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AQuery/server/dragonbox/dragonbox.h

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// Copyright 2020-2022 Junekey Jeon
//
// The contents of this file may be used under the terms of
// the Apache License v2.0 with LLVM Exceptions.
//
// (See accompanying file LICENSE-Apache or copy at
// https://llvm.org/foundation/relicensing/LICENSE.txt)
//
// Alternatively, the contents of this file may be used under the terms of
// the Boost Software License, Version 1.0.
// (See accompanying file LICENSE-Boost or copy at
// https://www.boost.org/LICENSE_1_0.txt)
//
// Unless required by applicable law or agreed to in writing, this software
// is distributed on an "AS IS" BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY
// KIND, either express or implied.
#ifndef JKJ_HEADER_DRAGONBOX
#define JKJ_HEADER_DRAGONBOX
#include <cassert>
#include <cstdint>
#include <cstring>
#include <limits>
#include <type_traits>
// Suppress additional buffer overrun check.
// I have no idea why MSVC thinks some functions here are vulnerable to the buffer overrun
// attacks. No, they aren't.
#if defined(__GNUC__) || defined(__clang__)
#define JKJ_SAFEBUFFERS
#define JKJ_FORCEINLINE inline __attribute__((always_inline))
#elif defined(_MSC_VER)
#define JKJ_SAFEBUFFERS __declspec(safebuffers)
#define JKJ_FORCEINLINE __forceinline
#else
#define JKJ_SAFEBUFFERS
#define JKJ_FORCEINLINE inline
#endif
#if defined(__has_builtin)
#define JKJ_DRAGONBOX_HAS_BUILTIN(x) __has_builtin(x)
#else
#define JKJ_DRAGONBOX_HAS_BUILTIN(x) false
#endif
#if defined(_MSC_VER)
#include <intrin.h>
#endif
namespace jkj::dragonbox {
namespace detail {
template <class T>
constexpr std::size_t
physical_bits = sizeof(T) * std::numeric_limits<unsigned char>::digits;
template <class T>
constexpr std::size_t value_bits =
std::numeric_limits<std::enable_if_t<std::is_unsigned_v<T>, T>>::digits;
}
// These classes expose encoding specs of IEEE-754-like floating-point formats.
// Currently available formats are IEEE754-binary32 & IEEE754-binary64.
struct ieee754_binary32 {
static constexpr int significand_bits = 23;
static constexpr int exponent_bits = 8;
static constexpr int min_exponent = -126;
static constexpr int max_exponent = 127;
static constexpr int exponent_bias = -127;
static constexpr int decimal_digits = 9;
};
struct ieee754_binary64 {
static constexpr int significand_bits = 52;
static constexpr int exponent_bits = 11;
static constexpr int min_exponent = -1022;
static constexpr int max_exponent = 1023;
static constexpr int exponent_bias = -1023;
static constexpr int decimal_digits = 17;
};
// A floating-point traits class defines ways to interpret a bit pattern of given size as an
// encoding of floating-point number. This is a default implementation of such a traits class,
// supporting ways to interpret 32-bits into a binary32-encoded floating-point number and to
// interpret 64-bits into a binary64-encoded floating-point number. Users might specialize this
// class to change the default behavior for certain types.
template <class T>
struct default_float_traits {
// I don't know if there is a truly reliable way of detecting
// IEEE-754 binary32/binary64 formats; I just did my best here.
static_assert(std::numeric_limits<T>::is_iec559 && std::numeric_limits<T>::radix == 2 &&
(detail::physical_bits<T> == 32 || detail::physical_bits<T> == 64),
"default_ieee754_traits only works for 32-bits or 64-bits types "
"supporting binary32 or binary64 formats!");
// The type that is being viewed.
using type = T;
// Refers to the format specification class.
using format =
std::conditional_t<detail::physical_bits<T> == 32, ieee754_binary32, ieee754_binary64>;
// Defines an unsigned integer type that is large enough to carry a variable of type T.
// Most of the operations will be done on this integer type.
using carrier_uint =
std::conditional_t<detail::physical_bits<T> == 32, std::uint32_t, std::uint64_t>;
static_assert(sizeof(carrier_uint) == sizeof(T));
// Number of bits in the above unsigned integer type.
static constexpr int carrier_bits = int(detail::physical_bits<carrier_uint>);
// Convert from carrier_uint into the original type.
// Depending on the floating-point encoding format, this operation might not be possible for
// some specific bit patterns. However, the contract is that u always denotes a
// valid bit pattern, so this function must be assumed to be noexcept.
static T carrier_to_float(carrier_uint u) noexcept {
T x;
std::memcpy(&x, &u, sizeof(carrier_uint));
return x;
}
// Same as above.
static carrier_uint float_to_carrier(T x) noexcept {
carrier_uint u;
std::memcpy(&u, &x, sizeof(carrier_uint));
return u;
}
// Extract exponent bits from a bit pattern.
// The result must be aligned to the LSB so that there is no additional zero paddings
// on the right. This function does not do bias adjustment.
static constexpr unsigned int extract_exponent_bits(carrier_uint u) noexcept {
constexpr int significand_bits = format::significand_bits;
constexpr int exponent_bits = format::exponent_bits;
static_assert(detail::value_bits<unsigned int> > exponent_bits);
constexpr auto exponent_bits_mask =
(unsigned int)(((unsigned int)(1) << exponent_bits) - 1);
return (unsigned int)(u >> significand_bits) & exponent_bits_mask;
}
// Extract significand bits from a bit pattern.
// The result must be aligned to the LSB so that there is no additional zero paddings
// on the right. The result does not contain the implicit bit.
static constexpr carrier_uint extract_significand_bits(carrier_uint u) noexcept {
constexpr auto mask = carrier_uint((carrier_uint(1) << format::significand_bits) - 1);
return carrier_uint(u & mask);
}
// Remove the exponent bits and extract significand bits together with the sign bit.
static constexpr carrier_uint remove_exponent_bits(carrier_uint u,
unsigned int exponent_bits) noexcept {
return u ^ (carrier_uint(exponent_bits) << format::significand_bits);
}
// Shift the obtained signed significand bits to the left by 1 to remove the sign bit.
static constexpr carrier_uint remove_sign_bit_and_shift(carrier_uint u) noexcept {
return carrier_uint(carrier_uint(u) << 1);
}
// The actual value of exponent is obtained by adding this value to the extracted exponent
// bits.
static constexpr int exponent_bias =
1 - (1 << (carrier_bits - format::significand_bits - 2));
// Obtain the actual value of the binary exponent from the extracted exponent bits.
static constexpr int binary_exponent(unsigned int exponent_bits) noexcept {
if (exponent_bits == 0) {
return format::min_exponent;
}
else {
return int(exponent_bits) + format::exponent_bias;
}
}
// Obtain the actual value of the binary exponent from the extracted significand bits and
// exponent bits.
static constexpr carrier_uint binary_significand(carrier_uint significand_bits,
unsigned int exponent_bits) noexcept {
if (exponent_bits == 0) {
return significand_bits;
}
else {
return significand_bits | (carrier_uint(1) << format::significand_bits);
}
}
/* Various boolean observer functions */
static constexpr bool is_nonzero(carrier_uint u) noexcept { return (u << 1) != 0; }
static constexpr bool is_positive(carrier_uint u) noexcept {
constexpr auto sign_bit = carrier_uint(1)
<< (format::significand_bits + format::exponent_bits);
return u < sign_bit;
}
static constexpr bool is_negative(carrier_uint u) noexcept { return !is_positive(u); }
static constexpr bool is_finite(unsigned int exponent_bits) noexcept {
constexpr unsigned int exponent_bits_all_set = (1u << format::exponent_bits) - 1;
return exponent_bits != exponent_bits_all_set;
}
static constexpr bool has_all_zero_significand_bits(carrier_uint u) noexcept {
return (u << 1) == 0;
}
static constexpr bool has_even_significand_bits(carrier_uint u) noexcept {
return u % 2 == 0;
}
};
// Convenient wrappers for floating-point traits classes.
// In order to reduce the argument passing overhead, these classes should be as simple as
// possible (e.g., no inheritance, no private non-static data member, etc.; this is an
// unfortunate fact about common ABI convention).
template <class T, class Traits = default_float_traits<T>>
struct float_bits;
template <class T, class Traits = default_float_traits<T>>
struct signed_significand_bits;
template <class T, class Traits>
struct float_bits {
using type = T;
using traits_type = Traits;
using carrier_uint = typename traits_type::carrier_uint;
carrier_uint u;
float_bits() = default;
constexpr explicit float_bits(carrier_uint bit_pattern) noexcept : u{bit_pattern} {}
constexpr explicit float_bits(T float_value) noexcept
: u{traits_type::float_to_carrier(float_value)} {}
constexpr T to_float() const noexcept { return traits_type::carrier_to_float(u); }
// Extract exponent bits from a bit pattern.
// The result must be aligned to the LSB so that there is no additional zero paddings
// on the right. This function does not do bias adjustment.
constexpr unsigned int extract_exponent_bits() const noexcept {
return traits_type::extract_exponent_bits(u);
}
// Extract significand bits from a bit pattern.
