libcxx/include/cmath

// -*- C++ -*-
//===----------------------------------------------------------------------===//
//
// Part of the LLVM Project, under the Apache License v2.0 with LLVM Exceptions.
// See https://llvm.org/LICENSE.txt for license information.
// SPDX-License-Identifier: Apache-2.0 WITH LLVM-exception
//
//===----------------------------------------------------------------------===//

#ifndef _LIBCPP_CMATH
#define _LIBCPP_CMATH

/*
    cmath synopsis

Macros:

    HUGE_VAL
    HUGE_VALF               // C99
    HUGE_VALL               // C99
    INFINITY                // C99
    NAN                     // C99
    FP_INFINITE             // C99
    FP_NAN                  // C99
    FP_NORMAL               // C99
    FP_SUBNORMAL            // C99
    FP_ZERO                 // C99
    FP_FAST_FMA             // C99
    FP_FAST_FMAF            // C99
    FP_FAST_FMAL            // C99
    FP_ILOGB0               // C99
    FP_ILOGBNAN             // C99
    MATH_ERRNO              // C99
    MATH_ERREXCEPT          // C99
    math_errhandling        // C99

namespace std
{

Types:

    float_t                 // C99
    double_t                // C99

// C90

floating_point abs(floating_point x);

floating_point acos (arithmetic x);
float          acosf(float x);
long double    acosl(long double x);

floating_point asin (arithmetic x);
float          asinf(float x);
long double    asinl(long double x);

floating_point atan (arithmetic x);
float          atanf(float x);
long double    atanl(long double x);

floating_point atan2 (arithmetic y, arithmetic x);
float          atan2f(float y, float x);
long double    atan2l(long double y, long double x);

floating_point ceil (arithmetic x);
float          ceilf(float x);
long double    ceill(long double x);

floating_point cos (arithmetic x);
float          cosf(float x);
long double    cosl(long double x);

floating_point cosh (arithmetic x);
float          coshf(float x);
long double    coshl(long double x);

floating_point exp (arithmetic x);
float          expf(float x);
long double    expl(long double x);

floating_point fabs (arithmetic x);
float          fabsf(float x);
long double    fabsl(long double x);

floating_point floor (arithmetic x);
float          floorf(float x);
long double    floorl(long double x);

floating_point fmod (arithmetic x, arithmetic y);
float          fmodf(float x, float y);
long double    fmodl(long double x, long double y);

floating_point frexp (arithmetic value, int* exp);
float          frexpf(float value, int* exp);
long double    frexpl(long double value, int* exp);

floating_point ldexp (arithmetic value, int exp);
float          ldexpf(float value, int exp);
long double    ldexpl(long double value, int exp);

floating_point log (arithmetic x);
float          logf(float x);
long double    logl(long double x);

floating_point log10 (arithmetic x);
float          log10f(float x);
long double    log10l(long double x);

floating_point modf (floating_point value, floating_point* iptr);
float          modff(float value, float* iptr);
long double    modfl(long double value, long double* iptr);

floating_point pow (arithmetic x, arithmetic y);
float          powf(float x, float y);
long double    powl(long double x, long double y);

floating_point sin (arithmetic x);
float          sinf(float x);
long double    sinl(long double x);

floating_point sinh (arithmetic x);
float          sinhf(float x);
long double    sinhl(long double x);

floating_point sqrt (arithmetic x);
float          sqrtf(float x);
long double    sqrtl(long double x);

floating_point tan (arithmetic x);
float          tanf(float x);
long double    tanl(long double x);

floating_point tanh (arithmetic x);
float          tanhf(float x);
long double    tanhl(long double x);

//  C99

bool signbit(arithmetic x);

int fpclassify(arithmetic x);

bool isfinite(arithmetic x);
bool isinf(arithmetic x);
bool isnan(arithmetic x);
bool isnormal(arithmetic x);

bool isgreater(arithmetic x, arithmetic y);
bool isgreaterequal(arithmetic x, arithmetic y);
bool isless(arithmetic x, arithmetic y);
bool islessequal(arithmetic x, arithmetic y);
bool islessgreater(arithmetic x, arithmetic y);
bool isunordered(arithmetic x, arithmetic y);

floating_point acosh (arithmetic x);
float          acoshf(float x);
long double    acoshl(long double x);

floating_point asinh (arithmetic x);
float          asinhf(float x);
long double    asinhl(long double x);

