//  (C) Copyright John Maddock 2005.

 //  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_COMPLEX_ATANH_INCLUDED

 #define BOOST_MATH_COMPLEX_ATANH_INCLUDED

 #ifndef BOOST_MATH_COMPLEX_DETAILS_INCLUDED

 #  include <boost/math/complex/details.hpp>

 #endif

 #ifndef BOOST_MATH_LOG1P_INCLUDED

 #  include <boost/math/special_functions/log1p.hpp>

 #endif

 #include <boost/assert.hpp>

 #ifdef BOOST_NO_STDC_NAMESPACE

 namespace std{ using ::sqrt; using ::fabs; using ::acos; using ::asin; using ::atan; using ::atan2; }

 #endif

 namespace boost{ namespace math{

 template<class T> 

 std::complex<T> atanh(const std::complex<T>& z)

 {

    //

    // References:

    //

    // Eric W. Weisstein. "Inverse Hyperbolic Tangent." 

    // From MathWorld--A Wolfram Web Resource. 

    // http://mathworld.wolfram.com/InverseHyperbolicTangent.html

    //

    // Also: The Wolfram Functions Site,

    // http://functions.wolfram.com/ElementaryFunctions/ArcTanh/

    //

    // Also "Abramowitz and Stegun. Handbook of Mathematical Functions."

    // at : http://jove.prohosting.com/~skripty/toc.htm

    //

    static const T half_pi = static_cast<T>(1.57079632679489661923132169163975144L);

    static const T pi = static_cast<T>(3.141592653589793238462643383279502884197L);

    static const T one = static_cast<T>(1.0L);

    static const T two = static_cast<T>(2.0L);

    static const T four = static_cast<T>(4.0L);

    static const T zero = static_cast<T>(0);

    static const T a_crossover = static_cast<T>(0.3L);

    T x = std::fabs(z.real());

    T y = std::fabs(z.imag());

    T real, imag;  // our results

    T safe_upper = detail::safe_max(two);

    T safe_lower = detail::safe_min(static_cast<T>(2));

    //

    // Begin by handling the special cases specified in C99:

    //

    if(detail::test_is_nan(x))

    {

       if(detail::test_is_nan(y))

          return std::complex<T>(x, x);

       else if(std::numeric_limits<T>::has_infinity && (y == std::numeric_limits<T>::infinity()))

          return std::complex<T>(0, ((z.imag() < 0) ? -half_pi : half_pi));

       else

          return std::complex<T>(x, x);

    }

    else if(detail::test_is_nan(y))

    {

       if(x == 0)

          return std::complex<T>(x, y);

       if(std::numeric_limits<T>::has_infinity && (x == std::numeric_limits<T>::infinity()))

          return std::complex<T>(0, y);

       else

          return std::complex<T>(y, y);

    }

    else if((x > safe_lower) && (x < safe_upper) && (y > safe_lower) && (y < safe_upper))

    {

       T xx = x*x;

       T yy = y*y;

       T x2 = x * two;

       ///

       // The real part is given by:

       // 

       // real(atanh(z)) == log((1 + x^2 + y^2 + 2x) / (1 + x^2 + y^2 - 2x))

       // 

       // However, when x is either large (x > 1/E) or very small

       // (x < E) then this effectively simplifies

       // to log(1), leading to wildly inaccurate results.  

       // By dividing the above (top and bottom) by (1 + x^2 + y^2) we get:

       //

       // real(atanh(z)) == log((1 + (2x / (1 + x^2 + y^2))) / (1 - (-2x / (1 + x^2 + y^2))))

       //

       // which is much more sensitive to the value of x, when x is not near 1

       // (remember we can compute log(1+x) for small x very accurately).

       //

       // The cross-over from one method to the other has to be determined

       // experimentally, the value used below appears correct to within a 

       // factor of 2 (and there are larger errors from other parts

       // of the input domain anyway).

       //

       T alpha = two*x / (one + xx + yy);

       if(alpha < a_crossover)

       {

          real = boost::math::log1p(alpha) - boost::math::log1p(-alpha);

       }

       else

       {

          T xm1 = x - one;

          real = boost::math::log1p(x2 + xx + yy) - std::log(xm1*xm1 + yy);

       }

       real /= four;

       if(z.real() < 0)

          real = -real;

       imag = std::atan2((y * two), (one - xx - yy));

       imag /= two;

       if(z.imag() < 0)

          imag = -imag;

    }

    else

    {

       //

       // This section handles exception cases that would normally cause

       // underflow or overflow in the main formulas.

       //

       // Begin by working out the real part, we need to approximate

       //    alpha = 2x / (1 + x^2 + y^2)

       // without either overflow or underflow in the squared terms.

       //

       T alpha = 0;

       if(x >= safe_upper)

       {

          // this is really a test for infinity, 

          // but we may not have the necessary numeric_limits support:

          if((x > (std::numeric_limits<T>::max)()) || (y > (std::numeric_limits<T>::max)()))

          {

             alpha = 0;

          }

          else if(y >= safe_upper)

          {

             // Big x and y: divide alpha through by x*y:

             alpha = (two/y) / (x/y + y/x);

          }

          else if(y > one)

          {

             // Big x: divide through by x:

             alpha = two / (x + y*y/x);

          }

          else

          {

             // Big x small y, as above but neglect y^2/x:

             alpha = two/x;

          }

       }

       else if(y >= safe_upper)

       {

          if(x > one)

          {

             // Big y, medium x, divide through by y:

             alpha = (two*x/y) / (y + x*x/y);

          }

          else

          {

             // Small x and y, whatever alpha is, it's too small to calculate:

             alpha = 0;

          }

       }

       else

       {

          // one or both of x and y are small, calculate divisor carefully:

          T div = one;

          if(x > safe_lower)

             div += x*x;

          if(y > safe_lower)

             div += y*y;

          alpha = two*x/div;

       }

       if(alpha < a_crossover)

       {

          real = boost::math::log1p(alpha) - boost::math::log1p(-alpha);

       }

       else

       {

          // We can only get here as a result of small y and medium sized x,

          // we can simply neglect the y^2 terms:

          BOOST_ASSERT(x >= safe_lower);

          BOOST_ASSERT(x <= safe_upper);

          //BOOST_ASSERT(y <= safe_lower);

          T xm1 = x - one;

          real = std::log(1 + two*x + x*x) - std::log(xm1*xm1);

       }

       real /= four;

       if(z.real() < 0)

          real = -real;

       //

       // Now handle imaginary part, this is much easier,

       // if x or y are large, then the formula:

       //    atan2(2y, 1 - x^2 - y^2)

       // evaluates to +-(PI - theta) where theta is negligible compared to PI.

       //

       if((x >= safe_upper) || (y >= safe_upper))

       {

          imag = pi;

       }

       else if(x <= safe_lower)

       {

          //

          // If both x and y are small then atan(2y),

          // otherwise just x^2 is negligible in the divisor:

          //

          if(y <= safe_lower)

             imag = std::atan2(two*y, one);

          else

          {

             if((y == zero) && (x == zero))

                imag = 0;

             else

                imag = std::atan2(two*y, one - y*y);

          }

       }

       else

       {

          //

          // y^2 is negligible:

          //

          if((y == zero) && (x == one))

             imag = 0;

          else

             imag = std::atan2(two*y, 1 - x*x);

       }

       imag /= two;

       if(z.imag() < 0)

          imag = -imag;

    }

    return std::complex<T>(real, imag);

 }

 } } // namespaces

 #endif // BOOST_MATH_COMPLEX_ATANH_INCLUDED