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This is the documentation for a snapshot of the develop branch, built from commit 87fa3edb00.

boost/math/special_functions/log1p.hpp

//  (C) Copyright John Maddock 2005-2006.
//  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_LOG1P_INCLUDED
#define BOOST_MATH_LOG1P_INCLUDED

#ifdef _MSC_VER
#pragma once
#pragma warning(push)
#pragma warning(disable:4702) // Unreachable code (release mode only warning)
#endif

#include <cmath>
#include <cstdint>
#include <limits>
#include <boost/math/tools/config.hpp>
#include <boost/math/tools/series.hpp>
#include <boost/math/tools/rational.hpp>
#include <boost/math/tools/big_constant.hpp>
#include <boost/math/policies/error_handling.hpp>
#include <boost/math/special_functions/math_fwd.hpp>
#include <boost/math/tools/assert.hpp>
#include <boost/math/special_functions/fpclassify.hpp>

#if defined(__GNUC__) && defined(BOOST_MATH_USE_FLOAT128)
//
// This is the only way we can avoid
// warning: non-standard suffix on floating constant [-Wpedantic]
// when building with -Wall -pedantic.  Neither __extension__
// nor #pragma diagnostic ignored work :(
//
#pragma GCC system_header
#endif

namespace boost{ namespace math{

namespace detail
{
  // Functor log1p_series returns the next term in the Taylor series
  //   pow(-1, k-1)*pow(x, k) / k
  // each time that operator() is invoked.
  //
  template <class T>
  struct log1p_series
  {
     typedef T result_type;

     log1p_series(T x)
        : k(0), m_mult(-x), m_prod(-1){}

     T operator()()
     {
        m_prod *= m_mult;
        return m_prod / ++k;
     }

     int count()const
     {
        return k;
     }

  private:
     int k;
     const T m_mult;
     T m_prod;
     log1p_series(const log1p_series&);
     log1p_series& operator=(const log1p_series&);
  };

// Algorithm log1p is part of C99, but is not yet provided by many compilers.
//
// This version uses a Taylor series expansion for 0.5 > x > epsilon, which may
// require up to std::numeric_limits<T>::digits+1 terms to be calculated.
// It would be much more efficient to use the equivalence:
//   log(1+x) == (log(1+x) * x) / ((1-x) - 1)
// Unfortunately many optimizing compilers make such a mess of this, that
// it performs no better than log(1+x): which is to say not very well at all.
//
template <class T, class Policy>
T log1p_imp(T const & x, const Policy& pol, const std::integral_constant<int, 0>&)
{ // The function returns the natural logarithm of 1 + x.
   typedef typename tools::promote_args<T>::type result_type;
   BOOST_MATH_STD_USING

   static const char* function = "boost::math::log1p<%1%>(%1%)";

   if((x < -1) || (boost::math::isnan)(x))
      return policies::raise_domain_error<T>(
         function, "log1p(x) requires x > -1, but got x = %1%.", x, pol);
   if(x == -1)
      return -policies::raise_overflow_error<T>(
         function, nullptr, pol);

   result_type a = abs(result_type(x));
   if(a > result_type(0.5f))
      return log(1 + result_type(x));
   // Note that without numeric_limits specialisation support,
   // epsilon just returns zero, and our "optimisation" will always fail:
   if(a < tools::epsilon<result_type>())
      return x;
   detail::log1p_series<result_type> s(x);
   std::uintmax_t max_iter = policies::get_max_series_iterations<Policy>();

   result_type result = tools::sum_series(s, policies::get_epsilon<result_type, Policy>(), max_iter);

   policies::check_series_iterations<T>(function, max_iter, pol);
   return result;
}

template <class T, class Policy>
T log1p_imp(T const& x, const Policy& pol, const std::integral_constant<int, 53>&)
{ // The function returns the natural logarithm of 1 + x.
   BOOST_MATH_STD_USING

   static const char* function = "boost::math::log1p<%1%>(%1%)";

   if(x < -1)
      return policies::raise_domain_error<T>(
         function, "log1p(x) requires x > -1, but got x = %1%.", x, pol);
   if(x == -1)
      return -policies::raise_overflow_error<T>(
         function, nullptr, pol);

