text: //  Copyright John Maddock 2010, 2012.
  //  Copyright Paul A. Bristow 2011, 2012.
  
  //  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_CALCULATE_CONSTANTS_CONSTANTS_INCLUDED
  #define BOOST_MATH_CALCULATE_CONSTANTS_CONSTANTS_INCLUDED
  
  #include <boost/math/special_functions/trunc.hpp>
  
  namespace boost{ namespace math{ namespace constants{ namespace detail{
  
  template <class T>
  template<int N>
  inline T constant_pi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
  {
     BOOST_MATH_STD_USING
  
     return ldexp(acos(T(0)), 1);
  
     /*
     // Although this code works well, it's usually more accurate to just call acos
     // and access the number types own representation of PI which is usually calculated
     // at slightly higher precision...
  
     T result;
     T a = 1;
     T b;
     T A(a);
     T B = 0.5f;
     T D = 0.25f;
  
     T lim;
     lim = boost::math::tools::epsilon<T>();
  
     unsigned k = 1;
  
     do
     {
        result = A + B;
        result = ldexp(result, -2);
        b = sqrt(B);
        a += b;
        a = ldexp(a, -1);
        A = a * a;
        B = A - result;
        B = ldexp(B, 1);
        result = A - B;
        bool neg = boost::math::sign(result) < 0;
        if(neg)
           result = -result;
        if(result <= lim)
           break;
        if(neg)
           result = -result;
        result = ldexp(result, k - 1);
        D -= result;
        ++k;
        lim = ldexp(lim, 1);
     }
     while(true);
  
     result = B / D;
     return result;
     */
  }
  
  template <class T>
  template<int N>
  inline T constant_two_pi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
  {
     return 2 * pi<T, policies::policy<policies::digits2<N> > >();
  }
  
  template <class T> // 2 / pi
  template<int N>
  inline T constant_two_div_pi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
  {
     return 2 / pi<T, policies::policy<policies::digits2<N> > >();
  }
  
  template <class T>  // sqrt(2/pi)
  template <int N>
  inline T constant_root_two_div_pi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
  {
     BOOST_MATH_STD_USING
     return sqrt((2 / pi<T, policies::policy<policies::digits2<N> > >()));
  }
  
  template <class T>
  template<int N>
  inline T constant_one_div_two_pi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
  {
     return 1 / two_pi<T, policies::policy<policies::digits2<N> > >();
  }
  
  template <class T>
  template<int N>
  inline T constant_root_pi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
  {
     BOOST_MATH_STD_USING
     return sqrt(pi<T, policies::policy<policies::digits2<N> > >());
  }
  
  template <class T>
  template<int N>
  inline T constant_root_half_pi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
  {
     BOOST_MATH_STD_USING
     return sqrt(pi<T, policies::policy<policies::digits2<N> > >() / 2);
  }
  
  template <class T>
  template<int N>
  inline T constant_root_two_pi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
  {
     BOOST_MATH_STD_USING
     return sqrt(two_pi<T, policies::policy<policies::digits2<N> > >());
  }
  
  template <class T>
  template<int N>
  inline T constant_log_root_two_pi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
  {
     BOOST_MATH_STD_USING
     return log(root_two_pi<T, policies::policy<policies::digits2<N> > >());
  }
  
  template <class T>
  template<int N>
  inline T constant_root_ln_four<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
  {
     BOOST_MATH_STD_USING
     return sqrt(log(static_cast<T>(4)));
  }
  
  template <class T>
  template<int N>
  inline T constant_e<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
  {
     //
     // Although we can clearly calculate this from first principles, this hooks into
     // T's own notion of e, which hopefully will more accurate than one calculated to
     // a few epsilon:
     //
     BOOST_MATH_STD_USING
     return exp(static_cast<T>(1));
  }
  
  template <class T>
  template<int N>
  inline T constant_half<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
  {
     return static_cast<T>(1) / static_cast<T>(2);
  }
  
