simd_math.h
// Math overloads for simd -*- C++ -*-
// Copyright (C) 2020-2021 Free Software Foundation, Inc.
//
// This file is part of the GNU ISO C++ Library. This library is free
// software; you can redistribute it and/or modify it under the
// terms of the GNU General Public License as published by the
// Free Software Foundation; either version 3, or (at your option)
// any later version.
// This library is distributed in the hope that it will be useful,
// but WITHOUT ANY WARRANTY; without even the implied warranty of
// MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
// GNU General Public License for more details.
// Under Section 7 of GPL version 3, you are granted additional
// permissions described in the GCC Runtime Library Exception, version
// 3.1, as published by the Free Software Foundation.
// You should have received a copy of the GNU General Public License and
// a copy of the GCC Runtime Library Exception along with this program;
// see the files COPYING3 and COPYING.RUNTIME respectively. If not, see
// <http://www.gnu.org/licenses/>.
#ifndef _GLIBCXX_EXPERIMENTAL_SIMD_MATH_H_
#define _GLIBCXX_EXPERIMENTAL_SIMD_MATH_H_
#if __cplusplus >= 201703L
#include <utility>
#include <iomanip>
_GLIBCXX_SIMD_BEGIN_NAMESPACE
template <typename _Tp, typename _V>
using _Samesize = fixed_size_simd<_Tp, _V::size()>;
// _Math_return_type {{{
template <typename _DoubleR, typename _Tp, typename _Abi>
struct _Math_return_type;
template <typename _DoubleR, typename _Tp, typename _Abi>
using _Math_return_type_t =
typename _Math_return_type<_DoubleR, _Tp, _Abi>::type;
template <typename _Tp, typename _Abi>
struct _Math_return_type<double, _Tp, _Abi>
{ using type = simd<_Tp, _Abi>; };
template <typename _Tp, typename _Abi>
struct _Math_return_type<bool, _Tp, _Abi>
{ using type = simd_mask<_Tp, _Abi>; };
template <typename _DoubleR, typename _Tp, typename _Abi>
struct _Math_return_type
{ using type = fixed_size_simd<_DoubleR, simd_size_v<_Tp, _Abi>>; };
//}}}
// _GLIBCXX_SIMD_MATH_CALL_ {{{
#define _GLIBCXX_SIMD_MATH_CALL_(__name) \
template <typename _Tp, typename _Abi, typename..., \
typename _R = _Math_return_type_t< \
decltype(std::__name(declval<double>())), _Tp, _Abi>> \
enable_if_t<is_floating_point_v<_Tp>, _R> \
__name(simd<_Tp, _Abi> __x) \
{ return {__private_init, _Abi::_SimdImpl::_S_##__name(__data(__x))}; }
// }}}
//_Extra_argument_type{{{
template <typename _Up, typename _Tp, typename _Abi>
struct _Extra_argument_type;
template <typename _Tp, typename _Abi>
struct _Extra_argument_type<_Tp*, _Tp, _Abi>
{
using type = simd<_Tp, _Abi>*;
static constexpr double* declval();
static constexpr bool __needs_temporary_scalar = true;
_GLIBCXX_SIMD_INTRINSIC static constexpr auto _S_data(type __x)
{ return &__data(*__x); }
};
template <typename _Up, typename _Tp, typename _Abi>
struct _Extra_argument_type<_Up*, _Tp, _Abi>
{
static_assert(is_integral_v<_Up>);
using type = fixed_size_simd<_Up, simd_size_v<_Tp, _Abi>>*;
static constexpr _Up* declval();
static constexpr bool __needs_temporary_scalar = true;
_GLIBCXX_SIMD_INTRINSIC static constexpr auto _S_data(type __x)
{ return &__data(*__x); }
};
template <typename _Tp, typename _Abi>
struct _Extra_argument_type<_Tp, _Tp, _Abi>
{
using type = simd<_Tp, _Abi>;
static constexpr double declval();
static constexpr bool __needs_temporary_scalar = false;
_GLIBCXX_SIMD_INTRINSIC static constexpr decltype(auto)
_S_data(const type& __x)
{ return __data(__x); }
};
template <typename _Up, typename _Tp, typename _Abi>
struct _Extra_argument_type
{
static_assert(is_integral_v<_Up>);
using type = fixed_size_simd<_Up, simd_size_v<_Tp, _Abi>>;
static constexpr _Up declval();
static constexpr bool __needs_temporary_scalar = false;
_GLIBCXX_SIMD_INTRINSIC static constexpr decltype(auto)
_S_data(const type& __x)
{ return __data(__x); }
};
//}}}
// _GLIBCXX_SIMD_MATH_CALL2_ {{{
#define _GLIBCXX_SIMD_MATH_CALL2_(__name, arg2_) \
template < \
typename _Tp, typename _Abi, typename..., \
typename _Arg2 = _Extra_argument_type<arg2_, _Tp, _Abi>, \
typename _R = _Math_return_type_t< \
decltype(std::__name(declval<double>(), _Arg2::declval())), _Tp, _Abi>> \
enable_if_t<is_floating_point_v<_Tp>, _R> \
__name(const simd<_Tp, _Abi>& __x, const typename _Arg2::type& __y) \
{ \
return {__private_init, \
_Abi::_SimdImpl::_S_##__name(__data(__x), _Arg2::_S_data(__y))}; \
} \
template <typename _Up, typename _Tp, typename _Abi> \
_GLIBCXX_SIMD_INTRINSIC _Math_return_type_t< \
decltype(std::__name( \
declval<double>(), \
declval<enable_if_t< \
conjunction_v< \
is_same<arg2_, _Tp>, \
negation<is_same<__remove_cvref_t<_Up>, simd<_Tp, _Abi>>>, \
is_convertible<_Up, simd<_Tp, _Abi>>, is_floating_point<_Tp>>, \
double>>())), \
_Tp, _Abi> \
__name(_Up&& __xx, const simd<_Tp, _Abi>& __yy) \
{ return __name(simd<_Tp, _Abi>(static_cast<_Up&&>(__xx)), __yy); }
// }}}
// _GLIBCXX_SIMD_MATH_CALL3_ {{{
#define _GLIBCXX_SIMD_MATH_CALL3_(__name, arg2_, arg3_) \
template <typename _Tp, typename _Abi, typename..., \
typename _Arg2 = _Extra_argument_type<arg2_, _Tp, _Abi>, \
typename _Arg3 = _Extra_argument_type<arg3_, _Tp, _Abi>, \
typename _R = _Math_return_type_t< \
decltype(std::__name(declval<double>(), _Arg2::declval(), \
_Arg3::declval())), \
_Tp, _Abi>> \
enable_if_t<is_floating_point_v<_Tp>, _R> \
__name(const simd<_Tp, _Abi>& __x, const typename _Arg2::type& __y, \
const typename _Arg3::type& __z) \
{ \
return {__private_init, \
_Abi::_SimdImpl::_S_##__name(__data(__x), _Arg2::_S_data(__y), \
_Arg3::_S_data(__z))}; \
} \
template < \
typename _T0, typename _T1, typename _T2, typename..., \
typename _U0 = __remove_cvref_t<_T0>, \
typename _U1 = __remove_cvref_t<_T1>, \
typename _U2 = __remove_cvref_t<_T2>, \
typename _Simd = conditional_t<is_simd_v<_U1>, _U1, _U2>, \
typename = enable_if_t<conjunction_v< \
is_simd<_Simd>, is_convertible<_T0&&, _Simd>, \
is_convertible<_T1&&, _Simd>, is_convertible<_T2&&, _Simd>, \
negation<conjunction< \
is_simd<_U0>, is_floating_point<__value_type_or_identity_t<_U0>>>>>>> \
_GLIBCXX_SIMD_INTRINSIC decltype(__name(declval<const _Simd&>(), \
declval<const _Simd&>(), \
declval<const _Simd&>())) \
__name(_T0&& __xx, _T1&& __yy, _T2&& __zz) \
{ \
return __name(_Simd(static_cast<_T0&&>(__xx)), \
_Simd(static_cast<_T1&&>(__yy)), \
_Simd(static_cast<_T2&&>(__zz))); \
}
// }}}
// __cosSeries {{{
template <typename _Abi>
_GLIBCXX_SIMD_ALWAYS_INLINE static simd<float, _Abi>
__cosSeries(const simd<float, _Abi>& __x)
{
const simd<float, _Abi> __x2 = __x * __x;
simd<float, _Abi> __y;
__y = 0x1.ap-16f; // 1/8!
__y = __y * __x2 - 0x1.6c1p-10f; // -1/6!
__y = __y * __x2 + 0x1.555556p-5f; // 1/4!
return __y * (__x2 * __x2) - .5f * __x2 + 1.f;
}
template <typename _Abi>
_GLIBCXX_SIMD_ALWAYS_INLINE static simd<double, _Abi>
__cosSeries(const simd<double, _Abi>& __x)
{
const simd<double, _Abi> __x2 = __x * __x;
simd<double, _Abi> __y;
