1// Copyright John Maddock 2007, 2014.
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#ifndef BOOST_MATH_ZETA_HPP
7#define BOOST_MATH_ZETA_HPP
8
9#ifdef _MSC_VER
10#pragma once
11#endif
12
13#include <boost/math/special_functions/math_fwd.hpp>
14#include <boost/math/tools/precision.hpp>
15#include <boost/math/tools/series.hpp>
16#include <boost/math/tools/big_constant.hpp>
17#include <boost/math/policies/error_handling.hpp>
18#include <boost/math/special_functions/gamma.hpp>
19#include <boost/math/special_functions/factorials.hpp>
20#include <boost/math/special_functions/sin_pi.hpp>
21
22#if defined(__GNUC__) && defined(BOOST_MATH_USE_FLOAT128)
23//
24// This is the only way we can avoid
25// warning: non-standard suffix on floating constant [-Wpedantic]
26// when building with -Wall -pedantic. Neither __extension__
27// nor #pragma diagnostic ignored work :(
28//
29#pragma GCC system_header
30#endif
31
32namespace boost{ namespace math{ namespace detail{
33
34#if 0
35//
36// This code is commented out because we have a better more rapidly converging series
37// now. Retained for future reference and in case the new code causes any issues down the line....
38//
39
40template <class T, class Policy>
41struct zeta_series_cache_size
42{
43 //
44 // Work how large to make our cache size when evaluating the series
45 // evaluation: normally this is just large enough for the series
46 // to have converged, but for arbitrary precision types we need a
47 // really large cache to achieve reasonable precision in a reasonable
48 // time. This is important when constructing rational approximations
49 // to zeta for example.
50 //
51 typedef typename boost::math::policies::precision<T,Policy>::type precision_type;
52 typedef typename mpl::if_<
53 mpl::less_equal<precision_type, boost::integral_constant<int, 0> >,
54 boost::integral_constant<int, 5000>,
55 typename mpl::if_<
56 mpl::less_equal<precision_type, boost::integral_constant<int, 64> >,
57 boost::integral_constant<int, 70>,
58 typename mpl::if_<
59 mpl::less_equal<precision_type, boost::integral_constant<int, 113> >,
60 boost::integral_constant<int, 100>,
61 boost::integral_constant<int, 5000>
62 >::type
63 >::type
64 >::type type;
65};
66
67template <class T, class Policy>
68T zeta_series_imp(T s, T sc, const Policy&)
69{
70 //
71 // Series evaluation from:
72 // Havil, J. Gamma: Exploring Euler's Constant.
73 // Princeton, NJ: Princeton University Press, 2003.
74 //
75 // See also http://mathworld.wolfram.com/RiemannZetaFunction.html
76 //
77 BOOST_MATH_STD_USING
78 T sum = 0;
79 T mult = 0.5;
80 T change;
81 typedef typename zeta_series_cache_size<T,Policy>::type cache_size;
82 T powers[cache_size::value] = { 0, };
83 unsigned n = 0;
84 do{
85 T binom = -static_cast<T>(n);
86 T nested_sum = 1;
87 if(n < sizeof(powers) / sizeof(powers[0]))
88 powers[n] = pow(static_cast<T>(n + 1), -s);
89 for(unsigned k = 1; k <= n; ++k)
90 {
91 T p;
92 if(k < sizeof(powers) / sizeof(powers[0]))
93 {
94 p = powers[k];
95 //p = pow(k + 1, -s);
96 }
97 else
98 p = pow(static_cast<T>(k + 1), -s);
99 nested_sum += binom * p;
100 binom *= (k - static_cast<T>(n)) / (k + 1);
101 }
102 change = mult * nested_sum;
103 sum += change;
104 mult /= 2;
105 ++n;
106 }while(fabs(change / sum) > tools::epsilon<T>());
107
108 return sum * 1 / -boost::math::powm1(T(2), sc);
109}
110
111//
112// Classical p-series:
113//
114template <class T>
115struct zeta_series2
116{
117 typedef T result_type;
118 zeta_series2(T _s) : s(-_s), k(1){}
119 T operator()()
120 {
121 BOOST_MATH_STD_USING
122 return pow(static_cast<T>(k++), s);
123 }
124private:
125 T s;
126 unsigned k;
127};
128
129template <class T, class Policy>
130inline T zeta_series2_imp(T s, const Policy& pol)
131{
132 boost::uintmax_t max_iter = policies::get_max_series_iterations<Policy>();;
133 zeta_series2<T> f(s);
134 T result = tools::sum_series(
135 f,
136 policies::get_epsilon<T, Policy>(),
137 max_iter);
138 policies::check_series_iterations<T>("boost::math::zeta_series2<%1%>(%1%)", max_iter, pol);
139 return result;
140}
141#endif
142
143template <class T, class Policy>
144T zeta_polynomial_series(T s, T sc, Policy const &)
145{
146 //
147 // This is algorithm 3 from:
148 //
149 // "An Efficient Algorithm for the Riemann Zeta Function", P. Borwein,
150 // Canadian Mathematical Society, Conference Proceedings.
151 // See: http://www.cecm.sfu.ca/personal/pborwein/PAPERS/P155.pdf
152 //
153 BOOST_MATH_STD_USING
154 int n = itrunc(T(log(boost::math::tools::epsilon<T>()) / -2));
155 T sum = 0;
156 T two_n = ldexp(T(1), n);
157 int ej_sign = 1;
158 for(int j = 0; j < n; ++j)
159 {
160 sum += ej_sign * -two_n / pow(T(j + 1), s);
161 ej_sign = -ej_sign;
162 }
163 T ej_sum = 1;
164 T ej_term = 1;
165 for(int j = n; j <= 2 * n - 1; ++j)
166 {
167 sum += ej_sign * (ej_sum - two_n) / pow(T(j + 1), s);
168 ej_sign = -ej_sign;
169 ej_term *= 2 * n - j;
170 ej_term /= j - n + 1;
171 ej_sum += ej_term;
172 }
173 return -sum / (two_n * (-powm1(T(2), sc)));
174}
175
176template <class T, class Policy>
177T zeta_imp_prec(T s, T sc, const Policy& pol, const boost::integral_constant<int, 0>&)
178{
179 BOOST_MATH_STD_USING
180 T result;
181 if(s >= policies::digits<T, Policy>())
182 return 1;
183 result = zeta_polynomial_series(s, sc, pol);
184#if 0
185 // Old code archived for future reference:
186
187 //
188 // Only use power series if it will converge in 100
189 // iterations or less: the more iterations it consumes
190 // the slower convergence becomes so we have to be very
191 // careful in it's usage.
