boost/math/complex/asin.hpp
// (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: // 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 pi = static_cast<T>(3.141592653589793238462643383279502884197L); static const T half_pi = static_cast<T>(1.57079632679489661923132169163975144L); static const T log_two = static_cast<T>(0.69314718055994530941723212145817657L); static const T quarter_pi = static_cast<T>(0.78539816339744830961566084581987572L); // // 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(detail::test_is_nan(x)) { if(detail::test_is_nan(y)) return std::complex<T>(x, x); if(std::numeric_limits<T>::has_infinity && (y == std::numeric_limits<T>::infinity())) { real = x; imag = std::numeric_limits<T>::infinity(); } else return std::complex<T>(x, x); } else if(detail::test_is_nan(y)) { if(x == 0) { real = 0; imag = y; } else if(std::numeric_limits<T>::has_infinity && (x == std::numeric_limits<T>::infinity())) { real = y; imag = std::numeric_limits<T>::infinity(); } else return std::complex<T>(y, y); } else if(std::numeric_limits<T>::has_infinity && (x == std::numeric_limits<T>::infinity())) { if(y == std::numeric_limits<T>::infinity()) { real = quarter_pi; imag = std::numeric_limits<T>::infinity(); } else { real = half_pi; imag = std::numeric_limits<T>::infinity(); } } else if(std::numeric_limits<T>::has_infinity && (y == std::numeric_limits<T>::infinity())) { 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())); // // 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(z.real() < 0) real = -real; if(z.imag() < 0) imag = -imag; return std::complex<T>(real, imag); } } } // namespaces #endif // BOOST_MATH_COMPLEX_ASIN_INCLUDED