  template <class T>
  template<int M>
  inline T constant_euler<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<M>))
  {
     BOOST_MATH_STD_USING
     //
     // This is the method described in:
     // "Some New Algorithms for High-Precision Computation of Euler's Constant"
     // Richard P Brent and Edwin M McMillan.
     // Mathematics of Computation, Volume 34, Number 149, Jan 1980, pages 305-312.
     // See equation 17 with p = 2.
     //
     T n = 3 + (M ? (std::min)(M, tools::digits<T>()) : tools::digits<T>()) / 4;
     T lim = M ? ldexp(T(1), 1 - (std::min)(M, tools::digits<T>())) : tools::epsilon<T>();
     T lnn = log(n);
     T term = 1;
     T N = -lnn;
     T D = 1;
     T Hk = 0;
     T one = 1;
  
     for(unsigned k = 1;; ++k)
     {
        term *= n * n;
        term /= k * k;
        Hk += one / k;
        N += term * (Hk - lnn);
        D += term;
  
        if(term < D * lim)
           break;
     }
     return N / D;
  }
  
  template <class T>
  template<int N>
  inline T constant_euler_sqr<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
  {
    BOOST_MATH_STD_USING
    return euler<T, policies::policy<policies::digits2<N> > >()
       * euler<T, policies::policy<policies::digits2<N> > >();
  }
  
  template <class T>
  template<int N>
  inline T constant_one_div_euler<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
  {
    BOOST_MATH_STD_USING
    return static_cast<T>(1)
       / euler<T, policies::policy<policies::digits2<N> > >();
  }
  
  
  template <class T>
  template<int N>
  inline T constant_root_two<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
  {
     BOOST_MATH_STD_USING
     return sqrt(static_cast<T>(2));
  }
  
  
  template <class T>
  template<int N>
  inline T constant_root_three<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
  {
     BOOST_MATH_STD_USING
     return sqrt(static_cast<T>(3));
  }
  
  template <class T>
  template<int N>
  inline T constant_half_root_two<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
  {
     BOOST_MATH_STD_USING
     return sqrt(static_cast<T>(2)) / 2;
  }
  
  template <class T>
  template<int N>
  inline T constant_ln_two<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
  {
     //
     // Although there are good ways to calculate this from scratch, this hooks into
     // T's own notion of log(2) which will hopefully be accurate to the full precision
     // of T:
     //
     BOOST_MATH_STD_USING
     return log(static_cast<T>(2));
  }
  
  template <class T>
  template<int N>
  inline T constant_ln_ten<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
  {
     BOOST_MATH_STD_USING
     return log(static_cast<T>(10));
  }
  
  template <class T>
  template<int N>
  inline T constant_ln_ln_two<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
  {
     BOOST_MATH_STD_USING
     return log(log(static_cast<T>(2)));
  }
  
  template <class T>
  template<int N>
  inline T constant_third<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
  {
     BOOST_MATH_STD_USING
     return static_cast<T>(1) / static_cast<T>(3);
  }
  
  template <class T>
  template<int N>
  inline T constant_twothirds<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
  {
     BOOST_MATH_STD_USING
     return static_cast<T>(2) / static_cast<T>(3);
  }
  
  template <class T>
  template<int N>
  inline T constant_two_thirds<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
  {
     BOOST_MATH_STD_USING
     return static_cast<T>(2) / static_cast<T>(3);
  }
  
  template <class T>
  template<int N>
  inline T constant_three_quarters<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
  {
     BOOST_MATH_STD_USING
     return static_cast<T>(3) / static_cast<T>(4);
  }
  
  template <class T>
  template<int N>
  inline T constant_pi_minus_three<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
  {
     return pi<T, policies::policy<policies::digits2<N> > >() - static_cast<T>(3);
  }
  
  template <class T>
  template<int N>
  inline T constant_four_minus_pi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
  {
     return static_cast<T>(4) - pi<T, policies::policy<policies::digits2<N> > >();
  }
  
  //template <class T>
  //template<int N>
  //inline T constant_pow23_four_minus_pi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
  //{
  //   BOOST_MATH_STD_USING
  //   return pow(four_minus_pi<T, policies::policy<policies::digits2<N> > >(), static_cast<T>(1.5));
  //}
  
  template <class T>
  template<int N>
  inline T constant_exp_minus_half<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
  {
     BOOST_MATH_STD_USING
     return exp(static_cast<T>(-0.5));
  }
  
