Chris@16: //  (C) Copyright John Maddock 2005.
Chris@16: //  Distributed under the Boost Software License, Version 1.0. (See accompanying
Chris@16: //  file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)
Chris@16: 
Chris@16: #ifndef BOOST_MATH_COMPLEX_ASIN_INCLUDED
Chris@16: #define BOOST_MATH_COMPLEX_ASIN_INCLUDED
Chris@16: 
Chris@16: #ifndef BOOST_MATH_COMPLEX_DETAILS_INCLUDED
Chris@16: #  include <boost/math/complex/details.hpp>
Chris@16: #endif
Chris@16: #ifndef BOOST_MATH_LOG1P_INCLUDED
Chris@16: #  include <boost/math/special_functions/log1p.hpp>
Chris@16: #endif
Chris@16: #include <boost/assert.hpp>
Chris@16: 
Chris@16: #ifdef BOOST_NO_STDC_NAMESPACE
Chris@16: namespace std{ using ::sqrt; using ::fabs; using ::acos; using ::asin; using ::atan; using ::atan2; }
Chris@16: #endif
Chris@16: 
Chris@16: namespace boost{ namespace math{
Chris@16: 
Chris@16: template<class T> 
Chris@16: inline std::complex<T> asin(const std::complex<T>& z)
Chris@16: {
Chris@16:    //
Chris@16:    // This implementation is a transcription of the pseudo-code in:
Chris@16:    //
Chris@16:    // "Implementing the complex Arcsine and Arccosine Functions using Exception Handling."
Chris@16:    // T E Hull, Thomas F Fairgrieve and Ping Tak Peter Tang.
Chris@16:    // ACM Transactions on Mathematical Software, Vol 23, No 3, Sept 1997.
Chris@16:    //
Chris@16: 
Chris@16:    //
Chris@16:    // These static constants should really be in a maths constants library,
Chris@16:    // note that we have tweaked the value of a_crossover as per https://svn.boost.org/trac/boost/ticket/7290:
Chris@16:    //
Chris@16:    static const T one = static_cast<T>(1);
Chris@16:    //static const T two = static_cast<T>(2);
Chris@16:    static const T half = static_cast<T>(0.5L);
Chris@16:    static const T a_crossover = static_cast<T>(10);
Chris@16:    static const T b_crossover = static_cast<T>(0.6417L);
Chris@16:    static const T s_pi = boost::math::constants::pi<T>();
Chris@16:    static const T half_pi = s_pi / 2;
Chris@16:    static const T log_two = boost::math::constants::ln_two<T>();
Chris@16:    static const T quarter_pi = s_pi / 4;
Chris@16: #ifdef BOOST_MSVC
Chris@16: #pragma warning(push)
Chris@16: #pragma warning(disable:4127)
Chris@16: #endif
Chris@16:    //
Chris@16:    // Get real and imaginary parts, discard the signs as we can 
Chris@16:    // figure out the sign of the result later:
Chris@16:    //
Chris@16:    T x = std::fabs(z.real());
Chris@16:    T y = std::fabs(z.imag());
Chris@16:    T real, imag;  // our results
Chris@16: 
Chris@16:    //
Chris@16:    // Begin by handling the special cases for infinities and nan's
Chris@16:    // specified in C99, most of this is handled by the regular logic
Chris@16:    // below, but handling it as a special case prevents overflow/underflow
Chris@16:    // arithmetic which may trip up some machines:
Chris@16:    //
Chris@16:    if((boost::math::isnan)(x))
Chris@16:    {
Chris@16:       if((boost::math::isnan)(y))
Chris@16:          return std::complex<T>(x, x);
Chris@16:       if((boost::math::isinf)(y))
Chris@16:       {
Chris@16:          real = x;
Chris@16:          imag = std::numeric_limits<T>::infinity();
Chris@16:       }
Chris@16:       else
Chris@16:          return std::complex<T>(x, x);
Chris@16:    }
Chris@16:    else if((boost::math::isnan)(y))
Chris@16:    {
Chris@16:       if(x == 0)
Chris@16:       {
Chris@16:          real = 0;
Chris@16:          imag = y;
Chris@16:       }
Chris@16:       else if((boost::math::isinf)(x))
Chris@16:       {
Chris@16:          real = y;
Chris@16:          imag = std::numeric_limits<T>::infinity();
Chris@16:       }
Chris@16:       else
Chris@16:          return std::complex<T>(y, y);
Chris@16:    }
Chris@16:    else if((boost::math::isinf)(x))
Chris@16:    {
Chris@16:       if((boost::math::isinf)(y))
Chris@16:       {
Chris@16:          real = quarter_pi;
Chris@16:          imag = std::numeric_limits<T>::infinity();
Chris@16:       }
Chris@16:       else
Chris@16:       {
Chris@16:          real = half_pi;
Chris@16:          imag = std::numeric_limits<T>::infinity();
Chris@16:       }
Chris@16:    }
Chris@16:    else if((boost::math::isinf)(y))
Chris@16:    {
Chris@16:       real = 0;
Chris@16:       imag = std::numeric_limits<T>::infinity();
Chris@16:    }
Chris@16:    else
Chris@16:    {
Chris@16:       //
Chris@16:       // special case for real numbers:
Chris@16:       //
Chris@16:       if((y == 0) && (x <= one))
Chris@16:          return std::complex<T>(std::asin(z.real()), z.imag());
Chris@16:       //
Chris@16:       // Figure out if our input is within the "safe area" identified by Hull et al.
Chris@16:       // This would be more efficient with portable floating point exception handling;
Chris@16:       // fortunately the quantities M and u identified by Hull et al (figure 3), 
Chris@16:       // match with the max and min methods of numeric_limits<T>.
