compare

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// -*- C++ -*- operator<=> three-way comparison support.
 
// Copyright (C) 2019-2025 Free Software Foundation, Inc.
//
// This file is part of GCC.
//
// GCC 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.
//
// GCC 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/>.
 
/** @file compare
 *  This is a Standard C++ Library header.
 */
 
#ifndef _COMPARE
#define _COMPARE
 
#ifdef _GLIBCXX_SYSHDR
#pragma GCC system_header
#endif
 
#define __glibcxx_want_three_way_comparison
#define __glibcxx_want_type_order
#include <bits/version.h>
 
#if __cplusplus > 201703L && __cpp_impl_three_way_comparison >= 201907L
 
#include <concepts>
 
#pragma GCC diagnostic push
#pragma GCC diagnostic ignored "-Wpedantic" // __int128
#pragma GCC diagnostic ignored "-Wzero-as-null-pointer-constant"
 
namespace std _GLIBCXX_VISIBILITY(default)
{
  // [cmp.categories], comparison category types
 
  namespace __cmp_cat
  {
    using type = signed char;
 
    enum class _Ord : type
    {
      equivalent = 0, less = -1, greater = 1,
      // value remains unchanged when negated
      unordered = -__SCHAR_MAX__ - 1
    };
 
    template<typename _Ordering>
      [[__gnu__::__always_inline__]]
      constexpr _Ord
      __ord(_Ordering __o) noexcept
      { return _Ord(__o._M_value); }
 
    template<typename _Ordering>
      [[__gnu__::__always_inline__]]
      constexpr _Ordering
      __make(_Ord __o) noexcept
      { return _Ordering(__o); }
 
    struct __literal_zero
    {
      consteval __literal_zero(__literal_zero*) noexcept { }
    };
  }
 
  class partial_ordering
  {
    // less=0xff, equiv=0x00, greater=0x01, unordered=0x80
    __cmp_cat::type _M_value;
 
    constexpr explicit
    partial_ordering(__cmp_cat::_Ord __v) noexcept
    : _M_value(__cmp_cat::type(__v))
    { }
 
    [[__gnu__::__always_inline__]]
    constexpr __cmp_cat::type
    _M_reverse() const
    {
      // leaves _Ord::unordered unchanged
      return static_cast<__cmp_cat::type>(-_M_value);
    }
 
    friend constexpr __cmp_cat::_Ord
    __cmp_cat::__ord<partial_ordering>(partial_ordering) noexcept;
    friend constexpr partial_ordering
    __cmp_cat::__make<partial_ordering>(__cmp_cat::_Ord) noexcept;
 
  public:
    // valid values
    static const partial_ordering less;
    static const partial_ordering equivalent;
    static const partial_ordering greater;
    static const partial_ordering unordered;
 
    // comparisons
    [[nodiscard]]
    friend constexpr bool
    operator==(partial_ordering __v, __cmp_cat::__literal_zero) noexcept
    { return __v._M_value == 0; }
 
    [[nodiscard]]
    friend constexpr bool
    operator==(partial_ordering, partial_ordering) noexcept = default;
 
    [[nodiscard]]
    friend constexpr bool
    operator< (partial_ordering __v, __cmp_cat::__literal_zero) noexcept
    { return __v._M_value == -1; }
 
    [[nodiscard]]
    friend constexpr bool
    operator> (partial_ordering __v, __cmp_cat::__literal_zero) noexcept
    { return __v._M_value == 1; }
 
    [[nodiscard]]
    friend constexpr bool
    operator<=(partial_ordering __v, __cmp_cat::__literal_zero) noexcept
    { return __v._M_reverse() >= 0; }
 
    [[nodiscard]]
    friend constexpr bool
    operator>=(partial_ordering __v, __cmp_cat::__literal_zero) noexcept
    { return __v._M_value >= 0; }
 
    [[nodiscard]]
    friend constexpr bool
    operator< (__cmp_cat::__literal_zero, partial_ordering __v) noexcept
    { return __v._M_value == 1; }
 
    [[nodiscard]]
    friend constexpr bool
    operator> (__cmp_cat::__literal_zero, partial_ordering __v) noexcept
    { return __v._M_value == -1; }
 
    [[nodiscard]]
    friend constexpr bool
    operator<=(__cmp_cat::__literal_zero, partial_ordering __v) noexcept
    { return 0 <= __v._M_value; }
 
    [[nodiscard]]
    friend constexpr bool
    operator>=(__cmp_cat::__literal_zero, partial_ordering __v) noexcept
    { return 0 <= __v._M_reverse(); }
 
    [[nodiscard]]
    friend constexpr partial_ordering
    operator<=>(partial_ordering __v, __cmp_cat::__literal_zero) noexcept
    { return __v; }
 
    [[nodiscard]]
    friend constexpr partial_ordering
    operator<=>(__cmp_cat::__literal_zero, partial_ordering __v) noexcept
    { return partial_ordering(__cmp_cat::_Ord(__v._M_reverse())); }
  };
 
  // valid values' definitions
  inline constexpr partial_ordering
  partial_ordering::less(__cmp_cat::_Ord::less);
 
  inline constexpr partial_ordering
  partial_ordering::equivalent(__cmp_cat::_Ord::equivalent);
 
  inline constexpr partial_ordering
  partial_ordering::greater(__cmp_cat::_Ord::greater);
 
  inline constexpr partial_ordering
  partial_ordering::unordered(__cmp_cat::_Ord::unordered);
 
  class weak_ordering
  {
    __cmp_cat::type _M_value;
 
    constexpr explicit
    weak_ordering(__cmp_cat::_Ord __v) noexcept : _M_value(__cmp_cat::type(__v))
    { }
 
    friend constexpr __cmp_cat::_Ord
    __cmp_cat::__ord<weak_ordering>(weak_ordering) noexcept;
    friend constexpr weak_ordering
    __cmp_cat::__make<weak_ordering>(__cmp_cat::_Ord) noexcept;
 
  public:
    // valid values
    static const weak_ordering less;
    static const weak_ordering equivalent;
    static const weak_ordering greater;
 