  // Pi
  template <class T>
  template<int N>
  inline T constant_one_div_root_two<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
  {
     return static_cast<T>(1) / root_two<T, policies::policy<policies::digits2<N> > >();
  }
  
  template <class T>
  template<int N>
  inline T constant_one_div_root_pi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
  {
     return static_cast<T>(1) / root_pi<T, policies::policy<policies::digits2<N> > >();
  }
  
  template <class T>
  template<int N>
  inline T constant_one_div_root_two_pi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
  {
     return static_cast<T>(1) / root_two_pi<T, policies::policy<policies::digits2<N> > >();
  }
  
  template <class T>
  template<int N>
  inline T constant_root_one_div_pi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
  {
     BOOST_MATH_STD_USING
     return sqrt(static_cast<T>(1) / pi<T, policies::policy<policies::digits2<N> > >());
  }
  
  
  template <class T>
  template<int N>
  inline T constant_four_thirds_pi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
  {
     BOOST_MATH_STD_USING
     return pi<T, policies::policy<policies::digits2<N> > >() * static_cast<T>(4) / static_cast<T>(3);
  }
  
  template <class T>
  template<int N>
  inline T constant_half_pi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
  {
     BOOST_MATH_STD_USING
     return pi<T, policies::policy<policies::digits2<N> > >()  / static_cast<T>(2);
  }
  
  
  template <class T>
  template<int N>
  inline T constant_third_pi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
  {
     BOOST_MATH_STD_USING
     return pi<T, policies::policy<policies::digits2<N> > >()  / static_cast<T>(3);
  }
  
  template <class T>
  template<int N>
  inline T constant_sixth_pi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
  {
     BOOST_MATH_STD_USING
     return pi<T, policies::policy<policies::digits2<N> > >()  / static_cast<T>(6);
  }
  
  template <class T>
  template<int N>
  inline T constant_two_thirds_pi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
  {
     BOOST_MATH_STD_USING
     return pi<T, policies::policy<policies::digits2<N> > >() * static_cast<T>(2) / static_cast<T>(3);
  }
  
  template <class T>
  template<int N>
  inline T constant_three_quarters_pi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
  {
     BOOST_MATH_STD_USING
     return pi<T, policies::policy<policies::digits2<N> > >() * static_cast<T>(3) / static_cast<T>(4);
  }
  
  template <class T>
  template<int N>
  inline T constant_pi_pow_e<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
  {
     BOOST_MATH_STD_USING
     return pow(pi<T, policies::policy<policies::digits2<N> > >(), e<T, policies::policy<policies::digits2<N> > >()); //
  }
  
  template <class T>
  template<int N>
  inline T constant_pi_sqr<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
  {
     BOOST_MATH_STD_USING
     return pi<T, policies::policy<policies::digits2<N> > >()
     *      pi<T, policies::policy<policies::digits2<N> > >() ; //
  }
  
  template <class T>
  template<int N>
  inline T constant_pi_sqr_div_six<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
  {
     BOOST_MATH_STD_USING
     return pi<T, policies::policy<policies::digits2<N> > >()
     *      pi<T, policies::policy<policies::digits2<N> > >()
     / static_cast<T>(6); //
  }
  
  
  template <class T>
  template<int N>
  inline T constant_pi_cubed<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
  {
     BOOST_MATH_STD_USING
     return pi<T, policies::policy<policies::digits2<N> > >()
     *      pi<T, policies::policy<policies::digits2<N> > >()
     *      pi<T, policies::policy<policies::digits2<N> > >()
     ; //
  }
  
  template <class T>
  template<int N>
  inline T constant_cbrt_pi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
  {
     BOOST_MATH_STD_USING
     return pow(pi<T, policies::policy<policies::digits2<N> > >(), static_cast<T>(1)/ static_cast<T>(3));
  }
  
  template <class T>
  template<int N>
  inline T constant_one_div_cbrt_pi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
  {
     BOOST_MATH_STD_USING
     return static_cast<T>(1)
     / pow(pi<T, policies::policy<policies::digits2<N> > >(), static_cast<T>(1)/ static_cast<T>(3));
  }
  
