1// Copyright John Maddock 2006.
2// Use, modification and distribution are subject to the
3// Boost Software License, Version 1.0. (See accompanying file
4// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)
5//
6// This file implements the asymptotic expansions of the incomplete
7// gamma functions P(a, x) and Q(a, x), used when a is large and
8// x ~ a.
9//
10// The primary reference is:
11//
12// "The Asymptotic Expansion of the Incomplete Gamma Functions"
13// N. M. Temme.
14// Siam J. Math Anal. Vol 10 No 4, July 1979, p757.
15//
16// A different way of evaluating these expansions,
17// plus a lot of very useful background information is in:
18//
19// "A Set of Algorithms For the Incomplete Gamma Functions."
20// N. M. Temme.
21// Probability in the Engineering and Informational Sciences,
22// 8, 1994, 291.
23//
24// An alternative implementation is in:
25//
26// "Computation of the Incomplete Gamma Function Ratios and their Inverse."
27// A. R. Didonato and A. H. Morris.
28// ACM TOMS, Vol 12, No 4, Dec 1986, p377.
29//
30// There are various versions of the same code below, each accurate
31// to a different precision. To understand the code, refer to Didonato
32// and Morris, from Eq 17 and 18 onwards.
33//
34// The coefficients used here are not taken from Didonato and Morris:
35// the domain over which these expansions are used is slightly different
36// to theirs, and their constants are not quite accurate enough for
37// 128-bit long double's. Instead the coefficients were calculated
38// using the methods described by Temme p762 from Eq 3.8 onwards.
39// The values obtained agree with those obtained by Didonato and Morris
40// (at least to the first 30 digits that they provide).
41// At double precision the degrees of polynomial required for full
42// machine precision are close to those recommended to Didonato and Morris,
43// but of course many more terms are needed for larger types.
44//
45#ifndef BOOST_MATH_DETAIL_IGAMMA_LARGE
46#define BOOST_MATH_DETAIL_IGAMMA_LARGE
47
48#ifdef _MSC_VER
49#pragma once
50#endif
51
52#if defined(__GNUC__) && defined(BOOST_MATH_USE_FLOAT128)
53//
54// This is the only way we can avoid
55// warning: non-standard suffix on floating constant [-Wpedantic]
56// when building with -Wall -pedantic. Neither __extension__
57// nor #pragma diagnostic ignored work :(
58//
59#pragma GCC system_header
60#endif
61
62namespace boost{ namespace math{ namespace detail{
63
64// This version will never be called (at runtime), it's a stub used
65// when T is unsuitable to be passed to these routines:
66//
67template <class T, class Policy>
68inline T igamma_temme_large(T, T, const Policy& /* pol */, boost::integral_constant<int, 0> const *)
69{
70 // stub function, should never actually be called
71 BOOST_ASSERT(0);
72 return 0;
73}
74//
75// This version is accurate for up to 64-bit mantissa's,
76// (80-bit long double, or 10^-20).
77//
78template <class T, class Policy>
79T igamma_temme_large(T a, T x, const Policy& pol, boost::integral_constant<int, 64> const *)
80{
81 BOOST_MATH_STD_USING // ADL of std functions
82 T sigma = (x - a) / a;
83 T phi = -boost::math::log1pmx(sigma, pol);
84 T y = a * phi;
85 T z = sqrt(2 * phi);
86 if(x < a)
87 z = -z;
88
89 T workspace[13];
90
91 static const T C0[] = {
92 BOOST_MATH_BIG_CONSTANT(T, 64, -0.333333333333333333333),
93 BOOST_MATH_BIG_CONSTANT(T, 64, 0.0833333333333333333333),
94 BOOST_MATH_BIG_CONSTANT(T, 64, -0.0148148148148148148148),
95 BOOST_MATH_BIG_CONSTANT(T, 64, 0.00115740740740740740741),
96 BOOST_MATH_BIG_CONSTANT(T, 64, 0.000352733686067019400353),
97 BOOST_MATH_BIG_CONSTANT(T, 64, -0.0001787551440329218107),
98 BOOST_MATH_BIG_CONSTANT(T, 64, 0.39192631785224377817e-4),
99 BOOST_MATH_BIG_CONSTANT(T, 64, -0.218544851067999216147e-5),
100 BOOST_MATH_BIG_CONSTANT(T, 64, -0.18540622107151599607e-5),
101 BOOST_MATH_BIG_CONSTANT(T, 64, 0.829671134095308600502e-6),
102 BOOST_MATH_BIG_CONSTANT(T, 64, -0.176659527368260793044e-6),
103 BOOST_MATH_BIG_CONSTANT(T, 64, 0.670785354340149858037e-8),
104 BOOST_MATH_BIG_CONSTANT(T, 64, 0.102618097842403080426e-7),
105 BOOST_MATH_BIG_CONSTANT(T, 64, -0.438203601845335318655e-8),
106 BOOST_MATH_BIG_CONSTANT(T, 64, 0.914769958223679023418e-9),
107 BOOST_MATH_BIG_CONSTANT(T, 64, -0.255141939949462497669e-10),
108 BOOST_MATH_BIG_CONSTANT(T, 64, -0.583077213255042506746e-10),
109 BOOST_MATH_BIG_CONSTANT(T, 64, 0.243619480206674162437e-10),
110 BOOST_MATH_BIG_CONSTANT(T, 64, -0.502766928011417558909e-11),
111 };
112 workspace[0] = tools::evaluate_polynomial(C0, z);
113
114 static const T C1[] = {
115 BOOST_MATH_BIG_CONSTANT(T, 64, -0.00185185185185185185185),
116 BOOST_MATH_BIG_CONSTANT(T, 64, -0.00347222222222222222222),
117 BOOST_MATH_BIG_CONSTANT(T, 64, 0.00264550264550264550265),
118 BOOST_MATH_BIG_CONSTANT(T, 64, -0.000990226337448559670782),
119 BOOST_MATH_BIG_CONSTANT(T, 64, 0.000205761316872427983539),
120 BOOST_MATH_BIG_CONSTANT(T, 64, -0.40187757201646090535e-6),
