// -*- 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_LIMITS
#define _LIBCPP_LIMITS

/*
    limits synopsis

namespace std
{

template<class T>
class numeric_limits
{
public:
    static constexpr bool is_specialized = false;
    static constexpr T min() noexcept;
    static constexpr T max() noexcept;
    static constexpr T lowest() noexcept;

    static constexpr int  digits = 0;
    static constexpr int  digits10 = 0;
    static constexpr int  max_digits10 = 0;
    static constexpr bool is_signed = false;
    static constexpr bool is_integer = false;
    static constexpr bool is_exact = false;
    static constexpr int  radix = 0;
    static constexpr T epsilon() noexcept;
    static constexpr T round_error() noexcept;

    static constexpr int  min_exponent = 0;
    static constexpr int  min_exponent10 = 0;
    static constexpr int  max_exponent = 0;
    static constexpr int  max_exponent10 = 0;

    static constexpr bool has_infinity = false;
    static constexpr bool has_quiet_NaN = false;
    static constexpr bool has_signaling_NaN = false;
    static constexpr float_denorm_style has_denorm = denorm_absent;
    static constexpr bool has_denorm_loss = false;
    static constexpr T infinity() noexcept;
    static constexpr T quiet_NaN() noexcept;
    static constexpr T signaling_NaN() noexcept;
    static constexpr T denorm_min() noexcept;

    static constexpr bool is_iec559 = false;
    static constexpr bool is_bounded = false;
    static constexpr bool is_modulo = false;

    static constexpr bool traps = false;
    static constexpr bool tinyness_before = false;
    static constexpr float_round_style round_style = round_toward_zero;
};

enum float_round_style
{
    round_indeterminate       = -1,
    round_toward_zero         =  0,
    round_to_nearest          =  1,
    round_toward_infinity     =  2,
    round_toward_neg_infinity =  3
};

enum float_denorm_style
{
    denorm_indeterminate = -1,
    denorm_absent = 0,
    denorm_present = 1
};

template<> class numeric_limits<cv bool>;

template<> class numeric_limits<cv char>;
template<> class numeric_limits<cv signed char>;
template<> class numeric_limits<cv unsigned char>;
template<> class numeric_limits<cv wchar_t>;
template<> class numeric_limits<cv char8_t>; // C++20
template<> class numeric_limits<cv char16_t>;
template<> class numeric_limits<cv char32_t>;

template<> class numeric_limits<cv short>;
template<> class numeric_limits<cv int>;
template<> class numeric_limits<cv long>;
template<> class numeric_limits<cv long long>;
template<> class numeric_limits<cv unsigned short>;
template<> class numeric_limits<cv unsigned int>;
template<> class numeric_limits<cv unsigned long>;
template<> class numeric_limits<cv unsigned long long>;

template<> class numeric_limits<cv float>;
template<> class numeric_limits<cv double>;
template<> class numeric_limits<cv long double>;

}  // std

*/

#include <__assert> // all public C++ headers provide the assertion handler
#include <__config>
#include <__type_traits/is_arithmetic.h>
#include <__type_traits/is_signed.h>
#include <__type_traits/remove_cv.h>

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

_LIBCPP_PUSH_MACROS
#include <__undef_macros>
#include <version>


_LIBCPP_BEGIN_NAMESPACE_STD

enum float_round_style
{
    round_indeterminate       = -1,
    round_toward_zero         =  0,
    round_to_nearest          =  1,
    round_toward_infinity     =  2,
    round_toward_neg_infinity =  3
};

enum float_denorm_style
{
    denorm_indeterminate = -1,
    denorm_absent = 0,
    denorm_present = 1
};

template <class _Tp, bool = is_arithmetic<_Tp>::value>
class __libcpp_numeric_limits
{
protected:
    typedef _Tp type;

    static _LIBCPP_CONSTEXPR const  bool is_specialized = false;
    _LIBCPP_INLINE_VISIBILITY static _LIBCPP_CONSTEXPR type min() _NOEXCEPT {return type();}
    _LIBCPP_INLINE_VISIBILITY static _LIBCPP_CONSTEXPR type max() _NOEXCEPT {return type();}
    _LIBCPP_INLINE_VISIBILITY static _LIBCPP_CONSTEXPR type lowest() _NOEXCEPT {return type();}

    static _LIBCPP_CONSTEXPR const int  digits = 0;
    static _LIBCPP_CONSTEXPR const int  digits10 = 0;
    static _LIBCPP_CONSTEXPR const int  max_digits10 = 0;
    static _LIBCPP_CONSTEXPR const bool is_signed = false;
    static _LIBCPP_CONSTEXPR const bool is_integer = false;
    static _LIBCPP_CONSTEXPR const bool is_exact = false;
    static _LIBCPP_CONSTEXPR const int  radix = 0;
    _LIBCPP_INLINE_VISIBILITY static _LIBCPP_CONSTEXPR type epsilon() _NOEXCEPT {return type();}
    _LIBCPP_INLINE_VISIBILITY static _LIBCPP_CONSTEXPR type round_error() _NOEXCEPT {return type();}

    static _LIBCPP_CONSTEXPR const int  min_exponent = 0;
    static _LIBCPP_CONSTEXPR const int  min_exponent10 = 0;
    static _LIBCPP_CONSTEXPR const int  max_exponent = 0;
    static _LIBCPP_CONSTEXPR const int  max_exponent10 = 0;

    static _LIBCPP_CONSTEXPR const bool has_infinity = false;
    static _LIBCPP_CONSTEXPR const bool has_quiet_NaN = false;
    static _LIBCPP_CONSTEXPR const bool has_signaling_NaN = false;
    static _LIBCPP_CONSTEXPR const float_denorm_style has_denorm = denorm_absent;
    static _LIBCPP_CONSTEXPR const bool has_denorm_loss = false;
    _LIBCPP_INLINE_VISIBILITY static _LIBCPP_CONSTEXPR type infinity() _NOEXCEPT {return type();}
    _LIBCPP_INLINE_VISIBILITY static _LIBCPP_CONSTEXPR type quiet_NaN() _NOEXCEPT {return type();}
    _LIBCPP_INLINE_VISIBILITY static _LIBCPP_CONSTEXPR type signaling_NaN() _NOEXCEPT {return type();}
    _LIBCPP_INLINE_VISIBILITY static _LIBCPP_CONSTEXPR type denorm_min() _NOEXCEPT {return type();}

    static _LIBCPP_CONSTEXPR const bool is_iec559 = false;
    static _LIBCPP_CONSTEXPR const bool is_bounded = false;
    static _LIBCPP_CONSTEXPR const bool is_modulo = false;

