Chris@16: //  Copyright John Maddock 2010, 2012.
Chris@16: //  Copyright Paul A. Bristow 2011, 2012.
Chris@16: 
Chris@16: //  Use, modification and distribution are subject to the
Chris@16: //  Boost Software License, Version 1.0. (See accompanying file
Chris@16: //  LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)
Chris@16: 
Chris@16: #ifndef BOOST_MATH_CALCULATE_CONSTANTS_CONSTANTS_INCLUDED
Chris@16: #define BOOST_MATH_CALCULATE_CONSTANTS_CONSTANTS_INCLUDED
Chris@16: 
Chris@16: #include <boost/math/special_functions/trunc.hpp>
Chris@16: 
Chris@16: namespace boost{ namespace math{ namespace constants{ namespace detail{
Chris@16: 
Chris@16: template <class T>
Chris@16: template<int N>
Chris@16: inline T constant_pi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
Chris@16: {
Chris@16:    BOOST_MATH_STD_USING
Chris@16: 
Chris@16:    return ldexp(acos(T(0)), 1);
Chris@16: 
Chris@16:    /*
Chris@16:    // Although this code works well, it's usually more accurate to just call acos
Chris@16:    // and access the number types own representation of PI which is usually calculated
Chris@16:    // at slightly higher precision...
Chris@16: 
Chris@16:    T result;
Chris@16:    T a = 1;
Chris@16:    T b;
Chris@16:    T A(a);
Chris@16:    T B = 0.5f;
Chris@16:    T D = 0.25f;
Chris@16: 
Chris@16:    T lim;
Chris@16:    lim = boost::math::tools::epsilon<T>();
Chris@16: 
Chris@16:    unsigned k = 1;
Chris@16: 
Chris@16:    do
Chris@16:    {
Chris@16:       result = A + B;
Chris@16:       result = ldexp(result, -2);
Chris@16:       b = sqrt(B);
Chris@16:       a += b;
Chris@16:       a = ldexp(a, -1);
Chris@16:       A = a * a;
Chris@16:       B = A - result;
Chris@16:       B = ldexp(B, 1);
Chris@16:       result = A - B;
Chris@16:       bool neg = boost::math::sign(result) < 0;
Chris@16:       if(neg)
Chris@16:          result = -result;
Chris@16:       if(result <= lim)
Chris@16:          break;
Chris@16:       if(neg)
Chris@16:          result = -result;
Chris@16:       result = ldexp(result, k - 1);
Chris@16:       D -= result;
Chris@16:       ++k;
Chris@16:       lim = ldexp(lim, 1);
Chris@16:    }
Chris@16:    while(true);
Chris@16: 
Chris@16:    result = B / D;
Chris@16:    return result;
Chris@16:    */
Chris@16: }
Chris@16: 
Chris@16: template <class T>
Chris@16: template<int N>
Chris@16: inline T constant_two_pi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
Chris@16: {
Chris@16:    return 2 * pi<T, policies::policy<policies::digits2<N> > >();
Chris@16: }
Chris@16: 
Chris@16: template <class T> // 2 / pi
Chris@16: template<int N>
Chris@16: inline T constant_two_div_pi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
Chris@16: {
Chris@16:    return 2 / pi<T, policies::policy<policies::digits2<N> > >();
Chris@16: }
Chris@16: 
Chris@16: template <class T>  // sqrt(2/pi)
Chris@16: template <int N>
Chris@16: inline T constant_root_two_div_pi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
Chris@16: {
Chris@16:    BOOST_MATH_STD_USING
Chris@16:    return sqrt((2 / pi<T, policies::policy<policies::digits2<N> > >()));
Chris@16: }
Chris@16: 
Chris@16: template <class T>
Chris@16: template<int N>
Chris@16: inline T constant_one_div_two_pi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
Chris@16: {
Chris@16:    return 1 / two_pi<T, policies::policy<policies::digits2<N> > >();
Chris@16: }
Chris@16: 
Chris@16: template <class T>
Chris@16: template<int N>
Chris@16: inline T constant_root_pi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
Chris@16: {
Chris@16:    BOOST_MATH_STD_USING
Chris@16:    return sqrt(pi<T, policies::policy<policies::digits2<N> > >());
Chris@16: }
Chris@16: 
Chris@16: template <class T>
Chris@16: template<int N>
Chris@16: inline T constant_root_half_pi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
Chris@16: {
Chris@16:    BOOST_MATH_STD_USING
Chris@16:    return sqrt(pi<T, policies::policy<policies::digits2<N> > >() / 2);
Chris@16: }
Chris@16: 
Chris@16: template <class T>
Chris@16: template<int N>
Chris@16: inline T constant_root_two_pi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
Chris@16: {
Chris@16:    BOOST_MATH_STD_USING
Chris@16:    return sqrt(two_pi<T, policies::policy<policies::digits2<N> > >());
Chris@16: }
Chris@16: 
Chris@16: template <class T>
Chris@16: template<int N>
Chris@16: inline T constant_log_root_two_pi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
Chris@16: {
Chris@16:    BOOST_MATH_STD_USING
Chris@16:    return log(root_two_pi<T, policies::policy<policies::digits2<N> > >());
Chris@16: }
Chris@16: 
Chris@16: template <class T>
Chris@16: template<int N>
Chris@16: inline T constant_root_ln_four<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
Chris@16: {
Chris@16:    BOOST_MATH_STD_USING
Chris@16:    return sqrt(log(static_cast<T>(4)));
Chris@16: }
Chris@16: 
Chris@16: template <class T>
Chris@16: template<int N>
Chris@16: inline T constant_e<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
Chris@16: {
Chris@16:    //
Chris@16:    // Although we can clearly calculate this from first principles, this hooks into
Chris@16:    // T's own notion of e, which hopefully will more accurate than one calculated to
Chris@16:    // a few epsilon:
Chris@16:    //
Chris@16:    BOOST_MATH_STD_USING
Chris@16:    return exp(static_cast<T>(1));
Chris@16: }
Chris@16: 
Chris@16: template <class T>
Chris@16: template<int N>
Chris@16: inline T constant_half<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
Chris@16: {
Chris@16:    return static_cast<T>(1) / static_cast<T>(2);
Chris@16: }
Chris@16: 
Chris@16: template <class T>
Chris@16: template<int M>
Chris@16: inline T constant_euler<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<M>))
Chris@16: {
Chris@16:    BOOST_MATH_STD_USING
Chris@16:    //
Chris@16:    // This is the method described in:
Chris@16:    // "Some New Algorithms for High-Precision Computation of Euler's Constant"
Chris@16:    // Richard P Brent and Edwin M McMillan.
