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\n#include \n#include \n#include \n#include \n#include \n#include \n#include \n#include \n\n\nnamespace detail {\n // Trait for excluding incomplete types: See https://stackoverflow.com/a/44229779/9938686\n template \n consteval std::true_type is_complete(T*) noexcept;\n\n consteval std::false_type is_complete(...) noexcept;\n}\n\ntemplate \nusing is_complete = decltype(detail::is_complete(std::declval()));\n\ntemplate \nstatic constexpr bool is_complete_v = is_complete::value;\n\nnamespace detail {\n\n // Class for determining the corresponding unsigned integer type with equal length to the floating type\n template \n class UIntEquiv {\n protected:\n UIntEquiv() = delete;\n UIntEquiv(UIntEquiv const&) = delete;\n UIntEquiv(UIntEquiv&&) = delete;\n UIntEquiv& operator= (UIntEquiv const&) = delete;\n UIntEquiv& operator= (UIntEquiv&&) = delete;\n\n template* = nullptr>\n static consteval std::uint8_t determineUIntType() noexcept;\n template* = nullptr>\n static consteval std::uint16_t determineUIntType() noexcept;\n template* = nullptr>\n static consteval std::uint32_t determineUIntType() noexcept;\n template* = nullptr>\n static consteval std::uint64_t determineUIntType() noexcept;\n\n public:\n using type = decltype(determineUIntType());\n };\n \n // You can potentially add specialisation of UIntEquiv for longer unsigned integer types here (e.g. for long double support).\n // e.g. GCC's __uint128_t: https://gcc.gnu.org/onlinedocs/gcc/_005f_005fint128.html\n // or Boost: https://www.boost.org/doc/libs/1_67_0/libs/multiprecision/doc/html/boost_multiprecision/tut/ints/cpp_int.html\n //\n // template <>\n // class UIntEquiv {\n // public:\n // using type = __uint128_t;\n // };\n //\n // As long as std::numeric_limits is specialized for the corresponding floating type and your architecture respects IEEE754 and stores\n // your floating point numbers with little-endian the code should compile correctly.\n // Therefore in case you have particular proprietary floating types with a different mantissa and exponent such as \n // e.g. GCC's __float128: https://gcc.gnu.org/onlinedocs/gcc/Floating-Types.html\n // the fastest solution is probably to specialize the std::numeric_limits trait yourself.\n // Boost should already provide the fully specialized traits: https://www.boost.org/doc/libs/1_65_1/libs/multiprecision/doc/html/boost_multiprecision/tut/floats/float128.html\n\n template \n using UIntEquiv_t = typename UIntEquiv::type;\n\n // In case your floating type does not respect IEEE754 or is not stored with little endian you will have to specialise the entire \n // FloatTrait yourself:\n template \n class FloatTrait;\n\n // Specialised trait for floating point number types according to IEEE754 stored with little endian\n template \n requires std::is_floating_point_v && std::numeric_limits::is_iec559 && (std::endian::native == std::endian::little)\n class FloatTrait {\n public:\n static constexpr std::size_t number_of_bytes {sizeof(T)};\n static constexpr std::size_t number_of_bits {number_of_bytes*std::numeric_limits::digits};\n using Bytes = UIntEquiv_t;\n\n static constexpr std::size_t number_of_sign_bits {1};\n static constexpr std::size_t number_of_fraction_bits {std::numeric_limits::digits-1};\n static constexpr std::size_t number_of_exponent_bits {number_of_bits - number_of_sign_bits - number_of_fraction_bits};\n\n static constexpr Bytes sign_mask {Bytes{1} << (number_of_bits - 1)};\n static constexpr Bytes fraction_mask {~Bytes{0} >> (number_of_exponent_bits + 1)};\n static constexpr Bytes exponent_mask {~(sign_mask | fraction_mask)};\n\n static constexpr bool isNan(T const t) noexcept {\n auto const bytes {std::bit_cast(t)};\n auto const exponent_bytes {extractExponent(bytes)};\n auto const