    static _LIBCPP_CONSTEXPR const bool traps = false;
    static _LIBCPP_CONSTEXPR const bool tinyness_before = false;
    static _LIBCPP_CONSTEXPR const float_round_style round_style = round_toward_zero;
};

template <class _Tp, int __digits, bool _IsSigned>
struct __libcpp_compute_min
{
    static _LIBCPP_CONSTEXPR const _Tp value = _Tp(_Tp(1) << __digits);
};

template <class _Tp, int __digits>
struct __libcpp_compute_min<_Tp, __digits, false>
{
    static _LIBCPP_CONSTEXPR const _Tp value = _Tp(0);
};

template <class _Tp>
class __libcpp_numeric_limits<_Tp, true>
{
protected:
    typedef _Tp type;

    static _LIBCPP_CONSTEXPR const bool is_specialized = true;

    static _LIBCPP_CONSTEXPR const bool is_signed = type(-1) < type(0);
    static _LIBCPP_CONSTEXPR const int  digits = static_cast<int>(sizeof(type) * __CHAR_BIT__ - is_signed);
    static _LIBCPP_CONSTEXPR const int  digits10 = digits * 3 / 10;
    static _LIBCPP_CONSTEXPR const int  max_digits10 = 0;
    static _LIBCPP_CONSTEXPR const type __min = __libcpp_compute_min<type, digits, is_signed>::value;
    static _LIBCPP_CONSTEXPR const type __max = is_signed ? type(type(~0) ^ __min) : type(~0);
    _LIBCPP_INLINE_VISIBILITY static _LIBCPP_CONSTEXPR type min() _NOEXCEPT {return __min;}
    _LIBCPP_INLINE_VISIBILITY static _LIBCPP_CONSTEXPR type max() _NOEXCEPT {return __max;}
    _LIBCPP_INLINE_VISIBILITY static _LIBCPP_CONSTEXPR type lowest() _NOEXCEPT {return min();}

    static _LIBCPP_CONSTEXPR const bool is_integer = true;
    static _LIBCPP_CONSTEXPR const bool is_exact = true;
    static _LIBCPP_CONSTEXPR const int  radix = 2;
    _LIBCPP_INLINE_VISIBILITY static _LIBCPP_CONSTEXPR type epsilon() _NOEXCEPT {return type(0);}
    _LIBCPP_INLINE_VISIBILITY static _LIBCPP_CONSTEXPR type round_error() _NOEXCEPT {return type(0);}

    static _LIBCPP_CONSTEXPR const int  min_exponent = 0;
    static _LIBCPP_CONSTEXPR const int  min_exponent10 = 0;
    static _LIBCPP_CONSTEXPR const int  max_exponent = 0;
    static _LIBCPP_CONSTEXPR const int  max_exponent10 = 0;

    static _LIBCPP_CONSTEXPR const bool has_infinity = false;
    static _LIBCPP_CONSTEXPR const bool has_quiet_NaN = false;
    static _LIBCPP_CONSTEXPR const bool has_signaling_NaN = false;
    static _LIBCPP_CONSTEXPR const float_denorm_style has_denorm = denorm_absent;
    static _LIBCPP_CONSTEXPR const bool has_denorm_loss = false;
    _LIBCPP_INLINE_VISIBILITY static _LIBCPP_CONSTEXPR type infinity() _NOEXCEPT {return type(0);}
    _LIBCPP_INLINE_VISIBILITY static _LIBCPP_CONSTEXPR type quiet_NaN() _NOEXCEPT {return type(0);}
    _LIBCPP_INLINE_VISIBILITY static _LIBCPP_CONSTEXPR type signaling_NaN() _NOEXCEPT {return type(0);}
    _LIBCPP_INLINE_VISIBILITY static _LIBCPP_CONSTEXPR type denorm_min() _NOEXCEPT {return type(0);}

    static _LIBCPP_CONSTEXPR const bool is_iec559 = false;
    static _LIBCPP_CONSTEXPR const bool is_bounded = true;
    static _LIBCPP_CONSTEXPR const bool is_modulo = !_VSTD::is_signed<_Tp>::value;

#if defined(__i386__) || defined(__x86_64__) || defined(__pnacl__) || \
    defined(__wasm__)
    static _LIBCPP_CONSTEXPR const bool traps = true;
#else
    static _LIBCPP_CONSTEXPR const bool traps = false;
#endif
    static _LIBCPP_CONSTEXPR const bool tinyness_before = false;
    static _LIBCPP_CONSTEXPR const float_round_style round_style = round_toward_zero;
};

template <>
class __libcpp_numeric_limits<bool, true>
{
protected:
    typedef bool type;

    static _LIBCPP_CONSTEXPR const bool is_specialized = true;

    static _LIBCPP_CONSTEXPR const bool is_signed = false;
    static _LIBCPP_CONSTEXPR const int  digits = 1;
    static _LIBCPP_CONSTEXPR const int  digits10 = 0;
    static _LIBCPP_CONSTEXPR const int  max_digits10 = 0;
    static _LIBCPP_CONSTEXPR const type __min = false;
    static _LIBCPP_CONSTEXPR const type __max = true;
    _LIBCPP_INLINE_VISIBILITY static _LIBCPP_CONSTEXPR type min() _NOEXCEPT {return __min;}
    _LIBCPP_INLINE_VISIBILITY static _LIBCPP_CONSTEXPR type max() _NOEXCEPT {return __max;}
    _LIBCPP_INLINE_VISIBILITY static _LIBCPP_CONSTEXPR type lowest() _NOEXCEPT {return min();}

    static _LIBCPP_CONSTEXPR const bool is_integer = true;
    static _LIBCPP_CONSTEXPR const bool is_exact = true;
    static _LIBCPP_CONSTEXPR const int  radix = 2;
    _LIBCPP_INLINE_VISIBILITY static _LIBCPP_CONSTEXPR type epsilon() _NOEXCEPT {return type(0);}
    _LIBCPP_INLINE_VISIBILITY static _LIBCPP_CONSTEXPR type round_error() _NOEXCEPT {return type(0);}

    static _LIBCPP_CONSTEXPR const int  min_exponent = 0;
    static _LIBCPP_CONSTEXPR const int  min_exponent10 = 0;
    static _LIBCPP_CONSTEXPR const int  max_exponent = 0;
    static _LIBCPP_CONSTEXPR const int  max_exponent10 = 0;

