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// Copyright Christopher Kormanyos 2002 - 2011.
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// Copyright 2011 John Maddock. Distributed under the Boost
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// Distributed under the Boost Software License, Version 1.0.
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// (See accompanying file LICENSE_1_0.txt or copy at
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// http://www.boost.org/LICENSE_1_0.txt)
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// This work is based on an earlier work:
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// "Algorithm 910: A Portable C++ Multiple-Precision System for Special-Function Calculations",
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// in ACM TOMS, {VOL 37, ISSUE 4, (February 2011)} (C) ACM, 2011. http://doi.acm.org/10.1145/1916461.1916469
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//
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// This file has no include guards or namespaces - it's expanded inline inside default_ops.hpp
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//
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#ifdef BOOST_MSVC
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#pragma warning(push)
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#pragma warning(disable : 6326) // comparison of two constants
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#endif
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template <class T>
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void hyp0F1(T& result, const T& b, const T& x)
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{
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typedef typename boost::multiprecision::detail::canonical<boost::int32_t, T>::type si_type;
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typedef typename boost::multiprecision::detail::canonical<boost::uint32_t, T>::type ui_type;
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// Compute the series representation of Hypergeometric0F1 taken from
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// http://functions.wolfram.com/HypergeometricFunctions/Hypergeometric0F1/06/01/01/
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// There are no checks on input range or parameter boundaries.
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T x_pow_n_div_n_fact(x);
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T pochham_b(b);
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T bp(b);
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eval_divide(result, x_pow_n_div_n_fact, pochham_b);
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eval_add(result, ui_type(1));
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si_type n;
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T tol;
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tol = ui_type(1);
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eval_ldexp(tol, tol, 1 - boost::multiprecision::detail::digits2<number<T, et_on> >::value());
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eval_multiply(tol, result);
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if (eval_get_sign(tol) < 0)
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tol.negate();
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T term;
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const int series_limit =
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boost::multiprecision::detail::digits2<number<T, et_on> >::value() < 100
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? 100
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: boost::multiprecision::detail::digits2<number<T, et_on> >::value();
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// Series expansion of hyperg_0f1(; b; x).
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for (n = 2; n < series_limit; ++n)
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{
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eval_multiply(x_pow_n_div_n_fact, x);
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eval_divide(x_pow_n_div_n_fact, n);
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eval_increment(bp);
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eval_multiply(pochham_b, bp);
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eval_divide(term, x_pow_n_div_n_fact, pochham_b);
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eval_add(result, term);
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bool neg_term = eval_get_sign(term) < 0;
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if (neg_term)
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term.negate();
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if (term.compare(tol) <= 0)
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break;
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}
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if (n >= series_limit)
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BOOST_THROW_EXCEPTION(std::runtime_error("H0F1 Failed to Converge"));
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}
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template <class T, unsigned N, bool b = boost::multiprecision::detail::is_variable_precision<boost::multiprecision::number<T> >::value>
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struct scoped_N_precision
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{
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template <class U>
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scoped_N_precision(U const&) {}
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template <class U>
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void reduce(U&) {}
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};
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template <class T, unsigned N>
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struct scoped_N_precision<T, N, true>
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{
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unsigned old_precision, old_arg_precision;
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scoped_N_precision(T& arg)
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{
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old_precision = T::default_precision();
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old_arg_precision = arg.precision();
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T::default_precision(old_arg_precision * N);
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arg.precision(old_arg_precision * N);
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}
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~scoped_N_precision()
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{
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T::default_precision(old_precision);
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}
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void reduce(T& arg)
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{
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arg.precision(old_arg_precision);
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}
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};
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template <class T>
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void reduce_n_half_pi(T& arg, const T& n, bool go_down)
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{
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//
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// We need to perform argument reduction at 3 times the precision of arg
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// in order to ensure a correct result up to arg = 1/epsilon. Beyond that
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// the value of n will have been incorrectly calculated anyway since it will
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// have a value greater than 1/epsilon and no longer be an exact integer value.
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//
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// More information in ARGUMENT REDUCTION FOR HUGE ARGUMENTS. K C Ng.
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//
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// There are two mutually exclusive ways to achieve this, both of which are
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// supported here:
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// 1) To define a fixed precision type with 3 times the precision for the calculation.
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// 2) To dynamically increase the precision of the variables.
