1 // (C) 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)
6 #ifndef BOOST_MATH_SPECIAL_FUNCTIONS_DETAIL_LGAMMA_SMALL
7 #define BOOST_MATH_SPECIAL_FUNCTIONS_DETAIL_LGAMMA_SMALL
13 #include <boost/math/tools/big_constant.hpp>
15 #if defined(__GNUC__) && defined(BOOST_MATH_USE_FLOAT128)
17 // This is the only way we can avoid
18 // warning: non-standard suffix on floating constant [-Wpedantic]
19 // when building with -Wall -pedantic. Neither __extension__
20 // nor #pragma dianostic ignored work :(
22 #pragma GCC system_header
25 namespace boost{ namespace math{ namespace detail{
28 // These need forward declaring to keep GCC happy:
30 template <class T, class Policy, class Lanczos>
31 T gamma_imp(T z, const Policy& pol, const Lanczos& l);
32 template <class T, class Policy>
33 T gamma_imp(T z, const Policy& pol, const lanczos::undefined_lanczos& l);
36 // lgamma for small arguments:
38 template <class T, class Policy, class Lanczos>
39 T lgamma_small_imp(T z, T zm1, T zm2, const mpl::int_<64>&, const Policy& /* l */, const Lanczos&)
41 // This version uses rational approximations for small
42 // values of z accurate enough for 64-bit mantissas
43 // (80-bit long doubles), works well for 53-bit doubles as well.
44 // Lanczos is only used to select the Lanczos function.
46 BOOST_MATH_STD_USING // for ADL of std names
48 if(z < tools::epsilon<T>())
52 else if((zm1 == 0) || (zm2 == 0))
54 // nothing to do, result is zero....
59 // Begin by performing argument reduction until
70 // Update zm2, we need it below:
75 // Use the following form:
77 // lgamma(z) = (z-2)(z+1)(Y + R(z-2))
79 // where R(z-2) is a rational approximation optimised for
80 // low absolute error - as long as it's absolute error
81 // is small compared to the constant Y - then any rounding
82 // error in it's computation will get wiped out.
84 // R(z-2) has the following properties:
86 // At double: Max error found: 4.231e-18
87 // At long double: Max error found: 1.987e-21
88 // Maximum Deviation Found (approximation error): 5.900e-24
90 static const T P[] = {
91 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -0.180355685678449379109e-1)),
92 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 0.25126649619989678683e-1)),
93 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 0.494103151567532234274e-1)),
94 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 0.172491608709613993966e-1)),
95 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -0.259453563205438108893e-3)),
96 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -0.541009869215204396339e-3)),
97 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -0.324588649825948492091e-4))
99 static const T Q[] = {
100 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 0.1e1)),
101 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 0.196202987197795200688e1)),
102 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 0.148019669424231326694e1)),
103 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 0.541391432071720958364e0)),
104 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 0.988504251128010129477e-1)),
105 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 0.82130967464889339326e-2)),
106 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 0.224936291922115757597e-3)),
107 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -0.223352763208617092964e-6))
110 static const float Y = 0.158963680267333984375e0f;
113 T R = tools::evaluate_polynomial(P, zm2);
114 R /= tools::evaluate_polynomial(Q, zm2);
116 result += r * Y + r * R;
121 // If z is less than 1 use recurrance to shift to
122 // z in the interval [1,2]:
132 // Two approximations, on for z in [1,1.5] and
133 // one for z in [1.5,2]:
138 // Use the following form:
140 // lgamma(z) = (z-1)(z-2)(Y + R(z-1))
142 // where R(z-1) is a rational approximation optimised for
143 // low absolute error - as long as it's absolute error
144 // is small compared to the constant Y - then any rounding
145 // error in it's computation will get wiped out.