    static _LIBCPP_CONSTEXPR const bool has_infinity = false;
    static _LIBCPP_CONSTEXPR const bool has_quiet_NaN = false;
    static _LIBCPP_CONSTEXPR const bool has_signaling_NaN = false;
    static _LIBCPP_CONSTEXPR const float_denorm_style has_denorm = denorm_absent;
    static _LIBCPP_CONSTEXPR const bool has_denorm_loss = false;
    _LIBCPP_INLINE_VISIBILITY static _LIBCPP_CONSTEXPR type infinity() _NOEXCEPT {return type(0);}
    _LIBCPP_INLINE_VISIBILITY static _LIBCPP_CONSTEXPR type quiet_NaN() _NOEXCEPT {return type(0);}
    _LIBCPP_INLINE_VISIBILITY static _LIBCPP_CONSTEXPR type signaling_NaN() _NOEXCEPT {return type(0);}
    _LIBCPP_INLINE_VISIBILITY static _LIBCPP_CONSTEXPR type denorm_min() _NOEXCEPT {return type(0);}

    static _LIBCPP_CONSTEXPR const bool is_iec559 = false;
    static _LIBCPP_CONSTEXPR const bool is_bounded = true;
    static _LIBCPP_CONSTEXPR const bool is_modulo = false;

    static _LIBCPP_CONSTEXPR const bool traps = false;
    static _LIBCPP_CONSTEXPR const bool tinyness_before = false;
    static _LIBCPP_CONSTEXPR const float_round_style round_style = round_toward_zero;
};

template <>
class __libcpp_numeric_limits<float, true>
{
protected:
    typedef float type;

    static _LIBCPP_CONSTEXPR const bool is_specialized = true;

    static _LIBCPP_CONSTEXPR const bool is_signed = true;
    static _LIBCPP_CONSTEXPR const int  digits = __FLT_MANT_DIG__;
    static _LIBCPP_CONSTEXPR const int  digits10 = __FLT_DIG__;
    static _LIBCPP_CONSTEXPR const int  max_digits10 = 2+(digits * 30103l)/100000l;
    _LIBCPP_INLINE_VISIBILITY static _LIBCPP_CONSTEXPR type min() _NOEXCEPT {return __FLT_MIN__;}
    _LIBCPP_INLINE_VISIBILITY static _LIBCPP_CONSTEXPR type max() _NOEXCEPT {return __FLT_MAX__;}
    _LIBCPP_INLINE_VISIBILITY static _LIBCPP_CONSTEXPR type lowest() _NOEXCEPT {return -max();}

    static _LIBCPP_CONSTEXPR const bool is_integer = false;
    static _LIBCPP_CONSTEXPR const bool is_exact = false;
    static _LIBCPP_CONSTEXPR const int  radix = __FLT_RADIX__;
    _LIBCPP_INLINE_VISIBILITY static _LIBCPP_CONSTEXPR type epsilon() _NOEXCEPT {return __FLT_EPSILON__;}
    _LIBCPP_INLINE_VISIBILITY static _LIBCPP_CONSTEXPR type round_error() _NOEXCEPT {return 0.5F;}

    static _LIBCPP_CONSTEXPR const int  min_exponent = __FLT_MIN_EXP__;
    static _LIBCPP_CONSTEXPR const int  min_exponent10 = __FLT_MIN_10_EXP__;
    static _LIBCPP_CONSTEXPR const int  max_exponent = __FLT_MAX_EXP__;
    static _LIBCPP_CONSTEXPR const int  max_exponent10 = __FLT_MAX_10_EXP__;

    static _LIBCPP_CONSTEXPR const bool has_infinity = true;
    static _LIBCPP_CONSTEXPR const bool has_quiet_NaN = true;
    static _LIBCPP_CONSTEXPR const bool has_signaling_NaN = true;
    static _LIBCPP_CONSTEXPR const float_denorm_style has_denorm = denorm_present;
    static _LIBCPP_CONSTEXPR const bool has_denorm_loss = false;
    _LIBCPP_INLINE_VISIBILITY static _LIBCPP_CONSTEXPR type infinity() _NOEXCEPT {return __builtin_huge_valf();}
    _LIBCPP_INLINE_VISIBILITY static _LIBCPP_CONSTEXPR type quiet_NaN() _NOEXCEPT {return __builtin_nanf("");}
    _LIBCPP_INLINE_VISIBILITY static _LIBCPP_CONSTEXPR type signaling_NaN() _NOEXCEPT {return __builtin_nansf("");}
    _LIBCPP_INLINE_VISIBILITY static _LIBCPP_CONSTEXPR type denorm_min() _NOEXCEPT {return __FLT_DENORM_MIN__;}

    static _LIBCPP_CONSTEXPR const bool is_iec559 = true;
    static _LIBCPP_CONSTEXPR const bool is_bounded = true;
    static _LIBCPP_CONSTEXPR const bool is_modulo = false;

    static _LIBCPP_CONSTEXPR const bool traps = false;
#if (defined(__arm__) || defined(__aarch64__))
    static _LIBCPP_CONSTEXPR const bool tinyness_before = true;
#else
    static _LIBCPP_CONSTEXPR const bool tinyness_before = false;
#endif
    static _LIBCPP_CONSTEXPR const float_round_style round_style = round_to_nearest;
};

template <>
class __libcpp_numeric_limits<double, true>
{
protected:
    typedef double type;

    static _LIBCPP_CONSTEXPR const bool is_specialized = true;

    static _LIBCPP_CONSTEXPR const bool is_signed = true;
    static _LIBCPP_CONSTEXPR const int  digits = __DBL_MANT_DIG__;
    static _LIBCPP_CONSTEXPR const int  digits10 = __DBL_DIG__;
    static _LIBCPP_CONSTEXPR const int  max_digits10 = 2+(digits * 30103l)/100000l;
    _LIBCPP_INLINE_VISIBILITY static _LIBCPP_CONSTEXPR type min() _NOEXCEPT {return __DBL_MIN__;}
    _LIBCPP_INLINE_VISIBILITY static _LIBCPP_CONSTEXPR type max() _NOEXCEPT {return __DBL_MAX__;}
    _LIBCPP_INLINE_VISIBILITY static _LIBCPP_CONSTEXPR type lowest() _NOEXCEPT {return -max();}

    static _LIBCPP_CONSTEXPR const bool is_integer = false;
    static _LIBCPP_CONSTEXPR const bool is_exact = false;
    static _LIBCPP_CONSTEXPR const int  radix = __FLT_RADIX__;
    _LIBCPP_INLINE_VISIBILITY static _LIBCPP_CONSTEXPR type epsilon() _NOEXCEPT {return __DBL_EPSILON__;}
    _LIBCPP_INLINE_VISIBILITY static _LIBCPP_CONSTEXPR type round_error() _NOEXCEPT {return 0.5;}