   T a = fabs(x);
   if(a > 0.5f)
      return log(1 + x);
   // Note that without numeric_limits specialisation support,
   // epsilon just returns zero, and our "optimisation" will always fail:
   if(a < tools::epsilon<T>())
      return x;

   // Maximum Deviation Found:                     1.846e-017
   // Expected Error Term:                         1.843e-017
   // Maximum Relative Change in Control Points:   8.138e-004
   // Max Error found at double precision =        3.250766e-016
   static const T P[] = {
       0.15141069795941984e-16L,
       0.35495104378055055e-15L,
       0.33333333333332835L,
       0.99249063543365859L,
       1.1143969784156509L,
       0.58052937949269651L,
       0.13703234928513215L,
       0.011294864812099712L
     };
   static const T Q[] = {
       1L,
       3.7274719063011499L,
       5.5387948649720334L,
       4.159201143419005L,
       1.6423855110312755L,
       0.31706251443180914L,
       0.022665554431410243L,
       -0.29252538135177773e-5L
     };

   T result = 1 - x / 2 + tools::evaluate_polynomial(P, x) / tools::evaluate_polynomial(Q, x);
   result *= x;

   return result;
}

template <class T, class Policy>
T log1p_imp(T const& x, const Policy& pol, const std::integral_constant<int, 64>&)
{ // The function returns the natural logarithm of 1 + x.
   BOOST_MATH_STD_USING

   static const char* function = "boost::math::log1p<%1%>(%1%)";

   if(x < -1)
      return policies::raise_domain_error<T>(
         function, "log1p(x) requires x > -1, but got x = %1%.", x, pol);
   if(x == -1)
      return -policies::raise_overflow_error<T>(
         function, nullptr, pol);

   T a = fabs(x);
   if(a > 0.5f)
      return log(1 + x);
   // Note that without numeric_limits specialisation support,
   // epsilon just returns zero, and our "optimisation" will always fail:
   if(a < tools::epsilon<T>())
      return x;

   // Maximum Deviation Found:                     8.089e-20
   // Expected Error Term:                         8.088e-20
   // Maximum Relative Change in Control Points:   9.648e-05
   // Max Error found at long double precision =   2.242324e-19
   static const T P[] = {
      BOOST_MATH_BIG_CONSTANT(T, 64, -0.807533446680736736712e-19),
      BOOST_MATH_BIG_CONSTANT(T, 64, -0.490881544804798926426e-18),
      BOOST_MATH_BIG_CONSTANT(T, 64, 0.333333333333333373941),
      BOOST_MATH_BIG_CONSTANT(T, 64, 1.17141290782087994162),
      BOOST_MATH_BIG_CONSTANT(T, 64, 1.62790522814926264694),
      BOOST_MATH_BIG_CONSTANT(T, 64, 1.13156411870766876113),
      BOOST_MATH_BIG_CONSTANT(T, 64, 0.408087379932853785336),
      BOOST_MATH_BIG_CONSTANT(T, 64, 0.0706537026422828914622),
      BOOST_MATH_BIG_CONSTANT(T, 64, 0.00441709903782239229447)
   };
   static const T Q[] = {
      BOOST_MATH_BIG_CONSTANT(T, 64, 1.0),
      BOOST_MATH_BIG_CONSTANT(T, 64, 4.26423872346263928361),
      BOOST_MATH_BIG_CONSTANT(T, 64, 7.48189472704477708962),
      BOOST_MATH_BIG_CONSTANT(T, 64, 6.94757016732904280913),
      BOOST_MATH_BIG_CONSTANT(T, 64, 3.6493508622280767304),
      BOOST_MATH_BIG_CONSTANT(T, 64, 1.06884863623790638317),
      BOOST_MATH_BIG_CONSTANT(T, 64, 0.158292216998514145947),
      BOOST_MATH_BIG_CONSTANT(T, 64, 0.00885295524069924328658),
      BOOST_MATH_BIG_CONSTANT(T, 64, -0.560026216133415663808e-6)
   };

