williamr@2: //  (C) Copyright John Maddock 2005.
williamr@2: //  Distributed under the Boost Software License, Version 1.0. (See accompanying
williamr@2: //  file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)
williamr@2: 
williamr@2: #ifndef BOOST_MATH_COMPLEX_ACOS_INCLUDED
williamr@2: #define BOOST_MATH_COMPLEX_ACOS_INCLUDED
williamr@2: 
williamr@2: #ifndef BOOST_MATH_COMPLEX_DETAILS_INCLUDED
williamr@2: #  include <boost/math/complex/details.hpp>
williamr@2: #endif
williamr@2: #ifndef BOOST_MATH_LOG1P_INCLUDED
williamr@2: #  include <boost/math/special_functions/log1p.hpp>
williamr@2: #endif
williamr@2: #include <boost/assert.hpp>
williamr@2: 
williamr@2: #ifdef BOOST_NO_STDC_NAMESPACE
williamr@2: namespace std{ using ::sqrt; using ::fabs; using ::acos; using ::asin; using ::atan; using ::atan2; }
williamr@2: #endif
williamr@2: 
williamr@2: namespace boost{ namespace math{
williamr@2: 
williamr@2: template<class T> 
williamr@2: std::complex<T> acos(const std::complex<T>& z)
williamr@2: {
williamr@2:    //
williamr@2:    // This implementation is a transcription of the pseudo-code in:
williamr@2:    //
williamr@2:    // "Implementing the Complex Arcsine and Arccosine Functions using Exception Handling."
williamr@2:    // T E Hull, Thomas F Fairgrieve and Ping Tak Peter Tang.
williamr@2:    // ACM Transactions on Mathematical Software, Vol 23, No 3, Sept 1997.
williamr@2:    //
williamr@2: 
williamr@2:    //
williamr@2:    // These static constants should really be in a maths constants library:
williamr@2:    //
williamr@2:    static const T one = static_cast<T>(1);
williamr@2:    //static const T two = static_cast<T>(2);
williamr@2:    static const T half = static_cast<T>(0.5L);
williamr@2:    static const T a_crossover = static_cast<T>(1.5L);
williamr@2:    static const T b_crossover = static_cast<T>(0.6417L);
williamr@2:    static const T s_pi = static_cast<T>(3.141592653589793238462643383279502884197L);
williamr@2:    static const T half_pi = static_cast<T>(1.57079632679489661923132169163975144L);
williamr@2:    static const T log_two = static_cast<T>(0.69314718055994530941723212145817657L);
williamr@2:    static const T quarter_pi = static_cast<T>(0.78539816339744830961566084581987572L);
williamr@2:    
williamr@2:    //
williamr@2:    // Get real and imaginary parts, discard the signs as we can 
williamr@2:    // figure out the sign of the result later:
williamr@2:    //
williamr@2:    T x = std::fabs(z.real());
williamr@2:    T y = std::fabs(z.imag());
williamr@2: 
williamr@2:    T real, imag; // these hold our result
williamr@2: 
williamr@2:    // 
williamr@2:    // Handle special cases specified by the C99 standard,
williamr@2:    // many of these special cases aren't really needed here,
williamr@2:    // but doing it this way prevents overflow/underflow arithmetic
williamr@2:    // in the main body of the logic, which may trip up some machines:
williamr@2:    //
williamr@2:    if(std::numeric_limits<T>::has_infinity && (x == std::numeric_limits<T>::infinity()))
williamr@2:    {
williamr@2:       if(y == std::numeric_limits<T>::infinity())
williamr@2:       {
williamr@2:          real = quarter_pi;
williamr@2:          imag = std::numeric_limits<T>::infinity();
williamr@2:       }
williamr@2:       else if(detail::test_is_nan(y))
williamr@2:       {
williamr@2:          return std::complex<T>(y, -std::numeric_limits<T>::infinity());
williamr@2:       }
williamr@2:       else
williamr@2:       {
williamr@2:          // y is not infinity or nan:
williamr@2:          real = 0;
williamr@2:          imag = std::numeric_limits<T>::infinity();
williamr@2:       }
williamr@2:    }
williamr@2:    else if(detail::test_is_nan(x))
williamr@2:    {
williamr@2:       if(y == std::numeric_limits<T>::infinity())
williamr@2:          return std::complex<T>(x, (z.imag() < 0) ? std::numeric_limits<T>::infinity() :  -std::numeric_limits<T>::infinity());
williamr@2:       return std::complex<T>(x, x);
williamr@2:    }
williamr@2:    else if(std::numeric_limits<T>::has_infinity && (y == std::numeric_limits<T>::infinity()))
williamr@2:    {
williamr@2:       real = half_pi;
williamr@2:       imag = std::numeric_limits<T>::infinity();
williamr@2:    }
williamr@2:    else if(detail::test_is_nan(y))
williamr@2:    {
williamr@2:       return std::complex<T>((x == 0) ? half_pi : y, y);
williamr@2:    }
williamr@2:    else
williamr@2:    {
williamr@2:       //
williamr@2:       // What follows is the regular Hull et al code,
williamr@2:       // begin with the special case for real numbers:
williamr@2:       //
williamr@2:       if((y == 0) && (x <= one))
williamr@2:          return std::complex<T>((x == 0) ? half_pi : std::acos(z.real()));
williamr@2:       //
williamr@2:       // Figure out if our input is within the "safe area" identified by Hull et al.
williamr@2:       // This would be more efficient with portable floating point exception handling;
williamr@2:       // fortunately the quantities M and u identified by Hull et al (figure 3), 
williamr@2:       // match with the max and min methods of numeric_limits<T>.