Chris@16:    // Mathematics of Computation, Volume 34, Number 149, Jan 1980, pages 305-312.
Chris@16:    // See equation 17 with p = 2.
Chris@16:    //
Chris@16:    T n = 3 + (M ? (std::min)(M, tools::digits<T>()) : tools::digits<T>()) / 4;
Chris@16:    T lim = M ? ldexp(T(1), 1 - (std::min)(M, tools::digits<T>())) : tools::epsilon<T>();
Chris@16:    T lnn = log(n);
Chris@16:    T term = 1;
Chris@16:    T N = -lnn;
Chris@16:    T D = 1;
Chris@16:    T Hk = 0;
Chris@16:    T one = 1;
Chris@16: 
Chris@16:    for(unsigned k = 1;; ++k)
Chris@16:    {
Chris@16:       term *= n * n;
Chris@16:       term /= k * k;
Chris@16:       Hk += one / k;
Chris@16:       N += term * (Hk - lnn);
Chris@16:       D += term;
Chris@16: 
Chris@16:       if(term < D * lim)
Chris@16:          break;
Chris@16:    }
Chris@16:    return N / D;
Chris@16: }
Chris@16: 
Chris@16: template <class T>
Chris@16: template<int N>
Chris@16: inline T constant_euler_sqr<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
Chris@16: {
Chris@16:   BOOST_MATH_STD_USING
Chris@16:   return euler<T, policies::policy<policies::digits2<N> > >()
Chris@16:      * euler<T, policies::policy<policies::digits2<N> > >();
Chris@16: }
Chris@16: 
Chris@16: template <class T>
Chris@16: template<int N>
Chris@16: inline T constant_one_div_euler<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
Chris@16: {
Chris@16:   BOOST_MATH_STD_USING
Chris@16:   return static_cast<T>(1)
Chris@16:      / euler<T, policies::policy<policies::digits2<N> > >();
Chris@16: }
Chris@16: 
Chris@16: 
Chris@16: template <class T>
Chris@16: template<int N>
Chris@16: inline T constant_root_two<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
Chris@16: {
Chris@16:    BOOST_MATH_STD_USING
Chris@16:    return sqrt(static_cast<T>(2));
Chris@16: }
Chris@16: 
Chris@16: 
Chris@16: template <class T>
Chris@16: template<int N>
Chris@16: inline T constant_root_three<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
Chris@16: {
Chris@16:    BOOST_MATH_STD_USING
Chris@16:    return sqrt(static_cast<T>(3));
Chris@16: }
Chris@16: 
Chris@16: template <class T>
Chris@16: template<int N>
Chris@16: inline T constant_half_root_two<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
Chris@16: {
Chris@16:    BOOST_MATH_STD_USING
Chris@16:    return sqrt(static_cast<T>(2)) / 2;
Chris@16: }
Chris@16: 
Chris@16: template <class T>
Chris@16: template<int N>
Chris@16: inline T constant_ln_two<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
Chris@16: {
Chris@16:    //
Chris@16:    // Although there are good ways to calculate this from scratch, this hooks into
Chris@16:    // T's own notion of log(2) which will hopefully be accurate to the full precision
Chris@16:    // of T:
Chris@16:    //
Chris@16:    BOOST_MATH_STD_USING
Chris@16:    return log(static_cast<T>(2));
Chris@16: }
Chris@16: 
Chris@16: template <class T>
Chris@16: template<int N>
Chris@16: inline T constant_ln_ten<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
Chris@16: {
Chris@16:    BOOST_MATH_STD_USING
Chris@16:    return log(static_cast<T>(10));
Chris@16: }
Chris@16: 
Chris@16: template <class T>
Chris@16: template<int N>
Chris@16: inline T constant_ln_ln_two<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
Chris@16: {
Chris@16:    BOOST_MATH_STD_USING
Chris@16:    return log(log(static_cast<T>(2)));
Chris@16: }
Chris@16: 
Chris@16: template <class T>
Chris@16: template<int N>
Chris@16: inline T constant_third<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
Chris@16: {
Chris@16:    BOOST_MATH_STD_USING
Chris@16:    return static_cast<T>(1) / static_cast<T>(3);
Chris@16: }
Chris@16: 
Chris@16: template <class T>
Chris@16: template<int N>
Chris@16: inline T constant_twothirds<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
Chris@16: {
Chris@16:    BOOST_MATH_STD_USING
Chris@16:    return static_cast<T>(2) / static_cast<T>(3);
Chris@16: }
Chris@16: 
Chris@16: template <class T>
Chris@16: template<int N>
Chris@16: inline T constant_two_thirds<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
Chris@16: {
Chris@16:    BOOST_MATH_STD_USING
Chris@16:    return static_cast<T>(2) / static_cast<T>(3);
Chris@16: }
Chris@16: 
Chris@16: template <class T>
Chris@16: template<int N>
Chris@16: inline T constant_three_quarters<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
Chris@16: {
Chris@16:    BOOST_MATH_STD_USING
Chris@16:    return static_cast<T>(3) / static_cast<T>(4);
Chris@16: }
Chris@16: 
Chris@16: template <class T>
Chris@16: template<int N>
Chris@16: inline T constant_pi_minus_three<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
Chris@16: {
Chris@16:    return pi<T, policies::policy<policies::digits2<N> > >() - static_cast<T>(3);
Chris@16: }
Chris@16: 
Chris@16: template <class T>
Chris@16: template<int N>
Chris@16: inline T constant_four_minus_pi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
