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ecency-mobile/ios/Pods/boost-for-react-native/boost/multiprecision/detail/functions/pow.hpp

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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
//
#ifdef BOOST_MSVC
#pragma warning(push)
#pragma warning(disable:6326) // comparison of two constants
#endif
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;
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;
const si_type 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;
if(std::numeric_limits<number<T, et_on> >::is_specialized)
lim = std::numeric_limits<number<T, et_on> >::epsilon().backend();
else
{
result = ui_type(1);
eval_ldexp(lim, result, 1 - boost::multiprecision::detail::digits2<number<T, et_on> >::value());
}
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.
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);
if(std::numeric_limits<number<T, et_on> >::is_specialized)
eval_multiply(lim, result, std::numeric_limits<number<T, et_on> >::epsilon().backend());
else
eval_ldexp(lim, result, 1 - boost::multiprecision::detail::digits2<number<T, et_on> >::value());
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 BOOST_MP_THREAD_LOCAL T result;
static BOOST_MP_THREAD_LOCAL bool b = false;
static BOOST_MP_THREAD_LOCAL long digits = boost::multiprecision::detail::digits2<number<T> >::value();
if(!b || (digits != boost::multiprecision::detail::digits2<number<T> >::value()))
{
typedef typename boost::multiprecision::detail::canonical<unsigned, T>::type ui_type;
T ten;
ten = ui_type(10u);
eval_log(result, ten);
b = true;
digits = boost::multiprecision::detail::digits2<number<T> >::value();
}
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 <class R, class T>
inline void eval_log2(R& result, const T& a)
{
eval_log(result, a);
eval_divide(result, get_constant_ln2<R>());
}
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;
#ifndef BOOST_NO_EXCEPTIONS
try
{
#endif
eval_convert_to(&an, a);
if(a.compare(an) == 0)
{
detail::pow_imp(result, x, an, mpl::true_());
return;
}
#ifndef BOOST_NO_EXCEPTIONS
}
catch(const std::exception&)
{
// conversion failed, just fall through, value is not an integer.
an = (std::numeric_limits<boost::intmax_t>::max)();
}
#endif
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
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