text: //  Copyright Paul A. Bristow 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_STATS_rayleigh_HPP
  #define BOOST_STATS_rayleigh_HPP
  
  #include <boost/math/distributions/fwd.hpp>
  #include <boost/math/constants/constants.hpp>
  #include <boost/math/special_functions/log1p.hpp>
  #include <boost/math/special_functions/expm1.hpp>
  #include <boost/math/distributions/complement.hpp>
  #include <boost/math/distributions/detail/common_error_handling.hpp>
  #include <boost/config/no_tr1/cmath.hpp>
  
  #ifdef BOOST_MSVC
  # pragma warning(push)
  # pragma warning(disable: 4702) // unreachable code (return after domain_error throw).
  #endif
  
  #include <utility>
  
  namespace boost{ namespace math{
  
  namespace detail
  { // Error checks:
    template <class RealType, class Policy>
    inline bool verify_sigma(const char* function, RealType sigma, RealType* presult, const Policy& pol)
    {
       if((sigma <= 0) || (!(boost::math::isfinite)(sigma)))
       {
          *presult = policies::raise_domain_error<RealType>(
             function,
             "The scale parameter \"sigma\" must be > 0 and finite, but was: %1%.", sigma, pol);
          return false;
       }
       return true;
    } // bool verify_sigma
  
    template <class RealType, class Policy>
    inline bool verify_rayleigh_x(const char* function, RealType x, RealType* presult, const Policy& pol)
    {
       if((x < 0) || (boost::math::isnan)(x))
       {
          *presult = policies::raise_domain_error<RealType>(
             function,
             "The random variable must be >= 0, but was: %1%.", x, pol);
          return false;
       }
       return true;
    } // bool verify_rayleigh_x
  } // namespace detail
  
  template <class RealType = double, class Policy = policies::policy<> >
  class rayleigh_distribution
  {
  public:
     typedef RealType value_type;
     typedef Policy policy_type;
  
     rayleigh_distribution(RealType l_sigma = 1)
        : m_sigma(l_sigma)
     {
        RealType err;
        detail::verify_sigma("boost::math::rayleigh_distribution<%1%>::rayleigh_distribution", l_sigma, &err, Policy());
     } // rayleigh_distribution
  
     RealType sigma()const
     { // Accessor.
       return m_sigma;
     }
  
  private:
     RealType m_sigma;
  }; // class rayleigh_distribution
  
  typedef rayleigh_distribution<double> rayleigh;
  
  template <class RealType, class Policy>
  inline const std::pair<RealType, RealType> range(const rayleigh_distribution<RealType, Policy>& /*dist*/)
  { // Range of permissible values for random variable x.
     using boost::math::tools::max_value;
     return std::pair<RealType, RealType>(static_cast<RealType>(0), std::numeric_limits<RealType>::has_infinity ? std::numeric_limits<RealType>::infinity() : max_value<RealType>());
  }
  
  template <class RealType, class Policy>
  inline const std::pair<RealType, RealType> support(const rayleigh_distribution<RealType, Policy>& /*dist*/)
  { // Range of supported values for random variable x.
     // This is range where cdf rises from 0 to 1, and outside it, the pdf is zero.
     using boost::math::tools::max_value;
     return std::pair<RealType, RealType>(static_cast<RealType>(0),  max_value<RealType>());
  }
  
  template <class RealType, class Policy>
  inline RealType pdf(const rayleigh_distribution<RealType, Policy>& dist, const RealType& x)
  {
     BOOST_MATH_STD_USING // for ADL of std function exp.
  
     RealType sigma = dist.sigma();
     RealType result = 0;
     static const char* function = "boost::math::pdf(const rayleigh_distribution<%1%>&, %1%)";
     if(false == detail::verify_sigma(function, sigma, &result, Policy()))
     {
        return result;
     }
     if(false == detail::verify_rayleigh_x(function, x, &result, Policy()))
     {
        return result;
     }
     if((boost::math::isinf)(x))
     {
        return 0;
     }
     RealType sigmasqr = sigma * sigma;
     result = x * (exp(-(x * x) / ( 2 * sigmasqr))) / sigmasqr;
     return result;
  } // pdf
  
  template <class RealType, class Policy>
  inline RealType cdf(const rayleigh_distribution<RealType, Policy>& dist, const RealType& x)
  {
     BOOST_MATH_STD_USING // for ADL of std functions
  
     RealType result = 0;
     RealType sigma = dist.sigma();
     static const char* function = "boost::math::cdf(const rayleigh_distribution<%1%>&, %1%)";
     if(false == detail::verify_sigma(function, sigma, &result, Policy()))
     {
        return result;
     }
     if(false == detail::verify_rayleigh_x(function, x, &result, Policy()))
     {
        return result;
     }
     result = -boost::math::expm1(-x * x / ( 2 * sigma * sigma), Policy());
     return result;
  } // cdf
  
  template <class RealType, class Policy>
  inline RealType quantile(const rayleigh_distribution<RealType, Policy>& dist, const RealType& p)
  {
     BOOST_MATH_STD_USING // for ADL of std functions
  