192 //
193 if (s > -log(tools::epsilon<T>()) / 4.5)
194 result = detail::zeta_series2_imp(s, pol);
195 else
196 result = detail::zeta_series_imp(s, sc, pol);
197#endif
198 return result;
199}
200
201template <class T, class Policy>
202inline T zeta_imp_prec(T s, T sc, const Policy&, const boost::integral_constant<int, 53>&)
203{
204 BOOST_MATH_STD_USING
205 T result;
206 if(s < 1)
207 {
208 // Rational Approximation
209 // Maximum Deviation Found: 2.020e-18
210 // Expected Error Term: -2.020e-18
211 // Max error found at double precision: 3.994987e-17
212 static const T P[6] = {
213 static_cast<T>(0.24339294433593750202L),
214 static_cast<T>(-0.49092470516353571651L),
215 static_cast<T>(0.0557616214776046784287L),
216 static_cast<T>(-0.00320912498879085894856L),
217 static_cast<T>(0.000451534528645796438704L),
218 static_cast<T>(-0.933241270357061460782e-5L),
219 };
220 static const T Q[6] = {
221 static_cast<T>(1L),
222 static_cast<T>(-0.279960334310344432495L),
223 static_cast<T>(0.0419676223309986037706L),
224 static_cast<T>(-0.00413421406552171059003L),
225 static_cast<T>(0.00024978985622317935355L),
226 static_cast<T>(-0.101855788418564031874e-4L),
227 };
228 result = tools::evaluate_polynomial(P, sc) / tools::evaluate_polynomial(Q, sc);
229 result -= 1.2433929443359375F;
230 result += (sc);
231 result /= (sc);
232 }
233 else if(s <= 2)
234 {
235 // Maximum Deviation Found: 9.007e-20
236 // Expected Error Term: 9.007e-20
237 static const T P[6] = {
238 static_cast<T>(0.577215664901532860516L),
239 static_cast<T>(0.243210646940107164097L),
240 static_cast<T>(0.0417364673988216497593L),
241 static_cast<T>(0.00390252087072843288378L),
242 static_cast<T>(0.000249606367151877175456L),
243 static_cast<T>(0.110108440976732897969e-4L),
244 };
245 static const T Q[6] = {
246 static_cast<T>(1.0),
247 static_cast<T>(0.295201277126631761737L),
248 static_cast<T>(0.043460910607305495864L),
249 static_cast<T>(0.00434930582085826330659L),
250 static_cast<T>(0.000255784226140488490982L),
251 static_cast<T>(0.10991819782396112081e-4L),
252 };
253 result = tools::evaluate_polynomial(P, T(-sc)) / tools::evaluate_polynomial(Q, T(-sc));
254 result += 1 / (-sc);
255 }
256 else if(s <= 4)
257 {
258 // Maximum Deviation Found: 5.946e-22
259 // Expected Error Term: -5.946e-22
260 static const float Y = 0.6986598968505859375;
261 static const T P[6] = {
262 static_cast<T>(-0.0537258300023595030676L),
263 static_cast<T>(0.0445163473292365591906L),
264 static_cast<T>(0.0128677673534519952905L),
265 static_cast<T>(0.00097541770457391752726L),
266 static_cast<T>(0.769875101573654070925e-4L),
267 static_cast<T>(0.328032510000383084155e-5L),
268 };
269 static const T Q[7] = {
270 1.0f,
271 static_cast<T>(0.33383194553034051422L),
272 static_cast<T>(0.0487798431291407621462L),
273 static_cast<T>(0.00479039708573558490716L),
274 static_cast<T>(0.000270776703956336357707L),
275 static_cast<T>(0.106951867532057341359e-4L),
276 static_cast<T>(0.236276623974978646399e-7L),
277 };
278 result = tools::evaluate_polynomial(P, T(s - 2)) / tools::evaluate_polynomial(Q, T(s - 2));
279 result += Y + 1 / (-sc);
280 }
281 else if(s <= 7)
282 {
283 // Maximum Deviation Found: 2.955e-17
284 // Expected Error Term: 2.955e-17
285 // Max error found at double precision: 2.009135e-16
286
287 static const T P[6] = {
288 static_cast<T>(-2.49710190602259410021L),
289 static_cast<T>(-2.60013301809475665334L),
290 static_cast<T>(-0.939260435377109939261L),
291 static_cast<T>(-0.138448617995741530935L),
292 static_cast<T>(-0.00701721240549802377623L),
293 static_cast<T>(-0.229257310594893932383e-4L),
294 };
295 static const T Q[9] = {
296 1.0f,
297 static_cast<T>(0.706039025937745133628L),
298 static_cast<T>(0.15739599649558626358L),
299 static_cast<T>(0.0106117950976845084417L),
300 static_cast<T>(-0.36910273311764618902e-4L),
301 static_cast<T>(0.493409563927590008943e-5L),
302 static_cast<T>(-0.234055487025287216506e-6L),
303 static_cast<T>(0.718833729365459760664e-8L),
304 static_cast<T>(-0.1129200113474947419e-9L),
305 };
306 result = tools::evaluate_polynomial(P, T(s - 4)) / tools::evaluate_polynomial(Q, T(s - 4));
307 result = 1 + exp(result);
308 }
309 else if(s < 15)
310 {
311 // Maximum Deviation Found: 7.117e-16
312 // Expected Error Term: 7.117e-16
313 // Max error found at double precision: 9.387771e-16
314 static const T P[7] = {
315 static_cast<T>(-4.78558028495135619286L),
316 static_cast<T>(-1.89197364881972536382L),
317 static_cast<T>(-0.211407134874412820099L),
318 static_cast<T>(-0.000189204758260076688518L),
319 static_cast<T>(0.00115140923889178742086L),
320 static_cast<T>(0.639949204213164496988e-4L),
321 static_cast<T>(0.139348932445324888343e-5L),
322 };
323 static const T Q[9] = {
324 1.0f,
325 static_cast<T>(0.244345337378188557777L),
326 static_cast<T>(0.00873370754492288653669L),
327 static_cast<T>(-0.00117592765334434471562L),
328 static_cast<T>(-0.743743682899933180415e-4L),
329 static_cast<T>(-0.21750464515767984778e-5L),
330 static_cast<T>(0.471001264003076486547e-8L),
331 static_cast<T>(-0.833378440625385520576e-10L),
332 static_cast<T>(0.699841545204845636531e-12L),
333 };
334 result = tools::evaluate_polynomial(P, T(s - 7)) / tools::evaluate_polynomial(Q, T(s - 7));
335 result = 1 + exp(result);
336 }
337 else if(s < 36)
338 {
339 // Max error in interpolated form: 1.668e-17
340 // Max error found at long double precision: 1.669714e-17
341 static const T P[8] = {
342 static_cast<T>(-10.3948950573308896825L),
343 static_cast<T>(-2.85827219671106697179L),
344 static_cast<T>(-0.347728266539245787271L),
345 static_cast<T>(-0.0251156064655346341766L),
346 static_cast<T>(-0.00119459173416968685689L),
347 static_cast<T>(-0.382529323507967522614e-4L),
348 static_cast<T>(-0.785523633796723466968e-6L),
349 static_cast<T>(-0.821465709095465524192e-8L),
350 };
351 static const T Q[10] = {
352 1.0f,
353 static_cast<T>(0.208196333572671890965L),
354 static_cast<T>(0.0195687657317205033485L),
355 static_cast<T>(0.00111079638102485921877L),
356 static_cast<T>(0.408507746266039256231e-4L),
357 static_cast<T>(0.955561123065693483991e-6L),
358 static_cast<T>(0.118507153474022900583e-7L),
359 static_cast<T>(0.222609483627352615142e-14L),
360 };
361 result = tools::evaluate_polynomial(P, T(s - 15)) / tools::evaluate_polynomial(Q, T(s - 15));
362 result = 1 + exp(result);
363 }
364 else if(s < 56)
365 {
366 result = 1 + pow(T(2), -s);
367 }
368 else
369 {
370 result = 1;
371 }
372 return result;
373}
374
375template <class T, class Policy>
