// Copyright 2020-2022 Junekey Jeon
//
// The contents of this file may be used under the terms of
// the Apache License v2.0 with LLVM Exceptions.
//
//    (See accompanying file LICENSE-Apache or copy at
//     https://llvm.org/foundation/relicensing/LICENSE.txt)
//
// Alternatively, the contents of this file may be used under the terms of
// the Boost Software License, Version 1.0.
//    (See accompanying file LICENSE-Boost or copy at
//     https://www.boost.org/LICENSE_1_0.txt)
//
// Unless required by applicable law or agreed to in writing, this software
// is distributed on an "AS IS" BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY
// KIND, either express or implied.


#ifndef JKJ_HEADER_DRAGONBOX
#define JKJ_HEADER_DRAGONBOX

#include <cassert>
#include <cstdint>
#include <cstring>
#include <limits>
#include <type_traits>

// Suppress additional buffer overrun check.
// I have no idea why MSVC thinks some functions here are vulnerable to the buffer overrun
// attacks. No, they aren't.
#if defined(__GNUC__) || defined(__clang__)
    #define JKJ_SAFEBUFFERS
    #define JKJ_FORCEINLINE inline __attribute__((always_inline))
#elif defined(_MSC_VER)
    #define JKJ_SAFEBUFFERS __declspec(safebuffers)
    #define JKJ_FORCEINLINE __forceinline
#else
    #define JKJ_SAFEBUFFERS
    #define JKJ_FORCEINLINE inline
#endif

#if defined(__has_builtin)
    #define JKJ_DRAGONBOX_HAS_BUILTIN(x) __has_builtin(x)
#else
    #define JKJ_DRAGONBOX_HAS_BUILTIN(x) false
#endif

#if defined(_MSC_VER)
    #include <intrin.h>
#endif

namespace jkj::dragonbox {
    namespace detail {
        template <class T>
        constexpr std::size_t
            physical_bits = sizeof(T) * std::numeric_limits<unsigned char>::digits;

        template <class T>
        constexpr std::size_t value_bits =
            std::numeric_limits<std::enable_if_t<std::is_unsigned_v<T>, T>>::digits;
    }

    // These classes expose encoding specs of IEEE-754-like floating-point formats.
    // Currently available formats are IEEE754-binary32 & IEEE754-binary64.

    struct ieee754_binary32 {
        static constexpr int significand_bits = 23;
        static constexpr int exponent_bits = 8;
        static constexpr int min_exponent = -126;
        static constexpr int max_exponent = 127;
        static constexpr int exponent_bias = -127;
        static constexpr int decimal_digits = 9;
    };
    struct ieee754_binary64 {
        static constexpr int significand_bits = 52;
        static constexpr int exponent_bits = 11;
        static constexpr int min_exponent = -1022;
        static constexpr int max_exponent = 1023;
        static constexpr int exponent_bias = -1023;
        static constexpr int decimal_digits = 17;
    };

    // A floating-point traits class defines ways to interpret a bit pattern of given size as an
    // encoding of floating-point number. This is a default implementation of such a traits class,
    // supporting ways to interpret 32-bits into a binary32-encoded floating-point number and to
    // interpret 64-bits into a binary64-encoded floating-point number. Users might specialize this
    // class to change the default behavior for certain types.
    template <class T>
    struct default_float_traits {
        // I don't know if there is a truly reliable way of detecting
        // IEEE-754 binary32/binary64 formats; I just did my best here.
        static_assert(std::numeric_limits<T>::is_iec559 && std::numeric_limits<T>::radix == 2 &&
                          (detail::physical_bits<T> == 32 || detail::physical_bits<T> == 64),
                      "default_ieee754_traits only works for 32-bits or 64-bits types "
                      "supporting binary32 or binary64 formats!");

        // The type that is being viewed.
        using type = T;

        // Refers to the format specification class.
        using format =
            std::conditional_t<detail::physical_bits<T> == 32, ieee754_binary32, ieee754_binary64>;

        // Defines an unsigned integer type that is large enough to carry a variable of type T.
        // Most of the operations will be done on this integer type.
        using carrier_uint =
            std::conditional_t<detail::physical_bits<T> == 32, std::uint32_t, std::uint64_t>;
        static_assert(sizeof(carrier_uint) == sizeof(T));

        // Number of bits in the above unsigned integer type.
        static constexpr int carrier_bits = int(detail::physical_bits<carrier_uint>);

        // Convert from carrier_uint into the original type.
        // Depending on the floating-point encoding format, this operation might not be possible for
        // some specific bit patterns. However, the contract is that u always denotes a
        // valid bit pattern, so this function must be assumed to be noexcept.
        static T carrier_to_float(carrier_uint u) noexcept {
            T x;
            std::memcpy(&x, &u, sizeof(carrier_uint));
            return x;
        }

        // Same as above.
        static carrier_uint float_to_carrier(T x) noexcept {
            carrier_uint u;
            std::memcpy(&u, &x, sizeof(carrier_uint));
            return u;
        }

        // Extract exponent bits from a bit pattern.
        // The result must be aligned to the LSB so that there is no additional zero paddings
        // on the right. This function does not do bias adjustment.
        static constexpr unsigned int extract_exponent_bits(carrier_uint u) noexcept {
            constexpr int significand_bits = format::significand_bits;
            constexpr int exponent_bits = format::exponent_bits;
            static_assert(detail::value_bits<unsigned int> > exponent_bits);
            constexpr auto exponent_bits_mask =
                (unsigned int)(((unsigned int)(1) << exponent_bits) - 1);
            return (unsigned int)(u >> significand_bits) & exponent_bits_mask;
        }

        // Extract significand bits from a bit pattern.
        // The result must be aligned to the LSB so that there is no additional zero paddings
        // on the right. The result does not contain the implicit bit.
        static constexpr carrier_uint extract_significand_bits(carrier_uint u) noexcept {
            constexpr auto mask = carrier_uint((carrier_uint(1) << format::significand_bits) - 1);
            return carrier_uint(u & mask);
        }

        // Remove the exponent bits and extract significand bits together with the sign bit.
        static constexpr carrier_uint remove_exponent_bits(carrier_uint u,
                                                           unsigned int exponent_bits) noexcept {
            return u ^ (carrier_uint(exponent_bits) << format::significand_bits);
        }

        // Shift the obtained signed significand bits to the left by 1 to remove the sign bit.
        static constexpr carrier_uint remove_sign_bit_and_shift(carrier_uint u) noexcept {
            return carrier_uint(carrier_uint(u) << 1);
        }

        // The actual value of exponent is obtained by adding this value to the extracted exponent
        // bits.
        static constexpr int exponent_bias =
            1 - (1 << (carrier_bits - format::significand_bits - 2));

        // Obtain the actual value of the binary exponent from the extracted exponent bits.
        static constexpr int binary_exponent(unsigned int exponent_bits) noexcept {
            if (exponent_bits == 0) {
                return format::min_exponent;
            }
            else {
                return int(exponent_bits) + format::exponent_bias;
            }
        }

        // Obtain the actual value of the binary exponent from the extracted significand bits and
        // exponent bits.
        static constexpr carrier_uint binary_significand(carrier_uint significand_bits,
                                                         unsigned int exponent_bits) noexcept {
            if (exponent_bits == 0) {
                return significand_bits;
            }
            else {
                return significand_bits | (carrier_uint(1) << format::significand_bits);
            }
        }


        /* Various boolean observer functions */

        static constexpr bool is_nonzero(carrier_uint u) noexcept { return (u << 1) != 0; }
        static constexpr bool is_positive(carrier_uint u) noexcept {
            constexpr auto sign_bit = carrier_uint(1)
                                      << (format::significand_bits + format::exponent_bits);
            return u < sign_bit;
        }
        static constexpr bool is_negative(carrier_uint u) noexcept { return !is_positive(u); }
        static constexpr bool is_finite(unsigned int exponent_bits) noexcept {
            constexpr unsigned int exponent_bits_all_set = (1u << format::exponent_bits) - 1;
            return exponent_bits != exponent_bits_all_set;
        }
        static constexpr bool has_all_zero_significand_bits(carrier_uint u) noexcept {
            return (u << 1) == 0;
        }
        static constexpr bool has_even_significand_bits(carrier_uint u) noexcept {
            return u % 2 == 0;
        }
    };

    // Convenient wrappers for floating-point traits classes.
    // In order to reduce the argument passing overhead, these classes should be as simple as
    // possible (e.g., no inheritance, no private non-static data member, etc.; this is an
    // unfortunate fact about common ABI convention).

    template <class T, class Traits = default_float_traits<T>>
    struct float_bits;

    template <class T, class Traits = default_float_traits<T>>
    struct signed_significand_bits;

    template <class T, class Traits>
    struct float_bits {
        using type = T;
        using traits_type = Traits;
        using carrier_uint = typename traits_type::carrier_uint;

        carrier_uint u;

        float_bits() = default;
        constexpr explicit float_bits(carrier_uint bit_pattern) noexcept : u{bit_pattern} {}
        constexpr explicit float_bits(T float_value) noexcept
            : u{traits_type::float_to_carrier(float_value)} {}

        constexpr T to_float() const noexcept { return traits_type::carrier_to_float(u); }

        // Extract exponent bits from a bit pattern.
        // The result must be aligned to the LSB so that there is no additional zero paddings
        // on the right. This function does not do bias adjustment.
        constexpr unsigned int extract_exponent_bits() const noexcept {
            return traits_type::extract_exponent_bits(u);
        }

        // Extract significand bits from a bit pattern.
        // The result must be aligned to the LSB so that there is no additional zero paddings
        // on the right. The result does not contain the implicit bit.
        constexpr carrier_uint extract_significand_bits() const noexcept {
            return traits_type::extract_significand_bits(u);
        }

        // Remove the exponent bits and extract significand bits together with the sign bit.
        constexpr auto remove_exponent_bits(unsigned int exponent_bits) const noexcept {
            return signed_significand_bits<type, traits_type>(
                traits_type::remove_exponent_bits(u, exponent_bits));
        }

        // Obtain the actual value of the binary exponent from the extracted exponent bits.
        static constexpr int binary_exponent(unsigned int exponent_bits) noexcept {
            return traits_type::binary_exponent(exponent_bits);
        }
        constexpr int binary_exponent() const noexcept {
            return binary_exponent(extract_exponent_bits());
        }

        // Obtain the actual value of the binary exponent from the extracted significand bits and
        // exponent bits.
        static constexpr carrier_uint binary_significand(carrier_uint significand_bits,
                                                         unsigned int exponent_bits) noexcept {
            return traits_type::binary_significand(significand_bits, exponent_bits);
        }
        constexpr carrier_uint binary_significand() const noexcept {
            return binary_significand(extract_significand_bits(), extract_exponent_bits());
        }

        constexpr bool is_nonzero() const noexcept { return traits_type::is_nonzero(u); }
        constexpr bool is_positive() const noexcept { return traits_type::is_positive(u); }
        constexpr bool is_negative() const noexcept { return traits_type::is_negative(u); }
        constexpr bool is_finite(unsigned int exponent_bits) const noexcept {
            return traits_type::is_finite(exponent_bits);
        }
        constexpr bool is_finite() const noexcept {
            return traits_type::is_finite(extract_exponent_bits());
        }
        constexpr bool has_even_significand_bits() const noexcept {
            return traits_type::has_even_significand_bits(u);
        }
    };

    template <class T, class Traits>
    struct signed_significand_bits {
        using type = T;
        using traits_type = Traits;
        using carrier_uint = typename traits_type::carrier_uint;

        carrier_uint u;

        signed_significand_bits() = default;
        constexpr explicit signed_significand_bits(carrier_uint bit_pattern) noexcept
            : u{bit_pattern} {}

        // Shift the obtained signed significand bits to the left by 1 to remove the sign bit.
        constexpr carrier_uint remove_sign_bit_and_shift() const noexcept {
            return traits_type::remove_sign_bit_and_shift(u);
        }

        constexpr bool is_positive() const noexcept { return traits_type::is_positive(u); }
        constexpr bool is_negative() const noexcept { return traits_type::is_negative(u); }
        constexpr bool has_all_zero_significand_bits() const noexcept {
            return traits_type::has_all_zero_significand_bits(u);
        }
        constexpr bool has_even_significand_bits() const noexcept {
            return traits_type::has_even_significand_bits(u);
        }
    };

    namespace detail {
        ////////////////////////////////////////////////////////////////////////////////////////
        // Bit operation intrinsics.
        ////////////////////////////////////////////////////////////////////////////////////////

        namespace bits {
            // Most compilers should be able to optimize this into the ROR instruction.
            inline std::uint32_t rotr(std::uint32_t n, std::uint32_t r) noexcept {
                r &= 31;
                return (n >> r) | (n << (32 - r));
            }
            inline std::uint64_t rotr(std::uint64_t n, std::uint32_t r) noexcept {
                r &= 63;
                return (n >> r) | (n << (64 - r));
            }
        }

        ////////////////////////////////////////////////////////////////////////////////////////
        // Utilities for wide unsigned integer arithmetic.
        ////////////////////////////////////////////////////////////////////////////////////////

        namespace wuint {
            // Compilers might support built-in 128-bit integer types. However, it seems that
            // emulating them with a pair of 64-bit integers actually produces a better code,
            // so we avoid using those built-ins. That said, they are still useful for
            // implementing 64-bit x 64-bit -> 128-bit multiplication.