// The result must be aligned to the LSB so that there is no additional zero paddings
// on the right. The result does not contain the implicit bit.
constexpr carrier_uint extract_significand_bits() const noexcept {
return traits_type::extract_significand_bits(u);
}
// Remove the exponent bits and extract significand bits together with the sign bit.
constexpr auto remove_exponent_bits(unsigned int exponent_bits) const noexcept {
return signed_significand_bits<type, traits_type>(
traits_type::remove_exponent_bits(u, exponent_bits));
}
// Obtain the actual value of the binary exponent from the extracted exponent bits.
static constexpr int binary_exponent(unsigned int exponent_bits) noexcept {
return traits_type::binary_exponent(exponent_bits);
}
constexpr int binary_exponent() const noexcept {
return binary_exponent(extract_exponent_bits());
}
// Obtain the actual value of the binary exponent from the extracted significand bits and
// exponent bits.
static constexpr carrier_uint binary_significand(carrier_uint significand_bits,
unsigned int exponent_bits) noexcept {
return traits_type::binary_significand(significand_bits, exponent_bits);
}
constexpr carrier_uint binary_significand() const noexcept {
return binary_significand(extract_significand_bits(), extract_exponent_bits());
}
constexpr bool is_nonzero() const noexcept { return traits_type::is_nonzero(u); }
constexpr bool is_positive() const noexcept { return traits_type::is_positive(u); }
constexpr bool is_negative() const noexcept { return traits_type::is_negative(u); }
constexpr bool is_finite(unsigned int exponent_bits) const noexcept {
return traits_type::is_finite(exponent_bits);
}
constexpr bool is_finite() const noexcept {
return traits_type::is_finite(extract_exponent_bits());
}
constexpr bool has_even_significand_bits() const noexcept {
return traits_type::has_even_significand_bits(u);
}
};
template <class T, class Traits>
struct signed_significand_bits {
using type = T;
using traits_type = Traits;
using carrier_uint = typename traits_type::carrier_uint;
carrier_uint u;
signed_significand_bits() = default;
constexpr explicit signed_significand_bits(carrier_uint bit_pattern) noexcept
: u{bit_pattern} {}
// Shift the obtained signed significand bits to the left by 1 to remove the sign bit.
constexpr carrier_uint remove_sign_bit_and_shift() const noexcept {
return traits_type::remove_sign_bit_and_shift(u);
}
constexpr bool is_positive() const noexcept { return traits_type::is_positive(u); }
constexpr bool is_negative() const noexcept { return traits_type::is_negative(u); }
constexpr bool has_all_zero_significand_bits() const noexcept {
return traits_type::has_all_zero_significand_bits(u);
}
constexpr bool has_even_significand_bits() const noexcept {
return traits_type::has_even_significand_bits(u);
}
};
namespace detail {
////////////////////////////////////////////////////////////////////////////////////////
// Bit operation intrinsics.
////////////////////////////////////////////////////////////////////////////////////////
namespace bits {
// Most compilers should be able to optimize this into the ROR instruction.
inline std::uint32_t rotr(std::uint32_t n, std::uint32_t r) noexcept {
r &= 31;
return (n >> r) | (n << (32 - r));
}
inline std::uint64_t rotr(std::uint64_t n, std::uint32_t r) noexcept {
r &= 63;
return (n >> r) | (n << (64 - r));
}
}
////////////////////////////////////////////////////////////////////////////////////////
// Utilities for wide unsigned integer arithmetic.
////////////////////////////////////////////////////////////////////////////////////////
namespace wuint {
// Compilers might support built-in 128-bit integer types. However, it seems that
// emulating them with a pair of 64-bit integers actually produces a better code,
// so we avoid using those built-ins. That said, they are still useful for
// implementing 64-bit x 64-bit -> 128-bit multiplication.
// clang-format off
#if defined(__SIZEOF_INT128__)
// To silence "error: ISO C++ does not support '__int128' for 'type name'
// [-Wpedantic]"
#if defined(__GNUC__)
__extension__
#endif
using builtin_uint128_t = unsigned __int128;
#endif
// clang-format on
struct uint128 {
uint128() = default;
std::uint64_t high_;
std::uint64_t low_;
constexpr uint128(std::uint64_t high, std::uint64_t low) noexcept
: high_{high}, low_{low} {}
constexpr std::uint64_t high() const noexcept { return high_; }
constexpr std::uint64_t low() const noexcept { return low_; }
uint128& operator+=(std::uint64_t n) & noexcept {
#if JKJ_DRAGONBOX_HAS_BUILTIN(__builtin_addcll)
unsigned long long carry;
low_ = __builtin_addcll(low_, n, 0, &carry);
high_ = __builtin_addcll(high_, 0, carry, &carry);
#elif JKJ_DRAGONBOX_HAS_BUILTIN(__builtin_ia32_addcarryx_u64)
unsigned long long result;
auto carry = __builtin_ia32_addcarryx_u64(0, low_, n, &result);
low_ = result;
__builtin_ia32_addcarryx_u64(carry, high_, 0, &result);
high_ = result;
#elif defined(_MSC_VER) && defined(_M_X64)
auto carry = _addcarry_u64(0, low_, n, &low_);
_addcarry_u64(carry, high_, 0, &high_);
#else
auto sum = low_ + n;
high_ += (sum < low_ ? 1 : 0);
low_ = sum;
#endif
return *this;
}
};
static inline std::uint64_t umul64(std::uint32_t x, std::uint32_t y) noexcept {
#if defined(_MSC_VER) && defined(_M_IX86)
return __emulu(x, y);
#else
return x * std::uint64_t(y);
#endif
}
// Get 128-bit result of multiplication of two 64-bit unsigned integers.
JKJ_SAFEBUFFERS inline uint128 umul128(std::uint64_t x, std::uint64_t y) noexcept {
#if defined(__SIZEOF_INT128__)
auto result = builtin_uint128_t(x) * builtin_uint128_t(y);
return {std::uint64_t(result >> 64), std::uint64_t(result)};
#elif defined(_MSC_VER) && defined(_M_X64)
uint128 result;
result.low_ = _umul128(x, y, &result.high_);
return result;
#else
auto a = std::uint32_t(x >> 32);
auto b = std::uint32_t(x);
auto c = std::uint32_t(y >> 32);
auto d = std::uint32_t(y);
auto ac = umul64(a, c);
auto bc = umul64(b, c);
auto ad = umul64(a, d);
auto bd = umul64(b, d);
auto intermediate = (bd >> 32) + std::uint32_t(ad) + std::uint32_t(bc);
return {ac + (intermediate >> 32) + (ad >> 32) + (bc >> 32),
(intermediate << 32) + std::uint32_t(bd)};
#endif
}
JKJ_SAFEBUFFERS inline std::uint64_t umul128_upper64(std::uint64_t x,
std::uint64_t y) noexcept {
#if defined(__SIZEOF_INT128__)
auto result = builtin_uint128_t(x) * builtin_uint128_t(y);
return std::uint64_t(result >> 64);
#elif defined(_MSC_VER) && defined(_M_X64)
return __umulh(x, y);
#else
auto a = std::uint32_t(x >> 32);
auto b = std::uint32_t(x);
auto c = std::uint32_t(y >> 32);
auto d = std::uint32_t(y);
auto ac = umul64(a, c);
auto bc = umul64(b, c);
auto ad = umul64(a, d);
auto bd = umul64(b, d);
auto intermediate = (bd >> 32) + std::uint32_t(ad) + std::uint32_t(bc);
return ac + (intermediate >> 32) + (ad >> 32) + (bc >> 32);
#endif
}
// Get upper 128-bits of multiplication of a 64-bit unsigned integer and a 128-bit
// unsigned integer.
JKJ_SAFEBUFFERS inline uint128 umul192_upper128(std::uint64_t x, uint128 y) noexcept {
auto r = umul128(x, y.high());
r += umul128_upper64(x, y.low());
return r;
}
// Get upper 64-bits of multiplication of a 32-bit unsigned integer and a 64-bit
// unsigned integer.
inline std::uint64_t umul96_upper64(std::uint32_t x, std::uint64_t y) noexcept {
#if defined(__SIZEOF_INT128__) || (defined(_MSC_VER) && defined(_M_X64))
return umul128_upper64(std::uint64_t(x) << 32, y);
#else
auto yh = std::uint32_t(y >> 32);
auto yl = std::uint32_t(y);
auto xyh = umul64(x, yh);
auto xyl = umul64(x, yl);
return xyh + (xyl >> 32);
#endif
}
// Get lower 128-bits of multiplication of a 64-bit unsigned integer and a 128-bit
// unsigned integer.
JKJ_SAFEBUFFERS inline uint128 umul192_lower128(std::uint64_t x, uint128 y) noexcept {
auto high = x * y.high();
auto high_low = umul128(x, y.low());
return {high + high_low.high(), high_low.low()};
}
// Get lower 64-bits of multiplication of a 32-bit unsigned integer and a 64-bit
// unsigned integer.
inline std::uint64_t umul96_lower64(std::uint32_t x, std::uint64_t y) noexcept {
return x * y;
}
}
////////////////////////////////////////////////////////////////////////////////////////
// Some simple utilities for constexpr computation.