floating_point atanh (arithmetic x);
float          atanhf(float x);
long double    atanhl(long double x);

floating_point cbrt (arithmetic x);
float          cbrtf(float x);
long double    cbrtl(long double x);

floating_point copysign (arithmetic x, arithmetic y);
float          copysignf(float x, float y);
long double    copysignl(long double x, long double y);

floating_point erf (arithmetic x);
float          erff(float x);
long double    erfl(long double x);

floating_point erfc (arithmetic x);
float          erfcf(float x);
long double    erfcl(long double x);

floating_point exp2 (arithmetic x);
float          exp2f(float x);
long double    exp2l(long double x);

floating_point expm1 (arithmetic x);
float          expm1f(float x);
long double    expm1l(long double x);

floating_point fdim (arithmetic x, arithmetic y);
float          fdimf(float x, float y);
long double    fdiml(long double x, long double y);

floating_point fma (arithmetic x, arithmetic y, arithmetic z);
float          fmaf(float x, float y, float z);
long double    fmal(long double x, long double y, long double z);

floating_point fmax (arithmetic x, arithmetic y);
float          fmaxf(float x, float y);
long double    fmaxl(long double x, long double y);

floating_point fmin (arithmetic x, arithmetic y);
float          fminf(float x, float y);
long double    fminl(long double x, long double y);

floating_point hypot (arithmetic x, arithmetic y);
float          hypotf(float x, float y);
long double    hypotl(long double x, long double y);

double       hypot(double x, double y, double z);                // C++17
float        hypot(float x, float y, float z);                   // C++17
long double  hypot(long double x, long double y, long double z); // C++17

int ilogb (arithmetic x);
int ilogbf(float x);
int ilogbl(long double x);

floating_point lgamma (arithmetic x);
float          lgammaf(float x);
long double    lgammal(long double x);

long long llrint (arithmetic x);
long long llrintf(float x);
long long llrintl(long double x);

long long llround (arithmetic x);
long long llroundf(float x);
long long llroundl(long double x);

floating_point log1p (arithmetic x);
float          log1pf(float x);
long double    log1pl(long double x);

floating_point log2 (arithmetic x);
float          log2f(float x);
long double    log2l(long double x);

floating_point logb (arithmetic x);
float          logbf(float x);
long double    logbl(long double x);

long lrint (arithmetic x);
long lrintf(float x);
long lrintl(long double x);

long lround (arithmetic x);
long lroundf(float x);
long lroundl(long double x);

double      nan (const char* str);
float       nanf(const char* str);
long double nanl(const char* str);

floating_point nearbyint (arithmetic x);
float          nearbyintf(float x);
long double    nearbyintl(long double x);

floating_point nextafter (arithmetic x, arithmetic y);
float          nextafterf(float x, float y);
long double    nextafterl(long double x, long double y);

floating_point nexttoward (arithmetic x, long double y);
float          nexttowardf(float x, long double y);
long double    nexttowardl(long double x, long double y);

floating_point remainder (arithmetic x, arithmetic y);
float          remainderf(float x, float y);
long double    remainderl(long double x, long double y);

floating_point remquo (arithmetic x, arithmetic y, int* pquo);
float          remquof(float x, float y, int* pquo);
long double    remquol(long double x, long double y, int* pquo);

floating_point rint (arithmetic x);
float          rintf(float x);
long double    rintl(long double x);

floating_point round (arithmetic x);
float          roundf(float x);
long double    roundl(long double x);

floating_point scalbln (arithmetic x, long ex);
float          scalblnf(float x, long ex);
long double    scalblnl(long double x, long ex);

floating_point scalbn (arithmetic x, int ex);
float          scalbnf(float x, int ex);
long double    scalbnl(long double x, int ex);

floating_point tgamma (arithmetic x);
float          tgammaf(float x);
long double    tgammal(long double x);

floating_point trunc (arithmetic x);
float          truncf(float x);
long double    truncl(long double x);

constexpr float       lerp(float a, float b, float t) noexcept;                   // C++20
constexpr double      lerp(double a, double b, double t) noexcept;                // C++20
constexpr long double lerp(long double a, long double b, long double t) noexcept; // C++20

}  // std

*/

#include <__assert> // all public C++ headers provide the assertion handler
#include <__config>
#include <__type_traits/enable_if.h>
#include <__type_traits/is_arithmetic.h>
#include <__type_traits/is_constant_evaluated.h>
#include <__type_traits/is_floating_point.h>
#include <__type_traits/is_same.h>
#include <__type_traits/promote.h>
#include <__type_traits/remove_cv.h>
#include <version>

