third_party/boost/include/boost/math/special_functions/zeta.hpp

//  Copyright John Maddock 2007.
//  Use, modification and distribution are subject to the
//  Boost Software License, Version 1.0. (See accompanying file
//  LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)

#ifndef BOOST_MATH_ZETA_HPP
#define BOOST_MATH_ZETA_HPP

#ifdef _MSC_VER
#pragma once
#endif

#include <boost/math/tools/precision.hpp>
#include <boost/math/tools/series.hpp>
#include <boost/math/policies/error_handling.hpp>
#include <boost/math/special_functions/gamma.hpp>
#include <boost/math/special_functions/sin_pi.hpp>

namespace boost{ namespace math{ namespace detail{

template <class T, class Policy>
struct zeta_series_cache_size
{
   //
   // Work how large to make our cache size when evaluating the series 
   // evaluation:  normally this is just large enough for the series
   // to have converged, but for arbitrary precision types we need a 
   // really large cache to achieve reasonable precision in a reasonable 
   // time.  This is important when constructing rational approximations
   // to zeta for example.
   //
   typedef typename boost::math::policies::precision<T,Policy>::type precision_type;
   typedef typename mpl::if_<
      mpl::less_equal<precision_type, mpl::int_<0> >,
      mpl::int_<5000>,
      typename mpl::if_<
         mpl::less_equal<precision_type, mpl::int_<64> >,
         mpl::int_<70>,
         typename mpl::if_<
            mpl::less_equal<precision_type, mpl::int_<113> >,
            mpl::int_<100>,
            mpl::int_<5000>
         >::type
      >::type
   >::type type;
};

template <class T, class Policy>
T zeta_series_imp(T s, T sc, const Policy&)
{
   //
   // Series evaluation from:
   // Havil, J. Gamma: Exploring Euler's Constant. 
   // Princeton, NJ: Princeton University Press, 2003.
   //
   // See also http://mathworld.wolfram.com/RiemannZetaFunction.html
   //
   BOOST_MATH_STD_USING
   T sum = 0;
   T mult = 0.5;
   T change;
   typedef typename zeta_series_cache_size<T,Policy>::type cache_size;
   T powers[cache_size::value] = { 0, };
   unsigned n = 0;
   do{
      T binom = -static_cast<T>(n);
      T nested_sum = 1;
      if(n < sizeof(powers) / sizeof(powers[0]))
         powers[n] = pow(static_cast<T>(n + 1), -s);
      for(unsigned k = 1; k <= n; ++k)
      {
         T p;
         if(k < sizeof(powers) / sizeof(powers[0]))
         {
            p = powers[k];
            //p = pow(k + 1, -s);
         }
         else
            p = pow(static_cast<T>(k + 1), -s);
         nested_sum += binom * p;
        binom *= (k - static_cast<T>(n)) / (k + 1);
      }
      change = mult * nested_sum;
      sum += change;
      mult /= 2;
      ++n;
   }while(fabs(change / sum) > tools::epsilon<T>());

   return sum * 1 / -boost::math::powm1(T(2), sc);
}
//
// Classical p-series:
//
template <class T>
struct zeta_series2
{
   typedef T result_type;
   zeta_series2(T _s) : s(-_s), k(1){}
   T operator()()
   {
      BOOST_MATH_STD_USING
      return pow(static_cast<T>(k++), s);
   }
private:
   T s;
   unsigned k;
};

template <class T, class Policy>
inline T zeta_series2_imp(T s, const Policy& pol)
{
   boost::uintmax_t max_iter = policies::get_max_series_iterations<Policy>();;
   zeta_series2<T> f(s);
   T result = tools::sum_series(
      f, 
      policies::get_epsilon<T, Policy>(),
      max_iter);
   policies::check_series_iterations("boost::math::zeta_series2<%1%>(%1%)", max_iter, pol);
   return result;
}

