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 namespace boost{ namespace math{ namespace detail{
23 
24 #if 0
25 //
26 // This code is commented out because we have a better more rapidly converging series
27 // now.  Retained for future reference and in case the new code causes any issues down the line....
28 //
29 
30 template <class T, class Policy>
31 struct zeta_series_cache_size
32 {
33    //
34    // Work how large to make our cache size when evaluating the series
35    // evaluation:  normally this is just large enough for the series
36    // to have converged, but for arbitrary precision types we need a
37    // really large cache to achieve reasonable precision in a reasonable
38    // time.  This is important when constructing rational approximations
39    // to zeta for example.
40    //
41    typedef typename boost::math::policies::precision<T,Policy>::type precision_type;
42    typedef typename mpl::if_<
43       mpl::less_equal<precision_type, mpl::int_<0> >,
44       mpl::int_<5000>,
45       typename mpl::if_<
46          mpl::less_equal<precision_type, mpl::int_<64> >,
47          mpl::int_<70>,
48          typename mpl::if_<
49             mpl::less_equal<precision_type, mpl::int_<113> >,
50             mpl::int_<100>,
51             mpl::int_<5000>
52          >::type
53       >::type
54    >::type type;
55 };
56 
57 template <class T, class Policy>
58 T zeta_series_imp(T s, T sc, const Policy&)
59 {
60    //
61    // Series evaluation from:
62    // Havil, J. Gamma: Exploring Euler's Constant.
63    // Princeton, NJ: Princeton University Press, 2003.
64    //
65    // See also http://mathworld.wolfram.com/RiemannZetaFunction.html
66    //
67    BOOST_MATH_STD_USING
68    T sum = 0;
69    T mult = 0.5;
70    T change;
71    typedef typename zeta_series_cache_size<T,Policy>::type cache_size;
72    T powers[cache_size::value] = { 0, };
73    unsigned n = 0;
74    do{
75       T binom = -static_cast<T>(n);
76       T nested_sum = 1;
77       if(n < sizeof(powers) / sizeof(powers[0]))
78          powers[n] = pow(static_cast<T>(n + 1), -s);
79       for(unsigned k = 1; k <= n; ++k)
80       {
81          T p;
82          if(k < sizeof(powers) / sizeof(powers[0]))
83          {
84             p = powers[k];
85             //p = pow(k + 1, -s);
86          }
87          else
88             p = pow(static_cast<T>(k + 1), -s);
89          nested_sum += binom * p;
90         binom *= (k - static_cast<T>(n)) / (k + 1);
91       }
92       change = mult * nested_sum;
93       sum += change;
94       mult /= 2;
95       ++n;
96    }while(fabs(change / sum) > tools::epsilon<T>());
97 
98    return sum * 1 / -boost::math::powm1(T(2), sc);
99 }
100 
101 //
102 // Classical p-series:
103 //
104 template <class T>
105 struct zeta_series2
106 {
107    typedef T result_type;
108    zeta_series2(T _s) : s(-_s), k(1){}
109    T operator()()
110    {
111       BOOST_MATH_STD_USING
112       return pow(static_cast<T>(k++), s);
113    }
114 private:
115    T s;
116    unsigned k;
117 };
118 
119 template <class T, class Policy>
120 inline T zeta_series2_imp(T s, const Policy& pol)
121 {
122    boost::uintmax_t max_iter = policies::get_max_series_iterations<Policy>();;
123    zeta_series2<T> f(s);
124    T result = tools::sum_series(
125       f,
126       policies::get_epsilon<T, Policy>(),
127       max_iter);
128    policies::check_series_iterations<T>("boost::math::zeta_series2<%1%>(%1%)", max_iter, pol);
129    return result;
130 }
131 #endif
132 
133 template <class T, class Policy>
134 T zeta_polynomial_series(T s, T sc, Policy const &)
135 {
136    //
137    // This is algorithm 3 from:
138    //
139    // "An Efficient Algorithm for the Riemann Zeta Function", P. Borwein,
140    // Canadian Mathematical Society, Conference Proceedings.
141    // See: http://www.cecm.sfu.ca/personal/pborwein/PAPERS/P155.pdf
142    //
143    BOOST_MATH_STD_USING
144    int n = itrunc(T(log(boost::math::tools::epsilon<T>()) / -2));
145    T sum = 0;
146    T two_n = ldexp(T(1), n);
147    int ej_sign = 1;
148    for(int j = 0; j < n; ++j)
149    {
150       sum += ej_sign * -two_n / pow(T(j + 1), s);
151       ej_sign = -ej_sign;
152    }
153    T ej_sum = 1;
154    T ej_term = 1;
155    for(int j = n; j <= 2 * n - 1; ++j)
156    {
157       sum += ej_sign * (ej_sum - two_n) / pow(T(j + 1), s);
158       ej_sign = -ej_sign;
159       ej_term *= 2 * n - j;
160       ej_term /= j - n + 1;
161       ej_sum += ej_term;
162    }
163    return -sum / (two_n * (-powm1(T(2), sc)));
164 }
165 
166 template <class T, class Policy>
167 T zeta_imp_prec(T s, T sc, const Policy& pol, const mpl::int_<0>&)
168 {
169    BOOST_MATH_STD_USING
170    T result;
171    if(s >= policies::digits<T, Policy>())
172       return 1;
173    result = zeta_polynomial_series(s, sc, pol);
174 #if 0
175    // Old code archived for future reference:
176 
177    //
178    // Only use power series if it will converge in 100
179    // iterations or less: the more iterations it consumes
180    // the slower convergence becomes so we have to be very
181    // careful in it's usage.
