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+
+// Copyright Christopher Kormanyos 2002 - 2013.
+// Copyright 2011 - 2013 John Maddock. Distributed under the Boost
+// Distributed under the Boost Software License, Version 1.0.
+// (See accompanying file LICENSE_1_0.txt or copy at
+// http://www.boost.org/LICENSE_1_0.txt)
+
+// This work is based on an earlier work:
+// "Algorithm 910: A Portable C++ Multiple-Precision System for Special-Function Calculations",
+// in ACM TOMS, {VOL 37, ISSUE 4, (February 2011)} (C) ACM, 2011. http://doi.acm.org/10.1145/1916461.1916469
+//
+// This file has no include guards or namespaces - it's expanded inline inside default_ops.hpp
+//
+
+namespace detail{
+
+template<typename T, typename U>
+inline void pow_imp(T& result, const T& t, const U& p, const mpl::false_&)
+{
+ // Compute the pure power of typename T t^p.
+ // Use the S-and-X binary method, as described in
+ // D. E. Knuth, "The Art of Computer Programming", Vol. 2,
+ // Section 4.6.3 . The resulting computational complexity
+ // is order log2[abs(p)].
+
+ typedef typename boost::multiprecision::detail::canonical<U, T>::type int_type;
+
+ if(&result == &t)
+ {
+ T temp;
+ pow_imp(temp, t, p, mpl::false_());
+ result = temp;
+ return;
+ }
+
+ // This will store the result.
+ if(U(p % U(2)) != U(0))
+ {
+ result = t;
+ }
+ else
+ result = int_type(1);
+
+ U p2(p);
+
+ // The variable x stores the binary powers of t.
+ T x(t);
+
+ while(U(p2 /= 2) != U(0))
+ {
+ // Square x for each binary power.
+ eval_multiply(x, x);
+
+ const bool has_binary_power = (U(p2 % U(2)) != U(0));
+
+ if(has_binary_power)
+ {
+ // Multiply the result with each binary power contained in the exponent.
+ eval_multiply(result, x);
+ }
+ }
+}
+
+template<typename T, typename U>
+inline void pow_imp(T& result, const T& t, const U& p, const mpl::true_&)
+{
+ // Signed integer power, just take care of the sign then call the unsigned version:
+ typedef typename boost::multiprecision::detail::canonical<U, T>::type int_type;
+ typedef typename make_unsigned<U>::type ui_type;
+
+ if(p < 0)
+ {
+ T temp;
+ temp = static_cast<int_type>(1);
+ T denom;
+ pow_imp(denom, t, static_cast<ui_type>(-p), mpl::false_());
+ eval_divide(result, temp, denom);
+ return;
+ }
+ pow_imp(result, t, static_cast<ui_type>(p), mpl::false_());
+}
+
+} // namespace detail
+
+template<typename T, typename U>
+inline typename enable_if<is_integral<U> >::type eval_pow(T& result, const T& t, const U& p)
+{
+ detail::pow_imp(result, t, p, boost::is_signed<U>());
+}
+
+template <class T>
+void hyp0F0(T& H0F0, const T& x)
+{
+ // Compute the series representation of Hypergeometric0F0 taken from
+ // http://functions.wolfram.com/HypergeometricFunctions/Hypergeometric0F0/06/01/
+ // There are no checks on input range or parameter boundaries.
+
+ typedef typename mpl::front<typename T::unsigned_types>::type ui_type;
+
+ BOOST_ASSERT(&H0F0 != &x);
+ long tol = boost::multiprecision::detail::digits2<number<T, et_on> >::value;
+ T t;
+
+ T x_pow_n_div_n_fact(x);
+
+ eval_add(H0F0, x_pow_n_div_n_fact, ui_type(1));
+
+ T lim;
+ eval_ldexp(lim, H0F0, 1 - tol);
+ if(eval_get_sign(lim) < 0)
+ lim.negate();
+
+ ui_type n;
+
+ static const unsigned series_limit =
+ boost::multiprecision::detail::digits2<number<T, et_on> >::value < 100
+ ? 100 : boost::multiprecision::detail::digits2<number<T, et_on> >::value;
+ // Series expansion of hyperg_0f0(; ; x).