fraction_bytes {extractFraction(bytes)};\n return (exponent_bytes == exponent_mask) && (fraction_bytes != 0);\n }\n\n static constexpr bool isPosInf(T const t) noexcept {\n return isPos(t) && isInf(t);\n }\n\n static constexpr bool isNegInf(T const t) noexcept {\n return isNeg(t) && isInf(t);\n }\n\n static constexpr bool isNeg(T const t) noexcept {\n auto const bytes {std::bit_cast(t)};\n auto const sign_bytes {extractSign(bytes)};\n return sign_bytes != 0;\n }\n\n // Optional helper functions\n static constexpr bool isPos(T const t) noexcept {\n auto const bytes {std::bit_cast(t)};\n auto const sign_bytes {extractSign(bytes)};\n return sign_bytes == 0;\n }\n\n static constexpr bool isInf(T const t) noexcept {\n auto const bytes {std::bit_cast(t)};\n auto const exponent_bytes {extractExponent(bytes)};\n auto const fraction_bytes {extractFraction(bytes)};\n return (exponent_bytes == exponent_mask) && (fraction_bytes == 0);\n }\n\n static constexpr Bytes extractSign(Bytes const bytes) noexcept {\n return bytes & sign_mask;\n }\n\n static constexpr Bytes extractExponent(Bytes const bytes) noexcept {\n return bytes & exponent_mask;\n }\n\n static constexpr Bytes extractFraction(Bytes const bytes) noexcept {\n return bytes & fraction_mask;\n }\n\n protected:\n FloatTrait() = delete;\n FloatTrait(FloatTrait const&) = delete;\n FloatTrait(FloatTrait&&) = delete;\n FloatTrait& operator= (FloatTrait const&) = delete;\n FloatTrait& operator= (FloatTrait&&) = delete;\n };\n\n template \n requires is_complete_v>\n class FloatView {\n public:\n using Trait = FloatTrait;\n using Bytes = typename FloatTrait::Bytes;\n\n explicit constexpr FloatView(T const t) noexcept\n : value{t} {\n return;\n }\n FloatView() = default;\n FloatView(FloatView const&) = default;\n FloatView(FloatView&&) = default;\n FloatView& operator= (FloatView const&) = default;\n FloatView& operator= (FloatView&&) = default;\n\n constexpr bool isAlmostEqual(FloatView const rhs, std::uint8_t const max_distance = 4) const noexcept {\n if (Trait::isNan(value) || Trait::isNan(rhs.value)) {\n return false;\n } else if (Trait::isNegInf(value) != Trait::isNegInf(rhs.value)) {\n return false;\n } else if (Trait::isPosInf(value) != Trait::isPosInf(rhs.value)) {\n return false;\n }\n return computeDistance(value, rhs.value) <= max_distance;\n }\n\n protected:\n T value;\n\n static constexpr Bytes signMagnitudeToBiased(T const t) noexcept {\n auto const b {std::bit_cast(t)};\n if (Trait::isNeg(t)) {\n return ~b + Bytes{1};\n } else {\n return Trait::sign_mask | b;\n }\n }\n\n static constexpr Bytes computeDistance(T const a, T const b) noexcept {\n auto const biased1 = signMagnitudeToBiased(a);\n auto const biased2 = signMagnitudeToBiased(b);\n return (biased1 >= biased2) ? (biased1 - biased2) : (biased2 - biased1);\n }\n };\n\n}\n\ntemplate \nconstexpr bool isAlmostEqual(T const lhs, T const rhs, std::uint8_t const max_distance = 4) noexcept {\n detail::FloatView const a {lhs};\n detail::FloatView const b {rhs};\n return a.isAlmostEqual(b, max_distance);\n}\n\n\nint main() {\n std::vector> const vec {\n std::make_pair(+0.0, -0.0),\n std::make_pair(1.0, 1.0+1e-16),\n std::make_pair(1.0, 1.0+1e-15),\n std::make_pair(5.2, 5.2+1e-15),\n std::make_pair(-6045.0, -6045.0+1e-13),\n std::make_pair(std::numeric_limits::quiet_NaN(), std::numeric_limits::quiet_NaN()),\n std::make_pair(-std::numeric_limits::infinity(), -std::numeric_limits::infinity()),\n std::make_pair(std::numeric_limits::max(), std::numeric_limits::infinity())\n };\n\n for (auto const& [a, b]: vec) {\n bool const is_equal {isAlmostEqual(a, b)};\n std::cout << std::setprecision(std::numeric_limits::max_digits10) << a << \" == \" << b << \": \" << std::boolalpha << is_equal << std::endl;\n }\n\n 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