376T zeta_imp_prec(T s, T sc, const Policy&, const boost::integral_constant<int, 64>&)
377{
378 BOOST_MATH_STD_USING
379 T result;
380 if(s < 1)
381 {
382 // Rational Approximation
383 // Maximum Deviation Found: 3.099e-20
384 // Expected Error Term: 3.099e-20
385 // Max error found at long double precision: 5.890498e-20
386 static const T P[6] = {
387 BOOST_MATH_BIG_CONSTANT(T, 64, 0.243392944335937499969),
388 BOOST_MATH_BIG_CONSTANT(T, 64, -0.496837806864865688082),
389 BOOST_MATH_BIG_CONSTANT(T, 64, 0.0680008039723709987107),
390 BOOST_MATH_BIG_CONSTANT(T, 64, -0.00511620413006619942112),
391 BOOST_MATH_BIG_CONSTANT(T, 64, 0.000455369899250053003335),
392 BOOST_MATH_BIG_CONSTANT(T, 64, -0.279496685273033761927e-4),
393 };
394 static const T Q[7] = {
395 BOOST_MATH_BIG_CONSTANT(T, 64, 1.0),
396 BOOST_MATH_BIG_CONSTANT(T, 64, -0.30425480068225790522),
397 BOOST_MATH_BIG_CONSTANT(T, 64, 0.050052748580371598736),
398 BOOST_MATH_BIG_CONSTANT(T, 64, -0.00519355671064700627862),
399 BOOST_MATH_BIG_CONSTANT(T, 64, 0.000360623385771198350257),
400 BOOST_MATH_BIG_CONSTANT(T, 64, -0.159600883054550987633e-4),
401 BOOST_MATH_BIG_CONSTANT(T, 64, 0.339770279812410586032e-6),
402 };
403 result = tools::evaluate_polynomial(P, sc) / tools::evaluate_polynomial(Q, sc);
404 result -= 1.2433929443359375F;
405 result += (sc);
406 result /= (sc);
407 }
408 else if(s <= 2)
409 {
410 // Maximum Deviation Found: 1.059e-21
411 // Expected Error Term: 1.059e-21
412 // Max error found at long double precision: 1.626303e-19
413
414 static const T P[6] = {
415 BOOST_MATH_BIG_CONSTANT(T, 64, 0.577215664901532860605),
416 BOOST_MATH_BIG_CONSTANT(T, 64, 0.222537368917162139445),
417 BOOST_MATH_BIG_CONSTANT(T, 64, 0.0356286324033215682729),
418 BOOST_MATH_BIG_CONSTANT(T, 64, 0.00304465292366350081446),
419 BOOST_MATH_BIG_CONSTANT(T, 64, 0.000178102511649069421904),
420 BOOST_MATH_BIG_CONSTANT(T, 64, 0.700867470265983665042e-5),
421 };
422 static const T Q[7] = {
423 BOOST_MATH_BIG_CONSTANT(T, 64, 1.0),
424 BOOST_MATH_BIG_CONSTANT(T, 64, 0.259385759149531030085),
425 BOOST_MATH_BIG_CONSTANT(T, 64, 0.0373974962106091316854),
426 BOOST_MATH_BIG_CONSTANT(T, 64, 0.00332735159183332820617),
427 BOOST_MATH_BIG_CONSTANT(T, 64, 0.000188690420706998606469),
428 BOOST_MATH_BIG_CONSTANT(T, 64, 0.635994377921861930071e-5),
429 BOOST_MATH_BIG_CONSTANT(T, 64, 0.226583954978371199405e-7),
430 };
431 result = tools::evaluate_polynomial(P, T(-sc)) / tools::evaluate_polynomial(Q, T(-sc));
432 result += 1 / (-sc);
433 }
434 else if(s <= 4)
435 {
436 // Maximum Deviation Found: 5.946e-22
437 // Expected Error Term: -5.946e-22
438 static const float Y = 0.6986598968505859375;
439 static const T P[7] = {
440 BOOST_MATH_BIG_CONSTANT(T, 64, -0.053725830002359501027),
441 BOOST_MATH_BIG_CONSTANT(T, 64, 0.0470551187571475844778),
442 BOOST_MATH_BIG_CONSTANT(T, 64, 0.0101339410415759517471),
443 BOOST_MATH_BIG_CONSTANT(T, 64, 0.00100240326666092854528),
444 BOOST_MATH_BIG_CONSTANT(T, 64, 0.685027119098122814867e-4),
445 BOOST_MATH_BIG_CONSTANT(T, 64, 0.390972820219765942117e-5),
446 BOOST_MATH_BIG_CONSTANT(T, 64, 0.540319769113543934483e-7),
447 };
448 static const T Q[8] = {
449 BOOST_MATH_BIG_CONSTANT(T, 64, 1.0),
450 BOOST_MATH_BIG_CONSTANT(T, 64, 0.286577739726542730421),
451 BOOST_MATH_BIG_CONSTANT(T, 64, 0.0447355811517733225843),
452 BOOST_MATH_BIG_CONSTANT(T, 64, 0.00430125107610252363302),
453 BOOST_MATH_BIG_CONSTANT(T, 64, 0.000284956969089786662045),
454 BOOST_MATH_BIG_CONSTANT(T, 64, 0.116188101609848411329e-4),
455 BOOST_MATH_BIG_CONSTANT(T, 64, 0.278090318191657278204e-6),
456 BOOST_MATH_BIG_CONSTANT(T, 64, -0.19683620233222028478e-8),
457 };
458 result = tools::evaluate_polynomial(P, T(s - 2)) / tools::evaluate_polynomial(Q, T(s - 2));
459 result += Y + 1 / (-sc);
460 }
461 else if(s <= 7)
462 {
463 // Max error found at long double precision: 8.132216e-19
464 static const T P[8] = {
465 BOOST_MATH_BIG_CONSTANT(T, 64, -2.49710190602259407065),
466 BOOST_MATH_BIG_CONSTANT(T, 64, -3.36664913245960625334),
467 BOOST_MATH_BIG_CONSTANT(T, 64, -1.77180020623777595452),
468 BOOST_MATH_BIG_CONSTANT(T, 64, -0.464717885249654313933),
469 BOOST_MATH_BIG_CONSTANT(T, 64, -0.0643694921293579472583),
470 BOOST_MATH_BIG_CONSTANT(T, 64, -0.00464265386202805715487),
471 BOOST_MATH_BIG_CONSTANT(T, 64, -0.000165556579779704340166),
472 BOOST_MATH_BIG_CONSTANT(T, 64, -0.252884970740994069582e-5),
473 };
474 static const T Q[9] = {
475 BOOST_MATH_BIG_CONSTANT(T, 64, 1.0),
476 BOOST_MATH_BIG_CONSTANT(T, 64, 1.01300131390690459085),
477 BOOST_MATH_BIG_CONSTANT(T, 64, 0.387898115758643503827),
478 BOOST_MATH_BIG_CONSTANT(T, 64, 0.0695071490045701135188),
479 BOOST_MATH_BIG_CONSTANT(T, 64, 0.00586908595251442839291),
480 BOOST_MATH_BIG_CONSTANT(T, 64, 0.000217752974064612188616),
481 BOOST_MATH_BIG_CONSTANT(T, 64, 0.397626583349419011731e-5),
482 BOOST_MATH_BIG_CONSTANT(T, 64, -0.927884739284359700764e-8),
483 BOOST_MATH_BIG_CONSTANT(T, 64, 0.119810501805618894381e-9),
484 };
485 result = tools::evaluate_polynomial(P, T(s - 4)) / tools::evaluate_polynomial(Q, T(s - 4));
486 result = 1 + exp(result);
487 }
488 else if(s < 15)
489 {
490 // Max error in interpolated form: 1.133e-18
491 // Max error found at long double precision: 2.183198e-18
492 static const T P[9] = {
493 BOOST_MATH_BIG_CONSTANT(T, 64, -4.78558028495135548083),
494 BOOST_MATH_BIG_CONSTANT(T, 64, -3.23873322238609358947),
495 BOOST_MATH_BIG_CONSTANT(T, 64, -0.892338582881021799922),
496 BOOST_MATH_BIG_CONSTANT(T, 64, -0.131326296217965913809),
497 BOOST_MATH_BIG_CONSTANT(T, 64, -0.0115651591773783712996),
498 BOOST_MATH_BIG_CONSTANT(T, 64, -0.000657728968362695775205),
499 BOOST_MATH_BIG_CONSTANT(T, 64, -0.252051328129449973047e-4),
500 BOOST_MATH_BIG_CONSTANT(T, 64, -0.626503445372641798925e-6),
501 BOOST_MATH_BIG_CONSTANT(T, 64, -0.815696314790853893484e-8),
502 };
503 static const T Q[9] = {
504 BOOST_MATH_BIG_CONSTANT(T, 64, 1.0),
505 BOOST_MATH_BIG_CONSTANT(T, 64, 0.525765665400123515036),
506 BOOST_MATH_BIG_CONSTANT(T, 64, 0.10852641753657122787),
507 BOOST_MATH_BIG_CONSTANT(T, 64, 0.0115669945375362045249),
508 BOOST_MATH_BIG_CONSTANT(T, 64, 0.000732896513858274091966),
509 BOOST_MATH_BIG_CONSTANT(T, 64, 0.30683952282420248448e-4),
510 BOOST_MATH_BIG_CONSTANT(T, 64, 0.819649214609633126119e-6),
511 BOOST_MATH_BIG_CONSTANT(T, 64, 0.117957556472335968146e-7),
512 BOOST_MATH_BIG_CONSTANT(T, 64, -0.193432300973017671137e-12),
513 };
514 result = tools::evaluate_polynomial(P, T(s - 7)) / tools::evaluate_polynomial(Q, T(s - 7));