121 BOOST_MATH_BIG_CONSTANT(T, 64, -0.18098550334489977837e-4),
122 BOOST_MATH_BIG_CONSTANT(T, 64, 0.764916091608111008464e-5),
123 BOOST_MATH_BIG_CONSTANT(T, 64, -0.161209008945634460038e-5),
124 BOOST_MATH_BIG_CONSTANT(T, 64, 0.464712780280743434226e-8),
125 BOOST_MATH_BIG_CONSTANT(T, 64, 0.137863344691572095931e-6),
126 BOOST_MATH_BIG_CONSTANT(T, 64, -0.575254560351770496402e-7),
127 BOOST_MATH_BIG_CONSTANT(T, 64, 0.119516285997781473243e-7),
128 BOOST_MATH_BIG_CONSTANT(T, 64, -0.175432417197476476238e-10),
129 BOOST_MATH_BIG_CONSTANT(T, 64, -0.100915437106004126275e-8),
130 BOOST_MATH_BIG_CONSTANT(T, 64, 0.416279299184258263623e-9),
131 BOOST_MATH_BIG_CONSTANT(T, 64, -0.856390702649298063807e-10),
132 };
133 workspace[1] = tools::evaluate_polynomial(C1, z);
134
135 static const T C2[] = {
136 BOOST_MATH_BIG_CONSTANT(T, 64, 0.00413359788359788359788),
137 BOOST_MATH_BIG_CONSTANT(T, 64, -0.00268132716049382716049),
138 BOOST_MATH_BIG_CONSTANT(T, 64, 0.000771604938271604938272),
139 BOOST_MATH_BIG_CONSTANT(T, 64, 0.200938786008230452675e-5),
140 BOOST_MATH_BIG_CONSTANT(T, 64, -0.000107366532263651605215),
141 BOOST_MATH_BIG_CONSTANT(T, 64, 0.529234488291201254164e-4),
142 BOOST_MATH_BIG_CONSTANT(T, 64, -0.127606351886187277134e-4),
143 BOOST_MATH_BIG_CONSTANT(T, 64, 0.342357873409613807419e-7),
144 BOOST_MATH_BIG_CONSTANT(T, 64, 0.137219573090629332056e-5),
145 BOOST_MATH_BIG_CONSTANT(T, 64, -0.629899213838005502291e-6),
146 BOOST_MATH_BIG_CONSTANT(T, 64, 0.142806142060642417916e-6),
147 BOOST_MATH_BIG_CONSTANT(T, 64, -0.204770984219908660149e-9),
148 BOOST_MATH_BIG_CONSTANT(T, 64, -0.140925299108675210533e-7),
149 BOOST_MATH_BIG_CONSTANT(T, 64, 0.622897408492202203356e-8),
150 BOOST_MATH_BIG_CONSTANT(T, 64, -0.136704883966171134993e-8),
151 };
152 workspace[2] = tools::evaluate_polynomial(C2, z);
153
154 static const T C3[] = {
155 BOOST_MATH_BIG_CONSTANT(T, 64, 0.000649434156378600823045),
156 BOOST_MATH_BIG_CONSTANT(T, 64, 0.000229472093621399176955),
157 BOOST_MATH_BIG_CONSTANT(T, 64, -0.000469189494395255712128),
158 BOOST_MATH_BIG_CONSTANT(T, 64, 0.000267720632062838852962),
159 BOOST_MATH_BIG_CONSTANT(T, 64, -0.756180167188397641073e-4),
160 BOOST_MATH_BIG_CONSTANT(T, 64, -0.239650511386729665193e-6),
161 BOOST_MATH_BIG_CONSTANT(T, 64, 0.110826541153473023615e-4),
162 BOOST_MATH_BIG_CONSTANT(T, 64, -0.56749528269915965675e-5),
163 BOOST_MATH_BIG_CONSTANT(T, 64, 0.142309007324358839146e-5),
164 BOOST_MATH_BIG_CONSTANT(T, 64, -0.278610802915281422406e-10),
165 BOOST_MATH_BIG_CONSTANT(T, 64, -0.169584040919302772899e-6),
166 BOOST_MATH_BIG_CONSTANT(T, 64, 0.809946490538808236335e-7),
167 BOOST_MATH_BIG_CONSTANT(T, 64, -0.191111684859736540607e-7),
168 };
169 workspace[3] = tools::evaluate_polynomial(C3, z);
170
171 static const T C4[] = {
172 BOOST_MATH_BIG_CONSTANT(T, 64, -0.000861888290916711698605),
173 BOOST_MATH_BIG_CONSTANT(T, 64, 0.000784039221720066627474),
174 BOOST_MATH_BIG_CONSTANT(T, 64, -0.000299072480303190179733),
175 BOOST_MATH_BIG_CONSTANT(T, 64, -0.146384525788434181781e-5),
176 BOOST_MATH_BIG_CONSTANT(T, 64, 0.664149821546512218666e-4),
177 BOOST_MATH_BIG_CONSTANT(T, 64, -0.396836504717943466443e-4),
178 BOOST_MATH_BIG_CONSTANT(T, 64, 0.113757269706784190981e-4),
179 BOOST_MATH_BIG_CONSTANT(T, 64, 0.250749722623753280165e-9),
180 BOOST_MATH_BIG_CONSTANT(T, 64, -0.169541495365583060147e-5),
181 BOOST_MATH_BIG_CONSTANT(T, 64, 0.890750753220530968883e-6),
182 BOOST_MATH_BIG_CONSTANT(T, 64, -0.229293483400080487057e-6),
183 };
184 workspace[4] = tools::evaluate_polynomial(C4, z);
185
186 static const T C5[] = {
187 BOOST_MATH_BIG_CONSTANT(T, 64, -0.000336798553366358150309),
188 BOOST_MATH_BIG_CONSTANT(T, 64, -0.697281375836585777429e-4),
189 BOOST_MATH_BIG_CONSTANT(T, 64, 0.000277275324495939207873),
190 BOOST_MATH_BIG_CONSTANT(T, 64, -0.000199325705161888477003),
191 BOOST_MATH_BIG_CONSTANT(T, 64, 0.679778047793720783882e-4),
192 BOOST_MATH_BIG_CONSTANT(T, 64, 0.141906292064396701483e-6),
193 BOOST_MATH_BIG_CONSTANT(T, 64, -0.135940481897686932785e-4),
194 BOOST_MATH_BIG_CONSTANT(T, 64, 0.801847025633420153972e-5),
195 BOOST_MATH_BIG_CONSTANT(T, 64, -0.229148117650809517038e-5),
196 };
197 workspace[5] = tools::evaluate_polynomial(C5, z);
198
199 static const T C6[] = {
200 BOOST_MATH_BIG_CONSTANT(T, 64, 0.000531307936463992223166),
201 BOOST_MATH_BIG_CONSTANT(T, 64, -0.000592166437353693882865),
202 BOOST_MATH_BIG_CONSTANT(T, 64, 0.000270878209671804482771),
203 BOOST_MATH_BIG_CONSTANT(T, 64, 0.790235323266032787212e-6),
204 BOOST_MATH_BIG_CONSTANT(T, 64, -0.815396936756196875093e-4),
205 BOOST_MATH_BIG_CONSTANT(T, 64, 0.561168275310624965004e-4),
206 BOOST_MATH_BIG_CONSTANT(T, 64, -0.183291165828433755673e-4),
207 BOOST_MATH_BIG_CONSTANT(T, 64, -0.307961345060330478256e-8),
208 BOOST_MATH_BIG_CONSTANT(T, 64, 0.346515536880360908674e-5),
209 BOOST_MATH_BIG_CONSTANT(T, 64, -0.20291327396058603727e-5),