  // Euler's e
  
  template <class T>
  template<int N>
  inline T constant_e_pow_pi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
  {
     BOOST_MATH_STD_USING
     return pow(e<T, policies::policy<policies::digits2<N> > >(), pi<T, policies::policy<policies::digits2<N> > >()); //
  }
  
  template <class T>
  template<int N>
  inline T constant_root_e<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
  {
     BOOST_MATH_STD_USING
     return sqrt(e<T, policies::policy<policies::digits2<N> > >());
  }
  
  template <class T>
  template<int N>
  inline T constant_log10_e<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
  {
     BOOST_MATH_STD_USING
     return log10(e<T, policies::policy<policies::digits2<N> > >());
  }
  
  template <class T>
  template<int N>
  inline T constant_one_div_log10_e<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
  {
     BOOST_MATH_STD_USING
     return  static_cast<T>(1) /
       log10(e<T, policies::policy<policies::digits2<N> > >());
  }
  
  // Trigonometric
  
  template <class T>
  template<int N>
  inline T constant_degree<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
  {
     BOOST_MATH_STD_USING
     return pi<T, policies::policy<policies::digits2<N> > >()
     / static_cast<T>(180)
     ; //
  }
  
  template <class T>
  template<int N>
  inline T constant_radian<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
  {
     BOOST_MATH_STD_USING
     return static_cast<T>(180)
     / pi<T, policies::policy<policies::digits2<N> > >()
     ; //
  }
  
  template <class T>
  template<int N>
  inline T constant_sin_one<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
  {
     BOOST_MATH_STD_USING
     return sin(static_cast<T>(1)) ; //
  }
  
  template <class T>
  template<int N>
  inline T constant_cos_one<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
  {
     BOOST_MATH_STD_USING
     return cos(static_cast<T>(1)) ; //
  }
  
  template <class T>
  template<int N>
  inline T constant_sinh_one<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
  {
     BOOST_MATH_STD_USING
     return sinh(static_cast<T>(1)) ; //
  }
  
  template <class T>
  template<int N>
  inline T constant_cosh_one<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
  {
     BOOST_MATH_STD_USING
     return cosh(static_cast<T>(1)) ; //
  }
  
  template <class T>
  template<int N>
  inline T constant_phi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
  {
     BOOST_MATH_STD_USING
     return (static_cast<T>(1) + sqrt(static_cast<T>(5)) )/static_cast<T>(2) ; //
  }
  
  template <class T>
  template<int N>
  inline T constant_ln_phi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
  {
     BOOST_MATH_STD_USING
     //return  log(phi<T, policies::policy<policies::digits2<N> > >()); // ???
     return log((static_cast<T>(1) + sqrt(static_cast<T>(5)) )/static_cast<T>(2) );
  }
  template <class T>
  template<int N>
  inline T constant_one_div_ln_phi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
  {
     BOOST_MATH_STD_USING
     return static_cast<T>(1) /
       log((static_cast<T>(1) + sqrt(static_cast<T>(5)) )/static_cast<T>(2) );
  }
  
  // Zeta
  
  template <class T>
  template<int N>
  inline T constant_zeta_two<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
  {
     BOOST_MATH_STD_USING
  
       return pi<T, policies::policy<policies::digits2<N> > >()
       *  pi<T, policies::policy<policies::digits2<N> > >()
       /static_cast<T>(6);
  }
  
  template <class T>
  template<int N>
  inline T constant_zeta_three<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
  {
     // http://mathworld.wolfram.com/AperysConstant.html
     // http://en.wikipedia.org/wiki/Mathematical_constant
  
     // http://oeis.org/A002117/constant
     //T zeta3("1.20205690315959428539973816151144999076"
     // "4986292340498881792271555341838205786313"
     // "09018645587360933525814619915");
  