__y = 0x1.AC00000000000p-45; // 1/16!
__y = __y * __x2 - 0x1.9394000000000p-37; // -1/14!
__y = __y * __x2 + 0x1.1EED8C0000000p-29; // 1/12!
__y = __y * __x2 - 0x1.27E4FB7400000p-22; // -1/10!
__y = __y * __x2 + 0x1.A01A01A018000p-16; // 1/8!
__y = __y * __x2 - 0x1.6C16C16C16C00p-10; // -1/6!
__y = __y * __x2 + 0x1.5555555555554p-5; // 1/4!
return (__y * __x2 - .5f) * __x2 + 1.f;
}
// }}}
// __sinSeries {{{
template <typename _Abi>
_GLIBCXX_SIMD_ALWAYS_INLINE static simd<float, _Abi>
__sinSeries(const simd<float, _Abi>& __x)
{
const simd<float, _Abi> __x2 = __x * __x;
simd<float, _Abi> __y;
__y = -0x1.9CC000p-13f; // -1/7!
__y = __y * __x2 + 0x1.111100p-7f; // 1/5!
__y = __y * __x2 - 0x1.555556p-3f; // -1/3!
return __y * (__x2 * __x) + __x;
}
template <typename _Abi>
_GLIBCXX_SIMD_ALWAYS_INLINE static simd<double, _Abi>
__sinSeries(const simd<double, _Abi>& __x)
{
// __x = [0, 0.7854 = pi/4]
// __x² = [0, 0.6169 = pi²/8]
const simd<double, _Abi> __x2 = __x * __x;
simd<double, _Abi> __y;
__y = -0x1.ACF0000000000p-41; // -1/15!
__y = __y * __x2 + 0x1.6124400000000p-33; // 1/13!
__y = __y * __x2 - 0x1.AE64567000000p-26; // -1/11!
__y = __y * __x2 + 0x1.71DE3A5540000p-19; // 1/9!
__y = __y * __x2 - 0x1.A01A01A01A000p-13; // -1/7!
__y = __y * __x2 + 0x1.1111111111110p-7; // 1/5!
__y = __y * __x2 - 0x1.5555555555555p-3; // -1/3!
return __y * (__x2 * __x) + __x;
}
// }}}
// __zero_low_bits {{{
template <int _Bits, typename _Tp, typename _Abi>
_GLIBCXX_SIMD_INTRINSIC simd<_Tp, _Abi>
__zero_low_bits(simd<_Tp, _Abi> __x)
{
const simd<_Tp, _Abi> __bitmask
= __bit_cast<_Tp>(~make_unsigned_t<__int_for_sizeof_t<_Tp>>() << _Bits);
return {__private_init,
_Abi::_SimdImpl::_S_bit_and(__data(__x), __data(__bitmask))};
}
// }}}
// __fold_input {{{
/**@internal
* Fold @p x into [-¼π, ¼π] and remember the quadrant it came from:
* quadrant 0: [-¼π, ¼π]
* quadrant 1: [ ¼π, ¾π]
* quadrant 2: [ ¾π, 1¼π]
* quadrant 3: [1¼π, 1¾π]
*
* The algorithm determines `y` as the multiple `x - y * ¼π = [-¼π, ¼π]`. Using
* a bitmask, `y` is reduced to `quadrant`. `y` can be calculated as
* ```
* y = trunc(x / ¼π);
* y += fmod(y, 2);
* ```
* This can be simplified by moving the (implicit) division by 2 into the
* truncation expression. The `+= fmod` effect can the be achieved by using
* rounding instead of truncation: `y = round(x / ½π) * 2`. If precision allows,
* `2/π * x` is better (faster).
*/
template <typename _Tp, typename _Abi>
struct _Folded
{
simd<_Tp, _Abi> _M_x;
rebind_simd_t<int, simd<_Tp, _Abi>> _M_quadrant;
};
namespace __math_float {
inline constexpr float __pi_over_4 = 0x1.921FB6p-1f; // π/4
inline constexpr float __2_over_pi = 0x1.45F306p-1f; // 2/π
inline constexpr float __pi_2_5bits0
= 0x1.921fc0p0f; // π/2, 5 0-bits (least significant)
inline constexpr float __pi_2_5bits0_rem
= -0x1.5777a6p-21f; // π/2 - __pi_2_5bits0
} // namespace __math_float
namespace __math_double {
inline constexpr double __pi_over_4 = 0x1.921fb54442d18p-1; // π/4
inline constexpr double __2_over_pi = 0x1.45F306DC9C883p-1; // 2/π
inline constexpr double __pi_2 = 0x1.921fb54442d18p0; // π/2
} // namespace __math_double
template <typename _Abi>
_GLIBCXX_SIMD_ALWAYS_INLINE _Folded<float, _Abi>
__fold_input(const simd<float, _Abi>& __x)
{
using _V = simd<float, _Abi>;
using _IV = rebind_simd_t<int, _V>;
using namespace __math_float;
_Folded<float, _Abi> __r;
__r._M_x = abs(__x);
#if 0
// zero most mantissa bits:
constexpr float __1_over_pi = 0x1.45F306p-2f; // 1/π
const auto __y = (__r._M_x * __1_over_pi + 0x1.8p23f) - 0x1.8p23f;
// split π into 4 parts, the first three with 13 trailing zeros (to make the
// following multiplications precise):
constexpr float __pi0 = 0x1.920000p1f;
constexpr float __pi1 = 0x1.fb4000p-11f;
constexpr float __pi2 = 0x1.444000p-23f;
constexpr float __pi3 = 0x1.68c234p-38f;
__r._M_x - __y*__pi0 - __y*__pi1 - __y*__pi2 - __y*__pi3
#else
if (_GLIBCXX_SIMD_IS_UNLIKELY(all_of(__r._M_x < __pi_over_4)))
__r._M_quadrant = 0;
else if (_GLIBCXX_SIMD_IS_LIKELY(all_of(__r._M_x < 6 * __pi_over_4)))
{
const _V __y = nearbyint(__r._M_x * __2_over_pi);
__r._M_quadrant = static_simd_cast<_IV>(__y) & 3; // __y mod 4
__r._M_x -= __y * __pi_2_5bits0;
__r._M_x -= __y * __pi_2_5bits0_rem;
}
else
{
using __math_double::__2_over_pi;
using __math_double::__pi_2;
using _VD = rebind_simd_t<double, _V>;
_VD __xd = static_simd_cast<_VD>(__r._M_x);
_VD __y = nearbyint(__xd * __2_over_pi);
__r._M_quadrant = static_simd_cast<_IV>(__y) & 3; // = __y mod 4
__r._M_x = static_simd_cast<_V>(__xd - __y * __pi_2);
}
#endif
return __r;
}
template <typename _Abi>
_GLIBCXX_SIMD_ALWAYS_INLINE _Folded<double, _Abi>
__fold_input(const simd<double, _Abi>& __x)
{
using _V = simd<double, _Abi>;
using _IV = rebind_simd_t<int, _V>;
using namespace __math_double;
_Folded<double, _Abi> __r;
__r._M_x = abs(__x);
if (_GLIBCXX_SIMD_IS_UNLIKELY(all_of(__r._M_x < __pi_over_4)))
{
__r._M_quadrant = 0;
return __r;
}
const _V __y = nearbyint(__r._M_x / (2 * __pi_over_4));
__r._M_quadrant = static_simd_cast<_IV>(__y) & 3;
if (_GLIBCXX_SIMD_IS_LIKELY(all_of(__r._M_x < 1025 * __pi_over_4)))
{
// x - y * pi/2, y uses no more than 11 mantissa bits
__r._M_x -= __y * 0x1.921FB54443000p0;
__r._M_x -= __y * -0x1.73DCB3B39A000p-43;
__r._M_x -= __y * 0x1.45C06E0E68948p-86;
}
else if (_GLIBCXX_SIMD_IS_LIKELY(all_of(__y <= 0x1.0p30)))
{
// x - y * pi/2, y uses no more than 29 mantissa bits
__r._M_x -= __y * 0x1.921FB40000000p0;
__r._M_x -= __y * 0x1.4442D00000000p-24;
__r._M_x -= __y * 0x1.8469898CC5170p-48;
}
else
{
// x - y * pi/2, y may require all mantissa bits
const _V __y_hi = __zero_low_bits<26>(__y);
const _V __y_lo = __y - __y_hi;
const auto __pi_2_1 = 0x1.921FB50000000p0;
const auto __pi_2_2 = 0x1.110B460000000p-26;
const auto __pi_2_3 = 0x1.1A62630000000p-54;
const auto __pi_2_4 = 0x1.8A2E03707344Ap-81;
__r._M_x = __r._M_x - __y_hi * __pi_2_1
- max(__y_hi * __pi_2_2, __y_lo * __pi_2_1)
- min(__y_hi * __pi_2_2, __y_lo * __pi_2_1)
- max(__y_hi * __pi_2_3, __y_lo * __pi_2_2)
- min(__y_hi * __pi_2_3, __y_lo * __pi_2_2)
- max(__y * __pi_2_4, __y_lo * __pi_2_3)
- min(__y * __pi_2_4, __y_lo * __pi_2_3);
}
return __r;
}
// }}}
// __extract_exponent_as_int {{{
template <typename _Tp, typename _Abi>
rebind_simd_t<int, simd<_Tp, _Abi>>
__extract_exponent_as_int(const simd<_Tp, _Abi>& __v)
{
using _Vp = simd<_Tp, _Abi>;
using _Up = make_unsigned_t<__int_for_sizeof_t<_Tp>>;
using namespace std::experimental::__float_bitwise_operators;
const _Vp __exponent_mask
= __infinity_v<_Tp>; // 0x7f800000 or 0x7ff0000000000000
return static_simd_cast<rebind_simd_t<int, _Vp>>(
__bit_cast<rebind_simd_t<_Up, _Vp>>(__v & __exponent_mask)
>> (__digits_v<_Tp> - 1));
}
// }}}
// __impl_or_fallback {{{
template <typename ImplFun, typename FallbackFun, typename... _Args>
_GLIBCXX_SIMD_INTRINSIC auto
__impl_or_fallback_dispatch(int, ImplFun&& __impl_fun, FallbackFun&&,
_Args&&... __args)
-> decltype(__impl_fun(static_cast<_Args&&>(__args)...))