    static _LIBCPP_CONSTEXPR const int  min_exponent = __DBL_MIN_EXP__;
    static _LIBCPP_CONSTEXPR const int  min_exponent10 = __DBL_MIN_10_EXP__;
    static _LIBCPP_CONSTEXPR const int  max_exponent = __DBL_MAX_EXP__;
    static _LIBCPP_CONSTEXPR const int  max_exponent10 = __DBL_MAX_10_EXP__;

    static _LIBCPP_CONSTEXPR const bool has_infinity = true;
    static _LIBCPP_CONSTEXPR const bool has_quiet_NaN = true;
    static _LIBCPP_CONSTEXPR const bool has_signaling_NaN = true;
    static _LIBCPP_CONSTEXPR const float_denorm_style has_denorm = denorm_present;
    static _LIBCPP_CONSTEXPR const bool has_denorm_loss = false;
    _LIBCPP_INLINE_VISIBILITY static _LIBCPP_CONSTEXPR type infinity() _NOEXCEPT {return __builtin_huge_val();}
    _LIBCPP_INLINE_VISIBILITY static _LIBCPP_CONSTEXPR type quiet_NaN() _NOEXCEPT {return __builtin_nan("");}
    _LIBCPP_INLINE_VISIBILITY static _LIBCPP_CONSTEXPR type signaling_NaN() _NOEXCEPT {return __builtin_nans("");}
    _LIBCPP_INLINE_VISIBILITY static _LIBCPP_CONSTEXPR type denorm_min() _NOEXCEPT {return __DBL_DENORM_MIN__;}

    static _LIBCPP_CONSTEXPR const bool is_iec559 = true;
    static _LIBCPP_CONSTEXPR const bool is_bounded = true;
    static _LIBCPP_CONSTEXPR const bool is_modulo = false;

    static _LIBCPP_CONSTEXPR const bool traps = false;
#if (defined(__arm__) || defined(__aarch64__))
    static _LIBCPP_CONSTEXPR const bool tinyness_before = true;
#else
    static _LIBCPP_CONSTEXPR const bool tinyness_before = false;
#endif
    static _LIBCPP_CONSTEXPR const float_round_style round_style = round_to_nearest;
};

template <>
class __libcpp_numeric_limits<long double, true>
{
protected:
    typedef long double type;

    static _LIBCPP_CONSTEXPR const bool is_specialized = true;

    static _LIBCPP_CONSTEXPR const bool is_signed = true;
    static _LIBCPP_CONSTEXPR const int  digits = __LDBL_MANT_DIG__;
    static _LIBCPP_CONSTEXPR const int  digits10 = __LDBL_DIG__;
    static _LIBCPP_CONSTEXPR const int  max_digits10 = 2+(digits * 30103l)/100000l;
    _LIBCPP_INLINE_VISIBILITY static _LIBCPP_CONSTEXPR type min() _NOEXCEPT {return __LDBL_MIN__;}
    _LIBCPP_INLINE_VISIBILITY static _LIBCPP_CONSTEXPR type max() _NOEXCEPT {return __LDBL_MAX__;}
    _LIBCPP_INLINE_VISIBILITY static _LIBCPP_CONSTEXPR type lowest() _NOEXCEPT {return -max();}

    static _LIBCPP_CONSTEXPR const bool is_integer = false;
    static _LIBCPP_CONSTEXPR const bool is_exact = false;
    static _LIBCPP_CONSTEXPR const int  radix = __FLT_RADIX__;
    _LIBCPP_INLINE_VISIBILITY static _LIBCPP_CONSTEXPR type epsilon() _NOEXCEPT {return __LDBL_EPSILON__;}
    _LIBCPP_INLINE_VISIBILITY static _LIBCPP_CONSTEXPR type round_error() _NOEXCEPT {return 0.5L;}

    static _LIBCPP_CONSTEXPR const int  min_exponent = __LDBL_MIN_EXP__;
    static _LIBCPP_CONSTEXPR const int  min_exponent10 = __LDBL_MIN_10_EXP__;
    static _LIBCPP_CONSTEXPR const int  max_exponent = __LDBL_MAX_EXP__;
    static _LIBCPP_CONSTEXPR const int  max_exponent10 = __LDBL_MAX_10_EXP__;

    static _LIBCPP_CONSTEXPR const bool has_infinity = true;
    static _LIBCPP_CONSTEXPR const bool has_quiet_NaN = true;
    static _LIBCPP_CONSTEXPR const bool has_signaling_NaN = true;
    static _LIBCPP_CONSTEXPR const float_denorm_style has_denorm = denorm_present;
    static _LIBCPP_CONSTEXPR const bool has_denorm_loss = false;
    _LIBCPP_INLINE_VISIBILITY static _LIBCPP_CONSTEXPR type infinity() _NOEXCEPT {return __builtin_huge_vall();}
    _LIBCPP_INLINE_VISIBILITY static _LIBCPP_CONSTEXPR type quiet_NaN() _NOEXCEPT {return __builtin_nanl("");}
    _LIBCPP_INLINE_VISIBILITY static _LIBCPP_CONSTEXPR type signaling_NaN() _NOEXCEPT {return __builtin_nansl("");}
    _LIBCPP_INLINE_VISIBILITY static _LIBCPP_CONSTEXPR type denorm_min() _NOEXCEPT {return __LDBL_DENORM_MIN__;}

#if defined(__powerpc__) && defined(__LONG_DOUBLE_IBM128__)
    static _LIBCPP_CONSTEXPR const bool is_iec559 = false;
#else
    static _LIBCPP_CONSTEXPR const bool is_iec559 = true;
#endif
    static _LIBCPP_CONSTEXPR const bool is_bounded = true;
    static _LIBCPP_CONSTEXPR const bool is_modulo = false;

    static _LIBCPP_CONSTEXPR const bool traps = false;
#if (defined(__arm__) || defined(__aarch64__))
    static _LIBCPP_CONSTEXPR const bool tinyness_before = true;
#else
    static _LIBCPP_CONSTEXPR const bool tinyness_before = false;
#endif
    static _LIBCPP_CONSTEXPR const float_round_style round_style = round_to_nearest;
};

template <class _Tp>
class _LIBCPP_TEMPLATE_VIS numeric_limits
    : private __libcpp_numeric_limits<__remove_cv_t<_Tp> >
{
    typedef __libcpp_numeric_limits<__remove_cv_t<_Tp> > __base;
    typedef typename __base::type type;
public:
    static _LIBCPP_CONSTEXPR const bool is_specialized = __base::is_specialized;
    _LIBCPP_INLINE_VISIBILITY static _LIBCPP_CONSTEXPR type min() _NOEXCEPT {return __base::min();}
    _LIBCPP_INLINE_VISIBILITY static _LIBCPP_CONSTEXPR type max() _NOEXCEPT {return __base::max();}
    _LIBCPP_INLINE_VISIBILITY static _LIBCPP_CONSTEXPR type lowest() _NOEXCEPT {return __base::lowest();}