    [[nodiscard]]
    constexpr operator partial_ordering() const noexcept
    { return __cmp_cat::__make<partial_ordering>(__cmp_cat::_Ord(_M_value)); }
 
    // comparisons
    [[nodiscard]]
    friend constexpr bool
    operator==(weak_ordering __v, __cmp_cat::__literal_zero) noexcept
    { return __v._M_value == 0; }
 
    [[nodiscard]]
    friend constexpr bool
    operator==(weak_ordering, weak_ordering) noexcept = default;
 
    [[nodiscard]]
    friend constexpr bool
    operator< (weak_ordering __v, __cmp_cat::__literal_zero) noexcept
    { return __v._M_value < 0; }
 
    [[nodiscard]]
    friend constexpr bool
    operator> (weak_ordering __v, __cmp_cat::__literal_zero) noexcept
    { return __v._M_value > 0; }
 
    [[nodiscard]]
    friend constexpr bool
    operator<=(weak_ordering __v, __cmp_cat::__literal_zero) noexcept
    { return __v._M_value <= 0; }
 
    [[nodiscard]]
    friend constexpr bool
    operator>=(weak_ordering __v, __cmp_cat::__literal_zero) noexcept
    { return __v._M_value >= 0; }
 
    [[nodiscard]]
    friend constexpr bool
    operator< (__cmp_cat::__literal_zero, weak_ordering __v) noexcept
    { return 0 < __v._M_value; }
 
    [[nodiscard]]
    friend constexpr bool
    operator> (__cmp_cat::__literal_zero, weak_ordering __v) noexcept
    { return 0 > __v._M_value; }
 
    [[nodiscard]]
    friend constexpr bool
    operator<=(__cmp_cat::__literal_zero, weak_ordering __v) noexcept
    { return 0 <= __v._M_value; }
 
    [[nodiscard]]
    friend constexpr bool
    operator>=(__cmp_cat::__literal_zero, weak_ordering __v) noexcept
    { return 0 >= __v._M_value; }
 
    [[nodiscard]]
    friend constexpr weak_ordering
    operator<=>(weak_ordering __v, __cmp_cat::__literal_zero) noexcept
    { return __v; }
 
    [[nodiscard]]
    friend constexpr weak_ordering
    operator<=>(__cmp_cat::__literal_zero, weak_ordering __v) noexcept
    { return weak_ordering(__cmp_cat::_Ord(-__v._M_value)); }
  };
 
  // valid values' definitions
  inline constexpr weak_ordering
  weak_ordering::less(__cmp_cat::_Ord::less);
 
  inline constexpr weak_ordering
  weak_ordering::equivalent(__cmp_cat::_Ord::equivalent);
 
  inline constexpr weak_ordering
  weak_ordering::greater(__cmp_cat::_Ord::greater);
 
  class strong_ordering
  {
    __cmp_cat::type _M_value;
 
    constexpr explicit
    strong_ordering(__cmp_cat::_Ord __v) noexcept
    : _M_value(__cmp_cat::type(__v))
    { }
 
    friend constexpr __cmp_cat::_Ord
    __cmp_cat::__ord<strong_ordering>(strong_ordering) noexcept;
    friend constexpr strong_ordering
    __cmp_cat::__make<strong_ordering>(__cmp_cat::_Ord) noexcept;
 
  public:
    // valid values
    static const strong_ordering less;
    static const strong_ordering equal;
    static const strong_ordering equivalent;
    static const strong_ordering greater;
 
    [[nodiscard]]
    constexpr operator partial_ordering() const noexcept
    { return __cmp_cat::__make<partial_ordering>(__cmp_cat::_Ord(_M_value)); }
 
    [[nodiscard]]
    constexpr operator weak_ordering() const noexcept
    { return __cmp_cat::__make<weak_ordering>(__cmp_cat::_Ord(_M_value)); }
 
    // comparisons
    [[nodiscard]]
    friend constexpr bool
    operator==(strong_ordering __v, __cmp_cat::__literal_zero) noexcept
    { return __v._M_value == 0; }
 
    [[nodiscard]]
    friend constexpr bool
    operator==(strong_ordering, strong_ordering) noexcept = default;
 
    [[nodiscard]]
    friend constexpr bool
    operator< (strong_ordering __v, __cmp_cat::__literal_zero) noexcept
    { return __v._M_value < 0; }
 
    [[nodiscard]]
    friend constexpr bool
    operator> (strong_ordering __v, __cmp_cat::__literal_zero) noexcept
    { return __v._M_value > 0; }
 
    [[nodiscard]]
    friend constexpr bool
    operator<=(strong_ordering __v, __cmp_cat::__literal_zero) noexcept
    { return __v._M_value <= 0; }
 
    [[nodiscard]]
    friend constexpr bool
    operator>=(strong_ordering __v, __cmp_cat::__literal_zero) noexcept
    { return __v._M_value >= 0; }
 
    [[nodiscard]]
    friend constexpr bool
    operator< (__cmp_cat::__literal_zero, strong_ordering __v) noexcept
    { return 0 < __v._M_value; }
 
    [[nodiscard]]
    friend constexpr bool
    operator> (__cmp_cat::__literal_zero, strong_ordering __v) noexcept
    { return 0 > __v._M_value; }
 
    [[nodiscard]]
    friend constexpr bool
    operator<=(__cmp_cat::__literal_zero, strong_ordering __v) noexcept
    { return 0 <= __v._M_value; }
 
    [[nodiscard]]
    friend constexpr bool
    operator>=(__cmp_cat::__literal_zero, strong_ordering __v) noexcept
    { return 0 >= __v._M_value; }
 
    [[nodiscard]]
    friend constexpr strong_ordering
    operator<=>(strong_ordering __v, __cmp_cat::__literal_zero) noexcept
    { return __v; }
 
    [[nodiscard]]
    friend constexpr strong_ordering
    operator<=>(__cmp_cat::__literal_zero, strong_ordering __v) noexcept
    { return strong_ordering(__cmp_cat::_Ord(-__v._M_value)); }
  };
 
