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

Add boost headers

+//  (C) Copyright John Maddock 2005.
+//  Use, modification and distribution are subject to 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_ATANH_INCLUDED
+#define BOOST_MATH_COMPLEX_ATANH_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> atanh(const std::complex<T>& z)
+{
+   //
+   // References:
+   //
+   // Eric W. Weisstein. "Inverse Hyperbolic Tangent." 
+   // From MathWorld--A Wolfram Web Resource. 
+   // http://mathworld.wolfram.com/InverseHyperbolicTangent.html
+   //
+   // Also: The Wolfram Functions Site,
+   // http://functions.wolfram.com/ElementaryFunctions/ArcTanh/
+   //
+   // Also "Abramowitz and Stegun. Handbook of Mathematical Functions."
+   // at : http://jove.prohosting.com/~skripty/toc.htm
+   //
+   // See also: https://svn.boost.org/trac/boost/ticket/7291
+   //
+   
+   static const T pi = boost::math::constants::pi<T>();
+   static const T half_pi = pi / 2;
+   static const T one = static_cast<T>(1.0L);
+   static const T two = static_cast<T>(2.0L);
+   static const T four = static_cast<T>(4.0L);
+   static const T zero = static_cast<T>(0);
+   static const T log_two = boost::math::constants::ln_two<T>();
+
+#ifdef BOOST_MSVC
+#pragma warning(push)
+#pragma warning(disable:4127)
+#endif
+
+   T x = std::fabs(z.real());
+   T y = std::fabs(z.imag());
+
+   T real, imag;  // our results
+
+   T safe_upper = detail::safe_max(two);
+   T safe_lower = detail::safe_min(static_cast<T>(2));
+
+   //
+   // Begin by handling the special cases specified in C99:
+   //
+   if((boost::math::isnan)(x))
+   {
+      if((boost::math::isnan)(y))
+         return std::complex<T>(x, x);
+      else if((boost::math::isinf)(y))
+         return std::complex<T>(0, ((boost::math::signbit)(z.imag()) ? -half_pi : half_pi));
+      else
+         return std::complex<T>(x, x);
+   }
+   else if((boost::math::isnan)(y))
+   {
+      if(x == 0)
+         return std::complex<T>(x, y);
+      if((boost::math::isinf)(x))
+         return std::complex<T>(0, y);
+      else
+         return std::complex<T>(y, y);
+   }
+   else if((x > safe_lower) && (x < safe_upper) && (y > safe_lower) && (y < safe_upper))
+   {
+
+      T yy = y*y;
+      T mxm1 = one - x;
+      ///
+      // The real part is given by:
+      // 
+      // real(atanh(z)) == log1p(4*x / ((x-1)*(x-1) + y^2))
+      // 
+      real = boost::math::log1p(four * x / (mxm1*mxm1 + yy));
+      real /= four;
+      if((boost::math::signbit)(z.real()))
+         real = (boost::math::changesign)(real);
+
+      imag = std::atan2((y * two), (mxm1*(one+x) - yy));
+      imag /= two;
+      if(z.imag() < 0)
+         imag = (boost::math::changesign)(imag);
+   }
+   else
+   {
+      //
+      // This section handles exception cases that would normally cause
+      // underflow or overflow in the main formulas.
+      //
+      // Begin by working out the real part, we need to approximate
+      //    real = boost::math::log1p(4x / ((x-1)^2 + y^2))
+      // without either overflow or underflow in the squared terms.
+      //
+      T mxm1 = one - x;
+      if(x >= safe_upper)
+      {
+         // x-1 = x to machine precision:
+         if((boost::math::isinf)(x) || (boost::math::isinf)(y))
+         {
+            real = 0;
+         }
+         else if(y >= safe_upper)
+         {
+            // Big x and y: divide through by x*y:
+            real = boost::math::log1p((four/y) / (x/y + y/x));
+         }
+         else if(y > one)
+         {
+            // Big x: divide through by x:
+            real = boost::math::log1p(four / (x + y*y/x));
+         }
+         else
+         {
+            // Big x small y, as above but neglect y^2/x:
+            real = boost::math::log1p(four/x);
+         }
+      }
+      else if(y >= safe_upper)
+      {
+         if(x > one)
+         {
+            // Big y, medium x, divide through by y:
+            real = boost::math::log1p((four*x/y) / (y + mxm1*mxm1/y));
+         }
+         else
+         {
+            // Small or medium x, large y:
+            real = four*x/y/y;
+         }
+      }
+      else if (x != one)
+      {
+         // y is small, calculate divisor carefully:
+         T div = mxm1*mxm1;
+         if(y > safe_lower)
+            div += y*y;
+         real = boost::math::log1p(four*x/div);
+      }
+      else
+         real = boost::math::changesign(two * (std::log(y) - log_two));
+
+      real /= four;
+      if((boost::math::signbit)(z.real()))
+         real = (boost::math::changesign)(real);
+
+      //
+      // Now handle imaginary part, this is much easier,
+      // if x or y are large, then the formula:
+      //    atan2(2y, (1-x)*(1+x) - y^2)
+      // evaluates to +-(PI - theta) where theta is negligible compared to PI.
+      //
+      if((x >= safe_upper) || (y >= safe_upper))
+      {
+         imag = pi;
+      }
+      else if(x <= safe_lower)
+      {
+         //
+         // If both x and y are small then atan(2y),
+         // otherwise just x^2 is negligible in the divisor:
+         //
+         if(y <= safe_lower)
+            imag = std::atan2(two*y, one);
+         else
+         {
+            if((y == zero) && (x == zero))
+               imag = 0;
+            else
+               imag = std::atan2(two*y, one - y*y);
+         }
+      }
+      else
+      {
+         //
+         // y^2 is negligible:
+         //
+         if((y == zero) && (x == one))
+            imag = 0;
+         else
+            imag = std::atan2(two*y, mxm1*(one+x));
+      }
+      imag /= two;
+      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_ATANH_INCLUDED