sl@0: //  (C) Copyright John Maddock 2005.
sl@0: //  Distributed under the Boost Software License, Version 1.0. (See accompanying
sl@0: //  file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)
sl@0: 
sl@0: #ifndef BOOST_MATH_COMPLEX_ASIN_INCLUDED
sl@0: #define BOOST_MATH_COMPLEX_ASIN_INCLUDED
sl@0: 
sl@0: #ifndef BOOST_MATH_COMPLEX_DETAILS_INCLUDED
sl@0: #  include <boost/math/complex/details.hpp>
sl@0: #endif
sl@0: #ifndef BOOST_MATH_LOG1P_INCLUDED
sl@0: #  include <boost/math/special_functions/log1p.hpp>
sl@0: #endif
sl@0: #include <boost/assert.hpp>
sl@0: 
sl@0: #ifdef BOOST_NO_STDC_NAMESPACE
sl@0: namespace std{ using ::sqrt; using ::fabs; using ::acos; using ::asin; using ::atan; using ::atan2; }
sl@0: #endif
sl@0: 
sl@0: namespace boost{ namespace math{
sl@0: 
sl@0: template<class T> 
sl@0: inline std::complex<T> asin(const std::complex<T>& z)
sl@0: {
sl@0:    //
sl@0:    // This implementation is a transcription of the pseudo-code in:
sl@0:    //
sl@0:    // "Implementing the complex Arcsine and Arccosine Functions using Exception Handling."
sl@0:    // T E Hull, Thomas F Fairgrieve and Ping Tak Peter Tang.
sl@0:    // ACM Transactions on Mathematical Software, Vol 23, No 3, Sept 1997.
sl@0:    //
sl@0: 
sl@0:    //
sl@0:    // These static constants should really be in a maths constants library:
sl@0:    //
sl@0:    static const T one = static_cast<T>(1);
sl@0:    //static const T two = static_cast<T>(2);
sl@0:    static const T half = static_cast<T>(0.5L);
sl@0:    static const T a_crossover = static_cast<T>(1.5L);
sl@0:    static const T b_crossover = static_cast<T>(0.6417L);
sl@0:    //static const T pi = static_cast<T>(3.141592653589793238462643383279502884197L);
sl@0:    static const T half_pi = static_cast<T>(1.57079632679489661923132169163975144L);
sl@0:    static const T log_two = static_cast<T>(0.69314718055994530941723212145817657L);
sl@0:    static const T quarter_pi = static_cast<T>(0.78539816339744830961566084581987572L);
sl@0:    
sl@0:    //
sl@0:    // Get real and imaginary parts, discard the signs as we can 
sl@0:    // figure out the sign of the result later:
sl@0:    //
sl@0:    T x = std::fabs(z.real());
sl@0:    T y = std::fabs(z.imag());
sl@0:    T real, imag;  // our results
sl@0: 
sl@0:    //
sl@0:    // Begin by handling the special cases for infinities and nan's
sl@0:    // specified in C99, most of this is handled by the regular logic
sl@0:    // below, but handling it as a special case prevents overflow/underflow
sl@0:    // arithmetic which may trip up some machines:
sl@0:    //
sl@0:    if(detail::test_is_nan(x))
sl@0:    {
sl@0:       if(detail::test_is_nan(y))
sl@0:          return std::complex<T>(x, x);
sl@0:       if(std::numeric_limits<T>::has_infinity && (y == std::numeric_limits<T>::infinity()))
sl@0:       {
sl@0:          real = x;
sl@0:          imag = std::numeric_limits<T>::infinity();
sl@0:       }
sl@0:       else
sl@0:          return std::complex<T>(x, x);
sl@0:    }
sl@0:    else if(detail::test_is_nan(y))
sl@0:    {
sl@0:       if(x == 0)
sl@0:       {
sl@0:          real = 0;
sl@0:          imag = y;
sl@0:       }
sl@0:       else if(std::numeric_limits<T>::has_infinity && (x == std::numeric_limits<T>::infinity()))
sl@0:       {
sl@0:          real = y;
sl@0:          imag = std::numeric_limits<T>::infinity();
sl@0:       }
sl@0:       else
sl@0:          return std::complex<T>(y, y);
sl@0:    }
sl@0:    else if(std::numeric_limits<T>::has_infinity && (x == std::numeric_limits<T>::infinity()))
sl@0:    {
sl@0:       if(y == std::numeric_limits<T>::infinity())
sl@0:       {
sl@0:          real = quarter_pi;
sl@0:          imag = std::numeric_limits<T>::infinity();
sl@0:       }
sl@0:       else
sl@0:       {
sl@0:          real = half_pi;
sl@0:          imag = std::numeric_limits<T>::infinity();
sl@0:       }
sl@0:    }
sl@0:    else if(std::numeric_limits<T>::has_infinity && (y == std::numeric_limits<T>::infinity()))
sl@0:    {
sl@0:       real = 0;
sl@0:       imag = std::numeric_limits<T>::infinity();
sl@0:    }
sl@0:    else
sl@0:    {
sl@0:       //
sl@0:       // special case for real numbers:
sl@0:       //
sl@0:       if((y == 0) && (x <= one))
sl@0:          return std::complex<T>(std::asin(z.real()));
sl@0:       //
sl@0:       // Figure out if our input is within the "safe area" identified by Hull et al.