////////////////////////////////////////////////////////////////////////////////////////
template <int k, class Int>
constexpr Int compute_power(Int a) noexcept {
static_assert(k >= 0);
Int p = 1;
for (int i = 0; i < k; ++i) {
p *= a;
}
return p;
}
template <int a, class UInt>
constexpr int count_factors(UInt n) noexcept {
static_assert(a > 1);
int c = 0;
while (n % a == 0) {
n /= a;
++c;
}
return c;
}
////////////////////////////////////////////////////////////////////////////////////////
// Utilities for fast/constexpr log computation.
////////////////////////////////////////////////////////////////////////////////////////
namespace log {
static_assert((-1 >> 1) == -1, "right-shift for signed integers must be arithmetic");
// Compute floor(e * c - s).
enum class multiply : std::uint32_t {};
enum class subtract : std::uint32_t {};
enum class shift : std::size_t {};
enum class min_exponent : std::int32_t {};
enum class max_exponent : std::int32_t {};
template <multiply m, subtract f, shift k, min_exponent e_min, max_exponent e_max>
constexpr int compute(int e) noexcept {
assert(std::int32_t(e_min) <= e && e <= std::int32_t(e_max));
return int((std::int32_t(e) * std::int32_t(m) - std::int32_t(f)) >> std::size_t(k));
}
// For constexpr computation.
// Returns -1 when n = 0.
template <class UInt>
constexpr int floor_log2(UInt n) noexcept {
int count = -1;
while (n != 0) {
++count;
n >>= 1;
}
return count;
}
static constexpr int floor_log10_pow2_min_exponent = -2620;
static constexpr int floor_log10_pow2_max_exponent = 2620;
constexpr int floor_log10_pow2(int e) noexcept {
using namespace log;
return compute<multiply(315653), subtract(0), shift(20),
min_exponent(floor_log10_pow2_min_exponent),
max_exponent(floor_log10_pow2_max_exponent)>(e);
}
static constexpr int floor_log2_pow10_min_exponent = -1233;
static constexpr int floor_log2_pow10_max_exponent = 1233;
constexpr int floor_log2_pow10(int e) noexcept {
using namespace log;
return compute<multiply(1741647), subtract(0), shift(19),
min_exponent(floor_log2_pow10_min_exponent),
max_exponent(floor_log2_pow10_max_exponent)>(e);
}
static constexpr int floor_log10_pow2_minus_log10_4_over_3_min_exponent = -2985;
static constexpr int floor_log10_pow2_minus_log10_4_over_3_max_exponent = 2936;
constexpr int floor_log10_pow2_minus_log10_4_over_3(int e) noexcept {
using namespace log;
return compute<multiply(631305), subtract(261663), shift(21),
min_exponent(floor_log10_pow2_minus_log10_4_over_3_min_exponent),
max_exponent(floor_log10_pow2_minus_log10_4_over_3_max_exponent)>(e);
}
static constexpr int floor_log5_pow2_min_exponent = -1831;
static constexpr int floor_log5_pow2_max_exponent = 1831;
constexpr int floor_log5_pow2(int e) noexcept {
using namespace log;
return compute<multiply(225799), subtract(0), shift(19),
min_exponent(floor_log5_pow2_min_exponent),
max_exponent(floor_log5_pow2_max_exponent)>(e);
}
static constexpr int floor_log5_pow2_minus_log5_3_min_exponent = -3543;
static constexpr int floor_log5_pow2_minus_log5_3_max_exponent = 2427;
constexpr int floor_log5_pow2_minus_log5_3(int e) noexcept {
using namespace log;
return compute<multiply(451597), subtract(715764), shift(20),
min_exponent(floor_log5_pow2_minus_log5_3_min_exponent),
max_exponent(floor_log5_pow2_minus_log5_3_max_exponent)>(e);
}
}
////////////////////////////////////////////////////////////////////////////////////////
// Utilities for fast divisibility tests.
////////////////////////////////////////////////////////////////////////////////////////
namespace div {
// Replace n by floor(n / 10^N).
// Returns true if and only if n is divisible by 10^N.
// Precondition: n <= 10^(N+1)
// !!It takes an in-out parameter!!
template <int N>
struct divide_by_pow10_info;
template <>
struct divide_by_pow10_info<1> {
static constexpr std::uint32_t magic_number = 6554;
static constexpr int shift_amount = 16;
};
template <>
struct divide_by_pow10_info<2> {
static constexpr std::uint32_t magic_number = 656;
static constexpr int shift_amount = 16;
};
template <int N>
constexpr bool check_divisibility_and_divide_by_pow10(std::uint32_t& n) noexcept {
// Make sure the computation for max_n does not overflow.
static_assert(N + 1 <= log::floor_log10_pow2(31));
assert(n <= compute_power<N + 1>(std::uint32_t(10)));
using info = divide_by_pow10_info<N>;
n *= info::magic_number;
constexpr auto mask = std::uint32_t(std::uint32_t(1) << info::shift_amount) - 1;
bool result = ((n & mask) < info::magic_number);
n >>= info::shift_amount;
return result;
}
// Compute floor(n / 10^N) for small n and N.
// Precondition: n <= 10^(N+1)
template <int N>
constexpr std::uint32_t small_division_by_pow10(std::uint32_t n) noexcept {
// Make sure the computation for max_n does not overflow.
static_assert(N + 1 <= log::floor_log10_pow2(31));
assert(n <= compute_power<N + 1>(std::uint32_t(10)));
return (n * divide_by_pow10_info<N>::magic_number) >>
divide_by_pow10_info<N>::shift_amount;
}
// Compute floor(n / 10^N) for small N.
// Precondition: n <= n_max
template <int N, class UInt, UInt n_max>
constexpr UInt divide_by_pow10(UInt n) noexcept {
static_assert(N >= 0);
// Specialize for 32-bit division by 100.
// Compiler is supposed to generate the identical code for just writing
// "n / 100", but for some reason MSVC generates an inefficient code
// (mul + mov for no apparent reason, instead of single imul),
// so we does this manually.
if constexpr (std::is_same_v<UInt, std::uint32_t> && N == 2) {
return std::uint32_t(wuint::umul64(n, std::uint32_t(1374389535)) >> 37);
}
// Specialize for 64-bit division by 1000.
// Ensure that the correctness condition is met.
if constexpr (std::is_same_v<UInt, std::uint64_t> && N == 3 &&
n_max <= std::uint64_t(15534100272597517998ull)) {
return wuint::umul128_upper64(n, std::uint64_t(2361183241434822607ull)) >> 7;
}
else {
constexpr auto divisor = compute_power<N>(UInt(10));
return n / divisor;
}
}
}
}
////////////////////////////////////////////////////////////////////////////////////////
// Return types for the main interface function.
////////////////////////////////////////////////////////////////////////////////////////
template <class UInt, bool is_signed, bool trailing_zero_flag>
struct decimal_fp;
template <class UInt>
struct decimal_fp<UInt, false, false> {
using carrier_uint = UInt;
carrier_uint significand;
int exponent;
};
template <class UInt>
struct decimal_fp<UInt, true, false> {
using carrier_uint = UInt;
carrier_uint significand;
int exponent;
bool is_negative;
};
template <class UInt>
struct decimal_fp<UInt, false, true> {
using carrier_uint = UInt;
carrier_uint significand;
int exponent;
bool may_have_trailing_zeros;
};
template <class UInt>
struct decimal_fp<UInt, true, true> {
using carrier_uint = UInt;
carrier_uint significand;
int exponent;
bool is_negative;
bool may_have_trailing_zeros;
};
template <class UInt>
using unsigned_decimal_fp = decimal_fp<UInt, false, false>;
template <class UInt>
using signed_decimal_fp = decimal_fp<UInt, true, false>;
////////////////////////////////////////////////////////////////////////////////////////
// Computed cache entries.