147 // R(z-1) has the following properties:
149 // At double precision: Max error found: 1.230011e-17
150 // At 80-bit long double precision: Max error found: 5.631355e-21
151 // Maximum Deviation Found: 3.139e-021
152 // Expected Error Term: 3.139e-021
155 static const float Y = 0.52815341949462890625f;
157 static const T P[] = {
158 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 0.490622454069039543534e-1)),
159 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -0.969117530159521214579e-1)),
160 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -0.414983358359495381969e0)),
161 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -0.406567124211938417342e0)),
162 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -0.158413586390692192217e0)),
163 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -0.240149820648571559892e-1)),
164 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -0.100346687696279557415e-2))
166 static const T Q[] = {
167 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 0.1e1)),
168 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 0.302349829846463038743e1)),
169 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 0.348739585360723852576e1)),
170 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 0.191415588274426679201e1)),
171 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 0.507137738614363510846e0)),
172 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 0.577039722690451849648e-1)),
173 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 0.195768102601107189171e-2))
176 T r = tools::evaluate_polynomial(P, zm1) / tools::evaluate_polynomial(Q, zm1);
177 T prefix = zm1 * zm2;
179 result += prefix * Y + prefix * r;
184 // Use the following form:
186 // lgamma(z) = (2-z)(1-z)(Y + R(2-z))
188 // where R(2-z) is a rational approximation optimised for
189 // low absolute error - as long as it's absolute error
190 // is small compared to the constant Y - then any rounding
191 // error in it's computation will get wiped out.
193 // R(2-z) has the following properties:
195 // At double precision, max error found: 1.797565e-17
196 // At 80-bit long double precision, max error found: 9.306419e-21
197 // Maximum Deviation Found: 2.151e-021
198 // Expected Error Term: 2.150e-021
200 static const float Y = 0.452017307281494140625f;
202 static const T P[] = {
203 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -0.292329721830270012337e-1)),
204 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 0.144216267757192309184e0)),
205 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -0.142440390738631274135e0)),
206 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 0.542809694055053558157e-1)),
207 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -0.850535976868336437746e-2)),
208 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 0.431171342679297331241e-3))
210 static const T Q[] = {
211 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 0.1e1)),
212 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -0.150169356054485044494e1)),
213 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 0.846973248876495016101e0)),
214 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -0.220095151814995745555e0)),
215 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 0.25582797155975869989e-1)),
216 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -0.100666795539143372762e-2)),
217 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -0.827193521891290553639e-6))
220 T R = tools::evaluate_polynomial(P, T(-zm2)) / tools::evaluate_polynomial(Q, T(-zm2));
222 result += r * Y + r * R;
227 template <class T, class Policy, class Lanczos>
228 T lgamma_small_imp(T z, T zm1, T zm2, const mpl::int_<113>&, const Policy& /* l */, const Lanczos&)
231 // This version uses rational approximations for small
232 // values of z accurate enough for 113-bit mantissas
233 // (128-bit long doubles).
235 BOOST_MATH_STD_USING // for ADL of std names
237 if(z < tools::epsilon<T>())
240 BOOST_MATH_INSTRUMENT_CODE(result);