sl@0:       // This would be more efficient with portable floating point exception handling;
sl@0:       // fortunately the quantities M and u identified by Hull et al (figure 3), 
sl@0:       // match with the max and min methods of numeric_limits<T>.
sl@0:       //
sl@0:       T safe_max = detail::safe_max(static_cast<T>(8));
sl@0:       T safe_min = detail::safe_min(static_cast<T>(4));
sl@0: 
sl@0:       T xp1 = one + x;
sl@0:       T xm1 = x - one;
sl@0: 
sl@0:       if((x < safe_max) && (x > safe_min) && (y < safe_max) && (y > safe_min))
sl@0:       {
sl@0:          T yy = y * y;
sl@0:          T r = std::sqrt(xp1*xp1 + yy);
sl@0:          T s = std::sqrt(xm1*xm1 + yy);
sl@0:          T a = half * (r + s);
sl@0:          T b = x / a;
sl@0: 
sl@0:          if(b <= b_crossover)
sl@0:          {
sl@0:             real = std::asin(b);
sl@0:          }
sl@0:          else
sl@0:          {
sl@0:             T apx = a + x;
sl@0:             if(x <= one)
sl@0:             {
sl@0:                real = std::atan(x/std::sqrt(half * apx * (yy /(r + xp1) + (s-xm1))));
sl@0:             }
sl@0:             else
sl@0:             {
sl@0:                real = std::atan(x/(y * std::sqrt(half * (apx/(r + xp1) + apx/(s+xm1)))));
sl@0:             }
sl@0:          }
sl@0: 
sl@0:          if(a <= a_crossover)
sl@0:          {
sl@0:             T am1;
sl@0:             if(x < one)
sl@0:             {
sl@0:                am1 = half * (yy/(r + xp1) + yy/(s - xm1));
sl@0:             }
sl@0:             else
sl@0:             {
sl@0:                am1 = half * (yy/(r + xp1) + (s + xm1));
sl@0:             }
sl@0:             imag = boost::math::log1p(am1 + std::sqrt(am1 * (a + one)));
sl@0:          }
sl@0:          else
sl@0:          {
sl@0:             imag = std::log(a + std::sqrt(a*a - one));
sl@0:          }
sl@0:       }
sl@0:       else
sl@0:       {
sl@0:          //
sl@0:          // This is the Hull et al exception handling code from Fig 3 of their paper:
sl@0:          //
sl@0:          if(y <= (std::numeric_limits<T>::epsilon() * std::fabs(xm1)))
sl@0:          {
sl@0:             if(x < one)
sl@0:             {
sl@0:                real = std::asin(x);
sl@0:                imag = y / std::sqrt(xp1*xm1);
sl@0:             }
sl@0:             else
sl@0:             {
sl@0:                real = half_pi;
sl@0:                if(((std::numeric_limits<T>::max)() / xp1) > xm1)
sl@0:                {
sl@0:                   // xp1 * xm1 won't overflow:
sl@0:                   imag = boost::math::log1p(xm1 + std::sqrt(xp1*xm1));
sl@0:                }
sl@0:                else
sl@0:                {
sl@0:                   imag = log_two + std::log(x);
sl@0:                }
sl@0:             }
sl@0:          }
sl@0:          else if(y <= safe_min)
sl@0:          {
sl@0:             // There is an assumption in Hull et al's analysis that
sl@0:             // if we get here then x == 1.  This is true for all "good"
sl@0:             // machines where :
sl@0:             // 
sl@0:             // E^2 > 8*sqrt(u); with:
sl@0:             //
sl@0:             // E =  std::numeric_limits<T>::epsilon()
sl@0:             // u = (std::numeric_limits<T>::min)()
sl@0:             //
sl@0:             // Hull et al provide alternative code for "bad" machines
sl@0:             // but we have no way to test that here, so for now just assert
sl@0:             // on the assumption:
sl@0:             //
sl@0:             BOOST_ASSERT(x == 1);
sl@0:             real = half_pi - std::sqrt(y);
sl@0:             imag = std::sqrt(y);
sl@0:          }
sl@0:          else if(std::numeric_limits<T>::epsilon() * y - one >= x)
sl@0:          {
sl@0:             real = x/y; // This can underflow!
sl@0:             imag = log_two + std::log(y);
sl@0:          }
sl@0:          else if(x > one)
sl@0:          {
sl@0:             real = std::atan(x/y);
sl@0:             T xoy = x/y;
sl@0:             imag = log_two + std::log(y) + half * boost::math::log1p(xoy*xoy);
sl@0:          }
sl@0:          else
sl@0:          {
sl@0:             T a = std::sqrt(one + y*y);
sl@0:             real = x/a; // This can underflow!
sl@0:             imag = half * boost::math::log1p(static_cast<T>(2)*y*(y+a));
sl@0:          }
sl@0:       }
sl@0:    }
sl@0: 
sl@0:    //
sl@0:    // Finish off by working out the sign of the result:
sl@0:    //
sl@0:    if(z.real() < 0)
sl@0:       real = -real;
sl@0:    if(z.imag() < 0)
sl@0:       imag = -imag;
sl@0: 
sl@0:    return std::complex<T>(real, imag);
sl@0: }
sl@0: 
sl@0: } } // namespaces
sl@0: 
sl@0: #endif // BOOST_MATH_COMPLEX_ASIN_INCLUDED