// Copyright 2020 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_DRAGONBOX
#define JKJ_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(_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;
}

enum class ieee754_format {
  binary32,
  binary64
};

template <ieee754_format format_>
struct ieee754_format_info;

template <>
struct ieee754_format_info<ieee754_format::binary32> {
  static constexpr auto format = ieee754_format::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;
};

template <>
struct ieee754_format_info<ieee754_format::binary64> {
  static constexpr auto format = ieee754_format::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;
};

// To reduce boilerplates
template <class T>
struct default_ieee754_traits {
  static_assert(detail::physical_bits<T> == 32 || detail::physical_bits<T> == 64);

  using type = T;
  static constexpr ieee754_format format =
      detail::physical_bits<T> == 32 ? ieee754_format::binary32 : ieee754_format::binary64;

  using carrier_uint = std::conditional_t<
      detail::physical_bits<T> == 32,
      std::uint32_t,
      std::uint64_t>;
  static_assert(sizeof(carrier_uint) == sizeof(T));

  static constexpr int carrier_bits = int(detail::physical_bits<carrier_uint>);

  static T carrier_to_float(carrier_uint u) noexcept {
    T x;
    std::memcpy(&x, &u, sizeof(carrier_uint));
    return x;
  }
  static carrier_uint float_to_carrier(T x) noexcept {
    carrier_uint u;
    std::memcpy(&u, &x, sizeof(carrier_uint));
    return u;
  }

  static constexpr unsigned int extract_exponent_bits(carrier_uint u) noexcept {
    constexpr int significand_bits = ieee754_format_info<format>::significand_bits;
    constexpr int exponent_bits = ieee754_format_info<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);
  }
  static constexpr carrier_uint extract_significand_bits(carrier_uint u) noexcept {
    constexpr int significand_bits = ieee754_format_info<format>::significand_bits;
    constexpr auto significand_bits_mask = carrier_uint((carrier_uint(1) << significand_bits) - 1);
    return carrier_uint(u & significand_bits_mask);
  }

  // Allows positive zero and positive NaN's, but not allow negative zero
  static constexpr bool is_positive(carrier_uint u) noexcept {
    return (u >> (carrier_bits - 1)) == 0;
  }
  // Allows negative zero and negative NaN's, but not allow positive zero
  static constexpr bool is_negative(carrier_uint u) noexcept {
    return (u >> (carrier_bits - 1)) != 0;
  }

  static constexpr int exponent_bias = 1 - (1 << (carrier_bits - ieee754_format_info<format>::significand_bits - 2));

  static constexpr bool is_finite(carrier_uint u) noexcept {
    constexpr int significand_bits = ieee754_format_info<format>::significand_bits;
    constexpr int exponent_bits = ieee754_format_info<format>::exponent_bits;
    constexpr auto exponent_bits_mask =
        carrier_uint(((carrier_uint(1) << exponent_bits) - 1) << significand_bits);

    return (u & exponent_bits_mask) != exponent_bits_mask;
  }
  static constexpr bool is_nonzero(carrier_uint u) noexcept {
    return (u << 1) != 0;
  }
  // Allows positive and negative zeros
  static constexpr bool is_subnormal(carrier_uint u) noexcept {
    constexpr int significand_bits = ieee754_format_info<format>::significand_bits;
    constexpr int exponent_bits = ieee754_format_info<format>::exponent_bits;
    constexpr auto exponent_bits_mask =
        carrier_uint(((carrier_uint(1) << exponent_bits) - 1) << significand_bits);

    return (u & exponent_bits_mask) == 0;
  }
  static constexpr bool is_positive_infinity(carrier_uint u) noexcept {
    constexpr int significand_bits = ieee754_format_info<format>::significand_bits;
    constexpr int exponent_bits = ieee754_format_info<format>::exponent_bits;
    constexpr auto positive_infinity =
        carrier_uint((carrier_uint(1) << exponent_bits) - 1) << significand_bits;
    return u == positive_infinity;
  }
  static constexpr bool is_negative_infinity(carrier_uint u) noexcept {
    constexpr int significand_bits = ieee754_format_info<format>::significand_bits;
    constexpr int exponent_bits = ieee754_format_info<format>::exponent_bits;
    constexpr auto negative_infinity =
        (carrier_uint((carrier_uint(1) << exponent_bits) - 1) << significand_bits)
        | (carrier_uint(1) << (carrier_bits - 1));
    return u == negative_infinity;
  }
  static constexpr bool is_infinity(carrier_uint u) noexcept {
    return is_positive_infinity(u) || is_negative_infinity(u);
  }
  static constexpr bool is_nan(carrier_uint u) noexcept {
    return !is_finite(u) && (extract_significand_bits(u) != 0);
  }
};

// Speciailze this class template for possible extensions
template <class T>
struct ieee754_traits : default_ieee754_traits<T> {
  // 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 worsk for 32-bits or 64-bits types "
                "supporting binary32 or binary64 formats!");
};

// Convenient wrapper for ieee754_traits
// In order to reduce the argument passing overhead,
// this class should be as simple as possible
// (e.g., no inheritance, no private non-static data member, etc.;
// this is an unfortunate fact about x64 calling convention)
template <class T>
struct ieee754_bits {
  using carrier_uint = typename ieee754_traits<T>::carrier_uint;
  carrier_uint u;

  ieee754_bits() = default;
  constexpr explicit ieee754_bits(carrier_uint bit_pattern) noexcept :
      u{ bit_pattern } {}
  constexpr explicit ieee754_bits(T float_value) noexcept :
      u{ ieee754_traits<T>::float_to_carrier(float_value) } {}

  constexpr T to_float() const noexcept {
    return ieee754_traits<T>::carrier_to_float(u);
  }

  constexpr carrier_uint extract_significand_bits() const noexcept {
    return ieee754_traits<T>::extract_significand_bits(u);
  }
  constexpr unsigned int extract_exponent_bits() const noexcept {
    return ieee754_traits<T>::extract_exponent_bits(u);
  }

  constexpr carrier_uint binary_significand() const noexcept {
    using format_info = ieee754_format_info<ieee754_traits<T>::format>;
    auto s = extract_significand_bits();
    if (extract_exponent_bits() == 0) {
      return s;
    }
    else {
      return s | (carrier_uint(1) << format_info::significand_bits);
    }
  }
  constexpr int binary_exponent() const noexcept {
    using format_info = ieee754_format_info<ieee754_traits<T>::format>;
    auto e = extract_exponent_bits();
    if (e == 0) {
      return format_info::min_exponent;
    }
    else {
      return e + format_info::exponent_bias;
    }
  }

  constexpr bool is_finite() const noexcept {
    return ieee754_traits<T>::is_finite(u);
  }
  constexpr bool is_nonzero() const noexcept {
    return ieee754_traits<T>::is_nonzero(u);
  }
  // Allows positive and negative zeros
  constexpr bool is_subnormal() const noexcept {
    return ieee754_traits<T>::is_subnormal(u);
  }
  // Allows positive zero and positive NaN's, but not allow negative zero
  constexpr bool is_positive() const noexcept {
    return ieee754_traits<T>::is_positive(u);
  }
  // Allows negative zero and negative NaN's, but not allow positive zero
  constexpr bool is_negative() const noexcept {
    return ieee754_traits<T>::is_negative(u);
  }
  constexpr bool is_positive_infinity() const noexcept {
    return ieee754_traits<T>::is_positive_infinity(u);
  }

  constexpr bool is_negative_infinity() const noexcept {
    return ieee754_traits<T>::is_negative_infinity(u);
  }
  // Allows both plus and minus infinities
  constexpr bool is_infinity() const noexcept {
    return ieee754_traits<T>::is_infinity(u);
  }
  constexpr bool is_nan() const noexcept {
    return ieee754_traits<T>::is_nan(u);
  }
};

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

namespace bits {
template <class UInt>
inline int countr_zero(UInt n) noexcept {
  static_assert(std::is_unsigned_v<UInt> && value_bits<UInt> <= 64);
#if defined(__GNUC__) || defined(__clang__)
#define JKJ_HAS_COUNTR_ZERO_INTRINSIC 1
  if constexpr (std::is_same_v<UInt, unsigned long>) {
    return __builtin_ctzl(n);
  }
  else if constexpr (std::is_same_v<UInt, unsigned long long>) {
    return __builtin_ctzll(n);
  }
  else {
    static_assert(sizeof(UInt) <= sizeof(unsigned int));
    return __builtin_ctz((unsigned int)n);
  }
#elif defined(_MSC_VER)
  #define JKJ_HAS_COUNTR_ZERO_INTRINSIC 1
				if constexpr (std::is_same_v<UInt, unsigned __int64>) {
#if defined(_M_X64)
					return int(_tzcnt_u64(n));
#else
					return ((unsigned int)(n) == 0) ?
						(32 + (_tzcnt_u32((unsigned int)(n >> 32)))) :
						(_tzcnt_u32((unsigned int)n));
#endif
				}
				else {
					static_assert(sizeof(UInt) <= sizeof(unsigned int));
					return int(_tzcnt_u32((unsigned int)n));
				}
#else
#define JKJ_HAS_COUNTR_ZERO_INTRINSIC 0
				int count;
				auto n32 = std::uint32_t(n);

				if constexpr (value_bits<UInt> > 32) {
					if (n32 != 0) {
						count = 31;
					}
					else {
						n32 = std::uint32_t(n >> 32);
						if constexpr (value_bits<UInt> == 64) {
							if (n32 != 0) {
								count = 63;
							}
							else {
								return 64;
							}
						}
						else {
							count = value_bits<UInt>;
						}
					}
				}
				else {
					if constexpr (value_bits<UInt> == 32) {
						if (n32 != 0) {
							count = 31;
						}
						else {
							return 32;
						}
					}
					else {
						count = value_bits<UInt>;
					}
				}

				n32 &= (0 - n32);
				if constexpr (value_bits<UInt> > 16) {
					if ((n32 & 0x0000ffff) != 0) count -= 16;
				}
				if constexpr (value_bits<UInt> > 8) {
					if ((n32 & 0x00ff00ff) != 0) count -= 8;
				}
				if ((n32 & 0x0f0f0f0f) != 0) count -= 4;
				if ((n32 & 0x33333333) != 0) count -= 2;
				if ((n32 & 0x55555555) != 0) count -= 1;

				return count;
#endif
}
}

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

namespace wuint {
struct uint128 {
  uint128() = default;