  // valid values' definitions
  inline constexpr strong_ordering
  strong_ordering::less(__cmp_cat::_Ord::less);
 
  inline constexpr strong_ordering
  strong_ordering::equal(__cmp_cat::_Ord::equivalent);
 
  inline constexpr strong_ordering
  strong_ordering::equivalent(__cmp_cat::_Ord::equivalent);
 
  inline constexpr strong_ordering
  strong_ordering::greater(__cmp_cat::_Ord::greater);
 
 
  // named comparison functions
  [[nodiscard]]
  constexpr bool
  is_eq(partial_ordering __cmp) noexcept
  { return __cmp == 0; }
 
  [[nodiscard]]
  constexpr bool
  is_neq(partial_ordering __cmp) noexcept
  { return __cmp != 0; }
 
  [[nodiscard]]
  constexpr bool
  is_lt  (partial_ordering __cmp) noexcept
  { return __cmp < 0; }
 
  [[nodiscard]]
  constexpr bool
  is_lteq(partial_ordering __cmp) noexcept
  { return __cmp <= 0; }
 
  [[nodiscard]]
  constexpr bool
  is_gt  (partial_ordering __cmp) noexcept
  { return __cmp > 0; }
 
  [[nodiscard]]
  constexpr bool
  is_gteq(partial_ordering __cmp) noexcept
  { return __cmp >= 0; }
 
  namespace __detail
  {
    template<typename _Tp>
      inline constexpr unsigned __cmp_cat_id = 1;
    template<>
      inline constexpr unsigned __cmp_cat_id<partial_ordering> = 2;
    template<>
      inline constexpr unsigned __cmp_cat_id<weak_ordering> = 4;
    template<>
      inline constexpr unsigned __cmp_cat_id<strong_ordering> = 8;
 
    template<typename... _Ts>
      constexpr auto __common_cmp_cat()
      {
        constexpr unsigned __cats = (__cmp_cat_id<_Ts> | ...);
        // If any Ti is not a comparison category type, U is void.
        if constexpr (__cats & 1)
          return;
        // Otherwise, if at least one Ti is std::partial_ordering,
        // U is std::partial_ordering.
        else if constexpr (bool(__cats & __cmp_cat_id<partial_ordering>))
          return partial_ordering::equivalent;
        // Otherwise, if at least one Ti is std::weak_ordering,
        // U is std::weak_ordering.
        else if constexpr (bool(__cats & __cmp_cat_id<weak_ordering>))
          return weak_ordering::equivalent;
        // Otherwise, U is std::strong_ordering.
        else
          return strong_ordering::equivalent;
      }
  } // namespace __detail
 
  // [cmp.common], common comparison category type
  template<typename... _Ts>
    struct common_comparison_category
    {
      using type = decltype(__detail::__common_cmp_cat<_Ts...>());
    };
 
  // Partial specializations for one and zero argument cases.
 
  template<typename _Tp>
    struct common_comparison_category<_Tp>
    { using type = void; };
 
  template<>
    struct common_comparison_category<partial_ordering>
    { using type = partial_ordering; };
 
  template<>
    struct common_comparison_category<weak_ordering>
    { using type = weak_ordering; };
 
  template<>
    struct common_comparison_category<strong_ordering>
    { using type = strong_ordering; };
 
  template<>
    struct common_comparison_category<>
    { using type = strong_ordering; };
 
  template<typename... _Ts>
    using common_comparison_category_t
      = typename common_comparison_category<_Ts...>::type;
 
#if __cpp_lib_three_way_comparison >= 201907L
  // C++ >= 20 && impl_3way_comparison >= 201907 && lib_concepts
  namespace __detail
  {
    template<typename _Tp, typename _Cat>
      concept __compares_as
        = same_as<common_comparison_category_t<_Tp, _Cat>, _Cat>;
  } // namespace __detail
 
  // [cmp.concept], concept three_way_comparable
  template<typename _Tp, typename _Cat = partial_ordering>
    concept three_way_comparable
      = __detail::__weakly_eq_cmp_with<_Tp, _Tp>
      && __detail::__partially_ordered_with<_Tp, _Tp>
      && requires(const remove_reference_t<_Tp>& __a,
                  const remove_reference_t<_Tp>& __b)
      {
        { __a <=> __b } -> __detail::__compares_as<_Cat>;
      };
 
  template<typename _Tp, typename _Up, typename _Cat = partial_ordering>
    concept three_way_comparable_with
      = three_way_comparable<_Tp, _Cat>
      && three_way_comparable<_Up, _Cat>
      && common_reference_with<const remove_reference_t<_Tp>&,
                               const remove_reference_t<_Up>&>
      && three_way_comparable<
          common_reference_t<const remove_reference_t<_Tp>&,
                             const remove_reference_t<_Up>&>, _Cat>
      && __detail::__weakly_eq_cmp_with<_Tp, _Up>
      && __detail::__partially_ordered_with<_Tp, _Up>
      && requires(const remove_reference_t<_Tp>& __t,
                  const remove_reference_t<_Up>& __u)
      {
        { __t <=> __u } -> __detail::__compares_as<_Cat>;
        { __u <=> __t } -> __detail::__compares_as<_Cat>;
      };
 
  namespace __detail
  {
    template<typename _Tp, typename _Up>
      using __cmp3way_res_t
        = decltype(std::declval<_Tp>() <=> std::declval<_Up>());
 
    // Implementation of std::compare_three_way_result.
    // It is undefined for a program to add specializations of
    // std::compare_three_way_result, so the std::compare_three_way_result_t
    // alias ignores std::compare_three_way_result and uses
    // __detail::__cmp3way_res_impl directly instead.
    template<typename _Tp, typename _Up>
      struct __cmp3way_res_impl
      { };
 
    template<typename _Tp, typename _Up>
      requires requires { typename __cmp3way_res_t<__cref<_Tp>, __cref<_Up>>; }
      struct __cmp3way_res_impl<_Tp, _Up>
      {
        using type = __cmp3way_res_t<__cref<_Tp>, __cref<_Up>>;
      };
  } // namespace __detail
 