template <class T, class Policy>
T zeta_imp_prec(T s, T sc, const Policy& pol, const mpl::int_<0>&)
{
   BOOST_MATH_STD_USING
   T result; 
   //
   // Only use power series if it will converge in 100 
   // iterations or less: the more iterations it consumes
   // the slower convergence becomes so we have to be very 
   // careful in it's usage.
   //
   if (s > -log(tools::epsilon<T>()) / 4.5)
      result = detail::zeta_series2_imp(s, pol);
   else
      result = detail::zeta_series_imp(s, sc, pol);
   return result;
}

template <class T, class Policy>
inline T zeta_imp_prec(T s, T sc, const Policy&, const mpl::int_<53>&)
{
   BOOST_MATH_STD_USING
   T result;
   if(s < 1)
   {
      // Rational Approximation
      // Maximum Deviation Found:                     2.020e-18
      // Expected Error Term:                         -2.020e-18
      // Max error found at double precision:         3.994987e-17
      static const T P[6] = {    
         0.24339294433593750202L,
         -0.49092470516353571651L,
         0.0557616214776046784287L,
         -0.00320912498879085894856L,
         0.000451534528645796438704L,
         -0.933241270357061460782e-5L,
        };
      static const T Q[6] = {    
         1L,
         -0.279960334310344432495L,
         0.0419676223309986037706L,
         -0.00413421406552171059003L,
         0.00024978985622317935355L,
         -0.101855788418564031874e-4L,
      };
      result = tools::evaluate_polynomial(P, sc) / tools::evaluate_polynomial(Q, sc);
      result -= 1.2433929443359375F;
      result += (sc);
      result /= (sc);
   }
   else if(s <= 2)
   {
      // Maximum Deviation Found:        9.007e-20
      // Expected Error Term:            9.007e-20
      static const T P[6] = {    
         0.577215664901532860516,
         0.243210646940107164097,
         0.0417364673988216497593,
         0.00390252087072843288378,
         0.000249606367151877175456,
         0.110108440976732897969e-4,
      };
      static const T Q[6] = {    
         1,
         0.295201277126631761737,
         0.043460910607305495864,
         0.00434930582085826330659,
         0.000255784226140488490982,
         0.10991819782396112081e-4,
      };
      result = tools::evaluate_polynomial(P, -sc) / tools::evaluate_polynomial(Q, -sc);
      result += 1 / (-sc);
   }
   else if(s <= 4)
   {
      // Maximum Deviation Found:          5.946e-22
      // Expected Error Term:              -5.946e-22
      static const float Y = 0.6986598968505859375;
      static const T P[6] = {    
         -0.0537258300023595030676,
         0.0445163473292365591906,
         0.0128677673534519952905,
         0.00097541770457391752726,
         0.769875101573654070925e-4,
         0.328032510000383084155e-5,
      };
      static const T Q[7] = {    
         1,
         0.33383194553034051422,
         0.0487798431291407621462,
         0.00479039708573558490716,
         0.000270776703956336357707,
         0.106951867532057341359e-4,
         0.236276623974978646399e-7,
      };
      result = tools::evaluate_polynomial(P, s - 2) / tools::evaluate_polynomial(Q, s - 2);
      result += Y + 1 / (-sc);
   }
   else if(s <= 7)
   {
      // Maximum Deviation Found:                     2.955e-17
      // Expected Error Term:                         2.955e-17
      // Max error found at double precision:         2.009135e-16