#include <math.h>

#ifndef _LIBCPP_MATH_H
#   error <cmath> tried including <math.h> but didn't find libc++'s <math.h> header. \
          This usually means that your header search paths are not configured properly. \
          The header search paths should contain the C++ Standard Library headers before \
          any C Standard Library, and you are probably using compiler flags that make that \
          not be the case.
#endif

#if !defined(_LIBCPP_HAS_NO_PRAGMA_SYSTEM_HEADER)
#  pragma GCC system_header
#endif

_LIBCPP_PUSH_MACROS
#include <__undef_macros>

_LIBCPP_BEGIN_NAMESPACE_STD

using ::signbit _LIBCPP_USING_IF_EXISTS;
using ::fpclassify _LIBCPP_USING_IF_EXISTS;
using ::isfinite _LIBCPP_USING_IF_EXISTS;
using ::isinf _LIBCPP_USING_IF_EXISTS;
using ::isnan _LIBCPP_USING_IF_EXISTS;
using ::isnormal _LIBCPP_USING_IF_EXISTS;
using ::isgreater _LIBCPP_USING_IF_EXISTS;
using ::isgreaterequal _LIBCPP_USING_IF_EXISTS;
using ::isless _LIBCPP_USING_IF_EXISTS;
using ::islessequal _LIBCPP_USING_IF_EXISTS;
using ::islessgreater _LIBCPP_USING_IF_EXISTS;
using ::isunordered _LIBCPP_USING_IF_EXISTS;
using ::isunordered _LIBCPP_USING_IF_EXISTS;

using ::float_t _LIBCPP_USING_IF_EXISTS;
using ::double_t _LIBCPP_USING_IF_EXISTS;

using ::abs _LIBCPP_USING_IF_EXISTS;

using ::acos _LIBCPP_USING_IF_EXISTS;
using ::acosf _LIBCPP_USING_IF_EXISTS;
using ::asin _LIBCPP_USING_IF_EXISTS;
using ::asinf _LIBCPP_USING_IF_EXISTS;
using ::atan _LIBCPP_USING_IF_EXISTS;
using ::atanf _LIBCPP_USING_IF_EXISTS;
using ::atan2 _LIBCPP_USING_IF_EXISTS;
using ::atan2f _LIBCPP_USING_IF_EXISTS;
using ::ceil _LIBCPP_USING_IF_EXISTS;
using ::ceilf _LIBCPP_USING_IF_EXISTS;
using ::cos _LIBCPP_USING_IF_EXISTS;
using ::cosf _LIBCPP_USING_IF_EXISTS;
using ::cosh _LIBCPP_USING_IF_EXISTS;
using ::coshf _LIBCPP_USING_IF_EXISTS;

using ::exp _LIBCPP_USING_IF_EXISTS;
using ::expf _LIBCPP_USING_IF_EXISTS;

using ::fabs _LIBCPP_USING_IF_EXISTS;
using ::fabsf _LIBCPP_USING_IF_EXISTS;
using ::floor _LIBCPP_USING_IF_EXISTS;
using ::floorf _LIBCPP_USING_IF_EXISTS;

using ::fmod _LIBCPP_USING_IF_EXISTS;
using ::fmodf _LIBCPP_USING_IF_EXISTS;

using ::frexp _LIBCPP_USING_IF_EXISTS;
using ::frexpf _LIBCPP_USING_IF_EXISTS;
using ::ldexp _LIBCPP_USING_IF_EXISTS;
using ::ldexpf _LIBCPP_USING_IF_EXISTS;

using ::log _LIBCPP_USING_IF_EXISTS;
using ::logf _LIBCPP_USING_IF_EXISTS;

using ::log10 _LIBCPP_USING_IF_EXISTS;
using ::log10f _LIBCPP_USING_IF_EXISTS;
using ::modf _LIBCPP_USING_IF_EXISTS;
using ::modff _LIBCPP_USING_IF_EXISTS;

using ::pow _LIBCPP_USING_IF_EXISTS;
using ::powf _LIBCPP_USING_IF_EXISTS;

using ::sin _LIBCPP_USING_IF_EXISTS;
using ::sinf _LIBCPP_USING_IF_EXISTS;
using ::sinh _LIBCPP_USING_IF_EXISTS;
using ::sinhf _LIBCPP_USING_IF_EXISTS;