242 else if((zm1 == 0) || (zm2 == 0))
244 // nothing to do, result is zero....
249 // Begin by performing argument reduction until
261 BOOST_MATH_INSTRUMENT_CODE(zm2);
262 BOOST_MATH_INSTRUMENT_CODE(z);
263 BOOST_MATH_INSTRUMENT_CODE(result);
266 // Use the following form:
268 // lgamma(z) = (z-2)(z+1)(Y + R(z-2))
270 // where R(z-2) is a rational approximation optimised for
271 // low absolute error - as long as it's absolute error
272 // is small compared to the constant Y - then any rounding
273 // error in it's computation will get wiped out.
275 // Maximum Deviation Found (approximation error) 3.73e-37
277 static const T P[] = {
278 BOOST_MATH_BIG_CONSTANT(T, 113, -0.018035568567844937910504030027467476655),
279 BOOST_MATH_BIG_CONSTANT(T, 113, 0.013841458273109517271750705401202404195),
280 BOOST_MATH_BIG_CONSTANT(T, 113, 0.062031842739486600078866923383017722399),
281 BOOST_MATH_BIG_CONSTANT(T, 113, 0.052518418329052161202007865149435256093),
282 BOOST_MATH_BIG_CONSTANT(T, 113, 0.01881718142472784129191838493267755758),
283 BOOST_MATH_BIG_CONSTANT(T, 113, 0.0025104830367021839316463675028524702846),
284 BOOST_MATH_BIG_CONSTANT(T, 113, -0.00021043176101831873281848891452678568311),
285 BOOST_MATH_BIG_CONSTANT(T, 113, -0.00010249622350908722793327719494037981166),
286 BOOST_MATH_BIG_CONSTANT(T, 113, -0.11381479670982006841716879074288176994e-4),
287 BOOST_MATH_BIG_CONSTANT(T, 113, -0.49999811718089980992888533630523892389e-6),
288 BOOST_MATH_BIG_CONSTANT(T, 113, -0.70529798686542184668416911331718963364e-8)
290 static const T Q[] = {
291 BOOST_MATH_BIG_CONSTANT(T, 113, 1.0),
292 BOOST_MATH_BIG_CONSTANT(T, 113, 2.5877485070422317542808137697939233685),
293 BOOST_MATH_BIG_CONSTANT(T, 113, 2.8797959228352591788629602533153837126),
294 BOOST_MATH_BIG_CONSTANT(T, 113, 1.8030885955284082026405495275461180977),
295 BOOST_MATH_BIG_CONSTANT(T, 113, 0.69774331297747390169238306148355428436),
296 BOOST_MATH_BIG_CONSTANT(T, 113, 0.17261566063277623942044077039756583802),
297 BOOST_MATH_BIG_CONSTANT(T, 113, 0.02729301254544230229429621192443000121),
298 BOOST_MATH_BIG_CONSTANT(T, 113, 0.0026776425891195270663133581960016620433),
299 BOOST_MATH_BIG_CONSTANT(T, 113, 0.00015244249160486584591370355730402168106),
300 BOOST_MATH_BIG_CONSTANT(T, 113, 0.43997034032479866020546814475414346627e-5),
301 BOOST_MATH_BIG_CONSTANT(T, 113, 0.46295080708455613044541885534408170934e-7),
302 BOOST_MATH_BIG_CONSTANT(T, 113, -0.93326638207459533682980757982834180952e-11),
303 BOOST_MATH_BIG_CONSTANT(T, 113, 0.42316456553164995177177407325292867513e-13)
306 T R = tools::evaluate_polynomial(P, zm2);
307 R /= tools::evaluate_polynomial(Q, zm2);
309 static const float Y = 0.158963680267333984375F;
313 result += r * Y + r * R;
314 BOOST_MATH_INSTRUMENT_CODE(result);
319 // If z is less than 1 use recurrance to shift to
320 // z in the interval [1,2]:
329 BOOST_MATH_INSTRUMENT_CODE(result);
330 BOOST_MATH_INSTRUMENT_CODE(z);
331 BOOST_MATH_INSTRUMENT_CODE(zm2);
333 // Three approximations, on for z in [1,1.35], [1.35,1.625] and [1.625,1]
338 // Use the following form:
340 // lgamma(z) = (z-1)(z-2)(Y + R(z-1))
342 // where R(z-1) is a rational approximation optimised for
343 // low absolute error - as long as it's absolute error
344 // is small compared to the constant Y - then any rounding
345 // error in it's computation will get wiped out.
347 // R(z-1) has the following properties:
349 // Maximum Deviation Found (approximation error) 1.659e-36
350 // Expected Error Term (theoretical error) 1.343e-36
351 // Max error found at 128-bit long double precision 1.007e-35
353 static const float Y = 0.54076099395751953125f;
355 static const T P[] = {
356 BOOST_MATH_BIG_CONSTANT(T, 113, 0.036454670944013329356512090082402429697),
357 BOOST_MATH_BIG_CONSTANT(T, 113, -0.066235835556476033710068679907798799959),
358 BOOST_MATH_BIG_CONSTANT(T, 113, -0.67492399795577182387312206593595565371),
359 BOOST_MATH_BIG_CONSTANT(T, 113, -1.4345555263962411429855341651960000166),