  /// [cmp.result], result of three-way comparison
  template<typename _Tp, typename _Up = _Tp>
    struct compare_three_way_result
    : __detail::__cmp3way_res_impl<_Tp, _Up>
    { };
 
  /// [cmp.result], result of three-way comparison
  template<typename _Tp, typename _Up = _Tp>
    using compare_three_way_result_t
      = typename __detail::__cmp3way_res_impl<_Tp, _Up>::type;
 
  namespace __detail
  {
    // BUILTIN-PTR-THREE-WAY(T, U)
    // This determines whether t <=> u results in a call to a built-in
    // operator<=> comparing pointers. It doesn't work for function pointers
    // (PR 93628).
    template<typename _Tp, typename _Up>
      concept __3way_builtin_ptr_cmp
        = requires(_Tp&& __t, _Up&& __u)
          { static_cast<_Tp&&>(__t) <=> static_cast<_Up&&>(__u); }
          && convertible_to<_Tp, const volatile void*>
          && convertible_to<_Up, const volatile void*>
          && ! requires(_Tp&& __t, _Up&& __u)
          { operator<=>(static_cast<_Tp&&>(__t), static_cast<_Up&&>(__u)); }
          && ! requires(_Tp&& __t, _Up&& __u)
          { static_cast<_Tp&&>(__t).operator<=>(static_cast<_Up&&>(__u)); };
  } // namespace __detail
 
  // _GLIBCXX_RESOLVE_LIB_DEFECTS
  // 3530 BUILTIN-PTR-MEOW should not opt the type out of syntactic checks
 
  // [cmp.object], typename compare_three_way
  struct compare_three_way
  {
#pragma GCC diagnostic push
#pragma GCC diagnostic ignored "-Wc++23-extensions" // static operator()
    template<typename _Tp, typename _Up>
      requires three_way_comparable_with<_Tp, _Up>
      static constexpr auto
      operator() [[nodiscard]] (_Tp&& __t, _Up&& __u)
      noexcept(noexcept(std::declval<_Tp>() <=> std::declval<_Up>()))
      {
        if constexpr (__detail::__3way_builtin_ptr_cmp<_Tp, _Up>)
          {
            auto __pt = static_cast<const volatile void*>(__t);
            auto __pu = static_cast<const volatile void*>(__u);
            if (std::__is_constant_evaluated())
              return __pt <=> __pu;
            auto __it = reinterpret_cast<__UINTPTR_TYPE__>(__pt);
            auto __iu = reinterpret_cast<__UINTPTR_TYPE__>(__pu);
            return __it <=> __iu;
          }
        else
          return static_cast<_Tp&&>(__t) <=> static_cast<_Up&&>(__u);
      }
#pragma GCC diagnostic pop
 
    using is_transparent = void;
  };
 
  /// @cond undocumented
  // Namespace for helpers for the <compare> customization points.
  namespace __compare
  {
    template<floating_point _Tp>
      constexpr weak_ordering
      __fp_weak_ordering(_Tp __e, _Tp __f)
      {
        auto __po = __cmp_cat::__ord(__e <=> __f);
        if (__po != __cmp_cat::_Ord::unordered)
          return __cmp_cat::__make<weak_ordering>(__po);
        else  // unordered, at least one argument is NaN
          {
            // return -1 for negative nan, +1 for positive nan, 0 otherwise.
            auto __isnan_sign = [](_Tp __fp) -> int {
              return __builtin_isnan(__fp)
                ? __builtin_signbit(__fp) ? -1 : 1
                : 0;
            };
            return __isnan_sign(__e) <=> __isnan_sign(__f);
          }
      }
 
    void strong_order() = delete;
 
    template<typename _Tp, typename _Up>
      concept __adl_strong = requires(_Tp&& __t, _Up&& __u)
        {
          strong_ordering(strong_order(static_cast<_Tp&&>(__t),
                                       static_cast<_Up&&>(__u)));
        };
 
    void weak_order() = delete;
 
    template<typename _Tp, typename _Up>
      concept __adl_weak = requires(_Tp&& __t, _Up&& __u)
        {
          weak_ordering(weak_order(static_cast<_Tp&&>(__t),
                                   static_cast<_Up&&>(__u)));
        };
 
    void partial_order() = delete;
 
    template<typename _Tp, typename _Up>
      concept __adl_partial = requires(_Tp&& __t, _Up&& __u)
        {
          partial_ordering(partial_order(static_cast<_Tp&&>(__t),
                                         static_cast<_Up&&>(__u)));
        };
 
    template<typename _Ord, typename _Tp, typename _Up>
      concept __cmp3way = requires(_Tp&& __t, _Up&& __u, compare_three_way __c)
        {
          _Ord(__c(static_cast<_Tp&&>(__t), static_cast<_Up&&>(__u)));
        };
 
    template<typename _Tp, typename _Up>
      concept __strongly_ordered
        = __adl_strong<_Tp, _Up>
          || floating_point<remove_reference_t<_Tp>>
          || __cmp3way<strong_ordering, _Tp, _Up>;
 
    template<typename _Tp, typename _Up>
      concept __decayed_same_as = same_as<decay_t<_Tp>, decay_t<_Up>>;
 
    class _Strong_order
    {
      template<typename _Tp, typename _Up>
        static constexpr bool
        _S_noexcept()
        {
          if constexpr (floating_point<decay_t<_Tp>>)
            return true;
          else if constexpr (__adl_strong<_Tp, _Up>)
            return noexcept(strong_ordering(strong_order(std::declval<_Tp>(),
                                                         std::declval<_Up>())));
          else if constexpr (__cmp3way<strong_ordering, _Tp, _Up>)
            return noexcept(compare_three_way()(std::declval<_Tp>(),
                                                std::declval<_Up>()));
        }
 
      friend class _Weak_order;
      friend class _Strong_fallback;
 