Chris@16: {
Chris@16:    return static_cast<T>(4) - pi<T, policies::policy<policies::digits2<N> > >();
Chris@16: }
Chris@16: 
Chris@101: //template <class T>
Chris@101: //template<int N>
Chris@101: //inline T constant_pow23_four_minus_pi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
Chris@101: //{
Chris@101: //   BOOST_MATH_STD_USING
Chris@101: //   return pow(four_minus_pi<T, policies::policy<policies::digits2<N> > >(), static_cast<T>(1.5));
Chris@101: //}
Chris@16: 
Chris@16: template <class T>
Chris@16: template<int N>
Chris@16: inline T constant_exp_minus_half<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
Chris@16: {
Chris@16:    BOOST_MATH_STD_USING
Chris@16:    return exp(static_cast<T>(-0.5));
Chris@16: }
Chris@16: 
Chris@16: // Pi
Chris@16: template <class T>
Chris@16: template<int N>
Chris@16: inline T constant_one_div_root_two<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
Chris@16: {
Chris@16:    return static_cast<T>(1) / root_two<T, policies::policy<policies::digits2<N> > >();
Chris@16: }
Chris@16: 
Chris@16: template <class T>
Chris@16: template<int N>
Chris@16: inline T constant_one_div_root_pi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
Chris@16: {
Chris@16:    return static_cast<T>(1) / root_pi<T, policies::policy<policies::digits2<N> > >();
Chris@16: }
Chris@16: 
Chris@16: template <class T>
Chris@16: template<int N>
Chris@16: inline T constant_one_div_root_two_pi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
Chris@16: {
Chris@16:    return static_cast<T>(1) / root_two_pi<T, policies::policy<policies::digits2<N> > >();
Chris@16: }
Chris@16: 
Chris@16: template <class T>
Chris@16: template<int N>
Chris@16: inline T constant_root_one_div_pi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
Chris@16: {
Chris@16:    BOOST_MATH_STD_USING
Chris@16:    return sqrt(static_cast<T>(1) / pi<T, policies::policy<policies::digits2<N> > >());
Chris@16: }
Chris@16: 
Chris@16: 
Chris@16: template <class T>
Chris@16: template<int N>
Chris@16: inline T constant_four_thirds_pi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
Chris@16: {
Chris@16:    BOOST_MATH_STD_USING
Chris@16:    return pi<T, policies::policy<policies::digits2<N> > >() * static_cast<T>(4) / static_cast<T>(3);
Chris@16: }
Chris@16: 
Chris@16: template <class T>
Chris@16: template<int N>
Chris@16: inline T constant_half_pi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
Chris@16: {
Chris@16:    BOOST_MATH_STD_USING
Chris@16:    return pi<T, policies::policy<policies::digits2<N> > >()  / static_cast<T>(2);
Chris@16: }
Chris@16: 
Chris@16: 
Chris@16: template <class T>
Chris@16: template<int N>
Chris@16: inline T constant_third_pi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
Chris@16: {
Chris@16:    BOOST_MATH_STD_USING
Chris@16:    return pi<T, policies::policy<policies::digits2<N> > >()  / static_cast<T>(3);
Chris@16: }
Chris@16: 
Chris@16: template <class T>
Chris@16: template<int N>
Chris@16: inline T constant_sixth_pi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
Chris@16: {
Chris@16:    BOOST_MATH_STD_USING
Chris@16:    return pi<T, policies::policy<policies::digits2<N> > >()  / static_cast<T>(6);
Chris@16: }
Chris@16: 
Chris@16: template <class T>
Chris@16: template<int N>
Chris@16: inline T constant_two_thirds_pi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
Chris@16: {
Chris@16:    BOOST_MATH_STD_USING
Chris@16:    return pi<T, policies::policy<policies::digits2<N> > >() * static_cast<T>(2) / static_cast<T>(3);
Chris@16: }
Chris@16: 
Chris@16: template <class T>
Chris@16: template<int N>
Chris@16: inline T constant_three_quarters_pi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
Chris@16: {
Chris@16:    BOOST_MATH_STD_USING
Chris@16:    return pi<T, policies::policy<policies::digits2<N> > >() * static_cast<T>(3) / static_cast<T>(4);
Chris@16: }
Chris@16: 
Chris@16: template <class T>
Chris@16: template<int N>
Chris@16: inline T constant_pi_pow_e<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
Chris@16: {
Chris@16:    BOOST_MATH_STD_USING
Chris@16:    return pow(pi<T, policies::policy<policies::digits2<N> > >(), e<T, policies::policy<policies::digits2<N> > >()); //
Chris@16: }
Chris@16: 
Chris@16: template <class T>
Chris@16: template<int N>
Chris@16: inline T constant_pi_sqr<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
Chris@16: {
Chris@16:    BOOST_MATH_STD_USING
Chris@16:    return pi<T, policies::policy<policies::digits2<N> > >()
Chris@16:    *      pi<T, policies::policy<policies::digits2<N> > >() ; //
Chris@16: }
Chris@16: 
Chris@16: template <class T>
Chris@16: template<int N>
Chris@16: inline T constant_pi_sqr_div_six<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
Chris@16: {
Chris@16:    BOOST_MATH_STD_USING
Chris@16:    return pi<T, policies::policy<policies::digits2<N> > >()