515 result = 1 + exp(result);
516 }
517 else if(s < 42)
518 {
519 // Max error in interpolated form: 1.668e-17
520 // Max error found at long double precision: 1.669714e-17
521 static const T P[9] = {
522 BOOST_MATH_BIG_CONSTANT(T, 64, -10.3948950573308861781),
523 BOOST_MATH_BIG_CONSTANT(T, 64, -2.82646012777913950108),
524 BOOST_MATH_BIG_CONSTANT(T, 64, -0.342144362739570333665),
525 BOOST_MATH_BIG_CONSTANT(T, 64, -0.0249285145498722647472),
526 BOOST_MATH_BIG_CONSTANT(T, 64, -0.00122493108848097114118),
527 BOOST_MATH_BIG_CONSTANT(T, 64, -0.423055371192592850196e-4),
528 BOOST_MATH_BIG_CONSTANT(T, 64, -0.1025215577185967488e-5),
529 BOOST_MATH_BIG_CONSTANT(T, 64, -0.165096762663509467061e-7),
530 BOOST_MATH_BIG_CONSTANT(T, 64, -0.145392555873022044329e-9),
531 };
532 static const T Q[10] = {
533 BOOST_MATH_BIG_CONSTANT(T, 64, 1.0),
534 BOOST_MATH_BIG_CONSTANT(T, 64, 0.205135978585281988052),
535 BOOST_MATH_BIG_CONSTANT(T, 64, 0.0192359357875879453602),
536 BOOST_MATH_BIG_CONSTANT(T, 64, 0.00111496452029715514119),
537 BOOST_MATH_BIG_CONSTANT(T, 64, 0.434928449016693986857e-4),
538 BOOST_MATH_BIG_CONSTANT(T, 64, 0.116911068726610725891e-5),
539 BOOST_MATH_BIG_CONSTANT(T, 64, 0.206704342290235237475e-7),
540 BOOST_MATH_BIG_CONSTANT(T, 64, 0.209772836100827647474e-9),
541 BOOST_MATH_BIG_CONSTANT(T, 64, -0.939798249922234703384e-16),
542 BOOST_MATH_BIG_CONSTANT(T, 64, 0.264584017421245080294e-18),
543 };
544 result = tools::evaluate_polynomial(P, T(s - 15)) / tools::evaluate_polynomial(Q, T(s - 15));
545 result = 1 + exp(result);
546 }
547 else if(s < 63)
548 {
549 result = 1 + pow(T(2), -s);
550 }
551 else
552 {
553 result = 1;
554 }
555 return result;
556}
557
558template <class T, class Policy>
559T zeta_imp_prec(T s, T sc, const Policy&, const boost::integral_constant<int, 113>&)
560{
561 BOOST_MATH_STD_USING
562 T result;
563 if(s < 1)
564 {
565 // Rational Approximation
566 // Maximum Deviation Found: 9.493e-37
567 // Expected Error Term: 9.492e-37
568 // Max error found at long double precision: 7.281332e-31
569
570 static const T P[10] = {
571 BOOST_MATH_BIG_CONSTANT(T, 113, -1.0),
572 BOOST_MATH_BIG_CONSTANT(T, 113, -0.0353008629988648122808504280990313668),
573 BOOST_MATH_BIG_CONSTANT(T, 113, 0.0107795651204927743049369868548706909),
574 BOOST_MATH_BIG_CONSTANT(T, 113, 0.000523961870530500751114866884685172975),
575 BOOST_MATH_BIG_CONSTANT(T, 113, -0.661805838304910731947595897966487515e-4),
576 BOOST_MATH_BIG_CONSTANT(T, 113, -0.658932670403818558510656304189164638e-5),
577 BOOST_MATH_BIG_CONSTANT(T, 113, -0.103437265642266106533814021041010453e-6),
578 BOOST_MATH_BIG_CONSTANT(T, 113, 0.116818787212666457105375746642927737e-7),
579 BOOST_MATH_BIG_CONSTANT(T, 113, 0.660690993901506912123512551294239036e-9),
580 BOOST_MATH_BIG_CONSTANT(T, 113, 0.113103113698388531428914333768142527e-10),
581 };
582 static const T Q[11] = {
583 BOOST_MATH_BIG_CONSTANT(T, 113, 1.0),
584 BOOST_MATH_BIG_CONSTANT(T, 113, -0.387483472099602327112637481818565459),
585 BOOST_MATH_BIG_CONSTANT(T, 113, 0.0802265315091063135271497708694776875),
586 BOOST_MATH_BIG_CONSTANT(T, 113, -0.0110727276164171919280036408995078164),
587 BOOST_MATH_BIG_CONSTANT(T, 113, 0.00112552716946286252000434849173787243),
588 BOOST_MATH_BIG_CONSTANT(T, 113, -0.874554160748626916455655180296834352e-4),
589 BOOST_MATH_BIG_CONSTANT(T, 113, 0.530097847491828379568636739662278322e-5),
590 BOOST_MATH_BIG_CONSTANT(T, 113, -0.248461553590496154705565904497247452e-6),
591 BOOST_MATH_BIG_CONSTANT(T, 113, 0.881834921354014787309644951507523899e-8),
592 BOOST_MATH_BIG_CONSTANT(T, 113, -0.217062446168217797598596496310953025e-9),
593 BOOST_MATH_BIG_CONSTANT(T, 113, 0.315823200002384492377987848307151168e-11),
594 };
595 result = tools::evaluate_polynomial(P, sc) / tools::evaluate_polynomial(Q, sc);
596 result += (sc);
597 result /= (sc);
598 }
599 else if(s <= 2)
600 {
601 // Maximum Deviation Found: 1.616e-37
602 // Expected Error Term: -1.615e-37
603
604 static const T P[10] = {
605 BOOST_MATH_BIG_CONSTANT(T, 113, 0.577215664901532860606512090082402431),
606 BOOST_MATH_BIG_CONSTANT(T, 113, 0.255597968739771510415479842335906308),
607 BOOST_MATH_BIG_CONSTANT(T, 113, 0.0494056503552807274142218876983542205),
608 BOOST_MATH_BIG_CONSTANT(T, 113, 0.00551372778611700965268920983472292325),
609 BOOST_MATH_BIG_CONSTANT(T, 113, 0.00043667616723970574871427830895192731),
610 BOOST_MATH_BIG_CONSTANT(T, 113, 0.268562259154821957743669387915239528e-4),
611 BOOST_MATH_BIG_CONSTANT(T, 113, 0.109249633923016310141743084480436612e-5),
612 BOOST_MATH_BIG_CONSTANT(T, 113, 0.273895554345300227466534378753023924e-7),
613 BOOST_MATH_BIG_CONSTANT(T, 113, 0.583103205551702720149237384027795038e-9),
614 BOOST_MATH_BIG_CONSTANT(T, 113, -0.835774625259919268768735944711219256e-11),
615 };
616 static const T Q[11] = {
617 BOOST_MATH_BIG_CONSTANT(T, 113, 1.0),
618 BOOST_MATH_BIG_CONSTANT(T, 113, 0.316661751179735502065583176348292881),
619 BOOST_MATH_BIG_CONSTANT(T, 113, 0.0540401806533507064453851182728635272),
620 BOOST_MATH_BIG_CONSTANT(T, 113, 0.00598621274107420237785899476374043797),
621 BOOST_MATH_BIG_CONSTANT(T, 113, 0.000474907812321704156213038740142079615),
622 BOOST_MATH_BIG_CONSTANT(T, 113, 0.272125421722314389581695715835862418e-4),
623 BOOST_MATH_BIG_CONSTANT(T, 113, 0.112649552156479800925522445229212933e-5),
624 BOOST_MATH_BIG_CONSTANT(T, 113, 0.301838975502992622733000078063330461e-7),
625 BOOST_MATH_BIG_CONSTANT(T, 113, 0.422960728687211282539769943184270106e-9),
626 BOOST_MATH_BIG_CONSTANT(T, 113, -0.377105263588822468076813329270698909e-11),
627 BOOST_MATH_BIG_CONSTANT(T, 113, -0.581926559304525152432462127383600681e-13),
628 };
629 result = tools::evaluate_polynomial(P, T(-sc)) / tools::evaluate_polynomial(Q, T(-sc));
630 result += 1 / (-sc);
631 }
632 else if(s <= 4)
633 {
634 // Maximum Deviation Found: 1.891e-36
635 // Expected Error Term: -1.891e-36
636 // Max error found: 2.171527e-35
637
638 static const float Y = 0.6986598968505859375;
639 static const T P[11] = {
640 BOOST_MATH_BIG_CONSTANT(T, 113, -0.0537258300023595010275848333539748089),
641 BOOST_MATH_BIG_CONSTANT(T, 113, 0.0429086930802630159457448174466342553),
642 BOOST_MATH_BIG_CONSTANT(T, 113, 0.0136148228754303412510213395034056857),
643 BOOST_MATH_BIG_CONSTANT(T, 113, 0.00190231601036042925183751238033763915),
644 BOOST_MATH_BIG_CONSTANT(T, 113, 0.000186880390916311438818302549192456581),
645 BOOST_MATH_BIG_CONSTANT(T, 113, 0.145347370745893262394287982691323657e-4),