    static _LIBCPP_CONSTEXPR const int  digits = __base::digits;
    static _LIBCPP_CONSTEXPR const int  digits10 = __base::digits10;
    static _LIBCPP_CONSTEXPR const int  max_digits10 = __base::max_digits10;
    static _LIBCPP_CONSTEXPR const bool is_signed = __base::is_signed;
    static _LIBCPP_CONSTEXPR const bool is_integer = __base::is_integer;
    static _LIBCPP_CONSTEXPR const bool is_exact = __base::is_exact;
    static _LIBCPP_CONSTEXPR const int  radix = __base::radix;
    _LIBCPP_INLINE_VISIBILITY static _LIBCPP_CONSTEXPR type epsilon() _NOEXCEPT {return __base::epsilon();}
    _LIBCPP_INLINE_VISIBILITY static _LIBCPP_CONSTEXPR type round_error() _NOEXCEPT {return __base::round_error();}

    static _LIBCPP_CONSTEXPR const int  min_exponent = __base::min_exponent;
    static _LIBCPP_CONSTEXPR const int  min_exponent10 = __base::min_exponent10;
    static _LIBCPP_CONSTEXPR const int  max_exponent = __base::max_exponent;
    static _LIBCPP_CONSTEXPR const int  max_exponent10 = __base::max_exponent10;

    static _LIBCPP_CONSTEXPR const bool has_infinity = __base::has_infinity;
    static _LIBCPP_CONSTEXPR const bool has_quiet_NaN = __base::has_quiet_NaN;
    static _LIBCPP_CONSTEXPR const bool has_signaling_NaN = __base::has_signaling_NaN;
    static _LIBCPP_CONSTEXPR const float_denorm_style has_denorm = __base::has_denorm;
    static _LIBCPP_CONSTEXPR const bool has_denorm_loss = __base::has_denorm_loss;
    _LIBCPP_INLINE_VISIBILITY static _LIBCPP_CONSTEXPR type infinity() _NOEXCEPT {return __base::infinity();}
    _LIBCPP_INLINE_VISIBILITY static _LIBCPP_CONSTEXPR type quiet_NaN() _NOEXCEPT {return __base::quiet_NaN();}
    _LIBCPP_INLINE_VISIBILITY static _LIBCPP_CONSTEXPR type signaling_NaN() _NOEXCEPT {return __base::signaling_NaN();}
    _LIBCPP_INLINE_VISIBILITY static _LIBCPP_CONSTEXPR type denorm_min() _NOEXCEPT {return __base::denorm_min();}

    static _LIBCPP_CONSTEXPR const bool is_iec559 = __base::is_iec559;
    static _LIBCPP_CONSTEXPR const bool is_bounded = __base::is_bounded;
    static _LIBCPP_CONSTEXPR const bool is_modulo = __base::is_modulo;

    static _LIBCPP_CONSTEXPR const bool traps = __base::traps;
    static _LIBCPP_CONSTEXPR const bool tinyness_before = __base::tinyness_before;
    static _LIBCPP_CONSTEXPR const float_round_style round_style = __base::round_style;
};

template <class _Tp>
    _LIBCPP_CONSTEXPR const bool numeric_limits<_Tp>::is_specialized;
template <class _Tp>
    _LIBCPP_CONSTEXPR const int numeric_limits<_Tp>::digits;
template <class _Tp>
    _LIBCPP_CONSTEXPR const int numeric_limits<_Tp>::digits10;
template <class _Tp>
    _LIBCPP_CONSTEXPR const int numeric_limits<_Tp>::max_digits10;
template <class _Tp>
    _LIBCPP_CONSTEXPR const bool numeric_limits<_Tp>::is_signed;
template <class _Tp>
    _LIBCPP_CONSTEXPR const bool numeric_limits<_Tp>::is_integer;
template <class _Tp>
    _LIBCPP_CONSTEXPR const bool numeric_limits<_Tp>::is_exact;
template <class _Tp>
    _LIBCPP_CONSTEXPR const int numeric_limits<_Tp>::radix;
template <class _Tp>
    _LIBCPP_CONSTEXPR const int numeric_limits<_Tp>::min_exponent;
template <class _Tp>
    _LIBCPP_CONSTEXPR const int numeric_limits<_Tp>::min_exponent10;
template <class _Tp>
    _LIBCPP_CONSTEXPR const int numeric_limits<_Tp>::max_exponent;
template <class _Tp>
    _LIBCPP_CONSTEXPR const int numeric_limits<_Tp>::max_exponent10;
template <class _Tp>
    _LIBCPP_CONSTEXPR const bool numeric_limits<_Tp>::has_infinity;
template <class _Tp>
    _LIBCPP_CONSTEXPR const bool numeric_limits<_Tp>::has_quiet_NaN;
template <class _Tp>
    _LIBCPP_CONSTEXPR const bool numeric_limits<_Tp>::has_signaling_NaN;
template <class _Tp>
    _LIBCPP_CONSTEXPR const float_denorm_style numeric_limits<_Tp>::has_denorm;
template <class _Tp>
    _LIBCPP_CONSTEXPR const bool numeric_limits<_Tp>::has_denorm_loss;
template <class _Tp>
    _LIBCPP_CONSTEXPR const bool numeric_limits<_Tp>::is_iec559;
template <class _Tp>
    _LIBCPP_CONSTEXPR const bool numeric_limits<_Tp>::is_bounded;
template <class _Tp>
    _LIBCPP_CONSTEXPR const bool numeric_limits<_Tp>::is_modulo;
template <class _Tp>
    _LIBCPP_CONSTEXPR const bool numeric_limits<_Tp>::traps;
template <class _Tp>
    _LIBCPP_CONSTEXPR const bool numeric_limits<_Tp>::tinyness_before;
template <class _Tp>
    _LIBCPP_CONSTEXPR const float_round_style numeric_limits<_Tp>::round_style;

template <class _Tp>
class _LIBCPP_TEMPLATE_VIS numeric_limits<const _Tp>
    : private numeric_limits<_Tp>
{
    typedef numeric_limits<_Tp> __base;
    typedef _Tp type;
public:
    static _LIBCPP_CONSTEXPR const bool is_specialized = __base::is_specialized;
    _LIBCPP_INLINE_VISIBILITY static _LIBCPP_CONSTEXPR type min() _NOEXCEPT {return __base::min();}
    _LIBCPP_INLINE_VISIBILITY static _LIBCPP_CONSTEXPR type max() _NOEXCEPT {return __base::max();}
    _LIBCPP_INLINE_VISIBILITY static _LIBCPP_CONSTEXPR type lowest() _NOEXCEPT {return __base::lowest();}