#if defined(__SIZEOF_INT128__)
  unsigned __int128	internal_;

  constexpr uint128(std::uint64_t high, std::uint64_t low) noexcept :
      internal_{ ((unsigned __int128)low) | (((unsigned __int128)high) << 64) } {}

  constexpr uint128(unsigned __int128 u) noexcept : internal_{ u } {}

  constexpr std::uint64_t high() const noexcept {
    return std::uint64_t(internal_ >> 64);
  }
  constexpr std::uint64_t low() const noexcept {
    return std::uint64_t(internal_);
  }

  uint128& operator+=(std::uint64_t n) & noexcept {
    internal_ += n;
    return *this;
  }
#else
  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 defined(_MSC_VER) && defined(_M_X64)
					auto carry = _addcarry_u64(0, low_, n, &low_);
					_addcarry_u64(carry, high_, 0, &high_);
					return *this;
#else
					auto sum = low_ + n;
					high_ += (sum < low_ ? 1 : 0);
					low_ = sum;
					return *this;
#endif
				}
#endif
};

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__)
  return (unsigned __int128)(x) * (unsigned __int128)(y);
#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 p = (unsigned __int128)(x) * (unsigned __int128)(y);
  return std::uint64_t(p >> 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 64-bits of multiplication of a 64-bit unsigned integer and a 128-bit unsigned integer
JKJ_SAFEBUFFERS inline std::uint64_t umul192_upper64(std::uint64_t x, uint128 y) noexcept {
  auto g0 = umul128(x, y.high());
  g0 += umul128_upper64(x, y.low());
  return g0.high();
}

// Get upper 32-bits of multiplication of a 32-bit unsigned integer and a 64-bit unsigned integer
inline std::uint32_t umul96_upper32(std::uint32_t x, std::uint64_t y) noexcept {
#if defined(__SIZEOF_INT128__) || (defined(_MSC_VER) && defined(_M_X64))
  return std::uint32_t(umul128_upper64(x, y));
#else
  //std::uint32_t a = 0;
				auto b = x;
				auto c = std::uint32_t(y >> 32);
				auto d = std::uint32_t(y);

				//std::uint64_t ac = 0;
				auto bc = umul64(b, c);
				//std::uint64_t ad = 0;
				auto bd = umul64(b, d);

				auto intermediate = (bd >> 32) + bc;
				return std::uint32_t(intermediate >> 32);
#endif
}

// Get middle 64-bits of multiplication of a 64-bit unsigned integer and a 128-bit unsigned integer
JKJ_SAFEBUFFERS inline std::uint64_t umul192_middle64(std::uint64_t x, uint128 y) noexcept {
  auto g01 = x * y.high();
  auto g10 = umul128_upper64(x, y.low());
  return g01 + g10;
}

// Get middle 32-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;
}
}

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 {
constexpr std::int32_t floor_shift(
    std::uint32_t integer_part,
    std::uint64_t fractional_digits,
    std::size_t shift_amount) noexcept
{
  assert(shift_amount < 32);
  // Ensure no overflow
  assert(shift_amount == 0 || integer_part < (std::uint32_t(1) << (32 - shift_amount)));

  return shift_amount == 0 ? std::int32_t(integer_part) :
         std::int32_t(
             (integer_part << shift_amount) |
             (fractional_digits >> (64 - shift_amount)));
}

// Compute floor(e * c - s)
template <
    std::uint32_t c_integer_part,
    std::uint64_t c_fractional_digits,
    std::size_t shift_amount,
    std::int32_t max_exponent,
    std::uint32_t s_integer_part = 0,
    std::uint64_t s_fractional_digits = 0
>
constexpr int compute(int e) noexcept {
  assert(e <= max_exponent && e >= -max_exponent);
  constexpr auto c = floor_shift(c_integer_part, c_fractional_digits, shift_amount);
  constexpr auto s = floor_shift(s_integer_part, s_fractional_digits, shift_amount);
  return int((std::int32_t(e) * c - s) >> shift_amount);
}

inline constexpr std::uint64_t log10_2_fractional_digits{ 0x4d10'4d42'7de7'fbcc };
inline constexpr std::uint64_t log10_4_over_3_fractional_digits{ 0x1ffb'fc2b'bc78'0375 };
inline constexpr std::size_t floor_log10_pow2_shift_amount = 22;
inline constexpr int floor_log10_pow2_input_limit = 1700;
inline constexpr int floor_log10_pow2_minus_log10_4_over_3_input_limit = 1700;

inline constexpr std::uint64_t log2_10_fractional_digits{ 0x5269'e12f'346e'2bf9 };
inline constexpr std::size_t floor_log2_pow10_shift_amount = 19;
inline constexpr int floor_log2_pow10_input_limit = 1233;

inline constexpr std::uint64_t log5_2_fractional_digits{ 0x6e40'd1a4'143d'cb94 };
inline constexpr std::uint64_t log5_3_fractional_digits{ 0xaebf'4791'5d44'3b24 };
inline constexpr std::size_t floor_log5_pow2_shift_amount = 20;
inline constexpr int floor_log5_pow2_input_limit = 1492;
inline constexpr int floor_log5_pow2_minus_log5_3_input_limit = 2427;

// 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;
}

constexpr int floor_log10_pow2(int e) noexcept {
  using namespace log;
  return compute<
      0, log10_2_fractional_digits,
      floor_log10_pow2_shift_amount,
      floor_log10_pow2_input_limit>(e);
}

constexpr int floor_log2_pow10(int e) noexcept {
  using namespace log;
  return compute<
      3, log2_10_fractional_digits,
      floor_log2_pow10_shift_amount,
      floor_log2_pow10_input_limit>(e);
}

constexpr int floor_log5_pow2(int e) noexcept {
  using namespace log;
  return compute<
      0, log5_2_fractional_digits,
      floor_log5_pow2_shift_amount,
      floor_log5_pow2_input_limit>(e);
}

constexpr int floor_log5_pow2_minus_log5_3(int e) noexcept {
  using namespace log;
  return compute<
      0, log5_2_fractional_digits,
      floor_log5_pow2_shift_amount,
      floor_log5_pow2_minus_log5_3_input_limit,
      0, log5_3_fractional_digits>(e);
}

constexpr int floor_log10_pow2_minus_log10_4_over_3(int e) noexcept {
  using namespace log;
  return compute<
      0, log10_2_fractional_digits,
      floor_log10_pow2_shift_amount,
      floor_log10_pow2_minus_log10_4_over_3_input_limit,
      0, log10_4_over_3_fractional_digits>(e);
}
}

////////////////////////////////////////////////////////////////////////////////////////
// Utilities for fast divisibility test
////////////////////////////////////////////////////////////////////////////////////////

namespace div {
template <class UInt, UInt a>
constexpr UInt modular_inverse(unsigned int bit_width = unsigned(value_bits<UInt>)) noexcept {
  // By Euler's theorem, a^phi(2^n) == 1 (mod 2^n),
  // where phi(2^n) = 2^(n-1), so the modular inverse of a is
  // a^(2^(n-1) - 1) = a^(1 + 2 + 2^2 + ... + 2^(n-2))
  std::common_type_t<UInt, unsigned int> mod_inverse = 1;
  for (unsigned int i = 1; i < bit_width; ++i) {
    mod_inverse = mod_inverse * mod_inverse * a;
  }
  if (bit_width < value_bits<UInt>) {
    auto mask = UInt((UInt(1) << bit_width) - 1);
    return UInt(mod_inverse & mask);
  }
  else {
    return UInt(mod_inverse);
  }
}

template <class UInt, UInt a, std::size_t N>
struct table_t {
  static_assert(std::is_unsigned_v<UInt>);
  static_assert(a % 2 != 0);
  static_assert(N > 0);

  static constexpr std::size_t size = N;
  UInt mod_inv[N];
  UInt max_quotients[N];
};

template <class UInt, UInt a, std::size_t N>
struct table_holder {
  static constexpr table_t<UInt, a, N> table = [] {
    constexpr auto mod_inverse = modular_inverse<UInt, a>();
    table_t<UInt, a, N> table{};
    std::common_type_t<UInt, unsigned int> pow_of_mod_inverse = 1;
    UInt pow_of_a = 1;
    for (std::size_t i = 0; i < N; ++i) {
      table.mod_inv[i] = UInt(pow_of_mod_inverse);
      table.max_quotients[i] = UInt(std::numeric_limits<UInt>::max() / pow_of_a);

      pow_of_mod_inverse *= mod_inverse;
      pow_of_a *= a;
    }

    return table;
  }();
};

template <std::size_t table_size, class UInt>
constexpr bool divisible_by_power_of_5(UInt x, unsigned int exp) noexcept {
  auto const& table = table_holder<UInt, 5, table_size>::table;
  assert(exp < table.size);
  return (x * table.mod_inv[exp]) <= table.max_quotients[exp];
}

template <class UInt>
constexpr bool divisible_by_power_of_2(UInt x, unsigned int exp) noexcept {
  assert(exp >= 1);
  assert(x != 0);
#if JKJ_HAS_COUNTR_ZERO_INTRINSIC
  return bits::countr_zero(x) >= int(exp);
#else
  if (exp >= value_bits<UInt>) {
					return false;
				}
				auto mask = UInt((UInt(1) << exp) - 1);
				return (x & mask) == 0;
#endif
}