646 BOOST_MATH_BIG_CONSTANT(T, 113, 0.805843276446813106414036600485884885e-6),
647 BOOST_MATH_BIG_CONSTANT(T, 113, 0.340818159286739137503297172091882574e-7),
648 BOOST_MATH_BIG_CONSTANT(T, 113, 0.115762357488748996526167305116837246e-8),
649 BOOST_MATH_BIG_CONSTANT(T, 113, 0.231904754577648077579913403645767214e-10),
650 BOOST_MATH_BIG_CONSTANT(T, 113, 0.340169592866058506675897646629036044e-12),
651 };
652 static const T Q[12] = {
653 BOOST_MATH_BIG_CONSTANT(T, 113, 1.0),
654 BOOST_MATH_BIG_CONSTANT(T, 113, 0.363755247765087100018556983050520554),
655 BOOST_MATH_BIG_CONSTANT(T, 113, 0.0696581979014242539385695131258321598),
656 BOOST_MATH_BIG_CONSTANT(T, 113, 0.00882208914484611029571547753782014817),
657 BOOST_MATH_BIG_CONSTANT(T, 113, 0.000815405623261946661762236085660996718),
658 BOOST_MATH_BIG_CONSTANT(T, 113, 0.571366167062457197282642344940445452e-4),
659 BOOST_MATH_BIG_CONSTANT(T, 113, 0.309278269271853502353954062051797838e-5),
660 BOOST_MATH_BIG_CONSTANT(T, 113, 0.12822982083479010834070516053794262e-6),
661 BOOST_MATH_BIG_CONSTANT(T, 113, 0.397876357325018976733953479182110033e-8),
662 BOOST_MATH_BIG_CONSTANT(T, 113, 0.8484432107648683277598472295289279e-10),
663 BOOST_MATH_BIG_CONSTANT(T, 113, 0.105677416606909614301995218444080615e-11),
664 BOOST_MATH_BIG_CONSTANT(T, 113, 0.547223964564003701979951154093005354e-15),
665 };
666 result = tools::evaluate_polynomial(P, T(s - 2)) / tools::evaluate_polynomial(Q, T(s - 2));
667 result += Y + 1 / (-sc);
668 }
669 else if(s <= 6)
670 {
671 // Max error in interpolated form: 1.510e-37
672 // Max error found at long double precision: 2.769266e-34
673
674 static const T Y = 3.28348541259765625F;
675
676 static const T P[13] = {
677 BOOST_MATH_BIG_CONSTANT(T, 113, 0.786383506575062179339611614117697622),
678 BOOST_MATH_BIG_CONSTANT(T, 113, 0.495766593395271370974685959652073976),
679 BOOST_MATH_BIG_CONSTANT(T, 113, -0.409116737851754766422360889037532228),
680 BOOST_MATH_BIG_CONSTANT(T, 113, -0.57340744006238263817895456842655987),
681 BOOST_MATH_BIG_CONSTANT(T, 113, -0.280479899797421910694892949057963111),
682 BOOST_MATH_BIG_CONSTANT(T, 113, -0.0753148409447590257157585696212649869),
683 BOOST_MATH_BIG_CONSTANT(T, 113, -0.0122934003684672788499099362823748632),
684 BOOST_MATH_BIG_CONSTANT(T, 113, -0.00126148398446193639247961370266962927),
685 BOOST_MATH_BIG_CONSTANT(T, 113, -0.828465038179772939844657040917364896e-4),
686 BOOST_MATH_BIG_CONSTANT(T, 113, -0.361008916706050977143208468690645684e-5),
687 BOOST_MATH_BIG_CONSTANT(T, 113, -0.109879825497910544424797771195928112e-6),
688 BOOST_MATH_BIG_CONSTANT(T, 113, -0.214539416789686920918063075528797059e-8),
689 BOOST_MATH_BIG_CONSTANT(T, 113, -0.15090220092460596872172844424267351e-10),
690 };
691 static const T Q[14] = {
692 BOOST_MATH_BIG_CONSTANT(T, 113, 1.0),
693 BOOST_MATH_BIG_CONSTANT(T, 113, 1.69490865837142338462982225731926485),
694 BOOST_MATH_BIG_CONSTANT(T, 113, 1.22697696630994080733321401255942464),
695 BOOST_MATH_BIG_CONSTANT(T, 113, 0.495409420862526540074366618006341533),
696 BOOST_MATH_BIG_CONSTANT(T, 113, 0.122368084916843823462872905024259633),
697 BOOST_MATH_BIG_CONSTANT(T, 113, 0.0191412993625268971656513890888208623),
698 BOOST_MATH_BIG_CONSTANT(T, 113, 0.00191401538628980617753082598351559642),
699 BOOST_MATH_BIG_CONSTANT(T, 113, 0.000123318142456272424148930280876444459),
700 BOOST_MATH_BIG_CONSTANT(T, 113, 0.531945488232526067889835342277595709e-5),
701 BOOST_MATH_BIG_CONSTANT(T, 113, 0.161843184071894368337068779669116236e-6),
702 BOOST_MATH_BIG_CONSTANT(T, 113, 0.305796079600152506743828859577462778e-8),
703 BOOST_MATH_BIG_CONSTANT(T, 113, 0.233582592298450202680170811044408894e-10),
704 BOOST_MATH_BIG_CONSTANT(T, 113, -0.275363878344548055574209713637734269e-13),
705 BOOST_MATH_BIG_CONSTANT(T, 113, 0.221564186807357535475441900517843892e-15),
706 };
707 result = tools::evaluate_polynomial(P, T(s - 4)) / tools::evaluate_polynomial(Q, T(s - 4));
708 result -= Y;
709 result = 1 + exp(result);
710 }
711 else if(s < 10)
712 {
713 // Max error in interpolated form: 1.999e-34
714 // Max error found at long double precision: 2.156186e-33
715
716 static const T P[13] = {
717 BOOST_MATH_BIG_CONSTANT(T, 113, -4.0545627381873738086704293881227365),
718 BOOST_MATH_BIG_CONSTANT(T, 113, -4.70088348734699134347906176097717782),
719 BOOST_MATH_BIG_CONSTANT(T, 113, -2.36921550900925512951976617607678789),
720 BOOST_MATH_BIG_CONSTANT(T, 113, -0.684322583796369508367726293719322866),
721 BOOST_MATH_BIG_CONSTANT(T, 113, -0.126026534540165129870721937592996324),
722 BOOST_MATH_BIG_CONSTANT(T, 113, -0.015636903921778316147260572008619549),
723 BOOST_MATH_BIG_CONSTANT(T, 113, -0.00135442294754728549644376325814460807),
724 BOOST_MATH_BIG_CONSTANT(T, 113, -0.842793965853572134365031384646117061e-4),
725 BOOST_MATH_BIG_CONSTANT(T, 113, -0.385602133791111663372015460784978351e-5),
726 BOOST_MATH_BIG_CONSTANT(T, 113, -0.130458500394692067189883214401478539e-6),
727 BOOST_MATH_BIG_CONSTANT(T, 113, -0.315861074947230418778143153383660035e-8),
728 BOOST_MATH_BIG_CONSTANT(T, 113, -0.500334720512030826996373077844707164e-10),
729 BOOST_MATH_BIG_CONSTANT(T, 113, -0.420204769185233365849253969097184005e-12),
730 };
731 static const T Q[14] = {
732 BOOST_MATH_BIG_CONSTANT(T, 113, 1.0),
733 BOOST_MATH_BIG_CONSTANT(T, 113, 0.97663511666410096104783358493318814),
734 BOOST_MATH_BIG_CONSTANT(T, 113, 0.40878780231201806504987368939673249),
735 BOOST_MATH_BIG_CONSTANT(T, 113, 0.0963890666609396058945084107597727252),
736 BOOST_MATH_BIG_CONSTANT(T, 113, 0.0142207619090854604824116070866614505),
737 BOOST_MATH_BIG_CONSTANT(T, 113, 0.00139010220902667918476773423995750877),
738 BOOST_MATH_BIG_CONSTANT(T, 113, 0.940669540194694997889636696089994734e-4),
739 BOOST_MATH_BIG_CONSTANT(T, 113, 0.458220848507517004399292480807026602e-5),
740 BOOST_MATH_BIG_CONSTANT(T, 113, 0.16345521617741789012782420625435495e-6),
741 BOOST_MATH_BIG_CONSTANT(T, 113, 0.414007452533083304371566316901024114e-8),
742 BOOST_MATH_BIG_CONSTANT(T, 113, 0.68701473543366328016953742622661377e-10),
743 BOOST_MATH_BIG_CONSTANT(T, 113, 0.603461891080716585087883971886075863e-12),
744 BOOST_MATH_BIG_CONSTANT(T, 113, 0.294670713571839023181857795866134957e-16),
745 BOOST_MATH_BIG_CONSTANT(T, 113, -0.147003914536437243143096875069813451e-18),
746 };
747 result = tools::evaluate_polynomial(P, T(s - 6)) / tools::evaluate_polynomial(Q, T(s - 6));
748 result = 1 + exp(result);
749 }
750 else if(s < 17)
751 {
752 // Max error in interpolated form: 1.641e-32
753 // Max error found at long double precision: 1.696121e-32