210 BOOST_MATH_BIG_CONSTANT(T, 64, 0.57887928631490037089e-6),
211 };
212 workspace[6] = tools::evaluate_polynomial(C6, z);
213
214 static const T C7[] = {
215 BOOST_MATH_BIG_CONSTANT(T, 64, 0.000344367606892377671254),
216 BOOST_MATH_BIG_CONSTANT(T, 64, 0.517179090826059219337e-4),
217 BOOST_MATH_BIG_CONSTANT(T, 64, -0.000334931610811422363117),
218 BOOST_MATH_BIG_CONSTANT(T, 64, 0.000281269515476323702274),
219 BOOST_MATH_BIG_CONSTANT(T, 64, -0.000109765822446847310235),
220 BOOST_MATH_BIG_CONSTANT(T, 64, -0.127410090954844853795e-6),
221 BOOST_MATH_BIG_CONSTANT(T, 64, 0.277444515115636441571e-4),
222 BOOST_MATH_BIG_CONSTANT(T, 64, -0.182634888057113326614e-4),
223 BOOST_MATH_BIG_CONSTANT(T, 64, 0.578769494973505239894e-5),
224 };
225 workspace[7] = tools::evaluate_polynomial(C7, z);
226
227 static const T C8[] = {
228 BOOST_MATH_BIG_CONSTANT(T, 64, -0.000652623918595309418922),
229 BOOST_MATH_BIG_CONSTANT(T, 64, 0.000839498720672087279993),
230 BOOST_MATH_BIG_CONSTANT(T, 64, -0.000438297098541721005061),
231 BOOST_MATH_BIG_CONSTANT(T, 64, -0.696909145842055197137e-6),
232 BOOST_MATH_BIG_CONSTANT(T, 64, 0.000166448466420675478374),
233 BOOST_MATH_BIG_CONSTANT(T, 64, -0.000127835176797692185853),
234 BOOST_MATH_BIG_CONSTANT(T, 64, 0.462995326369130429061e-4),
235 };
236 workspace[8] = tools::evaluate_polynomial(C8, z);
237
238 static const T C9[] = {
239 BOOST_MATH_BIG_CONSTANT(T, 64, -0.000596761290192746250124),
240 BOOST_MATH_BIG_CONSTANT(T, 64, -0.720489541602001055909e-4),
241 BOOST_MATH_BIG_CONSTANT(T, 64, 0.000678230883766732836162),
242 BOOST_MATH_BIG_CONSTANT(T, 64, -0.0006401475260262758451),
243 BOOST_MATH_BIG_CONSTANT(T, 64, 0.000277501076343287044992),
244 };
245 workspace[9] = tools::evaluate_polynomial(C9, z);
246
247 static const T C10[] = {
248 BOOST_MATH_BIG_CONSTANT(T, 64, 0.00133244544948006563713),
249 BOOST_MATH_BIG_CONSTANT(T, 64, -0.0019144384985654775265),
250 BOOST_MATH_BIG_CONSTANT(T, 64, 0.00110893691345966373396),
251 };
252 workspace[10] = tools::evaluate_polynomial(C10, z);
253
254 static const T C11[] = {
255 BOOST_MATH_BIG_CONSTANT(T, 64, 0.00157972766073083495909),
256 BOOST_MATH_BIG_CONSTANT(T, 64, 0.000162516262783915816899),
257 BOOST_MATH_BIG_CONSTANT(T, 64, -0.00206334210355432762645),
258 BOOST_MATH_BIG_CONSTANT(T, 64, 0.00213896861856890981541),
259 BOOST_MATH_BIG_CONSTANT(T, 64, -0.00101085593912630031708),
260 };
261 workspace[11] = tools::evaluate_polynomial(C11, z);
262
263 static const T C12[] = {
264 BOOST_MATH_BIG_CONSTANT(T, 64, -0.00407251211951401664727),
265 BOOST_MATH_BIG_CONSTANT(T, 64, 0.00640336283380806979482),
266 BOOST_MATH_BIG_CONSTANT(T, 64, -0.00404101610816766177474),
267 };
268 workspace[12] = tools::evaluate_polynomial(C12, z);
269
270 T result = tools::evaluate_polynomial<13, T, T>(workspace, 1/a);
271 result *= exp(-y) / sqrt(2 * constants::pi<T>() * a);
272 if(x < a)
273 result = -result;
274
275 result += boost::math::erfc(sqrt(y), pol) / 2;
276
277 return result;
278}
279//
280// This one is accurate for 53-bit mantissa's
281// (IEEE double precision or 10^-17).
282//
283template <class T, class Policy>
284T igamma_temme_large(T a, T x, const Policy& pol, boost::integral_constant<int, 53> const *)
285{
286 BOOST_MATH_STD_USING // ADL of std functions
287 T sigma = (x - a) / a;
288 T phi = -boost::math::log1pmx(sigma, pol);
289 T y = a * phi;
290 T z = sqrt(2 * phi);
291 if(x < a)
292 z = -z;
293
294 T workspace[10];
295
296 static const T C0[] = {
297 static_cast<T>(-0.33333333333333333L),
298 static_cast<T>(0.083333333333333333L),
299 static_cast<T>(-0.014814814814814815L),
300 static_cast<T>(0.0011574074074074074L),
301 static_cast<T>(0.0003527336860670194L),
302 static_cast<T>(-0.00017875514403292181L),
303 static_cast<T>(0.39192631785224378e-4L),
304 static_cast<T>(-0.21854485106799922e-5L),
305 static_cast<T>(-0.185406221071516e-5L),
306 static_cast<T>(0.8296711340953086e-6L),
307 static_cast<T>(-0.17665952736826079e-6L),
308 static_cast<T>(0.67078535434014986e-8L),
309 static_cast<T>(0.10261809784240308e-7L),
310 static_cast<T>(-0.43820360184533532e-8L),
311 static_cast<T>(0.91476995822367902e-9L),
312 };
313 workspace[0] = tools::evaluate_polynomial(C0, z);
314
315 static const T C1[] = {
316 static_cast<T>(-0.0018518518518518519L),
317 static_cast<T>(-0.0034722222222222222L),
318 static_cast<T>(0.0026455026455026455L),
319 static_cast<T>(-0.00099022633744855967L),
320 static_cast<T>(0.00020576131687242798L),
321 static_cast<T>(-0.40187757201646091e-6L),
322 static_cast<T>(-0.18098550334489978e-4L),
323 static_cast<T>(0.76491609160811101e-5L),
324 static_cast<T>(-0.16120900894563446e-5L),
325 static_cast<T>(0.46471278028074343e-8L),
326 static_cast<T>(0.1378633446915721e-6L),
327 static_cast<T>(-0.5752545603517705e-7L),
328 static_cast<T>(0.11951628599778147e-7L),
329 };
330 workspace[1] = tools::evaluate_polynomial(C1, z);
331
332 static const T C2[] = {
333 static_cast<T>(0.0041335978835978836L),