// Replace n by floor(n / 5^N)
// Returns true if and only if n is divisible by 5^N
// Precondition: n <= 2 * 5^(N+1)
template <int N>
struct check_divisibility_and_divide_by_pow5_info;

template <>
struct check_divisibility_and_divide_by_pow5_info<1> {
  static constexpr std::uint32_t magic_number = 0xcccd;
  static constexpr int bits_for_comparison = 16;
  static constexpr std::uint32_t threshold = 0x3333;
  static constexpr int shift_amount = 18;
};

template <>
struct check_divisibility_and_divide_by_pow5_info<2> {
  static constexpr std::uint32_t magic_number = 0xa429;
  static constexpr int bits_for_comparison = 8;
  static constexpr std::uint32_t threshold = 0x0a;
  static constexpr int shift_amount = 20;
};

template <int N>
constexpr bool check_divisibility_and_divide_by_pow5(std::uint32_t& n) noexcept
{
  // Make sure the computation for max_n does not overflow
  static_assert(N + 1 <= log::floor_log5_pow2(31));
  assert(n <= compute_power<N + 1>(std::uint32_t(5)) * 2);

  using info = check_divisibility_and_divide_by_pow5_info<N>;
  n *= info::magic_number;
  constexpr std::uint32_t comparison_mask =
      info::bits_for_comparison >= 32 ? std::numeric_limits<std::uint32_t>::max() :
      std::uint32_t((std::uint32_t(1) << info::bits_for_comparison) - 1);

  if ((n & comparison_mask) <= info::threshold) {
    n >>= info::shift_amount;
    return true;
  }
  else {
    n >>= info::shift_amount;
    return false;
  }
}

// Compute floor(n / 10^N) for small n and N
// Precondition: n <= 10^(N+1)
template <int N>
struct small_division_by_pow10_info;

template <>
struct small_division_by_pow10_info<1> {
  static constexpr std::uint32_t magic_number = 0xcccd;
  static constexpr int shift_amount = 19;
};

template <>
struct small_division_by_pow10_info<2> {
  static constexpr std::uint32_t magic_number = 0xa3d8;
  static constexpr int shift_amount = 22;
};

template <int N>
constexpr std::uint32_t small_division_by_pow10(std::uint32_t n) noexcept
{
  assert(n <= compute_power<N + 1>(std::uint32_t(10)));
  return (n * small_division_by_pow10_info<N>::magic_number)
      >> small_division_by_pow10_info<N>::shift_amount;
}

// Compute floor(n / 10^N) for small N
// Precondition: n <= 2^a * 5^b (a = max_pow2, b = max_pow5)
template <int N, int max_pow2, int max_pow5, class UInt>
constexpr UInt divide_by_pow10(UInt n) noexcept
{
  static_assert(N >= 0);

  // Ensure no overflow
  static_assert(max_pow2 + (log::floor_log2_pow10(max_pow5) - max_pow5) < value_bits<UInt>);

  // Specialize for 64bit division by 1000
  // Ensure that the correctness condition is met
  if constexpr (std::is_same_v<UInt, std::uint64_t> && N == 3 &&
                max_pow2 + (log::floor_log2_pow10(N + max_pow5) - (N + max_pow5)) < 70)
  {
    return wuint::umul128_upper64(n, 0x8312'6e97'8d4f'df3c) >> 9;
  }
  else {
    constexpr auto divisor = compute_power<N>(UInt(10));
    return n / divisor;
  }
}
}
}

////////////////////////////////////////////////////////////////////////////////////////
// DIY floating-point data type
////////////////////////////////////////////////////////////////////////////////////////

template <class Float, bool is_signed, bool trailing_zero_flag>
struct fp_t;

template <class Float>
struct fp_t<Float, false, false> {
  using float_type = Float;
  using carrier_uint = typename ieee754_traits<Float>::carrier_uint;

  carrier_uint	significand;
  int				exponent;
};

template <class Float>
struct fp_t<Float, true, false> {
  using float_type = Float;
  using carrier_uint = typename ieee754_traits<Float>::carrier_uint;

  carrier_uint	significand;
  int				exponent;
  bool			is_negative;
};

template <class Float>
struct fp_t<Float, false, true> {
  using float_type = Float;
  using carrier_uint = typename ieee754_traits<Float>::carrier_uint;

  carrier_uint	significand;
  int				exponent;
  bool			may_have_trailing_zeros;
};

template <class Float>
struct fp_t<Float, true, true> {
  using float_type = Float;
  using carrier_uint = typename ieee754_traits<Float>::carrier_uint;

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

template <class Float>
using unsigned_fp_t = fp_t<Float, false, false>;

template <class Float>
using signed_fp_t = fp_t<Float, true, false>;


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

namespace detail {
template <ieee754_format format>
struct cache_holder;

template <>
struct cache_holder<ieee754_format::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,
      0x813f3978f8940984,
      0xa18f07d736b90be5,
      0xc9f2c9cd04674ede,
      0xfc6f7c4045812296,
      0x9dc5ada82b70b59d,
      0xc5371912364ce305,
      0xf684df56c3e01bc6,
      0x9a130b963a6c115c,
      0xc097ce7bc90715b3,
      0xf0bdc21abb48db20,
      0x96769950b50d88f4,
      0xbc143fa4e250eb31,
      0xeb194f8e1ae525fd,
      0x92efd1b8d0cf37be,
      0xb7abc627050305ad,
      0xe596b7b0c643c719,
      0x8f7e32ce7bea5c6f,
      0xb35dbf821ae4f38b,
      0xe0352f62a19e306e
  };
};