{ return __impl_fun(static_cast<_Args&&>(__args)...); }
template <typename ImplFun, typename FallbackFun, typename... _Args>
inline auto
__impl_or_fallback_dispatch(float, ImplFun&&, FallbackFun&& __fallback_fun,
_Args&&... __args)
-> decltype(__fallback_fun(static_cast<_Args&&>(__args)...))
{ return __fallback_fun(static_cast<_Args&&>(__args)...); }
template <typename... _Args>
_GLIBCXX_SIMD_INTRINSIC auto
__impl_or_fallback(_Args&&... __args)
{
return __impl_or_fallback_dispatch(int(), static_cast<_Args&&>(__args)...);
}
//}}}
// trigonometric functions {{{
_GLIBCXX_SIMD_MATH_CALL_(acos)
_GLIBCXX_SIMD_MATH_CALL_(asin)
_GLIBCXX_SIMD_MATH_CALL_(atan)
_GLIBCXX_SIMD_MATH_CALL2_(atan2, _Tp)
/*
* algorithm for sine and cosine:
*
* The result can be calculated with sine or cosine depending on the π/4 section
* the input is in. sine ≈ __x + __x³ cosine ≈ 1 - __x²
*
* sine:
* Map -__x to __x and invert the output
* Extend precision of __x - n * π/4 by calculating
* ((__x - n * p1) - n * p2) - n * p3 (p1 + p2 + p3 = π/4)
*
* Calculate Taylor series with tuned coefficients.
* Fix sign.
*/
// cos{{{
template <typename _Tp, typename _Abi>
enable_if_t<is_floating_point_v<_Tp>, simd<_Tp, _Abi>>
cos(const simd<_Tp, _Abi>& __x)
{
using _V = simd<_Tp, _Abi>;
if constexpr (__is_scalar_abi<_Abi>() || __is_fixed_size_abi_v<_Abi>)
return {__private_init, _Abi::_SimdImpl::_S_cos(__data(__x))};
else
{
if constexpr (is_same_v<_Tp, float>)
if (_GLIBCXX_SIMD_IS_UNLIKELY(any_of(abs(__x) >= 393382)))
return static_simd_cast<_V>(
cos(static_simd_cast<rebind_simd_t<double, _V>>(__x)));
const auto __f = __fold_input(__x);
// quadrant | effect
// 0 | cosSeries, +
// 1 | sinSeries, -
// 2 | cosSeries, -
// 3 | sinSeries, +
using namespace std::experimental::__float_bitwise_operators;
const _V __sign_flip
= _V(-0.f) & static_simd_cast<_V>((1 + __f._M_quadrant) << 30);
const auto __need_cos = (__f._M_quadrant & 1) == 0;
if (_GLIBCXX_SIMD_IS_UNLIKELY(all_of(__need_cos)))
return __sign_flip ^ __cosSeries(__f._M_x);
else if (_GLIBCXX_SIMD_IS_UNLIKELY(none_of(__need_cos)))
return __sign_flip ^ __sinSeries(__f._M_x);
else // some_of(__need_cos)
{
_V __r = __sinSeries(__f._M_x);
where(__need_cos.__cvt(), __r) = __cosSeries(__f._M_x);
return __r ^ __sign_flip;
}
}
}
template <typename _Tp>
_GLIBCXX_SIMD_ALWAYS_INLINE
enable_if_t<is_floating_point<_Tp>::value, simd<_Tp, simd_abi::scalar>>
cos(simd<_Tp, simd_abi::scalar> __x)
{ return std::cos(__data(__x)); }
//}}}
// sin{{{
template <typename _Tp, typename _Abi>
enable_if_t<is_floating_point_v<_Tp>, simd<_Tp, _Abi>>
sin(const simd<_Tp, _Abi>& __x)
{
using _V = simd<_Tp, _Abi>;
if constexpr (__is_scalar_abi<_Abi>() || __is_fixed_size_abi_v<_Abi>)
return {__private_init, _Abi::_SimdImpl::_S_sin(__data(__x))};
else
{
if constexpr (is_same_v<_Tp, float>)
if (_GLIBCXX_SIMD_IS_UNLIKELY(any_of(abs(__x) >= 527449)))
return static_simd_cast<_V>(
sin(static_simd_cast<rebind_simd_t<double, _V>>(__x)));
const auto __f = __fold_input(__x);
// quadrant | effect
// 0 | sinSeries
// 1 | cosSeries
// 2 | sinSeries, sign flip
// 3 | cosSeries, sign flip
using namespace std::experimental::__float_bitwise_operators;
const auto __sign_flip
= (__x ^ static_simd_cast<_V>(1 - __f._M_quadrant)) & _V(_Tp(-0.));
const auto __need_sin = (__f._M_quadrant & 1) == 0;
if (_GLIBCXX_SIMD_IS_UNLIKELY(all_of(__need_sin)))
return __sign_flip ^ __sinSeries(__f._M_x);
else if (_GLIBCXX_SIMD_IS_UNLIKELY(none_of(__need_sin)))
return __sign_flip ^ __cosSeries(__f._M_x);
else // some_of(__need_sin)
{
_V __r = __cosSeries(__f._M_x);
where(__need_sin.__cvt(), __r) = __sinSeries(__f._M_x);
return __sign_flip ^ __r;
}
}
}
template <typename _Tp>
_GLIBCXX_SIMD_ALWAYS_INLINE
enable_if_t<is_floating_point<_Tp>::value, simd<_Tp, simd_abi::scalar>>
sin(simd<_Tp, simd_abi::scalar> __x)
{ return std::sin(__data(__x)); }
//}}}
_GLIBCXX_SIMD_MATH_CALL_(tan)
_GLIBCXX_SIMD_MATH_CALL_(acosh)
_GLIBCXX_SIMD_MATH_CALL_(asinh)
_GLIBCXX_SIMD_MATH_CALL_(atanh)
_GLIBCXX_SIMD_MATH_CALL_(cosh)
_GLIBCXX_SIMD_MATH_CALL_(sinh)
_GLIBCXX_SIMD_MATH_CALL_(tanh)
// }}}
// exponential functions {{{
_GLIBCXX_SIMD_MATH_CALL_(exp)
_GLIBCXX_SIMD_MATH_CALL_(exp2)
_GLIBCXX_SIMD_MATH_CALL_(expm1)
// }}}
// frexp {{{
#if _GLIBCXX_SIMD_X86INTRIN
template <typename _Tp, size_t _Np>
_SimdWrapper<_Tp, _Np>
__getexp(_SimdWrapper<_Tp, _Np> __x)
{
if constexpr (__have_avx512vl && __is_sse_ps<_Tp, _Np>())
return __auto_bitcast(_mm_getexp_ps(__to_intrin(__x)));
else if constexpr (__have_avx512f && __is_sse_ps<_Tp, _Np>())
return __auto_bitcast(_mm512_getexp_ps(__auto_bitcast(__to_intrin(__x))));
else if constexpr (__have_avx512vl && __is_sse_pd<_Tp, _Np>())
return _mm_getexp_pd(__x);
else if constexpr (__have_avx512f && __is_sse_pd<_Tp, _Np>())
return __lo128(_mm512_getexp_pd(__auto_bitcast(__x)));
else if constexpr (__have_avx512vl && __is_avx_ps<_Tp, _Np>())
return _mm256_getexp_ps(__x);
else if constexpr (__have_avx512f && __is_avx_ps<_Tp, _Np>())
return __lo256(_mm512_getexp_ps(__auto_bitcast(__x)));
else if constexpr (__have_avx512vl && __is_avx_pd<_Tp, _Np>())
return _mm256_getexp_pd(__x);
else if constexpr (__have_avx512f && __is_avx_pd<_Tp, _Np>())
return __lo256(_mm512_getexp_pd(__auto_bitcast(__x)));
else if constexpr (__is_avx512_ps<_Tp, _Np>())
return _mm512_getexp_ps(__x);
else if constexpr (__is_avx512_pd<_Tp, _Np>())
return _mm512_getexp_pd(__x);
else
__assert_unreachable<_Tp>();
}
template <typename _Tp, size_t _Np>
_SimdWrapper<_Tp, _Np>
__getmant_avx512(_SimdWrapper<_Tp, _Np> __x)
{
if constexpr (__have_avx512vl && __is_sse_ps<_Tp, _Np>())
return __auto_bitcast(_mm_getmant_ps(__to_intrin(__x), _MM_MANT_NORM_p5_1,
_MM_MANT_SIGN_src));
else if constexpr (__have_avx512f && __is_sse_ps<_Tp, _Np>())
return __auto_bitcast(_mm512_getmant_ps(__auto_bitcast(__to_intrin(__x)),
_MM_MANT_NORM_p5_1,
_MM_MANT_SIGN_src));
else if constexpr (__have_avx512vl && __is_sse_pd<_Tp, _Np>())
return _mm_getmant_pd(__x, _MM_MANT_NORM_p5_1, _MM_MANT_SIGN_src);
else if constexpr (__have_avx512f && __is_sse_pd<_Tp, _Np>())
return __lo128(_mm512_getmant_pd(__auto_bitcast(__x), _MM_MANT_NORM_p5_1,
_MM_MANT_SIGN_src));
else if constexpr (__have_avx512vl && __is_avx_ps<_Tp, _Np>())
return _mm256_getmant_ps(__x, _MM_MANT_NORM_p5_1, _MM_MANT_SIGN_src);
else if constexpr (__have_avx512f && __is_avx_ps<_Tp, _Np>())
return __lo256(_mm512_getmant_ps(__auto_bitcast(__x), _MM_MANT_NORM_p5_1,
_MM_MANT_SIGN_src));
else if constexpr (__have_avx512vl && __is_avx_pd<_Tp, _Np>())
return _mm256_getmant_pd(__x, _MM_MANT_NORM_p5_1, _MM_MANT_SIGN_src);
else if constexpr (__have_avx512f && __is_avx_pd<_Tp, _Np>())
return __lo256(_mm512_getmant_pd(__auto_bitcast(__x), _MM_MANT_NORM_p5_1,
_MM_MANT_SIGN_src));
else if constexpr (__is_avx512_ps<_Tp, _Np>())
return _mm512_getmant_ps(__x, _MM_MANT_NORM_p5_1, _MM_MANT_SIGN_src);