334 static_cast<T>(-0.0026813271604938272L),
335 static_cast<T>(0.00077160493827160494L),
336 static_cast<T>(0.20093878600823045e-5L),
337 static_cast<T>(-0.00010736653226365161L),
338 static_cast<T>(0.52923448829120125e-4L),
339 static_cast<T>(-0.12760635188618728e-4L),
340 static_cast<T>(0.34235787340961381e-7L),
341 static_cast<T>(0.13721957309062933e-5L),
342 static_cast<T>(-0.6298992138380055e-6L),
343 static_cast<T>(0.14280614206064242e-6L),
344 };
345 workspace[2] = tools::evaluate_polynomial(C2, z);
346
347 static const T C3[] = {
348 static_cast<T>(0.00064943415637860082L),
349 static_cast<T>(0.00022947209362139918L),
350 static_cast<T>(-0.00046918949439525571L),
351 static_cast<T>(0.00026772063206283885L),
352 static_cast<T>(-0.75618016718839764e-4L),
353 static_cast<T>(-0.23965051138672967e-6L),
354 static_cast<T>(0.11082654115347302e-4L),
355 static_cast<T>(-0.56749528269915966e-5L),
356 static_cast<T>(0.14230900732435884e-5L),
357 };
358 workspace[3] = tools::evaluate_polynomial(C3, z);
359
360 static const T C4[] = {
361 static_cast<T>(-0.0008618882909167117L),
362 static_cast<T>(0.00078403922172006663L),
363 static_cast<T>(-0.00029907248030319018L),
364 static_cast<T>(-0.14638452578843418e-5L),
365 static_cast<T>(0.66414982154651222e-4L),
366 static_cast<T>(-0.39683650471794347e-4L),
367 static_cast<T>(0.11375726970678419e-4L),
368 };
369 workspace[4] = tools::evaluate_polynomial(C4, z);
370
371 static const T C5[] = {
372 static_cast<T>(-0.00033679855336635815L),
373 static_cast<T>(-0.69728137583658578e-4L),
374 static_cast<T>(0.00027727532449593921L),
375 static_cast<T>(-0.00019932570516188848L),
376 static_cast<T>(0.67977804779372078e-4L),
377 static_cast<T>(0.1419062920643967e-6L),
378 static_cast<T>(-0.13594048189768693e-4L),
379 static_cast<T>(0.80184702563342015e-5L),
380 static_cast<T>(-0.22914811765080952e-5L),
381 };
382 workspace[5] = tools::evaluate_polynomial(C5, z);
383
384 static const T C6[] = {
385 static_cast<T>(0.00053130793646399222L),
386 static_cast<T>(-0.00059216643735369388L),
387 static_cast<T>(0.00027087820967180448L),
388 static_cast<T>(0.79023532326603279e-6L),
389 static_cast<T>(-0.81539693675619688e-4L),
390 static_cast<T>(0.56116827531062497e-4L),
391 static_cast<T>(-0.18329116582843376e-4L),
392 };
393 workspace[6] = tools::evaluate_polynomial(C6, z);
394
395 static const T C7[] = {
396 static_cast<T>(0.00034436760689237767L),
397 static_cast<T>(0.51717909082605922e-4L),
398 static_cast<T>(-0.00033493161081142236L),
399 static_cast<T>(0.0002812695154763237L),
400 static_cast<T>(-0.00010976582244684731L),
401 };
402 workspace[7] = tools::evaluate_polynomial(C7, z);
403
404 static const T C8[] = {
405 static_cast<T>(-0.00065262391859530942L),
406 static_cast<T>(0.00083949872067208728L),
407 static_cast<T>(-0.00043829709854172101L),
408 };
409 workspace[8] = tools::evaluate_polynomial(C8, z);
410 workspace[9] = static_cast<T>(-0.00059676129019274625L);
411
412 T result = tools::evaluate_polynomial<10, T, T>(workspace, 1/a);
413 result *= exp(-y) / sqrt(2 * constants::pi<T>() * a);
414 if(x < a)
415 result = -result;
416
417 result += boost::math::erfc(sqrt(y), pol) / 2;
418
419 return result;
420}
421//
422// This one is accurate for 24-bit mantissa's
423// (IEEE float precision, or 10^-8)
424//
425template <class T, class Policy>
426T igamma_temme_large(T a, T x, const Policy& pol, boost::integral_constant<int, 24> const *)
427{
428 BOOST_MATH_STD_USING // ADL of std functions
429 T sigma = (x - a) / a;
430 T phi = -boost::math::log1pmx(sigma, pol);
431 T y = a * phi;
432 T z = sqrt(2 * phi);
433 if(x < a)
434 z = -z;
435
436 T workspace[3];
437
438 static const T C0[] = {
439 static_cast<T>(-0.333333333L),
440 static_cast<T>(0.0833333333L),
441 static_cast<T>(-0.0148148148L),
442 static_cast<T>(0.00115740741L),
443 static_cast<T>(0.000352733686L),
444 static_cast<T>(-0.000178755144L),
445 static_cast<T>(0.391926318e-4L),
446 };
447 workspace[0] = tools::evaluate_polynomial(C0, z);
448
449 static const T C1[] = {
450 static_cast<T>(-0.00185185185L),
451 static_cast<T>(-0.00347222222L),
452 static_cast<T>(0.00264550265L),
453 static_cast<T>(-0.000990226337L),
454 static_cast<T>(0.000205761317L),
455 };
456 workspace[1] = tools::evaluate_polynomial(C1, z);
457
458 static const T C2[] = {
459 static_cast<T>(0.00413359788L),
460 static_cast<T>(-0.00268132716L),
461 static_cast<T>(0.000771604938L),
462 };
463 workspace[2] = tools::evaluate_polynomial(C2, z);
464
465 T result = tools::evaluate_polynomial(workspace, 1/a);
466 result *= exp(-y) / sqrt(2 * constants::pi<T>() * a);
467 if(x < a)
468 result = -result;
469
470 result += boost::math::erfc(sqrt(y), pol) / 2;
471
472 return result;
473}
474//
475// And finally, a version for 113-bit mantissa's
476// (128-bit long doubles, or 10^-34).
477// Note this one has been optimised for a > 200
478// It's use for a < 200 is not recommended, that would
479// require many more terms in the polynomials.