            // clang-format off
#if defined(__SIZEOF_INT128__)
		// To silence "error: ISO C++ does not support '__int128' for 'type name'
		// [-Wpedantic]"
#if defined(__GNUC__)
			__extension__
#endif
				using builtin_uint128_t = unsigned __int128;
#endif
            // clang-format on

            struct uint128 {
                uint128() = default;

                std::uint64_t high_;
                std::uint64_t low_;

                constexpr uint128(std::uint64_t high, std::uint64_t low) noexcept
                    : high_{high}, low_{low} {}

                constexpr std::uint64_t high() const noexcept { return high_; }
                constexpr std::uint64_t low() const noexcept { return low_; }

                uint128& operator+=(std::uint64_t n) & noexcept {
#if JKJ_DRAGONBOX_HAS_BUILTIN(__builtin_addcll)
                    unsigned long long carry;
                    low_ = __builtin_addcll(low_, n, 0, &carry);
                    high_ = __builtin_addcll(high_, 0, carry, &carry);
#elif JKJ_DRAGONBOX_HAS_BUILTIN(__builtin_ia32_addcarryx_u64)
                    unsigned long long result;
                    auto carry = __builtin_ia32_addcarryx_u64(0, low_, n, &result);
                    low_ = result;
                    __builtin_ia32_addcarryx_u64(carry, high_, 0, &result);
                    high_ = result;
#elif defined(_MSC_VER) && defined(_M_X64)
                    auto carry = _addcarry_u64(0, low_, n, &low_);
                    _addcarry_u64(carry, high_, 0, &high_);
#else
                    auto sum = low_ + n;
                    high_ += (sum < low_ ? 1 : 0);
                    low_ = sum;
#endif
                    return *this;
                }
            };

            static inline std::uint64_t umul64(std::uint32_t x, std::uint32_t y) noexcept {
#if defined(_MSC_VER) && defined(_M_IX86)
                return __emulu(x, y);
#else
                return x * std::uint64_t(y);
#endif
            }

            // Get 128-bit result of multiplication of two 64-bit unsigned integers.
            JKJ_SAFEBUFFERS inline uint128 umul128(std::uint64_t x, std::uint64_t y) noexcept {
#if defined(__SIZEOF_INT128__)
                auto result = builtin_uint128_t(x) * builtin_uint128_t(y);
                return {std::uint64_t(result >> 64), std::uint64_t(result)};
#elif defined(_MSC_VER) && defined(_M_X64)
                uint128 result;
                result.low_ = _umul128(x, y, &result.high_);
                return result;
#else
                auto a = std::uint32_t(x >> 32);
                auto b = std::uint32_t(x);
                auto c = std::uint32_t(y >> 32);
                auto d = std::uint32_t(y);

                auto ac = umul64(a, c);
                auto bc = umul64(b, c);
                auto ad = umul64(a, d);
                auto bd = umul64(b, d);

                auto intermediate = (bd >> 32) + std::uint32_t(ad) + std::uint32_t(bc);

                return {ac + (intermediate >> 32) + (ad >> 32) + (bc >> 32),
                        (intermediate << 32) + std::uint32_t(bd)};
#endif
            }

            JKJ_SAFEBUFFERS inline std::uint64_t umul128_upper64(std::uint64_t x,
                                                                 std::uint64_t y) noexcept {
#if defined(__SIZEOF_INT128__)
                auto result = builtin_uint128_t(x) * builtin_uint128_t(y);
                return std::uint64_t(result >> 64);
#elif defined(_MSC_VER) && defined(_M_X64)
                return __umulh(x, y);
#else
                auto a = std::uint32_t(x >> 32);
                auto b = std::uint32_t(x);
                auto c = std::uint32_t(y >> 32);
                auto d = std::uint32_t(y);

                auto ac = umul64(a, c);
                auto bc = umul64(b, c);
                auto ad = umul64(a, d);
                auto bd = umul64(b, d);

                auto intermediate = (bd >> 32) + std::uint32_t(ad) + std::uint32_t(bc);

                return ac + (intermediate >> 32) + (ad >> 32) + (bc >> 32);
#endif
            }

            // Get upper 128-bits of multiplication of a 64-bit unsigned integer and a 128-bit
            // unsigned integer.
            JKJ_SAFEBUFFERS inline uint128 umul192_upper128(std::uint64_t x, uint128 y) noexcept {
                auto r = umul128(x, y.high());
                r += umul128_upper64(x, y.low());
                return r;
            }

            // Get upper 64-bits of multiplication of a 32-bit unsigned integer and a 64-bit
            // unsigned integer.
            inline std::uint64_t umul96_upper64(std::uint32_t x, std::uint64_t y) noexcept {
#if defined(__SIZEOF_INT128__) || (defined(_MSC_VER) && defined(_M_X64))
                return umul128_upper64(std::uint64_t(x) << 32, y);
#else
                auto yh = std::uint32_t(y >> 32);
                auto yl = std::uint32_t(y);

                auto xyh = umul64(x, yh);
                auto xyl = umul64(x, yl);

                return xyh + (xyl >> 32);
#endif
            }

            // Get lower 128-bits of multiplication of a 64-bit unsigned integer and a 128-bit
            // unsigned integer.
            JKJ_SAFEBUFFERS inline uint128 umul192_lower128(std::uint64_t x, uint128 y) noexcept {
                auto high = x * y.high();
                auto high_low = umul128(x, y.low());
                return {high + high_low.high(), high_low.low()};
            }

            // Get lower 64-bits of multiplication of a 32-bit unsigned integer and a 64-bit
            // unsigned integer.
            inline std::uint64_t umul96_lower64(std::uint32_t x, std::uint64_t y) noexcept {
                return x * y;
            }
        }

        ////////////////////////////////////////////////////////////////////////////////////////
        // Some simple utilities for constexpr computation.
        ////////////////////////////////////////////////////////////////////////////////////////

        template <int k, class Int>
        constexpr Int compute_power(Int a) noexcept {
            static_assert(k >= 0);
            Int p = 1;
            for (int i = 0; i < k; ++i) {
                p *= a;
            }
            return p;
        }

        template <int a, class UInt>
        constexpr int count_factors(UInt n) noexcept {
            static_assert(a > 1);
            int c = 0;
            while (n % a == 0) {
                n /= a;
                ++c;
            }
            return c;
        }

        ////////////////////////////////////////////////////////////////////////////////////////
        // Utilities for fast/constexpr log computation.
        ////////////////////////////////////////////////////////////////////////////////////////

        namespace log {
            static_assert((-1 >> 1) == -1, "right-shift for signed integers must be arithmetic");

            // Compute floor(e * c - s).
            enum class multiply : std::uint32_t {};
            enum class subtract : std::uint32_t {};
            enum class shift : std::size_t {};
            enum class min_exponent : std::int32_t {};
            enum class max_exponent : std::int32_t {};

            template <multiply m, subtract f, shift k, min_exponent e_min, max_exponent e_max>
            constexpr int compute(int e) noexcept {
                assert(std::int32_t(e_min) <= e && e <= std::int32_t(e_max));
                return int((std::int32_t(e) * std::int32_t(m) - std::int32_t(f)) >> std::size_t(k));
            }

            // For constexpr computation.
            // Returns -1 when n = 0.
            template <class UInt>
            constexpr int floor_log2(UInt n) noexcept {
                int count = -1;
                while (n != 0) {
                    ++count;
                    n >>= 1;
                }
                return count;
            }

            static constexpr int floor_log10_pow2_min_exponent = -2620;
            static constexpr int floor_log10_pow2_max_exponent = 2620;
            constexpr int floor_log10_pow2(int e) noexcept {
                using namespace log;
                return compute<multiply(315653), subtract(0), shift(20),
                               min_exponent(floor_log10_pow2_min_exponent),
                               max_exponent(floor_log10_pow2_max_exponent)>(e);
            }

            static constexpr int floor_log2_pow10_min_exponent = -1233;
            static constexpr int floor_log2_pow10_max_exponent = 1233;
            constexpr int floor_log2_pow10(int e) noexcept {
                using namespace log;
                return compute<multiply(1741647), subtract(0), shift(19),
                               min_exponent(floor_log2_pow10_min_exponent),
                               max_exponent(floor_log2_pow10_max_exponent)>(e);
            }

            static constexpr int floor_log10_pow2_minus_log10_4_over_3_min_exponent = -2985;
            static constexpr int floor_log10_pow2_minus_log10_4_over_3_max_exponent = 2936;
            constexpr int floor_log10_pow2_minus_log10_4_over_3(int e) noexcept {
                using namespace log;
                return compute<multiply(631305), subtract(261663), shift(21),
                               min_exponent(floor_log10_pow2_minus_log10_4_over_3_min_exponent),
                               max_exponent(floor_log10_pow2_minus_log10_4_over_3_max_exponent)>(e);
            }

            static constexpr int floor_log5_pow2_min_exponent = -1831;
            static constexpr int floor_log5_pow2_max_exponent = 1831;
            constexpr int floor_log5_pow2(int e) noexcept {
                using namespace log;
                return compute<multiply(225799), subtract(0), shift(19),
                               min_exponent(floor_log5_pow2_min_exponent),
                               max_exponent(floor_log5_pow2_max_exponent)>(e);
            }

            static constexpr int floor_log5_pow2_minus_log5_3_min_exponent = -3543;
            static constexpr int floor_log5_pow2_minus_log5_3_max_exponent = 2427;
            constexpr int floor_log5_pow2_minus_log5_3(int e) noexcept {
                using namespace log;
                return compute<multiply(451597), subtract(715764), shift(20),
                               min_exponent(floor_log5_pow2_minus_log5_3_min_exponent),
                               max_exponent(floor_log5_pow2_minus_log5_3_max_exponent)>(e);
            }
        }

        ////////////////////////////////////////////////////////////////////////////////////////
        // Utilities for fast divisibility tests.
        ////////////////////////////////////////////////////////////////////////////////////////

        namespace div {
            // Replace n by floor(n / 10^N).
            // Returns true if and only if n is divisible by 10^N.
            // Precondition: n <= 10^(N+1)
            // !!It takes an in-out parameter!!
            template <int N>
            struct divide_by_pow10_info;

            template <>
            struct divide_by_pow10_info<1> {
                static constexpr std::uint32_t magic_number = 6554;
                static constexpr int shift_amount = 16;
            };

            template <>
            struct divide_by_pow10_info<2> {
                static constexpr std::uint32_t magic_number = 656;
                static constexpr int shift_amount = 16;
            };

            template <int N>
            constexpr bool check_divisibility_and_divide_by_pow10(std::uint32_t& n) noexcept {
                // Make sure the computation for max_n does not overflow.
                static_assert(N + 1 <= log::floor_log10_pow2(31));
                assert(n <= compute_power<N + 1>(std::uint32_t(10)));

                using info = divide_by_pow10_info<N>;
                n *= info::magic_number;

                constexpr auto mask = std::uint32_t(std::uint32_t(1) << info::shift_amount) - 1;
                bool result = ((n & mask) < info::magic_number);

                n >>= info::shift_amount;
                return result;
            }

            // Compute floor(n / 10^N) for small n and N.
            // Precondition: n <= 10^(N+1)
            template <int N>
            constexpr std::uint32_t small_division_by_pow10(std::uint32_t n) noexcept {
                // Make sure the computation for max_n does not overflow.
                static_assert(N + 1 <= log::floor_log10_pow2(31));
                assert(n <= compute_power<N + 1>(std::uint32_t(10)));

                return (n * divide_by_pow10_info<N>::magic_number) >>
                       divide_by_pow10_info<N>::shift_amount;
            }

            // Compute floor(n / 10^N) for small N.
            // Precondition: n <= n_max
            template <int N, class UInt, UInt n_max>
            constexpr UInt divide_by_pow10(UInt n) noexcept {
                static_assert(N >= 0);

                // Specialize for 32-bit division by 100.
                // Compiler is supposed to generate the identical code for just writing
                // "n / 100", but for some reason MSVC generates an inefficient code
                // (mul + mov for no apparent reason, instead of single imul),
                // so we does this manually.
                if constexpr (std::is_same_v<UInt, std::uint32_t> && N == 2) {
                    return std::uint32_t(wuint::umul64(n, std::uint32_t(1374389535)) >> 37);
                }
                // Specialize for 64-bit division by 1000.
                // Ensure that the correctness condition is met.
                if constexpr (std::is_same_v<UInt, std::uint64_t> && N == 3 &&
                              n_max <= std::uint64_t(15534100272597517998ull)) {
                    return wuint::umul128_upper64(n, std::uint64_t(2361183241434822607ull)) >> 7;
                }
                else {
                    constexpr auto divisor = compute_power<N>(UInt(10));
                    return n / divisor;
                }
            }
        }
    }

    ////////////////////////////////////////////////////////////////////////////////////////
    // Return types for the main interface function.
    ////////////////////////////////////////////////////////////////////////////////////////

    template <class UInt, bool is_signed, bool trailing_zero_flag>
    struct decimal_fp;

    template <class UInt>
    struct decimal_fp<UInt, false, false> {
        using carrier_uint = UInt;

        carrier_uint significand;
        int exponent;
    };

    template <class UInt>
    struct decimal_fp<UInt, true, false> {
        using carrier_uint = UInt;

        carrier_uint significand;
        int exponent;
        bool is_negative;
    };

    template <class UInt>
    struct decimal_fp<UInt, false, true> {
        using carrier_uint = UInt;

        carrier_uint significand;
        int exponent;
        bool may_have_trailing_zeros;
    };

    template <class UInt>
    struct decimal_fp<UInt, true, true> {
        using carrier_uint = UInt;

        carrier_uint significand;
        int exponent;
        bool is_negative;
        bool may_have_trailing_zeros;
    };

    template <class UInt>
    using unsigned_decimal_fp = decimal_fp<UInt, false, false>;

    template <class UInt>
    using signed_decimal_fp = decimal_fp<UInt, true, false>;