      static const T P[6] = {    
         -2.49710190602259410021,
         -2.60013301809475665334,
         -0.939260435377109939261,
         -0.138448617995741530935,
         -0.00701721240549802377623,
         -0.229257310594893932383e-4,
      };
      static const T Q[9] = {    
         1,
         0.706039025937745133628,
         0.15739599649558626358,
         0.0106117950976845084417,
         -0.36910273311764618902e-4,
         0.493409563927590008943e-5,
         -0.234055487025287216506e-6,
         0.718833729365459760664e-8,
         -0.1129200113474947419e-9,
      };
      result = tools::evaluate_polynomial(P, s - 4) / tools::evaluate_polynomial(Q, s - 4);
      result = 1 + exp(result);
   }
   else if(s < 15)
   {
      // Maximum Deviation Found:                     7.117e-16
      // Expected Error Term:                         7.117e-16
      // Max error found at double precision:         9.387771e-16
      static const T P[7] = {    
         -4.78558028495135619286,
         -1.89197364881972536382,
         -0.211407134874412820099,
         -0.000189204758260076688518,
         0.00115140923889178742086,
         0.639949204213164496988e-4,
         0.139348932445324888343e-5,
        };
      static const T Q[9] = {    
         1,
         0.244345337378188557777,
         0.00873370754492288653669,
         -0.00117592765334434471562,
         -0.743743682899933180415e-4,
         -0.21750464515767984778e-5,
         0.471001264003076486547e-8,
         -0.833378440625385520576e-10,
         0.699841545204845636531e-12,
        };
      result = tools::evaluate_polynomial(P, s - 7) / tools::evaluate_polynomial(Q, s - 7);
      result = 1 + exp(result);
   }
   else if(s < 36)
   {
      // Max error in interpolated form:             1.668e-17
      // Max error found at long double precision:   1.669714e-17
      static const T P[8] = {    
         -10.3948950573308896825,
         -2.85827219671106697179,
         -0.347728266539245787271,
         -0.0251156064655346341766,
         -0.00119459173416968685689,
         -0.382529323507967522614e-4,
         -0.785523633796723466968e-6,
         -0.821465709095465524192e-8,
      };
      static const T Q[10] = {    
         1,
         0.208196333572671890965,
         0.0195687657317205033485,
         0.00111079638102485921877,
         0.408507746266039256231e-4,
         0.955561123065693483991e-6,
         0.118507153474022900583e-7,
         0.222609483627352615142e-14,
      };
      result = tools::evaluate_polynomial(P, s - 15) / tools::evaluate_polynomial(Q, s - 15);
      result = 1 + exp(result);
   }
   else if(s < 56)
   {
      result = 1 + pow(T(2), -s);
   }
   else
   {
      result = 1;
   }
   return result;
}

template <class T, class Policy>
T zeta_imp_prec(T s, T sc, const Policy&, const mpl::int_<64>&)
{
   BOOST_MATH_STD_USING
   T result;
   if(s < 1)
   {
      // Rational Approximation
      // Maximum Deviation Found:                     3.099e-20
      // Expected Error Term:                         3.099e-20
      // Max error found at long double precision:    5.890498e-20
      static const T P[6] = {    
         0.243392944335937499969L,
         -0.496837806864865688082L,
         0.0680008039723709987107L,
         -0.00511620413006619942112L,
         0.000455369899250053003335L,
         -0.279496685273033761927e-4L,
        };
      static const T Q[7] = {    
         1L,
         -0.30425480068225790522L,
         0.050052748580371598736L,
         -0.00519355671064700627862L,
         0.000360623385771198350257L,
         -0.159600883054550987633e-4L,
         0.339770279812410586032e-6L,
      };
      result = tools::evaluate_polynomial(P, sc) / tools::evaluate_polynomial(Q, sc);
      result -= 1.2433929443359375F;
      result += (sc);
      result /= (sc);
   }
   else if(s <= 2)
   {
      // Maximum Deviation Found:                     1.059e-21
      // Expected Error Term:                         1.059e-21
      // Max error found at long double precision:    1.626303e-19