182    //
183    if (s > -log(tools::epsilon<T>()) / 4.5)
184       result = detail::zeta_series2_imp(s, pol);
185    else
186       result = detail::zeta_series_imp(s, sc, pol);
187 #endif
188    return result;
189 }
190 
191 template <class T, class Policy>
zeta_imp_prec(T s,T sc,const Policy &,const mpl::int_<53> &)192 inline T zeta_imp_prec(T s, T sc, const Policy&, const mpl::int_<53>&)
193 {
194    BOOST_MATH_STD_USING
195    T result;
196    if(s < 1)
197    {
198       // Rational Approximation
199       // Maximum Deviation Found:                     2.020e-18
200       // Expected Error Term:                         -2.020e-18
201       // Max error found at double precision:         3.994987e-17
202       static const T P[6] = {
203          static_cast<T>(0.24339294433593750202L),
204          static_cast<T>(-0.49092470516353571651L),
205          static_cast<T>(0.0557616214776046784287L),
206          static_cast<T>(-0.00320912498879085894856L),
207          static_cast<T>(0.000451534528645796438704L),
208          static_cast<T>(-0.933241270357061460782e-5L),
209         };
210       static const T Q[6] = {
211          static_cast<T>(1L),
212          static_cast<T>(-0.279960334310344432495L),
213          static_cast<T>(0.0419676223309986037706L),
214          static_cast<T>(-0.00413421406552171059003L),
215          static_cast<T>(0.00024978985622317935355L),
216          static_cast<T>(-0.101855788418564031874e-4L),
217       };
218       result = tools::evaluate_polynomial(P, sc) / tools::evaluate_polynomial(Q, sc);
219       result -= 1.2433929443359375F;
220       result += (sc);
221       result /= (sc);
222    }
223    else if(s <= 2)
224    {
225       // Maximum Deviation Found:        9.007e-20
226       // Expected Error Term:            9.007e-20
227       static const T P[6] = {
228          static_cast<T>(0.577215664901532860516L),
229          static_cast<T>(0.243210646940107164097L),
230          static_cast<T>(0.0417364673988216497593L),
231          static_cast<T>(0.00390252087072843288378L),
232          static_cast<T>(0.000249606367151877175456L),
233          static_cast<T>(0.110108440976732897969e-4L),
234       };
235       static const T Q[6] = {
236          static_cast<T>(1.0),
237          static_cast<T>(0.295201277126631761737L),
238          static_cast<T>(0.043460910607305495864L),
239          static_cast<T>(0.00434930582085826330659L),
240          static_cast<T>(0.000255784226140488490982L),
241          static_cast<T>(0.10991819782396112081e-4L),
242       };
243       result = tools::evaluate_polynomial(P, T(-sc)) / tools::evaluate_polynomial(Q, T(-sc));
244       result += 1 / (-sc);
245    }
246    else if(s <= 4)
247    {
248       // Maximum Deviation Found:          5.946e-22
249       // Expected Error Term:              -5.946e-22
250       static const float Y = 0.6986598968505859375;
251       static const T P[6] = {
252          static_cast<T>(-0.0537258300023595030676L),
253          static_cast<T>(0.0445163473292365591906L),
254          static_cast<T>(0.0128677673534519952905L),
255          static_cast<T>(0.00097541770457391752726L),
256          static_cast<T>(0.769875101573654070925e-4L),
257          static_cast<T>(0.328032510000383084155e-5L),
258       };
259       static const T Q[7] = {
260          1.0f,
261          static_cast<T>(0.33383194553034051422L),
262          static_cast<T>(0.0487798431291407621462L),
263          static_cast<T>(0.00479039708573558490716L),
264          static_cast<T>(0.000270776703956336357707L),
265          static_cast<T>(0.106951867532057341359e-4L),
266          static_cast<T>(0.236276623974978646399e-7L),
267       };
268       result = tools::evaluate_polynomial(P, T(s - 2)) / tools::evaluate_polynomial(Q, T(s - 2));
269       result += Y + 1 / (-sc);
270    }
271    else if(s <= 7)
272    {
273       // Maximum Deviation Found:                     2.955e-17
274       // Expected Error Term:                         2.955e-17
275       // Max error found at double precision:         2.009135e-16
276 
277       static const T P[6] = {
278          static_cast<T>(-2.49710190602259410021L),
279          static_cast<T>(-2.60013301809475665334L),
280          static_cast<T>(-0.939260435377109939261L),
281          static_cast<T>(-0.138448617995741530935L),
282          static_cast<T>(-0.00701721240549802377623L),
283          static_cast<T>(-0.229257310594893932383e-4L),
284       };
285       static const T Q[9] = {
286          1.0f,
287          static_cast<T>(0.706039025937745133628L),
288          static_cast<T>(0.15739599649558626358L),
289          static_cast<T>(0.0106117950976845084417L),
290          static_cast<T>(-0.36910273311764618902e-4L),
291          static_cast<T>(0.493409563927590008943e-5L),
292          static_cast<T>(-0.234055487025287216506e-6L),
293          static_cast<T>(0.718833729365459760664e-8L),
294          static_cast<T>(-0.1129200113474947419e-9L),
295       };
296       result = tools::evaluate_polynomial(P, T(s - 4)) / tools::evaluate_polynomial(Q, T(s - 4));
297       result = 1 + exp(result);
298    }
299    else if(s < 15)
300    {
301       // Maximum Deviation Found:                     7.117e-16
302       // Expected Error Term:                         7.117e-16
303       // Max error found at double precision:         9.387771e-16
304       static const T P[7] = {
305          static_cast<T>(-4.78558028495135619286L),
306          static_cast<T>(-1.89197364881972536382L),
307          static_cast<T>(-0.211407134874412820099L),
308          static_cast<T>(-0.000189204758260076688518L),
309          static_cast<T>(0.00115140923889178742086L),
310          static_cast<T>(0.639949204213164496988e-4L),
311          static_cast<T>(0.139348932445324888343e-5L),
312         };
313       static const T Q[9] = {
314          1.0f,
315          static_cast<T>(0.244345337378188557777L),
316          static_cast<T>(0.00873370754492288653669L),
317          static_cast<T>(-0.00117592765334434471562L),
318          static_cast<T>(-0.743743682899933180415e-4L),
319          static_cast<T>(-0.21750464515767984778e-5L),
320          static_cast<T>(0.471001264003076486547e-8L),
321          static_cast<T>(-0.833378440625385520576e-10L),
322          static_cast<T>(0.699841545204845636531e-12L),
323         };
324       result = tools::evaluate_polynomial(P, T(s - 7)) / tools::evaluate_polynomial(Q, T(s - 7));
325       result = 1 + exp(result);
326    }
327    else if(s < 36)
328    {
329       // Max error in interpolated form:             1.668e-17
330       // Max error found at long double precision:   1.669714e-17
331       static const T P[8] = {
332          static_cast<T>(-10.3948950573308896825L),
333          static_cast<T>(-2.85827219671106697179L),
334          static_cast<T>(-0.347728266539245787271L),