    ////////////////////////////////////////////////////////////////////////////////////////
    // Computed cache entries.
    ////////////////////////////////////////////////////////////////////////////////////////

    namespace detail {
        template <class FloatFormat>
        struct cache_holder;

        template <>
        struct cache_holder<ieee754_binary32> {
            using cache_entry_type = std::uint64_t;
            static constexpr int cache_bits = 64;
            static constexpr int min_k = -31;
            static constexpr int max_k = 46;
            static constexpr cache_entry_type cache[] = {
                0x81ceb32c4b43fcf5, 0xa2425ff75e14fc32, 0xcad2f7f5359a3b3f, 0xfd87b5f28300ca0e,
                0x9e74d1b791e07e49, 0xc612062576589ddb, 0xf79687aed3eec552, 0x9abe14cd44753b53,
                0xc16d9a0095928a28, 0xf1c90080baf72cb2, 0x971da05074da7bef, 0xbce5086492111aeb,
                0xec1e4a7db69561a6, 0x9392ee8e921d5d08, 0xb877aa3236a4b44a, 0xe69594bec44de15c,
                0x901d7cf73ab0acda, 0xb424dc35095cd810, 0xe12e13424bb40e14, 0x8cbccc096f5088cc,
                0xafebff0bcb24aaff, 0xdbe6fecebdedd5bf, 0x89705f4136b4a598, 0xabcc77118461cefd,
                0xd6bf94d5e57a42bd, 0x8637bd05af6c69b6, 0xa7c5ac471b478424, 0xd1b71758e219652c,
                0x83126e978d4fdf3c, 0xa3d70a3d70a3d70b, 0xcccccccccccccccd, 0x8000000000000000,
                0xa000000000000000, 0xc800000000000000, 0xfa00000000000000, 0x9c40000000000000,
                0xc350000000000000, 0xf424000000000000, 0x9896800000000000, 0xbebc200000000000,
                0xee6b280000000000, 0x9502f90000000000, 0xba43b74000000000, 0xe8d4a51000000000,
                0x9184e72a00000000, 0xb5e620f480000000, 0xe35fa931a0000000, 0x8e1bc9bf04000000,
                0xb1a2bc2ec5000000, 0xde0b6b3a76400000, 0x8ac7230489e80000, 0xad78ebc5ac620000,
                0xd8d726b7177a8000, 0x878678326eac9000, 0xa968163f0a57b400, 0xd3c21bcecceda100,
                0x84595161401484a0, 0xa56fa5b99019a5c8, 0xcecb8f27f4200f3a, 0x813f3978f8940985,
                0xa18f07d736b90be6, 0xc9f2c9cd04674edf, 0xfc6f7c4045812297, 0x9dc5ada82b70b59e,
                0xc5371912364ce306, 0xf684df56c3e01bc7, 0x9a130b963a6c115d, 0xc097ce7bc90715b4,
                0xf0bdc21abb48db21, 0x96769950b50d88f5, 0xbc143fa4e250eb32, 0xeb194f8e1ae525fe,
                0x92efd1b8d0cf37bf, 0xb7abc627050305ae, 0xe596b7b0c643c71a, 0x8f7e32ce7bea5c70,
                0xb35dbf821ae4f38c, 0xe0352f62a19e306f};
        };

        template <>
        struct cache_holder<ieee754_binary64> {
            using cache_entry_type = wuint::uint128;
            static constexpr int cache_bits = 128;
            static constexpr int min_k = -292;
            static constexpr int max_k = 326;
            static constexpr cache_entry_type cache[] = {
                {0xff77b1fcbebcdc4f, 0x25e8e89c13bb0f7b}, {0x9faacf3df73609b1, 0x77b191618c54e9ad},
                {0xc795830d75038c1d, 0xd59df5b9ef6a2418}, {0xf97ae3d0d2446f25, 0x4b0573286b44ad1e},
                {0x9becce62836ac577, 0x4ee367f9430aec33}, {0xc2e801fb244576d5, 0x229c41f793cda740},
                {0xf3a20279ed56d48a, 0x6b43527578c11110}, {0x9845418c345644d6, 0x830a13896b78aaaa},
                {0xbe5691ef416bd60c, 0x23cc986bc656d554}, {0xedec366b11c6cb8f, 0x2cbfbe86b7ec8aa9},
                {0x94b3a202eb1c3f39, 0x7bf7d71432f3d6aa}, {0xb9e08a83a5e34f07, 0xdaf5ccd93fb0cc54},
                {0xe858ad248f5c22c9, 0xd1b3400f8f9cff69}, {0x91376c36d99995be, 0x23100809b9c21fa2},
                {0xb58547448ffffb2d, 0xabd40a0c2832a78b}, {0xe2e69915b3fff9f9, 0x16c90c8f323f516d},
                {0x8dd01fad907ffc3b, 0xae3da7d97f6792e4}, {0xb1442798f49ffb4a, 0x99cd11cfdf41779d},
                {0xdd95317f31c7fa1d, 0x40405643d711d584}, {0x8a7d3eef7f1cfc52, 0x482835ea666b2573},
                {0xad1c8eab5ee43b66, 0xda3243650005eed0}, {0xd863b256369d4a40, 0x90bed43e40076a83},
                {0x873e4f75e2224e68, 0x5a7744a6e804a292}, {0xa90de3535aaae202, 0x711515d0a205cb37},
                {0xd3515c2831559a83, 0x0d5a5b44ca873e04}, {0x8412d9991ed58091, 0xe858790afe9486c3},
                {0xa5178fff668ae0b6, 0x626e974dbe39a873}, {0xce5d73ff402d98e3, 0xfb0a3d212dc81290},
                {0x80fa687f881c7f8e, 0x7ce66634bc9d0b9a}, {0xa139029f6a239f72, 0x1c1fffc1ebc44e81},
                {0xc987434744ac874e, 0xa327ffb266b56221}, {0xfbe9141915d7a922, 0x4bf1ff9f0062baa9},
                {0x9d71ac8fada6c9b5, 0x6f773fc3603db4aa}, {0xc4ce17b399107c22, 0xcb550fb4384d21d4},
                {0xf6019da07f549b2b, 0x7e2a53a146606a49}, {0x99c102844f94e0fb, 0x2eda7444cbfc426e},
                {0xc0314325637a1939, 0xfa911155fefb5309}, {0xf03d93eebc589f88, 0x793555ab7eba27cb},
                {0x96267c7535b763b5, 0x4bc1558b2f3458df}, {0xbbb01b9283253ca2, 0x9eb1aaedfb016f17},
                {0xea9c227723ee8bcb, 0x465e15a979c1cadd}, {0x92a1958a7675175f, 0x0bfacd89ec191eca},
                {0xb749faed14125d36, 0xcef980ec671f667c}, {0xe51c79a85916f484, 0x82b7e12780e7401b},
                {0x8f31cc0937ae58d2, 0xd1b2ecb8b0908811}, {0xb2fe3f0b8599ef07, 0x861fa7e6dcb4aa16},
                {0xdfbdcece67006ac9, 0x67a791e093e1d49b}, {0x8bd6a141006042bd, 0xe0c8bb2c5c6d24e1},
                {0xaecc49914078536d, 0x58fae9f773886e19}, {0xda7f5bf590966848, 0xaf39a475506a899f},
                {0x888f99797a5e012d, 0x6d8406c952429604}, {0xaab37fd7d8f58178, 0xc8e5087ba6d33b84},
                {0xd5605fcdcf32e1d6, 0xfb1e4a9a90880a65}, {0x855c3be0a17fcd26, 0x5cf2eea09a550680},
                {0xa6b34ad8c9dfc06f, 0xf42faa48c0ea481f}, {0xd0601d8efc57b08b, 0xf13b94daf124da27},
                {0x823c12795db6ce57, 0x76c53d08d6b70859}, {0xa2cb1717b52481ed, 0x54768c4b0c64ca6f},
                {0xcb7ddcdda26da268, 0xa9942f5dcf7dfd0a}, {0xfe5d54150b090b02, 0xd3f93b35435d7c4d},
                {0x9efa548d26e5a6e1, 0xc47bc5014a1a6db0}, {0xc6b8e9b0709f109a, 0x359ab6419ca1091c},
                {0xf867241c8cc6d4c0, 0xc30163d203c94b63}, {0x9b407691d7fc44f8, 0x79e0de63425dcf1e},
                {0xc21094364dfb5636, 0x985915fc12f542e5}, {0xf294b943e17a2bc4, 0x3e6f5b7b17b2939e},
                {0x979cf3ca6cec5b5a, 0xa705992ceecf9c43}, {0xbd8430bd08277231, 0x50c6ff782a838354},
                {0xece53cec4a314ebd, 0xa4f8bf5635246429}, {0x940f4613ae5ed136, 0x871b7795e136be9a},
                {0xb913179899f68584, 0x28e2557b59846e40}, {0xe757dd7ec07426e5, 0x331aeada2fe589d0},
                {0x9096ea6f3848984f, 0x3ff0d2c85def7622}, {0xb4bca50b065abe63, 0x0fed077a756b53aa},
                {0xe1ebce4dc7f16dfb, 0xd3e8495912c62895}, {0x8d3360f09cf6e4bd, 0x64712dd7abbbd95d},
                {0xb080392cc4349dec, 0xbd8d794d96aacfb4}, {0xdca04777f541c567, 0xecf0d7a0fc5583a1},
                {0x89e42caaf9491b60, 0xf41686c49db57245}, {0xac5d37d5b79b6239, 0x311c2875c522ced6},
                {0xd77485cb25823ac7, 0x7d633293366b828c}, {0x86a8d39ef77164bc, 0xae5dff9c02033198},
                {0xa8530886b54dbdeb, 0xd9f57f830283fdfd}, {0xd267caa862a12d66, 0xd072df63c324fd7c},
                {0x8380dea93da4bc60, 0x4247cb9e59f71e6e}, {0xa46116538d0deb78, 0x52d9be85f074e609},
                {0xcd795be870516656, 0x67902e276c921f8c}, {0x806bd9714632dff6, 0x00ba1cd8a3db53b7},
                {0xa086cfcd97bf97f3, 0x80e8a40eccd228a5}, {0xc8a883c0fdaf7df0, 0x6122cd128006b2ce},
                {0xfad2a4b13d1b5d6c, 0x796b805720085f82}, {0x9cc3a6eec6311a63, 0xcbe3303674053bb1},
                {0xc3f490aa77bd60fc, 0xbedbfc4411068a9d}, {0xf4f1b4d515acb93b, 0xee92fb5515482d45},
                {0x991711052d8bf3c5, 0x751bdd152d4d1c4b}, {0xbf5cd54678eef0b6, 0xd262d45a78a0635e},
                {0xef340a98172aace4, 0x86fb897116c87c35}, {0x9580869f0e7aac0e, 0xd45d35e6ae3d4da1},
                {0xbae0a846d2195712, 0x8974836059cca10a}, {0xe998d258869facd7, 0x2bd1a438703fc94c},
                {0x91ff83775423cc06, 0x7b6306a34627ddd0}, {0xb67f6455292cbf08, 0x1a3bc84c17b1d543},
                {0xe41f3d6a7377eeca, 0x20caba5f1d9e4a94}, {0x8e938662882af53e, 0x547eb47b7282ee9d},
                {0xb23867fb2a35b28d, 0xe99e619a4f23aa44}, {0xdec681f9f4c31f31, 0x6405fa00e2ec94d5},
                {0x8b3c113c38f9f37e, 0xde83bc408dd3dd05}, {0xae0b158b4738705e, 0x9624ab50b148d446},
                {0xd98ddaee19068c76, 0x3badd624dd9b0958}, {0x87f8a8d4cfa417c9, 0xe54ca5d70a80e5d7},
                {0xa9f6d30a038d1dbc, 0x5e9fcf4ccd211f4d}, {0xd47487cc8470652b, 0x7647c32000696720},
                {0x84c8d4dfd2c63f3b, 0x29ecd9f40041e074}, {0xa5fb0a17c777cf09, 0xf468107100525891},
                {0xcf79cc9db955c2cc, 0x7182148d4066eeb5}, {0x81ac1fe293d599bf, 0xc6f14cd848405531},
                {0xa21727db38cb002f, 0xb8ada00e5a506a7d}, {0xca9cf1d206fdc03b, 0xa6d90811f0e4851d},
                {0xfd442e4688bd304a, 0x908f4a166d1da664}, {0x9e4a9cec15763e2e, 0x9a598e4e043287ff},
                {0xc5dd44271ad3cdba, 0x40eff1e1853f29fe}, {0xf7549530e188c128, 0xd12bee59e68ef47d},
                {0x9a94dd3e8cf578b9, 0x82bb74f8301958cf}, {0xc13a148e3032d6e7, 0xe36a52363c1faf02},
                {0xf18899b1bc3f8ca1, 0xdc44e6c3cb279ac2}, {0x96f5600f15a7b7e5, 0x29ab103a5ef8c0ba},
                {0xbcb2b812db11a5de, 0x7415d448f6b6f0e8}, {0xebdf661791d60f56, 0x111b495b3464ad22},
                {0x936b9fcebb25c995, 0xcab10dd900beec35}, {0xb84687c269ef3bfb, 0x3d5d514f40eea743},
                {0xe65829b3046b0afa, 0x0cb4a5a3112a5113}, {0x8ff71a0fe2c2e6dc, 0x47f0e785eaba72ac},
                {0xb3f4e093db73a093, 0x59ed216765690f57}, {0xe0f218b8d25088b8, 0x306869c13ec3532d},
                {0x8c974f7383725573, 0x1e414218c73a13fc}, {0xafbd2350644eeacf, 0xe5d1929ef90898fb},
                {0xdbac6c247d62a583, 0xdf45f746b74abf3a}, {0x894bc396ce5da772, 0x6b8bba8c328eb784},
                {0xab9eb47c81f5114f, 0x066ea92f3f326565}, {0xd686619ba27255a2, 0xc80a537b0efefebe},
                {0x8613fd0145877585, 0xbd06742ce95f5f37}, {0xa798fc4196e952e7, 0x2c48113823b73705},
                {0xd17f3b51fca3a7a0, 0xf75a15862ca504c6}, {0x82ef85133de648c4, 0x9a984d73dbe722fc},
                {0xa3ab66580d5fdaf5, 0xc13e60d0d2e0ebbb}, {0xcc963fee10b7d1b3, 0x318df905079926a9},
                {0xffbbcfe994e5c61f, 0xfdf17746497f7053}, {0x9fd561f1fd0f9bd3, 0xfeb6ea8bedefa634},
                {0xc7caba6e7c5382c8, 0xfe64a52ee96b8fc1}, {0xf9bd690a1b68637b, 0x3dfdce7aa3c673b1},
                {0x9c1661a651213e2d, 0x06bea10ca65c084f}, {0xc31bfa0fe5698db8, 0x486e494fcff30a63},
                {0xf3e2f893dec3f126, 0x5a89dba3c3efccfb}, {0x986ddb5c6b3a76b7, 0xf89629465a75e01d},
                {0xbe89523386091465, 0xf6bbb397f1135824}, {0xee2ba6c0678b597f, 0x746aa07ded582e2d},
                {0x94db483840b717ef, 0xa8c2a44eb4571cdd}, {0xba121a4650e4ddeb, 0x92f34d62616ce414},
                {0xe896a0d7e51e1566, 0x77b020baf9c81d18}, {0x915e2486ef32cd60, 0x0ace1474dc1d122f},
                {0xb5b5ada8aaff80b8, 0x0d819992132456bb}, {0xe3231912d5bf60e6, 0x10e1fff697ed6c6a},
                {0x8df5efabc5979c8f, 0xca8d3ffa1ef463c2}, {0xb1736b96b6fd83b3, 0xbd308ff8a6b17cb3},
                {0xddd0467c64bce4a0, 0xac7cb3f6d05ddbdf}, {0x8aa22c0dbef60ee4, 0x6bcdf07a423aa96c},
                {0xad4ab7112eb3929d, 0x86c16c98d2c953c7}, {0xd89d64d57a607744, 0xe871c7bf077ba8b8},
                {0x87625f056c7c4a8b, 0x11471cd764ad4973}, {0xa93af6c6c79b5d2d, 0xd598e40d3dd89bd0},
                {0xd389b47879823479, 0x4aff1d108d4ec2c4}, {0x843610cb4bf160cb, 0xcedf722a585139bb},
                {0xa54394fe1eedb8fe, 0xc2974eb4ee658829}, {0xce947a3da6a9273e, 0x733d226229feea33},
                {0x811ccc668829b887, 0x0806357d5a3f5260}, {0xa163ff802a3426a8, 0xca07c2dcb0cf26f8},
                {0xc9bcff6034c13052, 0xfc89b393dd02f0b6}, {0xfc2c3f3841f17c67, 0xbbac2078d443ace3},
                {0x9d9ba7832936edc0, 0xd54b944b84aa4c0e}, {0xc5029163f384a931, 0x0a9e795e65d4df12},
                {0xf64335bcf065d37d, 0x4d4617b5ff4a16d6}, {0x99ea0196163fa42e, 0x504bced1bf8e4e46},
                {0xc06481fb9bcf8d39, 0xe45ec2862f71e1d7}, {0xf07da27a82c37088, 0x5d767327bb4e5a4d},
                {0x964e858c91ba2655, 0x3a6a07f8d510f870}, {0xbbe226efb628afea, 0x890489f70a55368c},
                {0xeadab0aba3b2dbe5, 0x2b45ac74ccea842f}, {0x92c8ae6b464fc96f, 0x3b0b8bc90012929e},
                {0xb77ada0617e3bbcb, 0x09ce6ebb40173745}, {0xe55990879ddcaabd, 0xcc420a6a101d0516},
                {0x8f57fa54c2a9eab6, 0x9fa946824a12232e}, {0xb32df8e9f3546564, 0x47939822dc96abfa},
                {0xdff9772470297ebd, 0x59787e2b93bc56f8}, {0x8bfbea76c619ef36, 0x57eb4edb3c55b65b},
                {0xaefae51477a06b03, 0xede622920b6b23f2}, {0xdab99e59958885c4, 0xe95fab368e45ecee},
                {0x88b402f7fd75539b, 0x11dbcb0218ebb415}, {0xaae103b5fcd2a881, 0xd652bdc29f26a11a},
                {0xd59944a37c0752a2, 0x4be76d3346f04960}, {0x857fcae62d8493a5, 0x6f70a4400c562ddc},
                {0xa6dfbd9fb8e5b88e, 0xcb4ccd500f6bb953}, {0xd097ad07a71f26b2, 0x7e2000a41346a7a8},
                {0x825ecc24c873782f, 0x8ed400668c0c28c9}, {0xa2f67f2dfa90563b, 0x728900802f0f32fb},
                {0xcbb41ef979346bca, 0x4f2b40a03ad2ffba}, {0xfea126b7d78186bc, 0xe2f610c84987bfa9},
                {0x9f24b832e6b0f436, 0x0dd9ca7d2df4d7ca}, {0xc6ede63fa05d3143, 0x91503d1c79720dbc},
                {0xf8a95fcf88747d94, 0x75a44c6397ce912b}, {0x9b69dbe1b548ce7c, 0xc986afbe3ee11abb},
                {0xc24452da229b021b, 0xfbe85badce996169}, {0xf2d56790ab41c2a2, 0xfae27299423fb9c4},
                {0x97c560ba6b0919a5, 0xdccd879fc967d41b}, {0xbdb6b8e905cb600f, 0x5400e987bbc1c921},
                {0xed246723473e3813, 0x290123e9aab23b69}, {0x9436c0760c86e30b, 0xf9a0b6720aaf6522},
                {0xb94470938fa89bce, 0xf808e40e8d5b3e6a}, {0xe7958cb87392c2c2, 0xb60b1d1230b20e05},
                {0x90bd77f3483bb9b9, 0xb1c6f22b5e6f48c3}, {0xb4ecd5f01a4aa828, 0x1e38aeb6360b1af4},
                {0xe2280b6c20dd5232, 0x25c6da63c38de1b1}, {0x8d590723948a535f, 0x579c487e5a38ad0f},
                {0xb0af48ec79ace837, 0x2d835a9df0c6d852}, {0xdcdb1b2798182244, 0xf8e431456cf88e66},
                {0x8a08f0f8bf0f156b, 0x1b8e9ecb641b5900}, {0xac8b2d36eed2dac5, 0xe272467e3d222f40},
                {0xd7adf884aa879177, 0x5b0ed81dcc6abb10}, {0x86ccbb52ea94baea, 0x98e947129fc2b4ea},
                {0xa87fea27a539e9a5, 0x3f2398d747b36225}, {0xd29fe4b18e88640e, 0x8eec7f0d19a03aae},
                {0x83a3eeeef9153e89, 0x1953cf68300424ad}, {0xa48ceaaab75a8e2b, 0x5fa8c3423c052dd8},
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                {0x92746b9be2f8552c, 0x32fd3cf5b4e49bb5}, {0xb7118682dbb66a77, 0x3fbc8c33221dc2a2},
                {0xe4d5e82392a40515, 0x0fabaf3feaa5334b}, {0x8f05b1163ba6832d, 0x29cb4d87f2a7400f},
                {0xb2c71d5bca9023f8, 0x743e20e9ef511013}, {0xdf78e4b2bd342cf6, 0x914da9246b255417},
                {0x8bab8eefb6409c1a, 0x1ad089b6c2f7548f}, {0xae9672aba3d0c320, 0xa184ac2473b529b2},
                {0xda3c0f568cc4f3e8, 0xc9e5d72d90a2741f}, {0x8865899617fb1871, 0x7e2fa67c7a658893},
                {0xaa7eebfb9df9de8d, 0xddbb901b98feeab8}, {0xd51ea6fa85785631, 0x552a74227f3ea566},
                {0x8533285c936b35de, 0xd53a88958f872760}, {0xa67ff273b8460356, 0x8a892abaf368f138},
                {0xd01fef10a657842c, 0x2d2b7569b0432d86}, {0x8213f56a67f6b29b, 0x9c3b29620e29fc74},
                {0xa298f2c501f45f42, 0x8349f3ba91b47b90}, {0xcb3f2f7642717713, 0x241c70a936219a74},
                {0xfe0efb53d30dd4d7, 0xed238cd383aa0111}, {0x9ec95d1463e8a506, 0xf4363804324a40ab},
                {0xc67bb4597ce2ce48, 0xb143c6053edcd0d6}, {0xf81aa16fdc1b81da, 0xdd94b7868e94050b},
                {0x9b10a4e5e9913128, 0xca7cf2b4191c8327}, {0xc1d4ce1f63f57d72, 0xfd1c2f611f63a3f1},
                {0xf24a01a73cf2dccf, 0xbc633b39673c8ced}, {0x976e41088617ca01, 0xd5be0503e085d814},
                {0xbd49d14aa79dbc82, 0x4b2d8644d8a74e19}, {0xec9c459d51852ba2, 0xddf8e7d60ed1219f},
                {0x93e1ab8252f33b45, 0xcabb90e5c942b504}, {0xb8da1662e7b00a17, 0x3d6a751f3b936244},
                {0xe7109bfba19c0c9d, 0x0cc512670a783ad5}, {0x906a617d450187e2, 0x27fb2b80668b24c6},
                {0xb484f9dc9641e9da, 0xb1f9f660802dedf7}, {0xe1a63853bbd26451, 0x5e7873f8a0396974},
                {0x8d07e33455637eb2, 0xdb0b487b6423e1e9}, {0xb049dc016abc5e5f, 0x91ce1a9a3d2cda63},
                {0xdc5c5301c56b75f7, 0x7641a140cc7810fc}, {0x89b9b3e11b6329ba, 0xa9e904c87fcb0a9e},
                {0xac2820d9623bf429, 0x546345fa9fbdcd45}, {0xd732290fbacaf133, 0xa97c177947ad4096},
                {0x867f59a9d4bed6c0, 0x49ed8eabcccc485e}, {0xa81f301449ee8c70, 0x5c68f256bfff5a75},
                {0xd226fc195c6a2f8c, 0x73832eec6fff3112}, {0x83585d8fd9c25db7, 0xc831fd53c5ff7eac},
                {0xa42e74f3d032f525, 0xba3e7ca8b77f5e56}, {0xcd3a1230c43fb26f, 0x28ce1bd2e55f35ec},
                {0x80444b5e7aa7cf85, 0x7980d163cf5b81b4}, {0xa0555e361951c366, 0xd7e105bcc3326220},
                {0xc86ab5c39fa63440, 0x8dd9472bf3fefaa8}, {0xfa856334878fc150, 0xb14f98f6f0feb952},
                {0x9c935e00d4b9d8d2, 0x6ed1bf9a569f33d4}, {0xc3b8358109e84f07, 0x0a862f80ec4700c9},
                {0xf4a642e14c6262c8, 0xcd27bb612758c0fb}, {0x98e7e9cccfbd7dbd, 0x8038d51cb897789d},
                {0xbf21e44003acdd2c, 0xe0470a63e6bd56c4}, {0xeeea5d5004981478, 0x1858ccfce06cac75},
                {0x95527a5202df0ccb, 0x0f37801e0c43ebc9}, {0xbaa718e68396cffd, 0xd30560258f54e6bb},
                {0xe950df20247c83fd, 0x47c6b82ef32a206a}, {0x91d28b7416cdd27e, 0x4cdc331d57fa5442},
                {0xb6472e511c81471d, 0xe0133fe4adf8e953}, {0xe3d8f9e563a198e5, 0x58180fddd97723a7},
                {0x8e679c2f5e44ff8f, 0x570f09eaa7ea7649}, {0xb201833b35d63f73, 0x2cd2cc6551e513db},
                {0xde81e40a034bcf4f, 0xf8077f7ea65e58d2}, {0x8b112e86420f6191, 0xfb04afaf27faf783},
                {0xadd57a27d29339f6, 0x79c5db9af1f9b564}, {0xd94ad8b1c7380874, 0x18375281ae7822bd},
                {0x87cec76f1c830548, 0x8f2293910d0b15b6}, {0xa9c2794ae3a3c69a, 0xb2eb3875504ddb23},
                {0xd433179d9c8cb841, 0x5fa60692a46151ec}, {0x849feec281d7f328, 0xdbc7c41ba6bcd334},
                {0xa5c7ea73224deff3, 0x12b9b522906c0801}, {0xcf39e50feae16bef, 0xd768226b34870a01},
                {0x81842f29f2cce375, 0xe6a1158300d46641}, {0xa1e53af46f801c53, 0x60495ae3c1097fd1},
                {0xca5e89b18b602368, 0x385bb19cb14bdfc5}, {0xfcf62c1dee382c42, 0x46729e03dd9ed7b6},
                {0x9e19db92b4e31ba9, 0x6c07a2c26a8346d2}, {0xc5a05277621be293, 0xc7098b7305241886},
                {0xf70867153aa2db38, 0xb8cbee4fc66d1ea8}};
        };

