ceph/src/boost/boost/math/special_functions/zeta.hpp

   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 dianostic ignored work :(
  28 //
  29 #pragma GCC system_header
  30 #endif
  31
  32 namespace 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
  40 template <class T, class Policy>
  41 struct 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, mpl::int_<0> >,
  54       mpl::int_<5000>,
  55       typename mpl::if_<
  56          mpl::less_equal<precision_type, mpl::int_<64> >,
  57          mpl::int_<70>,
  58          typename mpl::if_<
  59             mpl::less_equal<precision_type, mpl::int_<113> >,
  60             mpl::int_<100>,
  61             mpl::int_<5000>
  62          >::type
  63       >::type
  64    >::type type;
  65 };
  66
  67 template <class T, class Policy>
  68 T 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 //
 114 template <class T>
 115 struct 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    }
 124 private:
 125    T s;
 126    unsigned k;
 127 };
 128
 129 template <class T, class Policy>
 130 inline 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
 143 template <class T, class Policy>
 144 T 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
 176 template <class T, class Policy>
 177 T zeta_imp_prec(T s, T sc, const Policy& pol, const mpl::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
 201 template <class T, class Policy>
 202 inline T zeta_imp_prec(T s, T sc, const Policy&, const mpl::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
 375 template <class T, class Policy>
 376 T zeta_imp_prec(T s, T sc, const Policy&, const mpl::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
 558 template <class T, class Policy>
 559 T zeta_imp_prec(T s, T sc, const Policy&, const mpl::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
 879 template <class T, class Policy>
 880 T zeta_imp_odd_integer(int s, const T&, const Policy&, const mpl::true_&)
 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
 888 template <class T, class Policy>
 889 T zeta_imp_odd_integer(int s, const T& sc, const Policy& pol, const mpl::false_&)
 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
 915 template <class T, class Policy, class Tag>
 916 T 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, mpl::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
1010 template <class T, class Policy, class tag>
1011 struct zeta_initializer
1012 {
1013    struct init
1014    {
1015       init()
1016       {
1017          do_init(tag());
1018       }
1019       static void do_init(const mpl::int_<0>&){ boost::math::zeta(static_cast<T>(5), Policy()); }
1020       static void do_init(const mpl::int_<53>&){ boost::math::zeta(static_cast<T>(5), Policy()); }
1021       static void do_init(const mpl::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 mpl::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
1054 template <class T, class Policy, class tag>
1055 const typename zeta_initializer<T, Policy, tag>::init zeta_initializer<T, Policy, tag>::initializer;
1056
1057 } // detail
1058
1059 template <class T, class Policy>
1060 inline 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 typename mpl::if_<
1072       mpl::less_equal<precision_type, mpl::int_<0> >,
1073       mpl::int_<0>,
1074       typename mpl::if_<
1075          mpl::less_equal<precision_type, mpl::int_<53> >,
1076          mpl::int_<53>,  // double
1077          typename mpl::if_<
1078             mpl::less_equal<precision_type, mpl::int_<64> >,
1079             mpl::int_<64>, // 80-bit long double
1080             typename mpl::if_<
1081                mpl::less_equal<precision_type, mpl::int_<113> >,
1082                mpl::int_<113>, // 128-bit long double
1083                mpl::int_<0> // too many bits, use generic version.
1084             >::type
1085          >::type
1086       >::type
1087    >::type tag_type;
1088    //typedef mpl::int_<0> tag_type;
1089
1090    detail::zeta_initializer<value_type, forwarding_policy, tag_type>::force_instantiate();
1091
1092    return policies::checked_narrowing_cast<result_type, forwarding_policy>(detail::zeta_imp(
1093       static_cast<value_type>(s),
1094       static_cast<value_type>(1 - static_cast<value_type>(s)),
1095       forwarding_policy(),
1096       tag_type()), "boost::math::zeta<%1%>(%1%)");
1097 }
1098
1099 template <class T>
1100 inline typename tools::promote_args<T>::type zeta(T s)
1101 {
1102    return zeta(s, policies::policy<>());
1103 }
1104
1105 }} // namespaces
1106
1107 #endif // BOOST_MATH_ZETA_HPP
1108
1109
1110