//  (C) Copyright John Maddock 2005.

//  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)

#ifndef BOOST_MATH_COMPLEX_ACOS_INCLUDED

#define BOOST_MATH_COMPLEX_ACOS_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> acos(const std::complex<T>& z)

{

   //

   // This implementation is a transcription of the pseudo-code in:

   //

   // "Implementing the Complex Arcsine and Arccosine Functions using Exception Handling."

   // T E Hull, Thomas F Fairgrieve and Ping Tak Peter Tang.

   // ACM Transactions on Mathematical Software, Vol 23, No 3, Sept 1997.

   //

   //

   // These static constants should really be in a maths constants library:

   //

   static const T one = static_cast<T>(1);

   //static const T two = static_cast<T>(2);

   static const T half = static_cast<T>(0.5L);

   static const T a_crossover = static_cast<T>(1.5L);

   static const T b_crossover = static_cast<T>(0.6417L);

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

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

   static const T log_two = static_cast<T>(0.69314718055994530941723212145817657L);

   static const T quarter_pi = static_cast<T>(0.78539816339744830961566084581987572L);

   //

   // Get real and imaginary parts, discard the signs as we can 

   // figure out the sign of the result later:

   //

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

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

   T real, imag; // these hold our result

   // 

   // Handle special cases specified by the C99 standard,

   // many of these special cases aren't really needed here,

   // but doing it this way prevents overflow/underflow arithmetic

   // in the main body of the logic, which may trip up some machines:

   //

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

   {

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

      {

         real = quarter_pi;

         imag = std::numeric_limits<T>::infinity();

      }

      else if(detail::test_is_nan(y))

      {

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

      }

      else

      {

         // y is not infinity or nan:

         real = 0;

         imag = std::numeric_limits<T>::infinity();

      }

   }

   else if(detail::test_is_nan(x))

   {

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

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

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

   }

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

   {

      real = half_pi;

      imag = std::numeric_limits<T>::infinity();

   }

   else if(detail::test_is_nan(y))

   {

      return std::complex<T>((x == 0) ? half_pi : y, y);

   }

   else

   {

      //

      // What follows is the regular Hull et al code,

      // begin with the special case for real numbers:

      //

      if((y == 0) && (x <= one))

         return std::complex<T>((x == 0) ? half_pi : std::acos(z.real()));

      //

      // Figure out if our input is within the "safe area" identified by Hull et al.

      // This would be more efficient with portable floating point exception handling;

      // fortunately the quantities M and u identified by Hull et al (figure 3), 

      // match with the max and min methods of numeric_limits<T>.

      //

      T safe_max = detail::safe_max(static_cast<T>(8));

      T safe_min = detail::safe_min(static_cast<T>(4));

      T xp1 = one + x;

      T xm1 = x - one;

      if((x < safe_max) && (x > safe_min) && (y < safe_max) && (y > safe_min))

      {

         T yy = y * y;

         T r = std::sqrt(xp1*xp1 + yy);

         T s = std::sqrt(xm1*xm1 + yy);

         T a = half * (r + s);

         T b = x / a;

         if(b <= b_crossover)

         {

            real = std::acos(b);

         }

         else

         {

            T apx = a + x;

            if(x <= one)

            {

               real = std::atan(std::sqrt(half * apx * (yy /(r + xp1) + (s-xm1)))/x);

            }

            else

            {

               real = std::atan((y * std::sqrt(half * (apx/(r + xp1) + apx/(s+xm1))))/x);

            }

         }

         if(a <= a_crossover)

         {

            T am1;

            if(x < one)

            {

               am1 = half * (yy/(r + xp1) + yy/(s - xm1));

            }

            else

            {

               am1 = half * (yy/(r + xp1) + (s + xm1));

            }

            imag = boost::math::log1p(am1 + std::sqrt(am1 * (a + one)));

         }

         else

         {

            imag = std::log(a + std::sqrt(a*a - one));

         }

      }

      else

      {

         //

         // This is the Hull et al exception handling code from Fig 6 of their paper:

         //

         if(y <= (std::numeric_limits<T>::epsilon() * std::fabs(xm1)))

         {

            if(x < one)

            {

               real = std::acos(x);

               imag = y / std::sqrt(xp1*(one-x));

            }

            else

            {

               real = 0;

               if(((std::numeric_limits<T>::max)() / xp1) > xm1)

               {

                  // xp1 * xm1 won't overflow:

                  imag = boost::math::log1p(xm1 + std::sqrt(xp1*xm1));

               }

               else

               {

                  imag = log_two + std::log(x);

               }

            }

         }

         else if(y <= safe_min)

         {

            // There is an assumption in Hull et al's analysis that

            // if we get here then x == 1.  This is true for all "good"

            // machines where :

            // 

            // E^2 > 8*sqrt(u); with:

            //

            // E =  std::numeric_limits<T>::epsilon()

            // u = (std::numeric_limits<T>::min)()

            //

            // Hull et al provide alternative code for "bad" machines

            // but we have no way to test that here, so for now just assert

            // on the assumption:

            //

            BOOST_ASSERT(x == 1);

            real = std::sqrt(y);

            imag = std::sqrt(y);

         }

         else if(std::numeric_limits<T>::epsilon() * y - one >= x)

         {

            real = half_pi;

            imag = log_two + std::log(y);

         }

         else if(x > one)

         {

            real = std::atan(y/x);

            T xoy = x/y;

            imag = log_two + std::log(y) + half * boost::math::log1p(xoy*xoy);

         }

         else

         {

            real = half_pi;

            T a = std::sqrt(one + y*y);

            imag = half * boost::math::log1p(static_cast<T>(2)*y*(y+a));

         }

      }

   }

   //

   // Finish off by working out the sign of the result:

   //

   if(z.real() < 0)

      real = s_pi - real;

   if(z.imag() > 0)

      imag = -imag;

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

}

} } // namespaces

#endif // BOOST_MATH_COMPLEX_ACOS_INCLUDED