--- a/epoc32/include/stdapis/boost/math/complex/atanh.hpp Wed Mar 31 12:27:01 2010 +0100
+++ b/epoc32/include/stdapis/boost/math/complex/atanh.hpp Wed Mar 31 12:33:34 2010 +0100
@@ -1,267 +1,245 @@
-// boost atanh.hpp header file
-
-// (C) Copyright Hubert Holin 2001.
-// 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)
-
-// See http://www.boost.org for updates, documentation, and revision history.
+// (C) Copyright John Maddock 2005.
+// Use, modification and distribution are subject to 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_ATANH_HPP
-#define BOOST_ATANH_HPP
-
-
-#include <cmath>
-#include <limits>
-#include <string>
-#include <stdexcept>
-
+#ifndef BOOST_MATH_COMPLEX_ATANH_INCLUDED
+#define BOOST_MATH_COMPLEX_ATANH_INCLUDED
-#include <boost/config.hpp>
-
-
-// This is the inverse of the hyperbolic tangent function.
+#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>
-namespace boost
-{
- namespace math
- {
-#if defined(__GNUC__) && (__GNUC__ < 3)
- // gcc 2.x ignores function scope using declarations,
- // put them in the scope of the enclosing namespace instead:
-
- using ::std::abs;
- using ::std::sqrt;
- using ::std::log;
-
- using ::std::numeric_limits;
+#ifdef BOOST_NO_STDC_NAMESPACE
+namespace std{ using ::sqrt; using ::fabs; using ::acos; using ::asin; using ::atan; using ::atan2; }
#endif
-
-#if defined(BOOST_NO_TEMPLATE_PARTIAL_SPECIALIZATION)
- // This is the main fare
-
- template<typename T>
- inline T atanh(const T x)
- {
- using ::std::abs;
- using ::std::sqrt;
- using ::std::log;
-
- using ::std::numeric_limits;
-
- T const one = static_cast<T>(1);
- T const two = static_cast<T>(2);
-
- static T const taylor_2_bound = sqrt(numeric_limits<T>::epsilon());
- static T const taylor_n_bound = sqrt(taylor_2_bound);
-
- if (x < -one)
- {
- if (numeric_limits<T>::has_quiet_NaN)
- {
- return(numeric_limits<T>::quiet_NaN());
- }
- else
- {
- ::std::string error_reporting("Argument to atanh is strictly greater than +1 or strictly smaller than -1!");
- ::std::domain_error bad_argument(error_reporting);
-
- throw(bad_argument);
- }
- }
- else if (x < -one+numeric_limits<T>::epsilon())
- {
- if (numeric_limits<T>::has_infinity)
- {
- return(-numeric_limits<T>::infinity());
- }
- else
- {
- ::std::string error_reporting("Argument to atanh is -1 (result: -Infinity)!");
- ::std::out_of_range bad_argument(error_reporting);
-
- throw(bad_argument);
- }
- }
- else if (x > +one-numeric_limits<T>::epsilon())
- {
- if (numeric_limits<T>::has_infinity)
- {
- return(+numeric_limits<T>::infinity());
- }
- else
- {
- ::std::string error_reporting("Argument to atanh is +1 (result: +Infinity)!");
- ::std::out_of_range bad_argument(error_reporting);
-
- throw(bad_argument);
- }
- }
- else if (x > +one)
- {
- if (numeric_limits<T>::has_quiet_NaN)
- {
- return(numeric_limits<T>::quiet_NaN());
- }
- else
- {
- ::std::string error_reporting("Argument to atanh is strictly greater than +1 or strictly smaller than -1!");
- ::std::domain_error bad_argument(error_reporting);
-
- throw(bad_argument);
- }
- }
- else if (abs(x) >= taylor_n_bound)
- {
- return(log( (one + x) / (one - x) ) / two);
- }
+
+namespace boost{ namespace math{
+
+template<class T>
+std::complex<T> atanh(const std::complex<T>& z)
+{
+ //
+ // References:
+ //
+ // Eric W. Weisstein. "Inverse Hyperbolic Tangent."
+ // From MathWorld--A Wolfram Web Resource.
+ // http://mathworld.wolfram.com/InverseHyperbolicTangent.html
+ //
+ // Also: The Wolfram Functions Site,
+ // http://functions.wolfram.com/ElementaryFunctions/ArcTanh/
+ //
+ // Also "Abramowitz and Stegun. Handbook of Mathematical Functions."