template <>
struct cache_holder<ieee754_format::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 },
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      { 0x847c9b5d7c2e09b7, 0x69956135febada11 },
      { 0xa59bc234db398c25, 0x43fab9837e699095 },
      { 0xcf02b2c21207ef2e, 0x94f967e45e03f4bb },
      { 0x8161afb94b44f57d, 0x1d1be0eebac278f5 },
      { 0xa1ba1ba79e1632dc, 0x6462d92a69731732 },
      { 0xca28a291859bbf93, 0x7d7b8f7503cfdcfe },
      { 0xfcb2cb35e702af78, 0x5cda735244c3d43e },
      { 0x9defbf01b061adab, 0x3a0888136afa64a7 },
      { 0xc56baec21c7a1916, 0x088aaa1845b8fdd0 },
      { 0xf6c69a72a3989f5b, 0x8aad549e57273d45 },
      { 0x9a3c2087a63f6399, 0x36ac54e2f678864b },
      { 0xc0cb28a98fcf3c7f, 0x84576a1bb416a7dd },
      { 0xf0fdf2d3f3c30b9f, 0x656d44a2a11c51d5 },
      { 0x969eb7c47859e743, 0x9f644ae5a4b1b325 },
      { 0xbc4665b596706114, 0x873d5d9f0dde1fee },
      { 0xeb57ff22fc0c7959, 0xa90cb506d155a7ea },
      { 0x9316ff75dd87cbd8, 0x09a7f12442d588f2 },
      { 0xb7dcbf5354e9bece, 0x0c11ed6d538aeb2f },
      { 0xe5d3ef282a242e81, 0x8f1668c8a86da5fa },
      { 0x8fa475791a569d10, 0xf96e017d694487bc },
      { 0xb38d92d760ec4455, 0x37c981dcc395a9ac },
      { 0xe070f78d3927556a, 0x85bbe253f47b1417 },
      { 0x8c469ab843b89562, 0x93956d7478ccec8e },
      { 0xaf58416654a6babb, 0x387ac8d1970027b2 },
      { 0xdb2e51bfe9d0696a, 0x06997b05fcc0319e },
      { 0x88fcf317f22241e2, 0x441fece3bdf81f03 },
      { 0xab3c2fddeeaad25a, 0xd527e81cad7626c3 },
      { 0xd60b3bd56a5586f1, 0x8a71e223d8d3b074 },
      { 0x85c7056562757456, 0xf6872d5667844e49 },
      { 0xa738c6bebb12d16c, 0xb428f8ac016561db },
      { 0xd106f86e69d785c7, 0xe13336d701beba52 },
      { 0x82a45b450226b39c, 0xecc0024661173473 },
      { 0xa34d721642b06084, 0x27f002d7f95d0190 },
      { 0xcc20ce9bd35c78a5, 0x31ec038df7b441f4 },
      { 0xff290242c83396ce, 0x7e67047175a15271 },
      { 0x9f79a169bd203e41, 0x0f0062c6e984d386 },
      { 0xc75809c42c684dd1, 0x52c07b78a3e60868 },
      { 0xf92e0c3537826145, 0xa7709a56ccdf8a82 },
      { 0x9bbcc7a142b17ccb, 0x88a66076400bb691 },
      { 0xc2abf989935ddbfe, 0x6acff893d00ea435 },
      { 0xf356f7ebf83552fe, 0x0583f6b8c4124d43 },
      { 0x98165af37b2153de, 0xc3727a337a8b704a },
      { 0xbe1bf1b059e9a8d6, 0x744f18c0592e4c5c },
      { 0xeda2ee1c7064130c, 0x1162def06f79df73 },
      { 0x9485d4d1c63e8be7, 0x8addcb5645ac2ba8 },
      { 0xb9a74a0637ce2ee1, 0x6d953e2bd7173692 },
      { 0xe8111c87c5c1ba99, 0xc8fa8db6ccdd0437 },
      { 0x910ab1d4db9914a0, 0x1d9c9892400a22a2 },
      { 0xb54d5e4a127f59c8, 0x2503beb6d00cab4b },
      { 0xe2a0b5dc971f303a, 0x2e44ae64840fd61d },
      { 0x8da471a9de737e24, 0x5ceaecfed289e5d2 },
      { 0xb10d8e1456105dad, 0x7425a83e872c5f47 },
      { 0xdd50f1996b947518, 0xd12f124e28f77719 },
      { 0x8a5296ffe33cc92f, 0x82bd6b70d99aaa6f },
      { 0xace73cbfdc0bfb7b, 0x636cc64d1001550b },
      { 0xd8210befd30efa5a, 0x3c47f7e05401aa4e },
      { 0x8714a775e3e95c78, 0x65acfaec34810a71 },
      { 0xa8d9d1535ce3b396, 0x7f1839a741a14d0d },
      { 0xd31045a8341ca07c, 0x1ede48111209a050 },
      { 0x83ea2b892091e44d, 0x934aed0aab460432 },
      { 0xa4e4b66b68b65d60, 0xf81da84d5617853f },
      { 0xce1de40642e3f4b9, 0x36251260ab9d668e },
      { 0x80d2ae83e9ce78f3, 0xc1d72b7c6b426019 },
      { 0xa1075a24e4421730, 0xb24cf65b8612f81f },
      { 0xc94930ae1d529cfc, 0xdee033f26797b627 },
      { 0xfb9b7cd9a4a7443c, 0x169840ef017da3b1 },
      { 0x9d412e0806e88aa5, 0x8e1f289560ee864e },
      { 0xc491798a08a2ad4e, 0xf1a6f2bab92a27e2 },
      { 0xf5b5d7ec8acb58a2, 0xae10af696774b1db },
      { 0x9991a6f3d6bf1765, 0xacca6da1e0a8ef29 },
      { 0xbff610b0cc6edd3f, 0x17fd090a58d32af3 },
      { 0xeff394dcff8a948e, 0xddfc4b4cef07f5b0 },
      { 0x95f83d0a1fb69cd9, 0x4abdaf101564f98e },
      { 0xbb764c4ca7a4440f, 0x9d6d1ad41abe37f1 },
      { 0xea53df5fd18d5513, 0x84c86189216dc5ed },
      { 0x92746b9be2f8552c, 0x32fd3cf5b4e49bb4 },
      { 0xb7118682dbb66a77, 0x3fbc8c33221dc2a1 },
      { 0xe4d5e82392a40515, 0x0fabaf3feaa5334a },
      { 0x8f05b1163ba6832d, 0x29cb4d87f2a7400e },
      { 0xb2c71d5bca9023f8, 0x743e20e9ef511012 },
      { 0xdf78e4b2bd342cf6, 0x914da9246b255416 },
      { 0x8bab8eefb6409c1a, 0x1ad089b6c2f7548e },
      { 0xae9672aba3d0c320, 0xa184ac2473b529b1 },
      { 0xda3c0f568cc4f3e8, 0xc9e5d72d90a2741e },
      { 0x8865899617fb1871, 0x7e2fa67c7a658892 },
      { 0xaa7eebfb9df9de8d, 0xddbb901b98feeab7 },
      { 0xd51ea6fa85785631, 0x552a74227f3ea565 },
      { 0x8533285c936b35de, 0xd53a88958f87275f },
      { 0xa67ff273b8460356, 0x8a892abaf368f137 },
      { 0xd01fef10a657842c, 0x2d2b7569b0432d85 },
      { 0x8213f56a67f6b29b, 0x9c3b29620e29fc73 },
      { 0xa298f2c501f45f42, 0x8349f3ba91b47b8f },
      { 0xcb3f2f7642717713, 0x241c70a936219a73 },
      { 0xfe0efb53d30dd4d7, 0xed238cd383aa0110 },
      { 0x9ec95d1463e8a506, 0xf4363804324a40aa },
      { 0xc67bb4597ce2ce48, 0xb143c6053edcd0d5 },
      { 0xf81aa16fdc1b81da, 0xdd94b7868e94050a },
      { 0x9b10a4e5e9913128, 0xca7cf2b4191c8326 },
      { 0xc1d4ce1f63f57d72, 0xfd1c2f611f63a3f0 },
      { 0xf24a01a73cf2dccf, 0xbc633b39673c8cec },
      { 0x976e41088617ca01, 0xd5be0503e085d813 },
      { 0xbd49d14aa79dbc82, 0x4b2d8644d8a74e18 },
      { 0xec9c459d51852ba2, 0xddf8e7d60ed1219e },
      { 0x93e1ab8252f33b45, 0xcabb90e5c942b503 },
      { 0xb8da1662e7b00a17, 0x3d6a751f3b936243 },
      { 0xe7109bfba19c0c9d, 0x0cc512670a783ad4 },
      { 0x906a617d450187e2, 0x27fb2b80668b24c5 },
      { 0xb484f9dc9641e9da, 0xb1f9f660802dedf6 },
      { 0xe1a63853bbd26451, 0x5e7873f8a0396973 },
      { 0x8d07e33455637eb2, 0xdb0b487b6423e1e8 },
      { 0xb049dc016abc5e5f, 0x91ce1a9a3d2cda62 },
      { 0xdc5c5301c56b75f7, 0x7641a140cc7810fb },
      { 0x89b9b3e11b6329ba, 0xa9e904c87fcb0a9d },
      { 0xac2820d9623bf429, 0x546345fa9fbdcd44 },
      { 0xd732290fbacaf133, 0xa97c177947ad4095 },
      { 0x867f59a9d4bed6c0, 0x49ed8eabcccc485d },
      { 0xa81f301449ee8c70, 0x5c68f256bfff5a74 },
      { 0xd226fc195c6a2f8c, 0x73832eec6fff3111 },
      { 0x83585d8fd9c25db7, 0xc831fd53c5ff7eab },
      { 0xa42e74f3d032f525, 0xba3e7ca8b77f5e55 },
      { 0xcd3a1230c43fb26f, 0x28ce1bd2e55f35eb },
      { 0x80444b5e7aa7cf85, 0x7980d163cf5b81b3 },
      { 0xa0555e361951c366, 0xd7e105bcc332621f },
      { 0xc86ab5c39fa63440, 0x8dd9472bf3fefaa7 },
      { 0xfa856334878fc150, 0xb14f98f6f0feb951 },
      { 0x9c935e00d4b9d8d2, 0x6ed1bf9a569f33d3 },
      { 0xc3b8358109e84f07, 0x0a862f80ec4700c8 },
      { 0xf4a642e14c6262c8, 0xcd27bb612758c0fa },
      { 0x98e7e9cccfbd7dbd, 0x8038d51cb897789c },
      { 0xbf21e44003acdd2c, 0xe0470a63e6bd56c3 },
      { 0xeeea5d5004981478, 0x1858ccfce06cac74 },
      { 0x95527a5202df0ccb, 0x0f37801e0c43ebc8 },
      { 0xbaa718e68396cffd, 0xd30560258f54e6ba },
      { 0xe950df20247c83fd, 0x47c6b82ef32a2069 },
      { 0x91d28b7416cdd27e, 0x4cdc331d57fa5441 },
      { 0xb6472e511c81471d, 0xe0133fe4adf8e952 },
      { 0xe3d8f9e563a198e5, 0x58180fddd97723a6 },
      { 0x8e679c2f5e44ff8f, 0x570f09eaa7ea7648 },
      { 0xb201833b35d63f73, 0x2cd2cc6551e513da },
      { 0xde81e40a034bcf4f, 0xf8077f7ea65e58d1 },
      { 0x8b112e86420f6191, 0xfb04afaf27faf782 },
      { 0xadd57a27d29339f6, 0x79c5db9af1f9b563 },
      { 0xd94ad8b1c7380874, 0x18375281ae7822bc },
      { 0x87cec76f1c830548, 0x8f2293910d0b15b5 },
      { 0xa9c2794ae3a3c69a, 0xb2eb3875504ddb22 },
      { 0xd433179d9c8cb841, 0x5fa60692a46151eb },
      { 0x849feec281d7f328, 0xdbc7c41ba6bcd333 },
      { 0xa5c7ea73224deff3, 0x12b9b522906c0800 },
      { 0xcf39e50feae16bef, 0xd768226b34870a00 },
      { 0x81842f29f2cce375, 0xe6a1158300d46640 },
      { 0xa1e53af46f801c53, 0x60495ae3c1097fd0 },
      { 0xca5e89b18b602368, 0x385bb19cb14bdfc4 },
      { 0xfcf62c1dee382c42, 0x46729e03dd9ed7b5 },
      { 0x9e19db92b4e31ba9, 0x6c07a2c26a8346d1 },
      { 0xc5a05277621be293, 0xc7098b7305241885 },
      { 0xf70867153aa2db38, 0xb8cbee4fc66d1ea7 }
  };
};