        // Compressed cache for double
        struct compressed_cache_detail {
            static constexpr int compression_ratio = 27;
            static constexpr std::size_t compressed_table_size =
                (cache_holder<ieee754_binary64>::max_k - cache_holder<ieee754_binary64>::min_k +
                 compression_ratio) /
                compression_ratio;

            struct cache_holder_t {
                wuint::uint128 table[compressed_table_size];
            };
            static constexpr cache_holder_t cache = [] {
                cache_holder_t res{};
                for (std::size_t i = 0; i < compressed_table_size; ++i) {
                    res.table[i] = cache_holder<ieee754_binary64>::cache[i * compression_ratio];
                }
                return res;
            }();

            struct pow5_holder_t {
                std::uint64_t table[compression_ratio];
            };
            static constexpr pow5_holder_t pow5 = [] {
                pow5_holder_t res{};
                std::uint64_t p = 1;
                for (std::size_t i = 0; i < compression_ratio; ++i) {
                    res.table[i] = p;
                    p *= 5;
                }
                return res;
            }();
        };
    }


    ////////////////////////////////////////////////////////////////////////////////////////
    // Policies.
    ////////////////////////////////////////////////////////////////////////////////////////

    namespace detail {
        // Forward declare the implementation class.
        template <class Float, class FloatTraits = default_float_traits<Float>>
        struct impl;

        namespace policy_impl {
            // Sign policies.
            namespace sign {
                struct base {};

                struct ignore : base {
                    using sign_policy = ignore;
                    static constexpr bool return_has_sign = false;

                    template <class SignedSignificandBits, class ReturnType>
                    static constexpr void handle_sign(SignedSignificandBits, ReturnType&) noexcept {
                    }
                };

                struct return_sign : base {
                    using sign_policy = return_sign;
                    static constexpr bool return_has_sign = true;

                    template <class SignedSignificandBits, class ReturnType>
                    static constexpr void handle_sign(SignedSignificandBits s,
                                                      ReturnType& r) noexcept {
                        r.is_negative = s.is_negative();
                    }
                };
            }

            // Trailing zero policies.
            namespace trailing_zero {
                struct base {};

                struct ignore : base {
                    using trailing_zero_policy = ignore;
                    static constexpr bool report_trailing_zeros = false;

                    template <class Impl, class ReturnType>
                    static constexpr void on_trailing_zeros(ReturnType&) noexcept {}

                    template <class Impl, class ReturnType>
                    static constexpr void no_trailing_zeros(ReturnType&) noexcept {}
                };

                struct remove : base {
                    using trailing_zero_policy = remove;
                    static constexpr bool report_trailing_zeros = false;

                    template <class Impl, class ReturnType>
                    JKJ_FORCEINLINE static constexpr void
                    on_trailing_zeros(ReturnType& r) noexcept {
                        r.exponent += Impl::remove_trailing_zeros(r.significand);
                    }

                    template <class Impl, class ReturnType>
                    static constexpr void no_trailing_zeros(ReturnType&) noexcept {}
                };

                struct report : base {
                    using trailing_zero_policy = report;
                    static constexpr bool report_trailing_zeros = true;

                    template <class Impl, class ReturnType>
                    static constexpr void on_trailing_zeros(ReturnType& r) noexcept {
                        r.may_have_trailing_zeros = true;
                    }

                    template <class Impl, class ReturnType>
                    static constexpr void no_trailing_zeros(ReturnType& r) noexcept {
                        r.may_have_trailing_zeros = false;
                    }
                };
            }

