751 lines
22 KiB
Plaintext
751 lines
22 KiB
Plaintext
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// Copyright Christopher Kormanyos 2002 - 2013.
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// Copyright 2011 - 2013 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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namespace detail{
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template<typename T, typename U>
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inline void pow_imp(T& result, const T& t, const U& p, const mpl::false_&)
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{
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// Compute the pure power of typename T t^p.
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// Use the S-and-X binary method, as described in
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// D. E. Knuth, "The Art of Computer Programming", Vol. 2,
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// Section 4.6.3 . The resulting computational complexity
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// is order log2[abs(p)].
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typedef typename boost::multiprecision::detail::canonical<U, T>::type int_type;
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if(&result == &t)
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{
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T temp;
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pow_imp(temp, t, p, mpl::false_());
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result = temp;
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return;
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}
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// This will store the result.
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if(U(p % U(2)) != U(0))
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{
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result = t;
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}
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else
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result = int_type(1);
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U p2(p);
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// The variable x stores the binary powers of t.
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T x(t);
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while(U(p2 /= 2) != U(0))
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{
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// Square x for each binary power.
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eval_multiply(x, x);
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const bool has_binary_power = (U(p2 % U(2)) != U(0));
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if(has_binary_power)
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{
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// Multiply the result with each binary power contained in the exponent.
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eval_multiply(result, x);
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}
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}
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}
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template<typename T, typename U>
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inline void pow_imp(T& result, const T& t, const U& p, const mpl::true_&)
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{
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// Signed integer power, just take care of the sign then call the unsigned version:
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typedef typename boost::multiprecision::detail::canonical<U, T>::type int_type;
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typedef typename make_unsigned<U>::type ui_type;
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if(p < 0)
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{
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T temp;
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temp = static_cast<int_type>(1);
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T denom;
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pow_imp(denom, t, static_cast<ui_type>(-p), mpl::false_());
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eval_divide(result, temp, denom);
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return;
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}
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pow_imp(result, t, static_cast<ui_type>(p), mpl::false_());
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}
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} // namespace detail
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template<typename T, typename U>
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inline typename enable_if<is_integral<U> >::type eval_pow(T& result, const T& t, const U& p)
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{
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detail::pow_imp(result, t, p, boost::is_signed<U>());
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}
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template <class T>
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void hyp0F0(T& H0F0, const T& x)
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{
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// Compute the series representation of Hypergeometric0F0 taken from
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// http://functions.wolfram.com/HypergeometricFunctions/Hypergeometric0F0/06/01/
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// There are no checks on input range or parameter boundaries.
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typedef typename mpl::front<typename T::unsigned_types>::type ui_type;
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BOOST_ASSERT(&H0F0 != &x);
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long tol = boost::multiprecision::detail::digits2<number<T, et_on> >::value();
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T t;
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T x_pow_n_div_n_fact(x);
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eval_add(H0F0, x_pow_n_div_n_fact, ui_type(1));
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T lim;
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eval_ldexp(lim, H0F0, 1 - tol);
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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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const unsigned series_limit =
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boost::multiprecision::detail::digits2<number<T, et_on> >::value() < 100
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? 100 : boost::multiprecision::detail::digits2<number<T, et_on> >::value();
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// Series expansion of hyperg_0f0(; ; 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_add(H0F0, x_pow_n_div_n_fact);
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bool neg = eval_get_sign(x_pow_n_div_n_fact) < 0;
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if(neg)
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x_pow_n_div_n_fact.negate();
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if(lim.compare(x_pow_n_div_n_fact) > 0)
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break;
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if(neg)
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x_pow_n_div_n_fact.negate();
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}
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if(n >= series_limit)
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BOOST_THROW_EXCEPTION(std::runtime_error("H0F0 failed to converge"));
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}
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template <class T>
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void hyp1F0(T& H1F0, const T& a, const T& x)
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{
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// Compute the series representation of Hypergeometric1F0 taken from
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// http://functions.wolfram.com/HypergeometricFunctions/Hypergeometric1F0/06/01/01/
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// and also see the corresponding section for the power function (i.e. x^a).
