD2, NRD2);
923 
            }
924 
          else
925 
            {
926 
              z = x - 2.5Q;
927 
              p = z * neval (z, RN2r5, NRN2r5) / deval (z, RD2r5, NRD2r5);
928 
              p += lgam2r5b;
929 
              p += lgam2r5a;
930 
            }
931 
          break;
932 
 
933 
        case 3:
934 
          if (x < 2.75)
935 
            {
936 
              z = x - 2.5Q;
937 
              p = z * neval (z, RN2r5, NRN2r5) / deval (z, RD2r5, NRD2r5);
938 
              p += lgam2r5b;
939 
              p += lgam2r5a;
940 
            }
941 
          else
942 
            {
943 
              z = x - 3.0Q;
944 
              p = z * neval (z, RN3, NRN3) / deval (z, RD3, NRD3);
945 
              p += lgam3b;
946 
              p += lgam3a;
947 
            }
948 
          break;
949 
 
950 
        case 4:
951 
          z = x - 4.0Q;
952 
          p = z * neval (z, RN4, NRN4) / deval (z, RD4, NRD4);
953 
          p += lgam4b;
954 
          p += lgam4a;
955 
          break;
956 
 
957 
        case 5:
958 
          z = x - 5.0Q;
959 
          p = z * neval (z, RN5, NRN5) / deval (z, RD5, NRD5);
960 
          p += lgam5b;
961 
          p += lgam5a;
962 
          break;
963 
 
964 
        case 6:
965 
          z = x - 6.0Q;
966 
          p = z * neval (z, RN6, NRN6) / deval (z, RD6, NRD6);
967 
          p += lgam6b;
968 
          p += lgam6a;
969 
          break;
970 
 
971 
        case 7:
972 
          z = x - 7.0Q;
973 
          p = z * neval (z, RN7, NRN7) / deval (z, RD7, NRD7);
974 
          p += lgam7b;
975 
          p += lgam7a;
976 
          break;
977 
 
978 
        case 8:
979 
          z = x - 8.0Q;
980 
          p = z * neval (z, RN8, NRN8) / deval (z, RD8, NRD8);
981 
          p += lgam8b;
982 
          p += lgam8a;
983 
          break;
984 
 
985 
        case 9:
986 
          z = x - 9.0Q;
987 
          p = z * neval (z, RN9, NRN9) / deval (z, RD9, NRD9);
988 
          p += lgam9b;
989 
          p += lgam9a;
990 
          break;
991 
 
992 
        case 10:
993 
          z = x - 10.0Q;
994 
          p = z * neval (z, RN10, NRN10) / deval (z, RD10, NRD10);
995 
          p += lgam10b;
996 
          p += lgam10a;
997 
          break;
998 
 
999 
        case 11:
1000 
          z = x - 11.0Q;
1001 
          p = z * neval (z, RN11, NRN11) / deval (z, RD11, NRD11);
1002 
          p += lgam11b;
1003 
          p += lgam11a;
1004 
          break;
1005 
 
1006 
        case 12:
1007 
          z = x - 12.0Q;
1008 
          p = z * neval (z, RN12, NRN12) / deval (z, RD12, NRD12);
1009 
          p += lgam12b;
1010 
          p += lgam12a;
1011 
          break;
1012 
 
1013 
        case 13:
1014 
          z = x - 13.0Q;
1015 
          p = z * neval (z, RN13, NRN13) / deval (z, RD13, NRD13);
1016 
          p += lgam13b;
1017 
          p += lgam13a;
1018 
          break;
1019 
        }
1020 
      return p;
1021 
    }
1022 
 
1023 
  if (x > MAXLGM)
1024 
    return (sign * huge * huge);
1025 
 
1026 
  q = ls2pi - x;
1027 
  q = (x - 0.5Q) * logq (x) + q;
1028 
  if (x > 1.0e18Q)
1029 
    return (q);
1030 
 
1031 
  p = 1.0Q / (x * x);
1032 
  q += neval (p, RASY, NRASY) / x;
1033 
  return (q);
1034 
}

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