   T result = 1 - x / 2 + tools::evaluate_polynomial(P, x) / tools::evaluate_polynomial(Q, x);
   result *= x;

   return result;
}

template <class T, class Policy>
T log1p_imp(T const& x, const Policy& pol, const std::integral_constant<int, 24>&)
{ // The function returns the natural logarithm of 1 + x.
   BOOST_MATH_STD_USING

   static const char* function = "boost::math::log1p<%1%>(%1%)";

   if(x < -1)
      return policies::raise_domain_error<T>(
         function, "log1p(x) requires x > -1, but got x = %1%.", x, pol);
   if(x == -1)
      return -policies::raise_overflow_error<T>(
         function, nullptr, pol);

   T a = fabs(x);
   if(a > 0.5f)
      return log(1 + x);
   // Note that without numeric_limits specialisation support,
   // epsilon just returns zero, and our "optimisation" will always fail:
   if(a < tools::epsilon<T>())
      return x;

   // Maximum Deviation Found:                     6.910e-08
   // Expected Error Term:                         6.910e-08
   // Maximum Relative Change in Control Points:   2.509e-04
   // Max Error found at double precision =        6.910422e-08
   // Max Error found at float precision =         8.357242e-08
   static const T P[] = {
      -0.671192866803148236519e-7L,
      0.119670999140731844725e-6L,
      0.333339469182083148598L,
      0.237827183019664122066L
   };
   static const T Q[] = {
      1L,
      1.46348272586988539733L,
      0.497859871350117338894L,
      -0.00471666268910169651936L
   };

   T result = 1 - x / 2 + tools::evaluate_polynomial(P, x) / tools::evaluate_polynomial(Q, x);
   result *= x;

   return result;
}

template <class T, class Policy, class tag>
struct log1p_initializer
{
   struct init
   {
      init()
      {
         do_init(tag());
      }
      template <int N>
      static void do_init(const std::integral_constant<int, N>&){}
      static void do_init(const std::integral_constant<int, 64>&)
      {
         boost::math::log1p(static_cast<T>(0.25), Policy());
      }
      void force_instantiate()const{}
   };
   static const init initializer;
   static void force_instantiate()
   {
      initializer.force_instantiate();
   }
};

template <class T, class Policy, class tag>
const typename log1p_initializer<T, Policy, tag>::init log1p_initializer<T, Policy, tag>::initializer;


} // namespace detail

template <class T, class Policy>
inline typename tools::promote_args<T>::type log1p(T x, const Policy&)
{
   typedef typename tools::promote_args<T>::type result_type;
   typedef typename policies::evaluation<result_type, Policy>::type value_type;
   typedef typename policies::precision<result_type, Policy>::type precision_type;
   typedef typename policies::normalise<
      Policy,
      policies::promote_float<false>,
      policies::promote_double<false>,
      policies::discrete_quantile<>,
      policies::assert_undefined<> >::type forwarding_policy;

   typedef std::integral_constant<int,
      precision_type::value <= 0 ? 0 :
      precision_type::value <= 53 ? 53 :
      precision_type::value <= 64 ? 64 : 0
   > tag_type;

   detail::log1p_initializer<value_type, forwarding_policy, tag_type>::force_instantiate();

   return policies::checked_narrowing_cast<result_type, forwarding_policy>(
      detail::log1p_imp(static_cast<value_type>(x), forwarding_policy(), tag_type()), "boost::math::log1p<%1%>(%1%)");
}

#ifdef log1p
#  ifndef BOOST_HAS_LOG1P
#     define BOOST_HAS_LOG1P
#  endif
#  undef log1p
#endif