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// There are no checks on input range or parameter boundaries.
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typedef typename boost::multiprecision::detail::canonical<int, T>::type si_type;
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BOOST_ASSERT(&H1F0 != &x);
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BOOST_ASSERT(&H1F0 != &a);
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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 ap (a);
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eval_multiply(H1F0, pochham_a, x_pow_n_div_n_fact);
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eval_add(H1F0, si_type(1));
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T lim;
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eval_ldexp(lim, H1F0, 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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si_type n;
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T term, part;
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const si_type series_limit =
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boost::multiprecision::detail::digits2<number<T, et_on> >::value() < 100
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? 100 : boost::multiprecision::detail::digits2<number<T, et_on> >::value();
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// Series expansion of hyperg_1f0(a; ; 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(ap);
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eval_multiply(pochham_a, ap);
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eval_multiply(term, pochham_a, x_pow_n_div_n_fact);
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eval_add(H1F0, term);
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if(eval_get_sign(term) < 0)
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term.negate();
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if(lim.compare(term) >= 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("H1F0 failed to converge"));
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}
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template <class T>
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void eval_exp(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 exp function is only valid for floating point types.");
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if(&x == &result)
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{
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T temp;
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eval_exp(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<unsigned, T>::type ui_type;
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typedef typename boost::multiprecision::detail::canonical<int, T>::type si_type;
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typedef typename T::exponent_type exp_type;
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typedef typename boost::multiprecision::detail::canonical<exp_type, T>::type canonical_exp_type;
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// Handle special arguments.
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int type = eval_fpclassify(x);
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bool isneg = eval_get_sign(x) < 0;
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if(type == (int)FP_NAN)
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{
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result = x;
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return;
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}
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else if(type == (int)FP_INFINITE)
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{
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result = x;
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if(isneg)
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result = ui_type(0u);
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else
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result = x;
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return;
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}
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else if(type == (int)FP_ZERO)
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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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// Get local copy of argument and force it to be positive.
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T xx = x;
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T exp_series;
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if(isneg)
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xx.negate();
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// Check the range of the argument.
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if(xx.compare(si_type(1)) <= 0)
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{
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//
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// Use series for exp(x) - 1:
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//
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T lim;
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if(std::numeric_limits<number<T, et_on> >::is_specialized)
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lim = std::numeric_limits<number<T, et_on> >::epsilon().backend();
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else
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{
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result = ui_type(1);
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eval_ldexp(lim, result, 1 - boost::multiprecision::detail::digits2<number<T, et_on> >::value());
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}
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unsigned k = 2;
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exp_series = xx;
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result = si_type(1);
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if(isneg)
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eval_subtract(result, exp_series);
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else
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eval_add(result, exp_series);
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eval_multiply(exp_series, xx);
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eval_divide(exp_series, ui_type(k));
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eval_add(result, exp_series);
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while(exp_series.compare(lim) > 0)
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{
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++k;
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eval_multiply(exp_series, xx);
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eval_divide(exp_series, ui_type(k));
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if(isneg && (k&1))
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eval_subtract(result, exp_series);
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else
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eval_add(result, exp_series);
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}
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return;
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}
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// Check for pure-integer arguments which can be either signed or unsigned.
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typename boost::multiprecision::detail::canonical<boost::intmax_t, T>::type ll;
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eval_trunc(exp_series, x);
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eval_convert_to(&ll, exp_series);
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if(x.compare(ll) == 0)
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{
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detail::pow_imp(result, get_constant_e<T>(), ll, mpl::true_());
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return;
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}
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// The algorithm for exp has been taken from MPFUN.