            // Decimal-to-binary rounding mode policies.
            namespace decimal_to_binary_rounding {
                struct base {};

                enum class tag_t { to_nearest, left_closed_directed, right_closed_directed };
                namespace interval_type {
                    struct symmetric_boundary {
                        static constexpr bool is_symmetric = true;
                        bool is_closed;
                        constexpr bool include_left_endpoint() const noexcept { return is_closed; }
                        constexpr bool include_right_endpoint() const noexcept { return is_closed; }
                    };
                    struct asymmetric_boundary {
                        static constexpr bool is_symmetric = false;
                        bool is_left_closed;
                        constexpr bool include_left_endpoint() const noexcept {
                            return is_left_closed;
                        }
                        constexpr bool include_right_endpoint() const noexcept {
                            return !is_left_closed;
                        }
                    };
                    struct closed {
                        static constexpr bool is_symmetric = true;
                        static constexpr bool include_left_endpoint() noexcept { return true; }
                        static constexpr bool include_right_endpoint() noexcept { return true; }
                    };
                    struct open {
                        static constexpr bool is_symmetric = true;
                        static constexpr bool include_left_endpoint() noexcept { return false; }
                        static constexpr bool include_right_endpoint() noexcept { return false; }
                    };
                    struct left_closed_right_open {
                        static constexpr bool is_symmetric = false;
                        static constexpr bool include_left_endpoint() noexcept { return true; }
                        static constexpr bool include_right_endpoint() noexcept { return false; }
                    };
                    struct right_closed_left_open {
                        static constexpr bool is_symmetric = false;
                        static constexpr bool include_left_endpoint() noexcept { return false; }
                        static constexpr bool include_right_endpoint() noexcept { return true; }
                    };
                }

                struct nearest_to_even : base {
                    using decimal_to_binary_rounding_policy = nearest_to_even;
                    static constexpr auto tag = tag_t::to_nearest;
                    using normal_interval_type = interval_type::symmetric_boundary;
                    using shorter_interval_type = interval_type::closed;

                    template <class SignedSignificandBits, class Func>
                    JKJ_FORCEINLINE static auto delegate(SignedSignificandBits, Func&& f) noexcept {
                        return f(nearest_to_even{});
                    }

                    template <class SignedSignificandBits, class Func>
                    JKJ_FORCEINLINE static constexpr auto
                    invoke_normal_interval_case(SignedSignificandBits s, Func&& f) noexcept {
                        return f(s.has_even_significand_bits());
                    }
                    template <class SignedSignificandBits, class Func>
                    JKJ_FORCEINLINE static constexpr auto
                    invoke_shorter_interval_case(SignedSignificandBits, Func&& f) noexcept {
                        return f();
                    }
                };
                struct nearest_to_odd : base {
                    using decimal_to_binary_rounding_policy = nearest_to_odd;
                    static constexpr auto tag = tag_t::to_nearest;
                    using normal_interval_type = interval_type::symmetric_boundary;
                    using shorter_interval_type = interval_type::open;

                    template <class SignedSignificandBits, class Func>
                    JKJ_FORCEINLINE static auto delegate(SignedSignificandBits, Func&& f) noexcept {
                        return f(nearest_to_odd{});
                    }

                    template <class SignedSignificandBits, class Func>
                    JKJ_FORCEINLINE static constexpr auto
                    invoke_normal_interval_case(SignedSignificandBits s, Func&& f) noexcept {
                        return f(!s.has_even_significand_bits());
                    }
                    template <class SignedSignificandBits, class Func>
                    JKJ_FORCEINLINE static constexpr auto
                    invoke_shorter_interval_case(SignedSignificandBits, Func&& f) noexcept {
                        return f();
                    }
                };
                struct nearest_toward_plus_infinity : base {
                    using decimal_to_binary_rounding_policy = nearest_toward_plus_infinity;
                    static constexpr auto tag = tag_t::to_nearest;
                    using normal_interval_type = interval_type::asymmetric_boundary;
                    using shorter_interval_type = interval_type::asymmetric_boundary;

                    template <class SignedSignificandBits, class Func>
                    JKJ_FORCEINLINE static auto delegate(SignedSignificandBits, Func&& f) noexcept {
                        return f(nearest_toward_plus_infinity{});
                    }

                    template <class SignedSignificandBits, class Func>
                    JKJ_FORCEINLINE static constexpr auto
                    invoke_normal_interval_case(SignedSignificandBits s, Func&& f) noexcept {
                        return f(!s.is_negative());
                    }
                    template <class SignedSignificandBits, class Func>
                    JKJ_FORCEINLINE static constexpr auto
                    invoke_shorter_interval_case(SignedSignificandBits s, Func&& f) noexcept {
                        return f(!s.is_negative());
                    }
                };
                struct nearest_toward_minus_infinity : base {
                    using decimal_to_binary_rounding_policy = nearest_toward_minus_infinity;
                    static constexpr auto tag = tag_t::to_nearest;
                    using normal_interval_type = interval_type::asymmetric_boundary;
                    using shorter_interval_type = interval_type::asymmetric_boundary;

                    template <class SignedSignificandBits, class Func>
                    JKJ_FORCEINLINE static auto delegate(SignedSignificandBits, Func&& f) noexcept {
                        return f(nearest_toward_minus_infinity{});
                    }

                    template <class SignedSignificandBits, class Func>
                    JKJ_FORCEINLINE static constexpr auto
                    invoke_normal_interval_case(SignedSignificandBits s, Func&& f) noexcept {
                        return f(s.is_negative());
                    }
                    template <class SignedSignificandBits, class Func>
                    JKJ_FORCEINLINE static constexpr auto
                    invoke_shorter_interval_case(SignedSignificandBits s, Func&& f) noexcept {
                        return f(s.is_negative());
                    }
                };
                struct nearest_toward_zero : base {
                    using decimal_to_binary_rounding_policy = nearest_toward_zero;
                    static constexpr auto tag = tag_t::to_nearest;
                    using normal_interval_type = interval_type::right_closed_left_open;
                    using shorter_interval_type = interval_type::right_closed_left_open;

                    template <class SignedSignificandBits, class Func>
                    JKJ_FORCEINLINE static auto delegate(SignedSignificandBits, Func&& f) noexcept {
                        return f(nearest_toward_zero{});
                    }

                    template <class SignedSignificandBits, class Func>
                    JKJ_FORCEINLINE static constexpr auto
                    invoke_normal_interval_case(SignedSignificandBits, Func&& f) noexcept {
                        return f();
                    }
                    template <class SignedSignificandBits, class Func>
                    JKJ_FORCEINLINE static constexpr auto
                    invoke_shorter_interval_case(SignedSignificandBits, Func&& f) noexcept {
                        return f();
                    }
                };
                struct nearest_away_from_zero : base {
                    using decimal_to_binary_rounding_policy = nearest_away_from_zero;
                    static constexpr auto tag = tag_t::to_nearest;
                    using normal_interval_type = interval_type::left_closed_right_open;
                    using shorter_interval_type = interval_type::left_closed_right_open;

                    template <class SignedSignificandBits, class Func>
                    JKJ_FORCEINLINE static auto delegate(SignedSignificandBits, Func&& f) noexcept {
                        return f(nearest_away_from_zero{});
                    }

                    template <class SignedSignificandBits, class Func>
                    JKJ_FORCEINLINE static constexpr auto
                    invoke_normal_interval_case(SignedSignificandBits, Func&& f) noexcept {
                        return f();
                    }
                    template <class SignedSignificandBits, class Func>
                    JKJ_FORCEINLINE static constexpr auto
                    invoke_shorter_interval_case(SignedSignificandBits, Func&& f) noexcept {
                        return f();
                    }
                };

                namespace detail {
                    struct nearest_always_closed {
                        static constexpr auto tag = tag_t::to_nearest;
                        using normal_interval_type = interval_type::closed;
                        using shorter_interval_type = interval_type::closed;

                        template <class SignedSignificandBits, class Func>
                        JKJ_FORCEINLINE static constexpr auto
                        invoke_normal_interval_case(SignedSignificandBits, Func&& f) noexcept {
                            return f();
                        }
                        template <class SignedSignificandBits, class Func>
                        JKJ_FORCEINLINE static constexpr auto
                        invoke_shorter_interval_case(SignedSignificandBits, Func&& f) noexcept {
                            return f();
                        }
                    };
                    struct nearest_always_open {
                        static constexpr auto tag = tag_t::to_nearest;
                        using normal_interval_type = interval_type::open;
                        using shorter_interval_type = interval_type::open;

                        template <class SignedSignificandBits, class Func>
                        JKJ_FORCEINLINE static constexpr auto
                        invoke_normal_interval_case(SignedSignificandBits, Func&& f) noexcept {
                            return f();
                        }
                        template <class SignedSignificandBits, class Func>
                        JKJ_FORCEINLINE static constexpr auto
                        invoke_shorter_interval_case(SignedSignificandBits, Func&& f) noexcept {
                            return f();
                        }
                    };
                }

                struct nearest_to_even_static_boundary : base {
                    using decimal_to_binary_rounding_policy = nearest_to_even_static_boundary;
                    template <class SignedSignificandBits, class Func>
                    JKJ_FORCEINLINE static auto delegate(SignedSignificandBits s,
                                                         Func&& f) noexcept {
                        if (s.has_even_significand_bits()) {
                            return f(detail::nearest_always_closed{});
                        }
                        else {
                            return f(detail::nearest_always_open{});
                        }
                    }
                };
                struct nearest_to_odd_static_boundary : base {
                    using decimal_to_binary_rounding_policy = nearest_to_odd_static_boundary;
                    template <class SignedSignificandBits, class Func>
                    JKJ_FORCEINLINE static auto delegate(SignedSignificandBits s,
                                                         Func&& f) noexcept {
                        if (s.has_even_significand_bits()) {
                            return f(detail::nearest_always_open{});
                        }
                        else {
                            return f(detail::nearest_always_closed{});
                        }
                    }
                };
                struct nearest_toward_plus_infinity_static_boundary : base {
                    using decimal_to_binary_rounding_policy =
                        nearest_toward_plus_infinity_static_boundary;
                    template <class SignedSignificandBits, class Func>
                    JKJ_FORCEINLINE static auto delegate(SignedSignificandBits s,
                                                         Func&& f) noexcept {
                        if (s.is_negative()) {
                            return f(nearest_toward_zero{});
                        }
                        else {
                            return f(nearest_away_from_zero{});
                        }
                    }
                };
                struct nearest_toward_minus_infinity_static_boundary : base {
                    using decimal_to_binary_rounding_policy =
                        nearest_toward_minus_infinity_static_boundary;
                    template <class SignedSignificandBits, class Func>
                    JKJ_FORCEINLINE static auto delegate(SignedSignificandBits s,
                                                         Func&& f) noexcept {
                        if (s.is_negative()) {
                            return f(nearest_away_from_zero{});
                        }
                        else {
                            return f(nearest_toward_zero{});
                        }
                    }
                };

                namespace detail {
                    struct left_closed_directed {
                        static constexpr auto tag = tag_t::left_closed_directed;
                    };
                    struct right_closed_directed {
                        static constexpr auto tag = tag_t::right_closed_directed;
                    };
                }

                struct toward_plus_infinity : base {
                    using decimal_to_binary_rounding_policy = toward_plus_infinity;
                    template <class SignedSignificandBits, class Func>
                    JKJ_FORCEINLINE static auto delegate(SignedSignificandBits s,
                                                         Func&& f) noexcept {
                        if (s.is_negative()) {
                            return f(detail::left_closed_directed{});
                        }
                        else {
                            return f(detail::right_closed_directed{});
                        }
                    }
                };
                struct toward_minus_infinity : base {
                    using decimal_to_binary_rounding_policy = toward_minus_infinity;
                    template <class SignedSignificandBits, class Func>
                    JKJ_FORCEINLINE static auto delegate(SignedSignificandBits s,
                                                         Func&& f) noexcept {
                        if (s.is_negative()) {
                            return f(detail::right_closed_directed{});
                        }
                        else {
                            return f(detail::left_closed_directed{});
                        }
                    }
                };
                struct toward_zero : base {
                    using decimal_to_binary_rounding_policy = toward_zero;
                    template <class SignedSignificandBits, class Func>
                    JKJ_FORCEINLINE static auto delegate(SignedSignificandBits, Func&& f) noexcept {
                        return f(detail::left_closed_directed{});
                    }
                };
                struct away_from_zero : base {
                    using decimal_to_binary_rounding_policy = away_from_zero;
                    template <class SignedSignificandBits, class Func>
                    JKJ_FORCEINLINE static auto delegate(SignedSignificandBits, Func&& f) noexcept {
                        return f(detail::right_closed_directed{});
                    }
                };
            }

            // Binary-to-decimal rounding policies.
            // (Always assumes nearest rounding modes.)
            namespace binary_to_decimal_rounding {
                struct base {};

                enum class tag_t { do_not_care, to_even, to_odd, away_from_zero, toward_zero };

                struct do_not_care : base {
                    using binary_to_decimal_rounding_policy = do_not_care;
                    static constexpr auto tag = tag_t::do_not_care;

                    template <class ReturnType>
                    static constexpr bool prefer_round_down(ReturnType const&) noexcept {
                        return false;
                    }
                };

                struct to_even : base {
                    using binary_to_decimal_rounding_policy = to_even;
                    static constexpr auto tag = tag_t::to_even;

                    template <class ReturnType>
                    static constexpr bool prefer_round_down(ReturnType const& r) noexcept {
                        return r.significand % 2 != 0;
                    }
                };

                struct to_odd : base {
                    using binary_to_decimal_rounding_policy = to_odd;
                    static constexpr auto tag = tag_t::to_odd;

                    template <class ReturnType>
                    static constexpr bool prefer_round_down(ReturnType const& r) noexcept {
                        return r.significand % 2 == 0;
                    }
                };

                struct away_from_zero : base {
                    using binary_to_decimal_rounding_policy = away_from_zero;
                    static constexpr auto tag = tag_t::away_from_zero;

                    template <class ReturnType>
                    static constexpr bool prefer_round_down(ReturnType const&) noexcept {
                        return false;
                    }
                };

                struct toward_zero : base {
                    using binary_to_decimal_rounding_policy = toward_zero;
                    static constexpr auto tag = tag_t::toward_zero;

                    template <class ReturnType>
                    static constexpr bool prefer_round_down(ReturnType const&) noexcept {
                        return true;
                    }
                };
            }

            // Cache policies.
            namespace cache {
                struct base {};

                struct full : base {
                    using cache_policy = full;
                    template <class FloatFormat>
                    static constexpr typename cache_holder<FloatFormat>::cache_entry_type
                    get_cache(int k) noexcept {
                        assert(k >= cache_holder<FloatFormat>::min_k &&
                               k <= cache_holder<FloatFormat>::max_k);
                        return cache_holder<FloatFormat>::cache[std::size_t(
                            k - cache_holder<FloatFormat>::min_k)];
                    }
                };

                struct compact : base {
                    using cache_policy = compact;
                    template <class FloatFormat>
                    static constexpr typename cache_holder<FloatFormat>::cache_entry_type
                    get_cache(int k) noexcept {
                        assert(k >= cache_holder<FloatFormat>::min_k &&
                               k <= cache_holder<FloatFormat>::max_k);