using ::sqrt _LIBCPP_USING_IF_EXISTS;
using ::sqrtf _LIBCPP_USING_IF_EXISTS;
using ::tan _LIBCPP_USING_IF_EXISTS;
using ::tanf _LIBCPP_USING_IF_EXISTS;

using ::tanh _LIBCPP_USING_IF_EXISTS;
using ::tanhf _LIBCPP_USING_IF_EXISTS;

using ::acosh _LIBCPP_USING_IF_EXISTS;
using ::acoshf _LIBCPP_USING_IF_EXISTS;
using ::asinh _LIBCPP_USING_IF_EXISTS;
using ::asinhf _LIBCPP_USING_IF_EXISTS;
using ::atanh _LIBCPP_USING_IF_EXISTS;
using ::atanhf _LIBCPP_USING_IF_EXISTS;
using ::cbrt _LIBCPP_USING_IF_EXISTS;
using ::cbrtf _LIBCPP_USING_IF_EXISTS;

using ::copysign _LIBCPP_USING_IF_EXISTS;
using ::copysignf _LIBCPP_USING_IF_EXISTS;

using ::erf _LIBCPP_USING_IF_EXISTS;
using ::erff _LIBCPP_USING_IF_EXISTS;
using ::erfc _LIBCPP_USING_IF_EXISTS;
using ::erfcf _LIBCPP_USING_IF_EXISTS;
using ::exp2 _LIBCPP_USING_IF_EXISTS;
using ::exp2f _LIBCPP_USING_IF_EXISTS;
using ::expm1 _LIBCPP_USING_IF_EXISTS;
using ::expm1f _LIBCPP_USING_IF_EXISTS;
using ::fdim _LIBCPP_USING_IF_EXISTS;
using ::fdimf _LIBCPP_USING_IF_EXISTS;
using ::fmaf _LIBCPP_USING_IF_EXISTS;
using ::fma _LIBCPP_USING_IF_EXISTS;
using ::fmax _LIBCPP_USING_IF_EXISTS;
using ::fmaxf _LIBCPP_USING_IF_EXISTS;
using ::fmin _LIBCPP_USING_IF_EXISTS;
using ::fminf _LIBCPP_USING_IF_EXISTS;
using ::hypot _LIBCPP_USING_IF_EXISTS;
using ::hypotf _LIBCPP_USING_IF_EXISTS;
using ::ilogb _LIBCPP_USING_IF_EXISTS;
using ::ilogbf _LIBCPP_USING_IF_EXISTS;
using ::lgamma _LIBCPP_USING_IF_EXISTS;
using ::lgammaf _LIBCPP_USING_IF_EXISTS;
using ::llrint _LIBCPP_USING_IF_EXISTS;
using ::llrintf _LIBCPP_USING_IF_EXISTS;
using ::llround _LIBCPP_USING_IF_EXISTS;
using ::llroundf _LIBCPP_USING_IF_EXISTS;
using ::log1p _LIBCPP_USING_IF_EXISTS;
using ::log1pf _LIBCPP_USING_IF_EXISTS;
using ::log2 _LIBCPP_USING_IF_EXISTS;
using ::log2f _LIBCPP_USING_IF_EXISTS;
using ::logb _LIBCPP_USING_IF_EXISTS;
using ::logbf _LIBCPP_USING_IF_EXISTS;
using ::lrint _LIBCPP_USING_IF_EXISTS;
using ::lrintf _LIBCPP_USING_IF_EXISTS;
using ::lround _LIBCPP_USING_IF_EXISTS;
using ::lroundf _LIBCPP_USING_IF_EXISTS;

using ::nan _LIBCPP_USING_IF_EXISTS;
using ::nanf _LIBCPP_USING_IF_EXISTS;

