Attempt to represent the S^2->S^3 header reorganisation as a series of "hg rename" operations
1 // (C) Copyright John Maddock 2005.
2 // Distributed under the Boost Software License, Version 1.0. (See accompanying
3 // file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)
5 #ifndef BOOST_MATH_COMPLEX_ASIN_INCLUDED
6 #define BOOST_MATH_COMPLEX_ASIN_INCLUDED
8 #ifndef BOOST_MATH_COMPLEX_DETAILS_INCLUDED
9 # include <boost/math/complex/details.hpp>
11 #ifndef BOOST_MATH_LOG1P_INCLUDED
12 # include <boost/math/special_functions/log1p.hpp>
14 #include <boost/assert.hpp>
16 #ifdef BOOST_NO_STDC_NAMESPACE
17 namespace std{ using ::sqrt; using ::fabs; using ::acos; using ::asin; using ::atan; using ::atan2; }
20 namespace boost{ namespace math{
23 inline std::complex<T> asin(const std::complex<T>& z)
26 // This implementation is a transcription of the pseudo-code in:
28 // "Implementing the complex Arcsine and Arccosine Functions using Exception Handling."
29 // T E Hull, Thomas F Fairgrieve and Ping Tak Peter Tang.
30 // ACM Transactions on Mathematical Software, Vol 23, No 3, Sept 1997.
34 // These static constants should really be in a maths constants library:
36 static const T one = static_cast<T>(1);
37 //static const T two = static_cast<T>(2);
38 static const T half = static_cast<T>(0.5L);
39 static const T a_crossover = static_cast<T>(1.5L);
40 static const T b_crossover = static_cast<T>(0.6417L);
41 //static const T pi = static_cast<T>(3.141592653589793238462643383279502884197L);
42 static const T half_pi = static_cast<T>(1.57079632679489661923132169163975144L);
43 static const T log_two = static_cast<T>(0.69314718055994530941723212145817657L);
44 static const T quarter_pi = static_cast<T>(0.78539816339744830961566084581987572L);
47 // Get real and imaginary parts, discard the signs as we can
48 // figure out the sign of the result later:
50 T x = std::fabs(z.real());
51 T y = std::fabs(z.imag());
52 T real, imag; // our results
55 // Begin by handling the special cases for infinities and nan's
56 // specified in C99, most of this is handled by the regular logic
57 // below, but handling it as a special case prevents overflow/underflow
58 // arithmetic which may trip up some machines:
60 if(detail::test_is_nan(x))
62 if(detail::test_is_nan(y))
63 return std::complex<T>(x, x);
64 if(std::numeric_limits<T>::has_infinity && (y == std::numeric_limits<T>::infinity()))
67 imag = std::numeric_limits<T>::infinity();
70 return std::complex<T>(x, x);
72 else if(detail::test_is_nan(y))
79 else if(std::numeric_limits<T>::has_infinity && (x == std::numeric_limits<T>::infinity()))
82 imag = std::numeric_limits<T>::infinity();
85 return std::complex<T>(y, y);
87 else if(std::numeric_limits<T>::has_infinity && (x == std::numeric_limits<T>::infinity()))
89 if(y == std::numeric_limits<T>::infinity())
92 imag = std::numeric_limits<T>::infinity();
97 imag = std::numeric_limits<T>::infinity();
100 else if(std::numeric_limits<T>::has_infinity && (y == std::numeric_limits<T>::infinity()))
103 imag = std::numeric_limits<T>::infinity();
108 // special case for real numbers:
110 if((y == 0) && (x <= one))
111 return std::complex<T>(std::asin(z.real()));
113 // Figure out if our input is within the "safe area" identified by Hull et al.
114 // This would be more efficient with portable floating point exception handling;
115 // fortunately the quantities M and u identified by Hull et al (figure 3),
116 // match with the max and min methods of numeric_limits<T>.
118 T safe_max = detail::safe_max(static_cast<T>(8));
119 T safe_min = detail::safe_min(static_cast<T>(4));
124 if((x < safe_max) && (x > safe_min) && (y < safe_max) && (y > safe_min))
127 T r = std::sqrt(xp1*xp1 + yy);
128 T s = std::sqrt(xm1*xm1 + yy);
129 T a = half * (r + s);
141 real = std::atan(x/std::sqrt(half * apx * (yy /(r + xp1) + (s-xm1))));
145 real = std::atan(x/(y * std::sqrt(half * (apx/(r + xp1) + apx/(s+xm1)))));
154 am1 = half * (yy/(r + xp1) + yy/(s - xm1));
158 am1 = half * (yy/(r + xp1) + (s + xm1));
160 imag = boost::math::log1p(am1 + std::sqrt(am1 * (a + one)));
164 imag = std::log(a + std::sqrt(a*a - one));
170 // This is the Hull et al exception handling code from Fig 3 of their paper:
172 if(y <= (std::numeric_limits<T>::epsilon() * std::fabs(xm1)))
177 imag = y / std::sqrt(xp1*xm1);
182 if(((std::numeric_limits<T>::max)() / xp1) > xm1)
184 // xp1 * xm1 won't overflow:
185 imag = boost::math::log1p(xm1 + std::sqrt(xp1*xm1));
189 imag = log_two + std::log(x);
193 else if(y <= safe_min)
195 // There is an assumption in Hull et al's analysis that
196 // if we get here then x == 1. This is true for all "good"
199 // E^2 > 8*sqrt(u); with:
201 // E = std::numeric_limits<T>::epsilon()
202 // u = (std::numeric_limits<T>::min)()
204 // Hull et al provide alternative code for "bad" machines
205 // but we have no way to test that here, so for now just assert
206 // on the assumption:
208 BOOST_ASSERT(x == 1);
209 real = half_pi - std::sqrt(y);
212 else if(std::numeric_limits<T>::epsilon() * y - one >= x)
214 real = x/y; // This can underflow!
215 imag = log_two + std::log(y);
219 real = std::atan(x/y);
221 imag = log_two + std::log(y) + half * boost::math::log1p(xoy*xoy);
225 T a = std::sqrt(one + y*y);
226 real = x/a; // This can underflow!
227 imag = half * boost::math::log1p(static_cast<T>(2)*y*(y+a));
233 // Finish off by working out the sign of the result:
240 return std::complex<T>(real, imag);
245 #endif // BOOST_MATH_COMPLEX_ASIN_INCLUDED