360 BOOST_MATH_BIG_CONSTANT(T, 113, -1.4894319559821365820516771951249649563),
361 BOOST_MATH_BIG_CONSTANT(T, 113, -0.87210277668067964629483299712322411566),
362 BOOST_MATH_BIG_CONSTANT(T, 113, -0.29602090537771744401524080430529369136),
363 BOOST_MATH_BIG_CONSTANT(T, 113, -0.0561832587517836908929331992218879676),
364 BOOST_MATH_BIG_CONSTANT(T, 113, -0.0053236785487328044334381502530383140443),
365 BOOST_MATH_BIG_CONSTANT(T, 113, -0.00018629360291358130461736386077971890789),
366 BOOST_MATH_BIG_CONSTANT(T, 113, -0.10164985672213178500790406939467614498e-6),
367 BOOST_MATH_BIG_CONSTANT(T, 113, 0.13680157145361387405588201461036338274e-8)
369 static const T Q[] = {
370 BOOST_MATH_BIG_CONSTANT(T, 113, 1.0),
371 BOOST_MATH_BIG_CONSTANT(T, 113, 4.9106336261005990534095838574132225599),
372 BOOST_MATH_BIG_CONSTANT(T, 113, 10.258804800866438510889341082793078432),
373 BOOST_MATH_BIG_CONSTANT(T, 113, 11.88588976846826108836629960537466889),
374 BOOST_MATH_BIG_CONSTANT(T, 113, 8.3455000546999704314454891036700998428),
375 BOOST_MATH_BIG_CONSTANT(T, 113, 3.6428823682421746343233362007194282703),
376 BOOST_MATH_BIG_CONSTANT(T, 113, 0.97465989807254572142266753052776132252),
377 BOOST_MATH_BIG_CONSTANT(T, 113, 0.15121052897097822172763084966793352524),
378 BOOST_MATH_BIG_CONSTANT(T, 113, 0.012017363555383555123769849654484594893),
379 BOOST_MATH_BIG_CONSTANT(T, 113, 0.0003583032812720649835431669893011257277)
382 T r = tools::evaluate_polynomial(P, zm1) / tools::evaluate_polynomial(Q, zm1);
383 T prefix = zm1 * zm2;
385 result += prefix * Y + prefix * r;
386 BOOST_MATH_INSTRUMENT_CODE(result);
391 // Use the following form:
393 // lgamma(z) = (2-z)(1-z)(Y + R(2-z))
395 // where R(2-z) is a rational approximation optimised for
396 // low absolute error - as long as it's absolute error
397 // is small compared to the constant Y - then any rounding
398 // error in it's computation will get wiped out.
400 // R(2-z) has the following properties:
402 // Max error found at 128-bit long double precision 9.634e-36
403 // Maximum Deviation Found (approximation error) 1.538e-37
404 // Expected Error Term (theoretical error) 2.350e-38
406 static const float Y = 0.483787059783935546875f;
408 static const T P[] = {
409 BOOST_MATH_BIG_CONSTANT(T, 113, -0.017977422421608624353488126610933005432),
410 BOOST_MATH_BIG_CONSTANT(T, 113, 0.18484528905298309555089509029244135703),
411 BOOST_MATH_BIG_CONSTANT(T, 113, -0.40401251514859546989565001431430884082),
412 BOOST_MATH_BIG_CONSTANT(T, 113, 0.40277179799147356461954182877921388182),
413 BOOST_MATH_BIG_CONSTANT(T, 113, -0.21993421441282936476709677700477598816),
414 BOOST_MATH_BIG_CONSTANT(T, 113, 0.069595742223850248095697771331107571011),
415 BOOST_MATH_BIG_CONSTANT(T, 113, -0.012681481427699686635516772923547347328),
416 BOOST_MATH_BIG_CONSTANT(T, 113, 0.0012489322866834830413292771335113136034),
417 BOOST_MATH_BIG_CONSTANT(T, 113, -0.57058739515423112045108068834668269608e-4),
418 BOOST_MATH_BIG_CONSTANT(T, 113, 0.8207548771933585614380644961342925976e-6)
420 static const T Q[] = {
421 BOOST_MATH_BIG_CONSTANT(T, 113, 1.0),
422 BOOST_MATH_BIG_CONSTANT(T, 113, -2.9629552288944259229543137757200262073),
423 BOOST_MATH_BIG_CONSTANT(T, 113, 3.7118380799042118987185957298964772755),
424 BOOST_MATH_BIG_CONSTANT(T, 113, -2.5569815272165399297600586376727357187),
425 BOOST_MATH_BIG_CONSTANT(T, 113, 1.0546764918220835097855665680632153367),
426 BOOST_MATH_BIG_CONSTANT(T, 113, -0.26574021300894401276478730940980810831),
427 BOOST_MATH_BIG_CONSTANT(T, 113, 0.03996289731752081380552901986471233462),
428 BOOST_MATH_BIG_CONSTANT(T, 113, -0.0033398680924544836817826046380586480873),
429 BOOST_MATH_BIG_CONSTANT(T, 113, 0.00013288854760548251757651556792598235735),
430 BOOST_MATH_BIG_CONSTANT(T, 113, -0.17194794958274081373243161848194745111e-5)
433 T R = tools::evaluate_polynomial(P, T(0.625 - zm1)) / tools::evaluate_polynomial(Q, T(0.625 - zm1));
435 result += r * Y + r * R;
436 BOOST_MATH_INSTRUMENT_CODE(result);
441 // Same form as above.