335          static_cast<T>(-0.0251156064655346341766L),
336          static_cast<T>(-0.00119459173416968685689L),
337          static_cast<T>(-0.382529323507967522614e-4L),
338          static_cast<T>(-0.785523633796723466968e-6L),
339          static_cast<T>(-0.821465709095465524192e-8L),
340       };
341       static const T Q[10] = {
342          1.0f,
343          static_cast<T>(0.208196333572671890965L),
344          static_cast<T>(0.0195687657317205033485L),
345          static_cast<T>(0.00111079638102485921877L),
346          static_cast<T>(0.408507746266039256231e-4L),
347          static_cast<T>(0.955561123065693483991e-6L),
348          static_cast<T>(0.118507153474022900583e-7L),
349          static_cast<T>(0.222609483627352615142e-14L),
350       };
351       result = tools::evaluate_polynomial(P, T(s - 15)) / tools::evaluate_polynomial(Q, T(s - 15));
352       result = 1 + exp(result);
353    }
354    else if(s < 56)
355    {
356       result = 1 + pow(T(2), -s);
357    }
358    else
359    {
360       result = 1;
361    }
362    return result;
363 }
364 
365 template <class T, class Policy>
366 T zeta_imp_prec(T s, T sc, const Policy&, const mpl::int_<64>&)
367 {
368    BOOST_MATH_STD_USING
369    T result;
370    if(s < 1)
371    {
372       // Rational Approximation
373       // Maximum Deviation Found:                     3.099e-20
374       // Expected Error Term:                         3.099e-20
375       // Max error found at long double precision:    5.890498e-20
376       static const T P[6] = {
377          BOOST_MATH_BIG_CONSTANT(T, 64, 0.243392944335937499969),
378          BOOST_MATH_BIG_CONSTANT(T, 64, -0.496837806864865688082),
379          BOOST_MATH_BIG_CONSTANT(T, 64, 0.0680008039723709987107),
380          BOOST_MATH_BIG_CONSTANT(T, 64, -0.00511620413006619942112),
381          BOOST_MATH_BIG_CONSTANT(T, 64, 0.000455369899250053003335),
382          BOOST_MATH_BIG_CONSTANT(T, 64, -0.279496685273033761927e-4),
383         };
384       static const T Q[7] = {
385          BOOST_MATH_BIG_CONSTANT(T, 64, 1.0),
386          BOOST_MATH_BIG_CONSTANT(T, 64, -0.30425480068225790522),
387          BOOST_MATH_BIG_CONSTANT(T, 64, 0.050052748580371598736),
388          BOOST_MATH_BIG_CONSTANT(T, 64, -0.00519355671064700627862),
389          BOOST_MATH_BIG_CONSTANT(T, 64, 0.000360623385771198350257),
390          BOOST_MATH_BIG_CONSTANT(T, 64, -0.159600883054550987633e-4),
391          BOOST_MATH_BIG_CONSTANT(T, 64, 0.339770279812410586032e-6),
392       };
393       result = tools::evaluate_polynomial(P, sc) / tools::evaluate_polynomial(Q, sc);
394       result -= 1.2433929443359375F;
395       result += (sc);
396       result /= (sc);
397    }
398    else if(s <= 2)
399    {
400       // Maximum Deviation Found:                     1.059e-21
401       // Expected Error Term:                         1.059e-21
402       // Max error found at long double precision:    1.626303e-19
403 
404       static const T P[6] = {
405          BOOST_MATH_BIG_CONSTANT(T, 64, 0.577215664901532860605),
406          BOOST_MATH_BIG_CONSTANT(T, 64, 0.222537368917162139445),
407          BOOST_MATH_BIG_CONSTANT(T, 64, 0.0356286324033215682729),
408          BOOST_MATH_BIG_CONSTANT(T, 64, 0.00304465292366350081446),
409          BOOST_MATH_BIG_CONSTANT(T, 64, 0.000178102511649069421904),
410          BOOST_MATH_BIG_CONSTANT(T, 64, 0.700867470265983665042e-5),
411       };
412       static const T Q[7] = {
413          BOOST_MATH_BIG_CONSTANT(T, 64, 1.0),
414          BOOST_MATH_BIG_CONSTANT(T, 64, 0.259385759149531030085),
415          BOOST_MATH_BIG_CONSTANT(T, 64, 0.0373974962106091316854),
416          BOOST_MATH_BIG_CONSTANT(T, 64, 0.00332735159183332820617),
417          BOOST_MATH_BIG_CONSTANT(T, 64, 0.000188690420706998606469),
418          BOOST_MATH_BIG_CONSTANT(T, 64, 0.635994377921861930071e-5),
419          BOOST_MATH_BIG_CONSTANT(T, 64, 0.226583954978371199405e-7),
420       };
421       result = tools::evaluate_polynomial(P, T(-sc)) / tools::evaluate_polynomial(Q, T(-sc));
422       result += 1 / (-sc);
423    }
424    else if(s <= 4)
425    {
426       // Maximum Deviation Found:          5.946e-22
427       // Expected Error Term:              -5.946e-22
428       static const float Y = 0.6986598968505859375;
429       static const T P[7] = {
430          BOOST_MATH_BIG_CONSTANT(T, 64, -0.053725830002359501027),
431          BOOST_MATH_BIG_CONSTANT(T, 64, 0.0470551187571475844778),
432          BOOST_MATH_BIG_CONSTANT(T, 64, 0.0101339410415759517471),
433          BOOST_MATH_BIG_CONSTANT(T, 64, 0.00100240326666092854528),
434          BOOST_MATH_BIG_CONSTANT(T, 64, 0.685027119098122814867e-4),
435          BOOST_MATH_BIG_CONSTANT(T, 64, 0.390972820219765942117e-5),
436          BOOST_MATH_BIG_CONSTANT(T, 64, 0.540319769113543934483e-7),
437       };
438       static const T Q[8] = {
439          BOOST_MATH_BIG_CONSTANT(T, 64, 1.0),
440          BOOST_MATH_BIG_CONSTANT(T, 64, 0.286577739726542730421),
441          BOOST_MATH_BIG_CONSTANT(T, 64, 0.0447355811517733225843),
442          BOOST_MATH_BIG_CONSTANT(T, 64, 0.00430125107610252363302),
443          BOOST_MATH_BIG_CONSTANT(T, 64, 0.000284956969089786662045),
444          BOOST_MATH_BIG_CONSTANT(T, 64, 0.116188101609848411329e-4),
445          BOOST_MATH_BIG_CONSTANT(T, 64, 0.278090318191657278204e-6),
446          BOOST_MATH_BIG_CONSTANT(T, 64, -0.19683620233222028478e-8),
447       };
448       result = tools::evaluate_polynomial(P, T(s - 2)) / tools::evaluate_polynomial(Q, T(s - 2));
449       result += Y + 1 / (-sc);
450    }
451    else if(s <= 7)
452    {
453       // Max error found at long double precision: 8.132216e-19
454       static const T P[8] = {
455          BOOST_MATH_BIG_CONSTANT(T, 64, -2.49710190602259407065),
456          BOOST_MATH_BIG_CONSTANT(T, 64, -3.36664913245960625334),
457          BOOST_MATH_BIG_CONSTANT(T, 64, -1.77180020623777595452),
458          BOOST_MATH_BIG_CONSTANT(T, 64, -0.464717885249654313933),
459          BOOST_MATH_BIG_CONSTANT(T, 64, -0.0643694921293579472583),
460          BOOST_MATH_BIG_CONSTANT(T, 64, -0.00464265386202805715487),
461          BOOST_MATH_BIG_CONSTANT(T, 64, -0.000165556579779704340166),
462          BOOST_MATH_BIG_CONSTANT(T, 64, -0.252884970740994069582e-5),
463       };
464       static const T Q[9] = {
465          BOOST_MATH_BIG_CONSTANT(T, 64, 1.0),
466          BOOST_MATH_BIG_CONSTANT(T, 64, 1.01300131390690459085),
467          BOOST_MATH_BIG_CONSTANT(T, 64, 0.387898115758643503827),
468          BOOST_MATH_BIG_CONSTANT(T, 64, 0.0695071490045701135188),
469          BOOST_MATH_BIG_CONSTANT(T, 64, 0.00586908595251442839291),
470          BOOST_MATH_BIG_CONSTANT(T, 64, 0.000217752974064612188616),
471          BOOST_MATH_BIG_CONSTANT(T, 64, 0.397626583349419011731e-5),
472          BOOST_MATH_BIG_CONSTANT(T, 64, -0.927884739284359700764e-8),