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//
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typedef typename boost::multiprecision::detail::transcendental_reduction_type<T>::type reduction_type;
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//
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// Make a copy of the arg at higher precision:
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//
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reduction_type big_arg(arg);
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//
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// Dynamically increase precision when supported, this increases the default
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// and ups the precision of big_arg to match:
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//
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scoped_N_precision<T, 3> scoped_precision(big_arg);
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//
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// High precision PI:
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//
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reduction_type reduction = get_constant_pi<reduction_type>();
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eval_ldexp(reduction, reduction, -1); // divide by 2
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eval_multiply(reduction, n);
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BOOST_MATH_INSTRUMENT_CODE(big_arg.str(10, std::ios_base::scientific));
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BOOST_MATH_INSTRUMENT_CODE(reduction.str(10, std::ios_base::scientific));
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if (go_down)
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eval_subtract(big_arg, reduction, big_arg);
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else
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eval_subtract(big_arg, reduction);
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arg = T(big_arg);
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//
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// If arg is a variable precision type, then we have just copied the
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// precision of big_arg s well it's value. Reduce the precision now:
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//
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scoped_precision.reduce(arg);
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BOOST_MATH_INSTRUMENT_CODE(big_arg.str(10, std::ios_base::scientific));
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BOOST_MATH_INSTRUMENT_CODE(arg.str(10, std::ios_base::scientific));
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}
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template <class T>
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void eval_sin(T& result, const T& x)
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{
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BOOST_STATIC_ASSERT_MSG(number_category<T>::value == number_kind_floating_point, "The sin function is only valid for floating point types.");
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BOOST_MATH_INSTRUMENT_CODE(x.str(0, std::ios_base::scientific));
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if (&result == &x)
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{
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T temp;
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eval_sin(temp, x);
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result = temp;
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return;
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}
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typedef typename boost::multiprecision::detail::canonical<boost::int32_t, T>::type si_type;
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typedef typename boost::multiprecision::detail::canonical<boost::uint32_t, T>::type ui_type;
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typedef typename mpl::front<typename T::float_types>::type fp_type;
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switch (eval_fpclassify(x))
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{
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case FP_INFINITE:
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case FP_NAN:
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if (std::numeric_limits<number<T, et_on> >::has_quiet_NaN)
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{
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result = std::numeric_limits<number<T, et_on> >::quiet_NaN().backend();
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errno = EDOM;
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}
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else
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BOOST_THROW_EXCEPTION(std::domain_error("Result is undefined or complex and there is no NaN for this number type."));
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return;
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case FP_ZERO:
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result = x;
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return;
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default:;
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}
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// Local copy of the argument
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T xx = x;
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// Analyze and prepare the phase of the argument.
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// Make a local, positive copy of the argument, xx.
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// The argument xx will be reduced to 0 <= xx <= pi/2.
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bool b_negate_sin = false;
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if (eval_get_sign(x) < 0)
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{
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xx.negate();
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b_negate_sin = !b_negate_sin;
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}
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T n_pi, t;
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T half_pi = get_constant_pi<T>();
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eval_ldexp(half_pi, half_pi, -1); // divide by 2
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// Remove multiples of pi/2.
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if (xx.compare(half_pi) > 0)
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{
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eval_divide(n_pi, xx, half_pi);
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eval_trunc(n_pi, n_pi);
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t = ui_type(4);
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eval_fmod(t, n_pi, t);
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bool b_go_down = false;
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if (t.compare(ui_type(1)) == 0)
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{
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b_go_down = true;
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}
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else if (t.compare(ui_type(2)) == 0)
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{
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b_negate_sin = !b_negate_sin;
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}
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else if (t.compare(ui_type(3)) == 0)
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{
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b_negate_sin = !b_negate_sin;
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b_go_down = true;
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}
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if (b_go_down)
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eval_increment(n_pi);
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//
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// If n_pi is > 1/epsilon, then it is no longer an exact integer value
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// but an approximation. As a result we can no longer reliably reduce
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// xx to 0 <= xx < pi/2, nor can we tell the sign of the result as we need
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// n_pi % 4 for that, but that will always be zero in this situation.
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// We could use a higher precision type for n_pi, along with division at
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// higher precision, but that's rather expensive. So for now we do not support
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// this, and will see if anyone complains and has a legitimate use case.