      static const T P[6] = {    
         0.577215664901532860605L,
         0.222537368917162139445L,
         0.0356286324033215682729L,
         0.00304465292366350081446L,
         0.000178102511649069421904L,
         0.700867470265983665042e-5L,
      };
      static const T Q[7] = {    
         1L,
         0.259385759149531030085L,
         0.0373974962106091316854L,
         0.00332735159183332820617L,
         0.000188690420706998606469L,
         0.635994377921861930071e-5L,
         0.226583954978371199405e-7L,
      };
      result = tools::evaluate_polynomial(P, -sc) / tools::evaluate_polynomial(Q, -sc);
      result += 1 / (-sc);
   }
   else if(s <= 4)
   {
      // Maximum Deviation Found:          5.946e-22
      // Expected Error Term:              -5.946e-22
      static const float Y = 0.6986598968505859375;
      static const T P[7] = {    
         -0.053725830002359501027L,
         0.0470551187571475844778L,
         0.0101339410415759517471L,
         0.00100240326666092854528L,
         0.685027119098122814867e-4L,
         0.390972820219765942117e-5L,
         0.540319769113543934483e-7L,
      };
      static const T Q[8] = {    
         1,
         0.286577739726542730421L,
         0.0447355811517733225843L,
         0.00430125107610252363302L,
         0.000284956969089786662045L,
         0.116188101609848411329e-4L,
         0.278090318191657278204e-6L,
         -0.19683620233222028478e-8L,
      };
      result = tools::evaluate_polynomial(P, s - 2) / tools::evaluate_polynomial(Q, s - 2);
      result += Y + 1 / (-sc);
   }
   else if(s <= 7)
   {
      // Max error found at long double precision: 8.132216e-19
      static const T P[8] = {    
         -2.49710190602259407065L,
         -3.36664913245960625334L,
         -1.77180020623777595452L,
         -0.464717885249654313933L,
         -0.0643694921293579472583L,
         -0.00464265386202805715487L,
         -0.000165556579779704340166L,
         -0.252884970740994069582e-5L,
      };
      static const T Q[9] = {    
         1,
         1.01300131390690459085L,
         0.387898115758643503827L,
         0.0695071490045701135188L,
         0.00586908595251442839291L,
         0.000217752974064612188616L,
         0.397626583349419011731e-5L,
         -0.927884739284359700764e-8L,
         0.119810501805618894381e-9L,
      };
      result = tools::evaluate_polynomial(P, s - 4) / tools::evaluate_polynomial(Q, s - 4);
      result = 1 + exp(result);
   }
   else if(s < 15)
   {
      // Max error in interpolated form:              1.133e-18
      // Max error found at long double precision:    2.183198e-18
      static const T P[9] = {    
         -4.78558028495135548083L,
         -3.23873322238609358947L,
         -0.892338582881021799922L,
         -0.131326296217965913809L,
         -0.0115651591773783712996L,
         -0.000657728968362695775205L,
         -0.252051328129449973047e-4L,
         -0.626503445372641798925e-6L,
         -0.815696314790853893484e-8L,
        };
      static const T Q[9] = {    
         1,
         0.525765665400123515036L,
         0.10852641753657122787L,
         0.0115669945375362045249L,
         0.000732896513858274091966L,
         0.30683952282420248448e-4L,
         0.819649214609633126119e-6L,
         0.117957556472335968146e-7L,
         -0.193432300973017671137e-12L,
        };
      result = tools::evaluate_polynomial(P, s - 7) / tools::evaluate_polynomial(Q, s - 7);
      result = 1 + exp(result);
   }
   else if(s < 42)
   {
      // Max error in interpolated form:             1.668e-17
      // Max error found at long double precision:   1.669714e-17
      static const T P[9] = {    
         -10.3948950573308861781L,
         -2.82646012777913950108L,
         -0.342144362739570333665L,
         -0.0249285145498722647472L,
         -0.00122493108848097114118L,
         -0.423055371192592850196e-4L,
         -0.1025215577185967488e-5L,
         -0.165096762663509467061e-7L,
         -0.145392555873022044329e-9L,
      };
      static const T Q[10] = {    
         1,
         0.205135978585281988052L,
         0.0192359357875879453602L,
         0.00111496452029715514119L,
         0.434928449016693986857e-4L,
         0.116911068726610725891e-5L,
         0.206704342290235237475e-7L,
         0.209772836100827647474e-9L,
         -0.939798249922234703384e-16L,
         0.264584017421245080294e-18L,
      };
      result = tools::evaluate_polynomial(P, s - 15) / tools::evaluate_polynomial(Q, s - 15);
      result = 1 + exp(result);
   }
   else if(s < 63)
   {
      result = 1 + pow(T(2), -s);
   }
   else
   {
      result = 1;
   }
   return result;
}