+ // at : http://jove.prohosting.com/~skripty/toc.htm
+ //
+
+ static const T half_pi = static_cast<T>(1.57079632679489661923132169163975144L);
+ static const T pi = static_cast<T>(3.141592653589793238462643383279502884197L);
+ static const T one = static_cast<T>(1.0L);
+ static const T two = static_cast<T>(2.0L);
+ static const T four = static_cast<T>(4.0L);
+ static const T zero = static_cast<T>(0);
+ static const T a_crossover = static_cast<T>(0.3L);
+
+ T x = std::fabs(z.real());
+ T y = std::fabs(z.imag());
+
+ T real, imag; // our results
+
+ T safe_upper = detail::safe_max(two);
+ T safe_lower = detail::safe_min(static_cast<T>(2));
+
+ //
+ // Begin by handling the special cases specified in C99:
+ //
+ if(detail::test_is_nan(x))
+ {
+ if(detail::test_is_nan(y))
+ return std::complex<T>(x, x);
+ else if(std::numeric_limits<T>::has_infinity && (y == std::numeric_limits<T>::infinity()))
+ return std::complex<T>(0, ((z.imag() < 0) ? -half_pi : half_pi));
+ else
+ return std::complex<T>(x, x);
+ }
+ else if(detail::test_is_nan(y))
+ {
+ if(x == 0)
+ return std::complex<T>(x, y);
+ if(std::numeric_limits<T>::has_infinity && (x == std::numeric_limits<T>::infinity()))
+ return std::complex<T>(0, y);
+ else
+ return std::complex<T>(y, y);
+ }
+ else if((x > safe_lower) && (x < safe_upper) && (y > safe_lower) && (y < safe_upper))
+ {
+
+ T xx = x*x;
+ T yy = y*y;
+ T x2 = x * two;
+
+ ///
+ // The real part is given by:
+ //
+ // real(atanh(z)) == log((1 + x^2 + y^2 + 2x) / (1 + x^2 + y^2 - 2x))
+ //
+ // However, when x is either large (x > 1/E) or very small
+ // (x < E) then this effectively simplifies
+ // to log(1), leading to wildly inaccurate results.
+ // By dividing the above (top and bottom) by (1 + x^2 + y^2) we get:
+ //
+ // real(atanh(z)) == log((1 + (2x / (1 + x^2 + y^2))) / (1 - (-2x / (1 + x^2 + y^2))))
+ //
+ // which is much more sensitive to the value of x, when x is not near 1
+ // (remember we can compute log(1+x) for small x very accurately).
+ //
+ // The cross-over from one method to the other has to be determined
+ // experimentally, the value used below appears correct to within a
+ // factor of 2 (and there are larger errors from other parts
+ // of the input domain anyway).
+ //
+ T alpha = two*x / (one + xx + yy);
+ if(alpha < a_crossover)
+ {
+ real = boost::math::log1p(alpha) - boost::math::log1p(-alpha);
+ }
+ else
+ {
+ T xm1 = x - one;
+ real = boost::math::log1p(x2 + xx + yy) - std::log(xm1*xm1 + yy);
+ }
+ real /= four;
+ if(z.real() < 0)
+ real = -real;
+
+ imag = std::atan2((y * two), (one - xx - yy));
+ imag /= two;
+ if(z.imag() < 0)
+ imag = -imag;
+ }
+ else
+ {
+ //
+ // This section handles exception cases that would normally cause
+ // underflow or overflow in the main formulas.
+ //
+ // Begin by working out the real part, we need to approximate
+ // alpha = 2x / (1 + x^2 + y^2)
+ // without either overflow or underflow in the squared terms.
+ //
+ T alpha = 0;
+ if(x >= safe_upper)
+ {
+ // this is really a test for infinity,
+ // but we may not have the necessary numeric_limits support:
+ if((x > (std::numeric_limits<T>::max)()) || (y > (std::numeric_limits<T>::max)()))
+ {
+ alpha = 0;
+ }
+ else if(y >= safe_upper)
+ {
+ // Big x and y: divide alpha through by x*y:
+ alpha = (two/y) / (x/y + y/x);
+ }
+ else if(y > one)
+ {
+ // Big x: divide through by x:
+ alpha = two / (x + y*y/x);
+ }
+ else
+ {
+ // Big x small y, as above but neglect y^2/x:
+ alpha = two/x;
+ }
+ }
+ else if(y >= safe_upper)
+ {
+ if(x > one)
+ {
+ // Big y, medium x, divide through by y:
+ alpha = (two*x/y) / (y + x*x/y);
+ }
+ else
+ {
+ // Small x and y, whatever alpha is, it's too small to calculate:
+ alpha = 0;
+ }
+ }
+ else
+ {
+ // one or both of x and y are small, calculate divisor carefully:
+ T div = one;
+ if(x > safe_lower)
+ div += x*x;
+ if(y > safe_lower)
+ div += y*y;
+ alpha = two*x/div;
+ }
+ if(alpha < a_crossover)
+ {
+ real = boost::math::log1p(alpha) - boost::math::log1p(-alpha);
+ }
+ else
+ {
+ // We can only get here as a result of small y and medium sized x,
+ // we can simply neglect the y^2 terms:
+ BOOST_ASSERT(x >= safe_lower);
+ BOOST_ASSERT(x <= safe_upper);
+ //BOOST_ASSERT(y <= safe_lower);
+ T xm1 = x - one;
+ real = std::log(1 + two*x + x*x) - std::log(xm1*xm1);
+ }
+
+ real /= four;
+ if(z.real() < 0)
+ real = -real;
+
+ //
+ // Now handle imaginary part, this is much easier,
+ // if x or y are large, then the formula:
+ // atan2(2y, 1 - x^2 - y^2)
+ // evaluates to +-(PI - theta) where theta is negligible compared to PI.