473          BOOST_MATH_BIG_CONSTANT(T, 64, 0.119810501805618894381e-9),
474       };
475       result = tools::evaluate_polynomial(P, T(s - 4)) / tools::evaluate_polynomial(Q, T(s - 4));
476       result = 1 + exp(result);
477    }
478    else if(s < 15)
479    {
480       // Max error in interpolated form:              1.133e-18
481       // Max error found at long double precision:    2.183198e-18
482       static const T P[9] = {
483          BOOST_MATH_BIG_CONSTANT(T, 64, -4.78558028495135548083),
484          BOOST_MATH_BIG_CONSTANT(T, 64, -3.23873322238609358947),
485          BOOST_MATH_BIG_CONSTANT(T, 64, -0.892338582881021799922),
486          BOOST_MATH_BIG_CONSTANT(T, 64, -0.131326296217965913809),
487          BOOST_MATH_BIG_CONSTANT(T, 64, -0.0115651591773783712996),
488          BOOST_MATH_BIG_CONSTANT(T, 64, -0.000657728968362695775205),
489          BOOST_MATH_BIG_CONSTANT(T, 64, -0.252051328129449973047e-4),
490          BOOST_MATH_BIG_CONSTANT(T, 64, -0.626503445372641798925e-6),
491          BOOST_MATH_BIG_CONSTANT(T, 64, -0.815696314790853893484e-8),
492         };
493       static const T Q[9] = {
494          BOOST_MATH_BIG_CONSTANT(T, 64, 1.0),
495          BOOST_MATH_BIG_CONSTANT(T, 64, 0.525765665400123515036),
496          BOOST_MATH_BIG_CONSTANT(T, 64, 0.10852641753657122787),
497          BOOST_MATH_BIG_CONSTANT(T, 64, 0.0115669945375362045249),
498          BOOST_MATH_BIG_CONSTANT(T, 64, 0.000732896513858274091966),
499          BOOST_MATH_BIG_CONSTANT(T, 64, 0.30683952282420248448e-4),
500          BOOST_MATH_BIG_CONSTANT(T, 64, 0.819649214609633126119e-6),
501          BOOST_MATH_BIG_CONSTANT(T, 64, 0.117957556472335968146e-7),
502          BOOST_MATH_BIG_CONSTANT(T, 64, -0.193432300973017671137e-12),
503         };
504       result = tools::evaluate_polynomial(P, T(s - 7)) / tools::evaluate_polynomial(Q, T(s - 7));
505       result = 1 + exp(result);
506    }
507    else if(s < 42)
508    {
509       // Max error in interpolated form:             1.668e-17
510       // Max error found at long double precision:   1.669714e-17
511       static const T P[9] = {
512          BOOST_MATH_BIG_CONSTANT(T, 64, -10.3948950573308861781),
513          BOOST_MATH_BIG_CONSTANT(T, 64, -2.82646012777913950108),
514          BOOST_MATH_BIG_CONSTANT(T, 64, -0.342144362739570333665),
515          BOOST_MATH_BIG_CONSTANT(T, 64, -0.0249285145498722647472),
516          BOOST_MATH_BIG_CONSTANT(T, 64, -0.00122493108848097114118),
517          BOOST_MATH_BIG_CONSTANT(T, 64, -0.423055371192592850196e-4),
518          BOOST_MATH_BIG_CONSTANT(T, 64, -0.1025215577185967488e-5),
519          BOOST_MATH_BIG_CONSTANT(T, 64, -0.165096762663509467061e-7),
520          BOOST_MATH_BIG_CONSTANT(T, 64, -0.145392555873022044329e-9),
521       };
522       static const T Q[10] = {
523          BOOST_MATH_BIG_CONSTANT(T, 64, 1.0),
524          BOOST_MATH_BIG_CONSTANT(T, 64, 0.205135978585281988052),
525          BOOST_MATH_BIG_CONSTANT(T, 64, 0.0192359357875879453602),
526          BOOST_MATH_BIG_CONSTANT(T, 64, 0.00111496452029715514119),
527          BOOST_MATH_BIG_CONSTANT(T, 64, 0.434928449016693986857e-4),
528          BOOST_MATH_BIG_CONSTANT(T, 64, 0.116911068726610725891e-5),
529          BOOST_MATH_BIG_CONSTANT(T, 64, 0.206704342290235237475e-7),
530          BOOST_MATH_BIG_CONSTANT(T, 64, 0.209772836100827647474e-9),
531          BOOST_MATH_BIG_CONSTANT(T, 64, -0.939798249922234703384e-16),
532          BOOST_MATH_BIG_CONSTANT(T, 64, 0.264584017421245080294e-18),
533       };
534       result = tools::evaluate_polynomial(P, T(s - 15)) / tools::evaluate_polynomial(Q, T(s - 15));
535       result = 1 + exp(result);
536    }
537    else if(s < 63)
538    {
539       result = 1 + pow(T(2), -s);
540    }
541    else
542    {
543       result = 1;
544    }
545    return result;
546 }
547 
548 template <class T, class Policy>
549 T zeta_imp_prec(T s, T sc, const Policy&, const mpl::int_<113>&)
550 {
551    BOOST_MATH_STD_USING
552    T result;
553    if(s < 1)
554    {
555       // Rational Approximation
556       // Maximum Deviation Found:                     9.493e-37
557       // Expected Error Term:                         9.492e-37
558       // Max error found at long double precision:    7.281332e-31
559 
560       static const T P[10] = {
561          BOOST_MATH_BIG_CONSTANT(T, 113, -1.0),
562          BOOST_MATH_BIG_CONSTANT(T, 113, -0.0353008629988648122808504280990313668),
563          BOOST_MATH_BIG_CONSTANT(T, 113, 0.0107795651204927743049369868548706909),
564          BOOST_MATH_BIG_CONSTANT(T, 113, 0.000523961870530500751114866884685172975),
565          BOOST_MATH_BIG_CONSTANT(T, 113, -0.661805838304910731947595897966487515e-4),
566          BOOST_MATH_BIG_CONSTANT(T, 113, -0.658932670403818558510656304189164638e-5),
567          BOOST_MATH_BIG_CONSTANT(T, 113, -0.103437265642266106533814021041010453e-6),
568          BOOST_MATH_BIG_CONSTANT(T, 113, 0.116818787212666457105375746642927737e-7),
569          BOOST_MATH_BIG_CONSTANT(T, 113, 0.660690993901506912123512551294239036e-9),
570          BOOST_MATH_BIG_CONSTANT(T, 113, 0.113103113698388531428914333768142527e-10),
571         };
572       static const T Q[11] = {
573          BOOST_MATH_BIG_CONSTANT(T, 113, 1.0),
574          BOOST_MATH_BIG_CONSTANT(T, 113, -0.387483472099602327112637481818565459),
575          BOOST_MATH_BIG_CONSTANT(T, 113, 0.0802265315091063135271497708694776875),
576          BOOST_MATH_BIG_CONSTANT(T, 113, -0.0110727276164171919280036408995078164),
577          BOOST_MATH_BIG_CONSTANT(T, 113, 0.00112552716946286252000434849173787243),
578          BOOST_MATH_BIG_CONSTANT(T, 113, -0.874554160748626916455655180296834352e-4),
579          BOOST_MATH_BIG_CONSTANT(T, 113, 0.530097847491828379568636739662278322e-5),
580          BOOST_MATH_BIG_CONSTANT(T, 113, -0.248461553590496154705565904497247452e-6),
581          BOOST_MATH_BIG_CONSTANT(T, 113, 0.881834921354014787309644951507523899e-8),
582          BOOST_MATH_BIG_CONSTANT(T, 113, -0.217062446168217797598596496310953025e-9),
583          BOOST_MATH_BIG_CONSTANT(T, 113, 0.315823200002384492377987848307151168e-11),
584       };
585       result = tools::evaluate_polynomial(P, sc) / tools::evaluate_polynomial(Q, sc);
586       result += (sc);
587       result /= (sc);
588    }
589    else if(s <= 2)
590    {
591       // Maximum Deviation Found:                     1.616e-37
592       // Expected Error Term:                         -1.615e-37
593 
594       static const T P[10] = {
595          BOOST_MATH_BIG_CONSTANT(T, 113, 0.577215664901532860606512090082402431),
596          BOOST_MATH_BIG_CONSTANT(T, 113, 0.255597968739771510415479842335906308),
597          BOOST_MATH_BIG_CONSTANT(T, 113, 0.0494056503552807274142218876983542205),
598          BOOST_MATH_BIG_CONSTANT(T, 113, 0.00551372778611700965268920983472292325),