using ::nearbyint _LIBCPP_USING_IF_EXISTS;
using ::nearbyintf _LIBCPP_USING_IF_EXISTS;
using ::nextafter _LIBCPP_USING_IF_EXISTS;
using ::nextafterf _LIBCPP_USING_IF_EXISTS;
using ::nexttoward _LIBCPP_USING_IF_EXISTS;
using ::nexttowardf _LIBCPP_USING_IF_EXISTS;
using ::remainder _LIBCPP_USING_IF_EXISTS;
using ::remainderf _LIBCPP_USING_IF_EXISTS;
using ::remquo _LIBCPP_USING_IF_EXISTS;
using ::remquof _LIBCPP_USING_IF_EXISTS;
using ::rint _LIBCPP_USING_IF_EXISTS;
using ::rintf _LIBCPP_USING_IF_EXISTS;
using ::round _LIBCPP_USING_IF_EXISTS;
using ::roundf _LIBCPP_USING_IF_EXISTS;
using ::scalbln _LIBCPP_USING_IF_EXISTS;
using ::scalblnf _LIBCPP_USING_IF_EXISTS;
using ::scalbn _LIBCPP_USING_IF_EXISTS;
using ::scalbnf _LIBCPP_USING_IF_EXISTS;
using ::tgamma _LIBCPP_USING_IF_EXISTS;
using ::tgammaf _LIBCPP_USING_IF_EXISTS;
using ::trunc _LIBCPP_USING_IF_EXISTS;
using ::truncf _LIBCPP_USING_IF_EXISTS;

using ::acosl _LIBCPP_USING_IF_EXISTS;
using ::asinl _LIBCPP_USING_IF_EXISTS;
using ::atanl _LIBCPP_USING_IF_EXISTS;
using ::atan2l _LIBCPP_USING_IF_EXISTS;
using ::ceill _LIBCPP_USING_IF_EXISTS;
using ::cosl _LIBCPP_USING_IF_EXISTS;
using ::coshl _LIBCPP_USING_IF_EXISTS;
using ::expl _LIBCPP_USING_IF_EXISTS;
using ::fabsl _LIBCPP_USING_IF_EXISTS;
using ::floorl _LIBCPP_USING_IF_EXISTS;
using ::fmodl _LIBCPP_USING_IF_EXISTS;
using ::frexpl _LIBCPP_USING_IF_EXISTS;
using ::ldexpl _LIBCPP_USING_IF_EXISTS;
using ::logl _LIBCPP_USING_IF_EXISTS;
using ::log10l _LIBCPP_USING_IF_EXISTS;
using ::modfl _LIBCPP_USING_IF_EXISTS;
using ::powl _LIBCPP_USING_IF_EXISTS;
using ::sinl _LIBCPP_USING_IF_EXISTS;
using ::sinhl _LIBCPP_USING_IF_EXISTS;
using ::sqrtl _LIBCPP_USING_IF_EXISTS;
using ::tanl _LIBCPP_USING_IF_EXISTS;

using ::tanhl _LIBCPP_USING_IF_EXISTS;
using ::acoshl _LIBCPP_USING_IF_EXISTS;
using ::asinhl _LIBCPP_USING_IF_EXISTS;
using ::atanhl _LIBCPP_USING_IF_EXISTS;
using ::cbrtl _LIBCPP_USING_IF_EXISTS;

using ::copysignl _LIBCPP_USING_IF_EXISTS;

using ::erfl _LIBCPP_USING_IF_EXISTS;
using ::erfcl _LIBCPP_USING_IF_EXISTS;
using ::exp2l _LIBCPP_USING_IF_EXISTS;
using ::expm1l _LIBCPP_USING_IF_EXISTS;
using ::fdiml _LIBCPP_USING_IF_EXISTS;
using ::fmal _LIBCPP_USING_IF_EXISTS;
using ::fmaxl _LIBCPP_USING_IF_EXISTS;
using ::fminl _LIBCPP_USING_IF_EXISTS;
using ::hypotl _LIBCPP_USING_IF_EXISTS;
using ::ilogbl _LIBCPP_USING_IF_EXISTS;
using ::lgammal _LIBCPP_USING_IF_EXISTS;
using ::llrintl _LIBCPP_USING_IF_EXISTS;
using ::llroundl _LIBCPP_USING_IF_EXISTS;
using ::log1pl _LIBCPP_USING_IF_EXISTS;
using ::log2l _LIBCPP_USING_IF_EXISTS;
using ::logbl _LIBCPP_USING_IF_EXISTS;
using ::lrintl _LIBCPP_USING_IF_EXISTS;
using ::lroundl _LIBCPP_USING_IF_EXISTS;
using ::nanl _LIBCPP_USING_IF_EXISTS;
using ::nearbyintl _LIBCPP_USING_IF_EXISTS;
using ::nextafterl _LIBCPP_USING_IF_EXISTS;
using ::nexttowardl _LIBCPP_USING_IF_EXISTS;
using ::remainderl _LIBCPP_USING_IF_EXISTS;
using ::remquol _LIBCPP_USING_IF_EXISTS;
using ::rintl _LIBCPP_USING_IF_EXISTS;
using ::roundl _LIBCPP_USING_IF_EXISTS;
using ::scalblnl _LIBCPP_USING_IF_EXISTS;
using ::scalbnl _LIBCPP_USING_IF_EXISTS;
using ::tgammal _LIBCPP_USING_IF_EXISTS;
using ::truncl _LIBCPP_USING_IF_EXISTS;