template <class T, class Policy>
T zeta_imp_prec(T s, T sc, const Policy& pol, const mpl::int_<113>&)
{
   BOOST_MATH_STD_USING
   T result;
   if(s < 1)
   {
      // Rational Approximation
      // Maximum Deviation Found:                     9.493e-37
      // Expected Error Term:                         9.492e-37
      // Max error found at long double precision:    7.281332e-31

      static const T P[10] = {    
         -1L,
         -0.0353008629988648122808504280990313668L,
         0.0107795651204927743049369868548706909L,
         0.000523961870530500751114866884685172975L,
         -0.661805838304910731947595897966487515e-4L,
         -0.658932670403818558510656304189164638e-5L,
         -0.103437265642266106533814021041010453e-6L,
         0.116818787212666457105375746642927737e-7L,
         0.660690993901506912123512551294239036e-9L,
         0.113103113698388531428914333768142527e-10L,
        };
      static const T Q[11] = {    
         1L,
         -0.387483472099602327112637481818565459L,
         0.0802265315091063135271497708694776875L,
         -0.0110727276164171919280036408995078164L,
         0.00112552716946286252000434849173787243L,
         -0.874554160748626916455655180296834352e-4L,
         0.530097847491828379568636739662278322e-5L,
         -0.248461553590496154705565904497247452e-6L,
         0.881834921354014787309644951507523899e-8L,
         -0.217062446168217797598596496310953025e-9L,
         0.315823200002384492377987848307151168e-11L,
      };
      result = tools::evaluate_polynomial(P, sc) / tools::evaluate_polynomial(Q, sc);
      result += (sc);
      result /= (sc);
   }
   else if(s <= 2)
   {
      // Maximum Deviation Found:                     1.616e-37
      // Expected Error Term:                         -1.615e-37

      static const T P[10] = {    
         0.577215664901532860606512090082402431L,
         0.255597968739771510415479842335906308L,
         0.0494056503552807274142218876983542205L,
         0.00551372778611700965268920983472292325L,
         0.00043667616723970574871427830895192731L,
         0.268562259154821957743669387915239528e-4L,
         0.109249633923016310141743084480436612e-5L,
         0.273895554345300227466534378753023924e-7L,
         0.583103205551702720149237384027795038e-9L,
         -0.835774625259919268768735944711219256e-11L,
      };
      static const T Q[11] = {    
         1L,
         0.316661751179735502065583176348292881L,
         0.0540401806533507064453851182728635272L,
         0.00598621274107420237785899476374043797L,
         0.000474907812321704156213038740142079615L,
         0.272125421722314389581695715835862418e-4L,
         0.112649552156479800925522445229212933e-5L,
         0.301838975502992622733000078063330461e-7L,
         0.422960728687211282539769943184270106e-9L,
         -0.377105263588822468076813329270698909e-11L,
         -0.581926559304525152432462127383600681e-13L,
      };
      result = tools::evaluate_polynomial(P, -sc) / tools::evaluate_polynomial(Q, -sc);
      result += 1 / (-sc);
   }
   else if(s <= 4)
   {
      // Maximum Deviation Found:                     1.891e-36
      // Expected Error Term:                         -1.891e-36
      // Max error found: 2.171527e-35

      static const float Y = 0.6986598968505859375;
      static const T P[11] = {    
         -0.0537258300023595010275848333539748089L,
         0.0429086930802630159457448174466342553L,
         0.0136148228754303412510213395034056857L,
         0.00190231601036042925183751238033763915L,
         0.000186880390916311438818302549192456581L,
         0.145347370745893262394287982691323657e-4L,
         0.805843276446813106414036600485884885e-6L,
         0.340818159286739137503297172091882574e-7L,
         0.115762357488748996526167305116837246e-8L,
         0.231904754577648077579913403645767214e-10L,
         0.340169592866058506675897646629036044e-12L,
      };
      static const T Q[12] = {    
         1L,
         0.363755247765087100018556983050520554L,
         0.0696581979014242539385695131258321598L,
         0.00882208914484611029571547753782014817L,
         0.000815405623261946661762236085660996718L,
         0.571366167062457197282642344940445452e-4L,
         0.309278269271853502353954062051797838e-5L,
         0.12822982083479010834070516053794262e-6L,
         0.397876357325018976733953479182110033e-8L,
         0.8484432107648683277598472295289279e-10L,
         0.105677416606909614301995218444080615e-11L,
         0.547223964564003701979951154093005354e-15L,
      };
      result = tools::evaluate_polynomial(P, s - 2) / tools::evaluate_polynomial(Q, s - 2);
      result += Y + 1 / (-sc);
   }
   else if(s <= 6)
   {
      // Max error in interpolated form:             1.510e-37
      // Max error found at long double precision:   2.769266e-34