    static _LIBCPP_CONSTEXPR const int  digits = __base::digits;
    static _LIBCPP_CONSTEXPR const int  digits10 = __base::digits10;
    static _LIBCPP_CONSTEXPR const int  max_digits10 = __base::max_digits10;
    static _LIBCPP_CONSTEXPR const bool is_signed = __base::is_signed;
    static _LIBCPP_CONSTEXPR const bool is_integer = __base::is_integer;
    static _LIBCPP_CONSTEXPR const bool is_exact = __base::is_exact;
    static _LIBCPP_CONSTEXPR const int  radix = __base::radix;
    _LIBCPP_INLINE_VISIBILITY static _LIBCPP_CONSTEXPR type epsilon() _NOEXCEPT {return __base::epsilon();}
    _LIBCPP_INLINE_VISIBILITY static _LIBCPP_CONSTEXPR type round_error() _NOEXCEPT {return __base::round_error();}

    static _LIBCPP_CONSTEXPR const int  min_exponent = __base::min_exponent;
    static _LIBCPP_CONSTEXPR const int  min_exponent10 = __base::min_exponent10;
    static _LIBCPP_CONSTEXPR const int  max_exponent = __base::max_exponent;
    static _LIBCPP_CONSTEXPR const int  max_exponent10 = __base::max_exponent10;

    static _LIBCPP_CONSTEXPR const bool has_infinity = __base::has_infinity;
    static _LIBCPP_CONSTEXPR const bool has_quiet_NaN = __base::has_quiet_NaN;
    static _LIBCPP_CONSTEXPR const bool has_signaling_NaN = __base::has_signaling_NaN;
    static _LIBCPP_CONSTEXPR const float_denorm_style has_denorm = __base::has_denorm;
    static _LIBCPP_CONSTEXPR const bool has_denorm_loss = __base::has_denorm_loss;
    _LIBCPP_INLINE_VISIBILITY static _LIBCPP_CONSTEXPR type infinity() _NOEXCEPT {return __base::infinity();}
    _LIBCPP_INLINE_VISIBILITY static _LIBCPP_CONSTEXPR type quiet_NaN() _NOEXCEPT {return __base::quiet_NaN();}
    _LIBCPP_INLINE_VISIBILITY static _LIBCPP_CONSTEXPR type signaling_NaN() _NOEXCEPT {return __base::signaling_NaN();}
    _LIBCPP_INLINE_VISIBILITY static _LIBCPP_CONSTEXPR type denorm_min() _NOEXCEPT {return __base::denorm_min();}

    static _LIBCPP_CONSTEXPR const bool is_iec559 = __base::is_iec559;
    static _LIBCPP_CONSTEXPR const bool is_bounded = __base::is_bounded;
    static _LIBCPP_CONSTEXPR const bool is_modulo = __base::is_modulo;

    static _LIBCPP_CONSTEXPR const bool traps = __base::traps;
    static _LIBCPP_CONSTEXPR const bool tinyness_before = __base::tinyness_before;
    static _LIBCPP_CONSTEXPR const float_round_style round_style = __base::round_style;
};

template <class _Tp>
    _LIBCPP_CONSTEXPR const bool numeric_limits<const _Tp>::is_specialized;
template <class _Tp>
    _LIBCPP_CONSTEXPR const int numeric_limits<const _Tp>::digits;
template <class _Tp>
    _LIBCPP_CONSTEXPR const int numeric_limits<const _Tp>::digits10;
template <class _Tp>
    _LIBCPP_CONSTEXPR const int numeric_limits<const _Tp>::max_digits10;
template <class _Tp>
    _LIBCPP_CONSTEXPR const bool numeric_limits<const _Tp>::is_signed;
template <class _Tp>
    _LIBCPP_CONSTEXPR const bool numeric_limits<const _Tp>::is_integer;
template <class _Tp>
    _LIBCPP_CONSTEXPR const bool numeric_limits<const _Tp>::is_exact;
template <class _Tp>
    _LIBCPP_CONSTEXPR const int numeric_limits<const _Tp>::radix;
template <class _Tp>
    _LIBCPP_CONSTEXPR const int numeric_limits<const _Tp>::min_exponent;
template <class _Tp>
    _LIBCPP_CONSTEXPR const int numeric_limits<const _Tp>::min_exponent10;
template <class _Tp>
    _LIBCPP_CONSTEXPR const int numeric_limits<const _Tp>::max_exponent;
template <class _Tp>
    _LIBCPP_CONSTEXPR const int numeric_limits<const _Tp>::max_exponent10;
template <class _Tp>
    _LIBCPP_CONSTEXPR const bool numeric_limits<const _Tp>::has_infinity;
template <class _Tp>
    _LIBCPP_CONSTEXPR const bool numeric_limits<const _Tp>::has_quiet_NaN;
template <class _Tp>
    _LIBCPP_CONSTEXPR const bool numeric_limits<const _Tp>::has_signaling_NaN;
template <class _Tp>
    _LIBCPP_CONSTEXPR const float_denorm_style numeric_limits<const _Tp>::has_denorm;
template <class _Tp>
    _LIBCPP_CONSTEXPR const bool numeric_limits<const _Tp>::has_denorm_loss;
template <class _Tp>
    _LIBCPP_CONSTEXPR const bool numeric_limits<const _Tp>::is_iec559;
template <class _Tp>
    _LIBCPP_CONSTEXPR const bool numeric_limits<const _Tp>::is_bounded;
template <class _Tp>
    _LIBCPP_CONSTEXPR const bool numeric_limits<const _Tp>::is_modulo;
template <class _Tp>
    _LIBCPP_CONSTEXPR const bool numeric_limits<const _Tp>::traps;
template <class _Tp>
    _LIBCPP_CONSTEXPR const bool numeric_limits<const _Tp>::tinyness_before;
template <class _Tp>
    _LIBCPP_CONSTEXPR const float_round_style numeric_limits<const _Tp>::round_style;

template <class _Tp>
class _LIBCPP_TEMPLATE_VIS numeric_limits<volatile _Tp>
    : private numeric_limits<_Tp>
{
    typedef numeric_limits<_Tp> __base;
    typedef _Tp type;
public:
    static _LIBCPP_CONSTEXPR const bool is_specialized = __base::is_specialized;
    _LIBCPP_INLINE_VISIBILITY static _LIBCPP_CONSTEXPR type min() _NOEXCEPT {return __base::min();}
    _LIBCPP_INLINE_VISIBILITY static _LIBCPP_CONSTEXPR type max() _NOEXCEPT {return __base::max();}
    _LIBCPP_INLINE_VISIBILITY static _LIBCPP_CONSTEXPR type lowest() _NOEXCEPT {return __base::lowest();}