// 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_format::binary64>::max_k -
       cache_holder<ieee754_format::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_format::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;
  }();

  static constexpr std::uint32_t errors[] = {
      0x50001400, 0x54044100, 0x54014555, 0x55954415, 0x54115555,
      0x00000001, 0x50000000, 0x00104000, 0x54010004, 0x05004001,
      0x55555544, 0x41545555, 0x54040551, 0x15445545, 0x51555514,
      0x10000015, 0x00101100, 0x01100015, 0x00000000, 0x00000000,
      0x00000000, 0x00000000, 0x04450514, 0x45414110, 0x55555145,
      0x50544050, 0x15040155, 0x11054140, 0x50111514, 0x11451454,
      0x00400541, 0x00000000, 0x55555450, 0x10056551, 0x10054011,
      0x55551014, 0x69514555, 0x05151109, 0x00155555
  };
};
}


////////////////////////////////////////////////////////////////////////////////////////
// Policies
////////////////////////////////////////////////////////////////////////////////////////

namespace detail {
// Forward declare the implementation class
template <class Float>
struct impl;

namespace policy_impl {
// Sign policy
namespace sign {
struct base {};

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

  template <class Float, class Fp>
  static constexpr void handle_sign(ieee754_bits<Float>, Fp&) noexcept {}
};

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

  template <class Float, class Fp>
  static constexpr void handle_sign(ieee754_bits<Float> br, Fp& fp) noexcept {
    fp.is_negative = br.is_negative();
  }
};
}

// Trailing zero policy
namespace trailing_zero {
struct base {};

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

  template <class Fp>
  static constexpr void on_trailing_zeros(Fp&) noexcept {}

  template <class Fp>
  static constexpr void no_trailing_zeros(Fp&) noexcept {}
};

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

  template <class Fp>
  static constexpr void on_trailing_zeros(Fp& fp) noexcept {
    fp.exponent +=
        impl<typename Fp::float_type>::remove_trailing_zeros(fp.significand);
  }

  template <class Fp>
  static constexpr void no_trailing_zeros(Fp&) noexcept {}
};

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

  template <class Fp>
  static constexpr void on_trailing_zeros(Fp& fp) noexcept {
    fp.may_have_trailing_zeros = true;
  }

  template <class Fp>
  static constexpr void no_trailing_zeros(Fp& fp) noexcept {
    fp.may_have_trailing_zeros = false;
  }
};
}

// Rounding mode policy
namespace rounding_mode {
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 rounding_mode_policy = nearest_to_even;
  static constexpr auto tag = tag_t::to_nearest;

  template <class Float, class Func>
  static auto delegate(ieee754_bits<Float>, Func&& f) noexcept {
    return f(nearest_to_even{});
  }

  template <class Float>
  static constexpr interval_type::symmetric_boundary
  interval_type_normal(ieee754_bits<Float> br) noexcept
  {
    return{ br.u % 2 == 0 };
  }
  template <class Float>
  static constexpr interval_type::closed
  interval_type_shorter(ieee754_bits<Float>) noexcept
  {
    return{};
  }
};
struct nearest_to_odd : base {
  using rounding_mode_policy = nearest_to_odd;
  static constexpr auto tag = tag_t::to_nearest;

  template <class Float, class Func>
  static auto delegate(ieee754_bits<Float>, Func&& f) noexcept {
    return f(nearest_to_odd{});
  }

  template <class Float>
  static constexpr interval_type::symmetric_boundary
  interval_type_normal(ieee754_bits<Float> br) noexcept
  {
    return{ br.u % 2 != 0 };
  }
  template <class Float>
  static constexpr interval_type::closed
  interval_type_shorter(ieee754_bits<Float>) noexcept
  {
    return{};
  }
};
struct nearest_toward_plus_infinity : base {
  using rounding_mode_policy = nearest_toward_plus_infinity;
  static constexpr auto tag = tag_t::to_nearest;

  template <class Float, class Func>
  static auto delegate(ieee754_bits<Float>, Func&& f) noexcept {
    return f(nearest_toward_plus_infinity{});
  }

  template <class Float>
  static constexpr interval_type::asymmetric_boundary
  interval_type_normal(ieee754_bits<Float> br) noexcept
  {
    return{ !br.is_negative() };
  }
  template <class Float>
  static constexpr interval_type::asymmetric_boundary
  interval_type_shorter(ieee754_bits<Float> br) noexcept
  {
    return{ !br.is_negative() };
  }
};
struct nearest_toward_minus_infinity : base {
  using rounding_mode_policy = nearest_toward_minus_infinity;
  static constexpr auto tag = tag_t::to_nearest;

  template <class Float, class Func>
  static auto delegate(ieee754_bits<Float>, Func&& f) noexcept {
    return f(nearest_toward_minus_infinity{});
  }

  template <class Float>
  static constexpr interval_type::asymmetric_boundary
  interval_type_normal(ieee754_bits<Float> br) noexcept
  {
    return{ br.is_negative() };
  }
  template <class Float>
  static constexpr interval_type::asymmetric_boundary
  interval_type_shorter(ieee754_bits<Float> br) noexcept
  {
    return{ br.is_negative() };
  }
};
struct nearest_toward_zero : base {
  using rounding_mode_policy = nearest_toward_zero;
  static constexpr auto tag = tag_t::to_nearest;

  template <class Float, class Func>
  static auto delegate(ieee754_bits<Float>, Func&& f) noexcept {
    return f(nearest_toward_zero{});
  }
  template <class Float>
  static constexpr interval_type::right_closed_left_open
  interval_type_normal(ieee754_bits<Float>) noexcept
  {
    return{};
  }
  template <class Float>
  static constexpr interval_type::right_closed_left_open
  interval_type_shorter(ieee754_bits<Float>) noexcept
  {
    return{};
  }
};
struct nearest_away_from_zero : base {
  using rounding_mode_policy = nearest_away_from_zero;
  static constexpr auto tag = tag_t::to_nearest;

  template <class Float, class Func>
  static auto delegate(ieee754_bits<Float>, Func&& f) noexcept {
    return f(nearest_away_from_zero{});
  }
  template <class Float>
  static constexpr interval_type::left_closed_right_open
  interval_type_normal(ieee754_bits<Float>) noexcept
  {
    return{};
  }
  template <class Float>
  static constexpr interval_type::left_closed_right_open
  interval_type_shorter(ieee754_bits<Float>) noexcept
  {
    return{};
  }
};

namespace detail {
struct nearest_always_closed {
  static constexpr auto tag = tag_t::to_nearest;

  template <class Float>
  static constexpr interval_type::closed
  interval_type_normal(ieee754_bits<Float>) noexcept
  {
    return{};
  }
  template <class Float>
  static constexpr interval_type::closed
  interval_type_shorter(ieee754_bits<Float>) noexcept
  {
    return{};
  }
};
struct nearest_always_open {
  static constexpr auto tag = tag_t::to_nearest;

  template <class Float>
  static constexpr interval_type::open
  interval_type_normal(ieee754_bits<Float>) noexcept
  {
    return{};
  }
  template <class Float>
  static constexpr interval_type::open
  interval_type_shorter(ieee754_bits<Float>) noexcept
  {
    return{};
  }
};
}

struct nearest_to_even_static_boundary : base {
  using rounding_mode_policy = nearest_to_even_static_boundary;
  template <class Float, class Func>
  static auto delegate(ieee754_bits<Float> br, Func&& f) noexcept {
    if (br.u % 2 == 0) {
      return f(detail::nearest_always_closed{});
    }
    else {
      return f(detail::nearest_always_open{});
    }
  }
};
struct nearest_to_odd_static_boundary : base {
  using rounding_mode_policy = nearest_to_odd_static_boundary;
  template <class Float, class Func>
  static auto delegate(ieee754_bits<Float> br, Func&& f) noexcept {
    if (br.u % 2 == 0) {
      return f(detail::nearest_always_open{});
    }
    else {
      return f(detail::nearest_always_closed{});
    }
  }
};
struct nearest_toward_plus_infinity_static_boundary : base {
  using rounding_mode_policy = nearest_toward_plus_infinity_static_boundary;
  template <class Float, class Func>
  static auto delegate(ieee754_bits<Float> br, Func&& f) noexcept {
    if (br.is_negative()) {
      return f(nearest_toward_zero{});
    }
    else {
      return f(nearest_away_from_zero{});
    }
  }
};
struct nearest_toward_minus_infinity_static_boundary : base {
  using rounding_mode_policy = nearest_toward_minus_infinity_static_boundary;
  template <class Float, class Func>
  static auto delegate(ieee754_bits<Float> br, Func&& f) noexcept {
    if (br.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;