      static const T Y = 3.28348541259765625F;

      static const T P[13] = {    
         0.786383506575062179339611614117697622L,
         0.495766593395271370974685959652073976L,
         -0.409116737851754766422360889037532228L,
         -0.57340744006238263817895456842655987L,
         -0.280479899797421910694892949057963111L,
         -0.0753148409447590257157585696212649869L,
         -0.0122934003684672788499099362823748632L,
         -0.00126148398446193639247961370266962927L,
         -0.828465038179772939844657040917364896e-4L,
         -0.361008916706050977143208468690645684e-5L,
         -0.109879825497910544424797771195928112e-6L,
         -0.214539416789686920918063075528797059e-8L,
         -0.15090220092460596872172844424267351e-10L,
      };
      static const T Q[14] = {    
         1L,
         1.69490865837142338462982225731926485L,
         1.22697696630994080733321401255942464L,
         0.495409420862526540074366618006341533L,
         0.122368084916843823462872905024259633L,
         0.0191412993625268971656513890888208623L,
         0.00191401538628980617753082598351559642L,
         0.000123318142456272424148930280876444459L,
         0.531945488232526067889835342277595709e-5L,
         0.161843184071894368337068779669116236e-6L,
         0.305796079600152506743828859577462778e-8L,
         0.233582592298450202680170811044408894e-10L,
         -0.275363878344548055574209713637734269e-13L,
         0.221564186807357535475441900517843892e-15L,
      };
      result = tools::evaluate_polynomial(P, s - 4) / tools::evaluate_polynomial(Q, s - 4);
      result -= Y;
      result = 1 + exp(result);
   }
   else if(s < 10)
   {
      // Max error in interpolated form:             1.999e-34
      // Max error found at long double precision:   2.156186e-33