    //"1.202056903159594285399738161511449990, 76498629234049888179227155534183820578631309018645587360933525814619915"  A002117
    // 1.202056903159594285399738161511449990, 76498629234049888179227155534183820578631309018645587360933525814619915780, +00);
    //"1.2020569031595942 double
    // http://www.spaennare.se/SSPROG/ssnum.pdf  // section 11, Algorithm for Apery's constant zeta(3).
    // Programs to Calculate some Mathematical Constants to Large Precision, Document Version 1.50
  
    // by Stefan Spannare  September 19, 2007
    // zeta(3) = 1/64 * sum
     BOOST_MATH_STD_USING
     T n_fact=static_cast<T>(1); // build n! for n = 0.
     T sum = static_cast<double>(77); // Start with n = 0 case.
     // for n = 0, (77/1) /64 = 1.203125
     //double lim = std::numeric_limits<double>::epsilon();
     T lim = N ? ldexp(T(1), 1 - (std::min)(N, tools::digits<T>())) : tools::epsilon<T>();
     for(unsigned int n = 1; n < 40; ++n)
     { // three to five decimal digits per term, so 40 should be plenty for 100 decimal digits.
        //cout << "n = " << n << endl;
        n_fact *= n; // n!
        T n_fact_p10 = n_fact * n_fact * n_fact * n_fact * n_fact * n_fact * n_fact * n_fact * n_fact * n_fact; // (n!)^10
        T num = ((205 * n * n) + (250 * n) + 77) * n_fact_p10; // 205n^2 + 250n + 77
        // int nn = (2 * n + 1);
        // T d = factorial(nn); // inline factorial.
        T d = 1;
        for(unsigned int i = 1; i <= (n+n + 1); ++i) // (2n + 1)
        {
          d *= i;
        }
        T den = d * d * d * d * d; // [(2n+1)!]^5
        //cout << "den = " << den << endl;
        T term = num/den;
        if (n % 2 != 0)
        { //term *= -1;
          sum -= term;
        }
        else
        {
          sum += term;
        }
        //cout << "term = " << term << endl;
        //cout << "sum/64  = " << sum/64 << endl;
        if(abs(term) < lim)
        {
           break;
        }
     }
     return sum / 64;
  }
  
  template <class T>
  template<int N>
  inline T constant_catalan<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
  { // http://oeis.org/A006752/constant
    //T c("0.915965594177219015054603514932384110774"
   //"149374281672134266498119621763019776254769479356512926115106248574");
  
    // 9.159655941772190150546035149323841107, 74149374281672134266498119621763019776254769479356512926115106248574422619, -01);
  
     // This is equation (entry) 31 from
     // http://www-2.cs.cmu.edu/~adamchik/articles/catalan/catalan.htm
     // See also http://www.mpfr.org/algorithms.pdf
     BOOST_MATH_STD_USING
     T k_fact = 1;
     T tk_fact = 1;
     T sum = 1;
     T term;
     T lim = N ? ldexp(T(1), 1 - (std::min)(N, tools::digits<T>())) : tools::epsilon<T>();
  
     for(unsigned k = 1;; ++k)
     {
        k_fact *= k;
        tk_fact *= (2 * k) * (2 * k - 1);
        term = k_fact * k_fact / (tk_fact * (2 * k + 1) * (2 * k + 1));
        sum += term;
        if(term < lim)
        {
           break;
        }
     }
     return boost::math::constants::pi<T, boost::math::policies::policy<> >()
        * log(2 + boost::math::constants::root_three<T, boost::math::policies::policy<> >())
         / 8
        + 3 * sum / 8;
  }
  