                        if constexpr (std::is_same_v<FloatFormat, ieee754_binary64>) {
                            // Compute the base index.
                            auto const cache_index =
                                int(std::uint32_t(k - cache_holder<FloatFormat>::min_k) /
                                    compressed_cache_detail::compression_ratio);
                            auto const kb =
                                cache_index * compressed_cache_detail::compression_ratio +
                                cache_holder<FloatFormat>::min_k;
                            auto const offset = k - kb;

                            // Get the base cache.
                            auto const base_cache =
                                compressed_cache_detail::cache.table[cache_index];

                            if (offset == 0) {
                                return base_cache;
                            }
                            else {
                                // Compute the required amount of bit-shift.
                                auto const alpha = log::floor_log2_pow10(kb + offset) -
                                                   log::floor_log2_pow10(kb) - offset;
                                assert(alpha > 0 && alpha < 64);

                                // Try to recover the real cache.
                                auto const pow5 = compressed_cache_detail::pow5.table[offset];
                                auto recovered_cache = wuint::umul128(base_cache.high(), pow5);
                                auto const middle_low = wuint::umul128(base_cache.low(), pow5);

                                recovered_cache += middle_low.high();

                                auto const high_to_middle = recovered_cache.high() << (64 - alpha);
                                auto const middle_to_low = recovered_cache.low() << (64 - alpha);

                                recovered_cache = wuint::uint128{
                                    (recovered_cache.low() >> alpha) | high_to_middle,
                                    ((middle_low.low() >> alpha) | middle_to_low)};

                                assert(recovered_cache.low() + 1 != 0);
                                recovered_cache = {recovered_cache.high(),
                                                   recovered_cache.low() + 1};

                                return recovered_cache;
                            }
                        }
                        else {
                            // Just use the full cache for anything other than binary64
                            return cache_holder<FloatFormat>::cache[std::size_t(
                                k - cache_holder<FloatFormat>::min_k)];
                        }
                    }
                };
            }
        }
    }

    namespace policy {
        namespace sign {
            inline constexpr auto ignore = detail::policy_impl::sign::ignore{};
            inline constexpr auto return_sign = detail::policy_impl::sign::return_sign{};
        }

        namespace trailing_zero {
            inline constexpr auto ignore = detail::policy_impl::trailing_zero::ignore{};
            inline constexpr auto remove = detail::policy_impl::trailing_zero::remove{};
            inline constexpr auto report = detail::policy_impl::trailing_zero::report{};
        }

        namespace decimal_to_binary_rounding {
            inline constexpr auto nearest_to_even =
                detail::policy_impl::decimal_to_binary_rounding::nearest_to_even{};
            inline constexpr auto nearest_to_odd =
                detail::policy_impl::decimal_to_binary_rounding::nearest_to_odd{};
            inline constexpr auto nearest_toward_plus_infinity =
                detail::policy_impl::decimal_to_binary_rounding::nearest_toward_plus_infinity{};
            inline constexpr auto nearest_toward_minus_infinity =
                detail::policy_impl::decimal_to_binary_rounding::nearest_toward_minus_infinity{};
            inline constexpr auto nearest_toward_zero =
                detail::policy_impl::decimal_to_binary_rounding::nearest_toward_zero{};
            inline constexpr auto nearest_away_from_zero =
                detail::policy_impl::decimal_to_binary_rounding::nearest_away_from_zero{};

            inline constexpr auto nearest_to_even_static_boundary =
                detail::policy_impl::decimal_to_binary_rounding::nearest_to_even_static_boundary{};
            inline constexpr auto nearest_to_odd_static_boundary =
                detail::policy_impl::decimal_to_binary_rounding::nearest_to_odd_static_boundary{};
            inline constexpr auto nearest_toward_plus_infinity_static_boundary =
                detail::policy_impl::decimal_to_binary_rounding::
                    nearest_toward_plus_infinity_static_boundary{};
            inline constexpr auto nearest_toward_minus_infinity_static_boundary =
                detail::policy_impl::decimal_to_binary_rounding::
                    nearest_toward_minus_infinity_static_boundary{};

            inline constexpr auto toward_plus_infinity =
                detail::policy_impl::decimal_to_binary_rounding::toward_plus_infinity{};
            inline constexpr auto toward_minus_infinity =
                detail::policy_impl::decimal_to_binary_rounding::toward_minus_infinity{};
            inline constexpr auto toward_zero =
                detail::policy_impl::decimal_to_binary_rounding::toward_zero{};
            inline constexpr auto away_from_zero =
                detail::policy_impl::decimal_to_binary_rounding::away_from_zero{};
        }

        namespace binary_to_decimal_rounding {
            inline constexpr auto do_not_care =
                detail::policy_impl::binary_to_decimal_rounding::do_not_care{};
            inline constexpr auto to_even =
                detail::policy_impl::binary_to_decimal_rounding::to_even{};
            inline constexpr auto to_odd =
                detail::policy_impl::binary_to_decimal_rounding::to_odd{};
            inline constexpr auto away_from_zero =
                detail::policy_impl::binary_to_decimal_rounding::away_from_zero{};
            inline constexpr auto toward_zero =
                detail::policy_impl::binary_to_decimal_rounding::toward_zero{};
        }

        namespace cache {
            inline constexpr auto full = detail::policy_impl::cache::full{};
            inline constexpr auto compact = detail::policy_impl::cache::compact{};
        }
    }

    namespace detail {
        ////////////////////////////////////////////////////////////////////////////////////////
        // The main algorithm.
        ////////////////////////////////////////////////////////////////////////////////////////

        template <class Float, class FloatTraits>
        struct impl : private FloatTraits, private FloatTraits::format {
            using format = typename FloatTraits::format;
            using carrier_uint = typename FloatTraits::carrier_uint;

            using FloatTraits::carrier_bits;
            using format::significand_bits;
            using format::min_exponent;
            using format::max_exponent;
            using format::exponent_bias;
            using format::decimal_digits;

            static constexpr int kappa = std::is_same_v<format, ieee754_binary32> ? 1 : 2;
            static_assert(kappa >= 1);
            static_assert(carrier_bits >= significand_bits + 2 + log::floor_log2_pow10(kappa + 1));

            static constexpr int min_k = [] {
                constexpr auto a = -log::floor_log10_pow2_minus_log10_4_over_3(
                    int(max_exponent - significand_bits));
                constexpr auto b =
                    -log::floor_log10_pow2(int(max_exponent - significand_bits)) + kappa;
                return a < b ? a : b;
            }();
            static_assert(min_k >= cache_holder<format>::min_k);

            static constexpr int max_k = [] {
                // We do invoke shorter_interval_case for exponent == min_exponent case,
                // so we should not add 1 here.
                constexpr auto a = -log::floor_log10_pow2_minus_log10_4_over_3(
                    int(min_exponent - significand_bits /*+ 1*/));
                constexpr auto b =
                    -log::floor_log10_pow2(int(min_exponent - significand_bits)) + kappa;
                return a > b ? a : b;
            }();
            static_assert(max_k <= cache_holder<format>::max_k);

            using cache_entry_type = typename cache_holder<format>::cache_entry_type;
            static constexpr auto cache_bits = cache_holder<format>::cache_bits;

            static constexpr int case_shorter_interval_left_endpoint_lower_threshold = 2;
            static constexpr int case_shorter_interval_left_endpoint_upper_threshold =
                2 +
                log::floor_log2(
                    compute_power<
                        count_factors<5>((carrier_uint(1) << (significand_bits + 2)) - 1) + 1>(10) /
                    3);

            static constexpr int case_shorter_interval_right_endpoint_lower_threshold = 0;
            static constexpr int case_shorter_interval_right_endpoint_upper_threshold =
                2 +
                log::floor_log2(
                    compute_power<
                        count_factors<5>((carrier_uint(1) << (significand_bits + 1)) + 1) + 1>(10) /
                    3);

            static constexpr int shorter_interval_tie_lower_threshold =
                -log::floor_log5_pow2_minus_log5_3(significand_bits + 4) - 2 - significand_bits;
            static constexpr int shorter_interval_tie_upper_threshold =
                -log::floor_log5_pow2(significand_bits + 2) - 2 - significand_bits;

            struct compute_mul_result {
                carrier_uint result;
                bool is_integer;
            };
            struct compute_mul_parity_result {
                bool parity;
                bool is_integer;
            };

            //// The main algorithm assumes the input is a normal/subnormal finite number

            template <class ReturnType, class IntervalType, class TrailingZeroPolicy,
                      class BinaryToDecimalRoundingPolicy, class CachePolicy,
                      class... AdditionalArgs>
            JKJ_SAFEBUFFERS static ReturnType
            compute_nearest_normal(carrier_uint const two_fc, int const exponent,
                                   AdditionalArgs... additional_args) noexcept {
                //////////////////////////////////////////////////////////////////////
                // Step 1: Schubfach multiplier calculation
                //////////////////////////////////////////////////////////////////////

                ReturnType ret_value;
                IntervalType interval_type{additional_args...};

                // Compute k and beta.
                int const minus_k = log::floor_log10_pow2(exponent) - kappa;
                auto const cache = CachePolicy::template get_cache<format>(-minus_k);
                int const beta = exponent + log::floor_log2_pow10(-minus_k);

                // Compute zi and deltai.
                // 10^kappa <= deltai < 10^(kappa + 1)
                auto const deltai = compute_delta(cache, beta);
                // For the case of binary32, the result of integer check is not correct for
                // 29711844 * 2^-82
                // = 6.1442653300000000008655037797566933477355632930994033813476... * 10^-18
                // and 29711844 * 2^-81
                // = 1.2288530660000000001731007559513386695471126586198806762695... * 10^-17,
                // and they are the unique counterexamples. However, since 29711844 is even,
                // this does not cause any problem for the endpoints calculations; it can only
                // cause a problem when we need to perform integer check for the center.
                // Fortunately, with these inputs, that branch is never executed, so we are fine.
                auto const [zi, is_z_integer] = compute_mul((two_fc | 1) << beta, cache);


                //////////////////////////////////////////////////////////////////////
                // Step 2: Try larger divisor; remove trailing zeros if necessary
                //////////////////////////////////////////////////////////////////////

                constexpr auto big_divisor = compute_power<kappa + 1>(std::uint32_t(10));
                constexpr auto small_divisor = compute_power<kappa>(std::uint32_t(10));

                // Using an upper bound on zi, we might be able to optimize the division
                // better than the compiler; we are computing zi / big_divisor here.
                ret_value.significand =
                    div::divide_by_pow10<kappa + 1, carrier_uint,
                                         (carrier_uint(1) << (significand_bits + 1)) * big_divisor -
                                             1>(zi);
                auto r = std::uint32_t(zi - big_divisor * ret_value.significand);

                if (r < deltai) {
                    // Exclude the right endpoint if necessary.
                    if (r == 0 && (is_z_integer & !interval_type.include_right_endpoint())) {
                        if constexpr (BinaryToDecimalRoundingPolicy::tag ==
                                      policy_impl::binary_to_decimal_rounding::tag_t::do_not_care) {
                            ret_value.significand *= 10;
                            ret_value.exponent = minus_k + kappa;
                            --ret_value.significand;
                            TrailingZeroPolicy::template no_trailing_zeros<impl>(ret_value);
                            return ret_value;
                        }
                        else {
                            --ret_value.significand;
                            r = big_divisor;
                            goto small_divisor_case_label;
                        }
                    }
                }
                else if (r > deltai) {
                    goto small_divisor_case_label;
                }
                else {
                    // r == deltai; compare fractional parts.
                    auto const [xi_parity, x_is_integer] =
                        compute_mul_parity(two_fc - 1, cache, beta);

                    if (!(xi_parity | (x_is_integer & interval_type.include_left_endpoint()))) {
                        goto small_divisor_case_label;
                    }
                }
                ret_value.exponent = minus_k + kappa + 1;

                // We may need to remove trailing zeros.
                TrailingZeroPolicy::template on_trailing_zeros<impl>(ret_value);
                return ret_value;


                //////////////////////////////////////////////////////////////////////
                // Step 3: Find the significand with the smaller divisor
                //////////////////////////////////////////////////////////////////////

            small_divisor_case_label:
                TrailingZeroPolicy::template no_trailing_zeros<impl>(ret_value);
                ret_value.significand *= 10;
                ret_value.exponent = minus_k + kappa;

                if constexpr (BinaryToDecimalRoundingPolicy::tag ==
                              policy_impl::binary_to_decimal_rounding::tag_t::do_not_care) {
                    // Normally, we want to compute
                    // ret_value.significand += r / small_divisor
                    // and return, but we need to take care of the case that the resulting
                    // value is exactly the right endpoint, while that is not included in the
                    // interval.
                    if (!interval_type.include_right_endpoint()) {
                        // Is r divisible by 10^kappa?
                        if (is_z_integer && div::check_divisibility_and_divide_by_pow10<kappa>(r)) {
                            // This should be in the interval.
                            ret_value.significand += r - 1;
                        }
                        else {
                            ret_value.significand += r;
                        }
                    }
                    else {
                        ret_value.significand += div::small_division_by_pow10<kappa>(r);
                    }
                }
                else {
                    auto dist = r - (deltai / 2) + (small_divisor / 2);
                    bool const approx_y_parity = ((dist ^ (small_divisor / 2)) & 1) != 0;

                    // Is dist divisible by 10^kappa?
                    bool const divisible_by_small_divisor =
                        div::check_divisibility_and_divide_by_pow10<kappa>(dist);

                    // Add dist / 10^kappa to the significand.
                    ret_value.significand += dist;

                    if (divisible_by_small_divisor) {
                        // Check z^(f) >= epsilon^(f).
                        // We have either yi == zi - epsiloni or yi == (zi - epsiloni) - 1,
                        // where yi == zi - epsiloni if and only if z^(f) >= epsilon^(f).
                        // Since there are only 2 possibilities, we only need to care about the
                        // parity. Also, zi and r should have the same parity since the divisor is
                        // an even number.
                        auto const [yi_parity, is_y_integer] =
                            compute_mul_parity(two_fc, cache, beta);
                        if (yi_parity != approx_y_parity) {
                            --ret_value.significand;
                        }
                        else {
                            // If z^(f) >= epsilon^(f), we might have a tie
                            // when z^(f) == epsilon^(f), or equivalently, when y is an integer.
                            // For tie-to-up case, we can just choose the upper one.
                            if (BinaryToDecimalRoundingPolicy::prefer_round_down(ret_value) &
                                is_y_integer) {
                                --ret_value.significand;
                            }
                        }
                    }
                }
                return ret_value;
            }

            template <class ReturnType, class IntervalType, class TrailingZeroPolicy,
                      class BinaryToDecimalRoundingPolicy, class CachePolicy,
                      class... AdditionalArgs>
            JKJ_SAFEBUFFERS static ReturnType
            compute_nearest_shorter(int const exponent,
                                    AdditionalArgs... additional_args) noexcept {
                ReturnType ret_value;
                IntervalType interval_type{additional_args...};