754 static const T P[13] = {
755 BOOST_MATH_BIG_CONSTANT(T, 113, -6.91319491921722925920883787894829678),
756 BOOST_MATH_BIG_CONSTANT(T, 113, -3.65491257639481960248690596951049048),
757 BOOST_MATH_BIG_CONSTANT(T, 113, -0.813557553449954526442644544105257881),
758 BOOST_MATH_BIG_CONSTANT(T, 113, -0.0994317301685870959473658713841138083),
759 BOOST_MATH_BIG_CONSTANT(T, 113, -0.00726896610245676520248617014211734906),
760 BOOST_MATH_BIG_CONSTANT(T, 113, -0.000317253318715075854811266230916762929),
761 BOOST_MATH_BIG_CONSTANT(T, 113, -0.66851422826636750855184211580127133e-5),
762 BOOST_MATH_BIG_CONSTANT(T, 113, 0.879464154730985406003332577806849971e-7),
763 BOOST_MATH_BIG_CONSTANT(T, 113, 0.113838903158254250631678791998294628e-7),
764 BOOST_MATH_BIG_CONSTANT(T, 113, 0.379184410304927316385211327537817583e-9),
765 BOOST_MATH_BIG_CONSTANT(T, 113, 0.612992858643904887150527613446403867e-11),
766 BOOST_MATH_BIG_CONSTANT(T, 113, 0.347873737198164757035457841688594788e-13),
767 BOOST_MATH_BIG_CONSTANT(T, 113, -0.289187187441625868404494665572279364e-15),
768 };
769 static const T Q[14] = {
770 BOOST_MATH_BIG_CONSTANT(T, 113, 1.0),
771 BOOST_MATH_BIG_CONSTANT(T, 113, 0.427310044448071818775721584949868806),
772 BOOST_MATH_BIG_CONSTANT(T, 113, 0.074602514873055756201435421385243062),
773 BOOST_MATH_BIG_CONSTANT(T, 113, 0.00688651562174480772901425121653945942),
774 BOOST_MATH_BIG_CONSTANT(T, 113, 0.000360174847635115036351323894321880445),
775 BOOST_MATH_BIG_CONSTANT(T, 113, 0.973556847713307543918865405758248777e-5),
776 BOOST_MATH_BIG_CONSTANT(T, 113, -0.853455848314516117964634714780874197e-8),
777 BOOST_MATH_BIG_CONSTANT(T, 113, -0.118203513654855112421673192194622826e-7),
778 BOOST_MATH_BIG_CONSTANT(T, 113, -0.462521662511754117095006543363328159e-9),
779 BOOST_MATH_BIG_CONSTANT(T, 113, -0.834212591919475633107355719369463143e-11),
780 BOOST_MATH_BIG_CONSTANT(T, 113, -0.5354594751002702935740220218582929e-13),
781 BOOST_MATH_BIG_CONSTANT(T, 113, 0.406451690742991192964889603000756203e-15),
782 BOOST_MATH_BIG_CONSTANT(T, 113, 0.887948682401000153828241615760146728e-19),
783 BOOST_MATH_BIG_CONSTANT(T, 113, -0.34980761098820347103967203948619072e-21),
784 };
785 result = tools::evaluate_polynomial(P, T(s - 10)) / tools::evaluate_polynomial(Q, T(s - 10));
786 result = 1 + exp(result);
787 }
788 else if(s < 30)
789 {
790 // Max error in interpolated form: 1.563e-31
791 // Max error found at long double precision: 1.562725e-31
792
793 static const T P[13] = {
794 BOOST_MATH_BIG_CONSTANT(T, 113, -11.7824798233959252791987402769438322),
795 BOOST_MATH_BIG_CONSTANT(T, 113, -4.36131215284987731928174218354118102),
796 BOOST_MATH_BIG_CONSTANT(T, 113, -0.732260980060982349410898496846972204),
797 BOOST_MATH_BIG_CONSTANT(T, 113, -0.0744985185694913074484248803015717388),
798 BOOST_MATH_BIG_CONSTANT(T, 113, -0.00517228281320594683022294996292250527),
799 BOOST_MATH_BIG_CONSTANT(T, 113, -0.000260897206152101522569969046299309939),
800 BOOST_MATH_BIG_CONSTANT(T, 113, -0.989553462123121764865178453128769948e-5),
801 BOOST_MATH_BIG_CONSTANT(T, 113, -0.286916799741891410827712096608826167e-6),
802 BOOST_MATH_BIG_CONSTANT(T, 113, -0.637262477796046963617949532211619729e-8),
803 BOOST_MATH_BIG_CONSTANT(T, 113, -0.106796831465628373325491288787760494e-9),
804 BOOST_MATH_BIG_CONSTANT(T, 113, -0.129343095511091870860498356205376823e-11),
805 BOOST_MATH_BIG_CONSTANT(T, 113, -0.102397936697965977221267881716672084e-13),
806 BOOST_MATH_BIG_CONSTANT(T, 113, -0.402663128248642002351627980255756363e-16),
807 };
808 static const T Q[14] = {
809 BOOST_MATH_BIG_CONSTANT(T, 113, 1.0),
810 BOOST_MATH_BIG_CONSTANT(T, 113, 0.311288325355705609096155335186466508),
811 BOOST_MATH_BIG_CONSTANT(T, 113, 0.0438318468940415543546769437752132748),
812 BOOST_MATH_BIG_CONSTANT(T, 113, 0.00374396349183199548610264222242269536),
813 BOOST_MATH_BIG_CONSTANT(T, 113, 0.000218707451200585197339671707189281302),
814 BOOST_MATH_BIG_CONSTANT(T, 113, 0.927578767487930747532953583797351219e-5),
815 BOOST_MATH_BIG_CONSTANT(T, 113, 0.294145760625753561951137473484889639e-6),
816 BOOST_MATH_BIG_CONSTANT(T, 113, 0.704618586690874460082739479535985395e-8),
817 BOOST_MATH_BIG_CONSTANT(T, 113, 0.126333332872897336219649130062221257e-9),
818 BOOST_MATH_BIG_CONSTANT(T, 113, 0.16317315713773503718315435769352765e-11),
819 BOOST_MATH_BIG_CONSTANT(T, 113, 0.137846712823719515148344938160275695e-13),
820 BOOST_MATH_BIG_CONSTANT(T, 113, 0.580975420554224366450994232723910583e-16),
821 BOOST_MATH_BIG_CONSTANT(T, 113, -0.291354445847552426900293580511392459e-22),
822 BOOST_MATH_BIG_CONSTANT(T, 113, 0.73614324724785855925025452085443636e-25),
823 };
824 result = tools::evaluate_polynomial(P, T(s - 17)) / tools::evaluate_polynomial(Q, T(s - 17));
825 result = 1 + exp(result);
826 }
827 else if(s < 74)
828 {
829 // Max error in interpolated form: 2.311e-27
830 // Max error found at long double precision: 2.297544e-27
831 static const T P[14] = {
832 BOOST_MATH_BIG_CONSTANT(T, 113, -20.7944102007844314586649688802236072),
833 BOOST_MATH_BIG_CONSTANT(T, 113, -4.95759941987499442499908748130192187),
834 BOOST_MATH_BIG_CONSTANT(T, 113, -0.563290752832461751889194629200298688),
835 BOOST_MATH_BIG_CONSTANT(T, 113, -0.0406197001137935911912457120706122877),
836 BOOST_MATH_BIG_CONSTANT(T, 113, -0.0020846534789473022216888863613422293),
837 BOOST_MATH_BIG_CONSTANT(T, 113, -0.808095978462109173749395599401375667e-4),
838 BOOST_MATH_BIG_CONSTANT(T, 113, -0.244706022206249301640890603610060959e-5),
839 BOOST_MATH_BIG_CONSTANT(T, 113, -0.589477682919645930544382616501666572e-7),
840 BOOST_MATH_BIG_CONSTANT(T, 113, -0.113699573675553496343617442433027672e-8),
841 BOOST_MATH_BIG_CONSTANT(T, 113, -0.174767860183598149649901223128011828e-10),
842 BOOST_MATH_BIG_CONSTANT(T, 113, -0.210051620306761367764549971980026474e-12),
843 BOOST_MATH_BIG_CONSTANT(T, 113, -0.189187969537370950337212675466400599e-14),
844 BOOST_MATH_BIG_CONSTANT(T, 113, -0.116313253429564048145641663778121898e-16),
845 BOOST_MATH_BIG_CONSTANT(T, 113, -0.376708747782400769427057630528578187e-19),
846 };
847 static const T Q[16] = {
848 BOOST_MATH_BIG_CONSTANT(T, 113, 1.0),
849 BOOST_MATH_BIG_CONSTANT(T, 113, 0.205076752981410805177554569784219717),
850 BOOST_MATH_BIG_CONSTANT(T, 113, 0.0202526722696670378999575738524540269),
851 BOOST_MATH_BIG_CONSTANT(T, 113, 0.001278305290005994980069466658219057),
852 BOOST_MATH_BIG_CONSTANT(T, 113, 0.576404779858501791742255670403304787e-4),
853 BOOST_MATH_BIG_CONSTANT(T, 113, 0.196477049872253010859712483984252067e-5),