  namespace khinchin_detail{
  
  template <class T>
  T zeta_polynomial_series(T s, T sc, int digits)
  {
     BOOST_MATH_STD_USING
     //
     // This is algorithm 3 from:
     //
     // "An Efficient Algorithm for the Riemann Zeta Function", P. Borwein,
     // Canadian Mathematical Society, Conference Proceedings, 2000.
     // See: http://www.cecm.sfu.ca/personal/pborwein/PAPERS/P155.pdf
     //
     BOOST_MATH_STD_USING
     int n = (digits * 19) / 53;
     T sum = 0;
     T two_n = ldexp(T(1), n);
     int ej_sign = 1;
     for(int j = 0; j < n; ++j)
     {
        sum += ej_sign * -two_n / pow(T(j + 1), s);
        ej_sign = -ej_sign;
     }
     T ej_sum = 1;
     T ej_term = 1;
     for(int j = n; j <= 2 * n - 1; ++j)
     {
        sum += ej_sign * (ej_sum - two_n) / pow(T(j + 1), s);
        ej_sign = -ej_sign;
        ej_term *= 2 * n - j;
        ej_term /= j - n + 1;
        ej_sum += ej_term;
     }
     return -sum / (two_n * (1 - pow(T(2), sc)));
  }
  
  template <class T>
  T khinchin(int digits)
  {
     BOOST_MATH_STD_USING
     T sum = 0;
     T term;
     T lim = ldexp(T(1), 1-digits);
     T factor = 0;
     unsigned last_k = 1;
     T num = 1;
     for(unsigned n = 1;; ++n)
     {
        for(unsigned k = last_k; k <= 2 * n - 1; ++k)
        {
           factor += num / k;
           num = -num;
        }
        last_k = 2 * n;
        term = (zeta_polynomial_series(T(2 * n), T(1 - T(2 * n)), digits) - 1) * factor / n;
        sum += term;
        if(term < lim)
           break;
     }
     return exp(sum / boost::math::constants::ln_two<T, boost::math::policies::policy<> >());
  }
  
  }
  
  template <class T>
  template<int N>
  inline T constant_khinchin<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
  {
     int n = N ? (std::min)(N, tools::digits<T>()) : tools::digits<T>();
     return khinchin_detail::khinchin<T>(n);
  }
  
  template <class T>
  template<int N>
  inline T constant_extreme_value_skewness<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
  { // from e_float constants.cpp
    // Mathematica: N[12 Sqrt[6]  Zeta[3]/Pi^3, 1101]
     BOOST_MATH_STD_USING
     T ev(12 * sqrt(static_cast<T>(6)) * zeta_three<T, policies::policy<policies::digits2<N> > >()
      / pi_cubed<T, policies::policy<policies::digits2<N> > >() );
  
  //T ev(
  //"1.1395470994046486574927930193898461120875997958365518247216557100852480077060706857071875468869385150"
  //"1894272048688553376986765366075828644841024041679714157616857834895702411080704529137366329462558680"
  //"2015498788776135705587959418756809080074611906006528647805347822929577145038743873949415294942796280"
  //"0895597703063466053535550338267721294164578901640163603544404938283861127819804918174973533694090594"
  //"3094963822672055237678432023017824416203652657301470473548274848068762500300316769691474974950757965"
  //"8640779777748741897542093874605477776538884083378029488863880220988107155275203245233994097178778984"
  //"3488995668362387892097897322246698071290011857605809901090220903955815127463328974447572119951192970"
  //"3684453635456559086126406960279692862247058250100678008419431185138019869693206366891639436908462809"
  //"9756051372711251054914491837034685476095423926553367264355374652153595857163724698198860485357368964"
  //"3807049634423621246870868566707915720704996296083373077647528285782964567312903914752617978405994377"
  //"9064157147206717895272199736902453130842229559980076472936976287378945035706933650987259357729800315");
  