else if constexpr (__is_avx512_pd<_Tp, _Np>())
return _mm512_getmant_pd(__x, _MM_MANT_NORM_p5_1, _MM_MANT_SIGN_src);
else
__assert_unreachable<_Tp>();
}
#endif // _GLIBCXX_SIMD_X86INTRIN
/**
* splits @p __v into exponent and mantissa, the sign is kept with the mantissa
*
* The return value will be in the range [0.5, 1.0[
* The @p __e value will be an integer defining the power-of-two exponent
*/
template <typename _Tp, typename _Abi>
enable_if_t<is_floating_point_v<_Tp>, simd<_Tp, _Abi>>
frexp(const simd<_Tp, _Abi>& __x, _Samesize<int, simd<_Tp, _Abi>>* __exp)
{
if constexpr (simd_size_v<_Tp, _Abi> == 1)
{
int __tmp;
const auto __r = std::frexp(__x[0], &__tmp);
(*__exp)[0] = __tmp;
return __r;
}
else if constexpr (__is_fixed_size_abi_v<_Abi>)
{
return {__private_init,
_Abi::_SimdImpl::_S_frexp(__data(__x), __data(*__exp))};
#if _GLIBCXX_SIMD_X86INTRIN
}
else if constexpr (__have_avx512f)
{
constexpr size_t _Np = simd_size_v<_Tp, _Abi>;
constexpr size_t _NI = _Np < 4 ? 4 : _Np;
const auto __v = __data(__x);
const auto __isnonzero
= _Abi::_SimdImpl::_S_isnonzerovalue_mask(__v._M_data);
const _SimdWrapper<int, _NI> __exp_plus1
= 1 + __convert<_SimdWrapper<int, _NI>>(__getexp(__v))._M_data;
const _SimdWrapper<int, _Np> __e = __wrapper_bitcast<int, _Np>(
_Abi::_CommonImpl::_S_blend(_SimdWrapper<bool, _NI>(__isnonzero),
_SimdWrapper<int, _NI>(), __exp_plus1));
simd_abi::deduce_t<int, _Np>::_CommonImpl::_S_store(__e, __exp);
return {__private_init,
_Abi::_CommonImpl::_S_blend(_SimdWrapper<bool, _Np>(
__isnonzero),
__v, __getmant_avx512(__v))};
#endif // _GLIBCXX_SIMD_X86INTRIN
}
else
{
// fallback implementation
static_assert(sizeof(_Tp) == 4 || sizeof(_Tp) == 8);
using _V = simd<_Tp, _Abi>;
using _IV = rebind_simd_t<int, _V>;
using namespace std::experimental::__proposed;
using namespace std::experimental::__float_bitwise_operators;
constexpr int __exp_adjust = sizeof(_Tp) == 4 ? 0x7e : 0x3fe;
constexpr int __exp_offset = sizeof(_Tp) == 4 ? 0x70 : 0x200;
constexpr _Tp __subnorm_scale = sizeof(_Tp) == 4 ? 0x1p112 : 0x1p512;
_GLIBCXX_SIMD_USE_CONSTEXPR_API _V __exponent_mask
= __infinity_v<_Tp>; // 0x7f800000 or 0x7ff0000000000000
_GLIBCXX_SIMD_USE_CONSTEXPR_API _V __p5_1_exponent
= -(2 - __epsilon_v<_Tp>) / 2; // 0xbf7fffff or 0xbfefffffffffffff
_V __mant = __p5_1_exponent & (__exponent_mask | __x); // +/-[.5, 1)
const _IV __exponent_bits = __extract_exponent_as_int(__x);
if (_GLIBCXX_SIMD_IS_LIKELY(all_of(isnormal(__x))))
{
*__exp
= simd_cast<_Samesize<int, _V>>(__exponent_bits - __exp_adjust);
return __mant;
}
#if __FINITE_MATH_ONLY__
// at least one element of __x is 0 or subnormal, the rest is normal
// (inf and NaN are excluded by -ffinite-math-only)
const auto __iszero_inf_nan = __x == 0;
#else
const auto __as_int
= __bit_cast<rebind_simd_t<__int_for_sizeof_t<_Tp>, _V>>(abs(__x));
const auto __inf
= __bit_cast<rebind_simd_t<__int_for_sizeof_t<_Tp>, _V>>(
_V(__infinity_v<_Tp>));
const auto __iszero_inf_nan = static_simd_cast<typename _V::mask_type>(
__as_int == 0 || __as_int >= __inf);
#endif
const _V __scaled_subnormal = __x * __subnorm_scale;
const _V __mant_subnormal
= __p5_1_exponent & (__exponent_mask | __scaled_subnormal);
where(!isnormal(__x), __mant) = __mant_subnormal;
where(__iszero_inf_nan, __mant) = __x;
_IV __e = __extract_exponent_as_int(__scaled_subnormal);
using _MaskType =
typename conditional_t<sizeof(typename _V::value_type) == sizeof(int),
_V, _IV>::mask_type;
const _MaskType __value_isnormal = isnormal(__x).__cvt();
where(__value_isnormal.__cvt(), __e) = __exponent_bits;
static_assert(sizeof(_IV) == sizeof(__value_isnormal));
const _IV __offset
= (__bit_cast<_IV>(__value_isnormal) & _IV(__exp_adjust))
| (__bit_cast<_IV>(static_simd_cast<_MaskType>(__exponent_bits == 0)
& static_simd_cast<_MaskType>(__x != 0))
& _IV(__exp_adjust + __exp_offset));
*__exp = simd_cast<_Samesize<int, _V>>(__e - __offset);
return __mant;
}
}
// }}}
_GLIBCXX_SIMD_MATH_CALL2_(ldexp, int)
_GLIBCXX_SIMD_MATH_CALL_(ilogb)
// logarithms {{{
_GLIBCXX_SIMD_MATH_CALL_(log)
_GLIBCXX_SIMD_MATH_CALL_(log10)
_GLIBCXX_SIMD_MATH_CALL_(log1p)
_GLIBCXX_SIMD_MATH_CALL_(log2)
//}}}
// logb{{{
template <typename _Tp, typename _Abi>
enable_if_t<is_floating_point<_Tp>::value, simd<_Tp, _Abi>>
logb(const simd<_Tp, _Abi>& __x)
{
constexpr size_t _Np = simd_size_v<_Tp, _Abi>;
if constexpr (_Np == 1)
return std::logb(__x[0]);
else if constexpr (__is_fixed_size_abi_v<_Abi>)
{
return {__private_init,
__data(__x)._M_apply_per_chunk([](auto __impl, auto __xx) {
using _V = typename decltype(__impl)::simd_type;
return __data(
std::experimental::logb(_V(__private_init, __xx)));
})};
}
#if _GLIBCXX_SIMD_X86INTRIN // {{{
else if constexpr (__have_avx512vl && __is_sse_ps<_Tp, _Np>())
return {__private_init,
__auto_bitcast(_mm_getexp_ps(__to_intrin(__as_vector(__x))))};
else if constexpr (__have_avx512vl && __is_sse_pd<_Tp, _Np>())
return {__private_init, _mm_getexp_pd(__data(__x))};
else if constexpr (__have_avx512vl && __is_avx_ps<_Tp, _Np>())
return {__private_init, _mm256_getexp_ps(__data(__x))};
else if constexpr (__have_avx512vl && __is_avx_pd<_Tp, _Np>())
return {__private_init, _mm256_getexp_pd(__data(__x))};
else if constexpr (__have_avx512f && __is_avx_ps<_Tp, _Np>())
return {__private_init,
__lo256(_mm512_getexp_ps(__auto_bitcast(__data(__x))))};
else if constexpr (__have_avx512f && __is_avx_pd<_Tp, _Np>())
return {__private_init,
__lo256(_mm512_getexp_pd(__auto_bitcast(__data(__x))))};
else if constexpr (__is_avx512_ps<_Tp, _Np>())
return {__private_init, _mm512_getexp_ps(__data(__x))};
else if constexpr (__is_avx512_pd<_Tp, _Np>())
return {__private_init, _mm512_getexp_pd(__data(__x))};
#endif // _GLIBCXX_SIMD_X86INTRIN }}}
else
{
using _V = simd<_Tp, _Abi>;
using namespace std::experimental::__proposed;
auto __is_normal = isnormal(__x);
// work on abs(__x) to reflect the return value on Linux for negative
// inputs (domain-error => implementation-defined value is returned)
const _V abs_x = abs(__x);
// __exponent(__x) returns the exponent value (bias removed) as
// simd<_Up> with integral _Up
auto&& __exponent = [](const _V& __v) {
using namespace std::experimental::__proposed;
using _IV = rebind_simd_t<
conditional_t<sizeof(_Tp) == sizeof(_LLong), _LLong, int>, _V>;
return (__bit_cast<_IV>(__v) >> (__digits_v<_Tp> - 1))
- (__max_exponent_v<_Tp> - 1);
};
_V __r = static_simd_cast<_V>(__exponent(abs_x));
if (_GLIBCXX_SIMD_IS_LIKELY(all_of(__is_normal)))
// without corner cases (nan, inf, subnormal, zero) we have our
// answer:
return __r;
const auto __is_zero = __x == 0;
const auto __is_nan = isnan(__x);
const auto __is_inf = isinf(__x);
where(__is_zero, __r) = -__infinity_v<_Tp>;
where(__is_nan, __r) = __x;
where(__is_inf, __r) = __infinity_v<_Tp>;