Chris@16:    *      pi<T, policies::policy<policies::digits2<N> > >()
Chris@16:    / static_cast<T>(6); //
Chris@16: }
Chris@16: 
Chris@16: 
Chris@16: template <class T>
Chris@16: template<int N>
Chris@16: inline T constant_pi_cubed<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
Chris@16: {
Chris@16:    BOOST_MATH_STD_USING
Chris@16:    return pi<T, policies::policy<policies::digits2<N> > >()
Chris@16:    *      pi<T, policies::policy<policies::digits2<N> > >()
Chris@16:    *      pi<T, policies::policy<policies::digits2<N> > >()
Chris@16:    ; //
Chris@16: }
Chris@16: 
Chris@16: template <class T>
Chris@16: template<int N>
Chris@16: inline T constant_cbrt_pi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
Chris@16: {
Chris@16:    BOOST_MATH_STD_USING
Chris@16:    return pow(pi<T, policies::policy<policies::digits2<N> > >(), static_cast<T>(1)/ static_cast<T>(3));
Chris@16: }
Chris@16: 
Chris@16: template <class T>
Chris@16: template<int N>
Chris@16: inline T constant_one_div_cbrt_pi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
Chris@16: {
Chris@16:    BOOST_MATH_STD_USING
Chris@16:    return static_cast<T>(1)
Chris@16:    / pow(pi<T, policies::policy<policies::digits2<N> > >(), static_cast<T>(1)/ static_cast<T>(3));
Chris@16: }
Chris@16: 
Chris@16: // Euler's e
Chris@16: 
Chris@16: template <class T>
Chris@16: template<int N>
Chris@16: inline T constant_e_pow_pi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
Chris@16: {
Chris@16:    BOOST_MATH_STD_USING
Chris@16:    return pow(e<T, policies::policy<policies::digits2<N> > >(), pi<T, policies::policy<policies::digits2<N> > >()); //
Chris@16: }
Chris@16: 
Chris@16: template <class T>
Chris@16: template<int N>
Chris@16: inline T constant_root_e<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
Chris@16: {
Chris@16:    BOOST_MATH_STD_USING
Chris@16:    return sqrt(e<T, policies::policy<policies::digits2<N> > >());
Chris@16: }
Chris@16: 
Chris@16: template <class T>
Chris@16: template<int N>
Chris@16: inline T constant_log10_e<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
Chris@16: {
Chris@16:    BOOST_MATH_STD_USING
Chris@16:    return log10(e<T, policies::policy<policies::digits2<N> > >());
Chris@16: }
Chris@16: 
Chris@16: template <class T>
Chris@16: template<int N>
Chris@16: inline T constant_one_div_log10_e<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
Chris@16: {
Chris@16:    BOOST_MATH_STD_USING
Chris@16:    return  static_cast<T>(1) /
Chris@16:      log10(e<T, policies::policy<policies::digits2<N> > >());
Chris@16: }
Chris@16: 
Chris@16: // Trigonometric
Chris@16: 
Chris@16: template <class T>
Chris@16: template<int N>
Chris@16: inline T constant_degree<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
Chris@16: {
Chris@16:    BOOST_MATH_STD_USING
Chris@16:    return pi<T, policies::policy<policies::digits2<N> > >()
Chris@16:    / static_cast<T>(180)
Chris@16:    ; //
Chris@16: }
Chris@16: 
Chris@16: template <class T>
Chris@16: template<int N>
Chris@16: inline T constant_radian<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
Chris@16: {
Chris@16:    BOOST_MATH_STD_USING
Chris@16:    return static_cast<T>(180)
Chris@16:    / pi<T, policies::policy<policies::digits2<N> > >()
Chris@16:    ; //
Chris@16: }
Chris@16: 
Chris@16: template <class T>
Chris@16: template<int N>
Chris@16: inline T constant_sin_one<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
Chris@16: {
Chris@16:    BOOST_MATH_STD_USING
Chris@16:    return sin(static_cast<T>(1)) ; //
Chris@16: }
Chris@16: 
Chris@16: template <class T>
Chris@16: template<int N>
Chris@16: inline T constant_cos_one<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
Chris@16: {
Chris@16:    BOOST_MATH_STD_USING
Chris@16:    return cos(static_cast<T>(1)) ; //
Chris@16: }
Chris@16: 
Chris@16: template <class T>
Chris@16: template<int N>
Chris@16: inline T constant_sinh_one<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
Chris@16: {
Chris@16:    BOOST_MATH_STD_USING
Chris@16:    return sinh(static_cast<T>(1)) ; //
Chris@16: }
Chris@16: 
Chris@16: template <class T>
Chris@16: template<int N>
Chris@16: inline T constant_cosh_one<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
Chris@16: {
Chris@16:    BOOST_MATH_STD_USING
Chris@16:    return cosh(static_cast<T>(1)) ; //
Chris@16: }
Chris@16: 
Chris@16: template <class T>
Chris@16: template<int N>
Chris@16: inline T constant_phi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
Chris@16: {
Chris@16:    BOOST_MATH_STD_USING
Chris@16:    return (static_cast<T>(1) + sqrt(static_cast<T>(5)) )/static_cast<T>(2) ; //
Chris@16: }
Chris@16: 
Chris@16: template <class T>
Chris@16: template<int N>
Chris@16: inline T constant_ln_phi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
Chris@16: {
Chris@16:    BOOST_MATH_STD_USING
Chris@16:    //return  log(phi<T, policies::policy<policies::digits2<N> > >()); // ???