#if defined(BOOST_HAS_LOG1P) && !(defined(__osf__) && defined(__DECCXX_VER))
#  ifdef BOOST_MATH_USE_C99
template <class Policy>
inline float log1p(float x, const Policy& pol)
{
   if(x < -1)
      return policies::raise_domain_error<float>(
         "log1p<%1%>(%1%)", "log1p(x) requires x > -1, but got x = %1%.", x, pol);
   if(x == -1)
      return -policies::raise_overflow_error<float>(
         "log1p<%1%>(%1%)", nullptr, pol);
   return ::log1pf(x);
}
#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS
template <class Policy>
inline long double log1p(long double x, const Policy& pol)
{
   if(x < -1)
      return policies::raise_domain_error<long double>(
         "log1p<%1%>(%1%)", "log1p(x) requires x > -1, but got x = %1%.", x, pol);
   if(x == -1)
      return -policies::raise_overflow_error<long double>(
         "log1p<%1%>(%1%)", nullptr, pol);
   return ::log1pl(x);
}
#endif
#else
template <class Policy>
inline float log1p(float x, const Policy& pol)
{
   if(x < -1)
      return policies::raise_domain_error<float>(
         "log1p<%1%>(%1%)", "log1p(x) requires x > -1, but got x = %1%.", x, pol);
   if(x == -1)
      return -policies::raise_overflow_error<float>(
         "log1p<%1%>(%1%)", nullptr, pol);
   return ::log1p(x);
}
#endif
template <class Policy>
inline double log1p(double x, const Policy& pol)
{
   if(x < -1)
      return policies::raise_domain_error<double>(
         "log1p<%1%>(%1%)", "log1p(x) requires x > -1, but got x = %1%.", x, pol);
   if(x == -1)
      return -policies::raise_overflow_error<double>(
         "log1p<%1%>(%1%)", nullptr, pol);
   return ::log1p(x);
}
#elif defined(_MSC_VER) && (BOOST_MSVC >= 1400)
//
// You should only enable this branch if you are absolutely sure
// that your compilers optimizer won't mess this code up!!
// Currently tested with VC8 and Intel 9.1.
//
template <class Policy>
inline double log1p(double x, const Policy& pol)
{
   if(x < -1)
      return policies::raise_domain_error<double>(
         "log1p<%1%>(%1%)", "log1p(x) requires x > -1, but got x = %1%.", x, pol);
   if(x == -1)
      return -policies::raise_overflow_error<double>(
         "log1p<%1%>(%1%)", nullptr, pol);
   double u = 1+x;
   if(u == 1.0)
      return x;
   else
      return ::log(u)*(x/(u-1.0));
}
template <class Policy>
inline float log1p(float x, const Policy& pol)
{
   return static_cast<float>(boost::math::log1p(static_cast<double>(x), pol));
}
#ifndef _WIN32_WCE
//
// For some reason this fails to compile under WinCE...
// Needs more investigation.
//
template <class Policy>
inline long double log1p(long double x, const Policy& pol)
{
   if(x < -1)
      return policies::raise_domain_error<long double>(
         "log1p<%1%>(%1%)", "log1p(x) requires x > -1, but got x = %1%.", x, pol);
   if(x == -1)
      return -policies::raise_overflow_error<long double>(
         "log1p<%1%>(%1%)", nullptr, pol);
   long double u = 1+x;
   if(u == 1.0)
      return x;
   else
      return ::logl(u)*(x/(u-1.0));
}
#endif
#endif

template <class T>
inline typename tools::promote_args<T>::type log1p(T x)
{
   return boost::math::log1p(x, policies::policy<>());
}
//
// Compute log(1+x)-x:
//
template <class T, class Policy>
inline typename tools::promote_args<T>::type
   log1pmx(T x, const Policy& pol)
{
   typedef typename tools::promote_args<T>::type result_type;
   BOOST_MATH_STD_USING
   static const char* function = "boost::math::log1pmx<%1%>(%1%)";

   if(x < -1)
      return policies::raise_domain_error<T>(
         function, "log1pmx(x) requires x > -1, but got x = %1%.", x, pol);
   if(x == -1)
      return -policies::raise_overflow_error<T>(
         function, nullptr, pol);

   result_type a = abs(result_type(x));
   if(a > result_type(0.95f))
      return log(1 + result_type(x)) - result_type(x);
   // Note that without numeric_limits specialisation support,
   // epsilon just returns zero, and our "optimisation" will always fail:
   if(a < tools::epsilon<result_type>())
      return -x * x / 2;
   boost::math::detail::log1p_series<T> s(x);
   s();
   std::uintmax_t max_iter = policies::get_max_series_iterations<Policy>();

   T result = boost::math::tools::sum_series(s, policies::get_epsilon<T, Policy>(), max_iter);

   policies::check_series_iterations<T>(function, max_iter, pol);
   return result;
}

template <class T>
inline typename tools::promote_args<T>::type log1pmx(T x)
{
   return log1pmx(x, policies::policy<>());
}

} // namespace math
} // namespace boost

#ifdef _MSC_VER
#pragma warning(pop)
#endif

#endif // BOOST_MATH_LOG1P_INCLUDED