    return ev;
  }
  
  namespace detail{
  //
  // Calculation of the Glaisher constant depends upon calculating the
  // derivative of the zeta function at 2, we can then use the relation:
  // zeta'(2) = 1/6 pi^2 [euler + ln(2pi)-12ln(A)]
  // To get the constant A.
  // See equation 45 at http://mathworld.wolfram.com/RiemannZetaFunction.html.
  //
  // The derivative of the zeta function is computed by direct differentiation
  // of the relation:
  // (1-2^(1-s))zeta(s) = SUM(n=0, INF){ (-n)^n / (n+1)^s  }
  // Which gives us 2 slowly converging but alternating sums to compute,
  // for this we use Algorithm 1 from "Convergent Acceleration of Alternating Series",
  // Henri Cohen, Fernando Rodriguez Villegas and Don Zagier, Experimental Mathematics 9:1 (1999).
  // See http://www.math.utexas.edu/users/villegas/publications/conv-accel.pdf
  //
  template <class T>
  T zeta_series_derivative_2(unsigned digits)
  {
     // Derivative of the series part, evaluated at 2:
     BOOST_MATH_STD_USING
     int n = digits * 301 * 13 / 10000;
     boost::math::itrunc((std::numeric_limits<T>::digits10 + 1) * 1.3);
     T d = pow(3 + sqrt(T(8)), n);
     d = (d + 1 / d) / 2;
     T b = -1;
     T c = -d;
     T s = 0;
     for(int k = 0; k < n; ++k)
     {
        T a = -log(T(k+1)) / ((k+1) * (k+1));
        c = b - c;
        s = s + c * a;
        b = (k + n) * (k - n) * b / ((k + T(0.5f)) * (k + 1));
     }
     return s / d;
  }
  
  template <class T>
  T zeta_series_2(unsigned digits)
  {
     // Series part of zeta at 2:
     BOOST_MATH_STD_USING
     int n = digits * 301 * 13 / 10000;
     T d = pow(3 + sqrt(T(8)), n);
     d = (d + 1 / d) / 2;
     T b = -1;
     T c = -d;
     T s = 0;
     for(int k = 0; k < n; ++k)
     {
        T a = T(1) / ((k + 1) * (k + 1));
        c = b - c;
        s = s + c * a;
        b = (k + n) * (k - n) * b / ((k + T(0.5f)) * (k + 1));
     }
     return s / d;
  }
  
  template <class T>
  inline T zeta_series_lead_2()
  {
     // lead part at 2:
     return 2;
  }
  
  template <class T>
  inline T zeta_series_derivative_lead_2()
  {
     // derivative of lead part at 2:
     return -2 * boost::math::constants::ln_two<T>();
  }
  
  template <class T>
  inline T zeta_derivative_2(unsigned n)
  {
     // zeta derivative at 2:
     return zeta_series_derivative_2<T>(n) * zeta_series_lead_2<T>()
        + zeta_series_derivative_lead_2<T>() * zeta_series_2<T>(n);
  }
  
  }  // namespace detail
  
  template <class T>
  template<int N>
  inline T constant_glaisher<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
  {
  
     BOOST_MATH_STD_USING
     typedef policies::policy<policies::digits2<N> > forwarding_policy;
     int n = N ? (std::min)(N, tools::digits<T>()) : tools::digits<T>();
     T v = detail::zeta_derivative_2<T>(n);
     v *= 6;
     v /= boost::math::constants::pi<T, forwarding_policy>() * boost::math::constants::pi<T, forwarding_policy>();
     v -= boost::math::constants::euler<T, forwarding_policy>();
     v -= log(2 * boost::math::constants::pi<T, forwarding_policy>());
     v /= -12;
     return exp(v);
  
   /*
     // from http://mpmath.googlecode.com/svn/data/glaisher.txt
       // 20,000 digits of the Glaisher-Kinkelin constant A = exp(1/2 - zeta'(-1))
       // Computed using A = exp((6 (-zeta'(2))/pi^2 + log 2 pi + gamma)/12)
    // with Euler-Maclaurin summation for zeta'(2).
    T g(
    "1.282427129100622636875342568869791727767688927325001192063740021740406308858826"
    "46112973649195820237439420646120399000748933157791362775280404159072573861727522"
    "14334327143439787335067915257366856907876561146686449997784962754518174312394652"
    "76128213808180219264516851546143919901083573730703504903888123418813674978133050"
    "93770833682222494115874837348064399978830070125567001286994157705432053927585405"
    "81731588155481762970384743250467775147374600031616023046613296342991558095879293"
    "36343887288701988953460725233184702489001091776941712153569193674967261270398013"
    "52652668868978218897401729375840750167472114895288815996668743164513890306962645"
    "59870469543740253099606800842447417554061490189444139386196089129682173528798629"
    "88434220366989900606980888785849587494085307347117090132667567503310523405221054"
    "14176776156308191919997185237047761312315374135304725819814797451761027540834943"
    "14384965234139453373065832325673954957601692256427736926358821692159870775858274"
    "69575162841550648585890834128227556209547002918593263079373376942077522290940187");
  