480//
481template <class T, class Policy>
482T igamma_temme_large(T a, T x, const Policy& pol, boost::integral_constant<int, 113> const *)
483{
484 BOOST_MATH_STD_USING // ADL of std functions
485 T sigma = (x - a) / a;
486 T phi = -boost::math::log1pmx(sigma, pol);
487 T y = a * phi;
488 T z = sqrt(2 * phi);
489 if(x < a)
490 z = -z;
491
492 T workspace[14];
493
494 static const T C0[] = {
495 BOOST_MATH_BIG_CONSTANT(T, 113, -0.333333333333333333333333333333333333),
496 BOOST_MATH_BIG_CONSTANT(T, 113, 0.0833333333333333333333333333333333333),
497 BOOST_MATH_BIG_CONSTANT(T, 113, -0.0148148148148148148148148148148148148),
498 BOOST_MATH_BIG_CONSTANT(T, 113, 0.00115740740740740740740740740740740741),
499 BOOST_MATH_BIG_CONSTANT(T, 113, 0.0003527336860670194003527336860670194),
500 BOOST_MATH_BIG_CONSTANT(T, 113, -0.000178755144032921810699588477366255144),
501 BOOST_MATH_BIG_CONSTANT(T, 113, 0.391926317852243778169704095630021556e-4),
502 BOOST_MATH_BIG_CONSTANT(T, 113, -0.218544851067999216147364295512443661e-5),
503 BOOST_MATH_BIG_CONSTANT(T, 113, -0.185406221071515996070179883622956325e-5),
504 BOOST_MATH_BIG_CONSTANT(T, 113, 0.829671134095308600501624213166443227e-6),
505 BOOST_MATH_BIG_CONSTANT(T, 113, -0.17665952736826079304360054245742403e-6),
506 BOOST_MATH_BIG_CONSTANT(T, 113, 0.670785354340149858036939710029613572e-8),
507 BOOST_MATH_BIG_CONSTANT(T, 113, 0.102618097842403080425739573227252951e-7),
508 BOOST_MATH_BIG_CONSTANT(T, 113, -0.438203601845335318655297462244719123e-8),
509 BOOST_MATH_BIG_CONSTANT(T, 113, 0.914769958223679023418248817633113681e-9),
510 BOOST_MATH_BIG_CONSTANT(T, 113, -0.255141939949462497668779537993887013e-10),
511 BOOST_MATH_BIG_CONSTANT(T, 113, -0.583077213255042506746408945040035798e-10),
512 BOOST_MATH_BIG_CONSTANT(T, 113, 0.243619480206674162436940696707789943e-10),
513 BOOST_MATH_BIG_CONSTANT(T, 113, -0.502766928011417558909054985925744366e-11),
514 BOOST_MATH_BIG_CONSTANT(T, 113, 0.110043920319561347708374174497293411e-12),
515 BOOST_MATH_BIG_CONSTANT(T, 113, 0.337176326240098537882769884169200185e-12),
516 BOOST_MATH_BIG_CONSTANT(T, 113, -0.13923887224181620659193661848957998e-12),
517 BOOST_MATH_BIG_CONSTANT(T, 113, 0.285348938070474432039669099052828299e-13),
518 BOOST_MATH_BIG_CONSTANT(T, 113, -0.513911183424257261899064580300494205e-15),
519 BOOST_MATH_BIG_CONSTANT(T, 113, -0.197522882943494428353962401580710912e-14),
520 BOOST_MATH_BIG_CONSTANT(T, 113, 0.809952115670456133407115668702575255e-15),
521 BOOST_MATH_BIG_CONSTANT(T, 113, -0.165225312163981618191514820265351162e-15),
522 BOOST_MATH_BIG_CONSTANT(T, 113, 0.253054300974788842327061090060267385e-17),
523 BOOST_MATH_BIG_CONSTANT(T, 113, 0.116869397385595765888230876507793475e-16),
524 BOOST_MATH_BIG_CONSTANT(T, 113, -0.477003704982048475822167804084816597e-17),
525 BOOST_MATH_BIG_CONSTANT(T, 113, 0.969912605905623712420709685898585354e-18),
526 };
527 workspace[0] = tools::evaluate_polynomial(C0, z);
528
529 static const T C1[] = {
530 BOOST_MATH_BIG_CONSTANT(T, 113, -0.00185185185185185185185185185185185185),
531 BOOST_MATH_BIG_CONSTANT(T, 113, -0.00347222222222222222222222222222222222),
532 BOOST_MATH_BIG_CONSTANT(T, 113, 0.0026455026455026455026455026455026455),
533 BOOST_MATH_BIG_CONSTANT(T, 113, -0.000990226337448559670781893004115226337),
534 BOOST_MATH_BIG_CONSTANT(T, 113, 0.000205761316872427983539094650205761317),
535 BOOST_MATH_BIG_CONSTANT(T, 113, -0.401877572016460905349794238683127572e-6),
536 BOOST_MATH_BIG_CONSTANT(T, 113, -0.180985503344899778370285914867533523e-4),
537 BOOST_MATH_BIG_CONSTANT(T, 113, 0.76491609160811100846374214980916921e-5),
538 BOOST_MATH_BIG_CONSTANT(T, 113, -0.16120900894563446003775221882217767e-5),
539 BOOST_MATH_BIG_CONSTANT(T, 113, 0.464712780280743434226135033938722401e-8),
540 BOOST_MATH_BIG_CONSTANT(T, 113, 0.137863344691572095931187533077488877e-6),
541 BOOST_MATH_BIG_CONSTANT(T, 113, -0.575254560351770496402194531835048307e-7),
542 BOOST_MATH_BIG_CONSTANT(T, 113, 0.119516285997781473243076536699698169e-7),
543 BOOST_MATH_BIG_CONSTANT(T, 113, -0.175432417197476476237547551202312502e-10),
544 BOOST_MATH_BIG_CONSTANT(T, 113, -0.100915437106004126274577504686681675e-8),
545 BOOST_MATH_BIG_CONSTANT(T, 113, 0.416279299184258263623372347219858628e-9),
546 BOOST_MATH_BIG_CONSTANT(T, 113, -0.856390702649298063807431562579670208e-10),
547 BOOST_MATH_BIG_CONSTANT(T, 113, 0.606721510160475861512701762169919581e-13),
548 BOOST_MATH_BIG_CONSTANT(T, 113, 0.716249896481148539007961017165545733e-11),
549 BOOST_MATH_BIG_CONSTANT(T, 113, -0.293318664377143711740636683615595403e-11),
550 BOOST_MATH_BIG_CONSTANT(T, 113, 0.599669636568368872330374527568788909e-12),
551 BOOST_MATH_BIG_CONSTANT(T, 113, -0.216717865273233141017100472779701734e-15),
552 BOOST_MATH_BIG_CONSTANT(T, 113, -0.497833997236926164052815522048108548e-13),