__is_normal |= __is_zero || __is_nan || __is_inf;
if (all_of(__is_normal))
// at this point everything but subnormals is handled
return __r;
// subnormals repeat the exponent extraction after multiplication of the
// input with __a floating point value that has 112 (0x70) in its exponent
// (not too big for sp and large enough for dp)
const _V __scaled = abs_x * _Tp(0x1p112);
_V __scaled_exp = static_simd_cast<_V>(__exponent(__scaled) - 112);
where(__is_normal, __scaled_exp) = __r;
return __scaled_exp;
}
}
//}}}
template <typename _Tp, typename _Abi>
enable_if_t<is_floating_point_v<_Tp>, simd<_Tp, _Abi>>
modf(const simd<_Tp, _Abi>& __x, simd<_Tp, _Abi>* __iptr)
{
if constexpr (__is_scalar_abi<_Abi>()
|| (__is_fixed_size_abi_v<
_Abi> && simd_size_v<_Tp, _Abi> == 1))
{
_Tp __tmp;
_Tp __r = std::modf(__x[0], &__tmp);
__iptr[0] = __tmp;
return __r;
}
else
{
const auto __integral = trunc(__x);
*__iptr = __integral;
auto __r = __x - __integral;
#if !__FINITE_MATH_ONLY__
where(isinf(__x), __r) = _Tp();
#endif
return copysign(__r, __x);
}
}
_GLIBCXX_SIMD_MATH_CALL2_(scalbn, int)
_GLIBCXX_SIMD_MATH_CALL2_(scalbln, long)
_GLIBCXX_SIMD_MATH_CALL_(cbrt)
_GLIBCXX_SIMD_MATH_CALL_(abs)
_GLIBCXX_SIMD_MATH_CALL_(fabs)
// [parallel.simd.math] only asks for is_floating_point_v<_Tp> and forgot to
// allow signed integral _Tp
template <typename _Tp, typename _Abi>
enable_if_t<!is_floating_point_v<_Tp> && is_signed_v<_Tp>, simd<_Tp, _Abi>>
abs(const simd<_Tp, _Abi>& __x)
{ return {__private_init, _Abi::_SimdImpl::_S_abs(__data(__x))}; }
template <typename _Tp, typename _Abi>
enable_if_t<!is_floating_point_v<_Tp> && is_signed_v<_Tp>, simd<_Tp, _Abi>>
fabs(const simd<_Tp, _Abi>& __x)
{ return {__private_init, _Abi::_SimdImpl::_S_abs(__data(__x))}; }
// the following are overloads for functions in <cstdlib> and not covered by
// [parallel.simd.math]. I don't see much value in making them work, though
/*
template <typename _Abi> simd<long, _Abi> labs(const simd<long, _Abi> &__x)
{ return {__private_init, _Abi::_SimdImpl::abs(__data(__x))}; }
template <typename _Abi> simd<long long, _Abi> llabs(const simd<long long, _Abi>
&__x)
{ return {__private_init, _Abi::_SimdImpl::abs(__data(__x))}; }
*/
#define _GLIBCXX_SIMD_CVTING2(_NAME) \
template <typename _Tp, typename _Abi> \
_GLIBCXX_SIMD_INTRINSIC simd<_Tp, _Abi> _NAME( \
const simd<_Tp, _Abi>& __x, const __type_identity_t<simd<_Tp, _Abi>>& __y) \
{ \
return _NAME(__x, __y); \
} \
\
template <typename _Tp, typename _Abi> \
_GLIBCXX_SIMD_INTRINSIC simd<_Tp, _Abi> _NAME( \
const __type_identity_t<simd<_Tp, _Abi>>& __x, const simd<_Tp, _Abi>& __y) \
{ \
return _NAME(__x, __y); \
}
#define _GLIBCXX_SIMD_CVTING3(_NAME) \
template <typename _Tp, typename _Abi> \
_GLIBCXX_SIMD_INTRINSIC simd<_Tp, _Abi> _NAME( \
const __type_identity_t<simd<_Tp, _Abi>>& __x, const simd<_Tp, _Abi>& __y, \
const simd<_Tp, _Abi>& __z) \
{ \
return _NAME(__x, __y, __z); \
} \
\
template <typename _Tp, typename _Abi> \
_GLIBCXX_SIMD_INTRINSIC simd<_Tp, _Abi> _NAME( \
const simd<_Tp, _Abi>& __x, const __type_identity_t<simd<_Tp, _Abi>>& __y, \
const simd<_Tp, _Abi>& __z) \
{ \
return _NAME(__x, __y, __z); \
} \
\
template <typename _Tp, typename _Abi> \
_GLIBCXX_SIMD_INTRINSIC simd<_Tp, _Abi> _NAME( \
const simd<_Tp, _Abi>& __x, const simd<_Tp, _Abi>& __y, \
const __type_identity_t<simd<_Tp, _Abi>>& __z) \
{ \
return _NAME(__x, __y, __z); \
} \
\
template <typename _Tp, typename _Abi> \
_GLIBCXX_SIMD_INTRINSIC simd<_Tp, _Abi> _NAME( \
const simd<_Tp, _Abi>& __x, const __type_identity_t<simd<_Tp, _Abi>>& __y, \
const __type_identity_t<simd<_Tp, _Abi>>& __z) \
{ \
return _NAME(__x, __y, __z); \
} \
\
template <typename _Tp, typename _Abi> \
_GLIBCXX_SIMD_INTRINSIC simd<_Tp, _Abi> _NAME( \
const __type_identity_t<simd<_Tp, _Abi>>& __x, const simd<_Tp, _Abi>& __y, \
const __type_identity_t<simd<_Tp, _Abi>>& __z) \
{ \
return _NAME(__x, __y, __z); \
} \
\
template <typename _Tp, typename _Abi> \
_GLIBCXX_SIMD_INTRINSIC simd<_Tp, _Abi> _NAME( \
const __type_identity_t<simd<_Tp, _Abi>>& __x, \
const __type_identity_t<simd<_Tp, _Abi>>& __y, const simd<_Tp, _Abi>& __z) \
{ \
return _NAME(__x, __y, __z); \
}
template <typename _R, typename _ToApply, typename _Tp, typename... _Tps>
_GLIBCXX_SIMD_INTRINSIC _R
__fixed_size_apply(_ToApply&& __apply, const _Tp& __arg0,
const _Tps&... __args)
{
return {__private_init,
__data(__arg0)._M_apply_per_chunk(
[&](auto __impl, const auto&... __inner) {
using _V = typename decltype(__impl)::simd_type;
return __data(__apply(_V(__private_init, __inner)...));
},
__data(__args)...)};
}
template <typename _VV>
__remove_cvref_t<_VV>
__hypot(_VV __x, _VV __y)
{
using _V = __remove_cvref_t<_VV>;
using _Tp = typename _V::value_type;
if constexpr (_V::size() == 1)
return std::hypot(_Tp(__x[0]), _Tp(__y[0]));
else if constexpr (__is_fixed_size_abi_v<typename _V::abi_type>)
{
return __fixed_size_apply<_V>([](auto __a,
auto __b) { return hypot(__a, __b); },
__x, __y);
}
else
{
// A simple solution for _Tp == float would be to cast to double and
// simply calculate sqrt(x²+y²) as it can't over-/underflow anymore with
// dp. It still needs the Annex F fixups though and isn't faster on
// Skylake-AVX512 (not even for SSE and AVX vectors, and really bad for
// AVX-512).
using namespace __float_bitwise_operators;
_V __absx = abs(__x); // no error
_V __absy = abs(__y); // no error
_V __hi = max(__absx, __absy); // no error
_V __lo = min(__absy, __absx); // no error
// round __hi down to the next power-of-2:
_GLIBCXX_SIMD_USE_CONSTEXPR_API _V __inf(__infinity_v<_Tp>);
#ifndef __FAST_MATH__
if constexpr (__have_neon && !__have_neon_a32)
{ // With ARMv7 NEON, we have no subnormals and must use slightly
// different strategy
const _V __hi_exp = __hi & __inf;
_V __scale_back = __hi_exp;
// For large exponents (max & max/2) the inversion comes too close
// to subnormals. Subtract 3 from the exponent:
where(__hi_exp > 1, __scale_back) = __hi_exp * _Tp(0.125);
// Invert and adjust for the off-by-one error of inversion via xor:
const _V __scale = (__scale_back ^ __inf) * _Tp(.5);
const _V __h1 = __hi * __scale;
const _V __l1 = __lo * __scale;
_V __r = __scale_back * sqrt(__h1 * __h1 + __l1 * __l1);
// Fix up hypot(0, 0) to not be NaN:
where(__hi == 0, __r) = 0;
return __r;
}
#endif
#ifdef __FAST_MATH__
// With fast-math, ignore precision of subnormals and inputs from
// __finite_max_v/2 to __finite_max_v. This removes all
// branching/masking.
if constexpr (true)
#else
if (_GLIBCXX_SIMD_IS_LIKELY(all_of(isnormal(__x))
&& all_of(isnormal(__y))))
#endif
{
const _V __hi_exp = __hi & __inf;
//((__hi + __hi) & __inf) ^ __inf almost works for computing
//__scale,
// except when (__hi + __hi) & __inf == __inf, in which case __scale
// becomes 0 (should be min/2 instead) and thus loses the
// information from __lo.