////////////////////////////////////////////////////////////////////////////////////////
namespace detail {
template <class FloatFormat>
struct cache_holder;
template <>
struct cache_holder<ieee754_binary32> {
using cache_entry_type = std::uint64_t;
static constexpr int cache_bits = 64;
static constexpr int min_k = -31;
static constexpr int max_k = 46;
static constexpr cache_entry_type cache[] = {
0x81ceb32c4b43fcf5, 0xa2425ff75e14fc32, 0xcad2f7f5359a3b3f, 0xfd87b5f28300ca0e,
0x9e74d1b791e07e49, 0xc612062576589ddb, 0xf79687aed3eec552, 0x9abe14cd44753b53,
0xc16d9a0095928a28, 0xf1c90080baf72cb2, 0x971da05074da7bef, 0xbce5086492111aeb,
0xec1e4a7db69561a6, 0x9392ee8e921d5d08, 0xb877aa3236a4b44a, 0xe69594bec44de15c,
0x901d7cf73ab0acda, 0xb424dc35095cd810, 0xe12e13424bb40e14, 0x8cbccc096f5088cc,
0xafebff0bcb24aaff, 0xdbe6fecebdedd5bf, 0x89705f4136b4a598, 0xabcc77118461cefd,
0xd6bf94d5e57a42bd, 0x8637bd05af6c69b6, 0xa7c5ac471b478424, 0xd1b71758e219652c,
0x83126e978d4fdf3c, 0xa3d70a3d70a3d70b, 0xcccccccccccccccd, 0x8000000000000000,
0xa000000000000000, 0xc800000000000000, 0xfa00000000000000, 0x9c40000000000000,
0xc350000000000000, 0xf424000000000000, 0x9896800000000000, 0xbebc200000000000,
0xee6b280000000000, 0x9502f90000000000, 0xba43b74000000000, 0xe8d4a51000000000,
0x9184e72a00000000, 0xb5e620f480000000, 0xe35fa931a0000000, 0x8e1bc9bf04000000,
0xb1a2bc2ec5000000, 0xde0b6b3a76400000, 0x8ac7230489e80000, 0xad78ebc5ac620000,
0xd8d726b7177a8000, 0x878678326eac9000, 0xa968163f0a57b400, 0xd3c21bcecceda100,
0x84595161401484a0, 0xa56fa5b99019a5c8, 0xcecb8f27f4200f3a, 0x813f3978f8940985,
0xa18f07d736b90be6, 0xc9f2c9cd04674edf, 0xfc6f7c4045812297, 0x9dc5ada82b70b59e,
0xc5371912364ce306, 0xf684df56c3e01bc7, 0x9a130b963a6c115d, 0xc097ce7bc90715b4,
0xf0bdc21abb48db21, 0x96769950b50d88f5, 0xbc143fa4e250eb32, 0xeb194f8e1ae525fe,
0x92efd1b8d0cf37bf, 0xb7abc627050305ae, 0xe596b7b0c643c71a, 0x8f7e32ce7bea5c70,
0xb35dbf821ae4f38c, 0xe0352f62a19e306f};
};
template <>
struct cache_holder<ieee754_binary64> {
using cache_entry_type = wuint::uint128;
static constexpr int cache_bits = 128;
static constexpr int min_k = -292;
static constexpr int max_k = 326;
static constexpr cache_entry_type cache[] = {
{0xff77b1fcbebcdc4f, 0x25e8e89c13bb0f7b}, {0x9faacf3df73609b1, 0x77b191618c54e9ad},
{0xc795830d75038c1d, 0xd59df5b9ef6a2418}, {0xf97ae3d0d2446f25, 0x4b0573286b44ad1e},
{0x9becce62836ac577, 0x4ee367f9430aec33}, {0xc2e801fb244576d5, 0x229c41f793cda740},
{0xf3a20279ed56d48a, 0x6b43527578c11110}, {0x9845418c345644d6, 0x830a13896b78aaaa},
{0xbe5691ef416bd60c, 0x23cc986bc656d554}, {0xedec366b11c6cb8f, 0x2cbfbe86b7ec8aa9},
{0x94b3a202eb1c3f39, 0x7bf7d71432f3d6aa}, {0xb9e08a83a5e34f07, 0xdaf5ccd93fb0cc54},
{0xe858ad248f5c22c9, 0xd1b3400f8f9cff69}, {0x91376c36d99995be, 0x23100809b9c21fa2},
{0xb58547448ffffb2d, 0xabd40a0c2832a78b}, {0xe2e69915b3fff9f9, 0x16c90c8f323f516d},
{0x8dd01fad907ffc3b, 0xae3da7d97f6792e4}, {0xb1442798f49ffb4a, 0x99cd11cfdf41779d},
{0xdd95317f31c7fa1d, 0x40405643d711d584}, {0x8a7d3eef7f1cfc52, 0x482835ea666b2573},
{0xad1c8eab5ee43b66, 0xda3243650005eed0}, {0xd863b256369d4a40, 0x90bed43e40076a83},
{0x873e4f75e2224e68, 0x5a7744a6e804a292}, {0xa90de3535aaae202, 0x711515d0a205cb37},
{0xd3515c2831559a83, 0x0d5a5b44ca873e04}, {0x8412d9991ed58091, 0xe858790afe9486c3},
{0xa5178fff668ae0b6, 0x626e974dbe39a873}, {0xce5d73ff402d98e3, 0xfb0a3d212dc81290},
{0x80fa687f881c7f8e, 0x7ce66634bc9d0b9a}, {0xa139029f6a239f72, 0x1c1fffc1ebc44e81},
{0xc987434744ac874e, 0xa327ffb266b56221}, {0xfbe9141915d7a922, 0x4bf1ff9f0062baa9},
{0x9d71ac8fada6c9b5, 0x6f773fc3603db4aa}, {0xc4ce17b399107c22, 0xcb550fb4384d21d4},
{0xf6019da07f549b2b, 0x7e2a53a146606a49}, {0x99c102844f94e0fb, 0x2eda7444cbfc426e},
{0xc0314325637a1939, 0xfa911155fefb5309}, {0xf03d93eebc589f88, 0x793555ab7eba27cb},
{0x96267c7535b763b5, 0x4bc1558b2f3458df}, {0xbbb01b9283253ca2, 0x9eb1aaedfb016f17},
{0xea9c227723ee8bcb, 0x465e15a979c1cadd}, {0x92a1958a7675175f, 0x0bfacd89ec191eca},
{0xb749faed14125d36, 0xcef980ec671f667c}, {0xe51c79a85916f484, 0x82b7e12780e7401b},
{0x8f31cc0937ae58d2, 0xd1b2ecb8b0908811}, {0xb2fe3f0b8599ef07, 0x861fa7e6dcb4aa16},
{0xdfbdcece67006ac9, 0x67a791e093e1d49b}, {0x8bd6a141006042bd, 0xe0c8bb2c5c6d24e1},
{0xaecc49914078536d, 0x58fae9f773886e19}, {0xda7f5bf590966848, 0xaf39a475506a899f},
{0x888f99797a5e012d, 0x6d8406c952429604}, {0xaab37fd7d8f58178, 0xc8e5087ba6d33b84},
{0xd5605fcdcf32e1d6, 0xfb1e4a9a90880a65}, {0x855c3be0a17fcd26, 0x5cf2eea09a550680},
{0xa6b34ad8c9dfc06f, 0xf42faa48c0ea481f}, {0xd0601d8efc57b08b, 0xf13b94daf124da27},
{0x823c12795db6ce57, 0x76c53d08d6b70859}, {0xa2cb1717b52481ed, 0x54768c4b0c64ca6f},
{0xcb7ddcdda26da268, 0xa9942f5dcf7dfd0a}, {0xfe5d54150b090b02, 0xd3f93b35435d7c4d},
{0x9efa548d26e5a6e1, 0xc47bc5014a1a6db0}, {0xc6b8e9b0709f109a, 0x359ab6419ca1091c},
{0xf867241c8cc6d4c0, 0xc30163d203c94b63}, {0x9b407691d7fc44f8, 0x79e0de63425dcf1e},
{0xc21094364dfb5636, 0x985915fc12f542e5}, {0xf294b943e17a2bc4, 0x3e6f5b7b17b2939e},
{0x979cf3ca6cec5b5a, 0xa705992ceecf9c43}, {0xbd8430bd08277231, 0x50c6ff782a838354},
{0xece53cec4a314ebd, 0xa4f8bf5635246429}, {0x940f4613ae5ed136, 0x871b7795e136be9a},
{0xb913179899f68584, 0x28e2557b59846e40}, {0xe757dd7ec07426e5, 0x331aeada2fe589d0},
{0x9096ea6f3848984f, 0x3ff0d2c85def7622}, {0xb4bca50b065abe63, 0x0fed077a756b53aa},
{0xe1ebce4dc7f16dfb, 0xd3e8495912c62895}, {0x8d3360f09cf6e4bd, 0x64712dd7abbbd95d},
{0xb080392cc4349dec, 0xbd8d794d96aacfb4}, {0xdca04777f541c567, 0xecf0d7a0fc5583a1},
{0x89e42caaf9491b60, 0xf41686c49db57245}, {0xac5d37d5b79b6239, 0x311c2875c522ced6},
{0xd77485cb25823ac7, 0x7d633293366b828c}, {0x86a8d39ef77164bc, 0xae5dff9c02033198},
{0xa8530886b54dbdeb, 0xd9f57f830283fdfd}, {0xd267caa862a12d66, 0xd072df63c324fd7c},
{0x8380dea93da4bc60, 0x4247cb9e59f71e6e}, {0xa46116538d0deb78, 0x52d9be85f074e609},