553 BOOST_MATH_BIG_CONSTANT(T, 113, 0.202916288237134247736694804325894226e-13),
554 BOOST_MATH_BIG_CONSTANT(T, 113, -0.413125571381061004935108332558187111e-14),
555 BOOST_MATH_BIG_CONSTANT(T, 113, 0.828651623988309644380188591057589316e-18),
556 BOOST_MATH_BIG_CONSTANT(T, 113, 0.341003088693333279336339355910600992e-15),
557 BOOST_MATH_BIG_CONSTANT(T, 113, -0.138541953028939715357034547426313703e-15),
558 BOOST_MATH_BIG_CONSTANT(T, 113, 0.281234665322887466568860332727259483e-16),
559 };
560 workspace[1] = tools::evaluate_polynomial(C1, z);
561
562 static const T C2[] = {
563 BOOST_MATH_BIG_CONSTANT(T, 113, 0.0041335978835978835978835978835978836),
564 BOOST_MATH_BIG_CONSTANT(T, 113, -0.00268132716049382716049382716049382716),
565 BOOST_MATH_BIG_CONSTANT(T, 113, 0.000771604938271604938271604938271604938),
566 BOOST_MATH_BIG_CONSTANT(T, 113, 0.200938786008230452674897119341563786e-5),
567 BOOST_MATH_BIG_CONSTANT(T, 113, -0.000107366532263651605215391223621676297),
568 BOOST_MATH_BIG_CONSTANT(T, 113, 0.529234488291201254164217127180090143e-4),
569 BOOST_MATH_BIG_CONSTANT(T, 113, -0.127606351886187277133779191392360117e-4),
570 BOOST_MATH_BIG_CONSTANT(T, 113, 0.34235787340961380741902003904747389e-7),
571 BOOST_MATH_BIG_CONSTANT(T, 113, 0.137219573090629332055943852926020279e-5),
572 BOOST_MATH_BIG_CONSTANT(T, 113, -0.629899213838005502290672234278391876e-6),
573 BOOST_MATH_BIG_CONSTANT(T, 113, 0.142806142060642417915846008822771748e-6),
574 BOOST_MATH_BIG_CONSTANT(T, 113, -0.204770984219908660149195854409200226e-9),
575 BOOST_MATH_BIG_CONSTANT(T, 113, -0.140925299108675210532930244154315272e-7),
576 BOOST_MATH_BIG_CONSTANT(T, 113, 0.622897408492202203356394293530327112e-8),
577 BOOST_MATH_BIG_CONSTANT(T, 113, -0.136704883966171134992724380284402402e-8),
578 BOOST_MATH_BIG_CONSTANT(T, 113, 0.942835615901467819547711211663208075e-12),
579 BOOST_MATH_BIG_CONSTANT(T, 113, 0.128722524000893180595479368872770442e-9),
580 BOOST_MATH_BIG_CONSTANT(T, 113, -0.556459561343633211465414765894951439e-10),
581 BOOST_MATH_BIG_CONSTANT(T, 113, 0.119759355463669810035898150310311343e-10),
582 BOOST_MATH_BIG_CONSTANT(T, 113, -0.416897822518386350403836626692480096e-14),
583 BOOST_MATH_BIG_CONSTANT(T, 113, -0.109406404278845944099299008640802908e-11),
584 BOOST_MATH_BIG_CONSTANT(T, 113, 0.4662239946390135746326204922464679e-12),
585 BOOST_MATH_BIG_CONSTANT(T, 113, -0.990510576390690597844122258212382301e-13),
586 BOOST_MATH_BIG_CONSTANT(T, 113, 0.189318767683735145056885183170630169e-16),
587 BOOST_MATH_BIG_CONSTANT(T, 113, 0.885922187259112726176031067028740667e-14),
588 BOOST_MATH_BIG_CONSTANT(T, 113, -0.373782039804640545306560251777191937e-14),
589 BOOST_MATH_BIG_CONSTANT(T, 113, 0.786883363903515525774088394065960751e-15),
590 };
591 workspace[2] = tools::evaluate_polynomial(C2, z);
592
593 static const T C3[] = {
594 BOOST_MATH_BIG_CONSTANT(T, 113, 0.000649434156378600823045267489711934156),
595 BOOST_MATH_BIG_CONSTANT(T, 113, 0.000229472093621399176954732510288065844),
596 BOOST_MATH_BIG_CONSTANT(T, 113, -0.000469189494395255712128140111679206329),
597 BOOST_MATH_BIG_CONSTANT(T, 113, 0.000267720632062838852962309752433209223),
598 BOOST_MATH_BIG_CONSTANT(T, 113, -0.756180167188397641072538191879755666e-4),
599 BOOST_MATH_BIG_CONSTANT(T, 113, -0.239650511386729665193314027333231723e-6),
600 BOOST_MATH_BIG_CONSTANT(T, 113, 0.110826541153473023614770299726861227e-4),
601 BOOST_MATH_BIG_CONSTANT(T, 113, -0.567495282699159656749963105701560205e-5),
602 BOOST_MATH_BIG_CONSTANT(T, 113, 0.14230900732435883914551894470580433e-5),
603 BOOST_MATH_BIG_CONSTANT(T, 113, -0.278610802915281422405802158211174452e-10),
604 BOOST_MATH_BIG_CONSTANT(T, 113, -0.16958404091930277289864168795820267e-6),
605 BOOST_MATH_BIG_CONSTANT(T, 113, 0.809946490538808236335278504852724081e-7),
606 BOOST_MATH_BIG_CONSTANT(T, 113, -0.191111684859736540606728140872727635e-7),
607 BOOST_MATH_BIG_CONSTANT(T, 113, 0.239286204398081179686413514022282056e-11),
608 BOOST_MATH_BIG_CONSTANT(T, 113, 0.206201318154887984369925818486654549e-8),
609 BOOST_MATH_BIG_CONSTANT(T, 113, -0.946049666185513217375417988510192814e-9),
610 BOOST_MATH_BIG_CONSTANT(T, 113, 0.215410497757749078380130268468744512e-9),
611 BOOST_MATH_BIG_CONSTANT(T, 113, -0.138882333681390304603424682490735291e-13),
612 BOOST_MATH_BIG_CONSTANT(T, 113, -0.218947616819639394064123400466489455e-10),
613 BOOST_MATH_BIG_CONSTANT(T, 113, 0.979099895117168512568262802255883368e-11),
614 BOOST_MATH_BIG_CONSTANT(T, 113, -0.217821918801809621153859472011393244e-11),
615 BOOST_MATH_BIG_CONSTANT(T, 113, 0.62088195734079014258166361684972205e-16),
616 BOOST_MATH_BIG_CONSTANT(T, 113, 0.212697836327973697696702537114614471e-12),
617 BOOST_MATH_BIG_CONSTANT(T, 113, -0.934468879151743333127396765626749473e-13),
618 BOOST_MATH_BIG_CONSTANT(T, 113, 0.204536712267828493249215913063207436e-13),