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//
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if (n_pi.compare(get_constant_one_over_epsilon<T>()) > 0)
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{
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result = ui_type(0);
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return;
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}
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reduce_n_half_pi(xx, n_pi, b_go_down);
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//
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// Post reduction we may be a few ulp below zero or above pi/2
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// given that n_pi was calculated at working precision and not
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// at the higher precision used for reduction. Correct that now:
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//
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if (eval_get_sign(xx) < 0)
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{
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xx.negate();
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b_negate_sin = !b_negate_sin;
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}
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if (xx.compare(half_pi) > 0)
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{
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eval_ldexp(half_pi, half_pi, 1);
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eval_subtract(xx, half_pi, xx);
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eval_ldexp(half_pi, half_pi, -1);
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b_go_down = !b_go_down;
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}
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BOOST_MATH_INSTRUMENT_CODE(xx.str(0, std::ios_base::scientific));
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BOOST_MATH_INSTRUMENT_CODE(n_pi.str(0, std::ios_base::scientific));
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BOOST_ASSERT(xx.compare(half_pi) <= 0);
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BOOST_ASSERT(xx.compare(ui_type(0)) >= 0);
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}
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t = half_pi;
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eval_subtract(t, xx);
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const bool b_zero = eval_get_sign(xx) == 0;
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const bool b_pi_half = eval_get_sign(t) == 0;
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BOOST_MATH_INSTRUMENT_CODE(xx.str(0, std::ios_base::scientific));
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BOOST_MATH_INSTRUMENT_CODE(t.str(0, std::ios_base::scientific));
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// Check if the reduced argument is very close to 0 or pi/2.
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const bool b_near_zero = xx.compare(fp_type(1e-1)) < 0;
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const bool b_near_pi_half = t.compare(fp_type(1e-1)) < 0;
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if (b_zero)
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{
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result = ui_type(0);
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}
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else if (b_pi_half)
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{
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result = ui_type(1);
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}
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else if (b_near_zero)
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{
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eval_multiply(t, xx, xx);
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eval_divide(t, si_type(-4));
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T t2;
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t2 = fp_type(1.5);
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hyp0F1(result, t2, t);
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BOOST_MATH_INSTRUMENT_CODE(result.str(0, std::ios_base::scientific));
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eval_multiply(result, xx);
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}
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else if (b_near_pi_half)
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{
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eval_multiply(t, t);
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eval_divide(t, si_type(-4));
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T t2;
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t2 = fp_type(0.5);
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hyp0F1(result, t2, t);
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BOOST_MATH_INSTRUMENT_CODE(result.str(0, std::ios_base::scientific));
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}
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else
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{
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// Scale to a small argument for an efficient Taylor series,
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// implemented as a hypergeometric function. Use a standard
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// divide by three identity a certain number of times.
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// Here we use division by 3^9 --> (19683 = 3^9).
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static const si_type n_scale = 9;
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static const si_type n_three_pow_scale = static_cast<si_type>(19683L);
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eval_divide(xx, n_three_pow_scale);
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// Now with small arguments, we are ready for a series expansion.
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eval_multiply(t, xx, xx);
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eval_divide(t, si_type(-4));
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T t2;
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t2 = fp_type(1.5);
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hyp0F1(result, t2, t);
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BOOST_MATH_INSTRUMENT_CODE(result.str(0, std::ios_base::scientific));
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eval_multiply(result, xx);
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// Convert back using multiple angle identity.
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for (boost::int32_t k = static_cast<boost::int32_t>(0); k < n_scale; k++)
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{
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// Rescale the cosine value using the multiple angle identity.
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eval_multiply(t2, result, ui_type(3));
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eval_multiply(t, result, result);
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eval_multiply(t, result);
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eval_multiply(t, ui_type(4));
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eval_subtract(result, t2, t);
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}
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}
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if (b_negate_sin)
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result.negate();
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BOOST_MATH_INSTRUMENT_CODE(result.str(0, std::ios_base::scientific));
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}
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template <class T>
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void eval_cos(T& result, const T& x)
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{
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BOOST_STATIC_ASSERT_MSG(number_category<T>::value == number_kind_floating_point, "The cos function is only valid for floating point types.");
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if (&result == &x)
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{
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T temp;
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eval_cos(temp, x);
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result = temp;
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return;
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}
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typedef typename boost::multiprecision::detail::canonical<boost::int32_t, T>::type si_type;
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typedef typename boost::multiprecision::detail::canonical<boost::uint32_t, T>::type ui_type;
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switch (eval_fpclassify(x))
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{
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case FP_INFINITE:
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case FP_NAN:
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if (std::numeric_limits<number<T, et_on> >::has_quiet_NaN)
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{
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result = std::numeric_limits<number<T, et_on> >::quiet_NaN().backend();
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errno = EDOM;
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}
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else
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BOOST_THROW_EXCEPTION(std::domain_error("Result is undefined or complex and there is no NaN for this number type."));
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return;
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case FP_ZERO:
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result = ui_type(1);
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return;
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default:;
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}
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// Local copy of the argument
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T xx = x;
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// Analyze and prepare the phase of the argument.
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// Make a local, positive copy of the argument, xx.