#ifdef __FAST_MATH__
using _Ip = __int_for_sizeof_t<_Tp>;
using _IV = rebind_simd_t<_Ip, _V>;
const auto __as_int = __bit_cast<_IV>(__hi_exp);
const _V __scale
= __bit_cast<_V>(2 * __bit_cast<_Ip>(_Tp(1)) - __as_int);
#else
const _V __scale = (__hi_exp ^ __inf) * _Tp(.5);
#endif
_GLIBCXX_SIMD_USE_CONSTEXPR_API _V __mant_mask
= __norm_min_v<_Tp> - __denorm_min_v<_Tp>;
const _V __h1 = (__hi & __mant_mask) | _V(1);
const _V __l1 = __lo * __scale;
return __hi_exp * sqrt(__h1 * __h1 + __l1 * __l1);
}
else
{
// slower path to support subnormals
// if __hi is subnormal, avoid scaling by inf & final mul by 0
// (which yields NaN) by using min()
_V __scale = _V(1 / __norm_min_v<_Tp>);
// invert exponent w/o error and w/o using the slow divider unit:
// xor inverts the exponent but off by 1. Multiplication with .5
// adjusts for the discrepancy.
where(__hi >= __norm_min_v<_Tp>, __scale)
= ((__hi & __inf) ^ __inf) * _Tp(.5);
// adjust final exponent for subnormal inputs
_V __hi_exp = __norm_min_v<_Tp>;
where(__hi >= __norm_min_v<_Tp>, __hi_exp)
= __hi & __inf; // no error
_V __h1 = __hi * __scale; // no error
_V __l1 = __lo * __scale; // no error
// sqrt(x²+y²) = e*sqrt((x/e)²+(y/e)²):
// this ensures no overflow in the argument to sqrt
_V __r = __hi_exp * sqrt(__h1 * __h1 + __l1 * __l1);
#ifdef __STDC_IEC_559__
// fixup for Annex F requirements
// the naive fixup goes like this:
//
// where(__l1 == 0, __r) = __hi;
// where(isunordered(__x, __y), __r) = __quiet_NaN_v<_Tp>;
// where(isinf(__absx) || isinf(__absy), __r) = __inf;
//
// The fixup can be prepared in parallel with the sqrt, requiring a
// single blend step after hi_exp * sqrt, reducing latency and
// throughput:
_V __fixup = __hi; // __lo == 0
where(isunordered(__x, __y), __fixup) = __quiet_NaN_v<_Tp>;
where(isinf(__absx) || isinf(__absy), __fixup) = __inf;
where(!(__lo == 0 || isunordered(__x, __y)
|| (isinf(__absx) || isinf(__absy))),
__fixup)
= __r;
__r = __fixup;
#endif
return __r;
}
}
}
template <typename _Tp, typename _Abi>
_GLIBCXX_SIMD_INTRINSIC simd<_Tp, _Abi>
hypot(const simd<_Tp, _Abi>& __x, const simd<_Tp, _Abi>& __y)
{
return __hypot<conditional_t<__is_fixed_size_abi_v<_Abi>,
const simd<_Tp, _Abi>&, simd<_Tp, _Abi>>>(__x,
__y);
}
_GLIBCXX_SIMD_CVTING2(hypot)
template <typename _VV>
__remove_cvref_t<_VV>
__hypot(_VV __x, _VV __y, _VV __z)
{
using _V = __remove_cvref_t<_VV>;
using _Abi = typename _V::abi_type;
using _Tp = typename _V::value_type;
/* FIXME: enable after PR77776 is resolved
if constexpr (_V::size() == 1)
return std::hypot(_Tp(__x[0]), _Tp(__y[0]), _Tp(__z[0]));
else
*/
if constexpr (__is_fixed_size_abi_v<_Abi> && _V::size() > 1)
{
return __fixed_size_apply<simd<_Tp, _Abi>>(
[](auto __a, auto __b, auto __c) { return hypot(__a, __b, __c); },
__x, __y, __z);
}
else
{
using namespace __float_bitwise_operators;
const _V __absx = abs(__x); // no error
const _V __absy = abs(__y); // no error
const _V __absz = abs(__z); // no error
_V __hi = max(max(__absx, __absy), __absz); // no error
_V __l0 = min(__absz, max(__absx, __absy)); // no error
_V __l1 = min(__absy, __absx); // no error
if constexpr (__digits_v<_Tp> == 64 && __max_exponent_v<_Tp> == 0x4000
&& __min_exponent_v<_Tp> == -0x3FFD && _V::size() == 1)
{ // Seems like x87 fp80, where bit 63 is always 1 unless subnormal or
// NaN. In this case the bit-tricks don't work, they require IEC559
// binary32 or binary64 format.
#ifdef __STDC_IEC_559__
// fixup for Annex F requirements
if (isinf(__absx[0]) || isinf(__absy[0]) || isinf(__absz[0]))
return __infinity_v<_Tp>;
else if (isunordered(__absx[0], __absy[0] + __absz[0]))
return __quiet_NaN_v<_Tp>;
else if (__l0[0] == 0 && __l1[0] == 0)
return __hi;
#endif
_V __hi_exp = __hi;
const _ULLong __tmp = 0x8000'0000'0000'0000ull;
__builtin_memcpy(&__data(__hi_exp), &__tmp, 8);
const _V __scale = 1 / __hi_exp;
__hi *= __scale;
__l0 *= __scale;
__l1 *= __scale;
return __hi_exp * sqrt((__l0 * __l0 + __l1 * __l1) + __hi * __hi);
}
else
{
// round __hi down to the next power-of-2:
_GLIBCXX_SIMD_USE_CONSTEXPR_API _V __inf(__infinity_v<_Tp>);
#ifndef __FAST_MATH__
if constexpr (_V::size() > 1 && __have_neon && !__have_neon_a32)
{ // With ARMv7 NEON, we have no subnormals and must use slightly
// different strategy
const _V __hi_exp = __hi & __inf;
_V __scale_back = __hi_exp;
// For large exponents (max & max/2) the inversion comes too
// close to subnormals. Subtract 3 from the exponent:
where(__hi_exp > 1, __scale_back) = __hi_exp * _Tp(0.125);
// Invert and adjust for the off-by-one error of inversion via
// xor:
const _V __scale = (__scale_back ^ __inf) * _Tp(.5);
const _V __h1 = __hi * __scale;
__l0 *= __scale;
__l1 *= __scale;
_V __lo = __l0 * __l0
+ __l1 * __l1; // add the two smaller values first
asm("" : "+m"(__lo));
_V __r = __scale_back * sqrt(__h1 * __h1 + __lo);
// Fix up hypot(0, 0, 0) to not be NaN:
where(__hi == 0, __r) = 0;
return __r;
}
#endif
#ifdef __FAST_MATH__
// With fast-math, ignore precision of subnormals and inputs from
// __finite_max_v/2 to __finite_max_v. This removes all
// branching/masking.
if constexpr (true)
#else
if (_GLIBCXX_SIMD_IS_LIKELY(all_of(isnormal(__x))
&& all_of(isnormal(__y))
&& all_of(isnormal(__z))))
#endif
{
const _V __hi_exp = __hi & __inf;
//((__hi + __hi) & __inf) ^ __inf almost works for computing
//__scale, except when (__hi + __hi) & __inf == __inf, in which
// case __scale
// becomes 0 (should be min/2 instead) and thus loses the
// information from __lo.
#ifdef __FAST_MATH__
using _Ip = __int_for_sizeof_t<_Tp>;
using _IV = rebind_simd_t<_Ip, _V>;
const auto __as_int = __bit_cast<_IV>(__hi_exp);
const _V __scale
= __bit_cast<_V>(2 * __bit_cast<_Ip>(_Tp(1)) - __as_int);
#else
const _V __scale = (__hi_exp ^ __inf) * _Tp(.5);
#endif
constexpr _Tp __mant_mask
= __norm_min_v<_Tp> - __denorm_min_v<_Tp>;
const _V __h1 = (__hi & _V(__mant_mask)) | _V(1);
__l0 *= __scale;
__l1 *= __scale;
const _V __lo
= __l0 * __l0
+ __l1 * __l1; // add the two smaller values first
return __hi_exp * sqrt(__lo + __h1 * __h1);
}
else
{
// slower path to support subnormals
// if __hi is subnormal, avoid scaling by inf & final mul by 0
// (which yields NaN) by using min()
_V __scale = _V(1 / __norm_min_v<_Tp>);
// invert exponent w/o error and w/o using the slow divider
// unit: xor inverts the exponent but off by 1. Multiplication
// with .5 adjusts for the discrepancy.
