comparison DEPENDENCIES/generic/include/boost/math/complex/asin.hpp @ 16:2665513ce2d3

Add boost headers

+//  (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_ASIN_INCLUDED
+#define BOOST_MATH_COMPLEX_ASIN_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> 
+inline std::complex<T> asin(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,
+   // note that we have tweaked the value of a_crossover as per https://svn.boost.org/trac/boost/ticket/7290:
+   //
+   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>(10);
+   static const T b_crossover = static_cast<T>(0.6417L);
+   static const T s_pi = boost::math::constants::pi<T>();
+   static const T half_pi = s_pi / 2;
+   static const T log_two = boost::math::constants::ln_two<T>();
+   static const T quarter_pi = s_pi / 4;
+#ifdef BOOST_MSVC
+#pragma warning(push)
+#pragma warning(disable:4127)
+#endif
+   //
+   // 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;  // our results
+
+   //
+   // Begin by handling the special cases for infinities and nan's
+   // specified in C99, most of this is handled by the regular logic
+   // below, but handling it as a special case prevents overflow/underflow
+   // arithmetic which may trip up some machines:
+   //
+   if((boost::math::isnan)(x))
+   {
+      if((boost::math::isnan)(y))
+         return std::complex<T>(x, x);
+      if((boost::math::isinf)(y))
+      {
+         real = x;
+         imag = std::numeric_limits<T>::infinity();
+      }
+      else
+         return std::complex<T>(x, x);
+   }
+   else if((boost::math::isnan)(y))
+   {
+      if(x == 0)
+      {
+         real = 0;
+         imag = y;
+      }
+      else if((boost::math::isinf)(x))
+      {
+         real = y;
+         imag = std::numeric_limits<T>::infinity();
+      }
+      else
+         return std::complex<T>(y, y);
+   }
+   else if((boost::math::isinf)(x))
+   {
+      if((boost::math::isinf)(y))
+      {
+         real = quarter_pi;
+         imag = std::numeric_limits<T>::infinity();
+      }
+      else
+      {
+         real = half_pi;
+         imag = std::numeric_limits<T>::infinity();
+      }
+   }
+   else if((boost::math::isinf)(y))
+   {
+      real = 0;
+      imag = std::numeric_limits<T>::infinity();
+   }
+   else
+   {
+      //
+      // special case for real numbers:
+      //
+      if((y == 0) && (x <= one))
+         return std::complex<T>(std::asin(z.real()), z.imag());
+      //
+      // 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::asin(b);
+         }
+         else
+         {
+            T apx = a + x;
+            if(x <= one)
+            {
+               real = std::atan(x/std::sqrt(half * apx * (yy /(r + xp1) + (s-xm1))));
+            }
+            else
+            {
+               real = std::atan(x/(y * std::sqrt(half * (apx/(r + xp1) + apx/(s+xm1)))));
+            }
+         }
+
+         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 3 of their paper:
+         //
+         if(y <= (std::numeric_limits<T>::epsilon() * std::fabs(xm1)))
+         {
+            if(x < one)
+            {
+               real = std::asin(x);
+               imag = y / std::sqrt(-xp1*xm1);
+            }
+            else
+            {
+               real = half_pi;
+               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 = half_pi - std::sqrt(y);
+            imag = std::sqrt(y);
+         }
+         else if(std::numeric_limits<T>::epsilon() * y - one >= x)
+         {
+            real = x/y; // This can underflow!
+            imag = log_two + std::log(y);
+         }
+         else if(x > one)
+         {
+            real = std::atan(x/y);
+            T xoy = x/y;
+            imag = log_two + std::log(y) + half * boost::math::log1p(xoy*xoy);
+         }
+         else
+         {
+            T a = std::sqrt(one + y*y);
+            real = x/a; // This can underflow!
+            imag = half * boost::math::log1p(static_cast<T>(2)*y*(y+a));
+         }
+      }
+   }
+
+   //
+   // Finish off by working out the sign of the result:
+   //
+   if((boost::math::signbit)(z.real()))
+      real = (boost::math::changesign)(real);
+   if((boost::math::signbit)(z.imag()))
+      imag = (boost::math::changesign)(imag);
+
+   return std::complex<T>(real, imag);
+#ifdef BOOST_MSVC
+#pragma warning(pop)
+#endif
+}
+
+} } // namespaces
+
+#endif // BOOST_MATH_COMPLEX_ASIN_INCLUDED