    static _LIBCPP_CONSTEXPR const int  digits = __base::digits;
    static _LIBCPP_CONSTEXPR const int  digits10 = __base::digits10;
    static _LIBCPP_CONSTEXPR const int  max_digits10 = __base::max_digits10;
    static _LIBCPP_CONSTEXPR const bool is_signed = __base::is_signed;
    static _LIBCPP_CONSTEXPR const bool is_integer = __base::is_integer;
    static _LIBCPP_CONSTEXPR const bool is_exact = __base::is_exact;
    static _LIBCPP_CONSTEXPR const int  radix = __base::radix;
    _LIBCPP_INLINE_VISIBILITY static _LIBCPP_CONSTEXPR type epsilon() _NOEXCEPT {return __base::epsilon();}
    _LIBCPP_INLINE_VISIBILITY static _LIBCPP_CONSTEXPR type round_error() _NOEXCEPT {return __base::round_error();}

    static _LIBCPP_CONSTEXPR const int  min_exponent = __base::min_exponent;
    static _LIBCPP_CONSTEXPR const int  min_exponent10 = __base::min_exponent10;
    static _LIBCPP_CONSTEXPR const int  max_exponent = __base::max_exponent;
    static _LIBCPP_CONSTEXPR const int  max_exponent10 = __base::max_exponent10;

    static _LIBCPP_CONSTEXPR const bool has_infinity = __base::has_infinity;
    static _LIBCPP_CONSTEXPR const bool has_quiet_NaN = __base::has_quiet_NaN;
    static _LIBCPP_CONSTEXPR const bool has_signaling_NaN = __base::has_signaling_NaN;
    static _LIBCPP_CONSTEXPR const float_denorm_style has_denorm = __base::has_denorm;
    static _LIBCPP_CONSTEXPR const bool has_denorm_loss = __base::has_denorm_loss;
    _LIBCPP_INLINE_VISIBILITY static _LIBCPP_CONSTEXPR type infinity() _NOEXCEPT {return __base::infinity();}
    _LIBCPP_INLINE_VISIBILITY static _LIBCPP_CONSTEXPR type quiet_NaN() _NOEXCEPT {return __base::quiet_NaN();}
    _LIBCPP_INLINE_VISIBILITY static _LIBCPP_CONSTEXPR type signaling_NaN() _NOEXCEPT {return __base::signaling_NaN();}
    _LIBCPP_INLINE_VISIBILITY static _LIBCPP_CONSTEXPR type denorm_min() _NOEXCEPT {return __base::denorm_min();}

    static _LIBCPP_CONSTEXPR const bool is_iec559 = __base::is_iec559;
    static _LIBCPP_CONSTEXPR const bool is_bounded = __base::is_bounded;
    static _LIBCPP_CONSTEXPR const bool is_modulo = __base::is_modulo;

    static _LIBCPP_CONSTEXPR const bool traps = __base::traps;
    static _LIBCPP_CONSTEXPR const bool tinyness_before = __base::tinyness_before;
    static _LIBCPP_CONSTEXPR const float_round_style round_style = __base::round_style;
};

template <class _Tp>
    _LIBCPP_CONSTEXPR const bool numeric_limits<volatile _Tp>::is_specialized;
template <class _Tp>
    _LIBCPP_CONSTEXPR const int numeric_limits<volatile _Tp>::digits;
template <class _Tp>
    _LIBCPP_CONSTEXPR const int numeric_limits<volatile _Tp>::digits10;
template <class _Tp>
    _LIBCPP_CONSTEXPR const int numeric_limits<volatile _Tp>::max_digits10;
template <class _Tp>
    _LIBCPP_CONSTEXPR const bool numeric_limits<volatile _Tp>::is_signed;
template <class _Tp>
    _LIBCPP_CONSTEXPR const bool numeric_limits<volatile _Tp>::is_integer;
template <class _Tp>
    _LIBCPP_CONSTEXPR const bool numeric_limits<volatile _Tp>::is_exact;
template <class _Tp>
    _LIBCPP_CONSTEXPR const int numeric_limits<volatile _Tp>::radix;
template <class _Tp>
    _LIBCPP_CONSTEXPR const int numeric_limits<volatile _Tp>::min_exponent;
template <class _Tp>
    _LIBCPP_CONSTEXPR const int numeric_limits<volatile _Tp>::min_exponent10;
template <class _Tp>
    _LIBCPP_CONSTEXPR const int numeric_limits<volatile _Tp>::max_exponent;
template <class _Tp>
    _LIBCPP_CONSTEXPR const int numeric_limits<volatile _Tp>::max_exponent10;
template <class _Tp>
    _LIBCPP_CONSTEXPR const bool numeric_limits<volatile _Tp>::has_infinity;
template <class _Tp>
    _LIBCPP_CONSTEXPR const bool numeric_limits<volatile _Tp>::has_quiet_NaN;
template <class _Tp>
    _LIBCPP_CONSTEXPR const bool numeric_limits<volatile _Tp>::has_signaling_NaN;
template <class _Tp>
    _LIBCPP_CONSTEXPR const float_denorm_style numeric_limits<volatile _Tp>::has_denorm;
template <class _Tp>
    _LIBCPP_CONSTEXPR const bool numeric_limits<volatile _Tp>::has_denorm_loss;
template <class _Tp>
    _LIBCPP_CONSTEXPR const bool numeric_limits<volatile _Tp>::is_iec559;
template <class _Tp>
    _LIBCPP_CONSTEXPR const bool numeric_limits<volatile _Tp>::is_bounded;
template <class _Tp>
    _LIBCPP_CONSTEXPR const bool numeric_limits<volatile _Tp>::is_modulo;
template <class _Tp>
    _LIBCPP_CONSTEXPR const bool numeric_limits<volatile _Tp>::traps;
template <class _Tp>
    _LIBCPP_CONSTEXPR const bool numeric_limits<volatile _Tp>::tinyness_before;
template <class _Tp>
    _LIBCPP_CONSTEXPR const float_round_style numeric_limits<volatile _Tp>::round_style;