#if _LIBCPP_STD_VER >= 17
inline _LIBCPP_INLINE_VISIBILITY float       hypot(       float __x,       float __y,       float __z ) { return sqrt(__x*__x + __y*__y + __z*__z); }
inline _LIBCPP_INLINE_VISIBILITY double      hypot(      double __x,      double __y,      double __z ) { return sqrt(__x*__x + __y*__y + __z*__z); }
inline _LIBCPP_INLINE_VISIBILITY long double hypot( long double __x, long double __y, long double __z ) { return sqrt(__x*__x + __y*__y + __z*__z); }

template <class _A1, class _A2, class _A3>
inline _LIBCPP_INLINE_VISIBILITY
typename enable_if_t
<
    is_arithmetic<_A1>::value &&
    is_arithmetic<_A2>::value &&
    is_arithmetic<_A3>::value,
    __promote<_A1, _A2, _A3>
>::type
hypot(_A1 __lcpp_x, _A2 __lcpp_y, _A3 __lcpp_z) _NOEXCEPT
{
    typedef typename __promote<_A1, _A2, _A3>::type __result_type;
    static_assert((!(is_same<_A1, __result_type>::value &&
                     is_same<_A2, __result_type>::value &&
                     is_same<_A3, __result_type>::value)), "");
    return std::hypot((__result_type)__lcpp_x, (__result_type)__lcpp_y, (__result_type)__lcpp_z);
}
#endif

template <class _A1>
_LIBCPP_INLINE_VISIBILITY
_LIBCPP_CONSTEXPR typename enable_if<is_floating_point<_A1>::value, bool>::type
__constexpr_isnan(_A1 __lcpp_x) _NOEXCEPT
{
#if __has_builtin(__builtin_isnan)
    return __builtin_isnan(__lcpp_x);
#else
    return isnan(__lcpp_x);
#endif
}

template <class _A1>
_LIBCPP_INLINE_VISIBILITY
_LIBCPP_CONSTEXPR typename enable_if<!is_floating_point<_A1>::value, bool>::type
__constexpr_isnan(_A1 __lcpp_x) _NOEXCEPT
{
    return std::isnan(__lcpp_x);
}

template <class _A1>
_LIBCPP_INLINE_VISIBILITY
_LIBCPP_CONSTEXPR typename enable_if<is_floating_point<_A1>::value, bool>::type
__constexpr_isinf(_A1 __lcpp_x) _NOEXCEPT
{
#if __has_builtin(__builtin_isinf)
    return __builtin_isinf(__lcpp_x);
#else
    return isinf(__lcpp_x);
#endif
}

template <class _A1>
_LIBCPP_INLINE_VISIBILITY
_LIBCPP_CONSTEXPR typename enable_if<!is_floating_point<_A1>::value, bool>::type
__constexpr_isinf(_A1 __lcpp_x) _NOEXCEPT
{
    return std::isinf(__lcpp_x);
}

template <class _A1>
_LIBCPP_INLINE_VISIBILITY
_LIBCPP_CONSTEXPR typename enable_if<is_floating_point<_A1>::value, bool>::type
__constexpr_isfinite(_A1 __lcpp_x) _NOEXCEPT
{
#if __has_builtin(__builtin_isfinite)
    return __builtin_isfinite(__lcpp_x);
#else
    return isfinite(__lcpp_x);
#endif
}

template <class _A1>
_LIBCPP_INLINE_VISIBILITY
_LIBCPP_CONSTEXPR typename enable_if<!is_floating_point<_A1>::value, bool>::type
__constexpr_isfinite(_A1 __lcpp_x) _NOEXCEPT
{
    return __builtin_isfinite(__lcpp_x);
}

_LIBCPP_CONSTEXPR inline _LIBCPP_HIDE_FROM_ABI float __constexpr_copysign(float __x, float __y) _NOEXCEPT {
    return __builtin_copysignf(__x, __y);
}