+ for(n = 2; n < series_limit; ++n)
+ {
+ eval_multiply(x_pow_n_div_n_fact, x);
+ eval_divide(x_pow_n_div_n_fact, n);
+ eval_add(H0F0, x_pow_n_div_n_fact);
+ bool neg = eval_get_sign(x_pow_n_div_n_fact) < 0;
+ if(neg)
+ x_pow_n_div_n_fact.negate();
+ if(lim.compare(x_pow_n_div_n_fact) > 0)
+ break;
+ if(neg)
+ x_pow_n_div_n_fact.negate();
+ }
+ if(n >= series_limit)
+ BOOST_THROW_EXCEPTION(std::runtime_error("H0F0 failed to converge"));
+}
+
+template <class T>
+void hyp1F0(T& H1F0, const T& a, const T& x)
+{
+ // Compute the series representation of Hypergeometric1F0 taken from
+ // http://functions.wolfram.com/HypergeometricFunctions/Hypergeometric1F0/06/01/01/
+ // and also see the corresponding section for the power function (i.e. x^a).
+ // There are no checks on input range or parameter boundaries.
+
+ typedef typename boost::multiprecision::detail::canonical<int, T>::type si_type;
+
+ BOOST_ASSERT(&H1F0 != &x);
+ BOOST_ASSERT(&H1F0 != &a);
+
+ T x_pow_n_div_n_fact(x);
+ T pochham_a (a);
+ T ap (a);
+
+ eval_multiply(H1F0, pochham_a, x_pow_n_div_n_fact);
+ eval_add(H1F0, si_type(1));
+ T lim;
+ eval_ldexp(lim, H1F0, 1 - boost::multiprecision::detail::digits2<number<T, et_on> >::value);
+ if(eval_get_sign(lim) < 0)
+ lim.negate();
+
+ si_type n;
+ T term, part;
+
+ static const unsigned series_limit =
+ boost::multiprecision::detail::digits2<number<T, et_on> >::value < 100
+ ? 100 : boost::multiprecision::detail::digits2<number<T, et_on> >::value;
+ // Series expansion of hyperg_1f0(a; ; x).
+ for(n = 2; n < series_limit; n++)
+ {
+ eval_multiply(x_pow_n_div_n_fact, x);
+ eval_divide(x_pow_n_div_n_fact, n);
+ eval_increment(ap);
+ eval_multiply(pochham_a, ap);
+ eval_multiply(term, pochham_a, x_pow_n_div_n_fact);
+ eval_add(H1F0, term);
+ if(eval_get_sign(term) < 0)
+ term.negate();
+ if(lim.compare(term) >= 0)
+ break;
+ }
+ if(n >= series_limit)
+ BOOST_THROW_EXCEPTION(std::runtime_error("H1F0 failed to converge"));
+}
+
+template <class T>
+void eval_exp(T& result, const T& x)
+{
+ BOOST_STATIC_ASSERT_MSG(number_category<T>::value == number_kind_floating_point, "The exp function is only valid for floating point types.");
+ if(&x == &result)
+ {
+ T temp;
+ eval_exp(temp, x);
+ result = temp;
+ return;
+ }
+ typedef typename boost::multiprecision::detail::canonical<unsigned, T>::type ui_type;
+ typedef typename boost::multiprecision::detail::canonical<int, T>::type si_type;
+ typedef typename T::exponent_type exp_type;
+ typedef typename boost::multiprecision::detail::canonical<exp_type, T>::type canonical_exp_type;
+
+ // Handle special arguments.
+ int type = eval_fpclassify(x);
+ bool isneg = eval_get_sign(x) < 0;
+ if(type == (int)FP_NAN)
+ {
+ result = x;
+ return;
+ }
+ else if(type == (int)FP_INFINITE)
+ {
+ result = x;
+ if(isneg)
+ result = ui_type(0u);
+ else
+ result = x;
+ return;
+ }
+ else if(type == (int)FP_ZERO)
+ {
+ result = ui_type(1);
+ return;
+ }
+
+ // Get local copy of argument and force it to be positive.
