215 lines
6.2 KiB
C++
215 lines
6.2 KiB
C++
// (C) Copyright John Maddock 2005.
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// Use, modification and distribution are subject to the
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// Boost Software License, Version 1.0. (See accompanying file
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// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)
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#ifndef BOOST_MATH_COMPLEX_ATANH_INCLUDED
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#define BOOST_MATH_COMPLEX_ATANH_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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std::complex<T> atanh(const std::complex<T>& z)
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{
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//
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// References:
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//
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// Eric W. Weisstein. "Inverse Hyperbolic Tangent."
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// From MathWorld--A Wolfram Web Resource.
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// http://mathworld.wolfram.com/InverseHyperbolicTangent.html
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//
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// Also: The Wolfram Functions Site,
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// http://functions.wolfram.com/ElementaryFunctions/ArcTanh/
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//
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// Also "Abramowitz and Stegun. Handbook of Mathematical Functions."
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// at : http://jove.prohosting.com/~skripty/toc.htm
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//
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// See also: https://svn.boost.org/trac/boost/ticket/7291
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//
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static const T pi = boost::math::constants::pi<T>();
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static const T half_pi = pi / 2;
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static const T one = static_cast<T>(1.0L);
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static const T two = static_cast<T>(2.0L);
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static const T four = static_cast<T>(4.0L);
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static const T zero = static_cast<T>(0);
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static const T log_two = boost::math::constants::ln_two<T>();
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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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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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T safe_upper = detail::safe_max(two);
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T safe_lower = detail::safe_min(static_cast<T>(2));
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//
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// Begin by handling the special cases specified in C99:
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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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else if((boost::math::isinf)(y))
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return std::complex<T>(0, ((boost::math::signbit)(z.imag()) ? -half_pi : half_pi));
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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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return std::complex<T>(x, y);
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if((boost::math::isinf)(x))
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return std::complex<T>(0, y);
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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((x > safe_lower) && (x < safe_upper) && (y > safe_lower) && (y < safe_upper))
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{
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T yy = y*y;
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T mxm1 = one - x;
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///
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// The real part is given by:
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//
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// real(atanh(z)) == log1p(4*x / ((x-1)*(x-1) + y^2))
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//
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real = boost::math::log1p(four * x / (mxm1*mxm1 + yy));
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real /= four;
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if((boost::math::signbit)(z.real()))
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real = (boost::math::changesign)(real);
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imag = std::atan2((y * two), (mxm1*(one+x) - yy));
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imag /= two;
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if(z.imag() < 0)
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imag = (boost::math::changesign)(imag);
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}
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else
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{
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//
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// This section handles exception cases that would normally cause
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// underflow or overflow in the main formulas.
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//
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// Begin by working out the real part, we need to approximate
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// real = boost::math::log1p(4x / ((x-1)^2 + y^2))
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// without either overflow or underflow in the squared terms.
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//
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T mxm1 = one - x;
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if(x >= safe_upper)
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{
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// x-1 = x to machine precision:
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if((boost::math::isinf)(x) || (boost::math::isinf)(y))
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{
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real = 0;
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}
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else if(y >= safe_upper)
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{
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// Big x and y: divide through by x*y:
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real = boost::math::log1p((four/y) / (x/y + y/x));
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}
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else if(y > one)
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{
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// Big x: divide through by x:
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real = boost::math::log1p(four / (x + y*y/x));
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}
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else
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{
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// Big x small y, as above but neglect y^2/x:
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real = boost::math::log1p(four/x);
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}
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}
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else if(y >= safe_upper)
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{
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if(x > one)
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{
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// Big y, medium x, divide through by y:
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real = boost::math::log1p((four*x/y) / (y + mxm1*mxm1/y));
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}
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else
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{
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// Small or medium x, large y:
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real = four*x/y/y;
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}
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}
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else if (x != one)
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{
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// y is small, calculate divisor carefully:
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T div = mxm1*mxm1;
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if(y > safe_lower)
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div += y*y;
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real = boost::math::log1p(four*x/div);
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}
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else
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real = boost::math::changesign(two * (std::log(y) - log_two));
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real /= four;
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if((boost::math::signbit)(z.real()))
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real = (boost::math::changesign)(real);
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//
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// Now handle imaginary part, this is much easier,
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// if x or y are large, then the formula:
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// atan2(2y, (1-x)*(1+x) - y^2)
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// evaluates to +-(PI - theta) where theta is negligible compared to PI.
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//
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if((x >= safe_upper) || (y >= safe_upper))
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{
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imag = pi;
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}
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else if(x <= safe_lower)
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{
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//
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// If both x and y are small then atan(2y),
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// otherwise just x^2 is negligible in the divisor:
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//
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if(y <= safe_lower)
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imag = std::atan2(two*y, one);
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else
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{
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if((y == zero) && (x == zero))
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imag = 0;
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else
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imag = std::atan2(two*y, one - y*y);
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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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// y^2 is negligible:
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//
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if((y == zero) && (x == one))
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imag = 0;
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else
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imag = std::atan2(two*y, mxm1*(one+x));
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}
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imag /= two;
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if((boost::math::signbit)(z.imag()))
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imag = (boost::math::changesign)(imag);
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}
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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_ATANH_INCLUDED
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