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// The argument xx will be reduced to 0 <= xx <= pi/2.
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bool b_negate_cos = false;
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if (eval_get_sign(x) < 0)
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{
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xx.negate();
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}
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BOOST_MATH_INSTRUMENT_CODE(xx.str(0, std::ios_base::scientific));
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T n_pi, t;
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T half_pi = get_constant_pi<T>();
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eval_ldexp(half_pi, half_pi, -1); // divide by 2
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// Remove even multiples of pi.
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if (xx.compare(half_pi) > 0)
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{
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eval_divide(t, xx, half_pi);
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eval_trunc(n_pi, t);
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BOOST_MATH_INSTRUMENT_CODE(n_pi.str(0, std::ios_base::scientific));
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t = ui_type(4);
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eval_fmod(t, n_pi, t);
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bool b_go_down = false;
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if (t.compare(ui_type(0)) == 0)
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{
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b_go_down = true;
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}
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else if (t.compare(ui_type(1)) == 0)
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{
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b_negate_cos = true;
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}
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else if (t.compare(ui_type(2)) == 0)
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{
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b_go_down = true;
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b_negate_cos = true;
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}
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else
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{
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BOOST_ASSERT(t.compare(ui_type(3)) == 0);
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}
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if (b_go_down)
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eval_increment(n_pi);
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//
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// If n_pi is > 1/epsilon, then it is no longer an exact integer value
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// but an approximation. As a result we can no longer reliably reduce
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// xx to 0 <= xx < pi/2, nor can we tell the sign of the result as we need
|
// n_pi % 4 for that, but that will always be zero in this situation.
|
// We could use a higher precision type for n_pi, along with division at
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// higher precision, but that's rather expensive. So for now we do not support
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// this, and will see if anyone complains and has a legitimate use case.
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//
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if (n_pi.compare(get_constant_one_over_epsilon<T>()) > 0)
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{
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result = ui_type(1);
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return;
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}
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reduce_n_half_pi(xx, n_pi, b_go_down);
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//
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// Post reduction we may be a few ulp below zero or above pi/2
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// given that n_pi was calculated at working precision and not
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// at the higher precision used for reduction. Correct that now:
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//
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if (eval_get_sign(xx) < 0)
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{
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xx.negate();
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b_negate_cos = !b_negate_cos;
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}
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if (xx.compare(half_pi) > 0)
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{
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eval_ldexp(half_pi, half_pi, 1);
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eval_subtract(xx, half_pi, xx);
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eval_ldexp(half_pi, half_pi, -1);
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}
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BOOST_ASSERT(xx.compare(half_pi) <= 0);
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BOOST_ASSERT(xx.compare(ui_type(0)) >= 0);
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}
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else
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{
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n_pi = ui_type(1);
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reduce_n_half_pi(xx, n_pi, true);
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}
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const bool b_zero = eval_get_sign(xx) == 0;
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if (b_zero)
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{
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result = si_type(0);
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}
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else
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{
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eval_sin(result, xx);
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}
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if (b_negate_cos)
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result.negate();
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BOOST_MATH_INSTRUMENT_CODE(result.str(0, std::ios_base::scientific));
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}
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template <class T>
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void eval_tan(T& result, const T& x)
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{
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BOOST_STATIC_ASSERT_MSG(number_category<T>::value == number_kind_floating_point, "The tan function is only valid for floating point types.");
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if (&result == &x)
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{
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T temp;
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eval_tan(temp, x);
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result = temp;
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return;
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}
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T t;
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eval_sin(result, x);
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eval_cos(t, x);
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eval_divide(result, t);
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}
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template <class T>
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void hyp2F1(T& result, const T& a, const T& b, const T& c, const T& x)
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{
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// Compute the series representation of hyperg_2f1 taken from
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// Abramowitz and Stegun 15.1.1.
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// There are no checks on input range or parameter boundaries.
|
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typedef typename boost::multiprecision::detail::canonical<boost::uint32_t, T>::type ui_type;
|
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T x_pow_n_div_n_fact(x);
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T pochham_a(a);
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T pochham_b(b);
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T pochham_c(c);
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T ap(a);
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T bp(b);
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T cp(c);
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eval_multiply(result, pochham_a, pochham_b);
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eval_divide(result, pochham_c);
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eval_multiply(result, x_pow_n_div_n_fact);
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eval_add(result, ui_type(1));
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T lim;
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eval_ldexp(lim, result, 1 - boost::multiprecision::detail::digits2<number<T, et_on> >::value());
|
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if (eval_get_sign(lim) < 0)
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lim.negate();
|
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ui_type n;
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T term;
|
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const unsigned series_limit =
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boost::multiprecision::detail::digits2<number<T, et_on> >::value() < 100
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? 100
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: boost::multiprecision::detail::digits2<number<T, et_on> >::value();
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// Series expansion of hyperg_2f1(a, b; c; x).