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// exp(t) = [ (1 + r + r^2/2! + r^3/3! + r^4/4! ...)^p2 ] * 2^n
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// where p2 is a power of 2 such as 2048, r = t_prime / p2, and
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// t_prime = t - n*ln2, with n chosen to minimize the absolute
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// value of t_prime. In the resulting Taylor series, which is
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// implemented as a hypergeometric function, |r| is bounded by
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// ln2 / p2. For small arguments, no scaling is done.
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// Compute the exponential series of the (possibly) scaled argument.
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eval_divide(result, xx, get_constant_ln2<T>());
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exp_type n;
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eval_convert_to(&n, result);
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// The scaling is 2^11 = 2048.
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const si_type p2 = static_cast<si_type>(si_type(1) << 11);
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eval_multiply(exp_series, get_constant_ln2<T>(), static_cast<canonical_exp_type>(n));
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eval_subtract(exp_series, xx);
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eval_divide(exp_series, p2);
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exp_series.negate();
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hyp0F0(result, exp_series);
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detail::pow_imp(exp_series, result, p2, mpl::true_());
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result = ui_type(1);
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eval_ldexp(result, result, n);
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eval_multiply(exp_series, result);
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if(isneg)
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eval_divide(result, ui_type(1), exp_series);
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else
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result = exp_series;
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}
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template <class T>
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void eval_log(T& result, const T& arg)
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{
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BOOST_STATIC_ASSERT_MSG(number_category<T>::value == number_kind_floating_point, "The log function is only valid for floating point types.");
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//
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// We use a variation of http://dlmf.nist.gov/4.45#i
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// using frexp to reduce the argument to x * 2^n,
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// then let y = x - 1 and compute:
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// log(x) = log(2) * n + log1p(1 + y)
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//
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typedef typename boost::multiprecision::detail::canonical<unsigned, T>::type ui_type;
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typedef typename T::exponent_type exp_type;
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typedef typename boost::multiprecision::detail::canonical<exp_type, T>::type canonical_exp_type;
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typedef typename mpl::front<typename T::float_types>::type fp_type;
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exp_type e;
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T t;
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eval_frexp(t, arg, &e);
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bool alternate = false;
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if(t.compare(fp_type(2) / fp_type(3)) <= 0)
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{
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alternate = true;
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eval_ldexp(t, t, 1);
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--e;
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}
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eval_multiply(result, get_constant_ln2<T>(), canonical_exp_type(e));
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INSTRUMENT_BACKEND(result);
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eval_subtract(t, ui_type(1)); /* -0.3 <= t <= 0.3 */
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if(!alternate)
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t.negate(); /* 0 <= t <= 0.33333 */
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T pow = t;
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T lim;
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T t2;
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if(alternate)
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eval_add(result, t);
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else
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eval_subtract(result, t);
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if(std::numeric_limits<number<T, et_on> >::is_specialized)
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eval_multiply(lim, result, std::numeric_limits<number<T, et_on> >::epsilon().backend());
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else
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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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INSTRUMENT_BACKEND(lim);
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ui_type k = 1;
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do
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{
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++k;
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eval_multiply(pow, t);
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eval_divide(t2, pow, k);
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INSTRUMENT_BACKEND(t2);
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if(alternate && ((k & 1) != 0))
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eval_add(result, t2);
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else
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eval_subtract(result, t2);
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INSTRUMENT_BACKEND(result);
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}while(lim.compare(t2) < 0);
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}
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template <class T>
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const T& get_constant_log10()
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{
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static BOOST_MP_THREAD_LOCAL T result;
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static BOOST_MP_THREAD_LOCAL bool b = false;
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static BOOST_MP_THREAD_LOCAL long digits = boost::multiprecision::detail::digits2<number<T> >::value();
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if(!b || (digits != boost::multiprecision::detail::digits2<number<T> >::value()))
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{
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typedef typename boost::multiprecision::detail::canonical<unsigned, T>::type ui_type;
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T ten;