      // Names for the supported floating-point representations.
      enum class _Fp_fmt
      {
        _Binary16, _Binary32, _Binary64, _Binary128, // IEEE
        _X86_80bit,  // x86 80-bit extended precision
        _M68k_80bit, // m68k 80-bit extended precision
        _Dbldbl, // IBM 128-bit double-double
        _Bfloat16,   // std::bfloat16_t
      };
 
#ifndef __cpp_using_enum
      // XXX Remove these once 'using enum' support is ubiquitous.
      static constexpr _Fp_fmt _Binary16 = _Fp_fmt::_Binary16;
      static constexpr _Fp_fmt _Binary32 = _Fp_fmt::_Binary32;
      static constexpr _Fp_fmt _Binary64 = _Fp_fmt::_Binary64;
      static constexpr _Fp_fmt _Binary128 = _Fp_fmt::_Binary128;
      static constexpr _Fp_fmt _X86_80bit = _Fp_fmt::_X86_80bit;
      static constexpr _Fp_fmt _M68k_80bit = _Fp_fmt::_M68k_80bit;
      static constexpr _Fp_fmt _Dbldbl = _Fp_fmt::_Dbldbl;
      static constexpr _Fp_fmt _Bfloat16 = _Fp_fmt::_Bfloat16;
#endif
 
      // Identify the format used by a floating-point type.
      template<typename _Tp>
        static consteval _Fp_fmt
        _S_fp_fmt() noexcept
        {
#ifdef __cpp_using_enum
          using enum _Fp_fmt;
#endif
 
          // Identify these formats first, then assume anything else is IEEE.
          // N.B. ARM __fp16 alternative format can be handled as binary16.
 
#ifdef __LONG_DOUBLE_IBM128__
          if constexpr (__is_same(_Tp, long double))
            return _Dbldbl;
#elif defined __LONG_DOUBLE_IEEE128__ && defined __SIZEOF_IBM128__
          if constexpr (__is_same(_Tp, __ibm128))
            return _Dbldbl;
#endif
 
#if __LDBL_MANT_DIG__ == 64
          if constexpr (__is_same(_Tp, long double))
            return __LDBL_MIN_EXP__ == -16381 ? _X86_80bit : _M68k_80bit;
#endif
#ifdef __SIZEOF_FLOAT80__
          if constexpr (__is_same(_Tp, __float80))
            return _X86_80bit;
#endif
#ifdef __STDCPP_BFLOAT16_T__
          if constexpr (__is_same(_Tp, decltype(0.0bf16)))
            return _Bfloat16;
#endif
 
          constexpr int __width = sizeof(_Tp) * __CHAR_BIT__;
 
          if constexpr (__width == 16)       // IEEE binary16 (or ARM fp16).
            return _Binary16;
          else if constexpr (__width == 32)  // IEEE binary32
            return _Binary32;
          else if constexpr (__width == 64)  // IEEE binary64
            return _Binary64;
          else if constexpr (__width == 128) // IEEE binary128
            return _Binary128;
        }
 
      // So we don't need to include <stdint.h> and pollute the namespace.
      using int64_t = __INT64_TYPE__;
      using int32_t = __INT32_TYPE__;
      using int16_t = __INT16_TYPE__;
      using uint64_t = __UINT64_TYPE__;
      using uint16_t = __UINT16_TYPE__;
 
      // Used to unpack floating-point types that do not fit into an integer.
      template<typename _Tp>
        struct _Int
        {
#if __BYTE_ORDER__ == __ORDER_LITTLE_ENDIAN__
          uint64_t _M_lo;
          _Tp _M_hi;
#else
          _Tp _M_hi;
          uint64_t _M_lo;
#endif
 
          constexpr explicit
          _Int(_Tp __hi, uint64_t __lo) noexcept : _M_hi(__hi)
          { _M_lo = __lo; }
 
          constexpr explicit
          _Int(uint64_t __lo) noexcept : _M_hi(0)
          { _M_lo = __lo; }
 
          constexpr bool operator==(const _Int&) const = default;
 
#if defined __hppa__ || (defined __mips__ && !defined __mips_nan2008)
          consteval _Int
          operator<<(int __n) const noexcept
          {
            // XXX this assumes n >= 64, which is true for the use below.
            return _Int(static_cast<_Tp>(_M_lo << (__n - 64)), 0);
          }
#endif
 
          constexpr _Int&
          operator^=(const _Int& __rhs) noexcept
          {
            _M_hi ^= __rhs._M_hi;
            _M_lo ^= __rhs._M_lo;
            return *this;
          }
 
          constexpr strong_ordering
          operator<=>(const _Int& __rhs) const noexcept
          {
            strong_ordering __cmp = _M_hi <=> __rhs._M_hi;
            if (__cmp != strong_ordering::equal)
              return __cmp;
            return _M_lo <=> __rhs._M_lo;
          }
        };
 
      template<typename _Tp>
        static constexpr _Tp
        _S_compl(_Tp __t) noexcept
        {
          constexpr int __width = sizeof(_Tp) * __CHAR_BIT__;
          // Sign extend to get all ones or all zeros.
          make_unsigned_t<_Tp> __sign = __t >> (__width - 1);
          // If the sign bit was set, this flips all bits below it.
          // This converts ones' complement to two's complement.
          return __t ^ (__sign >> 1);
        }
 
      // As above but works on both parts of _Int<T>.
      template<typename _Tp>
        static constexpr _Int<_Tp>
        _S_compl(_Int<_Tp> __t) noexcept
        {
          constexpr int __width = sizeof(_Tp) * __CHAR_BIT__;
          make_unsigned_t<_Tp> __sign = __t._M_hi >> (__width - 1);
          __t._M_hi ^= (__sign >> 1 );
          uint64_t __sign64 = (_Tp)__sign;
          __t._M_lo ^= __sign64;
          return __t;
        }
 
