boost/math/special_functions/asinh.hpp
// boost asinh.hpp header file
// (C) Copyright Eric Ford & Hubert Holin 2001.
// Distributed under 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)
// See http://www.boost.org for updates, documentation, and revision history.
#ifndef BOOST_ASINH_HPP
#define BOOST_ASINH_HPP
#include <cmath>
#include <limits>
#include <string>
#include <stdexcept>
#include <boost/config.hpp>
// This is the inverse of the hyperbolic sine function.
namespace boost
{
namespace math
{
#if defined(__GNUC__) && (__GNUC__ < 3)
// gcc 2.x ignores function scope using declarations,
// put them in the scope of the enclosing namespace instead:
using ::std::abs;
using ::std::sqrt;
using ::std::log;
using ::std::numeric_limits;
#endif
template<typename T>
inline T asinh(const T x)
{
using ::std::abs;
using ::std::sqrt;
using ::std::log;
using ::std::numeric_limits;
T const one = static_cast<T>(1);
T const two = static_cast<T>(2);
static T const taylor_2_bound = sqrt(numeric_limits<T>::epsilon());
static T const taylor_n_bound = sqrt(taylor_2_bound);
static T const upper_taylor_2_bound = one/taylor_2_bound;
static T const upper_taylor_n_bound = one/taylor_n_bound;
if (x >= +taylor_n_bound)
{
if (x > upper_taylor_n_bound)
{
if (x > upper_taylor_2_bound)
{
// approximation by laurent series in 1/x at 0+ order from -1 to 0
return( log( x * two) );
}
else
{
// approximation by laurent series in 1/x at 0+ order from -1 to 1
return( log( x*two + (one/(x*two)) ) );
}
}
else
{
return( log( x + sqrt(x*x+one) ) );
}
}
else if (x <= -taylor_n_bound)
{
return(-asinh(-x));
}
else
{
// approximation by taylor series in x at 0 up to order 2
T result = x;
if (abs(x) >= taylor_2_bound)
{
T x3 = x*x*x;
// approximation by taylor series in x at 0 up to order 4
result -= x3/static_cast<T>(6);
}
return(result);
}
}
}
}
#endif /* BOOST_ASINH_HPP */