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ten = ui_type(10u);
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eval_log(result, ten);
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b = true;
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digits = boost::multiprecision::detail::digits2<number<T> >::value();
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}
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constant_initializer<T, &get_constant_log10<T> >::do_nothing();
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return result;
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}
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template <class T>
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void eval_log10(T& result, const T& arg)
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{
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BOOST_STATIC_ASSERT_MSG(number_category<T>::value == number_kind_floating_point, "The log10 function is only valid for floating point types.");
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eval_log(result, arg);
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eval_divide(result, get_constant_log10<T>());
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}
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template <class R, class T>
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inline void eval_log2(R& result, const T& a)
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{
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eval_log(result, a);
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eval_divide(result, get_constant_ln2<R>());
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}
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template<typename T>
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inline void eval_pow(T& result, const T& x, const T& a)
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{
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BOOST_STATIC_ASSERT_MSG(number_category<T>::value == number_kind_floating_point, "The pow function is only valid for floating point types.");
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typedef typename boost::multiprecision::detail::canonical<int, T>::type si_type;
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typedef typename mpl::front<typename T::float_types>::type fp_type;
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if((&result == &x) || (&result == &a))
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{
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T t;
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eval_pow(t, x, a);
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result = t;
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return;
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}
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if(a.compare(si_type(1)) == 0)
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{
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result = x;
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return;
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}
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int type = eval_fpclassify(x);
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switch(type)
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{
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case FP_INFINITE:
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result = x;
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return;
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case FP_ZERO:
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switch(eval_fpclassify(a))
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{
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case FP_ZERO:
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result = si_type(1);
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break;
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case FP_NAN:
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result = a;
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break;
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default:
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result = x;
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break;
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}
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return;
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case FP_NAN:
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result = x;
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return;
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default: ;
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}
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int s = eval_get_sign(a);
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if(s == 0)
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{
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result = si_type(1);
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return;
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}
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if(s < 0)
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{
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T t, da;
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t = a;
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t.negate();
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eval_pow(da, x, t);
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eval_divide(result, si_type(1), da);
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return;
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}
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typename boost::multiprecision::detail::canonical<boost::intmax_t, T>::type an;
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T fa;
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#ifndef BOOST_NO_EXCEPTIONS
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try
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{
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#endif
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eval_convert_to(&an, a);
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if(a.compare(an) == 0)
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{
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detail::pow_imp(result, x, an, mpl::true_());
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return;
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}
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#ifndef BOOST_NO_EXCEPTIONS
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}
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catch(const std::exception&)
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{
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// conversion failed, just fall through, value is not an integer.
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an = (std::numeric_limits<boost::intmax_t>::max)();
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}
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#endif
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if((eval_get_sign(x) < 0))
|
|
{
|
|
typename boost::multiprecision::detail::canonical<boost::uintmax_t, T>::type aun;
|
|
#ifndef BOOST_NO_EXCEPTIONS
|
|
try
|
|
{
|
|
#endif
|
|
eval_convert_to(&aun, a);
|
|
if(a.compare(aun) == 0)
|
|
{
|
|
fa = x;
|
|
fa.negate();
|
|
eval_pow(result, fa, a);
|
|
if(aun & 1u)
|
|
result.negate();
|
|
return;
|
|
}
|
|
#ifndef BOOST_NO_EXCEPTIONS
|
|
}
|
|
catch(const std::exception&)
|
|
{
|
|
// conversion failed, just fall through, value is not an integer.