                // Compute k and beta.
                int const minus_k = log::floor_log10_pow2_minus_log10_4_over_3(exponent);
                int const beta = exponent + log::floor_log2_pow10(-minus_k);

                // Compute xi and zi.
                auto const cache = CachePolicy::template get_cache<format>(-minus_k);

                auto xi = compute_left_endpoint_for_shorter_interval_case(cache, beta);
                auto zi = compute_right_endpoint_for_shorter_interval_case(cache, beta);

                // If we don't accept the right endpoint and
                // if the right endpoint is an integer, decrease it.
                if (!interval_type.include_right_endpoint() &&
                    is_right_endpoint_integer_shorter_interval(exponent)) {
                    --zi;
                }
                // If we don't accept the left endpoint or
                // if the left endpoint is not an integer, increase it.
                if (!interval_type.include_left_endpoint() ||
                    !is_left_endpoint_integer_shorter_interval(exponent)) {
                    ++xi;
                }

                // Try bigger divisor.
                ret_value.significand = zi / 10;

                // If succeed, remove trailing zeros if necessary and return.
                if (ret_value.significand * 10 >= xi) {
                    ret_value.exponent = minus_k + 1;
                    TrailingZeroPolicy::template on_trailing_zeros<impl>(ret_value);
                    return ret_value;
                }

                // Otherwise, compute the round-up of y.
                TrailingZeroPolicy::template no_trailing_zeros<impl>(ret_value);
                ret_value.significand = compute_round_up_for_shorter_interval_case(cache, beta);
                ret_value.exponent = minus_k;

                // When tie occurs, choose one of them according to the rule.
                if (BinaryToDecimalRoundingPolicy::prefer_round_down(ret_value) &&
                    exponent >= shorter_interval_tie_lower_threshold &&
                    exponent <= shorter_interval_tie_upper_threshold) {
                    --ret_value.significand;
                }
                else if (ret_value.significand < xi) {
                    ++ret_value.significand;
                }
                return ret_value;
            }

            template <class ReturnType, class TrailingZeroPolicy, class CachePolicy>
            JKJ_SAFEBUFFERS static ReturnType
            compute_left_closed_directed(carrier_uint const two_fc, int exponent) noexcept {
                //////////////////////////////////////////////////////////////////////
                // Step 1: Schubfach multiplier calculation
                //////////////////////////////////////////////////////////////////////

                ReturnType ret_value;

                // Compute k and beta.
                int const minus_k = log::floor_log10_pow2(exponent) - kappa;
                auto const cache = CachePolicy::template get_cache<format>(-minus_k);
                int const beta = exponent + log::floor_log2_pow10(-minus_k);

                // Compute xi and deltai.
                // 10^kappa <= deltai < 10^(kappa + 1)
                auto const deltai = compute_delta(cache, beta);
                auto [xi, is_x_integer] = compute_mul(two_fc << beta, cache);

                // Deal with the unique exceptional cases
                // 29711844 * 2^-82
                // = 6.1442653300000000008655037797566933477355632930994033813476... * 10^-18
                // and 29711844 * 2^-81
                // = 1.2288530660000000001731007559513386695471126586198806762695... * 10^-17
                // for binary32.
                if constexpr (std::is_same_v<format, ieee754_binary32>) {
                    if (exponent <= -80) {
                        is_x_integer = false;
                    }
                }

                if (!is_x_integer) {
                    ++xi;
                }

                //////////////////////////////////////////////////////////////////////
                // Step 2: Try larger divisor; remove trailing zeros if necessary
                //////////////////////////////////////////////////////////////////////

                constexpr auto big_divisor = compute_power<kappa + 1>(std::uint32_t(10));

                // Using an upper bound on xi, we might be able to optimize the division
                // better than the compiler; we are computing xi / big_divisor here.
                ret_value.significand =
                    div::divide_by_pow10<kappa + 1, carrier_uint,
                                         (carrier_uint(1) << (significand_bits + 1)) * big_divisor -
                                             1>(xi);
                auto r = std::uint32_t(xi - big_divisor * ret_value.significand);

                if (r != 0) {
                    ++ret_value.significand;
                    r = big_divisor - r;
                }

                if (r > deltai) {
                    goto small_divisor_case_label;
                }
                else if (r == deltai) {
                    // Compare the fractional parts.
                    // This branch is never taken for the exceptional cases
                    // 2f_c = 29711482, e = -81
                    // (6.1442649164096937243516663440523473127541365101933479309082... * 10^-18)
                    // and 2f_c = 29711482, e = -80
                    // (1.2288529832819387448703332688104694625508273020386695861816... * 10^-17).
                    auto const [zi_parity, is_z_integer] =
                        compute_mul_parity(two_fc + 2, cache, beta);
                    if (zi_parity || is_z_integer) {
                        goto small_divisor_case_label;
                    }
                }

                // The ceiling is inside, so we are done.
                ret_value.exponent = minus_k + kappa + 1;
                TrailingZeroPolicy::template on_trailing_zeros<impl>(ret_value);
                return ret_value;


                //////////////////////////////////////////////////////////////////////
                // Step 3: Find the significand with the smaller divisor
                //////////////////////////////////////////////////////////////////////

            small_divisor_case_label:
                ret_value.significand *= 10;
                ret_value.significand -= div::small_division_by_pow10<kappa>(r);
                ret_value.exponent = minus_k + kappa;
                TrailingZeroPolicy::template no_trailing_zeros<impl>(ret_value);
                return ret_value;
            }

            template <class ReturnType, class TrailingZeroPolicy, class CachePolicy>
            JKJ_SAFEBUFFERS static ReturnType
            compute_right_closed_directed(carrier_uint const two_fc, int const exponent,
                                          bool shorter_interval) noexcept {
                //////////////////////////////////////////////////////////////////////
                // Step 1: Schubfach multiplier calculation
                //////////////////////////////////////////////////////////////////////

                ReturnType ret_value;

                // Compute k and beta.
                int const minus_k =
                    log::floor_log10_pow2(exponent - (shorter_interval ? 1 : 0)) - kappa;
                auto const cache = CachePolicy::template get_cache<format>(-minus_k);
                int const beta = exponent + log::floor_log2_pow10(-minus_k);

                // Compute zi and deltai.
                // 10^kappa <= deltai < 10^(kappa + 1)
                auto const deltai =
                    shorter_interval ? compute_delta(cache, beta - 1) : compute_delta(cache, beta);
                carrier_uint const zi = compute_mul(two_fc << beta, cache).result;


                //////////////////////////////////////////////////////////////////////
                // Step 2: Try larger divisor; remove trailing zeros if necessary
                //////////////////////////////////////////////////////////////////////

                constexpr auto big_divisor = compute_power<kappa + 1>(std::uint32_t(10));

                // Using an upper bound on zi, we might be able to optimize the division better than
                // the compiler; we are computing zi / big_divisor here.
                ret_value.significand =
                    div::divide_by_pow10<kappa + 1, carrier_uint,
                                         (carrier_uint(1) << (significand_bits + 1)) * big_divisor -
                                             1>(zi);
                auto const r = std::uint32_t(zi - big_divisor * ret_value.significand);

                if (r > deltai) {
                    goto small_divisor_case_label;
                }
                else if (r == deltai) {
                    // Compare the fractional parts.
                    if (!compute_mul_parity(two_fc - (shorter_interval ? 1 : 2), cache, beta)
                             .parity) {
                        goto small_divisor_case_label;
                    }
                }

                // The floor is inside, so we are done.
                ret_value.exponent = minus_k + kappa + 1;
                TrailingZeroPolicy::template on_trailing_zeros<impl>(ret_value);
                return ret_value;


                //////////////////////////////////////////////////////////////////////
                // Step 3: Find the significand with the small divisor
                //////////////////////////////////////////////////////////////////////

            small_divisor_case_label:
                ret_value.significand *= 10;
                ret_value.significand += div::small_division_by_pow10<kappa>(r);
                ret_value.exponent = minus_k + kappa;
                TrailingZeroPolicy::template no_trailing_zeros<impl>(ret_value);
                return ret_value;
            }

            // Remove trailing zeros from n and return the number of zeros removed.
            JKJ_FORCEINLINE static int remove_trailing_zeros(carrier_uint& n) noexcept {
                assert(n != 0);

                if constexpr (std::is_same_v<format, ieee754_binary32>) {
                    constexpr auto mod_inv_5 = std::uint32_t(0xcccc'cccd);
                    constexpr auto mod_inv_25 = mod_inv_5 * mod_inv_5;

                    int s = 0;
                    while (true) {
                        auto q = bits::rotr(n * mod_inv_25, 2);
                        if (q <= std::numeric_limits<std::uint32_t>::max() / 100) {
                            n = q;
                            s += 2;
                        }
                        else {
                            break;
                        }
                    }
                    auto q = bits::rotr(n * mod_inv_5, 1);
                    if (q <= std::numeric_limits<std::uint32_t>::max() / 10) {
                        n = q;
                        s |= 1;
                    }

                    return s;
                }
                else {
                    static_assert(std::is_same_v<format, ieee754_binary64>);

                    // Divide by 10^8 and reduce to 32-bits if divisible.
                    // Since ret_value.significand <= (2^53 * 1000 - 1) / 1000 < 10^16,
                    // n is at most of 16 digits.

                    // This magic number is ceil(2^90 / 10^8).
                    constexpr auto magic_number = std::uint64_t(12379400392853802749ull);
                    auto nm = wuint::umul128(n, magic_number);

                    // Is n is divisible by 10^8?
                    if ((nm.high() & ((std::uint64_t(1) << (90 - 64)) - 1)) == 0 &&
                        nm.low() < magic_number) {
                        // If yes, work with the quotient.
                        auto n32 = std::uint32_t(nm.high() >> (90 - 64));

                        constexpr auto mod_inv_5 = std::uint32_t(0xcccc'cccd);
                        constexpr auto mod_inv_25 = mod_inv_5 * mod_inv_5;

                        int s = 8;
                        while (true) {
                            auto q = bits::rotr(n32 * mod_inv_25, 2);
                            if (q <= std::numeric_limits<std::uint32_t>::max() / 100) {
                                n32 = q;
                                s += 2;
                            }
                            else {
                                break;
                            }
                        }
                        auto q = bits::rotr(n32 * mod_inv_5, 1);
                        if (q <= std::numeric_limits<std::uint32_t>::max() / 10) {
                            n32 = q;
                            s |= 1;
                        }

                        n = n32;
                        return s;
                    }

                    // If n is not divisible by 10^8, work with n itself.
                    constexpr auto mod_inv_5 = std::uint64_t(0xcccc'cccc'cccc'cccd);
                    constexpr auto mod_inv_25 = mod_inv_5 * mod_inv_5;

                    int s = 0;
                    while (true) {
                        auto q = bits::rotr(n * mod_inv_25, 2);
                        if (q <= std::numeric_limits<std::uint64_t>::max() / 100) {
                            n = q;
                            s += 2;
                        }
                        else {
                            break;
                        }
                    }
                    auto q = bits::rotr(n * mod_inv_5, 1);
                    if (q <= std::numeric_limits<std::uint64_t>::max() / 10) {
                        n = q;
                        s |= 1;
                    }

                    return s;
                }
            }

            static compute_mul_result compute_mul(carrier_uint u,
                                                  cache_entry_type const& cache) noexcept {
                if constexpr (std::is_same_v<format, ieee754_binary32>) {
                    auto r = wuint::umul96_upper64(u, cache);
                    return {carrier_uint(r >> 32), carrier_uint(r) == 0};
                }
                else {
                    static_assert(std::is_same_v<format, ieee754_binary64>);
                    auto r = wuint::umul192_upper128(u, cache);
                    return {r.high(), r.low() == 0};
                }
            }

            static constexpr std::uint32_t compute_delta(cache_entry_type const& cache,
                                                         int beta) noexcept {
                if constexpr (std::is_same_v<format, ieee754_binary32>) {
                    return std::uint32_t(cache >> (cache_bits - 1 - beta));
                }
                else {
                    static_assert(std::is_same_v<format, ieee754_binary64>);
                    return std::uint32_t(cache.high() >> (carrier_bits - 1 - beta));
                }
            }

            static compute_mul_parity_result compute_mul_parity(carrier_uint two_f,
                                                                cache_entry_type const& cache,
                                                                int beta) noexcept {
                assert(beta >= 1);
                assert(beta < 64);

                if constexpr (std::is_same_v<format, ieee754_binary32>) {
                    auto r = wuint::umul96_lower64(two_f, cache);
                    return {((r >> (64 - beta)) & 1) != 0, std::uint32_t(r >> (32 - beta)) == 0};
                }
                else {
                    static_assert(std::is_same_v<format, ieee754_binary64>);
                    auto r = wuint::umul192_lower128(two_f, cache);
                    return {((r.high() >> (64 - beta)) & 1) != 0,
                            ((r.high() << beta) | (r.low() >> (64 - beta))) == 0};
                }
            }

            static constexpr carrier_uint
            compute_left_endpoint_for_shorter_interval_case(cache_entry_type const& cache,
                                                            int beta) noexcept {
                if constexpr (std::is_same_v<format, ieee754_binary32>) {
                    return carrier_uint((cache - (cache >> (significand_bits + 2))) >>
                                        (cache_bits - significand_bits - 1 - beta));
                }
                else {
                    static_assert(std::is_same_v<format, ieee754_binary64>);
                    return (cache.high() - (cache.high() >> (significand_bits + 2))) >>
                           (carrier_bits - significand_bits - 1 - beta);
                }
            }

            static constexpr carrier_uint
            compute_right_endpoint_for_shorter_interval_case(cache_entry_type const& cache,
                                                             int beta) noexcept {
                if constexpr (std::is_same_v<format, ieee754_binary32>) {
                    return carrier_uint((cache + (cache >> (significand_bits + 1))) >>
                                        (cache_bits - significand_bits - 1 - beta));
                }
                else {
                    static_assert(std::is_same_v<format, ieee754_binary64>);
                    return (cache.high() + (cache.high() >> (significand_bits + 1))) >>
                           (carrier_bits - significand_bits - 1 - beta);
                }
            }

            static constexpr carrier_uint
            compute_round_up_for_shorter_interval_case(cache_entry_type const& cache,
                                                       int beta) noexcept {
                if constexpr (std::is_same_v<format, ieee754_binary32>) {
                    return (carrier_uint(cache >> (cache_bits - significand_bits - 2 - beta)) + 1) /
                           2;
                }
                else {
                    static_assert(std::is_same_v<format, ieee754_binary64>);
                    return ((cache.high() >> (carrier_bits - significand_bits - 2 - beta)) + 1) / 2;
                }
            }

            static constexpr bool
            is_right_endpoint_integer_shorter_interval(int exponent) noexcept {
                return exponent >= case_shorter_interval_right_endpoint_lower_threshold &&
                       exponent <= case_shorter_interval_right_endpoint_upper_threshold;
            }

            static constexpr bool is_left_endpoint_integer_shorter_interval(int exponent) noexcept {
                return exponent >= case_shorter_interval_left_endpoint_lower_threshold &&
                       exponent <= case_shorter_interval_left_endpoint_upper_threshold;
            }
        };


        ////////////////////////////////////////////////////////////////////////////////////////
        // Policy holder.
        ////////////////////////////////////////////////////////////////////////////////////////

        namespace policy_impl {
            // The library will specify a list of accepted kinds of policies and their defaults, and
            // the user will pass a list of policies. The aim of helper classes/functions here is to
            // do the following:
            //   1. Check if the policy parameters given by the user are all valid; that means,
            //      each of them should be of the kinds specified by the library.
            //      If that's not the case, then the compilation fails.
            //   2. Check if multiple policy parameters for the same kind is specified by the user.
            //      If that's the case, then the compilation fails.
            //   3. Build a class deriving from all policies the user have given, and also from
            //      the default policies if the user did not specify one for some kinds.
            // A policy belongs to a certain kind if it is deriving from a base class.