599          BOOST_MATH_BIG_CONSTANT(T, 113, 0.00043667616723970574871427830895192731),
600          BOOST_MATH_BIG_CONSTANT(T, 113, 0.268562259154821957743669387915239528e-4),
601          BOOST_MATH_BIG_CONSTANT(T, 113, 0.109249633923016310141743084480436612e-5),
602          BOOST_MATH_BIG_CONSTANT(T, 113, 0.273895554345300227466534378753023924e-7),
603          BOOST_MATH_BIG_CONSTANT(T, 113, 0.583103205551702720149237384027795038e-9),
604          BOOST_MATH_BIG_CONSTANT(T, 113, -0.835774625259919268768735944711219256e-11),
605       };
606       static const T Q[11] = {
607          BOOST_MATH_BIG_CONSTANT(T, 113, 1.0),
608          BOOST_MATH_BIG_CONSTANT(T, 113, 0.316661751179735502065583176348292881),
609          BOOST_MATH_BIG_CONSTANT(T, 113, 0.0540401806533507064453851182728635272),
610          BOOST_MATH_BIG_CONSTANT(T, 113, 0.00598621274107420237785899476374043797),
611          BOOST_MATH_BIG_CONSTANT(T, 113, 0.000474907812321704156213038740142079615),
612          BOOST_MATH_BIG_CONSTANT(T, 113, 0.272125421722314389581695715835862418e-4),
613          BOOST_MATH_BIG_CONSTANT(T, 113, 0.112649552156479800925522445229212933e-5),
614          BOOST_MATH_BIG_CONSTANT(T, 113, 0.301838975502992622733000078063330461e-7),
615          BOOST_MATH_BIG_CONSTANT(T, 113, 0.422960728687211282539769943184270106e-9),
616          BOOST_MATH_BIG_CONSTANT(T, 113, -0.377105263588822468076813329270698909e-11),
617          BOOST_MATH_BIG_CONSTANT(T, 113, -0.581926559304525152432462127383600681e-13),
618       };
619       result = tools::evaluate_polynomial(P, T(-sc)) / tools::evaluate_polynomial(Q, T(-sc));
620       result += 1 / (-sc);
621    }
622    else if(s <= 4)
623    {
624       // Maximum Deviation Found:                     1.891e-36
625       // Expected Error Term:                         -1.891e-36
626       // Max error found: 2.171527e-35
627 
628       static const float Y = 0.6986598968505859375;
629       static const T P[11] = {
630          BOOST_MATH_BIG_CONSTANT(T, 113, -0.0537258300023595010275848333539748089),
631          BOOST_MATH_BIG_CONSTANT(T, 113, 0.0429086930802630159457448174466342553),
632          BOOST_MATH_BIG_CONSTANT(T, 113, 0.0136148228754303412510213395034056857),
633          BOOST_MATH_BIG_CONSTANT(T, 113, 0.00190231601036042925183751238033763915),
634          BOOST_MATH_BIG_CONSTANT(T, 113, 0.000186880390916311438818302549192456581),
635          BOOST_MATH_BIG_CONSTANT(T, 113, 0.145347370745893262394287982691323657e-4),
636          BOOST_MATH_BIG_CONSTANT(T, 113, 0.805843276446813106414036600485884885e-6),
637          BOOST_MATH_BIG_CONSTANT(T, 113, 0.340818159286739137503297172091882574e-7),
638          BOOST_MATH_BIG_CONSTANT(T, 113, 0.115762357488748996526167305116837246e-8),
639          BOOST_MATH_BIG_CONSTANT(T, 113, 0.231904754577648077579913403645767214e-10),
640          BOOST_MATH_BIG_CONSTANT(T, 113, 0.340169592866058506675897646629036044e-12),
641       };
642       static const T Q[12] = {
643          BOOST_MATH_BIG_CONSTANT(T, 113, 1.0),
644          BOOST_MATH_BIG_CONSTANT(T, 113, 0.363755247765087100018556983050520554),
645          BOOST_MATH_BIG_CONSTANT(T, 113, 0.0696581979014242539385695131258321598),
646          BOOST_MATH_BIG_CONSTANT(T, 113, 0.00882208914484611029571547753782014817),
647          BOOST_MATH_BIG_CONSTANT(T, 113, 0.000815405623261946661762236085660996718),
648          BOOST_MATH_BIG_CONSTANT(T, 113, 0.571366167062457197282642344940445452e-4),
649          BOOST_MATH_BIG_CONSTANT(T, 113, 0.309278269271853502353954062051797838e-5),
650          BOOST_MATH_BIG_CONSTANT(T, 113, 0.12822982083479010834070516053794262e-6),
651          BOOST_MATH_BIG_CONSTANT(T, 113, 0.397876357325018976733953479182110033e-8),
652          BOOST_MATH_BIG_CONSTANT(T, 113, 0.8484432107648683277598472295289279e-10),
653          BOOST_MATH_BIG_CONSTANT(T, 113, 0.105677416606909614301995218444080615e-11),
654          BOOST_MATH_BIG_CONSTANT(T, 113, 0.547223964564003701979951154093005354e-15),
655       };
656       result = tools::evaluate_polynomial(P, T(s - 2)) / tools::evaluate_polynomial(Q, T(s - 2));
657       result += Y + 1 / (-sc);
658    }
659    else if(s <= 6)
660    {
661       // Max error in interpolated form:             1.510e-37
662       // Max error found at long double precision:   2.769266e-34
663 
664       static const T Y = 3.28348541259765625F;
665 
666       static const T P[13] = {
667          BOOST_MATH_BIG_CONSTANT(T, 113, 0.786383506575062179339611614117697622),
668          BOOST_MATH_BIG_CONSTANT(T, 113, 0.495766593395271370974685959652073976),
669          BOOST_MATH_BIG_CONSTANT(T, 113, -0.409116737851754766422360889037532228),
670          BOOST_MATH_BIG_CONSTANT(T, 113, -0.57340744006238263817895456842655987),
671          BOOST_MATH_BIG_CONSTANT(T, 113, -0.280479899797421910694892949057963111),
672          BOOST_MATH_BIG_CONSTANT(T, 113, -0.0753148409447590257157585696212649869),
673          BOOST_MATH_BIG_CONSTANT(T, 113, -0.0122934003684672788499099362823748632),
674          BOOST_MATH_BIG_CONSTANT(T, 113, -0.00126148398446193639247961370266962927),
675          BOOST_MATH_BIG_CONSTANT(T, 113, -0.828465038179772939844657040917364896e-4),
676          BOOST_MATH_BIG_CONSTANT(T, 113, -0.361008916706050977143208468690645684e-5),
677          BOOST_MATH_BIG_CONSTANT(T, 113, -0.109879825497910544424797771195928112e-6),
678          BOOST_MATH_BIG_CONSTANT(T, 113, -0.214539416789686920918063075528797059e-8),
679          BOOST_MATH_BIG_CONSTANT(T, 113, -0.15090220092460596872172844424267351e-10),
680       };
681       static const T Q[14] = {
682          BOOST_MATH_BIG_CONSTANT(T, 113, 1.0),
683          BOOST_MATH_BIG_CONSTANT(T, 113, 1.69490865837142338462982225731926485),
684          BOOST_MATH_BIG_CONSTANT(T, 113, 1.22697696630994080733321401255942464),
685          BOOST_MATH_BIG_CONSTANT(T, 113, 0.495409420862526540074366618006341533),
686          BOOST_MATH_BIG_CONSTANT(T, 113, 0.122368084916843823462872905024259633),
687          BOOST_MATH_BIG_CONSTANT(T, 113, 0.0191412993625268971656513890888208623),
688          BOOST_MATH_BIG_CONSTANT(T, 113, 0.00191401538628980617753082598351559642),
689          BOOST_MATH_BIG_CONSTANT(T, 113, 0.000123318142456272424148930280876444459),
690          BOOST_MATH_BIG_CONSTANT(T, 113, 0.531945488232526067889835342277595709e-5),
691          BOOST_MATH_BIG_CONSTANT(T, 113, 0.161843184071894368337068779669116236e-6),
692          BOOST_MATH_BIG_CONSTANT(T, 113, 0.305796079600152506743828859577462778e-8),
693          BOOST_MATH_BIG_CONSTANT(T, 113, 0.233582592298450202680170811044408894e-10),
694          BOOST_MATH_BIG_CONSTANT(T, 113, -0.275363878344548055574209713637734269e-13),
695          BOOST_MATH_BIG_CONSTANT(T, 113, 0.221564186807357535475441900517843892e-15),
696       };