+ T xx = x;
+ T exp_series;
+ if(isneg)
+ xx.negate();
+
+ // Check the range of the argument.
+ if(xx.compare(si_type(1)) <= 0)
+ {
+ //
+ // Use series for exp(x) - 1:
+ //
+ T lim = std::numeric_limits<number<T, et_on> >::epsilon().backend();
+ unsigned k = 2;
+ exp_series = xx;
+ result = si_type(1);
+ if(isneg)
+ eval_subtract(result, exp_series);
+ else
+ eval_add(result, exp_series);
+ eval_multiply(exp_series, xx);
+ eval_divide(exp_series, ui_type(k));
+ eval_add(result, exp_series);
+ while(exp_series.compare(lim) > 0)
+ {
+ ++k;
+ eval_multiply(exp_series, xx);
+ eval_divide(exp_series, ui_type(k));
+ if(isneg && (k&1))
+ eval_subtract(result, exp_series);
+ else
+ eval_add(result, exp_series);
+ }
+ return;
+ }
+
+ // Check for pure-integer arguments which can be either signed or unsigned.
+ typename boost::multiprecision::detail::canonical<boost::intmax_t, T>::type ll;
+ eval_trunc(exp_series, x);
+ eval_convert_to(&ll, exp_series);
+ if(x.compare(ll) == 0)
+ {
+ detail::pow_imp(result, get_constant_e<T>(), ll, mpl::true_());
+ return;
+ }
+
+ // The algorithm for exp has been taken from MPFUN.
+ // exp(t) = [ (1 + r + r^2/2! + r^3/3! + r^4/4! ...)^p2 ] * 2^n
+ // where p2 is a power of 2 such as 2048, r = t_prime / p2, and
+ // t_prime = t - n*ln2, with n chosen to minimize the absolute
+ // value of t_prime. In the resulting Taylor series, which is
+ // implemented as a hypergeometric function, |r| is bounded by
+ // ln2 / p2. For small arguments, no scaling is done.
+
+ // Compute the exponential series of the (possibly) scaled argument.
+
+ eval_divide(result, xx, get_constant_ln2<T>());
+ exp_type n;
+ eval_convert_to(&n, result);
+
+ // The scaling is 2^11 = 2048.
+ static const si_type p2 = static_cast<si_type>(si_type(1) << 11);
+
+ eval_multiply(exp_series, get_constant_ln2<T>(), static_cast<canonical_exp_type>(n));
+ eval_subtract(exp_series, xx);
+ eval_divide(exp_series, p2);
+ exp_series.negate();
+ hyp0F0(result, exp_series);
+
+ detail::pow_imp(exp_series, result, p2, mpl::true_());
+ result = ui_type(1);
+ eval_ldexp(result, result, n);
+ eval_multiply(exp_series, result);
+
+ if(isneg)
+ eval_divide(result, ui_type(1), exp_series);
+ else
+ result = exp_series;
+}
+
+template <class T>
+void eval_log(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.");
+ //
+ // We use a variation of http://dlmf.nist.gov/4.45#i
+ // using frexp to reduce the argument to x * 2^n,
+ // then let y = x - 1 and compute:
+ // log(x) = log(2) * n + log1p(1 + y)
+ //
+ typedef typename boost::multiprecision::detail::canonical<unsigned, T>::type ui_type;
+ typedef typename T::exponent_type exp_type;
+ typedef typename boost::multiprecision::detail::canonical<exp_type, T>::type canonical_exp_type;
+ typedef typename mpl::front<typename T::float_types>::type fp_type;
+
+ exp_type e;
+ T t;
+ eval_frexp(t, arg, &e);
+ bool alternate = false;
+
+ if(t.compare(fp_type(2) / fp_type(3)) <= 0)
+ {
+ alternate = true;
+ eval_ldexp(t, t, 1);
+ --e;
+ }
+
+ eval_multiply(result, get_constant_ln2<T>(), canonical_exp_type(e));
+ INSTRUMENT_BACKEND(result);
+ eval_subtract(t, ui_type(1)); /* -0.3 <= t <= 0.3 */
+ if(!alternate)