443 // Max error found (at 128-bit long double precision) 1.831e-35
444 // Maximum Deviation Found (approximation error) 8.588e-36
445 // Expected Error Term (theoretical error) 1.458e-36
447 static const float Y = 0.443811893463134765625f;
449 static const T P[] = {
450 BOOST_MATH_BIG_CONSTANT(T, 113, -0.021027558364667626231512090082402429494),
451 BOOST_MATH_BIG_CONSTANT(T, 113, 0.15128811104498736604523586803722368377),
452 BOOST_MATH_BIG_CONSTANT(T, 113, -0.26249631480066246699388544451126410278),
453 BOOST_MATH_BIG_CONSTANT(T, 113, 0.21148748610533489823742352180628489742),
454 BOOST_MATH_BIG_CONSTANT(T, 113, -0.093964130697489071999873506148104370633),
455 BOOST_MATH_BIG_CONSTANT(T, 113, 0.024292059227009051652542804957550866827),
456 BOOST_MATH_BIG_CONSTANT(T, 113, -0.0036284453226534839926304745756906117066),
457 BOOST_MATH_BIG_CONSTANT(T, 113, 0.0002939230129315195346843036254392485984),
458 BOOST_MATH_BIG_CONSTANT(T, 113, -0.11088589183158123733132268042570710338e-4),
459 BOOST_MATH_BIG_CONSTANT(T, 113, 0.13240510580220763969511741896361984162e-6)
461 static const T Q[] = {
462 BOOST_MATH_BIG_CONSTANT(T, 113, 1.0),
463 BOOST_MATH_BIG_CONSTANT(T, 113, -2.4240003754444040525462170802796471996),
464 BOOST_MATH_BIG_CONSTANT(T, 113, 2.4868383476933178722203278602342786002),
465 BOOST_MATH_BIG_CONSTANT(T, 113, -1.4047068395206343375520721509193698547),
466 BOOST_MATH_BIG_CONSTANT(T, 113, 0.47583809087867443858344765659065773369),
467 BOOST_MATH_BIG_CONSTANT(T, 113, -0.09865724264554556400463655444270700132),
468 BOOST_MATH_BIG_CONSTANT(T, 113, 0.012238223514176587501074150988445109735),
469 BOOST_MATH_BIG_CONSTANT(T, 113, -0.00084625068418239194670614419707491797097),
470 BOOST_MATH_BIG_CONSTANT(T, 113, 0.2796574430456237061420839429225710602e-4),
471 BOOST_MATH_BIG_CONSTANT(T, 113, -0.30202973883316730694433702165188835331e-6)
473 // (2 - x) * (1 - x) * (c + R(2 - x))
475 T R = tools::evaluate_polynomial(P, T(-zm2)) / tools::evaluate_polynomial(Q, T(-zm2));
477 result += r * Y + r * R;
478 BOOST_MATH_INSTRUMENT_CODE(result);
481 BOOST_MATH_INSTRUMENT_CODE(result);
484 template <class T, class Policy, class Lanczos>
485 T lgamma_small_imp(T z, T zm1, T zm2, const mpl::int_<0>&, const Policy& pol, const Lanczos&)
488 // No rational approximations are available because either
489 // T has no numeric_limits support (so we can't tell how
490 // many digits it has), or T has more digits than we know
491 // what to do with.... we do have a Lanczos approximation
492 // though, and that can be used to keep errors under control.
494 BOOST_MATH_STD_USING // for ADL of std names
496 if(z < tools::epsilon<T>())
502 // taking the log of tgamma reduces the error, no danger of overflow here:
503 result = log(gamma_imp(z, pol, Lanczos()));
507 // taking the log of tgamma reduces the error, no danger of overflow here:
508 result = log(gamma_imp(z, pol, Lanczos()));
512 // special case near 2:
514 result = dz * log((z + Lanczos::g() - T(0.5)) / boost::math::constants::e<T>());
515 result += boost::math::log1p(dz / (Lanczos::g() + T(1.5)), pol) * T(1.5);
516 result += boost::math::log1p(Lanczos::lanczos_sum_near_2(dz), pol);
520 // special case near 1:
522 result = dz * log((z + Lanczos::g() - T(0.5)) / boost::math::constants::e<T>());
523 result += boost::math::log1p(dz / (Lanczos::g() + T(0.5)), pol) / 2;
524 result += boost::math::log1p(Lanczos::lanczos_sum_near_1(dz), pol);
531 #endif // BOOST_MATH_SPECIAL_FUNCTIONS_DETAIL_LGAMMA_SMALL