text: //  (C) Copyright John Maddock 2006.
  //  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_EXPM1_INCLUDED
  #define BOOST_MATH_EXPM1_INCLUDED
  
  #ifdef _MSC_VER
  #pragma once
  #endif
  
  #include <boost/config/no_tr1/cmath.hpp>
  #include <math.h> // platform's ::expm1
  #include <boost/limits.hpp>
  #include <boost/math/tools/config.hpp>
  #include <boost/math/tools/series.hpp>
  #include <boost/math/tools/precision.hpp>
  #include <boost/math/policies/error_handling.hpp>
  #include <boost/math/tools/rational.hpp>
  #include <boost/math/special_functions/math_fwd.hpp>
  #include <boost/mpl/less_equal.hpp>
  
  #ifndef BOOST_NO_LIMITS_COMPILE_TIME_CONSTANTS
  #  include <boost/static_assert.hpp>
  #else
  #  include <boost/assert.hpp>
  #endif
  
  namespace boost{ namespace math{
  
  namespace detail
  {
    // Functor expm1_series returns the next term in the Taylor series
    // x^k / k!
    // each time that operator() is invoked.
    //
    template <class T>
    struct expm1_series
    {
       typedef T result_type;
  
       expm1_series(T x)
          : k(0), m_x(x), m_term(1) {}
  
       T operator()()
       {
          ++k;
          m_term *= m_x;
          m_term /= k;
          return m_term;
       }
  
       int count()const
       {
          return k;
       }
  
    private:
       int k;
       const T m_x;
       T m_term;
       expm1_series(const expm1_series&);
       expm1_series& operator=(const expm1_series&);
    };
  
  //
  // Algorithm expm1 is part of C99, but is not yet provided by many compilers.
  //
  // This version uses a Taylor series expansion for 0.5 > |x| > epsilon.
  //
  template <class T, class Policy>
  T expm1_imp(T x, const mpl::int_<0>&, const Policy& pol)
  {
     BOOST_MATH_STD_USING
  
     T a = fabs(x);
     if(a > T(0.5f))
     {
        if(a >= tools::log_max_value<T>())
        {
           if(x > 0)
              return policies::raise_overflow_error<T>("boost::math::expm1<%1%>(%1%)", 0, pol);
           return -1;
        }
        return exp(x) - T(1);
     }
     if(a < tools::epsilon<T>())
        return x;
     detail::expm1_series<T> s(x);
     boost::uintmax_t max_iter = policies::get_max_series_iterations<Policy>();
  #if !BOOST_WORKAROUND(__BORLANDC__, BOOST_TESTED_AT(0x582)) && !BOOST_WORKAROUND(__EDG_VERSION__, <= 245)
     T result = tools::sum_series(s, policies::digits<T, Policy>(), max_iter);
  #else
     T zero = 0;
     T result = tools::sum_series(s, policies::digits<T, Policy>(), max_iter, zero);
  #endif
     policies::check_series_iterations("boost::math::expm1<%1%>(%1%)", max_iter, pol);
     return result;
  }
  
  template <class T, class P>
  T expm1_imp(T x, const mpl::int_<53>&, const P& pol)
  {
     BOOST_MATH_STD_USING
  
     T a = fabs(x);
     if(a > T(0.5L))
     {
        if(a >= tools::log_max_value<T>())
        {
           if(x > 0)
              return policies::raise_overflow_error<T>("boost::math::expm1<%1%>(%1%)", 0, pol);
           return -1;
        }
        return exp(x) - T(1);
     }
     if(a < tools::epsilon<T>())
        return x;
  
     static const float Y = 0.10281276702880859e1f;
     static const T n[] = { -0.28127670288085937e-1, 0.51278186299064534e0, -0.6310029069350198e-1, 0.11638457975729296e-1, -0.52143390687521003e-3, 0.21491399776965688e-4 };
     static const T d[] = { 1, -0.45442309511354755e0, 0.90850389570911714e-1, -0.10088963629815502e-1, 0.63003407478692265e-3, -0.17976570003654402e-4 };
  
     T result = x * Y + x * tools::evaluate_polynomial(n, x) / tools::evaluate_polynomial(d, x);
     return result;
  }
  
  template <class T, class P>
  T expm1_imp(T x, const mpl::int_<64>&, const P& pol)
  {
     BOOST_MATH_STD_USING
  
     T a = fabs(x);
     if(a > T(0.5L))
     {
        if(a >= tools::log_max_value<T>())
        {
           if(x > 0)
              return policies::raise_overflow_error<T>("boost::math::expm1<%1%>(%1%)", 0, pol);
           return -1;
        }
        return exp(x) - T(1);
     }
     if(a < tools::epsilon<T>())
        return x;
  
     static const float Y = 0.10281276702880859375e1f;
     static const T n[] = { 
        -0.281276702880859375e-1L, 
         0.512980290285154286358e0L, 
         -0.667758794592881019644e-1L,
         0.131432469658444745835e-1L,
         -0.72303795326880286965e-3L,
         0.447441185192951335042e-4L,
         -0.714539134024984593011e-6L
     };
     static const T d[] = { 
        1, 
        -0.461477618025562520389e0L,
        0.961237488025708540713e-1L,
        -0.116483957658204450739e-1L,
        0.873308008461557544458e-3L,
        -0.387922804997682392562e-4L,
        0.807473180049193557294e-6L
     };
  