where(__hi >= __norm_min_v<_Tp>, __scale)
= ((__hi & __inf) ^ __inf) * _Tp(.5);
// adjust final exponent for subnormal inputs
_V __hi_exp = __norm_min_v<_Tp>;
where(__hi >= __norm_min_v<_Tp>, __hi_exp)
= __hi & __inf; // no error
_V __h1 = __hi * __scale; // no error
__l0 *= __scale; // no error
__l1 *= __scale; // no error
_V __lo = __l0 * __l0
+ __l1 * __l1; // add the two smaller values first
_V __r = __hi_exp * sqrt(__lo + __h1 * __h1);
#ifdef __STDC_IEC_559__
// fixup for Annex F requirements
_V __fixup = __hi; // __lo == 0
// where(__lo == 0, __fixup) = __hi;
where(isunordered(__x, __y + __z), __fixup)
= __quiet_NaN_v<_Tp>;
where(isinf(__absx) || isinf(__absy) || isinf(__absz), __fixup)
= __inf;
// Instead of __lo == 0, the following could depend on __h1² ==
// __h1² + __lo (i.e. __hi is so much larger than the other two
// inputs that the result is exactly __hi). While this may
// improve precision, it is likely to reduce efficiency if the
// ISA has FMAs (because __h1² + __lo is an FMA, but the
// intermediate
// __h1² must be kept)
where(!(__lo == 0 || isunordered(__x, __y + __z)
|| isinf(__absx) || isinf(__absy) || isinf(__absz)),
__fixup)
= __r;
__r = __fixup;
#endif
return __r;
}
}
}
}
template <typename _Tp, typename _Abi>
_GLIBCXX_SIMD_INTRINSIC simd<_Tp, _Abi>
hypot(const simd<_Tp, _Abi>& __x, const simd<_Tp, _Abi>& __y,
const simd<_Tp, _Abi>& __z)
{
return __hypot<conditional_t<__is_fixed_size_abi_v<_Abi>,
const simd<_Tp, _Abi>&, simd<_Tp, _Abi>>>(__x,
__y,
__z);
}
_GLIBCXX_SIMD_CVTING3(hypot)
_GLIBCXX_SIMD_MATH_CALL2_(pow, _Tp)
_GLIBCXX_SIMD_MATH_CALL_(sqrt)
_GLIBCXX_SIMD_MATH_CALL_(erf)
_GLIBCXX_SIMD_MATH_CALL_(erfc)
_GLIBCXX_SIMD_MATH_CALL_(lgamma)
_GLIBCXX_SIMD_MATH_CALL_(tgamma)
_GLIBCXX_SIMD_MATH_CALL_(ceil)
_GLIBCXX_SIMD_MATH_CALL_(floor)
_GLIBCXX_SIMD_MATH_CALL_(nearbyint)
_GLIBCXX_SIMD_MATH_CALL_(rint)
_GLIBCXX_SIMD_MATH_CALL_(lrint)
_GLIBCXX_SIMD_MATH_CALL_(llrint)
_GLIBCXX_SIMD_MATH_CALL_(round)
_GLIBCXX_SIMD_MATH_CALL_(lround)
_GLIBCXX_SIMD_MATH_CALL_(llround)
_GLIBCXX_SIMD_MATH_CALL_(trunc)
_GLIBCXX_SIMD_MATH_CALL2_(fmod, _Tp)
_GLIBCXX_SIMD_MATH_CALL2_(remainder, _Tp)
_GLIBCXX_SIMD_MATH_CALL3_(remquo, _Tp, int*)
template <typename _Tp, typename _Abi>
enable_if_t<is_floating_point_v<_Tp>, simd<_Tp, _Abi>>
copysign(const simd<_Tp, _Abi>& __x, const simd<_Tp, _Abi>& __y)
{
if constexpr (simd_size_v<_Tp, _Abi> == 1)
return std::copysign(__x[0], __y[0]);
else if constexpr (is_same_v<_Tp, long double> && sizeof(_Tp) == 12)
// Remove this case once __bit_cast is implemented via __builtin_bit_cast.
// It is necessary, because __signmask below cannot be computed at compile
// time.
return simd<_Tp, _Abi>(
[&](auto __i) { return std::copysign(__x[__i], __y[__i]); });
else
{
using _V = simd<_Tp, _Abi>;
using namespace std::experimental::__float_bitwise_operators;
_GLIBCXX_SIMD_USE_CONSTEXPR_API auto __signmask = _V(1) ^ _V(-1);
return (__x & (__x ^ __signmask)) | (__y & __signmask);
}
}
_GLIBCXX_SIMD_MATH_CALL2_(nextafter, _Tp)
// not covered in [parallel.simd.math]:
// _GLIBCXX_SIMD_MATH_CALL2_(nexttoward, long double)
_GLIBCXX_SIMD_MATH_CALL2_(fdim, _Tp)
_GLIBCXX_SIMD_MATH_CALL2_(fmax, _Tp)
_GLIBCXX_SIMD_MATH_CALL2_(fmin, _Tp)
_GLIBCXX_SIMD_MATH_CALL3_(fma, _Tp, _Tp)
_GLIBCXX_SIMD_MATH_CALL_(fpclassify)
_GLIBCXX_SIMD_MATH_CALL_(isfinite)
// isnan and isinf require special treatment because old glibc may declare
// `int isinf(double)`.
template <typename _Tp, typename _Abi, typename...,
typename _R = _Math_return_type_t<bool, _Tp, _Abi>>
enable_if_t<is_floating_point_v<_Tp>, _R>
isinf(simd<_Tp, _Abi> __x)
{ return {__private_init, _Abi::_SimdImpl::_S_isinf(__data(__x))}; }
template <typename _Tp, typename _Abi, typename...,
typename _R = _Math_return_type_t<bool, _Tp, _Abi>>
enable_if_t<is_floating_point_v<_Tp>, _R>
isnan(simd<_Tp, _Abi> __x)
{ return {__private_init, _Abi::_SimdImpl::_S_isnan(__data(__x))}; }
_GLIBCXX_SIMD_MATH_CALL_(isnormal)
template <typename..., typename _Tp, typename _Abi>
simd_mask<_Tp, _Abi>
signbit(simd<_Tp, _Abi> __x)
{
if constexpr (is_integral_v<_Tp>)
{
if constexpr (is_unsigned_v<_Tp>)
return simd_mask<_Tp, _Abi>{}; // false
else
return __x < 0;
}
else
return {__private_init, _Abi::_SimdImpl::_S_signbit(__data(__x))};
}
_GLIBCXX_SIMD_MATH_CALL2_(isgreater, _Tp)
_GLIBCXX_SIMD_MATH_CALL2_(isgreaterequal, _Tp)
_GLIBCXX_SIMD_MATH_CALL2_(isless, _Tp)
_GLIBCXX_SIMD_MATH_CALL2_(islessequal, _Tp)
_GLIBCXX_SIMD_MATH_CALL2_(islessgreater, _Tp)
_GLIBCXX_SIMD_MATH_CALL2_(isunordered, _Tp)
/* not covered in [parallel.simd.math]
template <typename _Abi> __doublev<_Abi> nan(const char* tagp);
template <typename _Abi> __floatv<_Abi> nanf(const char* tagp);
template <typename _Abi> __ldoublev<_Abi> nanl(const char* tagp);
template <typename _V> struct simd_div_t {
_V quot, rem;
};
template <typename _Abi>
simd_div_t<_SCharv<_Abi>> div(_SCharv<_Abi> numer,
_SCharv<_Abi> denom);
template <typename _Abi>
simd_div_t<__shortv<_Abi>> div(__shortv<_Abi> numer,
__shortv<_Abi> denom);
template <typename _Abi>
simd_div_t<__intv<_Abi>> div(__intv<_Abi> numer, __intv<_Abi> denom);
template <typename _Abi>
simd_div_t<__longv<_Abi>> div(__longv<_Abi> numer,
__longv<_Abi> denom);
template <typename _Abi>
simd_div_t<__llongv<_Abi>> div(__llongv<_Abi> numer,
__llongv<_Abi> denom);
*/
// special math {{{
template <typename _Tp, typename _Abi>
enable_if_t<is_floating_point_v<_Tp>, simd<_Tp, _Abi>>
assoc_laguerre(const fixed_size_simd<unsigned, simd_size_v<_Tp, _Abi>>& __n,
const fixed_size_simd<unsigned, simd_size_v<_Tp, _Abi>>& __m,
const simd<_Tp, _Abi>& __x)
{
return simd<_Tp, _Abi>([&](auto __i) {
return std::assoc_laguerre(__n[__i], __m[__i], __x[__i]);
});
}
template <typename _Tp, typename _Abi>
enable_if_t<is_floating_point_v<_Tp>, simd<_Tp, _Abi>>
assoc_legendre(const fixed_size_simd<unsigned, simd_size_v<_Tp, _Abi>>& __n,
const fixed_size_simd<unsigned, simd_size_v<_Tp, _Abi>>& __m,
const simd<_Tp, _Abi>& __x)
{
return simd<_Tp, _Abi>([&](auto __i) {
return std::assoc_legendre(__n[__i], __m[__i], __x[__i]);
});
}
_GLIBCXX_SIMD_MATH_CALL2_(beta, _Tp)
_GLIBCXX_SIMD_MATH_CALL_(comp_ellint_1)
_GLIBCXX_SIMD_MATH_CALL_(comp_ellint_2)
_GLIBCXX_SIMD_MATH_CALL2_(comp_ellint_3, _Tp)
_GLIBCXX_SIMD_MATH_CALL2_(cyl_bessel_i, _Tp)
_GLIBCXX_SIMD_MATH_CALL2_(cyl_bessel_j, _Tp)
_GLIBCXX_SIMD_MATH_CALL2_(cyl_bessel_k, _Tp)
_GLIBCXX_SIMD_MATH_CALL2_(cyl_neumann, _Tp)
_GLIBCXX_SIMD_MATH_CALL2_(ellint_1, _Tp)
_GLIBCXX_SIMD_MATH_CALL2_(ellint_2, _Tp)
_GLIBCXX_SIMD_MATH_CALL3_(ellint_3, _Tp, _Tp)
_GLIBCXX_SIMD_MATH_CALL_(expint)
template <typename _Tp, typename _Abi>
enable_if_t<is_floating_point_v<_Tp>, simd<_Tp, _Abi>>
hermite(const fixed_size_simd<unsigned, simd_size_v<_Tp, _Abi>>& __n,
const simd<_Tp, _Abi>& __x)
{
return simd<_Tp, _Abi>(
[&](auto __i) { return std::hermite(__n[__i], __x[__i]); });
}
template <typename _Tp, typename _Abi>
enable_if_t<is_floating_point_v<_Tp>, simd<_Tp, _Abi>>
laguerre(const fixed_size_simd<unsigned, simd_size_v<_Tp, _Abi>>& __n,
const simd<_Tp, _Abi>& __x)
{
return simd<_Tp, _Abi>(
[&](auto __i) { return std::laguerre(__n[__i], __x[__i]); });
}
template <typename _Tp, typename _Abi>
enable_if_t<is_floating_point_v<_Tp>, simd<_Tp, _Abi>>
legendre(const fixed_size_simd<unsigned, simd_size_v<_Tp, _Abi>>& __n,
const simd<_Tp, _Abi>& __x)
{
return simd<_Tp, _Abi>(
[&](auto __i) { return std::legendre(__n[__i], __x[__i]); });
}
_GLIBCXX_SIMD_MATH_CALL_(riemann_zeta)
template <typename _Tp, typename _Abi>
enable_if_t<is_floating_point_v<_Tp>, simd<_Tp, _Abi>>
sph_bessel(const fixed_size_simd<unsigned, simd_size_v<_Tp, _Abi>>& __n,
const simd<_Tp, _Abi>& __x)
{
return simd<_Tp, _Abi>(
[&](auto __i) { return std::sph_bessel(__n[__i], __x[__i]); });
}
template <typename _Tp, typename _Abi>
enable_if_t<is_floating_point_v<_Tp>, simd<_Tp, _Abi>>
sph_legendre(const fixed_size_simd<unsigned, simd_size_v<_Tp, _Abi>>& __l,
const fixed_size_simd<unsigned, simd_size_v<_Tp, _Abi>>& __m,
const simd<_Tp, _Abi>& theta)
{
return simd<_Tp, _Abi>([&](auto __i) {
return std::assoc_legendre(__l[__i], __m[__i], theta[__i]);
});
}
template <typename _Tp, typename _Abi>
enable_if_t<is_floating_point_v<_Tp>, simd<_Tp, _Abi>>
sph_neumann(const fixed_size_simd<unsigned, simd_size_v<_Tp, _Abi>>& __n,
const simd<_Tp, _Abi>& __x)
{
return simd<_Tp, _Abi>(
[&](auto __i) { return std::sph_neumann(__n[__i], __x[__i]); });
}
// }}}
#undef _GLIBCXX_SIMD_MATH_CALL_
#undef _GLIBCXX_SIMD_MATH_CALL2_
#undef _GLIBCXX_SIMD_MATH_CALL3_
_GLIBCXX_SIMD_END_NAMESPACE
#endif // __cplusplus >= 201703L
#endif // _GLIBCXX_EXPERIMENTAL_SIMD_MATH_H_
// vim: foldmethod=marker sw=2 ts=8 noet sts=2
iomanip
std::log10
complex< _Tp > log10(const complex< _Tp > &)
Return complex base 10 logarithm of z.