+ t.negate(); /* 0 <= t <= 0.33333 */
+ T pow = t;
+ T lim;
+ T t2;
+
+ if(alternate)
+ eval_add(result, t);
+ else
+ eval_subtract(result, t);
+
+ eval_multiply(lim, result, std::numeric_limits<number<T, et_on> >::epsilon().backend());
+ if(eval_get_sign(lim) < 0)
+ lim.negate();
+ INSTRUMENT_BACKEND(lim);
+
+ ui_type k = 1;
+ do
+ {
+ ++k;
+ eval_multiply(pow, t);
+ eval_divide(t2, pow, k);
+ INSTRUMENT_BACKEND(t2);
+ if(alternate && ((k & 1) != 0))
+ eval_add(result, t2);
+ else
+ eval_subtract(result, t2);
+ INSTRUMENT_BACKEND(result);
+ }while(lim.compare(t2) < 0);
+}
+
+template <class T>
+const T& get_constant_log10()
+{
+ static T result;
+ static bool b = false;
+ if(!b)
+ {
+ typedef typename boost::multiprecision::detail::canonical<unsigned, T>::type ui_type;
+ T ten;
+ ten = ui_type(10u);
+ eval_log(result, ten);
+ }
+
+ constant_initializer<T, &get_constant_log10<T> >::do_nothing();
+
+ return result;
+}
+
+template <class T>
+void eval_log10(T& result, const T& arg)
+{
+ BOOST_STATIC_ASSERT_MSG(number_category<T>::value == number_kind_floating_point, "The log10 function is only valid for floating point types.");
+ eval_log(result, arg);
+ eval_divide(result, get_constant_log10<T>());
+}
+
+template<typename T>
+inline void eval_pow(T& result, const T& x, const T& a)
+{
+ BOOST_STATIC_ASSERT_MSG(number_category<T>::value == number_kind_floating_point, "The pow function is only valid for floating point types.");
+ typedef typename boost::multiprecision::detail::canonical<int, T>::type si_type;
+ typedef typename mpl::front<typename T::float_types>::type fp_type;
+
+ if((&result == &x) || (&result == &a))
+ {
+ T t;
+ eval_pow(t, x, a);
+ result = t;
+ return;
+ }
+
+ if(a.compare(si_type(1)) == 0)
+ {
+ result = x;
+ return;
+ }
+
+ int type = eval_fpclassify(x);
+
+ switch(type)
+ {
+ case FP_INFINITE:
+ result = x;
+ return;
+ case FP_ZERO:
+ switch(eval_fpclassify(a))
+ {
+ case FP_ZERO:
+ result = si_type(1);
+ break;
+ case FP_NAN:
+ result = a;
+ break;
+ default:
+ result = x;
+ break;
+ }
+ return;
+ case FP_NAN:
+ result = x;
+ return;
+ default: ;
+ }
+
+ int s = eval_get_sign(a);
+ if(s == 0)
+ {
+ result = si_type(1);
+ return;
+ }
+
+ if(s < 0)
+ {
+ T t, da;
+ t = a;
+ t.negate();
+ eval_pow(da, x, t);
+ eval_divide(result, si_type(1), da);
+ return;
+ }
+
+ typename boost::multiprecision::detail::canonical<boost::intmax_t, T>::type an;
+ T fa;
+ try
+ {
+ eval_convert_to(&an, a);
+ if(a.compare(an) == 0)
+ {
+ detail::pow_imp(result, x, an, mpl::true_());
+ return;
+ }
+ }
+ catch(const std::exception&)
+ {
+ // conversion failed, just fall through, value is not an integer.
+ an = (std::numeric_limits<boost::intmax_t>::max)();
+ }
+
+ if((eval_get_sign(x) < 0))
+ {
+ typename boost::multiprecision::detail::canonical<boost::uintmax_t, T>::type aun;
+ try
+ {
+ 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;
+ }
+ }
+ catch(const std::exception&)
+ {
+ // conversion failed, just fall through, value is not an integer.
+ }
+ 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);
+}
+
+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);
+}
+
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