     RealType result = 0;
     RealType sigma = dist.sigma();
     static const char* function = "boost::math::quantile(const rayleigh_distribution<%1%>&, %1%)";
     if(false == detail::verify_sigma(function, sigma, &result, Policy()))
        return result;
     if(false == detail::check_probability(function, p, &result, Policy()))
        return result;
  
     if(p == 0)
     {
        return 0;
     }
     if(p == 1)
     {
       return policies::raise_overflow_error<RealType>(function, 0, Policy());
     }
     result = sqrt(-2 * sigma * sigma * boost::math::log1p(-p, Policy()));
     return result;
  } // quantile
  
  template <class RealType, class Policy>
  inline RealType cdf(const complemented2_type<rayleigh_distribution<RealType, Policy>, RealType>& c)
  {
     BOOST_MATH_STD_USING // for ADL of std functions
  
     RealType result = 0;
     RealType sigma = c.dist.sigma();
     static const char* function = "boost::math::cdf(const rayleigh_distribution<%1%>&, %1%)";
     if(false == detail::verify_sigma(function, sigma, &result, Policy()))
     {
        return result;
     }
     RealType x = c.param;
     if(false == detail::verify_rayleigh_x(function, x, &result, Policy()))
     {
        return result;
     }
     RealType ea = x * x / (2 * sigma * sigma);
     // Fix for VC11/12 x64 bug in exp(float):
     if (ea >= tools::max_value<RealType>())
        return 0;
     result =  exp(-ea);
     return result;
  } // cdf complement
  
  template <class RealType, class Policy>
  inline RealType quantile(const complemented2_type<rayleigh_distribution<RealType, Policy>, RealType>& c)
  {
     BOOST_MATH_STD_USING // for ADL of std functions, log & sqrt.
  
     RealType result = 0;
     RealType sigma = c.dist.sigma();
     static const char* function = "boost::math::quantile(const rayleigh_distribution<%1%>&, %1%)";
     if(false == detail::verify_sigma(function, sigma, &result, Policy()))
     {
        return result;
     }
     RealType q = c.param;
     if(false == detail::check_probability(function, q, &result, Policy()))
     {
        return result;
     }
     if(q == 1)
     {
        return 0;
     }
     if(q == 0)
     {
       return policies::raise_overflow_error<RealType>(function, 0, Policy());
     }
     result = sqrt(-2 * sigma * sigma * log(q));
     return result;
  } // quantile complement
  
  template <class RealType, class Policy>
  inline RealType mean(const rayleigh_distribution<RealType, Policy>& dist)
  {
     RealType result = 0;
     RealType sigma = dist.sigma();
     static const char* function = "boost::math::mean(const rayleigh_distribution<%1%>&, %1%)";
     if(false == detail::verify_sigma(function, sigma, &result, Policy()))
     {
        return result;
     }
     using boost::math::constants::root_half_pi;
     return sigma * root_half_pi<RealType>();
  } // mean
  
  template <class RealType, class Policy>
  inline RealType variance(const rayleigh_distribution<RealType, Policy>& dist)
  {
     RealType result = 0;
     RealType sigma = dist.sigma();
     static const char* function = "boost::math::variance(const rayleigh_distribution<%1%>&, %1%)";
     if(false == detail::verify_sigma(function, sigma, &result, Policy()))
     {
        return result;
     }
     using boost::math::constants::four_minus_pi;
     return four_minus_pi<RealType>() * sigma * sigma / 2;
  } // variance
  
  template <class RealType, class Policy>
  inline RealType mode(const rayleigh_distribution<RealType, Policy>& dist)
  {
     return dist.sigma();
  }
  
  template <class RealType, class Policy>
  inline RealType median(const rayleigh_distribution<RealType, Policy>& dist)
  {
     using boost::math::constants::root_ln_four;
     return root_ln_four<RealType>() * dist.sigma();
  }
  
  template <class RealType, class Policy>
  inline RealType skewness(const rayleigh_distribution<RealType, Policy>& /*dist*/)
  {
    // using namespace boost::math::constants;
    return static_cast<RealType>(0.63111065781893713819189935154422777984404221106391L);
    // Computed using NTL at 150 bit, about 50 decimal digits.
    // return 2 * root_pi<RealType>() * pi_minus_three<RealType>() / pow23_four_minus_pi<RealType>();
  }
  
  template <class RealType, class Policy>
  inline RealType kurtosis(const rayleigh_distribution<RealType, Policy>& /*dist*/)
  {
    // using namespace boost::math::constants;
    return static_cast<RealType>(3.2450893006876380628486604106197544154170667057995L);
    // Computed using NTL at 150 bit, about 50 decimal digits.
    // return 3 - (6 * pi<RealType>() * pi<RealType>() - 24 * pi<RealType>() + 16) /
    // (four_minus_pi<RealType>() * four_minus_pi<RealType>());
  }
  
  template <class RealType, class Policy>
  inline RealType kurtosis_excess(const rayleigh_distribution<RealType, Policy>& /*dist*/)
  {
    //using namespace boost::math::constants;
    // Computed using NTL at 150 bit, about 50 decimal digits.
    return static_cast<RealType>(0.2450893006876380628486604106197544154170667057995L);
    // return -(6 * pi<RealType>() * pi<RealType>() - 24 * pi<RealType>() + 16) /
    //   (four_minus_pi<RealType>() * four_minus_pi<RealType>());
  } // kurtosis
  
  } // namespace math
  } // namespace boost
  
  #ifdef BOOST_MSVC
  # pragma warning(pop)
  #endif
  
  // This include must be at the end, *after* the accessors
  // for this distribution have been defined, in order to
  // keep compilers that support two-phase lookup happy.
  #include <boost/math/distributions/detail/derived_accessors.hpp>
  
  #endif // BOOST_STATS_rayleigh_HPP
  