  template <class Float>
  static constexpr interval_type::left_closed_right_open
  interval_type_normal(ieee754_bits<Float>) noexcept
  {
    return{};
  }
};
struct right_closed_directed {
  static constexpr auto tag = tag_t::right_closed_directed;

  template <class Float>
  static constexpr interval_type::right_closed_left_open
  interval_type_normal(ieee754_bits<Float>) noexcept
  {
    return{};
  }
};
}

struct toward_plus_infinity : base {
  using rounding_mode_policy = toward_plus_infinity;
  template <class Float, class Func>
  static auto delegate(ieee754_bits<Float> br, Func&& f) noexcept {
    if (br.is_negative()) {
      return f(detail::left_closed_directed{});
    }
    else {
      return f(detail::right_closed_directed{});
    }
  }
};
struct toward_minus_infinity : base {
  using rounding_mode_policy = toward_minus_infinity;
  template <class Float, class Func>
  static auto delegate(ieee754_bits<Float> br, Func&& f) noexcept {
    if (br.is_negative()) {
      return f(detail::right_closed_directed{});
    }
    else {
      return f(detail::left_closed_directed{});
    }
  }
};
struct toward_zero : base {
  using rounding_mode_policy = toward_zero;
  template <class Float, class Func>
  static auto delegate(ieee754_bits<Float>, Func&& f) noexcept {
    return f(detail::left_closed_directed{});
  }
};
struct away_from_zero : base {
  using rounding_mode_policy = away_from_zero;
  template <class Float, class Func>
  static auto delegate(ieee754_bits<Float>, Func&& f) noexcept {
    return f(detail::right_closed_directed{});
  }
};
}

// Correct rounding policy
namespace correct_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 correct_rounding_policy = do_not_care;
  static constexpr auto tag = tag_t::do_not_care;

  template <class Fp>
  static constexpr void break_rounding_tie(Fp&) noexcept {}
};

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

  template <class Fp>
  static constexpr void break_rounding_tie(Fp& fp) noexcept
  {
    fp.significand = fp.significand % 2 == 0 ?
                     fp.significand : fp.significand - 1;
  }
};

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

  template <class Fp>
  static constexpr void break_rounding_tie(Fp& fp) noexcept
  {
    fp.significand = fp.significand % 2 != 0 ?
                     fp.significand : fp.significand - 1;
  }
};

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

  template <class Fp>
  static constexpr void break_rounding_tie(Fp& /*fp*/) noexcept {}
};

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

  template <class Fp>
  static constexpr void break_rounding_tie(Fp& fp) noexcept
  {
    --fp.significand;
  }
};
}

namespace cache {
struct base {};

struct normal : base {
  using cache_policy = normal;
  template <ieee754_format format>
  static constexpr typename cache_holder<format>::cache_entry_type get_cache(int k) noexcept {
    assert(k >= cache_holder<format>::min_k && k <= cache_holder<format>::max_k);
    return cache_holder<format>::cache[std::size_t(k - cache_holder<format>::min_k)];
  }
};

struct compressed : base {
  using cache_policy = compressed;
  template <ieee754_format format>
  static constexpr typename cache_holder<format>::cache_entry_type get_cache(int k) noexcept {
    assert(k >= cache_holder<format>::min_k && k <= cache_holder<format>::max_k);

    if constexpr (format == ieee754_format::binary64)
    {
      // Compute base index
      auto cache_index = (k - cache_holder<format>::min_k) /
                         compressed_cache_detail::compression_ratio;
      auto kb = cache_index * compressed_cache_detail::compression_ratio
                + cache_holder<format>::min_k;
      auto offset = k - kb;

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

      if (offset == 0) {
        return base_cache;
      }
      else {
        // Compute the required amount of bit-shift
        auto 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 pow5 = compressed_cache_detail::pow5.table[offset];
        auto recovered_cache = wuint::umul128(base_cache.high(), pow5);
        auto middle_low = wuint::umul128(base_cache.low() - (kb < 0 ? 1 : 0), pow5);

        recovered_cache += middle_low.high();

        auto high_to_middle = recovered_cache.high() << (64 - alpha);
        auto 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)
        };

        if (kb < 0) {
          recovered_cache += 1;
        }

        // Get error
        auto error_idx = (k - cache_holder<format>::min_k) / 16;
        auto error = (compressed_cache_detail::errors[error_idx] >>
                                                                 ((k - cache_holder<format>::min_k) % 16) * 2) & 0x3;

        // Add the error back
        assert(recovered_cache.low() + error >= recovered_cache.low());
        recovered_cache = {
            recovered_cache.high(),
            recovered_cache.low() + error
        };

        return recovered_cache;
      }
    }
    else
    {
      return cache_holder<format>::cache[std::size_t(k - cache_holder<format>::min_k)];
    }
  }
};
}

namespace input_validation {
struct base {};

struct assert_finite : base {
  using input_validation_policy = assert_finite;
  template <class Float>
  static void validate_input([[maybe_unused]] ieee754_bits<Float> br) noexcept
  {
    assert(br.is_finite());
  }
};

struct do_nothing : base {
  using input_validation_policy = do_nothing;
  template <class Float>
  static void validate_input(ieee754_bits<Float>) noexcept {}
};
}
}
}

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

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

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

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

namespace cache {
inline constexpr auto normal = detail::policy_impl::cache::normal{};
inline constexpr auto compressed = detail::policy_impl::cache::compressed{};
}

namespace input_validation {
inline constexpr auto assert_finite = detail::policy_impl::input_validation::assert_finite{};
inline constexpr auto do_nothing = detail::policy_impl::input_validation::do_nothing{};
}
}

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

// Get sign/decimal significand/decimal exponent from
// the bit representation of a floating-point number
template <class Float>
struct impl : private ieee754_traits<Float>,
              private ieee754_format_info<ieee754_traits<Float>::format>
{
  using carrier_uint = typename ieee754_traits<Float>::carrier_uint;

  using ieee754_traits<Float>::format;
  using ieee754_traits<Float>::carrier_bits;
  using ieee754_format_info<format>::significand_bits;
  using ieee754_format_info<format>::min_exponent;
  using ieee754_format_info<format>::max_exponent;
  using ieee754_format_info<format>::exponent_bias;
  using ieee754_format_info<format>::decimal_digits;

  static constexpr int kappa = format == ieee754_format::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 = [] {
    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 max_power_of_factor_of_5 = log::floor_log5_pow2(int(significand_bits + 2));
  static constexpr int divisibility_check_by_5_threshold =
      log::floor_log2_pow10(max_power_of_factor_of_5 + kappa + 1);

  static constexpr int case_fc_pm_half_lower_threshold = -kappa - log::floor_log5_pow2(kappa);
  static constexpr int case_fc_pm_half_upper_threshold = log::floor_log2_pow10(kappa + 1);

  static constexpr int case_fc_lower_threshold = -kappa - 1 - log::floor_log5_pow2(kappa + 1);
  static constexpr int case_fc_upper_threshold = log::floor_log2_pow10(kappa + 1);

  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;

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

  template <class ReturnType, class IntervalTypeProvider, class SignPolicy,
      class TrailingZeroPolicy, class CorrectRoundingPolicy, class CachePolicy>
  JKJ_SAFEBUFFERS static ReturnType compute_nearest(ieee754_bits<Float> const br) noexcept
  {
    //////////////////////////////////////////////////////////////////////
    // Step 1: integer promotion & Schubfach multiplier calculation
    //////////////////////////////////////////////////////////////////////

    ReturnType ret_value;

    SignPolicy::handle_sign(br, ret_value);

    auto significand = br.extract_significand_bits();
    auto exponent = int(br.extract_exponent_bits());

    // Deal with normal/subnormal dichotomy
    if (exponent != 0) {
      exponent += exponent_bias - significand_bits;

      // Shorter interval case; proceed like Schubfach
      if (significand == 0) {
        shorter_interval_case<TrailingZeroPolicy, CorrectRoundingPolicy, CachePolicy>(
            ret_value, exponent,
            IntervalTypeProvider::interval_type_shorter(br));
        return ret_value;
      }

      significand |= (carrier_uint(1) << significand_bits);
    }
      // Subnormal case; interval is always regular
    else {
      exponent = min_exponent - significand_bits;
    }

    auto const interval_type = IntervalTypeProvider::interval_type_normal(br);

    // 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_minus_1 = exponent + log::floor_log2_pow10(-minus_k);