      // Bit-cast a floating-point value to an unsigned integer.
      template<typename _Tp>
        constexpr static auto
        _S_fp_bits(_Tp __val) noexcept
        {
          if constexpr (sizeof(_Tp) == sizeof(int64_t))
            return __builtin_bit_cast(int64_t, __val);
          else if constexpr (sizeof(_Tp) == sizeof(int32_t))
            return __builtin_bit_cast(int32_t, __val);
          else if constexpr (sizeof(_Tp) == sizeof(int16_t))
            return __builtin_bit_cast(int16_t, __val);
          else
            {
#ifdef __cpp_using_enum
              using enum _Fp_fmt;
#endif
              constexpr auto __fmt = _S_fp_fmt<_Tp>();
              if constexpr (__fmt == _X86_80bit)
                {
                  if constexpr (sizeof(_Tp) == 3 * sizeof(int32_t))
                    {
                      auto __ival = __builtin_bit_cast(_Int<int32_t>, __val);
                      return _Int<int16_t>(__ival._M_hi, __ival._M_lo);
                    }
                  else
                    {
                      auto __ival = __builtin_bit_cast(_Int<int64_t>, __val);
                      return _Int<int16_t>(__ival._M_hi, __ival._M_lo);
                    }
                }
              else if constexpr (__fmt == _M68k_80bit)
                {
                  auto __ival = __builtin_bit_cast(_Int<int32_t>, __val);
                  return _Int<int16_t>(__ival._M_hi >> 16, __ival._M_lo);
                }
              else if constexpr (sizeof(_Tp) == 2 * sizeof(int64_t))
                {
#if __SIZEOF_INT128__
                  return __builtin_bit_cast(__int128, __val);
#else
                  return __builtin_bit_cast(_Int<int64_t>, __val);
#endif
                }
              else
                static_assert(sizeof(_Tp) == sizeof(int64_t),
                              "unsupported floating-point type");
            }
        }
 
      template<typename _Tp>
        static constexpr strong_ordering
        _S_fp_cmp(_Tp __x, _Tp __y) noexcept
        {
#ifdef __vax__
          if (__builtin_isnan(__x) || __builtin_isnan(__y))
            {
              int __ix = (bool) __builtin_isnan(__x);
              int __iy = (bool) __builtin_isnan(__y);
              __ix *= __builtin_signbit(__x) ? -1 : 1;
              __iy *= __builtin_signbit(__y) ? -1 : 1;
              return __ix <=> __iy;
            }
          else
            return __builtin_bit_cast(strong_ordering, __x <=> __y);
#endif
 
          auto __ix = _S_fp_bits(__x);
          auto __iy = _S_fp_bits(__y);
 
          if (__ix == __iy)
            return strong_ordering::equal; // All bits are equal, we're done.
 
#ifdef __cpp_using_enum
          using enum _Fp_fmt;
#endif
          constexpr auto __fmt = _S_fp_fmt<_Tp>();
 
          if constexpr (__fmt == _Dbldbl) // double-double
            {
              // Unpack the double-double into two parts.
              // We never inspect the low double as a double, cast to integer.
              struct _Unpacked { double _M_hi; int64_t _M_lo; };
              auto __x2 = __builtin_bit_cast(_Unpacked, __x);
              auto __y2 = __builtin_bit_cast(_Unpacked, __y);
 
              // Compare the high doubles first and use result if unequal.
              auto __cmp = _S_fp_cmp(__x2._M_hi, __y2._M_hi);
              if (__cmp != strong_ordering::equal)
                return __cmp;
 
              // For NaN the low double is unused, so if the high doubles
              // are the same NaN, we don't need to compare the low doubles.
              if (__builtin_isnan(__x2._M_hi))
                return strong_ordering::equal;
              // Similarly, if the low doubles are +zero or -zero (which is
              // true for all infinities and some other values), we're done.
              if (((__x2._M_lo | __y2._M_lo) & 0x7fffffffffffffffULL) == 0)
                return strong_ordering::equal;
 
              // Otherwise, compare the low parts.
              return _S_compl(__x2._M_lo) <=> _S_compl(__y2._M_lo);
            }
          else
            {
              if constexpr (__fmt == _M68k_80bit)
                {
                  // For m68k the MSB of the significand is ignored for the
                  // greatest exponent, so either 0 or 1 is valid there.
                  // Set it before comparing, so that we never have 0 there.
                  constexpr uint16_t __maxexp = 0x7fff;
                  if ((__ix._M_hi & __maxexp) == __maxexp)
                    __ix._M_lo |= 1ull << 63;
                  if ((__iy._M_hi & __maxexp) == __maxexp)
                    __iy._M_lo |= 1ull << 63;
                }
              else
                {
#if defined __hppa__ || (defined __mips__ && !defined __mips_nan2008)
                  // IEEE 754-1985 allowed the meaning of the quiet/signaling
                  // bit to be reversed. Flip that to give desired ordering.
                  if (__builtin_isnan(__x) && __builtin_isnan(__y))
                    {
                      using _Int = decltype(__ix);
 
                      constexpr int __nantype = __fmt == _Binary32  ?  22
                                              : __fmt == _Binary64  ?  51
                                              : __fmt == _Binary128 ? 111
                                              : -1;
                      constexpr _Int __bit = _Int(1) << __nantype;
                      __ix ^= __bit;
                      __iy ^= __bit;
                    }
#endif
                }
              return _S_compl(__ix) <=> _S_compl(__iy);
            }
        }
 
    public:
      template<typename _Tp, __decayed_same_as<_Tp> _Up>
        requires __strongly_ordered<_Tp, _Up>
        constexpr strong_ordering
        operator() [[nodiscard]] (_Tp&& __e, _Up&& __f) const
        noexcept(_S_noexcept<_Tp, _Up>())
        {
          if constexpr (floating_point<decay_t<_Tp>>)
            return _S_fp_cmp(__e, __f);
          else if constexpr (__adl_strong<_Tp, _Up>)
            return strong_ordering(strong_order(static_cast<_Tp&&>(__e),
                                                static_cast<_Up&&>(__f)));
          else if constexpr (__cmp3way<strong_ordering, _Tp, _Up>)
            return compare_three_way()(static_cast<_Tp&&>(__e),
                                       static_cast<_Up&&>(__f));
        }
    };
 