619 };
620 workspace[3] = tools::evaluate_polynomial(C3, z);
621
622 static const T C4[] = {
623 BOOST_MATH_BIG_CONSTANT(T, 113, -0.000861888290916711698604702719929057378),
624 BOOST_MATH_BIG_CONSTANT(T, 113, 0.00078403922172006662747403488144228885),
625 BOOST_MATH_BIG_CONSTANT(T, 113, -0.000299072480303190179733389609932819809),
626 BOOST_MATH_BIG_CONSTANT(T, 113, -0.146384525788434181781232535690697556e-5),
627 BOOST_MATH_BIG_CONSTANT(T, 113, 0.664149821546512218665853782451862013e-4),
628 BOOST_MATH_BIG_CONSTANT(T, 113, -0.396836504717943466443123507595386882e-4),
629 BOOST_MATH_BIG_CONSTANT(T, 113, 0.113757269706784190980552042885831759e-4),
630 BOOST_MATH_BIG_CONSTANT(T, 113, 0.250749722623753280165221942390057007e-9),
631 BOOST_MATH_BIG_CONSTANT(T, 113, -0.169541495365583060147164356781525752e-5),
632 BOOST_MATH_BIG_CONSTANT(T, 113, 0.890750753220530968882898422505515924e-6),
633 BOOST_MATH_BIG_CONSTANT(T, 113, -0.229293483400080487057216364891158518e-6),
634 BOOST_MATH_BIG_CONSTANT(T, 113, 0.295679413754404904696572852500004588e-10),
635 BOOST_MATH_BIG_CONSTANT(T, 113, 0.288658297427087836297341274604184504e-7),
636 BOOST_MATH_BIG_CONSTANT(T, 113, -0.141897394378032193894774303903982717e-7),
637 BOOST_MATH_BIG_CONSTANT(T, 113, 0.344635804994648970659527720474194356e-8),
638 BOOST_MATH_BIG_CONSTANT(T, 113, -0.230245171745280671320192735850147087e-12),
639 BOOST_MATH_BIG_CONSTANT(T, 113, -0.394092330280464052750697640085291799e-9),
640 BOOST_MATH_BIG_CONSTANT(T, 113, 0.186023389685045019134258533045185639e-9),
641 BOOST_MATH_BIG_CONSTANT(T, 113, -0.435632300505661804380678327446262424e-10),
642 BOOST_MATH_BIG_CONSTANT(T, 113, 0.127860010162962312660550463349930726e-14),
643 BOOST_MATH_BIG_CONSTANT(T, 113, 0.467927502665791946200382739991760062e-11),
644 BOOST_MATH_BIG_CONSTANT(T, 113, -0.214924647061348285410535341910721086e-11),
645 BOOST_MATH_BIG_CONSTANT(T, 113, 0.490881561480965216323649688463984082e-12),
646 };
647 workspace[4] = tools::evaluate_polynomial(C4, z);
648
649 static const T C5[] = {
650 BOOST_MATH_BIG_CONSTANT(T, 113, -0.000336798553366358150308767592718210002),
651 BOOST_MATH_BIG_CONSTANT(T, 113, -0.697281375836585777429398828575783308e-4),
652 BOOST_MATH_BIG_CONSTANT(T, 113, 0.00027727532449593920787336425196507501),
653 BOOST_MATH_BIG_CONSTANT(T, 113, -0.000199325705161888477003360405280844238),
654 BOOST_MATH_BIG_CONSTANT(T, 113, 0.679778047793720783881640176604435742e-4),
655 BOOST_MATH_BIG_CONSTANT(T, 113, 0.141906292064396701483392727105575757e-6),
656 BOOST_MATH_BIG_CONSTANT(T, 113, -0.135940481897686932784583938837504469e-4),
657 BOOST_MATH_BIG_CONSTANT(T, 113, 0.80184702563342015397192571980419684e-5),
658 BOOST_MATH_BIG_CONSTANT(T, 113, -0.229148117650809517038048790128781806e-5),
659 BOOST_MATH_BIG_CONSTANT(T, 113, -0.325247355129845395166230137750005047e-9),
660 BOOST_MATH_BIG_CONSTANT(T, 113, 0.346528464910852649559195496827579815e-6),
661 BOOST_MATH_BIG_CONSTANT(T, 113, -0.184471871911713432765322367374920978e-6),
662 BOOST_MATH_BIG_CONSTANT(T, 113, 0.482409670378941807563762631738989002e-7),
663 BOOST_MATH_BIG_CONSTANT(T, 113, -0.179894667217435153025754291716644314e-13),
664 BOOST_MATH_BIG_CONSTANT(T, 113, -0.630619450001352343517516981425944698e-8),
665 BOOST_MATH_BIG_CONSTANT(T, 113, 0.316241762877456793773762181540969623e-8),
666 BOOST_MATH_BIG_CONSTANT(T, 113, -0.784092425369742929000839303523267545e-9),
667 };
668 workspace[5] = tools::evaluate_polynomial(C5, z);
669
670 static const T C6[] = {
671 BOOST_MATH_BIG_CONSTANT(T, 113, 0.00053130793646399222316574854297762391),
672 BOOST_MATH_BIG_CONSTANT(T, 113, -0.000592166437353693882864836225604401187),
673 BOOST_MATH_BIG_CONSTANT(T, 113, 0.000270878209671804482771279183488328692),
674 BOOST_MATH_BIG_CONSTANT(T, 113, 0.790235323266032787212032944390816666e-6),
675 BOOST_MATH_BIG_CONSTANT(T, 113, -0.815396936756196875092890088464682624e-4),
676 BOOST_MATH_BIG_CONSTANT(T, 113, 0.561168275310624965003775619041471695e-4),
677 BOOST_MATH_BIG_CONSTANT(T, 113, -0.183291165828433755673259749374098313e-4),
678 BOOST_MATH_BIG_CONSTANT(T, 113, -0.307961345060330478256414192546677006e-8),
679 BOOST_MATH_BIG_CONSTANT(T, 113, 0.346515536880360908673728529745376913e-5),
680 BOOST_MATH_BIG_CONSTANT(T, 113, -0.202913273960586037269527254582695285e-5),
681 BOOST_MATH_BIG_CONSTANT(T, 113, 0.578879286314900370889997586203187687e-6),
682 BOOST_MATH_BIG_CONSTANT(T, 113, 0.233863067382665698933480579231637609e-12),
683 BOOST_MATH_BIG_CONSTANT(T, 113, -0.88286007463304835250508524317926246e-7),
684 BOOST_MATH_BIG_CONSTANT(T, 113, 0.474359588804081278032150770595852426e-7),
685 BOOST_MATH_BIG_CONSTANT(T, 113, -0.125454150207103824457130611214783073e-7),
686 };
687 workspace[6] = tools::evaluate_polynomial(C6, z);
688
689 static const T C7[] = {
690 BOOST_MATH_BIG_CONSTANT(T, 113, 0.000344367606892377671254279625108523655),