Chris@16:    return log((static_cast<T>(1) + sqrt(static_cast<T>(5)) )/static_cast<T>(2) );
Chris@16: }
Chris@16: template <class T>
Chris@16: template<int N>
Chris@16: inline T constant_one_div_ln_phi<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
Chris@16: {
Chris@16:    BOOST_MATH_STD_USING
Chris@16:    return static_cast<T>(1) /
Chris@16:      log((static_cast<T>(1) + sqrt(static_cast<T>(5)) )/static_cast<T>(2) );
Chris@16: }
Chris@16: 
Chris@16: // Zeta
Chris@16: 
Chris@16: template <class T>
Chris@16: template<int N>
Chris@16: inline T constant_zeta_two<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
Chris@16: {
Chris@16:    BOOST_MATH_STD_USING
Chris@16: 
Chris@16:      return pi<T, policies::policy<policies::digits2<N> > >()
Chris@16:      *  pi<T, policies::policy<policies::digits2<N> > >()
Chris@16:      /static_cast<T>(6);
Chris@16: }
Chris@16: 
Chris@16: template <class T>
Chris@16: template<int N>
Chris@16: inline T constant_zeta_three<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
Chris@16: {
Chris@16:    // http://mathworld.wolfram.com/AperysConstant.html
Chris@16:    // http://en.wikipedia.org/wiki/Mathematical_constant
Chris@16: 
Chris@16:    // http://oeis.org/A002117/constant
Chris@16:    //T zeta3("1.20205690315959428539973816151144999076"
Chris@16:    // "4986292340498881792271555341838205786313"
Chris@16:    // "09018645587360933525814619915");
Chris@16: 
Chris@16:   //"1.202056903159594285399738161511449990, 76498629234049888179227155534183820578631309018645587360933525814619915"  A002117
Chris@16:   // 1.202056903159594285399738161511449990, 76498629234049888179227155534183820578631309018645587360933525814619915780, +00);
Chris@16:   //"1.2020569031595942 double
Chris@16:   // http://www.spaennare.se/SSPROG/ssnum.pdf  // section 11, Algorithm for Apery's constant zeta(3).
Chris@16:   // Programs to Calculate some Mathematical Constants to Large Precision, Document Version 1.50
Chris@16: 
Chris@16:   // by Stefan Spannare  September 19, 2007
Chris@16:   // zeta(3) = 1/64 * sum
Chris@16:    BOOST_MATH_STD_USING
Chris@16:    T n_fact=static_cast<T>(1); // build n! for n = 0.
Chris@16:    T sum = static_cast<double>(77); // Start with n = 0 case.
Chris@16:    // for n = 0, (77/1) /64 = 1.203125
Chris@16:    //double lim = std::numeric_limits<double>::epsilon();
Chris@16:    T lim = N ? ldexp(T(1), 1 - (std::min)(N, tools::digits<T>())) : tools::epsilon<T>();
Chris@16:    for(unsigned int n = 1; n < 40; ++n)
Chris@16:    { // three to five decimal digits per term, so 40 should be plenty for 100 decimal digits.
Chris@16:       //cout << "n = " << n << endl;
Chris@16:       n_fact *= n; // n!
Chris@16:       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
Chris@16:       T num = ((205 * n * n) + (250 * n) + 77) * n_fact_p10; // 205n^2 + 250n + 77
Chris@16:       // int nn = (2 * n + 1);
Chris@16:       // T d = factorial(nn); // inline factorial.
Chris@16:       T d = 1;
Chris@16:       for(unsigned int i = 1; i <= (n+n + 1); ++i) // (2n + 1)
Chris@16:       {
Chris@16:         d *= i;
Chris@16:       }
Chris@16:       T den = d * d * d * d * d; // [(2n+1)!]^5
Chris@16:       //cout << "den = " << den << endl;
Chris@16:       T term = num/den;
Chris@16:       if (n % 2 != 0)
Chris@16:       { //term *= -1;
Chris@16:         sum -= term;
Chris@16:       }
Chris@16:       else
Chris@16:       {
Chris@16:         sum += term;
Chris@16:       }
Chris@16:       //cout << "term = " << term << endl;
Chris@16:       //cout << "sum/64  = " << sum/64 << endl;
Chris@16:       if(abs(term) < lim)
Chris@16:       {
Chris@16:          break;
Chris@16:       }
Chris@16:    }
Chris@16:    return sum / 64;
Chris@16: }
Chris@16: 
Chris@16: template <class T>
Chris@16: template<int N>
Chris@16: inline T constant_catalan<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
Chris@16: { // http://oeis.org/A006752/constant
Chris@16:   //T c("0.915965594177219015054603514932384110774"
Chris@16:  //"149374281672134266498119621763019776254769479356512926115106248574");
Chris@16: 
Chris@16:   // 9.159655941772190150546035149323841107, 74149374281672134266498119621763019776254769479356512926115106248574422619, -01);
Chris@16: 
Chris@16:    // This is equation (entry) 31 from
Chris@16:    // http://www-2.cs.cmu.edu/~adamchik/articles/catalan/catalan.htm
Chris@16:    // See also http://www.mpfr.org/algorithms.pdf
Chris@16:    BOOST_MATH_STD_USING
Chris@16:    T k_fact = 1;
Chris@16:    T tk_fact = 1;
Chris@16:    T sum = 1;
Chris@16:    T term;
Chris@16:    T lim = N ? ldexp(T(1), 1 - (std::min)(N, tools::digits<T>())) : tools::epsilon<T>();
Chris@16: 
Chris@16:    for(unsigned k = 1;; ++k)
Chris@16:    {
Chris@16:       k_fact *= k;
Chris@16:       tk_fact *= (2 * k) * (2 * k - 1);
Chris@16:       term = k_fact * k_fact / (tk_fact * (2 * k + 1) * (2 * k + 1));
Chris@16:       sum += term;
Chris@16:       if(term < lim)
Chris@16:       {
Chris@16:          break;
Chris@16:       }
Chris@16:    }
Chris@16:    return boost::math::constants::pi<T, boost::math::policies::policy<> >()
Chris@16:       * log(2 + boost::math::constants::root_three<T, boost::math::policies::policy<> >())
Chris@16:        / 8
Chris@16:       + 3 * sum / 8;
Chris@16: }
Chris@16: 
Chris@16: namespace khinchin_detail{
Chris@16: 
Chris@16: template <class T>
Chris@16: T zeta_polynomial_series(T s, T sc, int digits)
Chris@16: {
Chris@16:    BOOST_MATH_STD_USING
Chris@16:    //
Chris@16:    // This is algorithm 3 from:
Chris@16:    //
Chris@16:    // "An Efficient Algorithm for the Riemann Zeta Function", P. Borwein,
Chris@16:    // Canadian Mathematical Society, Conference Proceedings, 2000.