    template<typename _Tp, typename _Up>
      concept __weakly_ordered
        = floating_point<remove_reference_t<_Tp>>
          || __adl_weak<_Tp, _Up>
          || __cmp3way<weak_ordering, _Tp, _Up>
          || __strongly_ordered<_Tp, _Up>;
 
    class _Weak_order
    {
      template<typename _Tp, typename _Up>
        static constexpr bool
        _S_noexcept()
        {
          if constexpr (floating_point<decay_t<_Tp>>)
            return true;
          else if constexpr (__adl_weak<_Tp, _Up>)
            return noexcept(weak_ordering(weak_order(std::declval<_Tp>(),
                                                     std::declval<_Up>())));
          else if constexpr (__cmp3way<weak_ordering, _Tp, _Up>)
            return noexcept(compare_three_way()(std::declval<_Tp>(),
                                                std::declval<_Up>()));
          else if constexpr (__strongly_ordered<_Tp, _Up>)
            return _Strong_order::_S_noexcept<_Tp, _Up>();
        }
 
      friend class _Partial_order;
      friend class _Weak_fallback;
 
    public:
      template<typename _Tp, __decayed_same_as<_Tp> _Up>
        requires __weakly_ordered<_Tp, _Up>
        constexpr weak_ordering
        operator() [[nodiscard]] (_Tp&& __e, _Up&& __f) const
        noexcept(_S_noexcept<_Tp, _Up>())
        {
          if constexpr (floating_point<decay_t<_Tp>>)
            return __compare::__fp_weak_ordering(__e, __f);
          else if constexpr (__adl_weak<_Tp, _Up>)
            return weak_ordering(weak_order(static_cast<_Tp&&>(__e),
                                            static_cast<_Up&&>(__f)));
          else if constexpr (__cmp3way<weak_ordering, _Tp, _Up>)
            return compare_three_way()(static_cast<_Tp&&>(__e),
                                       static_cast<_Up&&>(__f));
          else if constexpr (__strongly_ordered<_Tp, _Up>)
            return _Strong_order{}(static_cast<_Tp&&>(__e),
                                   static_cast<_Up&&>(__f));
        }
    };
 
    template<typename _Tp, typename _Up>
      concept __partially_ordered
        = __adl_partial<_Tp, _Up>
        || __cmp3way<partial_ordering, _Tp, _Up>
        || __weakly_ordered<_Tp, _Up>;
 
    class _Partial_order
    {
      template<typename _Tp, typename _Up>
        static constexpr bool
        _S_noexcept()
        {
          if constexpr (__adl_partial<_Tp, _Up>)
            return noexcept(partial_ordering(partial_order(std::declval<_Tp>(),
                                                         std::declval<_Up>())));
          else if constexpr (__cmp3way<partial_ordering, _Tp, _Up>)
            return noexcept(compare_three_way()(std::declval<_Tp>(),
                                                std::declval<_Up>()));
          else if constexpr (__weakly_ordered<_Tp, _Up>)
            return _Weak_order::_S_noexcept<_Tp, _Up>();
        }
 
      friend class _Partial_fallback;
 
    public:
      template<typename _Tp, __decayed_same_as<_Tp> _Up>
        requires __partially_ordered<_Tp, _Up>
        constexpr partial_ordering
        operator() [[nodiscard]] (_Tp&& __e, _Up&& __f) const
        noexcept(_S_noexcept<_Tp, _Up>())
        {
          if constexpr (__adl_partial<_Tp, _Up>)
            return partial_ordering(partial_order(static_cast<_Tp&&>(__e),
                                                  static_cast<_Up&&>(__f)));
          else if constexpr (__cmp3way<partial_ordering, _Tp, _Up>)
            return compare_three_way()(static_cast<_Tp&&>(__e),
                                       static_cast<_Up&&>(__f));
          else if constexpr (__weakly_ordered<_Tp, _Up>)
            return _Weak_order{}(static_cast<_Tp&&>(__e),
                                 static_cast<_Up&&>(__f));
        }
    };
 
    template<typename _Tp, typename _Up>
      concept __op_eq_lt = requires(_Tp&& __t, _Up&& __u)
        {
          { static_cast<_Tp&&>(__t) == static_cast<_Up&&>(__u) }
            -> convertible_to<bool>;
          { static_cast<_Tp&&>(__t) < static_cast<_Up&&>(__u) }
            -> convertible_to<bool>;
        };
 
    class _Strong_fallback
    {
      template<typename _Tp, typename _Up>
        static constexpr bool
        _S_noexcept()
        {
          if constexpr (__strongly_ordered<_Tp, _Up>)
            return _Strong_order::_S_noexcept<_Tp, _Up>();
          else
            return noexcept(bool(std::declval<_Tp>() == std::declval<_Up>()))
              && noexcept(bool(std::declval<_Tp>() < std::declval<_Up>()));
        }
 
    public:
      template<typename _Tp, __decayed_same_as<_Tp> _Up>
        requires __strongly_ordered<_Tp, _Up> || __op_eq_lt<_Tp, _Up>
        constexpr strong_ordering
        operator() [[nodiscard]] (_Tp&& __e, _Up&& __f) const
        noexcept(_S_noexcept<_Tp, _Up>())
        {
          if constexpr (__strongly_ordered<_Tp, _Up>)
            return _Strong_order{}(static_cast<_Tp&&>(__e),
                                   static_cast<_Up&&>(__f));
          else // __op_eq_lt<_Tp, _Up>
            return static_cast<_Tp&&>(__e) == static_cast<_Up&&>(__f)
              ? strong_ordering::equal
              : static_cast<_Tp&&>(__e) < static_cast<_Up&&>(__f)
              ? strong_ordering::less
              : strong_ordering::greater;
        }
    };
 
    class _Weak_fallback
    {
      template<typename _Tp, typename _Up>
        static constexpr bool
        _S_noexcept()
        {
          if constexpr (__weakly_ordered<_Tp, _Up>)
            return _Weak_order::_S_noexcept<_Tp, _Up>();
          else
            return noexcept(bool(std::declval<_Tp>() == std::declval<_Up>()))
              && noexcept(bool(std::declval<_Tp>() < std::declval<_Up>()));
        }
 