{0xcd795be870516656, 0x67902e276c921f8c}, {0x806bd9714632dff6, 0x00ba1cd8a3db53b7},
{0xa086cfcd97bf97f3, 0x80e8a40eccd228a5}, {0xc8a883c0fdaf7df0, 0x6122cd128006b2ce},
{0xfad2a4b13d1b5d6c, 0x796b805720085f82}, {0x9cc3a6eec6311a63, 0xcbe3303674053bb1},
{0xc3f490aa77bd60fc, 0xbedbfc4411068a9d}, {0xf4f1b4d515acb93b, 0xee92fb5515482d45},
{0x991711052d8bf3c5, 0x751bdd152d4d1c4b}, {0xbf5cd54678eef0b6, 0xd262d45a78a0635e},
{0xef340a98172aace4, 0x86fb897116c87c35}, {0x9580869f0e7aac0e, 0xd45d35e6ae3d4da1},
{0xbae0a846d2195712, 0x8974836059cca10a}, {0xe998d258869facd7, 0x2bd1a438703fc94c},
{0x91ff83775423cc06, 0x7b6306a34627ddd0}, {0xb67f6455292cbf08, 0x1a3bc84c17b1d543},
{0xe41f3d6a7377eeca, 0x20caba5f1d9e4a94}, {0x8e938662882af53e, 0x547eb47b7282ee9d},
{0xb23867fb2a35b28d, 0xe99e619a4f23aa44}, {0xdec681f9f4c31f31, 0x6405fa00e2ec94d5},
{0x8b3c113c38f9f37e, 0xde83bc408dd3dd05}, {0xae0b158b4738705e, 0x9624ab50b148d446},
{0xd98ddaee19068c76, 0x3badd624dd9b0958}, {0x87f8a8d4cfa417c9, 0xe54ca5d70a80e5d7},
{0xa9f6d30a038d1dbc, 0x5e9fcf4ccd211f4d}, {0xd47487cc8470652b, 0x7647c32000696720},
{0x84c8d4dfd2c63f3b, 0x29ecd9f40041e074}, {0xa5fb0a17c777cf09, 0xf468107100525891},
{0xcf79cc9db955c2cc, 0x7182148d4066eeb5}, {0x81ac1fe293d599bf, 0xc6f14cd848405531},
{0xa21727db38cb002f, 0xb8ada00e5a506a7d}, {0xca9cf1d206fdc03b, 0xa6d90811f0e4851d},
{0xfd442e4688bd304a, 0x908f4a166d1da664}, {0x9e4a9cec15763e2e, 0x9a598e4e043287ff},
{0xc5dd44271ad3cdba, 0x40eff1e1853f29fe}, {0xf7549530e188c128, 0xd12bee59e68ef47d},
{0x9a94dd3e8cf578b9, 0x82bb74f8301958cf}, {0xc13a148e3032d6e7, 0xe36a52363c1faf02},
{0xf18899b1bc3f8ca1, 0xdc44e6c3cb279ac2}, {0x96f5600f15a7b7e5, 0x29ab103a5ef8c0ba},
{0xbcb2b812db11a5de, 0x7415d448f6b6f0e8}, {0xebdf661791d60f56, 0x111b495b3464ad22},
{0x936b9fcebb25c995, 0xcab10dd900beec35}, {0xb84687c269ef3bfb, 0x3d5d514f40eea743},
{0xe65829b3046b0afa, 0x0cb4a5a3112a5113}, {0x8ff71a0fe2c2e6dc, 0x47f0e785eaba72ac},
{0xb3f4e093db73a093, 0x59ed216765690f57}, {0xe0f218b8d25088b8, 0x306869c13ec3532d},
{0x8c974f7383725573, 0x1e414218c73a13fc}, {0xafbd2350644eeacf, 0xe5d1929ef90898fb},
{0xdbac6c247d62a583, 0xdf45f746b74abf3a}, {0x894bc396ce5da772, 0x6b8bba8c328eb784},
{0xab9eb47c81f5114f, 0x066ea92f3f326565}, {0xd686619ba27255a2, 0xc80a537b0efefebe},
{0x8613fd0145877585, 0xbd06742ce95f5f37}, {0xa798fc4196e952e7, 0x2c48113823b73705},
{0xd17f3b51fca3a7a0, 0xf75a15862ca504c6}, {0x82ef85133de648c4, 0x9a984d73dbe722fc},
{0xa3ab66580d5fdaf5, 0xc13e60d0d2e0ebbb}, {0xcc963fee10b7d1b3, 0x318df905079926a9},
{0xffbbcfe994e5c61f, 0xfdf17746497f7053}, {0x9fd561f1fd0f9bd3, 0xfeb6ea8bedefa634},
{0xc7caba6e7c5382c8, 0xfe64a52ee96b8fc1}, {0xf9bd690a1b68637b, 0x3dfdce7aa3c673b1},
{0x9c1661a651213e2d, 0x06bea10ca65c084f}, {0xc31bfa0fe5698db8, 0x486e494fcff30a63},
{0xf3e2f893dec3f126, 0x5a89dba3c3efccfb}, {0x986ddb5c6b3a76b7, 0xf89629465a75e01d},
{0xbe89523386091465, 0xf6bbb397f1135824}, {0xee2ba6c0678b597f, 0x746aa07ded582e2d},
{0x94db483840b717ef, 0xa8c2a44eb4571cdd}, {0xba121a4650e4ddeb, 0x92f34d62616ce414},
{0xe896a0d7e51e1566, 0x77b020baf9c81d18}, {0x915e2486ef32cd60, 0x0ace1474dc1d122f},
{0xb5b5ada8aaff80b8, 0x0d819992132456bb}, {0xe3231912d5bf60e6, 0x10e1fff697ed6c6a},
{0x8df5efabc5979c8f, 0xca8d3ffa1ef463c2}, {0xb1736b96b6fd83b3, 0xbd308ff8a6b17cb3},
{0xddd0467c64bce4a0, 0xac7cb3f6d05ddbdf}, {0x8aa22c0dbef60ee4, 0x6bcdf07a423aa96c},
{0xad4ab7112eb3929d, 0x86c16c98d2c953c7}, {0xd89d64d57a607744, 0xe871c7bf077ba8b8},
{0x87625f056c7c4a8b, 0x11471cd764ad4973}, {0xa93af6c6c79b5d2d, 0xd598e40d3dd89bd0},
{0xd389b47879823479, 0x4aff1d108d4ec2c4}, {0x843610cb4bf160cb, 0xcedf722a585139bb},
{0xa54394fe1eedb8fe, 0xc2974eb4ee658829}, {0xce947a3da6a9273e, 0x733d226229feea33},
{0x811ccc668829b887, 0x0806357d5a3f5260}, {0xa163ff802a3426a8, 0xca07c2dcb0cf26f8},
{0xc9bcff6034c13052, 0xfc89b393dd02f0b6}, {0xfc2c3f3841f17c67, 0xbbac2078d443ace3},
{0x9d9ba7832936edc0, 0xd54b944b84aa4c0e}, {0xc5029163f384a931, 0x0a9e795e65d4df12},
{0xf64335bcf065d37d, 0x4d4617b5ff4a16d6}, {0x99ea0196163fa42e, 0x504bced1bf8e4e46},
{0xc06481fb9bcf8d39, 0xe45ec2862f71e1d7}, {0xf07da27a82c37088, 0x5d767327bb4e5a4d},
{0x964e858c91ba2655, 0x3a6a07f8d510f870}, {0xbbe226efb628afea, 0x890489f70a55368c},
{0xeadab0aba3b2dbe5, 0x2b45ac74ccea842f}, {0x92c8ae6b464fc96f, 0x3b0b8bc90012929e},
{0xb77ada0617e3bbcb, 0x09ce6ebb40173745}, {0xe55990879ddcaabd, 0xcc420a6a101d0516},
{0x8f57fa54c2a9eab6, 0x9fa946824a12232e}, {0xb32df8e9f3546564, 0x47939822dc96abfa},
{0xdff9772470297ebd, 0x59787e2b93bc56f8}, {0x8bfbea76c619ef36, 0x57eb4edb3c55b65b},
{0xaefae51477a06b03, 0xede622920b6b23f2}, {0xdab99e59958885c4, 0xe95fab368e45ecee},
{0x88b402f7fd75539b, 0x11dbcb0218ebb415}, {0xaae103b5fcd2a881, 0xd652bdc29f26a11a},
{0xd59944a37c0752a2, 0x4be76d3346f04960}, {0x857fcae62d8493a5, 0x6f70a4400c562ddc},
{0xa6dfbd9fb8e5b88e, 0xcb4ccd500f6bb953}, {0xd097ad07a71f26b2, 0x7e2000a41346a7a8},
{0x825ecc24c873782f, 0x8ed400668c0c28c9}, {0xa2f67f2dfa90563b, 0x728900802f0f32fb},
{0xcbb41ef979346bca, 0x4f2b40a03ad2ffba}, {0xfea126b7d78186bc, 0xe2f610c84987bfa9},
{0x9f24b832e6b0f436, 0x0dd9ca7d2df4d7ca}, {0xc6ede63fa05d3143, 0x91503d1c79720dbc},
{0xf8a95fcf88747d94, 0x75a44c6397ce912b}, {0x9b69dbe1b548ce7c, 0xc986afbe3ee11abb},
{0xc24452da229b021b, 0xfbe85badce996169}, {0xf2d56790ab41c2a2, 0xfae27299423fb9c4},
{0x97c560ba6b0919a5, 0xdccd879fc967d41b}, {0xbdb6b8e905cb600f, 0x5400e987bbc1c921},
{0xed246723473e3813, 0x290123e9aab23b69}, {0x9436c0760c86e30b, 0xf9a0b6720aaf6522},
{0xb94470938fa89bce, 0xf808e40e8d5b3e6a}, {0xe7958cb87392c2c2, 0xb60b1d1230b20e05},
{0x90bd77f3483bb9b9, 0xb1c6f22b5e6f48c3}, {0xb4ecd5f01a4aa828, 0x1e38aeb6360b1af4},
{0xe2280b6c20dd5232, 0x25c6da63c38de1b1}, {0x8d590723948a535f, 0x579c487e5a38ad0f},
{0xb0af48ec79ace837, 0x2d835a9df0c6d852}, {0xdcdb1b2798182244, 0xf8e431456cf88e66},
{0x8a08f0f8bf0f156b, 0x1b8e9ecb641b5900}, {0xac8b2d36eed2dac5, 0xe272467e3d222f40},