691 BOOST_MATH_BIG_CONSTANT(T, 113, 0.517179090826059219337057843002058823e-4),
692 BOOST_MATH_BIG_CONSTANT(T, 113, -0.000334931610811422363116635090580012327),
693 BOOST_MATH_BIG_CONSTANT(T, 113, 0.000281269515476323702273722110707777978),
694 BOOST_MATH_BIG_CONSTANT(T, 113, -0.000109765822446847310235396824500789005),
695 BOOST_MATH_BIG_CONSTANT(T, 113, -0.127410090954844853794579954588107623e-6),
696 BOOST_MATH_BIG_CONSTANT(T, 113, 0.277444515115636441570715073933712622e-4),
697 BOOST_MATH_BIG_CONSTANT(T, 113, -0.182634888057113326614324442681892723e-4),
698 BOOST_MATH_BIG_CONSTANT(T, 113, 0.578769494973505239894178121070843383e-5),
699 BOOST_MATH_BIG_CONSTANT(T, 113, 0.493875893393627039981813418398565502e-9),
700 BOOST_MATH_BIG_CONSTANT(T, 113, -0.105953670140260427338098566209633945e-5),
701 BOOST_MATH_BIG_CONSTANT(T, 113, 0.616671437611040747858836254004890765e-6),
702 BOOST_MATH_BIG_CONSTANT(T, 113, -0.175629733590604619378669693914265388e-6),
703 };
704 workspace[7] = tools::evaluate_polynomial(C7, z);
705
706 static const T C8[] = {
707 BOOST_MATH_BIG_CONSTANT(T, 113, -0.000652623918595309418922034919726622692),
708 BOOST_MATH_BIG_CONSTANT(T, 113, 0.000839498720672087279993357516764983445),
709 BOOST_MATH_BIG_CONSTANT(T, 113, -0.000438297098541721005061087953050560377),
710 BOOST_MATH_BIG_CONSTANT(T, 113, -0.696909145842055197136911097362072702e-6),
711 BOOST_MATH_BIG_CONSTANT(T, 113, 0.00016644846642067547837384572662326101),
712 BOOST_MATH_BIG_CONSTANT(T, 113, -0.000127835176797692185853344001461664247),
713 BOOST_MATH_BIG_CONSTANT(T, 113, 0.462995326369130429061361032704489636e-4),
714 BOOST_MATH_BIG_CONSTANT(T, 113, 0.455790986792270771162749294232219616e-8),
715 BOOST_MATH_BIG_CONSTANT(T, 113, -0.105952711258051954718238500312872328e-4),
716 BOOST_MATH_BIG_CONSTANT(T, 113, 0.678334290486516662273073740749269432e-5),
717 BOOST_MATH_BIG_CONSTANT(T, 113, -0.210754766662588042469972680229376445e-5),
718 };
719 workspace[8] = tools::evaluate_polynomial(C8, z);
720
721 static const T C9[] = {
722 BOOST_MATH_BIG_CONSTANT(T, 113, -0.000596761290192746250124390067179459605),
723 BOOST_MATH_BIG_CONSTANT(T, 113, -0.720489541602001055908571930225015052e-4),
724 BOOST_MATH_BIG_CONSTANT(T, 113, 0.000678230883766732836161951166000673426),
725 BOOST_MATH_BIG_CONSTANT(T, 113, -0.000640147526026275845100045652582354779),
726 BOOST_MATH_BIG_CONSTANT(T, 113, 0.000277501076343287044992374518205845463),
727 BOOST_MATH_BIG_CONSTANT(T, 113, 0.181970083804651510461686554030325202e-6),
728 BOOST_MATH_BIG_CONSTANT(T, 113, -0.847950711706850318239732559632810086e-4),
729 BOOST_MATH_BIG_CONSTANT(T, 113, 0.610519208250153101764709122740859458e-4),
730 BOOST_MATH_BIG_CONSTANT(T, 113, -0.210739201834048624082975255893773306e-4),
731 };
732 workspace[9] = tools::evaluate_polynomial(C9, z);
733
734 static const T C10[] = {
735 BOOST_MATH_BIG_CONSTANT(T, 113, 0.00133244544948006563712694993432717968),
736 BOOST_MATH_BIG_CONSTANT(T, 113, -0.00191443849856547752650089885832852254),
737 BOOST_MATH_BIG_CONSTANT(T, 113, 0.0011089369134596637339607446329267522),
738 BOOST_MATH_BIG_CONSTANT(T, 113, 0.993240412264229896742295262075817566e-6),
739 BOOST_MATH_BIG_CONSTANT(T, 113, -0.000508745012930931989848393025305956774),
740 BOOST_MATH_BIG_CONSTANT(T, 113, 0.00042735056665392884328432271160040444),
741 BOOST_MATH_BIG_CONSTANT(T, 113, -0.000168588537679107988033552814662382059),
742 };
743 workspace[10] = tools::evaluate_polynomial(C10, z);
744
745 static const T C11[] = {
746 BOOST_MATH_BIG_CONSTANT(T, 113, 0.00157972766073083495908785631307733022),
747 BOOST_MATH_BIG_CONSTANT(T, 113, 0.000162516262783915816898635123980270998),
748 BOOST_MATH_BIG_CONSTANT(T, 113, -0.00206334210355432762645284467690276817),
749 BOOST_MATH_BIG_CONSTANT(T, 113, 0.00213896861856890981541061922797693947),
750 BOOST_MATH_BIG_CONSTANT(T, 113, -0.00101085593912630031708085801712479376),
751 };
752 workspace[11] = tools::evaluate_polynomial(C11, z);
753
754 static const T C12[] = {
755 BOOST_MATH_BIG_CONSTANT(T, 113, -0.00407251211951401664727281097914544601),
756 BOOST_MATH_BIG_CONSTANT(T, 113, 0.00640336283380806979482363809026579583),
757 BOOST_MATH_BIG_CONSTANT(T, 113, -0.00404101610816766177473974858518094879),
758 };
759 workspace[12] = tools::evaluate_polynomial(C12, z);
760 workspace[13] = -0.0059475779383993002845382844736066323L;
761
762 T result = tools::evaluate_polynomial(workspace, T(1/a));
763 result *= exp(-y) / sqrt(2 * constants::pi<T>() * a);
764 if(x < a)
765 result = -result;
766
767 result += boost::math::erfc(sqrt(y), pol) / 2;
768
769 return result;
770}
771
772} // namespace detail
773} // namespace math
774} // namespace math
775
776
777#endif // BOOST_MATH_DETAIL_IGAMMA_LARGE
778
779

source code of include/boost/math/special_functions/detail/igamma_large.hpp