697       result = tools::evaluate_polynomial(P, T(s - 4)) / tools::evaluate_polynomial(Q, T(s - 4));
698       result -= Y;
699       result = 1 + exp(result);
700    }
701    else if(s < 10)
702    {
703       // Max error in interpolated form:             1.999e-34
704       // Max error found at long double precision:   2.156186e-33
705 
706       static const T P[13] = {
707          BOOST_MATH_BIG_CONSTANT(T, 113, -4.0545627381873738086704293881227365),
708          BOOST_MATH_BIG_CONSTANT(T, 113, -4.70088348734699134347906176097717782),
709          BOOST_MATH_BIG_CONSTANT(T, 113, -2.36921550900925512951976617607678789),
710          BOOST_MATH_BIG_CONSTANT(T, 113, -0.684322583796369508367726293719322866),
711          BOOST_MATH_BIG_CONSTANT(T, 113, -0.126026534540165129870721937592996324),
712          BOOST_MATH_BIG_CONSTANT(T, 113, -0.015636903921778316147260572008619549),
713          BOOST_MATH_BIG_CONSTANT(T, 113, -0.00135442294754728549644376325814460807),
714          BOOST_MATH_BIG_CONSTANT(T, 113, -0.842793965853572134365031384646117061e-4),
715          BOOST_MATH_BIG_CONSTANT(T, 113, -0.385602133791111663372015460784978351e-5),
716          BOOST_MATH_BIG_CONSTANT(T, 113, -0.130458500394692067189883214401478539e-6),
717          BOOST_MATH_BIG_CONSTANT(T, 113, -0.315861074947230418778143153383660035e-8),
718          BOOST_MATH_BIG_CONSTANT(T, 113, -0.500334720512030826996373077844707164e-10),
719          BOOST_MATH_BIG_CONSTANT(T, 113, -0.420204769185233365849253969097184005e-12),
720         };
721       static const T Q[14] = {
722          BOOST_MATH_BIG_CONSTANT(T, 113, 1.0),
723          BOOST_MATH_BIG_CONSTANT(T, 113, 0.97663511666410096104783358493318814),
724          BOOST_MATH_BIG_CONSTANT(T, 113, 0.40878780231201806504987368939673249),
725          BOOST_MATH_BIG_CONSTANT(T, 113, 0.0963890666609396058945084107597727252),
726          BOOST_MATH_BIG_CONSTANT(T, 113, 0.0142207619090854604824116070866614505),
727          BOOST_MATH_BIG_CONSTANT(T, 113, 0.00139010220902667918476773423995750877),
728          BOOST_MATH_BIG_CONSTANT(T, 113, 0.940669540194694997889636696089994734e-4),
729          BOOST_MATH_BIG_CONSTANT(T, 113, 0.458220848507517004399292480807026602e-5),
730          BOOST_MATH_BIG_CONSTANT(T, 113, 0.16345521617741789012782420625435495e-6),
731          BOOST_MATH_BIG_CONSTANT(T, 113, 0.414007452533083304371566316901024114e-8),
732          BOOST_MATH_BIG_CONSTANT(T, 113, 0.68701473543366328016953742622661377e-10),
733          BOOST_MATH_BIG_CONSTANT(T, 113, 0.603461891080716585087883971886075863e-12),
734          BOOST_MATH_BIG_CONSTANT(T, 113, 0.294670713571839023181857795866134957e-16),
735          BOOST_MATH_BIG_CONSTANT(T, 113, -0.147003914536437243143096875069813451e-18),
736         };
737       result = tools::evaluate_polynomial(P, T(s - 6)) / tools::evaluate_polynomial(Q, T(s - 6));
738       result = 1 + exp(result);
739    }
740    else if(s < 17)
741    {
742       // Max error in interpolated form:             1.641e-32
743       // Max error found at long double precision:   1.696121e-32
744       static const T P[13] = {
745          BOOST_MATH_BIG_CONSTANT(T, 113, -6.91319491921722925920883787894829678),
746          BOOST_MATH_BIG_CONSTANT(T, 113, -3.65491257639481960248690596951049048),
747          BOOST_MATH_BIG_CONSTANT(T, 113, -0.813557553449954526442644544105257881),
748          BOOST_MATH_BIG_CONSTANT(T, 113, -0.0994317301685870959473658713841138083),
749          BOOST_MATH_BIG_CONSTANT(T, 113, -0.00726896610245676520248617014211734906),
750          BOOST_MATH_BIG_CONSTANT(T, 113, -0.000317253318715075854811266230916762929),
751          BOOST_MATH_BIG_CONSTANT(T, 113, -0.66851422826636750855184211580127133e-5),
752          BOOST_MATH_BIG_CONSTANT(T, 113, 0.879464154730985406003332577806849971e-7),
753          BOOST_MATH_BIG_CONSTANT(T, 113, 0.113838903158254250631678791998294628e-7),
754          BOOST_MATH_BIG_CONSTANT(T, 113, 0.379184410304927316385211327537817583e-9),
755          BOOST_MATH_BIG_CONSTANT(T, 113, 0.612992858643904887150527613446403867e-11),
756          BOOST_MATH_BIG_CONSTANT(T, 113, 0.347873737198164757035457841688594788e-13),
757          BOOST_MATH_BIG_CONSTANT(T, 113, -0.289187187441625868404494665572279364e-15),
758         };
759       static const T Q[14] = {
760          BOOST_MATH_BIG_CONSTANT(T, 113, 1.0),
761          BOOST_MATH_BIG_CONSTANT(T, 113, 0.427310044448071818775721584949868806),
762          BOOST_MATH_BIG_CONSTANT(T, 113, 0.074602514873055756201435421385243062),
763          BOOST_MATH_BIG_CONSTANT(T, 113, 0.00688651562174480772901425121653945942),
764          BOOST_MATH_BIG_CONSTANT(T, 113, 0.000360174847635115036351323894321880445),
765          BOOST_MATH_BIG_CONSTANT(T, 113, 0.973556847713307543918865405758248777e-5),
766          BOOST_MATH_BIG_CONSTANT(T, 113, -0.853455848314516117964634714780874197e-8),
767          BOOST_MATH_BIG_CONSTANT(T, 113, -0.118203513654855112421673192194622826e-7),
768          BOOST_MATH_BIG_CONSTANT(T, 113, -0.462521662511754117095006543363328159e-9),
769          BOOST_MATH_BIG_CONSTANT(T, 113, -0.834212591919475633107355719369463143e-11),
770          BOOST_MATH_BIG_CONSTANT(T, 113, -0.5354594751002702935740220218582929e-13),
771          BOOST_MATH_BIG_CONSTANT(T, 113, 0.406451690742991192964889603000756203e-15),
772          BOOST_MATH_BIG_CONSTANT(T, 113, 0.887948682401000153828241615760146728e-19),
773          BOOST_MATH_BIG_CONSTANT(T, 113, -0.34980761098820347103967203948619072e-21),
774         };
775       result = tools::evaluate_polynomial(P, T(s - 10)) / tools::evaluate_polynomial(Q, T(s - 10));
776       result = 1 + exp(result);
777    }
778    else if(s < 30)
779    {
780       // Max error in interpolated form:             1.563e-31
781       // Max error found at long double precision:   1.562725e-31
782 
783       static const T P[13] = {
784          BOOST_MATH_BIG_CONSTANT(T, 113, -11.7824798233959252791987402769438322),
785          BOOST_MATH_BIG_CONSTANT(T, 113, -4.36131215284987731928174218354118102),
786          BOOST_MATH_BIG_CONSTANT(T, 113, -0.732260980060982349410898496846972204),
787          BOOST_MATH_BIG_CONSTANT(T, 113, -0.0744985185694913074484248803015717388),
788          BOOST_MATH_BIG_CONSTANT(T, 113, -0.00517228281320594683022294996292250527),
789          BOOST_MATH_BIG_CONSTANT(T, 113, -0.000260897206152101522569969046299309939),
790          BOOST_MATH_BIG_CONSTANT(T, 113, -0.989553462123121764865178453128769948e-5),
791          BOOST_MATH_BIG_CONSTANT(T, 113, -0.286916799741891410827712096608826167e-6),
792          BOOST_MATH_BIG_CONSTANT(T, 113, -0.637262477796046963617949532211619729e-8),
793          BOOST_MATH_BIG_CONSTANT(T, 113, -0.106796831465628373325491288787760494e-9),
794          BOOST_MATH_BIG_CONSTANT(T, 113, -0.129343095511091870860498356205376823e-11),