williamr@2:       //
williamr@2:       T safe_max = detail::safe_max(static_cast<T>(8));
williamr@2:       T safe_min = detail::safe_min(static_cast<T>(4));
williamr@2: 
williamr@2:       T xp1 = one + x;
williamr@2:       T xm1 = x - one;
williamr@2: 
williamr@2:       if((x < safe_max) && (x > safe_min) && (y < safe_max) && (y > safe_min))
williamr@2:       {
williamr@2:          T yy = y * y;
williamr@2:          T r = std::sqrt(xp1*xp1 + yy);
williamr@2:          T s = std::sqrt(xm1*xm1 + yy);
williamr@2:          T a = half * (r + s);
williamr@2:          T b = x / a;
williamr@2: 
williamr@2:          if(b <= b_crossover)
williamr@2:          {
williamr@2:             real = std::acos(b);
williamr@2:          }
williamr@2:          else
williamr@2:          {
williamr@2:             T apx = a + x;
williamr@2:             if(x <= one)
williamr@2:             {
williamr@2:                real = std::atan(std::sqrt(half * apx * (yy /(r + xp1) + (s-xm1)))/x);
williamr@2:             }
williamr@2:             else
williamr@2:             {
williamr@2:                real = std::atan((y * std::sqrt(half * (apx/(r + xp1) + apx/(s+xm1))))/x);
williamr@2:             }
williamr@2:          }
williamr@2: 
williamr@2:          if(a <= a_crossover)
williamr@2:          {
williamr@2:             T am1;
williamr@2:             if(x < one)
williamr@2:             {
williamr@2:                am1 = half * (yy/(r + xp1) + yy/(s - xm1));
williamr@2:             }
williamr@2:             else
williamr@2:             {
williamr@2:                am1 = half * (yy/(r + xp1) + (s + xm1));
williamr@2:             }
williamr@2:             imag = boost::math::log1p(am1 + std::sqrt(am1 * (a + one)));
williamr@2:          }
williamr@2:          else
williamr@2:          {
williamr@2:             imag = std::log(a + std::sqrt(a*a - one));
williamr@2:          }
williamr@2:       }
williamr@2:       else
williamr@2:       {
williamr@2:          //
williamr@2:          // This is the Hull et al exception handling code from Fig 6 of their paper:
williamr@2:          //
williamr@2:          if(y <= (std::numeric_limits<T>::epsilon() * std::fabs(xm1)))
williamr@2:          {
williamr@2:             if(x < one)
williamr@2:             {
williamr@2:                real = std::acos(x);
williamr@2:                imag = y / std::sqrt(xp1*(one-x));
williamr@2:             }
williamr@2:             else
williamr@2:             {
williamr@2:                real = 0;
williamr@2:                if(((std::numeric_limits<T>::max)() / xp1) > xm1)
williamr@2:                {
williamr@2:                   // xp1 * xm1 won't overflow:
williamr@2:                   imag = boost::math::log1p(xm1 + std::sqrt(xp1*xm1));
williamr@2:                }
williamr@2:                else
williamr@2:                {
williamr@2:                   imag = log_two + std::log(x);
williamr@2:                }
williamr@2:             }
williamr@2:          }
williamr@2:          else if(y <= safe_min)
williamr@2:          {
williamr@2:             // There is an assumption in Hull et al's analysis that
williamr@2:             // if we get here then x == 1.  This is true for all "good"
williamr@2:             // machines where :
williamr@2:             // 
williamr@2:             // E^2 > 8*sqrt(u); with:
williamr@2:             //
williamr@2:             // E =  std::numeric_limits<T>::epsilon()
williamr@2:             // u = (std::numeric_limits<T>::min)()
williamr@2:             //
williamr@2:             // Hull et al provide alternative code for "bad" machines
williamr@2:             // but we have no way to test that here, so for now just assert
williamr@2:             // on the assumption:
williamr@2:             //
williamr@2:             BOOST_ASSERT(x == 1);
williamr@2:             real = std::sqrt(y);
williamr@2:             imag = std::sqrt(y);
williamr@2:          }
williamr@2:          else if(std::numeric_limits<T>::epsilon() * y - one >= x)
williamr@2:          {
williamr@2:             real = half_pi;
williamr@2:             imag = log_two + std::log(y);
williamr@2:          }
williamr@2:          else if(x > one)
williamr@2:          {
williamr@2:             real = std::atan(y/x);
williamr@2:             T xoy = x/y;
williamr@2:             imag = log_two + std::log(y) + half * boost::math::log1p(xoy*xoy);
williamr@2:          }
williamr@2:          else
williamr@2:          {
williamr@2:             real = half_pi;
williamr@2:             T a = std::sqrt(one + y*y);
williamr@2:             imag = half * boost::math::log1p(static_cast<T>(2)*y*(y+a));
williamr@2:          }
williamr@2:       }
williamr@2:    }
williamr@2: 
williamr@2:    //
williamr@2:    // Finish off by working out the sign of the result:
williamr@2:    //
williamr@2:    if(z.real() < 0)
williamr@2:       real = s_pi - real;
williamr@2:    if(z.imag() > 0)
williamr@2:       imag = -imag;
williamr@2: 
williamr@2:    return std::complex<T>(real, imag);
williamr@2: }
williamr@2: 
williamr@2: } } // namespaces
williamr@2: 
williamr@2: #endif // BOOST_MATH_COMPLEX_ACOS_INCLUDED