{0xd7adf884aa879177, 0x5b0ed81dcc6abb10}, {0x86ccbb52ea94baea, 0x98e947129fc2b4ea},
{0xa87fea27a539e9a5, 0x3f2398d747b36225}, {0xd29fe4b18e88640e, 0x8eec7f0d19a03aae},
{0x83a3eeeef9153e89, 0x1953cf68300424ad}, {0xa48ceaaab75a8e2b, 0x5fa8c3423c052dd8},
{0xcdb02555653131b6, 0x3792f412cb06794e}, {0x808e17555f3ebf11, 0xe2bbd88bbee40bd1},
{0xa0b19d2ab70e6ed6, 0x5b6aceaeae9d0ec5}, {0xc8de047564d20a8b, 0xf245825a5a445276},
{0xfb158592be068d2e, 0xeed6e2f0f0d56713}, {0x9ced737bb6c4183d, 0x55464dd69685606c},
{0xc428d05aa4751e4c, 0xaa97e14c3c26b887}, {0xf53304714d9265df, 0xd53dd99f4b3066a9},
{0x993fe2c6d07b7fab, 0xe546a8038efe402a}, {0xbf8fdb78849a5f96, 0xde98520472bdd034},
{0xef73d256a5c0f77c, 0x963e66858f6d4441}, {0x95a8637627989aad, 0xdde7001379a44aa9},
{0xbb127c53b17ec159, 0x5560c018580d5d53}, {0xe9d71b689dde71af, 0xaab8f01e6e10b4a7},
{0x9226712162ab070d, 0xcab3961304ca70e9}, {0xb6b00d69bb55c8d1, 0x3d607b97c5fd0d23},
{0xe45c10c42a2b3b05, 0x8cb89a7db77c506b}, {0x8eb98a7a9a5b04e3, 0x77f3608e92adb243},
{0xb267ed1940f1c61c, 0x55f038b237591ed4}, {0xdf01e85f912e37a3, 0x6b6c46dec52f6689},
{0x8b61313bbabce2c6, 0x2323ac4b3b3da016}, {0xae397d8aa96c1b77, 0xabec975e0a0d081b},
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{0xa3d70a3d70a3d70a, 0x3d70a3d70a3d70a4}, {0xcccccccccccccccc, 0xcccccccccccccccd},
{0x8000000000000000, 0x0000000000000000}, {0xa000000000000000, 0x0000000000000000},
{0xc800000000000000, 0x0000000000000000}, {0xfa00000000000000, 0x0000000000000000},
{0x9c40000000000000, 0x0000000000000000}, {0xc350000000000000, 0x0000000000000000},
{0xf424000000000000, 0x0000000000000000}, {0x9896800000000000, 0x0000000000000000},
{0xbebc200000000000, 0x0000000000000000}, {0xee6b280000000000, 0x0000000000000000},
{0x9502f90000000000, 0x0000000000000000}, {0xba43b74000000000, 0x0000000000000000},
{0xe8d4a51000000000, 0x0000000000000000}, {0x9184e72a00000000, 0x0000000000000000},
{0xb5e620f480000000, 0x0000000000000000}, {0xe35fa931a0000000, 0x0000000000000000},
{0x8e1bc9bf04000000, 0x0000000000000000}, {0xb1a2bc2ec5000000, 0x0000000000000000},
{0xde0b6b3a76400000, 0x0000000000000000}, {0x8ac7230489e80000, 0x0000000000000000},
{0xad78ebc5ac620000, 0x0000000000000000}, {0xd8d726b7177a8000, 0x0000000000000000},
{0x878678326eac9000, 0x0000000000000000}, {0xa968163f0a57b400, 0x0000000000000000},
{0xd3c21bcecceda100, 0x0000000000000000}, {0x84595161401484a0, 0x0000000000000000},
{0xa56fa5b99019a5c8, 0x0000000000000000}, {0xcecb8f27f4200f3a, 0x0000000000000000},
{0x813f3978f8940984, 0x4000000000000000}, {0xa18f07d736b90be5, 0x5000000000000000},
{0xc9f2c9cd04674ede, 0xa400000000000000}, {0xfc6f7c4045812296, 0x4d00000000000000},
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{0x93e1ab8252f33b45, 0xcabb90e5c942b504}, {0xb8da1662e7b00a17, 0x3d6a751f3b936244},
{0xe7109bfba19c0c9d, 0x0cc512670a783ad5}, {0x906a617d450187e2, 0x27fb2b80668b24c6},
{0xb484f9dc9641e9da, 0xb1f9f660802dedf7}, {0xe1a63853bbd26451, 0x5e7873f8a0396974},
{0x8d07e33455637eb2, 0xdb0b487b6423e1e9}, {0xb049dc016abc5e5f, 0x91ce1a9a3d2cda63},
{0xdc5c5301c56b75f7, 0x7641a140cc7810fc}, {0x89b9b3e11b6329ba, 0xa9e904c87fcb0a9e},
{0xac2820d9623bf429, 0x546345fa9fbdcd45}, {0xd732290fbacaf133, 0xa97c177947ad4096},
{0x867f59a9d4bed6c0, 0x49ed8eabcccc485e}, {0xa81f301449ee8c70, 0x5c68f256bfff5a75},
{0xd226fc195c6a2f8c, 0x73832eec6fff3112}, {0x83585d8fd9c25db7, 0xc831fd53c5ff7eac},
{0xa42e74f3d032f525, 0xba3e7ca8b77f5e56}, {0xcd3a1230c43fb26f, 0x28ce1bd2e55f35ec},
{0x80444b5e7aa7cf85, 0x7980d163cf5b81b4}, {0xa0555e361951c366, 0xd7e105bcc3326220},
{0xc86ab5c39fa63440, 0x8dd9472bf3fefaa8}, {0xfa856334878fc150, 0xb14f98f6f0feb952},
{0x9c935e00d4b9d8d2, 0x6ed1bf9a569f33d4}, {0xc3b8358109e84f07, 0x0a862f80ec4700c9},
{0xf4a642e14c6262c8, 0xcd27bb612758c0fb}, {0x98e7e9cccfbd7dbd, 0x8038d51cb897789d},
{0xbf21e44003acdd2c, 0xe0470a63e6bd56c4}, {0xeeea5d5004981478, 0x1858ccfce06cac75},
{0x95527a5202df0ccb, 0x0f37801e0c43ebc9}, {0xbaa718e68396cffd, 0xd30560258f54e6bb},
{0xe950df20247c83fd, 0x47c6b82ef32a206a}, {0x91d28b7416cdd27e, 0x4cdc331d57fa5442},
{0xb6472e511c81471d, 0xe0133fe4adf8e953}, {0xe3d8f9e563a198e5, 0x58180fddd97723a7},
{0x8e679c2f5e44ff8f, 0x570f09eaa7ea7649}, {0xb201833b35d63f73, 0x2cd2cc6551e513db},
{0xde81e40a034bcf4f, 0xf8077f7ea65e58d2}, {0x8b112e86420f6191, 0xfb04afaf27faf783},
{0xadd57a27d29339f6, 0x79c5db9af1f9b564}, {0xd94ad8b1c7380874, 0x18375281ae7822bd},
{0x87cec76f1c830548, 0x8f2293910d0b15b6}, {0xa9c2794ae3a3c69a, 0xb2eb3875504ddb23},
{0xd433179d9c8cb841, 0x5fa60692a46151ec}, {0x849feec281d7f328, 0xdbc7c41ba6bcd334},
{0xa5c7ea73224deff3, 0x12b9b522906c0801}, {0xcf39e50feae16bef, 0xd768226b34870a01},
{0x81842f29f2cce375, 0xe6a1158300d46641}, {0xa1e53af46f801c53, 0x60495ae3c1097fd1},
{0xca5e89b18b602368, 0x385bb19cb14bdfc5}, {0xfcf62c1dee382c42, 0x46729e03dd9ed7b6},
{0x9e19db92b4e31ba9, 0x6c07a2c26a8346d2}, {0xc5a05277621be293, 0xc7098b7305241886},
{0xf70867153aa2db38, 0xb8cbee4fc66d1ea8}};
};
// Compressed cache for double
struct compressed_cache_detail {
static constexpr int compression_ratio = 27;
static constexpr std::size_t compressed_table_size =
(cache_holder<ieee754_binary64>::max_k - cache_holder<ieee754_binary64>::min_k +
compression_ratio) /
compression_ratio;
struct cache_holder_t {
wuint::uint128 table[compressed_table_size];
};
static constexpr cache_holder_t cache = [] {
cache_holder_t res{};
for (std::size_t i = 0; i < compressed_table_size; ++i) {
res.table[i] = cache_holder<ieee754_binary64>::cache[i * compression_ratio];
}
return res;
}();
struct pow5_holder_t {
std::uint64_t table[compression_ratio];
};
static constexpr pow5_holder_t pow5 = [] {
pow5_holder_t res{};
std::uint64_t p = 1;
for (std::size_t i = 0; i < compression_ratio; ++i) {
res.table[i] = p;
p *= 5;
}
return res;
}();
};
}
////////////////////////////////////////////////////////////////////////////////////////
// Policies.