854 BOOST_MATH_BIG_CONSTANT(T, 113, 0.521863830500876189501054079974475762e-7),
855 BOOST_MATH_BIG_CONSTANT(T, 113, 0.109524209196868135198775445228552059e-8),
856 BOOST_MATH_BIG_CONSTANT(T, 113, 0.181698713448644481083966260949267825e-10),
857 BOOST_MATH_BIG_CONSTANT(T, 113, 0.234793316975091282090312036524695562e-12),
858 BOOST_MATH_BIG_CONSTANT(T, 113, 0.227490441461460571047545264251399048e-14),
859 BOOST_MATH_BIG_CONSTANT(T, 113, 0.151500292036937400913870642638520668e-16),
860 BOOST_MATH_BIG_CONSTANT(T, 113, 0.543475775154780935815530649335936121e-19),
861 BOOST_MATH_BIG_CONSTANT(T, 113, 0.241647013434111434636554455083309352e-28),
862 BOOST_MATH_BIG_CONSTANT(T, 113, -0.557103423021951053707162364713587374e-31),
863 BOOST_MATH_BIG_CONSTANT(T, 113, 0.618708773442584843384712258199645166e-34),
864 };
865 result = tools::evaluate_polynomial(P, T(s - 30)) / tools::evaluate_polynomial(Q, T(s - 30));
866 result = 1 + exp(result);
867 }
868 else if(s < 117)
869 {
870 result = 1 + pow(T(2), -s);
871 }
872 else
873 {
874 result = 1;
875 }
876 return result;
877}
878
879template <class T, class Policy>
880T zeta_imp_odd_integer(int s, const T&, const Policy&, const true_type&)
881{
882 static const T results[] = {
883 BOOST_MATH_BIG_CONSTANT(T, 113, 1.2020569031595942853997381615114500), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0369277551433699263313654864570342), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0083492773819228268397975498497968), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0020083928260822144178527692324121), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0004941886041194645587022825264699), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0001227133475784891467518365263574), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000305882363070204935517285106451), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000076371976378997622736002935630), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000019082127165539389256569577951), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000004769329867878064631167196044), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000001192199259653110730677887189), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000298035035146522801860637051), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000074507117898354294919810042), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000018626597235130490064039099), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000004656629065033784072989233), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000001164155017270051977592974), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000000291038504449709968692943), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000000072759598350574810145209), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000000018189896503070659475848), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000000004547473783042154026799), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000000001136868407680227849349), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000000000284217097688930185546), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000000000071054273952108527129), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000000000017763568435791203275), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000000000004440892103143813364), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000000000001110223025141066134), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000000000000277555756213612417), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000000000000069388939045441537), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000000000000017347234760475766), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000000000000004336808690020650), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000000000000001084202172494241), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000000000000000271050543122347), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000000000000000067762635780452), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000000000000000016940658945098), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000000000000000004235164736273), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000000000000000001058791184068), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000000000000000000264697796017), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000000000000000000066174449004), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000000000000000000016543612251), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000000000000000000004135903063), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000000000000000000001033975766), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000000000000000000000258493941), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000000000000000000000064623485), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000000000000000000000016155871), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000000000000000000000004038968), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000000000000000000000001009742), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000000000000000000000000252435), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000000000000000000000000063109), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000000000000000000000000015777), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000000000000000000000000003944), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000000000000000000000000000986), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000000000000000000000000000247), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000000000000000000000000000062), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000000000000000000000000000015), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000000000000000000000000000004), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000000000000000000000000000001),
884 };
885 return s > 113 ? 1 : results[(s - 3) / 2];
886}
887
888template <class T, class Policy>
889T zeta_imp_odd_integer(int s, const T& sc, const Policy& pol, const false_type&)
890{
891 static BOOST_MATH_THREAD_LOCAL bool is_init = false;
892 static BOOST_MATH_THREAD_LOCAL T results[50] = {};
893 static BOOST_MATH_THREAD_LOCAL int digits = tools::digits<T>();
894 int current_digits = tools::digits<T>();
895 if(digits != current_digits)
896 {
897 // Oh my precision has changed...