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for (n = 2; n < series_limit; ++n)
|
{
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eval_multiply(x_pow_n_div_n_fact, x);
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eval_divide(x_pow_n_div_n_fact, n);
|
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eval_increment(ap);
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eval_multiply(pochham_a, ap);
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eval_increment(bp);
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eval_multiply(pochham_b, bp);
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eval_increment(cp);
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eval_multiply(pochham_c, cp);
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eval_multiply(term, pochham_a, pochham_b);
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eval_divide(term, pochham_c);
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eval_multiply(term, x_pow_n_div_n_fact);
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eval_add(result, term);
|
|
if (eval_get_sign(term) < 0)
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term.negate();
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if (lim.compare(term) >= 0)
|
break;
|
}
|
if (n > series_limit)
|
BOOST_THROW_EXCEPTION(std::runtime_error("H2F1 failed to converge."));
|
}
|
|
template <class T>
|
void eval_asin(T& result, const T& x)
|
{
|
BOOST_STATIC_ASSERT_MSG(number_category<T>::value == number_kind_floating_point, "The asin function is only valid for floating point types.");
|
typedef typename boost::multiprecision::detail::canonical<boost::uint32_t, T>::type ui_type;
|
typedef typename mpl::front<typename T::float_types>::type fp_type;
|
|
if (&result == &x)
|
{
|
T t(x);
|
eval_asin(result, t);
|
return;
|
}
|
|
switch (eval_fpclassify(x))
|
{
|
case FP_NAN:
|
case FP_INFINITE:
|
if (std::numeric_limits<number<T, et_on> >::has_quiet_NaN)
|
{
|
result = std::numeric_limits<number<T, et_on> >::quiet_NaN().backend();
|
errno = EDOM;
|
}
|
else
|
BOOST_THROW_EXCEPTION(std::domain_error("Result is undefined or complex and there is no NaN for this number type."));
|
return;
|
case FP_ZERO:
|
result = x;
|
return;
|
default:;
|
}
|
|
const bool b_neg = eval_get_sign(x) < 0;
|
|
T xx(x);
|
if (b_neg)
|
xx.negate();
|
|
int c = xx.compare(ui_type(1));
|
if (c > 0)
|
{
|
if (std::numeric_limits<number<T, et_on> >::has_quiet_NaN)
|
{
|
result = std::numeric_limits<number<T, et_on> >::quiet_NaN().backend();
|
errno = EDOM;
|
}
|
else
|
BOOST_THROW_EXCEPTION(std::domain_error("Result is undefined or complex and there is no NaN for this number type."));
|
return;
|
}
|
else if (c == 0)
|
{
|
result = get_constant_pi<T>();
|
eval_ldexp(result, result, -1);
|
if (b_neg)
|
result.negate();
|
return;
|
}
|
|
if (xx.compare(fp_type(1e-3)) < 0)
|
{
|
// http://functions.wolfram.com/ElementaryFunctions/ArcSin/26/01/01/
|
eval_multiply(xx, xx);
|
T t1, t2;
|
t1 = fp_type(0.5f);
|
t2 = fp_type(1.5f);
|
hyp2F1(result, t1, t1, t2, xx);
|
eval_multiply(result, x);
|
return;
|
}
|
else if (xx.compare(fp_type(1 - 5e-2f)) > 0)
|
{
|
// http://functions.wolfram.com/ElementaryFunctions/ArcSin/26/01/01/
|
// This branch is simlilar in complexity to Newton iterations down to
|
// the above limit. It is *much* more accurate.
|
T dx1;
|
T t1, t2;
|
eval_subtract(dx1, ui_type(1), xx);
|
t1 = fp_type(0.5f);
|
t2 = fp_type(1.5f);
|
eval_ldexp(dx1, dx1, -1);
|
hyp2F1(result, t1, t1, t2, dx1);
|
eval_ldexp(dx1, dx1, 2);
|
eval_sqrt(t1, dx1);
|
eval_multiply(result, t1);
|
eval_ldexp(t1, get_constant_pi<T>(), -1);
|
result.negate();
|
eval_add(result, t1);
|
if (b_neg)
|
result.negate();
|
return;
|
}
|
#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS
|
typedef typename boost::multiprecision::detail::canonical<long double, T>::type guess_type;
|
#else
|
typedef fp_type guess_type;
|
#endif
|
// Get initial estimate using standard math function asin.