Chris@16:    // See: http://www.cecm.sfu.ca/personal/pborwein/PAPERS/P155.pdf
Chris@16:    //
Chris@16:    BOOST_MATH_STD_USING
Chris@16:    int n = (digits * 19) / 53;
Chris@16:    T sum = 0;
Chris@16:    T two_n = ldexp(T(1), n);
Chris@16:    int ej_sign = 1;
Chris@16:    for(int j = 0; j < n; ++j)
Chris@16:    {
Chris@16:       sum += ej_sign * -two_n / pow(T(j + 1), s);
Chris@16:       ej_sign = -ej_sign;
Chris@16:    }
Chris@16:    T ej_sum = 1;
Chris@16:    T ej_term = 1;
Chris@16:    for(int j = n; j <= 2 * n - 1; ++j)
Chris@16:    {
Chris@16:       sum += ej_sign * (ej_sum - two_n) / pow(T(j + 1), s);
Chris@16:       ej_sign = -ej_sign;
Chris@16:       ej_term *= 2 * n - j;
Chris@16:       ej_term /= j - n + 1;
Chris@16:       ej_sum += ej_term;
Chris@16:    }
Chris@16:    return -sum / (two_n * (1 - pow(T(2), sc)));
Chris@16: }
Chris@16: 
Chris@16: template <class T>
Chris@16: T khinchin(int digits)
Chris@16: {
Chris@16:    BOOST_MATH_STD_USING
Chris@16:    T sum = 0;
Chris@16:    T term;
Chris@16:    T lim = ldexp(T(1), 1-digits);
Chris@16:    T factor = 0;
Chris@16:    unsigned last_k = 1;
Chris@16:    T num = 1;
Chris@16:    for(unsigned n = 1;; ++n)
Chris@16:    {
Chris@16:       for(unsigned k = last_k; k <= 2 * n - 1; ++k)
Chris@16:       {
Chris@16:          factor += num / k;
Chris@16:          num = -num;
Chris@16:       }
Chris@16:       last_k = 2 * n;
Chris@16:       term = (zeta_polynomial_series(T(2 * n), T(1 - T(2 * n)), digits) - 1) * factor / n;
Chris@16:       sum += term;
Chris@16:       if(term < lim)
Chris@16:          break;
Chris@16:    }
Chris@16:    return exp(sum / boost::math::constants::ln_two<T, boost::math::policies::policy<> >());
Chris@16: }
Chris@16: 
Chris@16: }
Chris@16: 
Chris@16: template <class T>
Chris@16: template<int N>
Chris@16: inline T constant_khinchin<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
Chris@16: {
Chris@16:    int n = N ? (std::min)(N, tools::digits<T>()) : tools::digits<T>();
Chris@16:    return khinchin_detail::khinchin<T>(n);
Chris@16: }
Chris@16: 
Chris@16: template <class T>
Chris@16: template<int N>
Chris@16: inline T constant_extreme_value_skewness<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
Chris@16: { // from e_float constants.cpp
Chris@16:   // Mathematica: N[12 Sqrt[6]  Zeta[3]/Pi^3, 1101]
Chris@16:    BOOST_MATH_STD_USING
Chris@16:    T ev(12 * sqrt(static_cast<T>(6)) * zeta_three<T, policies::policy<policies::digits2<N> > >()
Chris@16:     / pi_cubed<T, policies::policy<policies::digits2<N> > >() );
Chris@16: 
Chris@16: //T ev(
Chris@16: //"1.1395470994046486574927930193898461120875997958365518247216557100852480077060706857071875468869385150"
Chris@16: //"1894272048688553376986765366075828644841024041679714157616857834895702411080704529137366329462558680"
Chris@16: //"2015498788776135705587959418756809080074611906006528647805347822929577145038743873949415294942796280"
Chris@16: //"0895597703063466053535550338267721294164578901640163603544404938283861127819804918174973533694090594"
Chris@16: //"3094963822672055237678432023017824416203652657301470473548274848068762500300316769691474974950757965"
Chris@16: //"8640779777748741897542093874605477776538884083378029488863880220988107155275203245233994097178778984"
Chris@16: //"3488995668362387892097897322246698071290011857605809901090220903955815127463328974447572119951192970"
Chris@16: //"3684453635456559086126406960279692862247058250100678008419431185138019869693206366891639436908462809"
Chris@16: //"9756051372711251054914491837034685476095423926553367264355374652153595857163724698198860485357368964"
Chris@16: //"3807049634423621246870868566707915720704996296083373077647528285782964567312903914752617978405994377"
Chris@16: //"9064157147206717895272199736902453130842229559980076472936976287378945035706933650987259357729800315");
Chris@16: 
Chris@16:   return ev;
Chris@16: }
Chris@16: 
Chris@16: namespace detail{
Chris@16: //
Chris@16: // Calculation of the Glaisher constant depends upon calculating the
Chris@16: // derivative of the zeta function at 2, we can then use the relation:
Chris@16: // zeta'(2) = 1/6 pi^2 [euler + ln(2pi)-12ln(A)]
Chris@16: // To get the constant A.