|
|
}
|
|
#endif
|
|
if(std::numeric_limits<number<T, et_on> >::has_quiet_NaN)
|
|
result = std::numeric_limits<number<T, et_on> >::quiet_NaN().backend();
|
|
else
|
|
{
|
|
BOOST_THROW_EXCEPTION(std::domain_error("Result of pow is undefined or non-real and there is no NaN for this number type."));
|
|
}
|
|
return;
|
|
}
|
|
|
|
T t, da;
|
|
|
|
eval_subtract(da, a, an);
|
|
|
|
if((x.compare(fp_type(0.5)) >= 0) && (x.compare(fp_type(0.9)) < 0))
|
|
{
|
|
if(a.compare(fp_type(1e-5f)) <= 0)
|
|
{
|
|
// Series expansion for small a.
|
|
eval_log(t, x);
|
|
eval_multiply(t, a);
|
|
hyp0F0(result, t);
|
|
return;
|
|
}
|
|
else
|
|
{
|
|
// Series expansion for moderately sized x. Note that for large power of a,
|
|
// the power of the integer part of a is calculated using the pown function.
|
|
if(an)
|
|
{
|
|
da.negate();
|
|
t = si_type(1);
|
|
eval_subtract(t, x);
|
|
hyp1F0(result, da, t);
|
|
detail::pow_imp(t, x, an, mpl::true_());
|
|
eval_multiply(result, t);
|
|
}
|
|
else
|
|
{
|
|
da = a;
|
|
da.negate();
|
|
t = si_type(1);
|
|
eval_subtract(t, x);
|
|
hyp1F0(result, da, t);
|
|
}
|
|
}
|
|
}
|
|
else
|
|
{
|
|
// Series expansion for pow(x, a). Note that for large power of a, the power
|
|
// of the integer part of a is calculated using the pown function.
|
|
if(an)
|
|
{
|
|
eval_log(t, x);
|
|
eval_multiply(t, da);
|
|
eval_exp(result, t);
|
|
detail::pow_imp(t, x, an, mpl::true_());
|
|
eval_multiply(result, t);
|
|
}
|
|
else
|
|
{
|
|
eval_log(t, x);
|
|
eval_multiply(t, a);
|
|
eval_exp(result, t);
|
|
}
|
|
}
|
|
}
|
|
|
|
template<class T, class A>
|
|
inline typename enable_if<is_floating_point<A>, void>::type eval_pow(T& result, const T& x, const A& a)
|
|
{
|
|
// Note this one is restricted to float arguments since pow.hpp already has a version for
|
|
// integer powers....
|
|
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_pow(result, x, c);
|
|
}
|
|
|
|
template<class T, class A>
|
|
inline typename enable_if<is_arithmetic<A>, void>::type eval_pow(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_pow(result, c, a);
|
|
}
|
|
|
|
template <class T>
|
|
void eval_exp2(T& result, const T& arg)
|
|
{
|
|
BOOST_STATIC_ASSERT_MSG(number_category<T>::value == number_kind_floating_point, "The log function is only valid for floating point types.");
|
|
|
|
// Check for pure-integer arguments which can be either signed or unsigned.