|
guess_type dd;
|
eval_convert_to(&dd, xx);
|
|
result = (guess_type)(std::asin(dd));
|
|
// Newton-Raphson iteration, we should double our precision with each iteration,
|
// in practice this seems to not quite work in all cases... so terminate when we
|
// have at least 2/3 of the digits correct on the assumption that the correction
|
// we've just added will finish the job...
|
|
boost::intmax_t current_precision = eval_ilogb(result);
|
boost::intmax_t target_precision = std::numeric_limits<number<T> >::is_specialized ?
|
current_precision - 1 - (std::numeric_limits<number<T> >::digits * 2) / 3
|
: current_precision - 1 - (boost::multiprecision::detail::digits2<number<T> >::value() * 2) / 3;
|
|
// Newton-Raphson iteration
|
while (current_precision > target_precision)
|
{
|
T sine, cosine;
|
eval_sin(sine, result);
|
eval_cos(cosine, result);
|
eval_subtract(sine, xx);
|
eval_divide(sine, cosine);
|
eval_subtract(result, sine);
|
current_precision = eval_ilogb(sine);
|
if (current_precision <= (std::numeric_limits<typename T::exponent_type>::min)() + 1)
|
break;
|
}
|
if (b_neg)
|
result.negate();
|
}
|
|
template <class T>
|
inline void eval_acos(T& result, const T& x)
|
{
|
BOOST_STATIC_ASSERT_MSG(number_category<T>::value == number_kind_floating_point, "The acos function is only valid for floating point types.");
|
typedef typename boost::multiprecision::detail::canonical<boost::uint32_t, T>::type ui_type;
|
|
switch (eval_fpclassify(x))
|
{
|
case FP_NAN:
|
case FP_INFINITE:
|
if (std::numeric_limits<number<T, et_on> >::has_quiet_NaN)
|
{
|
result = std::numeric_limits<number<T, et_on> >::quiet_NaN().backend();
|
errno = EDOM;
|
}
|
else
|
BOOST_THROW_EXCEPTION(std::domain_error("Result is undefined or complex and there is no NaN for this number type."));
|
return;
|
case FP_ZERO:
|
result = get_constant_pi<T>();
|
eval_ldexp(result, result, -1); // divide by two.
|
return;
|
}
|
|
T xx;
|
eval_abs(xx, x);
|
int c = xx.compare(ui_type(1));
|
|
if (c > 0)
|
{
|
if (std::numeric_limits<number<T, et_on> >::has_quiet_NaN)
|
{
|
result = std::numeric_limits<number<T, et_on> >::quiet_NaN().backend();
|
errno = EDOM;
|
}
|
else
|
BOOST_THROW_EXCEPTION(std::domain_error("Result is undefined or complex and there is no NaN for this number type."));
|
return;
|
}
|
else if (c == 0)
|
{
|
if (eval_get_sign(x) < 0)
|
result = get_constant_pi<T>();
|
else
|
result = ui_type(0);
|
return;
|
}
|
|
typedef typename mpl::front<typename T::float_types>::type fp_type;
|
|
if (xx.compare(fp_type(1e-3)) < 0)
|
{
|
// https://functions.wolfram.com/ElementaryFunctions/ArcCos/26/01/01/
|
eval_multiply(xx, xx);
|
T t1, t2;
|
t1 = fp_type(0.5f);
|
t2 = fp_type(1.5f);
|
hyp2F1(result, t1, t1, t2, xx);
|
eval_multiply(result, x);
|
eval_ldexp(t1, get_constant_pi<T>(), -1);
|
result.negate();
|
eval_add(result, t1);
|
return;
|
}
|
if (eval_get_sign(x) < 0)
|
{
|
eval_acos(result, xx);
|
result.negate();
|
eval_add(result, get_constant_pi<T>());
|
return;
|
}
|
else if (xx.compare(fp_type(0.85)) > 0)
|
{
|
// https://functions.wolfram.com/ElementaryFunctions/ArcCos/26/01/01/
|
// This branch is simlilar in complexity to Newton iterations down to
|
// the above limit. It is *much* more accurate.
|
T dx1;
|
T t1, t2;
|
eval_subtract(dx1, ui_type(1), xx);
|
t1 = fp_type(0.5f);
|
t2 = fp_type(1.5f);
|
eval_ldexp(dx1, dx1, -1);
|
hyp2F1(result, t1, t1, t2, dx1);
|
eval_ldexp(dx1, dx1, 2);
|
eval_sqrt(t1, dx1);
|
eval_multiply(result, t1);
|
return;
|
}
|
|
#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS
|
typedef typename boost::multiprecision::detail::canonical<long double, T>::type guess_type;
|
#else
|
typedef fp_type guess_type;
|
#endif
|
// Get initial estimate using standard math function asin.