Chris@16: // See equation 45 at http://mathworld.wolfram.com/RiemannZetaFunction.html.
Chris@16: //
Chris@16: // The derivative of the zeta function is computed by direct differentiation
Chris@16: // of the relation:
Chris@16: // (1-2^(1-s))zeta(s) = SUM(n=0, INF){ (-n)^n / (n+1)^s  }
Chris@16: // Which gives us 2 slowly converging but alternating sums to compute,
Chris@16: // for this we use Algorithm 1 from "Convergent Acceleration of Alternating Series",
Chris@16: // Henri Cohen, Fernando Rodriguez Villegas and Don Zagier, Experimental Mathematics 9:1 (1999).
Chris@16: // See http://www.math.utexas.edu/users/villegas/publications/conv-accel.pdf
Chris@16: //
Chris@16: template <class T>
Chris@16: T zeta_series_derivative_2(unsigned digits)
Chris@16: {
Chris@16:    // Derivative of the series part, evaluated at 2:
Chris@16:    BOOST_MATH_STD_USING
Chris@16:    int n = digits * 301 * 13 / 10000;
Chris@16:    boost::math::itrunc((std::numeric_limits<T>::digits10 + 1) * 1.3);
Chris@16:    T d = pow(3 + sqrt(T(8)), n);
Chris@16:    d = (d + 1 / d) / 2;
Chris@16:    T b = -1;
Chris@16:    T c = -d;
Chris@16:    T s = 0;
Chris@16:    for(int k = 0; k < n; ++k)
Chris@16:    {
Chris@16:       T a = -log(T(k+1)) / ((k+1) * (k+1));
Chris@16:       c = b - c;
Chris@16:       s = s + c * a;
Chris@16:       b = (k + n) * (k - n) * b / ((k + T(0.5f)) * (k + 1));
Chris@16:    }
Chris@16:    return s / d;
Chris@16: }
Chris@16: 
Chris@16: template <class T>
Chris@16: T zeta_series_2(unsigned digits)
Chris@16: {
Chris@16:    // Series part of zeta at 2:
Chris@16:    BOOST_MATH_STD_USING
Chris@16:    int n = digits * 301 * 13 / 10000;
Chris@16:    T d = pow(3 + sqrt(T(8)), n);
Chris@16:    d = (d + 1 / d) / 2;
Chris@16:    T b = -1;
Chris@16:    T c = -d;
Chris@16:    T s = 0;
Chris@16:    for(int k = 0; k < n; ++k)
Chris@16:    {
Chris@16:       T a = T(1) / ((k + 1) * (k + 1));
Chris@16:       c = b - c;
Chris@16:       s = s + c * a;
Chris@16:       b = (k + n) * (k - n) * b / ((k + T(0.5f)) * (k + 1));
Chris@16:    }
Chris@16:    return s / d;
Chris@16: }
Chris@16: 
Chris@16: template <class T>
Chris@16: inline T zeta_series_lead_2()
Chris@16: {
Chris@16:    // lead part at 2:
Chris@16:    return 2;
Chris@16: }
Chris@16: 
Chris@16: template <class T>
Chris@16: inline T zeta_series_derivative_lead_2()
Chris@16: {
Chris@16:    // derivative of lead part at 2:
Chris@16:    return -2 * boost::math::constants::ln_two<T>();
Chris@16: }
Chris@16: 
Chris@16: template <class T>
Chris@16: inline T zeta_derivative_2(unsigned n)
Chris@16: {
Chris@16:    // zeta derivative at 2:
Chris@16:    return zeta_series_derivative_2<T>(n) * zeta_series_lead_2<T>()
Chris@16:       + zeta_series_derivative_lead_2<T>() * zeta_series_2<T>(n);
Chris@16: }
Chris@16: 
Chris@16: }  // namespace detail
Chris@16: 
Chris@16: template <class T>
Chris@16: template<int N>
Chris@16: inline T constant_glaisher<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
Chris@16: {
Chris@16: 
Chris@16:    BOOST_MATH_STD_USING
Chris@16:    typedef policies::policy<policies::digits2<N> > forwarding_policy;
Chris@16:    int n = N ? (std::min)(N, tools::digits<T>()) : tools::digits<T>();
Chris@16:    T v = detail::zeta_derivative_2<T>(n);
Chris@16:    v *= 6;
Chris@16:    v /= boost::math::constants::pi<T, forwarding_policy>() * boost::math::constants::pi<T, forwarding_policy>();
Chris@16:    v -= boost::math::constants::euler<T, forwarding_policy>();
Chris@16:    v -= log(2 * boost::math::constants::pi<T, forwarding_policy>());
Chris@16:    v /= -12;
Chris@16:    return exp(v);
Chris@16: 
Chris@16:  /*
Chris@16:    // from http://mpmath.googlecode.com/svn/data/glaisher.txt
Chris@16:      // 20,000 digits of the Glaisher-Kinkelin constant A = exp(1/2 - zeta'(-1))
Chris@16:      // Computed using A = exp((6 (-zeta'(2))/pi^2 + log 2 pi + gamma)/12)
Chris@16:   // with Euler-Maclaurin summation for zeta'(2).