|
|
typename boost::multiprecision::detail::canonical<typename T::exponent_type, T>::type i;
|
|
T temp;
|
|
eval_trunc(temp, arg);
|
|
eval_convert_to(&i, temp);
|
|
if(arg.compare(i) == 0)
|
|
{
|
|
temp = static_cast<typename mpl::front<typename T::unsigned_types>::type>(1u);
|
|
eval_ldexp(result, temp, i);
|
|
return;
|
|
}
|
|
|
|
temp = static_cast<typename mpl::front<typename T::unsigned_types>::type>(2u);
|
|
eval_pow(result, temp, arg);
|
|
}
|
|
|
|
namespace detail{
|
|
|
|
template <class T>
|
|
void small_sinh_series(T x, T& result)
|
|
{
|
|
typedef typename boost::multiprecision::detail::canonical<unsigned, T>::type ui_type;
|
|
bool neg = eval_get_sign(x) < 0;
|
|
if(neg)
|
|
x.negate();
|
|
T p(x);
|
|
T mult(x);
|
|
eval_multiply(mult, x);
|
|
result = x;
|
|
ui_type k = 1;
|
|
|
|
T lim(x);
|
|
eval_ldexp(lim, lim, 1 - boost::multiprecision::detail::digits2<number<T, et_on> >::value());
|
|
|
|
do
|
|
{
|
|
eval_multiply(p, mult);
|
|
eval_divide(p, ++k);
|
|
eval_divide(p, ++k);
|
|
eval_add(result, p);
|
|
}while(p.compare(lim) >= 0);
|
|
if(neg)
|
|
result.negate();
|
|
}
|
|
|
|
template <class T>
|
|
void sinhcosh(const T& x, T* p_sinh, T* p_cosh)
|
|
{
|
|
typedef typename boost::multiprecision::detail::canonical<unsigned, T>::type ui_type;
|
|
typedef typename mpl::front<typename T::float_types>::type fp_type;
|
|
|
|
switch(eval_fpclassify(x))
|
|
{
|
|
case FP_NAN:
|
|
case FP_INFINITE:
|
|
if(p_sinh)
|
|
*p_sinh = x;
|
|
if(p_cosh)
|
|
{
|
|
*p_cosh = x;
|
|
if(eval_get_sign(x) < 0)
|
|
p_cosh->negate();
|
|
}
|
|
return;
|
|
case FP_ZERO:
|
|
if(p_sinh)
|
|
*p_sinh = x;
|
|
if(p_cosh)
|
|
*p_cosh = ui_type(1);
|
|
return;
|
|
default: ;
|
|
}
|
|
|
|
bool small_sinh = eval_get_sign(x) < 0 ? x.compare(fp_type(-0.5)) > 0 : x.compare(fp_type(0.5)) < 0;
|
|
|
|
if(p_cosh || !small_sinh)
|
|
{
|
|
T e_px, e_mx;
|
|
eval_exp(e_px, x);
|
|
eval_divide(e_mx, ui_type(1), e_px);
|
|
|
|
if(p_sinh)
|
|
{
|
|
if(small_sinh)
|
|
{
|
|
small_sinh_series(x, *p_sinh);
|
|
}
|
|
else
|
|
{
|
|
eval_subtract(*p_sinh, e_px, e_mx);
|
|
eval_ldexp(*p_sinh, *p_sinh, -1);
|
|
}
|
|
}
|
|
if(p_cosh)
|
|
{
|
|
eval_add(*p_cosh, e_px, e_mx);
|
|
eval_ldexp(*p_cosh, *p_cosh, -1);
|
|
}
|
|
}
|
|
else
|
|
{
|
|
small_sinh_series(x, *p_sinh);
|
|
}
|
|
}
|
|
|
|
} // namespace detail
|
|
|
|
template <class T>
|
|
inline void eval_sinh(T& result, const T& x)
|
|
{
|
|
BOOST_STATIC_ASSERT_MSG(number_category<T>::value == number_kind_floating_point, "The sinh function is only valid for floating point types.");
|
|
detail::sinhcosh(x, &result, static_cast<T*>(0));
|
|
}
|
|
|
|
template <class T>
|
|
inline void eval_cosh(T& result, const T& x)
|
|
{
|
|
BOOST_STATIC_ASSERT_MSG(number_category<T>::value == number_kind_floating_point, "The cosh function is only valid for floating point types.");
|
|
detail::sinhcosh(x, static_cast<T*>(0), &result);
|
|
}
|
|
|
|
template <class T>
|
|
inline void eval_tanh(T& result, const T& x)
|
|
{
|
|
BOOST_STATIC_ASSERT_MSG(number_category<T>::value == number_kind_floating_point, "The tanh function is only valid for floating point types.");
|
|
T c;
|
|
detail::sinhcosh(x, &result, &c);
|
|
eval_divide(result, c);
|
|
}
|
|
|
|
#ifdef BOOST_MSVC
|
|
#pragma warning(pop)
|
|
#endif
|