|
guess_type dd;
|
eval_convert_to(&dd, xx);
|
|
result = (guess_type)(std::acos(dd));
|
|
// Newton-Raphson iteration, we should double our precision with each iteration,
|
// in practice this seems to not quite work in all cases... so terminate when we
|
// have at least 2/3 of the digits correct on the assumption that the correction
|
// we've just added will finish the job...
|
|
boost::intmax_t current_precision = eval_ilogb(result);
|
boost::intmax_t target_precision = std::numeric_limits<number<T> >::is_specialized ?
|
current_precision - 1 - (std::numeric_limits<number<T> >::digits * 2) / 3
|
: current_precision - 1 - (boost::multiprecision::detail::digits2<number<T> >::value() * 2) / 3;
|
|
// Newton-Raphson iteration
|
while (current_precision > target_precision)
|
{
|
T sine, cosine;
|
eval_sin(sine, result);
|
eval_cos(cosine, result);
|
eval_subtract(cosine, xx);
|
cosine.negate();
|
eval_divide(cosine, sine);
|
eval_subtract(result, cosine);
|
current_precision = eval_ilogb(cosine);
|
if (current_precision <= (std::numeric_limits<typename T::exponent_type>::min)() + 1)
|
break;
|
}
|
}
|
|
template <class T>
|
void eval_atan(T& result, const T& x)
|
{
|
BOOST_STATIC_ASSERT_MSG(number_category<T>::value == number_kind_floating_point, "The atan function is only valid for floating point types.");
|
typedef typename boost::multiprecision::detail::canonical<boost::int32_t, T>::type si_type;
|
typedef typename boost::multiprecision::detail::canonical<boost::uint32_t, T>::type ui_type;
|
typedef typename mpl::front<typename T::float_types>::type fp_type;
|
|
switch (eval_fpclassify(x))
|
{
|
case FP_NAN:
|
result = x;
|
errno = EDOM;
|
return;
|
case FP_ZERO:
|
result = x;
|
return;
|
case FP_INFINITE:
|
if (eval_get_sign(x) < 0)
|
{
|
eval_ldexp(result, get_constant_pi<T>(), -1);
|
result.negate();
|
}
|
else
|
eval_ldexp(result, get_constant_pi<T>(), -1);
|
return;
|
default:;
|
}
|
|
const bool b_neg = eval_get_sign(x) < 0;
|
|
T xx(x);
|
if (b_neg)
|
xx.negate();
|
|
if (xx.compare(fp_type(0.1)) < 0)
|
{
|
T t1, t2, t3;
|
t1 = ui_type(1);
|
t2 = fp_type(0.5f);
|
t3 = fp_type(1.5f);
|
eval_multiply(xx, xx);
|
xx.negate();
|
hyp2F1(result, t1, t2, t3, xx);
|
eval_multiply(result, x);
|
return;
|
}
|
|
if (xx.compare(fp_type(10)) > 0)
|
{
|
T t1, t2, t3;
|
t1 = fp_type(0.5f);
|
t2 = ui_type(1u);
|
t3 = fp_type(1.5f);
|
eval_multiply(xx, xx);
|
eval_divide(xx, si_type(-1), xx);
|
hyp2F1(result, t1, t2, t3, xx);
|
eval_divide(result, x);
|
if (!b_neg)
|
result.negate();
|
eval_ldexp(t1, get_constant_pi<T>(), -1);
|
eval_add(result, t1);
|
if (b_neg)
|
result.negate();
|
return;
|
}
|
|
// Get initial estimate using standard math function atan.
|
fp_type d;
|
eval_convert_to(&d, xx);
|
result = fp_type(std::atan(d));
|
|
// Newton-Raphson iteration, we should double our precision with each iteration,
|
// in practice this seems to not quite work in all cases... so terminate when we
|
// have at least 2/3 of the digits correct on the assumption that the correction
|
// we've just added will finish the job...
|
|
boost::intmax_t current_precision = eval_ilogb(result);
|
boost::intmax_t target_precision = std::numeric_limits<number<T> >::is_specialized ?