            // For a given kind, find a policy belonging to that kind.
            // Check if there are more than one such policies.
            enum class policy_found_info { not_found, unique, repeated };
            template <class Policy, policy_found_info info>
            struct found_policy_pair {
                using policy = Policy;
                static constexpr auto found_info = info;
            };

            template <class Base, class DefaultPolicy>
            struct base_default_pair {
                using base = Base;

                template <class FoundPolicyInfo>
                static constexpr FoundPolicyInfo get_policy_impl(FoundPolicyInfo) {
                    return {};
                }
                template <class FoundPolicyInfo, class FirstPolicy, class... RemainingPolicies>
                static constexpr auto get_policy_impl(FoundPolicyInfo, FirstPolicy,
                                                      RemainingPolicies... remainings) {
                    if constexpr (std::is_base_of_v<Base, FirstPolicy>) {
                        if constexpr (FoundPolicyInfo::found_info == policy_found_info::not_found) {
                            return get_policy_impl(
                                found_policy_pair<FirstPolicy, policy_found_info::unique>{},
                                remainings...);
                        }
                        else {
                            return get_policy_impl(
                                found_policy_pair<FirstPolicy, policy_found_info::repeated>{},
                                remainings...);
                        }
                    }
                    else {
                        return get_policy_impl(FoundPolicyInfo{}, remainings...);
                    }
                }

                template <class... Policies>
                static constexpr auto get_policy(Policies... policies) {
                    return get_policy_impl(
                        found_policy_pair<DefaultPolicy, policy_found_info::not_found>{},
                        policies...);
                }
            };
            template <class... BaseDefaultPairs>
            struct base_default_pair_list {};

            // Check if a given policy belongs to one of the kinds specified by the library.
            template <class Policy>
            constexpr bool check_policy_validity(Policy, base_default_pair_list<>) {
                return false;
            }
            template <class Policy, class FirstBaseDefaultPair, class... RemainingBaseDefaultPairs>
            constexpr bool check_policy_validity(
                Policy,
                base_default_pair_list<FirstBaseDefaultPair, RemainingBaseDefaultPairs...>) {
                return std::is_base_of_v<typename FirstBaseDefaultPair::base, Policy> ||
                       check_policy_validity(
                           Policy{}, base_default_pair_list<RemainingBaseDefaultPairs...>{});
            }

            template <class BaseDefaultPairList>
            constexpr bool check_policy_list_validity(BaseDefaultPairList) {
                return true;
            }

            template <class BaseDefaultPairList, class FirstPolicy, class... RemainingPolicies>
            constexpr bool check_policy_list_validity(BaseDefaultPairList, FirstPolicy,
                                                      RemainingPolicies... remaining_policies) {
                return check_policy_validity(FirstPolicy{}, BaseDefaultPairList{}) &&
                       check_policy_list_validity(BaseDefaultPairList{}, remaining_policies...);
            }

            // Build policy_holder.
            template <bool repeated_, class... FoundPolicyPairs>
            struct found_policy_pair_list {
                static constexpr bool repeated = repeated_;
            };

            template <class... Policies>
            struct policy_holder : Policies... {};

            template <bool repeated, class... FoundPolicyPairs, class... Policies>
            constexpr auto
            make_policy_holder_impl(base_default_pair_list<>,
                                    found_policy_pair_list<repeated, FoundPolicyPairs...>,
                                    Policies...) {
                return found_policy_pair_list<repeated, FoundPolicyPairs...>{};
            }

            template <class FirstBaseDefaultPair, class... RemainingBaseDefaultPairs, bool repeated,
                      class... FoundPolicyPairs, class... Policies>
            constexpr auto make_policy_holder_impl(
                base_default_pair_list<FirstBaseDefaultPair, RemainingBaseDefaultPairs...>,
                found_policy_pair_list<repeated, FoundPolicyPairs...>, Policies... policies) {
                using new_found_policy_pair =
                    decltype(FirstBaseDefaultPair::get_policy(policies...));

                return make_policy_holder_impl(
                    base_default_pair_list<RemainingBaseDefaultPairs...>{},
                    found_policy_pair_list < repeated ||
                        new_found_policy_pair::found_info == policy_found_info::repeated,
                    new_found_policy_pair, FoundPolicyPairs... > {}, policies...);
            }

            template <bool repeated, class... RawPolicies>
            constexpr auto convert_to_policy_holder(found_policy_pair_list<repeated>,
                                                    RawPolicies...) {
                return policy_holder<RawPolicies...>{};
            }

            template <bool repeated, class FirstFoundPolicyPair, class... RemainingFoundPolicyPairs,
                      class... RawPolicies>
            constexpr auto
            convert_to_policy_holder(found_policy_pair_list<repeated, FirstFoundPolicyPair,
                                                            RemainingFoundPolicyPairs...>,
                                     RawPolicies... policies) {
                return convert_to_policy_holder(
                    found_policy_pair_list<repeated, RemainingFoundPolicyPairs...>{},
                    typename FirstFoundPolicyPair::policy{}, policies...);
            }

            template <class BaseDefaultPairList, class... Policies>
            constexpr auto make_policy_holder(BaseDefaultPairList, Policies... policies) {
                static_assert(check_policy_list_validity(BaseDefaultPairList{}, Policies{}...),
                              "jkj::dragonbox: an invalid policy is specified");

                using policy_pair_list = decltype(make_policy_holder_impl(
                    BaseDefaultPairList{}, found_policy_pair_list<false>{}, policies...));

                static_assert(!policy_pair_list::repeated,
                              "jkj::dragonbox: each policy should be specified at most once");

                return convert_to_policy_holder(policy_pair_list{});
            }
        }
    }


    ////////////////////////////////////////////////////////////////////////////////////////
    // The interface function.
    ////////////////////////////////////////////////////////////////////////////////////////

    template <class Float, class FloatTraits = default_float_traits<Float>, class... Policies>
    JKJ_FORCEINLINE JKJ_SAFEBUFFERS auto
    to_decimal(signed_significand_bits<Float, FloatTraits> signed_significand_bits,
               unsigned int exponent_bits, Policies... policies) noexcept {
        // Build policy holder type.
        using namespace detail::policy_impl;
        using policy_holder = decltype(make_policy_holder(
            base_default_pair_list<base_default_pair<sign::base, sign::return_sign>,
                                   base_default_pair<trailing_zero::base, trailing_zero::remove>,
                                   base_default_pair<decimal_to_binary_rounding::base,
                                                     decimal_to_binary_rounding::nearest_to_even>,
                                   base_default_pair<binary_to_decimal_rounding::base,
                                                     binary_to_decimal_rounding::to_even>,
                                   base_default_pair<cache::base, cache::full>>{},
            policies...));

        using return_type =
            decimal_fp<typename FloatTraits::carrier_uint, policy_holder::return_has_sign,
                       policy_holder::report_trailing_zeros>;

        return_type ret = policy_holder::delegate(
            signed_significand_bits,
            [exponent_bits, signed_significand_bits](auto interval_type_provider) {
                using format = typename FloatTraits::format;
                constexpr auto tag = decltype(interval_type_provider)::tag;

                auto two_fc = signed_significand_bits.remove_sign_bit_and_shift();
                auto exponent = int(exponent_bits);

                if constexpr (tag == decimal_to_binary_rounding::tag_t::to_nearest) {
                    // Is the input a normal number?
                    if (exponent != 0) {
                        exponent += format::exponent_bias - format::significand_bits;

                        // Shorter interval case; proceed like Schubfach.
                        // One might think this condition is wrong, since when exponent_bits == 1
                        // and two_fc == 0, the interval is actually regular. However, it turns out
                        // that this seemingly wrong condition is actually fine, because the end
                        // result is anyway the same.
                        //
                        // [binary32]
                        // (fc-1/2) * 2^e = 1.175'494'28... * 10^-38
                        // (fc-1/4) * 2^e = 1.175'494'31... * 10^-38
                        //    fc    * 2^e = 1.175'494'35... * 10^-38
                        // (fc+1/2) * 2^e = 1.175'494'42... * 10^-38
                        //
                        // Hence, shorter_interval_case will return 1.175'494'4 * 10^-38.
                        // 1.175'494'3 * 10^-38 is also a correct shortest representation that will
                        // be rejected if we assume shorter interval, but 1.175'494'4 * 10^-38 is
                        // closer to the true value so it doesn't matter.
                        //
                        // [binary64]
                        // (fc-1/2) * 2^e = 2.225'073'858'507'201'13... * 10^-308
                        // (fc-1/4) * 2^e = 2.225'073'858'507'201'25... * 10^-308
                        //    fc    * 2^e = 2.225'073'858'507'201'38... * 10^-308
                        // (fc+1/2) * 2^e = 2.225'073'858'507'201'63... * 10^-308
                        //
                        // Hence, shorter_interval_case will return 2.225'073'858'507'201'4 *
                        // 10^-308. This is indeed of the shortest length, and it is the unique one
                        // closest to the true value among valid representations of the same length.
                        static_assert(std::is_same_v<format, ieee754_binary32> ||
                                      std::is_same_v<format, ieee754_binary64>);

                        if (two_fc == 0) {
                            return decltype(interval_type_provider)::invoke_shorter_interval_case(
                                signed_significand_bits, [exponent](auto... additional_args) {
                                    return detail::impl<Float, FloatTraits>::
                                        template compute_nearest_shorter<
                                            return_type,
                                            typename decltype(interval_type_provider)::
                                                shorter_interval_type,
                                            typename policy_holder::trailing_zero_policy,
                                            typename policy_holder::
                                                binary_to_decimal_rounding_policy,
                                            typename policy_holder::cache_policy>(
                                            exponent, additional_args...);
                                });
                        }

                        two_fc |= (decltype(two_fc)(1) << (format::significand_bits + 1));
                    }
                    // Is the input a subnormal number?
                    else {
                        exponent = format::min_exponent - format::significand_bits;
                    }

                    return decltype(interval_type_provider)::invoke_normal_interval_case(
                        signed_significand_bits, [two_fc, exponent](auto... additional_args) {
                            return detail::impl<Float, FloatTraits>::
                                template compute_nearest_normal<
                                    return_type,
                                    typename decltype(interval_type_provider)::normal_interval_type,
                                    typename policy_holder::trailing_zero_policy,
                                    typename policy_holder::binary_to_decimal_rounding_policy,
                                    typename policy_holder::cache_policy>(two_fc, exponent,
                                                                          additional_args...);
                        });
                }
                else if constexpr (tag == decimal_to_binary_rounding::tag_t::left_closed_directed) {
                    // Is the input a normal number?
                    if (exponent != 0) {
                        exponent += format::exponent_bias - format::significand_bits;
                        two_fc |= (decltype(two_fc)(1) << (format::significand_bits + 1));
                    }
                    // Is the input a subnormal number?
                    else {
                        exponent = format::min_exponent - format::significand_bits;
                    }

                    return detail::impl<Float>::template compute_left_closed_directed<
                        return_type, typename policy_holder::trailing_zero_policy,
                        typename policy_holder::cache_policy>(two_fc, exponent);
                }
                else {
                    static_assert(tag == decimal_to_binary_rounding::tag_t::right_closed_directed);

                    bool shorter_interval = false;

                    // Is the input a normal number?
                    if (exponent != 0) {
                        if (two_fc == 0 && exponent != 1) {
                            shorter_interval = true;
                        }
                        exponent += format::exponent_bias - format::significand_bits;
                        two_fc |= (decltype(two_fc)(1) << (format::significand_bits + 1));
                    }
                    // Is the input a subnormal number?
                    else {
                        exponent = format::min_exponent - format::significand_bits;
                    }

                    return detail::impl<Float>::template compute_right_closed_directed<
                        return_type, typename policy_holder::trailing_zero_policy,
                        typename policy_holder::cache_policy>(two_fc, exponent, shorter_interval);
                }
            });

        policy_holder::handle_sign(signed_significand_bits, ret);
        return ret;
    }

    template <class Float, class FloatTraits = default_float_traits<Float>, class... Policies>
    JKJ_FORCEINLINE JKJ_SAFEBUFFERS auto to_decimal(Float x, Policies... policies) noexcept {
        auto const br = float_bits<Float, FloatTraits>(x);
        auto const exponent_bits = br.extract_exponent_bits();
        auto const s = br.remove_exponent_bits(exponent_bits);
        assert(br.is_finite());

        return to_decimal<Float, FloatTraits>(s, exponent_bits, policies...);
    }
}

#undef JKJ_FORCEINLINE
#undef JKJ_SAFEBUFFERS
#undef JKJ_DRAGONBOX_HAS_BUILTIN

#endif