Chris@16:   T g(
Chris@16:   "1.282427129100622636875342568869791727767688927325001192063740021740406308858826"
Chris@16:   "46112973649195820237439420646120399000748933157791362775280404159072573861727522"
Chris@16:   "14334327143439787335067915257366856907876561146686449997784962754518174312394652"
Chris@16:   "76128213808180219264516851546143919901083573730703504903888123418813674978133050"
Chris@16:   "93770833682222494115874837348064399978830070125567001286994157705432053927585405"
Chris@16:   "81731588155481762970384743250467775147374600031616023046613296342991558095879293"
Chris@16:   "36343887288701988953460725233184702489001091776941712153569193674967261270398013"
Chris@16:   "52652668868978218897401729375840750167472114895288815996668743164513890306962645"
Chris@16:   "59870469543740253099606800842447417554061490189444139386196089129682173528798629"
Chris@16:   "88434220366989900606980888785849587494085307347117090132667567503310523405221054"
Chris@16:   "14176776156308191919997185237047761312315374135304725819814797451761027540834943"
Chris@16:   "14384965234139453373065832325673954957601692256427736926358821692159870775858274"
Chris@16:   "69575162841550648585890834128227556209547002918593263079373376942077522290940187");
Chris@16: 
Chris@16:   return g;
Chris@16:   */
Chris@16: }
Chris@16: 
Chris@16: template <class T>
Chris@16: template<int N>
Chris@16: inline T constant_rayleigh_skewness<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
Chris@16: {  // From e_float
Chris@16:   // 1100 digits of the Rayleigh distribution skewness
Chris@16:   // Mathematica: N[2 Sqrt[Pi] (Pi - 3)/((4 - Pi)^(3/2)), 1100]
Chris@16: 
Chris@16:    BOOST_MATH_STD_USING
Chris@16:    T rs(2 * root_pi<T, policies::policy<policies::digits2<N> > >()
Chris@16:       * pi_minus_three<T, policies::policy<policies::digits2<N> > >()
Chris@16:       / pow(four_minus_pi<T, policies::policy<policies::digits2<N> > >(), static_cast<T>(3./2))
Chris@16:       );
Chris@16:  //   6.31110657818937138191899351544227779844042203134719497658094585692926819617473725459905027032537306794400047264,
Chris@16: 
Chris@16:   //"0.6311106578189371381918993515442277798440422031347194976580945856929268196174737254599050270325373067"
Chris@16:   //"9440004726436754739597525250317640394102954301685809920213808351450851396781817932734836994829371322"
Chris@16:   //"5797376021347531983451654130317032832308462278373358624120822253764532674177325950686466133508511968"
Chris@16:   //"2389168716630349407238090652663422922072397393006683401992961569208109477307776249225072042971818671"
Chris@16:   //"4058887072693437217879039875871765635655476241624825389439481561152126886932506682176611183750503553"
Chris@16:   //"1218982627032068396407180216351425758181396562859085306247387212297187006230007438534686340210168288"
Chris@16:   //"8956816965453815849613622117088096547521391672977226658826566757207615552041767516828171274858145957"
Chris@16:   //"6137539156656005855905288420585194082284972984285863898582313048515484073396332610565441264220790791"
Chris@16:   //"0194897267890422924599776483890102027823328602965235306539844007677157873140562950510028206251529523"
Chris@16:   //"7428049693650605954398446899724157486062545281504433364675815915402937209673727753199567661561209251"
Chris@16:   //"4695589950526053470201635372590001578503476490223746511106018091907936826431407434894024396366284848");  ;
Chris@16:   return rs;
Chris@16: }
Chris@16: 
Chris@16: template <class T>
Chris@16: template<int N>
Chris@16: inline T constant_rayleigh_kurtosis_excess<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
Chris@16: { // - (6 Pi^2 - 24 Pi + 16)/((Pi - 4)^2)
Chris@16:     // Might provide and calculate this using pi_minus_four.
Chris@16:    BOOST_MATH_STD_USING
Chris@16:    return - (((static_cast<T>(6) * pi<T, policies::policy<policies::digits2<N> > >()
Chris@16:         * pi<T, policies::policy<policies::digits2<N> > >())
Chris@16:    - (static_cast<T>(24) * pi<T, policies::policy<policies::digits2<N> > >()) + static_cast<T>(16) )
Chris@16:    /
Chris@16:    ((pi<T, policies::policy<policies::digits2<N> > >() - static_cast<T>(4))
Chris@16:    * (pi<T, policies::policy<policies::digits2<N> > >() - static_cast<T>(4)))
Chris@16:    );
Chris@16: }
Chris@16: 
Chris@16: template <class T>
Chris@16: template<int N>
Chris@16: inline T constant_rayleigh_kurtosis<T>::compute(BOOST_MATH_EXPLICIT_TEMPLATE_TYPE_SPEC(mpl::int_<N>))
Chris@16: { // 3 - (6 Pi^2 - 24 Pi + 16)/((Pi - 4)^2)
Chris@16:   // Might provide and calculate this using pi_minus_four.
Chris@16:    BOOST_MATH_STD_USING
Chris@16:    return static_cast<T>(3) - (((static_cast<T>(6) * pi<T, policies::policy<policies::digits2<N> > >()
Chris@16:         * pi<T, policies::policy<policies::digits2<N> > >())
Chris@16:    - (static_cast<T>(24) * pi<T, policies::policy<policies::digits2<N> > >()) + static_cast<T>(16) )
Chris@16:    /
Chris@16:    ((pi<T, policies::policy<policies::digits2<N> > >() - static_cast<T>(4))
Chris@16:    * (pi<T, policies::policy<policies::digits2<N> > >() - static_cast<T>(4)))
Chris@16:    );
Chris@16: }
Chris@16: 
Chris@16: }}}} // namespaces
Chris@16: 
Chris@16: #endif // BOOST_MATH_CALCULATE_CONSTANTS_CONSTANTS_INCLUDED