|
current_precision - 1 - (std::numeric_limits<number<T> >::digits * 2) / 3
|
: current_precision - 1 - (boost::multiprecision::detail::digits2<number<T> >::value() * 2) / 3;
|
|
T s, c, t;
|
while (current_precision > target_precision)
|
{
|
eval_sin(s, result);
|
eval_cos(c, result);
|
eval_multiply(t, xx, c);
|
eval_subtract(t, s);
|
eval_multiply(s, t, c);
|
eval_add(result, s);
|
current_precision = eval_ilogb(s);
|
if (current_precision <= (std::numeric_limits<typename T::exponent_type>::min)() + 1)
|
break;
|
}
|
if (b_neg)
|
result.negate();
|
}
|
|
template <class T>
|
void eval_atan2(T& result, const T& y, const T& x)
|
{
|
BOOST_STATIC_ASSERT_MSG(number_category<T>::value == number_kind_floating_point, "The atan2 function is only valid for floating point types.");
|
if (&result == &y)
|
{
|
T temp(y);
|
eval_atan2(result, temp, x);
|
return;
|
}
|
else if (&result == &x)
|
{
|
T temp(x);
|
eval_atan2(result, y, temp);
|
return;
|
}
|
|
typedef typename boost::multiprecision::detail::canonical<boost::uint32_t, T>::type ui_type;
|
|
switch (eval_fpclassify(y))
|
{
|
case FP_NAN:
|
result = y;
|
errno = EDOM;
|
return;
|
case FP_ZERO:
|
{
|
if (eval_signbit(x))
|
{
|
result = get_constant_pi<T>();
|
if (eval_signbit(y))
|
result.negate();
|
}
|
else
|
{
|
result = y; // Note we allow atan2(0,0) to be +-zero, even though it's mathematically undefined
|
}
|
return;
|
}
|
case FP_INFINITE:
|
{
|
if (eval_fpclassify(x) == FP_INFINITE)
|
{
|
if (eval_signbit(x))
|
{
|
// 3Pi/4
|
eval_ldexp(result, get_constant_pi<T>(), -2);
|
eval_subtract(result, get_constant_pi<T>());
|
if (eval_get_sign(y) >= 0)
|
result.negate();
|
}
|
else
|
{
|
// Pi/4
|
eval_ldexp(result, get_constant_pi<T>(), -2);
|
if (eval_get_sign(y) < 0)
|
result.negate();
|
}
|
}
|
else
|
{
|
eval_ldexp(result, get_constant_pi<T>(), -1);
|
if (eval_get_sign(y) < 0)
|
result.negate();
|
}
|
return;
|
}
|
}
|
|
switch (eval_fpclassify(x))
|
{
|
case FP_NAN:
|
result = x;
|
errno = EDOM;
|
return;
|
case FP_ZERO:
|
{
|
eval_ldexp(result, get_constant_pi<T>(), -1);
|
if (eval_get_sign(y) < 0)
|
result.negate();
|
return;
|
}
|
case FP_INFINITE:
|
if (eval_get_sign(x) > 0)
|
result = ui_type(0);
|
else
|
result = get_constant_pi<T>();
|
if (eval_get_sign(y) < 0)
|
result.negate();
|
return;
|
}
|
|
T xx;
|
eval_divide(xx, y, x);
|
if (eval_get_sign(xx) < 0)
|
xx.negate();
|
|
eval_atan(result, xx);
|
|
// Determine quadrant (sign) based on signs of x, y
|
const bool y_neg = eval_get_sign(y) < 0;
|
const bool x_neg = eval_get_sign(x) < 0;
|
|
if (y_neg != x_neg)
|
result.negate();
|
|
if (x_neg)
|
{
|
if (y_neg)
|
eval_subtract(result, get_constant_pi<T>());
|
else
|
eval_add(result, get_constant_pi<T>());
|
}
|
}
|
template <class T, class A>
|
inline typename enable_if<is_arithmetic<A>, void>::type eval_atan2(T& result, const T& x, const A& a)
|
{
|
typedef typename boost::multiprecision::detail::canonical<A, T>::type canonical_type;
|
typedef typename mpl::if_<is_same<A, canonical_type>, T, canonical_type>::type cast_type;
|
cast_type c;
|
c = a;
|
eval_atan2(result, x, c);
|
}
|
|
template <class T, class A>
|
inline typename enable_if<is_arithmetic<A>, void>::type eval_atan2(T& result, const A& x, const T& a)
|
{
|
typedef typename boost::multiprecision::detail::canonical<A, T>::type canonical_type;
|
typedef typename mpl::if_<is_same<A, canonical_type>, T, canonical_type>::type cast_type;
|
cast_type c;
|
c = x;
|
eval_atan2(result, c, a);
|
}
|
|
#ifdef BOOST_MSVC
|
#pragma warning(pop)
|
#endif
|