    return g;
    */
  }
  
  template <class T>
  template<int N>
  inline T constant_rayleigh_skewness<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
  {  // From e_float
    // 1100 digits of the Rayleigh distribution skewness
    // Mathematica: N[2 Sqrt[Pi] (Pi - 3)/((4 - Pi)^(3/2)), 1100]
  
     BOOST_MATH_STD_USING
     T rs(2 * root_pi<T, policies::policy<policies::digits2<N> > >()
        * pi_minus_three<T, policies::policy<policies::digits2<N> > >()
        / pow(four_minus_pi<T, policies::policy<policies::digits2<N> > >(), static_cast<T>(3./2))
        );
   //   6.31110657818937138191899351544227779844042203134719497658094585692926819617473725459905027032537306794400047264,
  
    //"0.6311106578189371381918993515442277798440422031347194976580945856929268196174737254599050270325373067"
    //"9440004726436754739597525250317640394102954301685809920213808351450851396781817932734836994829371322"
    //"5797376021347531983451654130317032832308462278373358624120822253764532674177325950686466133508511968"
    //"2389168716630349407238090652663422922072397393006683401992961569208109477307776249225072042971818671"
    //"4058887072693437217879039875871765635655476241624825389439481561152126886932506682176611183750503553"
    //"1218982627032068396407180216351425758181396562859085306247387212297187006230007438534686340210168288"
    //"8956816965453815849613622117088096547521391672977226658826566757207615552041767516828171274858145957"
    //"6137539156656005855905288420585194082284972984285863898582313048515484073396332610565441264220790791"
    //"0194897267890422924599776483890102027823328602965235306539844007677157873140562950510028206251529523"
    //"7428049693650605954398446899724157486062545281504433364675815915402937209673727753199567661561209251"
    //"4695589950526053470201635372590001578503476490223746511106018091907936826431407434894024396366284848");  ;
    return rs;
  }
  
  template <class T>
  template<int N>
  inline T constant_rayleigh_kurtosis_excess<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
  { // - (6 Pi^2 - 24 Pi + 16)/((Pi - 4)^2)
      // Might provide and calculate this using pi_minus_four.
     BOOST_MATH_STD_USING
     return - (((static_cast<T>(6) * pi<T, policies::policy<policies::digits2<N> > >()
          * pi<T, policies::policy<policies::digits2<N> > >())
     - (static_cast<T>(24) * pi<T, policies::policy<policies::digits2<N> > >()) + static_cast<T>(16) )
     /
     ((pi<T, policies::policy<policies::digits2<N> > >() - static_cast<T>(4))
     * (pi<T, policies::policy<policies::digits2<N> > >() - static_cast<T>(4)))
     );
  }
  
  template <class T>
  template<int N>
  inline T constant_rayleigh_kurtosis<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
  { // 3 - (6 Pi^2 - 24 Pi + 16)/((Pi - 4)^2)
    // Might provide and calculate this using pi_minus_four.
     BOOST_MATH_STD_USING
     return static_cast<T>(3) - (((static_cast<T>(6) * pi<T, policies::policy<policies::digits2<N> > >()
          * pi<T, policies::policy<policies::digits2<N> > >())
     - (static_cast<T>(24) * pi<T, policies::policy<policies::digits2<N> > >()) + static_cast<T>(16) )
     /
     ((pi<T, policies::policy<policies::digits2<N> > >() - static_cast<T>(4))
     * (pi<T, policies::policy<policies::digits2<N> > >() - static_cast<T>(4)))
     );
  }
  
  }}}} // namespaces
  
  #endif // BOOST_MATH_CALCULATE_CONSTANTS_CONSTANTS_INCLUDED
  