    public:
      template<typename _Tp, __decayed_same_as<_Tp> _Up>
        requires __weakly_ordered<_Tp, _Up> || __op_eq_lt<_Tp, _Up>
        constexpr weak_ordering
        operator() [[nodiscard]] (_Tp&& __e, _Up&& __f) const
        noexcept(_S_noexcept<_Tp, _Up>())
        {
          if constexpr (__weakly_ordered<_Tp, _Up>)
            return _Weak_order{}(static_cast<_Tp&&>(__e),
                                 static_cast<_Up&&>(__f));
          else // __op_eq_lt<_Tp, _Up>
            return static_cast<_Tp&&>(__e) == static_cast<_Up&&>(__f)
              ? weak_ordering::equivalent
              : static_cast<_Tp&&>(__e) < static_cast<_Up&&>(__f)
              ? weak_ordering::less
              : weak_ordering::greater;
        }
    };
 
    // _GLIBCXX_RESOLVE_LIB_DEFECTS
    // 3465. compare_partial_order_fallback requires F < E
    template<typename _Tp, typename _Up>
      concept __op_eq_lt_lt = __op_eq_lt<_Tp, _Up>
        && requires(_Tp&& __t, _Up&& __u)
        {
          { static_cast<_Up&&>(__u) < static_cast<_Tp&&>(__t) }
            -> convertible_to<bool>;
        };
 
    class _Partial_fallback
    {
      template<typename _Tp, typename _Up>
        static constexpr bool
        _S_noexcept()
        {
          if constexpr (__partially_ordered<_Tp, _Up>)
            return _Partial_order::_S_noexcept<_Tp, _Up>();
          else
            return noexcept(bool(std::declval<_Tp>() == std::declval<_Up>()))
              && noexcept(bool(std::declval<_Tp>() < std::declval<_Up>()));
        }
 
    public:
      template<typename _Tp, __decayed_same_as<_Tp> _Up>
        requires __partially_ordered<_Tp, _Up> || __op_eq_lt_lt<_Tp, _Up>
        constexpr partial_ordering
        operator() [[nodiscard]] (_Tp&& __e, _Up&& __f) const
        noexcept(_S_noexcept<_Tp, _Up>())
        {
          if constexpr (__partially_ordered<_Tp, _Up>)
            return _Partial_order{}(static_cast<_Tp&&>(__e),
                                    static_cast<_Up&&>(__f));
          else // __op_eq_lt_lt<_Tp, _Up>
            return static_cast<_Tp&&>(__e) == static_cast<_Up&&>(__f)
              ? partial_ordering::equivalent
              : static_cast<_Tp&&>(__e) < static_cast<_Up&&>(__f)
              ? partial_ordering::less
              : static_cast<_Up&&>(__f) < static_cast<_Tp&&>(__e)
              ? partial_ordering::greater
              : partial_ordering::unordered;
        }
    };
  } // namespace @endcond
 
  // [cmp.alg], comparison algorithms
 
  inline namespace _Cpo
  {
    inline constexpr __compare::_Strong_order strong_order{};
 
    inline constexpr __compare::_Weak_order weak_order{};
 
    inline constexpr __compare::_Partial_order partial_order{};
 
    inline constexpr __compare::_Strong_fallback
      compare_strong_order_fallback{};
 
    inline constexpr __compare::_Weak_fallback
      compare_weak_order_fallback{};
 
    inline constexpr __compare::_Partial_fallback
      compare_partial_order_fallback{};
  }
 
  /// @cond undocumented
  namespace __detail
  {
    // [expos.only.func] synth-three-way
    inline constexpr struct _Synth3way
    {
      template<typename _Tp, typename _Up>
        static constexpr bool
        _S_noexcept(const _Tp* __t = nullptr, const _Up* __u = nullptr)
        {
          if constexpr (three_way_comparable_with<_Tp, _Up>)
            return noexcept(*__t <=> *__u);
          else
            return noexcept(*__t < *__u) && noexcept(*__u < *__t);
        }
 
      template<typename _Tp, typename _Up>
        [[nodiscard]]
        constexpr auto
        operator()(const _Tp& __t, const _Up& __u) const
        noexcept(_S_noexcept<_Tp, _Up>())
        requires requires
        {
          { __t < __u } -> __boolean_testable;
          { __u < __t } -> __boolean_testable;
        }
        {
          if constexpr (three_way_comparable_with<_Tp, _Up>)
            return __t <=> __u;
          else
            {
              if (__t < __u)
                return weak_ordering::less;
              else if (__u < __t)
                return weak_ordering::greater;
              else
                return weak_ordering::equivalent;
            }
        }
    } __synth3way = {};
 
    // [expos.only.func] synth-three-way-result
    template<typename _Tp, typename _Up = _Tp>
      using __synth3way_t
        = decltype(__detail::__synth3way(std::declval<_Tp&>(),
                                         std::declval<_Up&>()));
  } // namespace __detail
  /// @endcond
 
#if __glibcxx_type_order >= 202506L // C++ >= 26
  /// Total ordering of types.
  /// @since C++26
 
  template<typename _Tp, typename _Up>
    struct type_order
    {
      static constexpr strong_ordering value = __builtin_type_order(_Tp, _Up);
      using value_type = strong_ordering;
      constexpr operator value_type() const noexcept { return value; }
      constexpr value_type operator()() const noexcept { return value; }
    };
 
  /// @ingroup variable_templates
  /// @since C++26
  template<typename _Tp, typename _Up>
    inline constexpr strong_ordering type_order_v
      = __builtin_type_order(_Tp, _Up);
#endif // __glibcxx_type_order >= 202506L
 
#endif // __cpp_lib_three_way_comparison >= 201907L
} // namespace std
 
#pragma GCC diagnostic pop
 
#endif // C++20
 
#endif // _COMPARE