898 is_init = false;
899 }
900 if(!is_init)
901 {
902 is_init = true;
903 digits = current_digits;
904 for(unsigned k = 0; k < sizeof(results) / sizeof(results[0]); ++k)
905 {
906 T arg = k * 2 + 3;
907 T c_arg = 1 - arg;
908 results[k] = zeta_polynomial_series(arg, c_arg, pol);
909 }
910 }
911 unsigned index = (s - 3) / 2;
912 return index >= sizeof(results) / sizeof(results[0]) ? zeta_polynomial_series(T(s), sc, pol): results[index];
913}
914
915template <class T, class Policy, class Tag>
916T zeta_imp(T s, T sc, const Policy& pol, const Tag& tag)
917{
918 BOOST_MATH_STD_USING
919 static const char* function = "boost::math::zeta<%1%>";
920 if(sc == 0)
921 return policies::raise_pole_error<T>(
922 function,
923 "Evaluation of zeta function at pole %1%",
924 s, pol);
925 T result;
926 //
927 // Trivial case:
928 //
929 if(s > policies::digits<T, Policy>())
930 return 1;
931 //
932 // Start by seeing if we have a simple closed form:
933 //
934 if(floor(s) == s)
935 {
936#ifndef BOOST_NO_EXCEPTIONS
937 // Without exceptions we expect itrunc to return INT_MAX on overflow
938 // and we fall through anyway.
939 try
940 {
941#endif
942 int v = itrunc(s);
943 if(v == s)
944 {
945 if(v < 0)
946 {
947 if(((-v) & 1) == 0)
948 return 0;
949 int n = (-v + 1) / 2;
950 if(n <= (int)boost::math::max_bernoulli_b2n<T>::value)
951 return T((-v & 1) ? -1 : 1) * boost::math::unchecked_bernoulli_b2n<T>(n) / (1 - v);
952 }
953 else if((v & 1) == 0)
954 {
955 if(((v / 2) <= (int)boost::math::max_bernoulli_b2n<T>::value) && (v <= (int)boost::math::max_factorial<T>::value))
956 return T(((v / 2 - 1) & 1) ? -1 : 1) * ldexp(T(1), v - 1) * pow(constants::pi<T, Policy>(), v) *
957 boost::math::unchecked_bernoulli_b2n<T>(v / 2) / boost::math::unchecked_factorial<T>(v);
958 return T(((v / 2 - 1) & 1) ? -1 : 1) * ldexp(T(1), v - 1) * pow(constants::pi<T, Policy>(), v) *
959 boost::math::bernoulli_b2n<T>(v / 2) / boost::math::factorial<T>(v);
960 }
961 else
962 return zeta_imp_odd_integer(v, sc, pol, boost::integral_constant<bool, (Tag::value <= 113) && Tag::value>());
963 }
964#ifndef BOOST_NO_EXCEPTIONS
965 }
966 catch(const boost::math::rounding_error&){} // Just fall through, s is too large to round
967 catch(const std::overflow_error&){}
968#endif
969 }
970
971 if(fabs(s) < tools::root_epsilon<T>())
972 {
973 result = -0.5f - constants::log_root_two_pi<T, Policy>() * s;
974 }
975 else if(s < 0)
976 {
977 std::swap(s, sc);
978 if(floor(sc/2) == sc/2)
979 result = 0;
980 else
981 {
982 if(s > max_factorial<T>::value)
983 {
984 T mult = boost::math::sin_pi(0.5f * sc, pol) * 2 * zeta_imp(s, sc, pol, tag);
985 result = boost::math::lgamma(s, pol);
986 result -= s * log(2 * constants::pi<T>());
987 if(result > tools::log_max_value<T>())
988 return sign(mult) * policies::raise_overflow_error<T>(function, 0, pol);
989 result = exp(result);
990 if(tools::max_value<T>() / fabs(mult) < result)
991 return boost::math::sign(mult) * policies::raise_overflow_error<T>(function, 0, pol);
992 result *= mult;
993 }
994 else
995 {
996 result = boost::math::sin_pi(0.5f * sc, pol)
997 * 2 * pow(2 * constants::pi<T>(), -s)
998 * boost::math::tgamma(s, pol)
999 * zeta_imp(s, sc, pol, tag);
1000 }
1001 }
1002 }
1003 else
1004 {
1005 result = zeta_imp_prec(s, sc, pol, tag);
1006 }
1007 return result;
1008}
1009
1010template <class T, class Policy, class tag>
1011struct zeta_initializer
1012{
1013 struct init
1014 {
1015 init()
1016 {
1017 do_init(tag());
1018 }
1019 static void do_init(const boost::integral_constant<int, 0>&){ boost::math::zeta(static_cast<T>(5), Policy()); }
1020 static void do_init(const boost::integral_constant<int, 53>&){ boost::math::zeta(static_cast<T>(5), Policy()); }
1021 static void do_init(const boost::integral_constant<int, 64>&)
1022 {
1023 boost::math::zeta(static_cast<T>(0.5), Policy());
1024 boost::math::zeta(static_cast<T>(1.5), Policy());
1025 boost::math::zeta(static_cast<T>(3.5), Policy());
1026 boost::math::zeta(static_cast<T>(6.5), Policy());
1027 boost::math::zeta(static_cast<T>(14.5), Policy());
1028 boost::math::zeta(static_cast<T>(40.5), Policy());
1029
1030 boost::math::zeta(static_cast<T>(5), Policy());
1031 }
1032 static void do_init(const boost::integral_constant<int, 113>&)
1033 {
1034 boost::math::zeta(static_cast<T>(0.5), Policy());
1035 boost::math::zeta(static_cast<T>(1.5), Policy());
1036 boost::math::zeta(static_cast<T>(3.5), Policy());
1037 boost::math::zeta(static_cast<T>(5.5), Policy());
1038 boost::math::zeta(static_cast<T>(9.5), Policy());
1039 boost::math::zeta(static_cast<T>(16.5), Policy());
1040 boost::math::zeta(static_cast<T>(25.5), Policy());
1041 boost::math::zeta(static_cast<T>(70.5), Policy());
1042
1043 boost::math::zeta(static_cast<T>(5), Policy());
1044 }
1045 void force_instantiate()const{}
1046 };
1047 static const init initializer;
1048 static void force_instantiate()
1049 {
1050 initializer.force_instantiate();
1051 }
1052};
1053
1054template <class T, class Policy, class tag>
1055const typename zeta_initializer<T, Policy, tag>::init zeta_initializer<T, Policy, tag>::initializer;
1056
1057} // detail
1058
1059template <class T, class Policy>
1060inline typename tools::promote_args<T>::type zeta(T s, const Policy&)
1061{
1062 typedef typename tools::promote_args<T>::type result_type;
1063 typedef typename policies::evaluation<result_type, Policy>::type value_type;
1064 typedef typename policies::precision<result_type, Policy>::type precision_type;
1065 typedef typename policies::normalise<
1066 Policy,
1067 policies::promote_float<false>,
1068 policies::promote_double<false>,
1069 policies::discrete_quantile<>,
1070 policies::assert_undefined<> >::type forwarding_policy;
1071 typedef boost::integral_constant<int,
1072 precision_type::value <= 0 ? 0 :
1073 precision_type::value <= 53 ? 53 :
1074 precision_type::value <= 64 ? 64 :
1075 precision_type::value <= 113 ? 113 : 0
1076 > tag_type;
1077
1078 detail::zeta_initializer<value_type, forwarding_policy, tag_type>::force_instantiate();
1079
1080 return policies::checked_narrowing_cast<result_type, forwarding_policy>(detail::zeta_imp(
1081 static_cast<value_type>(s),
1082 static_cast<value_type>(1 - static_cast<value_type>(s)),
1083 forwarding_policy(),
1084 tag_type()), "boost::math::zeta<%1%>(%1%)");
1085}
1086
1087template <class T>
1088inline typename tools::promote_args<T>::type zeta(T s)
1089{
1090 return zeta(s, policies::policy<>());
1091}
1092
1093}} // namespaces
1094
1095#endif // BOOST_MATH_ZETA_HPP
1096
1097
1098
1099

source code of include/boost/math/special_functions/zeta.hpp