    // Compute zi and deltai
    // 10^kappa <= deltai < 10^(kappa + 1)
    auto const deltai = compute_delta(cache, beta_minus_1);
    carrier_uint const two_fc = significand << 1;
    carrier_uint const two_fr = two_fc | 1;
    carrier_uint const zi = compute_mul(two_fr << beta_minus_1, 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,
        significand_bits + kappa + 2, kappa + 1>(zi);
    auto r = std::uint32_t(zi - big_divisor * ret_value.significand);

    if (r > deltai) {
      goto small_divisor_case_label;
    }
    else if (r < deltai) {
      // Exclude the right endpoint if necessary
      if (r == 0 && !interval_type.include_right_endpoint() &&
          is_product_integer<integer_check_case_id::fc_pm_half>(two_fr, exponent, minus_k))
      {
        if constexpr (CorrectRoundingPolicy::tag ==
                      policy_impl::correct_rounding::tag_t::do_not_care)
        {
          ret_value.significand *= 10;
          ret_value.exponent = minus_k + kappa;
          --ret_value.significand;
          return ret_value;
        }
        else {
          --ret_value.significand;
          r = big_divisor;
          goto small_divisor_case_label;
        }
      }
    }
    else {
      // r == deltai; compare fractional parts
      // Check conditions in the order different from the paper
      // to take advantage of short-circuiting
      auto const two_fl = two_fc - 1;
      if ((!interval_type.include_left_endpoint() ||
           !is_product_integer<integer_check_case_id::fc_pm_half>(
               two_fl, exponent, minus_k)) &&
          !compute_mul_parity(two_fl, cache, beta_minus_1))
      {
        goto small_divisor_case_label;
      }
    }
    ret_value.exponent = minus_k + kappa + 1;

    // We may need to remove trailing zeros
    TrailingZeroPolicy::on_trailing_zeros(ret_value);
    return ret_value;


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

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

    constexpr auto mask = (std::uint32_t(1) << kappa) - 1;

    if constexpr (CorrectRoundingPolicy::tag ==
                  policy_impl::correct_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 2^kappa?
        if ((r & mask) == 0) {
          r >>= kappa;

          // Is r divisible by 5^kappa?
          if (div::check_divisibility_and_divide_by_pow5<kappa>(r) &&
              is_product_integer<integer_check_case_id::fc_pm_half>(two_fr, exponent, minus_k))
          {
            // 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 {
        ret_value.significand += div::small_division_by_pow10<kappa>(r);
      }
    }
    else
    {
      auto dist = r - (deltai / 2) + (small_divisor / 2);

      // Is dist divisible by 2^kappa?
      if ((dist & mask) == 0) {
        bool const approx_y_parity = ((dist ^ (small_divisor / 2)) & 1) != 0;
        dist >>= kappa;

        // Is dist divisible by 5^kappa?
        if (div::check_divisibility_and_divide_by_pow5<kappa>(dist)) {
          ret_value.significand += dist;

          // 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
          if (compute_mul_parity(two_fc, cache, beta_minus_1) != 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 constexpr (CorrectRoundingPolicy::tag !=
                          policy_impl::correct_rounding::tag_t::away_from_zero)
            {
              if (is_product_integer<integer_check_case_id::fc>(
                  two_fc, exponent, minus_k))
              {
                CorrectRoundingPolicy::break_rounding_tie(ret_value);
              }
            }
          }
        }
          // Is dist not divisible by 5^kappa?
        else {
          ret_value.significand += dist;
        }
      }
        // Is dist not divisible by 2^kappa?
      else {
        // Since we know dist is small, we might be able to optimize the division
        // better than the compiler; we are computing dist / small_divisor here
        ret_value.significand += div::small_division_by_pow10<kappa>(dist);
      }
    }
    return ret_value;
  }

  template <class TrailingZeroPolicy, class CorrectRoundingPolicy,
      class CachePolicy, class ReturnType, class IntervalType>
  JKJ_FORCEINLINE JKJ_SAFEBUFFERS static void shorter_interval_case(
      ReturnType& ret_value, int const exponent, IntervalType const interval_type) noexcept
  {
    // Compute k and beta
    int const minus_k = log::floor_log10_pow2_minus_log10_4_over_3(exponent);
    int const beta_minus_1 = 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_minus_1);
    auto zi = compute_right_endpoint_for_shorter_interval_case(cache, beta_minus_1);

    // 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::on_trailing_zeros(ret_value);
      return;
    }

    // Otherwise, compute the round-up of y
    TrailingZeroPolicy::no_trailing_zeros(ret_value);
    ret_value.significand = compute_round_up_for_shorter_interval_case(cache, beta_minus_1);
    ret_value.exponent = minus_k;

    // When tie occurs, choose one of them according to the rule
    if constexpr (CorrectRoundingPolicy::tag !=
                  policy_impl::correct_rounding::tag_t::do_not_care &&
                  CorrectRoundingPolicy::tag !=
                  policy_impl::correct_rounding::tag_t::away_from_zero)
    {
      if (exponent >= shorter_interval_tie_lower_threshold &&
          exponent <= shorter_interval_tie_upper_threshold)
      {
        CorrectRoundingPolicy::break_rounding_tie(ret_value);
      }
      else if (ret_value.significand < xi) {
        ++ret_value.significand;
      }
    }
    else
    {
      if (ret_value.significand < xi) {
        ++ret_value.significand;
      }
    }
  }

  template <class ReturnType, class SignPolicy, class TrailingZeroPolicy, class CachePolicy>
  JKJ_SAFEBUFFERS static ReturnType
  compute_left_closed_directed(ieee754_bits<Float> const br) noexcept
  {
    //////////////////////////////////////////////////////////////////////
    // Step 1: integer promotion & Schubfach multiplier calculation
    //////////////////////////////////////////////////////////////////////

    ReturnType ret_value;

    SignPolicy::handle_sign(br, ret_value);

    auto significand = br.extract_significand_bits();
    auto exponent = int(br.extract_exponent_bits());

    // Deal with normal/subnormal dichotomy
    if (exponent != 0) {
      exponent += exponent_bias - significand_bits;
      significand |= (carrier_uint(1) << significand_bits);
    }
      // Subnormal case; interval is always regular
    else {
      exponent = min_exponent - significand_bits;
    }

    // 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) + 1;

    // Compute xi and deltai
    // 10^kappa <= deltai < 10^(kappa + 1)
    auto const deltai = compute_delta(cache, beta - 1);
    carrier_uint xi = compute_mul(significand << beta, cache);

    if (!is_product_integer<integer_check_case_id::fc>(significand, exponent + 1, minus_k)) {
      ++xi;
    }

    //////////////////////////////////////////////////////////////////////
    // 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 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,
        significand_bits + kappa + 2, kappa + 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
      if (compute_mul_parity(significand + 1, cache, beta) ||
          is_product_integer<integer_check_case_id::fc>(significand + 1, exponent + 1, minus_k))
      {
        goto small_divisor_case_label;
      }

    }

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

  template <class ReturnType, class SignPolicy, class TrailingZeroPolicy, class CachePolicy>
  JKJ_SAFEBUFFERS static ReturnType
  compute_right_closed_directed(ieee754_bits<Float> const br) noexcept
  {
    //////////////////////////////////////////////////////////////////////
    // Step 1: integer promotion & Schubfach multiplier calculation
    //////////////////////////////////////////////////////////////////////

    ReturnType ret_value;

    SignPolicy::handle_sign(br, ret_value);

    auto significand = br.extract_significand_bits();
    auto exponent = int(br.extract_exponent_bits());

    // Deal with normal/subnormal dichotomy
    bool closer_boundary = false;
    if (exponent != 0) {
      exponent += exponent_bias - significand_bits;
      if (significand == 0) {
        closer_boundary = true;
      }
      significand |= (carrier_uint(1) << significand_bits);
    }
      // Subnormal case; interval is always regular
    else {
      exponent = min_exponent - significand_bits;
    }

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

    // Compute zi and deltai
    // 10^kappa <= deltai < 10^(kappa + 1)
    auto const deltai = closer_boundary ?
                        compute_delta(cache, beta - 2) :
                        compute_delta(cache, beta - 1);
    carrier_uint const zi = compute_mul(significand << 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,
        significand_bits + kappa + 2, kappa + 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 (closer_boundary) {
        if (!compute_mul_parity((significand * 2) - 1, cache, beta - 1))
        {
          goto small_divisor_case_label;
        }
      }
      else {
        if (!compute_mul_parity(significand - 1, cache, beta))
        {
          goto small_divisor_case_label;
        }
      }
    }

    // The floor is inside, so we are done
    ret_value.exponent = minus_k + kappa + 1;
    TrailingZeroPolicy::on_trailing_zeros(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::no_trailing_zeros(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 {
    constexpr auto max_power = [] {
      auto max_possible_significand =
          std::numeric_limits<carrier_uint>::max() /
          compute_power<kappa + 1>(std::uint32_t(10));

      int k = 0;
      carrier_uint p = 1;
      while (p < max_possible_significand / 10) {
        p *= 10;
        ++k;
      }
      return k;
    }();

    auto t = bits::countr_zero(n);
    if (t > max_power) {
      t = max_power;
    }

    if constexpr (format == ieee754_format::binary32) {
      constexpr auto const& divtable =
          div::table_holder<carrier_uint, 5, decimal_digits>::table;

      int s = 0;
      for (; s < t - 1; s += 2) {
        if (n * divtable.mod_inv[2] > divtable.max_quotients[2]) {
          break;
        }
        n *= divtable.mod_inv[2];
      }
      if (s < t && n * divtable.mod_inv[1] <= divtable.max_quotients[1])
      {
        n *= divtable.mod_inv[1];
        ++s;
      }
      n >>= s;
      return s;
    }
    else {
      static_assert(format == ieee754_format::binary64);
      static_assert(kappa >= 2);