_LIBCPP_CONSTEXPR inline _LIBCPP_HIDE_FROM_ABI double __constexpr_copysign(double __x, double __y) _NOEXCEPT {
    return __builtin_copysign(__x, __y);
}

_LIBCPP_CONSTEXPR inline _LIBCPP_HIDE_FROM_ABI long double
__constexpr_copysign(long double __x, long double __y) _NOEXCEPT {
    return __builtin_copysignl(__x, __y);
}

template <class _A1, class _A2>
_LIBCPP_CONSTEXPR inline _LIBCPP_HIDE_FROM_ABI
    typename std::__enable_if_t<std::is_arithmetic<_A1>::value && std::is_arithmetic<_A2>::value,
                                std::__promote<_A1, _A2> >::type
    __constexpr_copysign(_A1 __x, _A2 __y) _NOEXCEPT {
    typedef typename std::__promote<_A1, _A2>::type __result_type;
    static_assert((!(std::_IsSame<_A1, __result_type>::value && std::_IsSame<_A2, __result_type>::value)), "");
    return __builtin_copysign((__result_type)__x, (__result_type)__y);
}

inline _LIBCPP_HIDE_FROM_ABI _LIBCPP_CONSTEXPR float __constexpr_fabs(float __x) _NOEXCEPT {
    return __builtin_fabsf(__x);
}

inline _LIBCPP_HIDE_FROM_ABI _LIBCPP_CONSTEXPR double __constexpr_fabs(double __x) _NOEXCEPT {
    return __builtin_fabs(__x);
}

inline _LIBCPP_HIDE_FROM_ABI _LIBCPP_CONSTEXPR long double __constexpr_fabs(long double __x) _NOEXCEPT {
    return __builtin_fabsl(__x);
}

template <class _Tp, __enable_if_t<is_integral<_Tp>::value, int> = 0>
_LIBCPP_HIDE_FROM_ABI _LIBCPP_CONSTEXPR double __constexpr_fabs(_Tp __x) _NOEXCEPT {
    return __builtin_fabs(static_cast<double>(__x));
}

inline _LIBCPP_HIDE_FROM_ABI _LIBCPP_CONSTEXPR_SINCE_CXX14 float __constexpr_fmax(float __x, float __y) _NOEXCEPT {
#if !__has_constexpr_builtin(__builtin_fmaxf)
  if (__libcpp_is_constant_evaluated()) {
    if (std::__constexpr_isnan(__x))
      return __y;
    if (std::__constexpr_isnan(__y))
      return __x;
    return __x < __y ? __y : __x;
  }
#endif
  return __builtin_fmaxf(__x, __y);
}

inline _LIBCPP_HIDE_FROM_ABI _LIBCPP_CONSTEXPR_SINCE_CXX14 double __constexpr_fmax(double __x, double __y) _NOEXCEPT {
#if !__has_constexpr_builtin(__builtin_fmax)
  if (__libcpp_is_constant_evaluated()) {
    if (std::__constexpr_isnan(__x))
      return __y;
    if (std::__constexpr_isnan(__y))
      return __x;
    return __x < __y ? __y : __x;
  }
#endif
  return __builtin_fmax(__x, __y);
}

inline _LIBCPP_HIDE_FROM_ABI _LIBCPP_CONSTEXPR_SINCE_CXX14 long double
__constexpr_fmax(long double __x, long double __y) _NOEXCEPT {
#if !__has_constexpr_builtin(__builtin_fmaxl)
  if (__libcpp_is_constant_evaluated()) {
    if (std::__constexpr_isnan(__x))
      return __y;
    if (std::__constexpr_isnan(__y))
      return __x;
    return __x < __y ? __y : __x;
  }
#endif
  return __builtin_fmaxl(__x, __y);
}

template <class _Tp, class _Up, __enable_if_t<is_arithmetic<_Tp>::value && is_arithmetic<_Up>::value, int> = 0>
_LIBCPP_HIDE_FROM_ABI _LIBCPP_CONSTEXPR_SINCE_CXX14 typename __promote<_Tp, _Up>::type
__constexpr_fmax(_Tp __x, _Up __y) _NOEXCEPT {
  using __result_type = typename __promote<_Tp, _Up>::type;
  return std::__constexpr_fmax(static_cast<__result_type>(__x), static_cast<__result_type>(__y));
}

template <class _Tp>
_LIBCPP_HIDE_FROM_ABI _LIBCPP_CONSTEXPR_SINCE_CXX14 _Tp __constexpr_logb(_Tp __x) {
#if !__has_constexpr_builtin(__builtin_logb)
  if (__libcpp_is_constant_evaluated()) {
    if (__x == _Tp(0)) {
      // raise FE_DIVBYZERO
      return -numeric_limits<_Tp>::infinity();
    }

    if (std::__constexpr_isinf(__x))
      return numeric_limits<_Tp>::infinity();

    if (std::__constexpr_isnan(__x))
      return numeric_limits<_Tp>::quiet_NaN();