      static const T P[13] = {    
         -4.0545627381873738086704293881227365L,
         -4.70088348734699134347906176097717782L,
         -2.36921550900925512951976617607678789L,
         -0.684322583796369508367726293719322866L,
         -0.126026534540165129870721937592996324L,
         -0.015636903921778316147260572008619549L,
         -0.00135442294754728549644376325814460807L,
         -0.842793965853572134365031384646117061e-4L,
         -0.385602133791111663372015460784978351e-5L,
         -0.130458500394692067189883214401478539e-6L,
         -0.315861074947230418778143153383660035e-8L,
         -0.500334720512030826996373077844707164e-10L,
         -0.420204769185233365849253969097184005e-12L,
        };
      static const T Q[14] = {    
         1L,
         0.97663511666410096104783358493318814L,
         0.40878780231201806504987368939673249L,
         0.0963890666609396058945084107597727252L,
         0.0142207619090854604824116070866614505L,
         0.00139010220902667918476773423995750877L,
         0.940669540194694997889636696089994734e-4L,
         0.458220848507517004399292480807026602e-5L,
         0.16345521617741789012782420625435495e-6L,
         0.414007452533083304371566316901024114e-8L,
         0.68701473543366328016953742622661377e-10L,
         0.603461891080716585087883971886075863e-12L,
         0.294670713571839023181857795866134957e-16L,
         -0.147003914536437243143096875069813451e-18L,
        };
      result = tools::evaluate_polynomial(P, s - 6) / tools::evaluate_polynomial(Q, s - 6);
      result = 1 + exp(result);
   }
   else if(s < 17)
   {
      // Max error in interpolated form:             1.641e-32
      // Max error found at long double precision:   1.696121e-32
      static const T P[13] = {    
         -6.91319491921722925920883787894829678L,
         -3.65491257639481960248690596951049048L,
         -0.813557553449954526442644544105257881L,
         -0.0994317301685870959473658713841138083L,
         -0.00726896610245676520248617014211734906L,
         -0.000317253318715075854811266230916762929L,
         -0.66851422826636750855184211580127133e-5L,
         0.879464154730985406003332577806849971e-7L,
         0.113838903158254250631678791998294628e-7L,
         0.379184410304927316385211327537817583e-9L,
         0.612992858643904887150527613446403867e-11L,
         0.347873737198164757035457841688594788e-13L,
         -0.289187187441625868404494665572279364e-15L,
        };
      static const T Q[14] = {    
         1L,
         0.427310044448071818775721584949868806L,
         0.074602514873055756201435421385243062L,
         0.00688651562174480772901425121653945942L,
         0.000360174847635115036351323894321880445L,
         0.973556847713307543918865405758248777e-5L,
         -0.853455848314516117964634714780874197e-8L,
         -0.118203513654855112421673192194622826e-7L,
         -0.462521662511754117095006543363328159e-9L,
         -0.834212591919475633107355719369463143e-11L,
         -0.5354594751002702935740220218582929e-13L,
         0.406451690742991192964889603000756203e-15L,
         0.887948682401000153828241615760146728e-19L,
         -0.34980761098820347103967203948619072e-21L,
        };
      result = tools::evaluate_polynomial(P, s - 10) / tools::evaluate_polynomial(Q, s - 10);
      result = 1 + exp(result);
   }
   else if(s < 30)
   {
      // Max error in interpolated form:             1.563e-31
      // Max error found at long double precision:   1.562725e-31

      static const T P[13] = {    
         -11.7824798233959252791987402769438322L,
         -4.36131215284987731928174218354118102L,
         -0.732260980060982349410898496846972204L,
         -0.0744985185694913074484248803015717388L,
         -0.00517228281320594683022294996292250527L,
         -0.000260897206152101522569969046299309939L,
         -0.989553462123121764865178453128769948e-5L,
         -0.286916799741891410827712096608826167e-6L,
         -0.637262477796046963617949532211619729e-8L,
         -0.106796831465628373325491288787760494e-9L,
         -0.129343095511091870860498356205376823e-11L,
         -0.102397936697965977221267881716672084e-13L,
         -0.402663128248642002351627980255756363e-16L,
      };
      static const T Q[14] = {    
         1L,
         0.311288325355705609096155335186466508L,
         0.0438318468940415543546769437752132748L,
         0.00374396349183199548610264222242269536L,
         0.000218707451200585197339671707189281302L,
         0.927578767487930747532953583797351219e-5L,
         0.294145760625753561951137473484889639e-6L,
         0.704618586690874460082739479535985395e-8L,
         0.126333332872897336219649130062221257e-9L,
         0.16317315713773503718315435769352765e-11L,
         0.137846712823719515148344938160275695e-13L,
         0.580975420554224366450994232723910583e-16L,
         -0.291354445847552426900293580511392459e-22L,
         0.73614324724785855925025452085443636e-25L,
      };
      result = tools::evaluate_polynomial(P, s - 17) / tools::evaluate_polynomial(Q, s - 17);
      result = 1 + exp(result);
   }
   else if(s < 74)
   {
      // Max error in interpolated form:             2.311e-27
      // Max error found at long double precision:   2.297544e-27
      static const T P[14] = {    
         -20.7944102007844314586649688802236072L,
         -4.95759941987499442499908748130192187L,
         -0.563290752832461751889194629200298688L,
         -0.0406197001137935911912457120706122877L,
         -0.0020846534789473022216888863613422293L,
         -0.808095978462109173749395599401375667e-4L,
         -0.244706022206249301640890603610060959e-5L,
         -0.589477682919645930544382616501666572e-7L,
         -0.113699573675553496343617442433027672e-8L,
         -0.174767860183598149649901223128011828e-10L,
         -0.210051620306761367764549971980026474e-12L,
         -0.189187969537370950337212675466400599e-14L,
         -0.116313253429564048145641663778121898e-16L,
         -0.376708747782400769427057630528578187e-19L,
      };
      static const T Q[16] = {    
         1L,
         0.205076752981410805177554569784219717L,
         0.0202526722696670378999575738524540269L,
         0.001278305290005994980069466658219057L,
         0.576404779858501791742255670403304787e-4L,
         0.196477049872253010859712483984252067e-5L,
         0.521863830500876189501054079974475762e-7L,
         0.109524209196868135198775445228552059e-8L,
         0.181698713448644481083966260949267825e-10L,
         0.234793316975091282090312036524695562e-12L,
         0.227490441461460571047545264251399048e-14L,
         0.151500292036937400913870642638520668e-16L,
         0.543475775154780935815530649335936121e-19L,
         0.241647013434111434636554455083309352e-28L,
         -0.557103423021951053707162364713587374e-31L,
         0.618708773442584843384712258199645166e-34L,
      };
      result = tools::evaluate_polynomial(P, s - 30) / tools::evaluate_polynomial(Q, s - 30);
      result = 1 + exp(result);
   }
   else if(s < 117)
   {
      result = 1 + pow(T(2), -s);
   }
   else
   {
      result = 1;
   }
   return result;
}

