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// (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_ACOS_INCLUDED
#define BOOST_MATH_COMPLEX_ACOS_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> acos(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:
//
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>(1.5L);
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; // these hold our result
//
// Handle special cases specified by the C99 standard,
// many of these special cases aren't really needed here,
// but doing it this way prevents overflow/underflow arithmetic
// in the main body of the logic, which may trip up some machines:
//
if((boost::math::isinf)(x))
{
if((boost::math::isinf)(y))
{
real = quarter_pi;
imag = std::numeric_limits<T>::infinity();
}
else if((boost::math::isnan)(y))
{
return std::complex<T>(y, -std::numeric_limits<T>::infinity());
}
else
{
// y is not infinity or nan:
real = 0;
imag = std::numeric_limits<T>::infinity();
}
}
else if((boost::math::isnan)(x))
{
if((boost::math::isinf)(y))
return std::complex<T>(x, ((boost::math::signbit)(z.imag())) ? std::numeric_limits<T>::infinity() : -std::numeric_limits<T>::infinity());
return std::complex<T>(x, x);
}
else if((boost::math::isinf)(y))
{
real = half_pi;
imag = std::numeric_limits<T>::infinity();
}
else if((boost::math::isnan)(y))
{
return std::complex<T>((x == 0) ? half_pi : y, y);
}
else
{
//
// What follows is the regular Hull et al code,
// begin with the special case for real numbers:
//
if((y == 0) && (x <= one))
return std::complex<T>((x == 0) ? half_pi : std::acos(z.real()), (boost::math::changesign)(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::acos(b);
}
else
{
T apx = a + x;
if(x <= one)
{
real = std::atan(std::sqrt(half * apx * (yy /(r + xp1) + (s-xm1)))/x);
}
else
{
real = std::atan((y * std::sqrt(half * (apx/(r + xp1) + apx/(s+xm1))))/x);
}
}
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 6 of their paper:
//
if(y <= (std::numeric_limits<T>::epsilon() * std::fabs(xm1)))
{
if(x < one)
{
real = std::acos(x);
imag = y / std::sqrt(xp1*(one-x));
}
else
{
real = 0;
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 = std::sqrt(y);
imag = std::sqrt(y);
}
else if(std::numeric_limits<T>::epsilon() * y - one >= x)
{
real = half_pi;
imag = log_two + std::log(y);
}
else if(x > one)
{
real = std::atan(y/x);
T xoy = x/y;
imag = log_two + std::log(y) + half * boost::math::log1p(xoy*xoy);
}
else
{
real = half_pi;
T a = std::sqrt(one + y*y);
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 = s_pi - 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_ACOS_INCLUDED
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