template <class _Tp>
class _LIBCPP_TEMPLATE_VIS numeric_limits<const volatile _Tp>
    : private numeric_limits<_Tp>
{
    typedef numeric_limits<_Tp> __base;
    typedef _Tp type;
public:
    static _LIBCPP_CONSTEXPR const bool is_specialized = __base::is_specialized;
    _LIBCPP_INLINE_VISIBILITY static _LIBCPP_CONSTEXPR type min() _NOEXCEPT {return __base::min();}
    _LIBCPP_INLINE_VISIBILITY static _LIBCPP_CONSTEXPR type max() _NOEXCEPT {return __base::max();}
    _LIBCPP_INLINE_VISIBILITY static _LIBCPP_CONSTEXPR type lowest() _NOEXCEPT {return __base::lowest();}

    static _LIBCPP_CONSTEXPR const int  digits = __base::digits;
    static _LIBCPP_CONSTEXPR const int  digits10 = __base::digits10;
    static _LIBCPP_CONSTEXPR const int  max_digits10 = __base::max_digits10;
    static _LIBCPP_CONSTEXPR const bool is_signed = __base::is_signed;
    static _LIBCPP_CONSTEXPR const bool is_integer = __base::is_integer;
    static _LIBCPP_CONSTEXPR const bool is_exact = __base::is_exact;
    static _LIBCPP_CONSTEXPR const int  radix = __base::radix;
    _LIBCPP_INLINE_VISIBILITY static _LIBCPP_CONSTEXPR type epsilon() _NOEXCEPT {return __base::epsilon();}
    _LIBCPP_INLINE_VISIBILITY static _LIBCPP_CONSTEXPR type round_error() _NOEXCEPT {return __base::round_error();}

    static _LIBCPP_CONSTEXPR const int  min_exponent = __base::min_exponent;
    static _LIBCPP_CONSTEXPR const int  min_exponent10 = __base::min_exponent10;
    static _LIBCPP_CONSTEXPR const int  max_exponent = __base::max_exponent;
    static _LIBCPP_CONSTEXPR const int  max_exponent10 = __base::max_exponent10;

    static _LIBCPP_CONSTEXPR const bool has_infinity = __base::has_infinity;
    static _LIBCPP_CONSTEXPR const bool has_quiet_NaN = __base::has_quiet_NaN;
    static _LIBCPP_CONSTEXPR const bool has_signaling_NaN = __base::has_signaling_NaN;
    static _LIBCPP_CONSTEXPR const float_denorm_style has_denorm = __base::has_denorm;
    static _LIBCPP_CONSTEXPR const bool has_denorm_loss = __base::has_denorm_loss;
    _LIBCPP_INLINE_VISIBILITY static _LIBCPP_CONSTEXPR type infinity() _NOEXCEPT {return __base::infinity();}
    _LIBCPP_INLINE_VISIBILITY static _LIBCPP_CONSTEXPR type quiet_NaN() _NOEXCEPT {return __base::quiet_NaN();}
    _LIBCPP_INLINE_VISIBILITY static _LIBCPP_CONSTEXPR type signaling_NaN() _NOEXCEPT {return __base::signaling_NaN();}
    _LIBCPP_INLINE_VISIBILITY static _LIBCPP_CONSTEXPR type denorm_min() _NOEXCEPT {return __base::denorm_min();}

    static _LIBCPP_CONSTEXPR const bool is_iec559 = __base::is_iec559;
    static _LIBCPP_CONSTEXPR const bool is_bounded = __base::is_bounded;
    static _LIBCPP_CONSTEXPR const bool is_modulo = __base::is_modulo;

    static _LIBCPP_CONSTEXPR const bool traps = __base::traps;
    static _LIBCPP_CONSTEXPR const bool tinyness_before = __base::tinyness_before;
    static _LIBCPP_CONSTEXPR const float_round_style round_style = __base::round_style;
};

template <class _Tp>
    _LIBCPP_CONSTEXPR const bool numeric_limits<const volatile _Tp>::is_specialized;
template <class _Tp>
    _LIBCPP_CONSTEXPR const int numeric_limits<const volatile _Tp>::digits;
template <class _Tp>
    _LIBCPP_CONSTEXPR const int numeric_limits<const volatile _Tp>::digits10;
template <class _Tp>
    _LIBCPP_CONSTEXPR const int numeric_limits<const volatile _Tp>::max_digits10;
template <class _Tp>
    _LIBCPP_CONSTEXPR const bool numeric_limits<const volatile _Tp>::is_signed;
template <class _Tp>
    _LIBCPP_CONSTEXPR const bool numeric_limits<const volatile _Tp>::is_integer;
template <class _Tp>
    _LIBCPP_CONSTEXPR const bool numeric_limits<const volatile _Tp>::is_exact;
template <class _Tp>
    _LIBCPP_CONSTEXPR const int numeric_limits<const volatile _Tp>::radix;
template <class _Tp>
    _LIBCPP_CONSTEXPR const int numeric_limits<const volatile _Tp>::min_exponent;
template <class _Tp>
    _LIBCPP_CONSTEXPR const int numeric_limits<const volatile _Tp>::min_exponent10;
template <class _Tp>
    _LIBCPP_CONSTEXPR const int numeric_limits<const volatile _Tp>::max_exponent;
template <class _Tp>
    _LIBCPP_CONSTEXPR const int numeric_limits<const volatile _Tp>::max_exponent10;
template <class _Tp>
    _LIBCPP_CONSTEXPR const bool numeric_limits<const volatile _Tp>::has_infinity;
template <class _Tp>
    _LIBCPP_CONSTEXPR const bool numeric_limits<const volatile _Tp>::has_quiet_NaN;
template <class _Tp>
    _LIBCPP_CONSTEXPR const bool numeric_limits<const volatile _Tp>::has_signaling_NaN;
template <class _Tp>
    _LIBCPP_CONSTEXPR const float_denorm_style numeric_limits<const volatile _Tp>::has_denorm;
template <class _Tp>
    _LIBCPP_CONSTEXPR const bool numeric_limits<const volatile _Tp>::has_denorm_loss;
template <class _Tp>
    _LIBCPP_CONSTEXPR const bool numeric_limits<const volatile _Tp>::is_iec559;
template <class _Tp>
    _LIBCPP_CONSTEXPR const bool numeric_limits<const volatile _Tp>::is_bounded;
template <class _Tp>
    _LIBCPP_CONSTEXPR const bool numeric_limits<const volatile _Tp>::is_modulo;
template <class _Tp>
    _LIBCPP_CONSTEXPR const bool numeric_limits<const volatile _Tp>::traps;
template <class _Tp>
    _LIBCPP_CONSTEXPR const bool numeric_limits<const volatile _Tp>::tinyness_before;
template <class _Tp>
    _LIBCPP_CONSTEXPR const float_round_style numeric_limits<const volatile _Tp>::round_style;

_LIBCPP_END_NAMESPACE_STD

_LIBCPP_POP_MACROS

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

#endif // _LIBCPP_LIMITS