795          BOOST_MATH_BIG_CONSTANT(T, 113, -0.102397936697965977221267881716672084e-13),
796          BOOST_MATH_BIG_CONSTANT(T, 113, -0.402663128248642002351627980255756363e-16),
797       };
798       static const T Q[14] = {
799          BOOST_MATH_BIG_CONSTANT(T, 113, 1.0),
800          BOOST_MATH_BIG_CONSTANT(T, 113, 0.311288325355705609096155335186466508),
801          BOOST_MATH_BIG_CONSTANT(T, 113, 0.0438318468940415543546769437752132748),
802          BOOST_MATH_BIG_CONSTANT(T, 113, 0.00374396349183199548610264222242269536),
803          BOOST_MATH_BIG_CONSTANT(T, 113, 0.000218707451200585197339671707189281302),
804          BOOST_MATH_BIG_CONSTANT(T, 113, 0.927578767487930747532953583797351219e-5),
805          BOOST_MATH_BIG_CONSTANT(T, 113, 0.294145760625753561951137473484889639e-6),
806          BOOST_MATH_BIG_CONSTANT(T, 113, 0.704618586690874460082739479535985395e-8),
807          BOOST_MATH_BIG_CONSTANT(T, 113, 0.126333332872897336219649130062221257e-9),
808          BOOST_MATH_BIG_CONSTANT(T, 113, 0.16317315713773503718315435769352765e-11),
809          BOOST_MATH_BIG_CONSTANT(T, 113, 0.137846712823719515148344938160275695e-13),
810          BOOST_MATH_BIG_CONSTANT(T, 113, 0.580975420554224366450994232723910583e-16),
811          BOOST_MATH_BIG_CONSTANT(T, 113, -0.291354445847552426900293580511392459e-22),
812          BOOST_MATH_BIG_CONSTANT(T, 113, 0.73614324724785855925025452085443636e-25),
813       };
814       result = tools::evaluate_polynomial(P, T(s - 17)) / tools::evaluate_polynomial(Q, T(s - 17));
815       result = 1 + exp(result);
816    }
817    else if(s < 74)
818    {
819       // Max error in interpolated form:             2.311e-27
820       // Max error found at long double precision:   2.297544e-27
821       static const T P[14] = {
822          BOOST_MATH_BIG_CONSTANT(T, 113, -20.7944102007844314586649688802236072),
823          BOOST_MATH_BIG_CONSTANT(T, 113, -4.95759941987499442499908748130192187),
824          BOOST_MATH_BIG_CONSTANT(T, 113, -0.563290752832461751889194629200298688),
825          BOOST_MATH_BIG_CONSTANT(T, 113, -0.0406197001137935911912457120706122877),
826          BOOST_MATH_BIG_CONSTANT(T, 113, -0.0020846534789473022216888863613422293),
827          BOOST_MATH_BIG_CONSTANT(T, 113, -0.808095978462109173749395599401375667e-4),
828          BOOST_MATH_BIG_CONSTANT(T, 113, -0.244706022206249301640890603610060959e-5),
829          BOOST_MATH_BIG_CONSTANT(T, 113, -0.589477682919645930544382616501666572e-7),
830          BOOST_MATH_BIG_CONSTANT(T, 113, -0.113699573675553496343617442433027672e-8),
831          BOOST_MATH_BIG_CONSTANT(T, 113, -0.174767860183598149649901223128011828e-10),
832          BOOST_MATH_BIG_CONSTANT(T, 113, -0.210051620306761367764549971980026474e-12),
833          BOOST_MATH_BIG_CONSTANT(T, 113, -0.189187969537370950337212675466400599e-14),
834          BOOST_MATH_BIG_CONSTANT(T, 113, -0.116313253429564048145641663778121898e-16),
835          BOOST_MATH_BIG_CONSTANT(T, 113, -0.376708747782400769427057630528578187e-19),
836       };
837       static const T Q[16] = {
838          BOOST_MATH_BIG_CONSTANT(T, 113, 1.0),
839          BOOST_MATH_BIG_CONSTANT(T, 113, 0.205076752981410805177554569784219717),
840          BOOST_MATH_BIG_CONSTANT(T, 113, 0.0202526722696670378999575738524540269),
841          BOOST_MATH_BIG_CONSTANT(T, 113, 0.001278305290005994980069466658219057),
842          BOOST_MATH_BIG_CONSTANT(T, 113, 0.576404779858501791742255670403304787e-4),
843          BOOST_MATH_BIG_CONSTANT(T, 113, 0.196477049872253010859712483984252067e-5),
844          BOOST_MATH_BIG_CONSTANT(T, 113, 0.521863830500876189501054079974475762e-7),
845          BOOST_MATH_BIG_CONSTANT(T, 113, 0.109524209196868135198775445228552059e-8),
846          BOOST_MATH_BIG_CONSTANT(T, 113, 0.181698713448644481083966260949267825e-10),
847          BOOST_MATH_BIG_CONSTANT(T, 113, 0.234793316975091282090312036524695562e-12),
848          BOOST_MATH_BIG_CONSTANT(T, 113, 0.227490441461460571047545264251399048e-14),
849          BOOST_MATH_BIG_CONSTANT(T, 113, 0.151500292036937400913870642638520668e-16),
850          BOOST_MATH_BIG_CONSTANT(T, 113, 0.543475775154780935815530649335936121e-19),
851          BOOST_MATH_BIG_CONSTANT(T, 113, 0.241647013434111434636554455083309352e-28),
852          BOOST_MATH_BIG_CONSTANT(T, 113, -0.557103423021951053707162364713587374e-31),
853          BOOST_MATH_BIG_CONSTANT(T, 113, 0.618708773442584843384712258199645166e-34),
854       };
855       result = tools::evaluate_polynomial(P, T(s - 30)) / tools::evaluate_polynomial(Q, T(s - 30));
856       result = 1 + exp(result);
857    }
858    else if(s < 117)
859    {
860       result = 1 + pow(T(2), -s);
861    }
862    else
863    {
864       result = 1;
865    }
866    return result;
867 }
868 
869 template <class T, class Policy>
870 T zeta_imp_odd_integer(int s, const T&, const Policy&, const mpl::true_&)
871 {
872    static const T results[] = {
873       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),
874    };
875    return s > 113 ? 1 : results[(s - 3) / 2];
876 }
877 
878 template <class T, class Policy>
879 T zeta_imp_odd_integer(int s, const T& sc, const Policy& pol, const mpl::false_&)
880 {
881    static BOOST_MATH_THREAD_LOCAL bool is_init = false;
882    static BOOST_MATH_THREAD_LOCAL T results[50] = {};
883    static BOOST_MATH_THREAD_LOCAL int digits = tools::digits<T>();
884    int current_digits = tools::digits<T>();
885    if(digits != current_digits)
886    {
887       // Oh my precision has changed...
888       is_init = false;
889    }
890    if(!is_init)
891    {
892       is_init = true;
893       digits = current_digits;
894       for(unsigned k = 0; k < sizeof(results) / sizeof(results[0]); ++k)
895       {
896          T arg = k * 2 + 3;
897          T c_arg = 1 - arg;
898          results[k] = zeta_polynomial_series(arg, c_arg, pol);
899       }
900    }
901    unsigned index = (s - 3) / 2;
902    return index >= sizeof(results) / sizeof(results[0]) ? zeta_polynomial_series(T(s), sc, pol): results[index];
903 }
904 
905 template <class T, class Policy, class Tag>
906 T zeta_imp(T s, T sc, const Policy& pol, const Tag& tag)
907 {
908    BOOST_MATH_STD_USING
909    static const char* function = "boost::math::zeta<%1%>";
910    if(sc == 0)
911       return policies::raise_pole_error<T>(
912          function,
913          "Evaluation of zeta function at pole %1%",
914          s, pol);
915    T result;
916    //
917    // Trivial case:
918    //
919    if(s > policies::digits<T, Policy>())
920       return 1;
921    //
922    // Start by seeing if we have a simple closed form:
923    //
924    if(floor(s) == s)
925    {
926 #ifndef BOOST_NO_EXCEPTIONS
927       // Without exceptions we expect itrunc to return INT_MAX on overflow
928       // and we fall through anyway.
