253 lines
7.5 KiB
Plaintext
253 lines
7.5 KiB
Plaintext
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// (C) Copyright John Maddock 2005.
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// Distributed under the Boost Software License, Version 1.0. (See accompanying
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// file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)
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#ifndef BOOST_MATH_COMPLEX_ASIN_INCLUDED
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#define BOOST_MATH_COMPLEX_ASIN_INCLUDED
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#ifndef BOOST_MATH_COMPLEX_DETAILS_INCLUDED
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# include <boost/math/complex/details.hpp>
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#endif
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#ifndef BOOST_MATH_LOG1P_INCLUDED
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# include <boost/math/special_functions/log1p.hpp>
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#endif
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#include <boost/assert.hpp>
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#ifdef BOOST_NO_STDC_NAMESPACE
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namespace std{ using ::sqrt; using ::fabs; using ::acos; using ::asin; using ::atan; using ::atan2; }
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#endif
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namespace boost{ namespace math{
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template<class T>
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inline std::complex<T> asin(const std::complex<T>& z)
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{
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//
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// This implementation is a transcription of the pseudo-code in:
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//
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// "Implementing the complex Arcsine and Arccosine Functions using Exception Handling."
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// T E Hull, Thomas F Fairgrieve and Ping Tak Peter Tang.
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// ACM Transactions on Mathematical Software, Vol 23, No 3, Sept 1997.
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//
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//
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// These static constants should really be in a maths constants library,
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// note that we have tweaked the value of a_crossover as per https://svn.boost.org/trac/boost/ticket/7290:
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//
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static const T one = static_cast<T>(1);
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//static const T two = static_cast<T>(2);
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static const T half = static_cast<T>(0.5L);
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static const T a_crossover = static_cast<T>(10);
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static const T b_crossover = static_cast<T>(0.6417L);
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static const T s_pi = boost::math::constants::pi<T>();
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static const T half_pi = s_pi / 2;
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static const T log_two = boost::math::constants::ln_two<T>();
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static const T quarter_pi = s_pi / 4;
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#ifdef BOOST_MSVC
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#pragma warning(push)
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#pragma warning(disable:4127)
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#endif
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//
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// Get real and imaginary parts, discard the signs as we can
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// figure out the sign of the result later:
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//
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T x = std::fabs(z.real());
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T y = std::fabs(z.imag());
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T real, imag; // our results
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//
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// Begin by handling the special cases for infinities and nan's
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// specified in C99, most of this is handled by the regular logic
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// below, but handling it as a special case prevents overflow/underflow
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// arithmetic which may trip up some machines:
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//
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if((boost::math::isnan)(x))
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{
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if((boost::math::isnan)(y))
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return std::complex<T>(x, x);
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if((boost::math::isinf)(y))
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{
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real = x;
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imag = std::numeric_limits<T>::infinity();
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}
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else
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return std::complex<T>(x, x);
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}
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else if((boost::math::isnan)(y))
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{
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if(x == 0)
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{
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real = 0;
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imag = y;
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}
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else if((boost::math::isinf)(x))
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{
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real = y;
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imag = std::numeric_limits<T>::infinity();
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}
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else
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return std::complex<T>(y, y);
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}
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else if((boost::math::isinf)(x))
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{
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if((boost::math::isinf)(y))
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{
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real = quarter_pi;
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imag = std::numeric_limits<T>::infinity();
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}
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else
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{
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real = half_pi;
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imag = std::numeric_limits<T>::infinity();
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}
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}
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else if((boost::math::isinf)(y))
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{
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real = 0;
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imag = std::numeric_limits<T>::infinity();
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}
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else
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{
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//
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// special case for real numbers:
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//
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if((y == 0) && (x <= one))
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return std::complex<T>(std::asin(z.real()), z.imag());
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//
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// Figure out if our input is within the "safe area" identified by Hull et al.
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// This would be more efficient with portable floating point exception handling;
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// fortunately the quantities M and u identified by Hull et al (figure 3),
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// match with the max and min methods of numeric_limits<T>.
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//
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T safe_max = detail::safe_max(static_cast<T>(8));
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T safe_min = detail::safe_min(static_cast<T>(4));
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T xp1 = one + x;
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T xm1 = x - one;
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if((x < safe_max) && (x > safe_min) && (y < safe_max) && (y > safe_min))
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{
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T yy = y * y;
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T r = std::sqrt(xp1*xp1 + yy);
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T s = std::sqrt(xm1*xm1 + yy);
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T a = half * (r + s);
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T b = x / a;
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if(b <= b_crossover)
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{
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real = std::asin(b);
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}
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else
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{
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T apx = a + x;
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if(x <= one)
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{
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real = std::atan(x/std::sqrt(half * apx * (yy /(r + xp1) + (s-xm1))));
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}
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else
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{
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real = std::atan(x/(y * std::sqrt(half * (apx/(r + xp1) + apx/(s+xm1)))));
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}
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}
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if(a <= a_crossover)
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{
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T am1;
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if(x < one)
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{
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am1 = half * (yy/(r + xp1) + yy/(s - xm1));
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}
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else
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{
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am1 = half * (yy/(r + xp1) + (s + xm1));
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}
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imag = boost::math::log1p(am1 + std::sqrt(am1 * (a + one)));
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}
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else
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{
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imag = std::log(a + std::sqrt(a*a - one));
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}
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}
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else
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{
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//
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// This is the Hull et al exception handling code from Fig 3 of their paper:
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//
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if(y <= (std::numeric_limits<T>::epsilon() * std::fabs(xm1)))
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{
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if(x < one)
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{
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real = std::asin(x);
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imag = y / std::sqrt(-xp1*xm1);
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}
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else
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{
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real = half_pi;
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if(((std::numeric_limits<T>::max)() / xp1) > xm1)
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{
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// xp1 * xm1 won't overflow:
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imag = boost::math::log1p(xm1 + std::sqrt(xp1*xm1));
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}
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else
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{
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imag = log_two + std::log(x);
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}
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}
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}
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else if(y <= safe_min)
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{
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// There is an assumption in Hull et al's analysis that
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// if we get here then x == 1. This is true for all "good"
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// machines where :
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//
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// E^2 > 8*sqrt(u); with:
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//
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// E = std::numeric_limits<T>::epsilon()
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// u = (std::numeric_limits<T>::min)()
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//
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// Hull et al provide alternative code for "bad" machines
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// but we have no way to test that here, so for now just assert
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// on the assumption:
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//
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BOOST_ASSERT(x == 1);
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real = half_pi - std::sqrt(y);
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imag = std::sqrt(y);
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}
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else if(std::numeric_limits<T>::epsilon() * y - one >= x)
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{
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real = x/y; // This can underflow!
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imag = log_two + std::log(y);
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}
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else if(x > one)
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{
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real = std::atan(x/y);
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T xoy = x/y;
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imag = log_two + std::log(y) + half * boost::math::log1p(xoy*xoy);
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}
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else
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{
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T a = std::sqrt(one + y*y);
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real = x/a; // This can underflow!
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imag = half * boost::math::log1p(static_cast<T>(2)*y*(y+a));
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}
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}
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}
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//
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// Finish off by working out the sign of the result:
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//
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if((boost::math::signbit)(z.real()))
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real = (boost::math::changesign)(real);
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if((boost::math::signbit)(z.imag()))
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imag = (boost::math::changesign)(imag);
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return std::complex<T>(real, imag);
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#ifdef BOOST_MSVC
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#pragma warning(pop)
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#endif
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}
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} } // namespaces
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#endif // BOOST_MATH_COMPLEX_ASIN_INCLUDED
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