Chris@16:       //
Chris@16:       T safe_max = detail::safe_max(static_cast<T>(8));
Chris@16:       T safe_min = detail::safe_min(static_cast<T>(4));
Chris@16: 
Chris@16:       T xp1 = one + x;
Chris@16:       T xm1 = x - one;
Chris@16: 
Chris@16:       if((x < safe_max) && (x > safe_min) && (y < safe_max) && (y > safe_min))
Chris@16:       {
Chris@16:          T yy = y * y;
Chris@16:          T r = std::sqrt(xp1*xp1 + yy);
Chris@16:          T s = std::sqrt(xm1*xm1 + yy);
Chris@16:          T a = half * (r + s);
Chris@16:          T b = x / a;
Chris@16: 
Chris@16:          if(b <= b_crossover)
Chris@16:          {
Chris@16:             real = std::asin(b);
Chris@16:          }
Chris@16:          else
Chris@16:          {
Chris@16:             T apx = a + x;
Chris@16:             if(x <= one)
Chris@16:             {
Chris@16:                real = std::atan(x/std::sqrt(half * apx * (yy /(r + xp1) + (s-xm1))));
Chris@16:             }
Chris@16:             else
Chris@16:             {
Chris@16:                real = std::atan(x/(y * std::sqrt(half * (apx/(r + xp1) + apx/(s+xm1)))));
Chris@16:             }
Chris@16:          }
Chris@16: 
Chris@16:          if(a <= a_crossover)
Chris@16:          {
Chris@16:             T am1;
Chris@16:             if(x < one)
Chris@16:             {
Chris@16:                am1 = half * (yy/(r + xp1) + yy/(s - xm1));
Chris@16:             }
Chris@16:             else
Chris@16:             {
Chris@16:                am1 = half * (yy/(r + xp1) + (s + xm1));
Chris@16:             }
Chris@16:             imag = boost::math::log1p(am1 + std::sqrt(am1 * (a + one)));
Chris@16:          }
Chris@16:          else
Chris@16:          {
Chris@16:             imag = std::log(a + std::sqrt(a*a - one));
Chris@16:          }
Chris@16:       }
Chris@16:       else
Chris@16:       {
Chris@16:          //
Chris@16:          // This is the Hull et al exception handling code from Fig 3 of their paper:
Chris@16:          //
Chris@16:          if(y <= (std::numeric_limits<T>::epsilon() * std::fabs(xm1)))
Chris@16:          {
Chris@16:             if(x < one)
Chris@16:             {
Chris@16:                real = std::asin(x);
Chris@16:                imag = y / std::sqrt(-xp1*xm1);
Chris@16:             }
Chris@16:             else
Chris@16:             {
Chris@16:                real = half_pi;
Chris@16:                if(((std::numeric_limits<T>::max)() / xp1) > xm1)
Chris@16:                {
Chris@16:                   // xp1 * xm1 won't overflow:
Chris@16:                   imag = boost::math::log1p(xm1 + std::sqrt(xp1*xm1));
Chris@16:                }
Chris@16:                else
Chris@16:                {
Chris@16:                   imag = log_two + std::log(x);
Chris@16:                }
Chris@16:             }
Chris@16:          }
Chris@16:          else if(y <= safe_min)
Chris@16:          {
Chris@16:             // There is an assumption in Hull et al's analysis that
Chris@16:             // if we get here then x == 1.  This is true for all "good"
Chris@16:             // machines where :
Chris@16:             // 
Chris@16:             // E^2 > 8*sqrt(u); with:
Chris@16:             //
Chris@16:             // E =  std::numeric_limits<T>::epsilon()
Chris@16:             // u = (std::numeric_limits<T>::min)()
Chris@16:             //
Chris@16:             // Hull et al provide alternative code for "bad" machines
Chris@16:             // but we have no way to test that here, so for now just assert
Chris@16:             // on the assumption:
Chris@16:             //
Chris@16:             BOOST_ASSERT(x == 1);
Chris@16:             real = half_pi - std::sqrt(y);
Chris@16:             imag = std::sqrt(y);
Chris@16:          }
Chris@16:          else if(std::numeric_limits<T>::epsilon() * y - one >= x)
Chris@16:          {
Chris@16:             real = x/y; // This can underflow!
Chris@16:             imag = log_two + std::log(y);
Chris@16:          }
Chris@16:          else if(x > one)
Chris@16:          {
Chris@16:             real = std::atan(x/y);
Chris@16:             T xoy = x/y;
Chris@16:             imag = log_two + std::log(y) + half * boost::math::log1p(xoy*xoy);
Chris@16:          }
Chris@16:          else
Chris@16:          {
Chris@16:             T a = std::sqrt(one + y*y);
Chris@16:             real = x/a; // This can underflow!
Chris@16:             imag = half * boost::math::log1p(static_cast<T>(2)*y*(y+a));
Chris@16:          }
Chris@16:       }
Chris@16:    }
Chris@16: 
Chris@16:    //
Chris@16:    // Finish off by working out the sign of the result:
Chris@16:    //
Chris@16:    if((boost::math::signbit)(z.real()))
Chris@16:       real = (boost::math::changesign)(real);
Chris@16:    if((boost::math::signbit)(z.imag()))
Chris@16:       imag = (boost::math::changesign)(imag);
Chris@16: 
Chris@16:    return std::complex<T>(real, imag);
Chris@16: #ifdef BOOST_MSVC
Chris@16: #pragma warning(pop)
Chris@16: #endif
Chris@16: }
Chris@16: 
Chris@16: } } // namespaces
Chris@16: 
Chris@16: #endif // BOOST_MATH_COMPLEX_ASIN_INCLUDED