+ //
+ if((x >= safe_upper) || (y >= safe_upper))
+ {
+ imag = pi;
+ }
+ else if(x <= safe_lower)
+ {
+ //
+ // If both x and y are small then atan(2y),
+ // otherwise just x^2 is negligible in the divisor:
+ //
+ if(y <= safe_lower)
+ imag = std::atan2(two*y, one);
+ else
+ {
+ if((y == zero) && (x == zero))
+ imag = 0;
else
- {
- // approximation by taylor series in x at 0 up to order 2
- T result = x;
-
- if (abs(x) >= taylor_2_bound)
- {
- T x3 = x*x*x;
-
- // approximation by taylor series in x at 0 up to order 4
- result += x3/static_cast<T>(3);
- }
-
- return(result);
- }
- }
-#else
- // These are implementation details (for main fare see below)
-
- namespace detail
- {
- template <
- typename T,
- bool InfinitySupported
- >
- struct atanh_helper1_t
- {
- static T get_pos_infinity()
- {
- return(+::std::numeric_limits<T>::infinity());
- }
-
- static T get_neg_infinity()
- {
- return(-::std::numeric_limits<T>::infinity());
- }
- }; // boost::math::detail::atanh_helper1_t
-
-
- template<typename T>
- struct atanh_helper1_t<T, false>
- {
- static T get_pos_infinity()
- {
- ::std::string error_reporting("Argument to atanh is +1 (result: +Infinity)!");
- ::std::out_of_range bad_argument(error_reporting);
-
- throw(bad_argument);
- }
-
- static T get_neg_infinity()
- {
- ::std::string error_reporting("Argument to atanh is -1 (result: -Infinity)!");
- ::std::out_of_range bad_argument(error_reporting);
-
- throw(bad_argument);
- }
- }; // boost::math::detail::atanh_helper1_t
-
-
- template <
- typename T,
- bool QuietNanSupported
- >
- struct atanh_helper2_t
- {
- static T get_NaN()
- {
- return(::std::numeric_limits<T>::quiet_NaN());
- }
- }; // boost::detail::atanh_helper2_t
-
-
- template<typename T>
- struct atanh_helper2_t<T, false>
- {
- static T get_NaN()
- {
- ::std::string error_reporting("Argument to atanh is strictly greater than +1 or strictly smaller than -1!");
- ::std::domain_error bad_argument(error_reporting);
-
- throw(bad_argument);
- }
- }; // boost::detail::atanh_helper2_t
- } // boost::detail
-
-
- // This is the main fare
-
- template<typename T>
- inline T atanh(const T x)
- {
- using ::std::abs;
- using ::std::sqrt;
- using ::std::log;
-
- using ::std::numeric_limits;
-
- typedef detail::atanh_helper1_t<T, ::std::numeric_limits<T>::has_infinity> helper1_type;
- typedef detail::atanh_helper2_t<T, ::std::numeric_limits<T>::has_quiet_NaN> helper2_type;
-
-
- T const one = static_cast<T>(1);
- T const two = static_cast<T>(2);
-
- static T const taylor_2_bound = sqrt(numeric_limits<T>::epsilon());
- static T const taylor_n_bound = sqrt(taylor_2_bound);
-
- if (x < -one)
- {
- return(helper2_type::get_NaN());
- }
- else if (x < -one+numeric_limits<T>::epsilon())
- {
- return(helper1_type::get_neg_infinity());
- }
- else if (x > +one-numeric_limits<T>::epsilon())
- {
- return(helper1_type::get_pos_infinity());
- }
- else if (x > +one)
- {
- return(helper2_type::get_NaN());
- }
- else if (abs(x) >= taylor_n_bound)
- {
- return(log( (one + x) / (one - x) ) / two);
- }
- else
- {
- // approximation by taylor series in x at 0 up to order 2
- T result = x;
-
- if (abs(x) >= taylor_2_bound)
- {
- T x3 = x*x*x;
-
- // approximation by taylor series in x at 0 up to order 4
- result += x3/static_cast<T>(3);
- }
-
- return(result);
- }
- }
-#endif /* defined(BOOST_NO_TEMPLATE_PARTIAL_SPECIALIZATION) */
- }
+ imag = std::atan2(two*y, one - y*y);
+ }
+ }
+ else
+ {
+ //
+ // y^2 is negligible:
+ //
+ if((y == zero) && (x == one))
+ imag = 0;
+ else
+ imag = std::atan2(two*y, 1 - x*x);
+ }
+ imag /= two;
+ if(z.imag() < 0)
+ imag = -imag;
+ }
+ return std::complex<T>(real, imag);
}
-#endif /* BOOST_ATANH_HPP */
+} } // namespaces
+#endif // BOOST_MATH_COMPLEX_ATANH_INCLUDED