////////////////////////////////////////////////////////////////////////////////////////
namespace detail {
// Forward declare the implementation class.
template <class Float, class FloatTraits = default_float_traits<Float>>
struct impl;
namespace policy_impl {
// Sign policies.
namespace sign {
struct base {};
struct ignore : base {
using sign_policy = ignore;
static constexpr bool return_has_sign = false;
template <class SignedSignificandBits, class ReturnType>
static constexpr void handle_sign(SignedSignificandBits, ReturnType&) noexcept {
}
};
struct return_sign : base {
using sign_policy = return_sign;
static constexpr bool return_has_sign = true;
template <class SignedSignificandBits, class ReturnType>
static constexpr void handle_sign(SignedSignificandBits s,
ReturnType& r) noexcept {
r.is_negative = s.is_negative();
}
};
}
// Trailing zero policies.
namespace trailing_zero {
struct base {};
struct ignore : base {
using trailing_zero_policy = ignore;
static constexpr bool report_trailing_zeros = false;
template <class Impl, class ReturnType>
static constexpr void on_trailing_zeros(ReturnType&) noexcept {}
template <class Impl, class ReturnType>
static constexpr void no_trailing_zeros(ReturnType&) noexcept {}
};
struct remove : base {
using trailing_zero_policy = remove;
static constexpr bool report_trailing_zeros = false;
template <class Impl, class ReturnType>
JKJ_FORCEINLINE static constexpr void
on_trailing_zeros(ReturnType& r) noexcept {
r.exponent += Impl::remove_trailing_zeros(r.significand);
}
template <class Impl, class ReturnType>
static constexpr void no_trailing_zeros(ReturnType&) noexcept {}
};
struct report : base {
using trailing_zero_policy = report;
static constexpr bool report_trailing_zeros = true;
template <class Impl, class ReturnType>
static constexpr void on_trailing_zeros(ReturnType& r) noexcept {
r.may_have_trailing_zeros = true;
}
template <class Impl, class ReturnType>
static constexpr void no_trailing_zeros(ReturnType& r) noexcept {
r.may_have_trailing_zeros = false;
}
};
}
// Decimal-to-binary rounding mode policies.
namespace decimal_to_binary_rounding {
struct base {};
enum class tag_t { to_nearest, left_closed_directed, right_closed_directed };
namespace interval_type {
struct symmetric_boundary {
static constexpr bool is_symmetric = true;
bool is_closed;
constexpr bool include_left_endpoint() const noexcept { return is_closed; }
constexpr bool include_right_endpoint() const noexcept { return is_closed; }
};
struct asymmetric_boundary {
static constexpr bool is_symmetric = false;
bool is_left_closed;
constexpr bool include_left_endpoint() const noexcept {
return is_left_closed;
}
constexpr bool include_right_endpoint() const noexcept {
return !is_left_closed;
}
};
struct closed {
static constexpr bool is_symmetric = true;
static constexpr bool include_left_endpoint() noexcept { return true; }
static constexpr bool include_right_endpoint() noexcept { return true; }
};
struct open {
static constexpr bool is_symmetric = true;
static constexpr bool include_left_endpoint() noexcept { return false; }
static constexpr bool include_right_endpoint() noexcept { return false; }
};
struct left_closed_right_open {
static constexpr bool is_symmetric = false;
static constexpr bool include_left_endpoint() noexcept { return true; }
static constexpr bool include_right_endpoint() noexcept { return false; }
};
struct right_closed_left_open {
static constexpr bool is_symmetric = false;
static constexpr bool include_left_endpoint() noexcept { return false; }
static constexpr bool include_right_endpoint() noexcept { return true; }
};
}
struct nearest_to_even : base {
using decimal_to_binary_rounding_policy = nearest_to_even;
static constexpr auto tag = tag_t::to_nearest;
using normal_interval_type = interval_type::symmetric_boundary;
using shorter_interval_type = interval_type::closed;
template <class SignedSignificandBits, class Func>
JKJ_FORCEINLINE static auto delegate(SignedSignificandBits, Func&& f) noexcept {
return f(nearest_to_even{});
}
template <class SignedSignificandBits, class Func>
JKJ_FORCEINLINE static constexpr auto
invoke_normal_interval_case(SignedSignificandBits s, Func&& f) noexcept {
return f(s.has_even_significand_bits());
}
template <class SignedSignificandBits, class Func>
JKJ_FORCEINLINE static constexpr auto
invoke_shorter_interval_case(SignedSignificandBits, Func&& f) noexcept {
return f();
}
};
struct nearest_to_odd : base {
using decimal_to_binary_rounding_policy = nearest_to_odd;
static constexpr auto tag = tag_t::to_nearest;
using normal_interval_type = interval_type::symmetric_boundary;
using shorter_interval_type = interval_type::open;
template <class SignedSignificandBits, class Func>
JKJ_FORCEINLINE static auto delegate(SignedSignificandBits, Func&& f) noexcept {
return f(nearest_to_odd{});
}
template <class SignedSignificandBits, class Func>
JKJ_FORCEINLINE static constexpr auto
invoke_normal_interval_case(SignedSignificandBits s, Func&& f) noexcept {
return f(!s.has_even_significand_bits());
}
template <class SignedSignificandBits, class Func>
JKJ_FORCEINLINE static constexpr auto
invoke_shorter_interval_case(SignedSignificandBits, Func&& f) noexcept {
return f();
}
};
struct nearest_toward_plus_infinity : base {
using decimal_to_binary_rounding_policy = nearest_toward_plus_infinity;
static constexpr auto tag = tag_t::to_nearest;
using normal_interval_type = interval_type::asymmetric_boundary;
using shorter_interval_type = interval_type::asymmetric_boundary;
template <class SignedSignificandBits, class Func>
JKJ_FORCEINLINE static auto delegate(SignedSignificandBits, Func&& f) noexcept {
return f(nearest_toward_plus_infinity{});
}
template <class SignedSignificandBits, class Func>
JKJ_FORCEINLINE static constexpr auto
invoke_normal_interval_case(SignedSignificandBits s, Func&& f) noexcept {
return f(!s.is_negative());
}
template <class SignedSignificandBits, class Func>
JKJ_FORCEINLINE static constexpr auto
invoke_shorter_interval_case(SignedSignificandBits s, Func&& f) noexcept {
return f(!s.is_negative());
}
};
struct nearest_toward_minus_infinity : base {
using decimal_to_binary_rounding_policy = nearest_toward_minus_infinity;
static constexpr auto tag = tag_t::to_nearest;
using normal_interval_type = interval_type::asymmetric_boundary;
using shorter_interval_type = interval_type::asymmetric_boundary;
template <class SignedSignificandBits, class Func>
JKJ_FORCEINLINE static auto delegate(SignedSignificandBits, Func&& f) noexcept {
return f(nearest_toward_minus_infinity{});
}
template <class SignedSignificandBits, class Func>
JKJ_FORCEINLINE static constexpr auto
invoke_normal_interval_case(SignedSignificandBits s, Func&& f) noexcept {
return f(s.is_negative());
}
template <class SignedSignificandBits, class Func>
JKJ_FORCEINLINE static constexpr auto
invoke_shorter_interval_case(SignedSignificandBits s, Func&& f) noexcept {
return f(s.is_negative());
}
};
struct nearest_toward_zero : base {
using decimal_to_binary_rounding_policy = nearest_toward_zero;
static constexpr auto tag = tag_t::to_nearest;
using normal_interval_type = interval_type::right_closed_left_open;
using shorter_interval_type = interval_type::right_closed_left_open;
template <class SignedSignificandBits, class Func>
JKJ_FORCEINLINE static auto delegate(SignedSignificandBits, Func&& f) noexcept {
return f(nearest_toward_zero{});
}
template <class SignedSignificandBits, class Func>
JKJ_FORCEINLINE static constexpr auto
invoke_normal_interval_case(SignedSignificandBits, Func&& f) noexcept {
return f();
}
template <class SignedSignificandBits, class Func>
JKJ_FORCEINLINE static constexpr auto
invoke_shorter_interval_case(SignedSignificandBits, Func&& f) noexcept {
return f();
}
};
struct nearest_away_from_zero : base {
using decimal_to_binary_rounding_policy = nearest_away_from_zero;
static constexpr auto tag = tag_t::to_nearest;
using normal_interval_type = interval_type::left_closed_right_open;
using shorter_interval_type = interval_type::left_closed_right_open;
template <class SignedSignificandBits, class Func>