929       try
930       {
931 #endif
932          int v = itrunc(s);
933          if(v == s)
934          {
935             if(v < 0)
936             {
937                if(((-v) & 1) == 0)
938                   return 0;
939                int n = (-v + 1) / 2;
940                if(n <= (int)boost::math::max_bernoulli_b2n<T>::value)
941                   return T((-v & 1) ? -1 : 1) * boost::math::unchecked_bernoulli_b2n<T>(n) / (1 - v);
942             }
943             else if((v & 1) == 0)
944             {
945                if(((v / 2) <= (int)boost::math::max_bernoulli_b2n<T>::value) && (v <= (int)boost::math::max_factorial<T>::value))
946                   return T(((v / 2 - 1) & 1) ? -1 : 1) * ldexp(T(1), v - 1) * pow(constants::pi<T, Policy>(), v) *
947                      boost::math::unchecked_bernoulli_b2n<T>(v / 2) / boost::math::unchecked_factorial<T>(v);
948                return T(((v / 2 - 1) & 1) ? -1 : 1) * ldexp(T(1), v - 1) * pow(constants::pi<T, Policy>(), v) *
949                   boost::math::bernoulli_b2n<T>(v / 2) / boost::math::factorial<T>(v);
950             }
951             else
952                return zeta_imp_odd_integer(v, sc, pol, mpl::bool_<(Tag::value <= 113) && Tag::value>());
953          }
954 #ifndef BOOST_NO_EXCEPTIONS
955       }
956       catch(const boost::math::rounding_error&){} // Just fall through, s is too large to round
957       catch(const std::overflow_error&){}
958 #endif
959    }
960 
961    if(fabs(s) < tools::root_epsilon<T>())
962    {
963       result = -0.5f - constants::log_root_two_pi<T, Policy>() * s;
964    }
965    else if(s < 0)
966    {
967       std::swap(s, sc);
968       if(floor(sc/2) == sc/2)
969          result = 0;
970       else
971       {
972          if(s > max_factorial<T>::value)
973          {
974             T mult = boost::math::sin_pi(0.5f * sc, pol) * 2 * zeta_imp(s, sc, pol, tag);
975             result = boost::math::lgamma(s, pol);
976             result -= s * log(2 * constants::pi<T>());
977             if(result > tools::log_max_value<T>())
978                return sign(mult) * policies::raise_overflow_error<T>(function, 0, pol);
979             result = exp(result);
980             if(tools::max_value<T>() / fabs(mult) < result)
981                return boost::math::sign(mult) * policies::raise_overflow_error<T>(function, 0, pol);
982             result *= mult;
983          }
984          else
985          {
986             result = boost::math::sin_pi(0.5f * sc, pol)
987                * 2 * pow(2 * constants::pi<T>(), -s)
988                * boost::math::tgamma(s, pol)
989                * zeta_imp(s, sc, pol, tag);
990          }
991       }
992    }
993    else
994    {
995       result = zeta_imp_prec(s, sc, pol, tag);
996    }
997    return result;
998 }
999 
1000 template <class T, class Policy, class tag>
1001 struct zeta_initializer
1002 {
1003    struct init
1004    {
initboost::math::detail::zeta_initializer::init1005       init()
1006       {
1007          do_init(tag());
1008       }
do_initboost::math::detail::zeta_initializer::init1009       static void do_init(const mpl::int_<0>&){ boost::math::zeta(static_cast<T>(5), Policy()); }
do_initboost::math::detail::zeta_initializer::init1010       static void do_init(const mpl::int_<53>&){ boost::math::zeta(static_cast<T>(5), Policy()); }
do_initboost::math::detail::zeta_initializer::init1011       static void do_init(const mpl::int_<64>&)
1012       {
1013          boost::math::zeta(static_cast<T>(0.5), Policy());
1014          boost::math::zeta(static_cast<T>(1.5), Policy());
1015          boost::math::zeta(static_cast<T>(3.5), Policy());
1016          boost::math::zeta(static_cast<T>(6.5), Policy());
1017          boost::math::zeta(static_cast<T>(14.5), Policy());
1018          boost::math::zeta(static_cast<T>(40.5), Policy());
1019 
1020          boost::math::zeta(static_cast<T>(5), Policy());
1021       }
do_initboost::math::detail::zeta_initializer::init1022       static void do_init(const mpl::int_<113>&)
1023       {
1024          boost::math::zeta(static_cast<T>(0.5), Policy());
1025          boost::math::zeta(static_cast<T>(1.5), Policy());
1026          boost::math::zeta(static_cast<T>(3.5), Policy());
1027          boost::math::zeta(static_cast<T>(5.5), Policy());
1028          boost::math::zeta(static_cast<T>(9.5), Policy());
1029          boost::math::zeta(static_cast<T>(16.5), Policy());
1030          boost::math::zeta(static_cast<T>(25.5), Policy());
1031          boost::math::zeta(static_cast<T>(70.5), Policy());
1032 
1033          boost::math::zeta(static_cast<T>(5), Policy());
1034       }
force_instantiateboost::math::detail::zeta_initializer::init1035       void force_instantiate()const{}
1036    };
1037    static const init initializer;
force_instantiateboost::math::detail::zeta_initializer1038    static void force_instantiate()
1039    {
1040       initializer.force_instantiate();
1041    }
1042 };
1043 
1044 template <class T, class Policy, class tag>
1045 const typename zeta_initializer<T, Policy, tag>::init zeta_initializer<T, Policy, tag>::initializer;
1046 
1047 } // detail
1048 
1049 template <class T, class Policy>
zeta(T s,const Policy &)1050 inline typename tools::promote_args<T>::type zeta(T s, const Policy&)
1051 {
1052    typedef typename tools::promote_args<T>::type result_type;
1053    typedef typename policies::evaluation<result_type, Policy>::type value_type;
1054    typedef typename policies::precision<result_type, Policy>::type precision_type;
1055    typedef typename policies::normalise<
1056       Policy,
1057       policies::promote_float<false>,
1058       policies::promote_double<false>,
1059       policies::discrete_quantile<>,
1060       policies::assert_undefined<> >::type forwarding_policy;
1061    typedef typename mpl::if_<
1062       mpl::less_equal<precision_type, mpl::int_<0> >,
1063       mpl::int_<0>,
1064       typename mpl::if_<
1065          mpl::less_equal<precision_type, mpl::int_<53> >,
1066          mpl::int_<53>,  // double
1067          typename mpl::if_<
1068             mpl::less_equal<precision_type, mpl::int_<64> >,
1069             mpl::int_<64>, // 80-bit long double
1070             typename mpl::if_<
1071                mpl::less_equal<precision_type, mpl::int_<113> >,
1072                mpl::int_<113>, // 128-bit long double
1073                mpl::int_<0> // too many bits, use generic version.
1074             >::type
1075          >::type
1076       >::type
1077    >::type tag_type;
1078    //typedef mpl::int_<0> tag_type;
1079 
1080    detail::zeta_initializer<value_type, forwarding_policy, tag_type>::force_instantiate();
1081 
1082    return policies::checked_narrowing_cast<result_type, forwarding_policy>(detail::zeta_imp(
1083       static_cast<value_type>(s),
1084       static_cast<value_type>(1 - static_cast<value_type>(s)),
1085       forwarding_policy(),
1086       tag_type()), "boost::math::zeta<%1%>(%1%)");
1087 }
1088 
1089 template <class T>
zeta(T s)1090 inline typename tools::promote_args<T>::type zeta(T s)
1091 {
1092    return zeta(s, policies::policy<>());
1093 }
1094 
1095 }} // namespaces
1096 
1097 #endif // BOOST_MATH_ZETA_HPP
1098 
1099 
1100 
1101