      // Divide by 10^8 and reduce to 32-bits
      // Since ret_value.significand <= (2^64 - 1) / 1000 < 10^17,
      // both of the quotient and the r should fit in 32-bits

      constexpr auto const& divtable =
          div::table_holder<carrier_uint, 5, decimal_digits>::table;

      // If the number is divisible by 1'0000'0000, work with the quotient
      if (t >= 8) {
        auto quotient_candidate = n * divtable.mod_inv[8];

        if (quotient_candidate <= divtable.max_quotients[8]) {
          auto quotient = std::uint32_t(quotient_candidate >> 8);

          constexpr auto mod_inverse = std::uint32_t(divtable.mod_inv[1]);
          constexpr auto max_quotient =
              std::numeric_limits<std::uint32_t>::max() / 5;

          int s = 8;
          for (; s < t; ++s) {
            if (quotient * mod_inverse > max_quotient) {
              break;
            }
            quotient *= mod_inverse;
          }
          quotient >>= (s - 8);
          n = quotient;
          return s;
        }
      }

      // Otherwise, work with the remainder
      auto quotient = std::uint32_t(div::divide_by_pow10<8, 54, 0>(n));
      auto remainder = std::uint32_t(n - 1'0000'0000 * quotient);

      constexpr auto mod_inverse = std::uint32_t(divtable.mod_inv[1]);
      constexpr auto max_quotient =
          std::numeric_limits<std::uint32_t>::max() / 5;

      if (t == 0 || remainder * mod_inverse > max_quotient) {
        return 0;
      }
      remainder *= mod_inverse;

      if (t == 1 || remainder * mod_inverse > max_quotient) {
        n = (remainder >> 1)
            + quotient * carrier_uint(1000'0000);
        return 1;
      }
      remainder *= mod_inverse;

      if (t == 2 || remainder * mod_inverse > max_quotient) {
        n = (remainder >> 2)
            + quotient * carrier_uint(100'0000);
        return 2;
      }
      remainder *= mod_inverse;

      if (t == 3 || remainder * mod_inverse > max_quotient) {
        n = (remainder >> 3)
            + quotient * carrier_uint(10'0000);
        return 3;
      }
      remainder *= mod_inverse;

      if (t == 4 || remainder * mod_inverse > max_quotient) {
        n = (remainder >> 4)
            + quotient * carrier_uint(1'0000);
        return 4;
      }
      remainder *= mod_inverse;

      if (t == 5 || remainder * mod_inverse > max_quotient) {
        n = (remainder >> 5)
            + quotient * carrier_uint(1000);
        return 5;
      }
      remainder *= mod_inverse;

      if (t == 6 || remainder * mod_inverse > max_quotient) {
        n = (remainder >> 6)
            + quotient * carrier_uint(100);
        return 6;
      }
      remainder *= mod_inverse;

      n = (remainder >> 7)
          + quotient * carrier_uint(10);
      return 7;
    }
  }

  static carrier_uint compute_mul(carrier_uint u, cache_entry_type const& cache) noexcept
  {
    if constexpr (format == ieee754_format::binary32) {
      return wuint::umul96_upper32(u, cache);
    }
    else {
      return wuint::umul192_upper64(u, cache);
    }
  }

  static std::uint32_t compute_delta(cache_entry_type const& cache, int beta_minus_1) noexcept
  {
    if constexpr (format == ieee754_format::binary32) {
      return std::uint32_t(cache >> (cache_bits - 1 - beta_minus_1));
    }
    else {
      return std::uint32_t(cache.high() >> (carrier_bits - 1 - beta_minus_1));
    }
  }

  static bool compute_mul_parity(carrier_uint two_f, cache_entry_type const& cache, int beta_minus_1) noexcept
  {
    assert(beta_minus_1 >= 1);
    assert(beta_minus_1 < 64);

    if constexpr (format == ieee754_format::binary32) {
      return ((wuint::umul96_lower64(two_f, cache) >>
                                                   (64 - beta_minus_1)) & 1) != 0;
    }
    else {
      return ((wuint::umul192_middle64(two_f, cache) >>
                                                     (64 - beta_minus_1)) & 1) != 0;
    }
  }

  static carrier_uint compute_left_endpoint_for_shorter_interval_case(
      cache_entry_type const& cache, int beta_minus_1) noexcept
  {
    if constexpr (format == ieee754_format::binary32) {
      return carrier_uint(
          (cache - (cache >> (significand_bits + 2))) >>
                                                      (cache_bits - significand_bits - 1 - beta_minus_1));
    }
    else {
      return (cache.high() - (cache.high() >> (significand_bits + 2))) >>
                                                                       (carrier_bits - significand_bits - 1 - beta_minus_1);
    }
  }

  static carrier_uint compute_right_endpoint_for_shorter_interval_case(
      cache_entry_type const& cache, int beta_minus_1) noexcept
  {
    if constexpr (format == ieee754_format::binary32) {
      return carrier_uint(
          (cache + (cache >> (significand_bits + 1))) >>
                                                      (cache_bits - significand_bits - 1 - beta_minus_1));
    }
    else {
      return (cache.high() + (cache.high() >> (significand_bits + 1))) >>
                                                                       (carrier_bits - significand_bits - 1 - beta_minus_1);
    }
  }

  static carrier_uint compute_round_up_for_shorter_interval_case(
      cache_entry_type const& cache, int beta_minus_1) noexcept
  {
    if constexpr (format == ieee754_format::binary32) {
      return (carrier_uint(cache >> (cache_bits - significand_bits - 2 - beta_minus_1)) + 1) / 2;
    }
    else {
      return ((cache.high() >> (carrier_bits - significand_bits - 2 - beta_minus_1)) + 1) / 2;
    }
  }

  static 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 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;
  }

  enum class integer_check_case_id {
    fc_pm_half,
    fc
  };
  template <integer_check_case_id case_id>
  static bool is_product_integer(carrier_uint two_f, int exponent, int minus_k) noexcept
  {
    // Case I: f = fc +- 1/2
    if constexpr (case_id == integer_check_case_id::fc_pm_half)
    {
      if (exponent < case_fc_pm_half_lower_threshold) {
        return false;
      }
        // For k >= 0
      else if (exponent <= case_fc_pm_half_upper_threshold) {
        return true;
      }
        // For k < 0
      else if (exponent > divisibility_check_by_5_threshold) {
        return false;
      }
      else {
        return div::divisible_by_power_of_5<max_power_of_factor_of_5 + 1>(two_f, minus_k);
      }
    }
      // Case II: f = fc + 1
      // Case III: f = fc
    else
    {
      // Exponent for 5 is negative
      if (exponent > divisibility_check_by_5_threshold) {
        return false;
      }
      else if (exponent > case_fc_upper_threshold) {
        return div::divisible_by_power_of_5<max_power_of_factor_of_5 + 1>(two_f, minus_k);
      }
        // Both exponents are nonnegative
      else if (exponent >= case_fc_lower_threshold) {
        return true;
      }
        // Exponent for 2 is negative
      else {
        return div::divisible_by_power_of_2(two_f, minus_k - exponent + 1);
      }
    }
  }
};


////////////////////////////////////////////////////////////////////////////////////////
// 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... Policies>
JKJ_SAFEBUFFERS JKJ_FORCEINLINE auto to_decimal(Float x, Policies... policies)
{
  // 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<rounding_mode::base, rounding_mode::nearest_to_even>,
          base_default_pair<correct_rounding::base, correct_rounding::to_even>,
          base_default_pair<cache::base, cache::normal>,
          base_default_pair<input_validation::base, input_validation::assert_finite>
      >{}, policies...));

  using return_type = fp_t<Float,
      policy_holder::return_has_sign,
      policy_holder::report_trailing_zeros>;

  auto br = ieee754_bits(x);
  policy_holder::validate_input(br);

  return policy_holder::delegate(br,
                                 [br](auto interval_type_provider) {
                                   constexpr auto tag = decltype(interval_type_provider)::tag;

                                   if constexpr (tag == rounding_mode::tag_t::to_nearest) {
                                     return detail::impl<Float>::template
                                     compute_nearest<return_type, decltype(interval_type_provider),
                                         typename policy_holder::sign_policy,
                                         typename policy_holder::trailing_zero_policy,
                                         typename policy_holder::correct_rounding_policy,
                                         typename policy_holder::cache_policy
                                     >(br);
                                   }
                                   else if constexpr (tag == rounding_mode::tag_t::left_closed_directed) {
                                     return detail::impl<Float>::template
                                     compute_left_closed_directed<return_type,
                                         typename policy_holder::sign_policy,
                                         typename policy_holder::trailing_zero_policy,
                                         typename policy_holder::cache_policy
                                     >(br);
                                   }
                                   else {
                                     return detail::impl<Float>::template
                                     compute_right_closed_directed<return_type,
                                         typename policy_holder::sign_policy,
                                         typename policy_holder::trailing_zero_policy,
                                         typename policy_holder::cache_policy
                                     >(br);
                                   }
                                 });
}
}

#undef JKJ_HAS_COUNTR_ZERO_INTRINSIC
#undef JKJ_FORCEINLINE
#undef JKJ_SAFEBUFFERS

#endif