     T result = x * Y + x * tools::evaluate_polynomial(n, x) / tools::evaluate_polynomial(d, x);
     return result;
  }
  
  template <class T, class P>
  T expm1_imp(T x, const mpl::int_<113>&, const P& pol)
  {
     BOOST_MATH_STD_USING
  
     T a = fabs(x);
     if(a > T(0.5L))
     {
        if(a >= tools::log_max_value<T>())
        {
           if(x > 0)
              return policies::raise_overflow_error<T>("boost::math::expm1<%1%>(%1%)", 0, pol);
           return -1;
        }
        return exp(x) - T(1);
     }
     if(a < tools::epsilon<T>())
        return x;
  
     static const float Y = 0.10281276702880859375e1f;
     static const T n[] = { 
        -0.28127670288085937499999999999999999854e-1L,
        0.51278156911210477556524452177540792214e0L,
        -0.63263178520747096729500254678819588223e-1L,
        0.14703285606874250425508446801230572252e-1L,
        -0.8675686051689527802425310407898459386e-3L,
        0.88126359618291165384647080266133492399e-4L,
        -0.25963087867706310844432390015463138953e-5L,
        0.14226691087800461778631773363204081194e-6L,
        -0.15995603306536496772374181066765665596e-8L,
        0.45261820069007790520447958280473183582e-10L
     };
     static const T d[] = { 
        1,
        -0.45441264709074310514348137469214538853e0L,
        0.96827131936192217313133611655555298106e-1L,
        -0.12745248725908178612540554584374876219e-1L,
        0.11473613871583259821612766907781095472e-2L,
        -0.73704168477258911962046591907690764416e-4L,
        0.34087499397791555759285503797256103259e-5L,
        -0.11114024704296196166272091230695179724e-6L,
        0.23987051614110848595909588343223896577e-8L,
        -0.29477341859111589208776402638429026517e-10L,
        0.13222065991022301420255904060628100924e-12L
     };
  
     T result = x * Y + x * tools::evaluate_polynomial(n, x) / tools::evaluate_polynomial(d, x);
     return result;
  }
  
  } // namespace detail
  
  template <class T, class Policy>
  inline typename tools::promote_args<T>::type expm1(T x, const Policy& /* pol */)
  {
     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_c<
        ::std::numeric_limits<result_type>::is_specialized == 0,
        mpl::int_<0>,  // no numeric_limits, use generic solution
        typename mpl::if_<
           typename mpl::less_equal<precision_type, mpl::int_<53> >::type,
           mpl::int_<53>,  // double
           typename mpl::if_<
              typename mpl::less_equal<precision_type, mpl::int_<64> >::type,
              mpl::int_<64>, // 80-bit long double
              typename mpl::if_<
                 typename mpl::less_equal<precision_type, mpl::int_<113> >::type,
                 mpl::int_<113>, // 128-bit long double
                 mpl::int_<0> // too many bits, use generic version.
              >::type
           >::type
        >::type
     >::type tag_type;
  
     return policies::checked_narrowing_cast<result_type, forwarding_policy>(detail::expm1_imp(
        static_cast<value_type>(x),
        tag_type(), forwarding_policy()), "boost::math::expm1<%1%>(%1%)");
  }
  
  #ifdef expm1
  #  ifndef BOOST_HAS_expm1
  #     define BOOST_HAS_expm1
  #  endif
  #  undef expm1
  #endif
  
  #if defined(BOOST_HAS_EXPM1) && !(defined(__osf__) && defined(__DECCXX_VER))
  #  ifdef BOOST_MATH_USE_C99
  inline float expm1(float x, const policies::policy<>&){ return ::expm1f(x); }
  #     ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS
  inline long double expm1(long double x, const policies::policy<>&){ return ::expm1l(x); }
  #     endif
  #  else
  inline float expm1(float x, const policies::policy<>&){ return ::expm1(x); }
  #  endif
  inline double expm1(double x, const policies::policy<>&){ return ::expm1(x); }
  #endif
  
  template <class T>
  inline typename tools::promote_args<T>::type expm1(T x)
  {
     return expm1(x, policies::policy<>());
  }
  
  #if BOOST_WORKAROUND(__BORLANDC__, BOOST_TESTED_AT(0x564))
  inline float expm1(float z)
  {
     return expm1<float>(z);
  }
  inline double expm1(double z)
  {
     return expm1<double>(z);
  }
  #ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS
  inline long double expm1(long double z)
  {
     return expm1<long double>(z);
  }
  #endif
  #endif
  
  } // namespace math
  } // namespace boost
  
  #endif // BOOST_MATH_HYPOT_INCLUDED
  
  
  
  
  