    __x = std::__constexpr_fabs(__x);
    unsigned long long __exp = 0;
    while (__x >= numeric_limits<_Tp>::radix) {
      __x /= numeric_limits<_Tp>::radix;
      __exp += 1;
    }
    return _Tp(__exp);
  }
#endif // !__has_constexpr_builtin(__builtin_logb)
  return __builtin_logb(__x);
}

template <class _Tp>
_LIBCPP_HIDE_FROM_ABI _LIBCPP_CONSTEXPR_SINCE_CXX20 _Tp __constexpr_scalbn(_Tp __x, int __exp) {
#if !__has_constexpr_builtin(__builtin_scalbln)
  if (__libcpp_is_constant_evaluated()) {
    if (__x == _Tp(0))
      return __x;

    if (std::__constexpr_isinf(__x))
      return __x;

    if (__exp == _Tp(0))
      return __x;

    if (std::__constexpr_isnan(__x))
      return numeric_limits<_Tp>::quiet_NaN();

    _Tp __mult(1);
    if (__exp > 0) {
      __mult = numeric_limits<_Tp>::radix;
      --__exp;
    } else {
      ++__exp;
      __exp = -__exp;
      __mult /= numeric_limits<_Tp>::radix;
    }

    while (__exp > 0) {
      if (!(__exp & 1)) {
        __mult *= __mult;
        __exp >>= 1;
      } else {
        __x *= __mult;
        --__exp;
      }
    }
    return __x;
  }
#endif // !__has_constexpr_builtin(__builtin_scalbln)
  return __builtin_scalbn(__x, __exp);
}

#if _LIBCPP_STD_VER >= 20
template <typename _Fp>
_LIBCPP_HIDE_FROM_ABI constexpr
_Fp __lerp(_Fp __a, _Fp __b, _Fp __t) noexcept {
    if ((__a <= 0 && __b >= 0) || (__a >= 0 && __b <= 0))
        return __t * __b + (1 - __t) * __a;

    if (__t == 1) return __b;
    const _Fp __x = __a + __t * (__b - __a);
    if ((__t > 1) == (__b > __a))
        return __b < __x ? __x : __b;
    else
        return __x < __b ? __x : __b;
}

_LIBCPP_HIDE_FROM_ABI constexpr float
lerp(float __a, float __b, float __t)                   _NOEXCEPT { return __lerp(__a, __b, __t); }

_LIBCPP_HIDE_FROM_ABI constexpr double
lerp(double __a, double __b, double __t)                _NOEXCEPT { return __lerp(__a, __b, __t); }

_LIBCPP_HIDE_FROM_ABI constexpr long double
lerp(long double __a, long double __b, long double __t) _NOEXCEPT { return __lerp(__a, __b, __t); }

template <class _A1, class _A2, class _A3>
inline _LIBCPP_HIDE_FROM_ABI
constexpr typename enable_if_t
<
    is_arithmetic<_A1>::value &&
    is_arithmetic<_A2>::value &&
    is_arithmetic<_A3>::value,
    __promote<_A1, _A2, _A3>
>::type
lerp(_A1 __a, _A2 __b, _A3 __t) noexcept
{
    typedef typename __promote<_A1, _A2, _A3>::type __result_type;
    static_assert(!(_IsSame<_A1, __result_type>::value &&
                    _IsSame<_A2, __result_type>::value &&
                    _IsSame<_A3, __result_type>::value));
    return std::__lerp((__result_type)__a, (__result_type)__b, (__result_type)__t);
}
#endif // _LIBCPP_STD_VER >= 20

_LIBCPP_END_NAMESPACE_STD

_LIBCPP_POP_MACROS

#if !defined(_LIBCPP_REMOVE_TRANSITIVE_INCLUDES) && _LIBCPP_STD_VER <= 20
#  include <type_traits>
#endif

#endif // _LIBCPP_CMATH