Definition: complex:829
std::sin
complex< _Tp > sin(const complex< _Tp > &)
Return complex sine of z.
Definition: complex:859
std::log
complex< _Tp > log(const complex< _Tp > &)
Return complex natural logarithm of z.
Definition: complex:824
std::tan
complex< _Tp > tan(const complex< _Tp > &)
Return complex tangent of z.
Definition: complex:960
std::abs
_Tp abs(const complex< _Tp > &)
Return magnitude of z.
Definition: complex:630
std::exp
complex< _Tp > exp(const complex< _Tp > &)
Return complex base e exponential of z.
Definition: complex:797
std::cosh
complex< _Tp > cosh(const complex< _Tp > &)
Return complex hyperbolic cosine of z.
Definition: complex:771
std::tanh
complex< _Tp > tanh(const complex< _Tp > &)
Return complex hyperbolic tangent of z.
Definition: complex:988
std::pow
complex< _Tp > pow(const complex< _Tp > &, int)
Return x to the y'th power.
Definition: complex:1019
std::sinh
complex< _Tp > sinh(const complex< _Tp > &)
Return complex hyperbolic sine of z.
Definition: complex:889
std::cos
complex< _Tp > cos(const complex< _Tp > &)
Return complex cosine of z.
Definition: complex:741
std::sqrt
complex< _Tp > sqrt(const complex< _Tp > &)
Return complex square root of z.
Definition: complex:933
std::make_unsigned_t
typename make_unsigned< _Tp >::type make_unsigned_t
Alias template for make_unsigned.
Definition: type_traits:1980
std::conditional_t
typename conditional< _Cond, _Iftrue, _Iffalse >::type conditional_t
Alias template for conditional.
Definition: type_traits:2589
std::declval
auto declval() noexcept -> decltype(__declval< _Tp >(0))
Definition: type_traits:2364
std::max
constexpr const _Tp & max(const _Tp &, const _Tp &)
This does what you think it does.
Definition: stl_algobase.h:254
std::min
constexpr const _Tp & min(const _Tp &, const _Tp &)
This does what you think it does.
Definition: stl_algobase.h:230
std::cyl_bessel_i
__gnu_cxx::__promote_2< _Tpnu, _Tp >::__type cyl_bessel_i(_Tpnu __nu, _Tp __x)
Definition: specfun.h:535
std::sph_neumann
__gnu_cxx::__promote< _Tp >::__type sph_neumann(unsigned int __n, _Tp __x)
Definition: specfun.h:1193
std::ellint_3
__gnu_cxx::__promote_3< _Tp, _Tpn, _Tpp >::__type ellint_3(_Tp __k, _Tpn __nu, _Tpp __phi)
Return the incomplete elliptic integral of the third kind .
Definition: specfun.h:830
std::comp_ellint_2
__gnu_cxx::__promote< _Tp >::__type comp_ellint_2(_Tp __k)
Definition: specfun.h:438
std::assoc_legendre
__gnu_cxx::__promote< _Tp >::__type assoc_legendre(unsigned int __l, unsigned int __m, _Tp __x)
Definition: specfun.h:298
std::assoc_laguerre
__gnu_cxx::__promote< _Tp >::__type assoc_laguerre(unsigned int __n, unsigned int __m, _Tp __x)
Definition: specfun.h:252
std::sph_bessel
__gnu_cxx::__promote< _Tp >::__type sph_bessel(unsigned int __n, _Tp __x)
Definition: specfun.h:1102
std::cyl_bessel_j
__gnu_cxx::__promote_2< _Tpnu, _Tp >::__type cyl_bessel_j(_Tpnu __nu, _Tp __x)
Definition: specfun.h:581
std::sph_legendre
__gnu_cxx::__promote< _Tp >::__type sph_legendre(unsigned int __l, unsigned int __m, _Tp __theta)
Definition: specfun.h:1149
std::cyl_neumann
__gnu_cxx::__promote_2< _Tpnu, _Tp >::__type cyl_neumann(_Tpnu __nu, _Tp __x)
Definition: specfun.h:681
std::riemann_zeta
__gnu_cxx::__promote< _Tp >::__type riemann_zeta(_Tp __s)
Definition: specfun.h:1058
std::beta
__gnu_cxx::__promote_2< _Tpa, _Tpb >::__type beta(_Tpa __a, _Tpb __b)
Definition: specfun.h:343
std::cyl_bessel_k
__gnu_cxx::__promote_2< _Tpnu, _Tp >::__type cyl_bessel_k(_Tpnu __nu, _Tp __x)
Definition: specfun.h:633
std::expint
__gnu_cxx::__promote< _Tp >::__type expint(_Tp __x)
Definition: specfun.h:870
std::hermite
__gnu_cxx::__promote< _Tp >::__type hermite(unsigned int __n, _Tp __x)
Definition: specfun.h:918
std::comp_ellint_1
__gnu_cxx::__promote< _Tp >::__type comp_ellint_1(_Tp __k)
Definition: specfun.h:391
std::laguerre
__gnu_cxx::__promote< _Tp >::__type laguerre(unsigned int __n, _Tp __x)
Definition: specfun.h:962
std::ellint_2
__gnu_cxx::__promote_2< _Tp, _Tpp >::__type ellint_2(_Tp __k, _Tpp __phi)
Definition: specfun.h:777
std::comp_ellint_3
__gnu_cxx::__promote_2< _Tp, _Tpn >::__type comp_ellint_3(_Tp __k, _Tpn __nu)
Definition: specfun.h:489
std::ellint_1
__gnu_cxx::__promote_2< _Tp, _Tpp >::__type ellint_1(_Tp __k, _Tpp __phi)
Definition: specfun.h:729
std::legendre
__gnu_cxx::__promote< _Tp >::__type legendre(unsigned int __l, _Tp __x)
Definition: specfun.h:1007
std
ISO C++ entities toplevel namespace is std.
std::fabs
_Tp fabs(const std::complex< _Tp > &)
fabs(__z) [8.1.8].
Definition: complex:1846
std::asinh
std::complex< _Tp > asinh(const std::complex< _Tp > &)
asinh(__z) [8.1.6].
Definition: complex:1793
std::atan
std::complex< _Tp > atan(const std::complex< _Tp > &)
atan(__z) [8.1.4].
Definition: complex:1718
std::size
constexpr auto size(const _Container &__cont) noexcept(noexcept(__cont.size())) -> decltype(__cont.size())
Return the size of a container.
Definition: range_access.h:245
std::atanh
std::complex< _Tp > atanh(const std::complex< _Tp > &)
atanh(__z) [8.1.7].
Definition: complex:1837
std::acosh
std::complex< _Tp > acosh(const std::complex< _Tp > &)
acosh(__z) [8.1.5].
Definition: complex:1754
std::acos
std::complex< _Tp > acos(const std::complex< _Tp > &)
acos(__z) [8.1.2].
Definition: complex:1638
std::asin
std::complex< _Tp > asin(const std::complex< _Tp > &)
asin(__z) [8.1.3].
Definition: complex:1674
utility