template <class T, class Policy, class Tag>
T zeta_imp(T s, T sc, const Policy& pol, const Tag& tag)
{
   BOOST_MATH_STD_USING
   if(s == 1)
      return policies::raise_pole_error<T>(
         "boost::math::zeta<%1%>", 
         "Evaluation of zeta function at pole %1%", 
         s, pol);
   T result;
   if(s == 0)
   {
      result = -0.5;
   }
   else if(s < 0)
   {
      std::swap(s, sc);
      if(floor(sc/2) == sc/2)
         result = 0;
      else
      {
         result = boost::math::sin_pi(0.5f * sc, pol)
            * 2 * pow(2 * constants::pi<T>(), -s) 
            * boost::math::tgamma(s, pol) 
            * zeta_imp(s, sc, pol, tag);
      }
   }
   else
   {
      result = zeta_imp_prec(s, sc, pol, tag);
   }
   return result;
}

} // detail

template <class T, class Policy>
inline typename tools::promote_args<T>::type zeta(T s, const Policy&)
{
   typedef typename tools::promote_args<T>::type result_type;
   typedef typename policies::evaluation<result_type, Policy>::type value_type;
   typedef typename policies::precision<result_type, Policy>::type precision_type;
   typedef typename policies::normalise<
      Policy, 
      policies::promote_float<false>, 
      policies::promote_double<false>, 
      policies::discrete_quantile<>,
      policies::assert_undefined<> >::type forwarding_policy;
   typedef typename mpl::if_<
      mpl::less_equal<precision_type, mpl::int_<0> >,
      mpl::int_<0>,
      typename mpl::if_<
         mpl::less_equal<precision_type, mpl::int_<53> >,
         mpl::int_<53>,  // double
         typename mpl::if_<
            mpl::less_equal<precision_type, mpl::int_<64> >,
            mpl::int_<64>, // 80-bit long double
            typename mpl::if_<
               mpl::less_equal<precision_type, mpl::int_<113> >,
               mpl::int_<113>, // 128-bit long double
               mpl::int_<0> // too many bits, use generic version.
            >::type
         >::type
      >::type
   >::type tag_type;
   //typedef mpl::int_<0> tag_type;

   return policies::checked_narrowing_cast<result_type, forwarding_policy>(detail::zeta_imp(
      static_cast<value_type>(s),
      static_cast<value_type>(1 - static_cast<value_type>(s)),
      forwarding_policy(),
      tag_type()), "boost::math::zeta<%1%>(%1%)");
}

template <class T>
inline typename tools::promote_args<T>::type zeta(T s)
{
   return zeta(s, policies::policy<>());
}

}} // namespaces

#endif // BOOST_MATH_ZETA_HPP