JKJ_FORCEINLINE static auto delegate(SignedSignificandBits, Func&& f) noexcept {
return f(nearest_away_from_zero{});
}
template <class SignedSignificandBits, class Func>
JKJ_FORCEINLINE static constexpr auto
invoke_normal_interval_case(SignedSignificandBits, Func&& f) noexcept {
return f();
}
template <class SignedSignificandBits, class Func>
JKJ_FORCEINLINE static constexpr auto
invoke_shorter_interval_case(SignedSignificandBits, Func&& f) noexcept {
return f();
}
};
namespace detail {
struct nearest_always_closed {
static constexpr auto tag = tag_t::to_nearest;
using normal_interval_type = interval_type::closed;
using shorter_interval_type = interval_type::closed;
template <class SignedSignificandBits, class Func>
JKJ_FORCEINLINE static constexpr auto
invoke_normal_interval_case(SignedSignificandBits, Func&& f) noexcept {
return f();
}
template <class SignedSignificandBits, class Func>
JKJ_FORCEINLINE static constexpr auto
invoke_shorter_interval_case(SignedSignificandBits, Func&& f) noexcept {
return f();
}
};
struct nearest_always_open {
static constexpr auto tag = tag_t::to_nearest;
using normal_interval_type = interval_type::open;
using shorter_interval_type = interval_type::open;
template <class SignedSignificandBits, class Func>
JKJ_FORCEINLINE static constexpr auto
invoke_normal_interval_case(SignedSignificandBits, Func&& f) noexcept {
return f();
}
template <class SignedSignificandBits, class Func>
JKJ_FORCEINLINE static constexpr auto
invoke_shorter_interval_case(SignedSignificandBits, Func&& f) noexcept {
return f();
}
};
}
struct nearest_to_even_static_boundary : base {
using decimal_to_binary_rounding_policy = nearest_to_even_static_boundary;
template <class SignedSignificandBits, class Func>
JKJ_FORCEINLINE static auto delegate(SignedSignificandBits s,
Func&& f) noexcept {
if (s.has_even_significand_bits()) {
return f(detail::nearest_always_closed{});
}
else {
return f(detail::nearest_always_open{});
}
}
};
struct nearest_to_odd_static_boundary : base {
using decimal_to_binary_rounding_policy = nearest_to_odd_static_boundary;
template <class SignedSignificandBits, class Func>
JKJ_FORCEINLINE static auto delegate(SignedSignificandBits s,
Func&& f) noexcept {
if (s.has_even_significand_bits()) {
return f(detail::nearest_always_open{});
}
else {
return f(detail::nearest_always_closed{});
}
}
};
struct nearest_toward_plus_infinity_static_boundary : base {
using decimal_to_binary_rounding_policy =
nearest_toward_plus_infinity_static_boundary;
template <class SignedSignificandBits, class Func>
JKJ_FORCEINLINE static auto delegate(SignedSignificandBits s,
Func&& f) noexcept {
if (s.is_negative()) {
return f(nearest_toward_zero{});
}
else {
return f(nearest_away_from_zero{});
}
}
};
struct nearest_toward_minus_infinity_static_boundary : base {
using decimal_to_binary_rounding_policy =
nearest_toward_minus_infinity_static_boundary;
template <class SignedSignificandBits, class Func>
JKJ_FORCEINLINE static auto delegate(SignedSignificandBits s,
Func&& f) noexcept {
if (s.is_negative()) {
return f(nearest_away_from_zero{});
}
else {
return f(nearest_toward_zero{});
}
}
};
namespace detail {
struct left_closed_directed {
static constexpr auto tag = tag_t::left_closed_directed;
};
struct right_closed_directed {
static constexpr auto tag = tag_t::right_closed_directed;
};
}
struct toward_plus_infinity : base {
using decimal_to_binary_rounding_policy = toward_plus_infinity;
template <class SignedSignificandBits, class Func>
JKJ_FORCEINLINE static auto delegate(SignedSignificandBits s,
Func&& f) noexcept {
if (s.is_negative()) {
return f(detail::left_closed_directed{});
}
else {
return f(detail::right_closed_directed{});
}
}
};
struct toward_minus_infinity : base {
using decimal_to_binary_rounding_policy = toward_minus_infinity;
template <class SignedSignificandBits, class Func>
JKJ_FORCEINLINE static auto delegate(SignedSignificandBits s,
Func&& f) noexcept {
if (s.is_negative()) {
return f(detail::right_closed_directed{});
}
else {
return f(detail::left_closed_directed{});
}
}
};
struct toward_zero : base {
using decimal_to_binary_rounding_policy = toward_zero;
template <class SignedSignificandBits, class Func>
JKJ_FORCEINLINE static auto delegate(SignedSignificandBits, Func&& f) noexcept {
return f(detail::left_closed_directed{});
}
};
struct away_from_zero : base {
using decimal_to_binary_rounding_policy = away_from_zero;
template <class SignedSignificandBits, class Func>
JKJ_FORCEINLINE static auto delegate(SignedSignificandBits, Func&& f) noexcept {
return f(detail::right_closed_directed{});
}
};
}
// Binary-to-decimal rounding policies.
// (Always assumes nearest rounding modes.)
namespace binary_to_decimal_rounding {
struct base {};
enum class tag_t { do_not_care, to_even, to_odd, away_from_zero, toward_zero };
struct do_not_care : base {
using binary_to_decimal_rounding_policy = do_not_care;
static constexpr auto tag = tag_t::do_not_care;
template <class ReturnType>
static constexpr bool prefer_round_down(ReturnType const&) noexcept {
return false;
}
};
struct to_even : base {
using binary_to_decimal_rounding_policy = to_even;
static constexpr auto tag = tag_t::to_even;
template <class ReturnType>
static constexpr bool prefer_round_down(ReturnType const& r) noexcept {
return r.significand % 2 != 0;
}
};
struct to_odd : base {
using binary_to_decimal_rounding_policy = to_odd;
static constexpr auto tag = tag_t::to_odd;
template <class ReturnType>
static constexpr bool prefer_round_down(ReturnType const& r) noexcept {
return r.significand % 2 == 0;
}
};
struct away_from_zero : base {
using binary_to_decimal_rounding_policy = away_from_zero;
static constexpr auto tag = tag_t::away_from_zero;
template <class ReturnType>
static constexpr bool prefer_round_down(ReturnType const&) noexcept {
return false;
}
};
struct toward_zero : base {
using binary_to_decimal_rounding_policy = toward_zero;
static constexpr auto tag = tag_t::toward_zero;
template <class ReturnType>
static constexpr bool prefer_round_down(ReturnType const&) noexcept {
return true;
}
};
}
// Cache policies.
namespace cache {
struct base {};
struct full : base {
using cache_policy = full;
template <class FloatFormat>
static constexpr typename cache_holder<FloatFormat>::cache_entry_type
get_cache(int k) noexcept {
assert(k >= cache_holder<FloatFormat>::min_k &&
k <= cache_holder<FloatFormat>::max_k);
return cache_holder<FloatFormat>::cache[std::size_t(
k - cache_holder<FloatFormat>::min_k)];
}
};
struct compact : base {
using cache_policy = compact;
template <class FloatFormat>
static constexpr typename cache_holder<FloatFormat>::cache_entry_type
get_cache(int k) noexcept {
assert(k >= cache_holder<FloatFormat>::min_k &&
k <= cache_holder<FloatFormat>::max_k);
if constexpr (std::is_same_v<FloatFormat, ieee754_binary64>) {
// Compute the base index.
auto const cache_index =
int(std::uint32_t(k - cache_holder<FloatFormat>::min_k) /
compressed_cache_detail::compression_ratio);
auto const kb =
cache_index * compressed_cache_detail::compression_ratio +
cache_holder<FloatFormat>::min_k;
auto const offset = k - kb;
// Get the base cache.
auto const base_cache =
compressed_cache_detail::cache.table[cache_index];
if (offset == 0) {
return base_cache;
}
else {
// Compute the required amount of bit-shift.
auto const alpha = log::floor_log2_pow10(kb + offset) -
log::floor_log2_pow10(kb) - offset;
assert(alpha > 0 && alpha < 64);
// Try to recover the real cache.
auto const pow5 = compressed_cache_detail::pow5.table[offset];
auto recovered_cache = wuint::umul128(base_cache.high(), pow5);
auto const middle_low = wuint::umul128(base_cache.low(), pow5);
recovered_cache += middle_low.high();
auto const high_to_middle = recovered_cache.high() << (64 - alpha);
auto const middle_to_low = recovered_cache.low() << (64 - alpha);
recovered_cache = wuint::uint128{
(recovered_cache.low() >> alpha) | high_to_middle,
((middle_low.low() >> alpha) | middle_to_low)};
assert(recovered_cache.low() + 1 != 0);
recovered_cache = {recovered_cache.high(),
recovered_cache.low() + 1};
return recovered_cache;
}
}
else {
// Just use the full cache for anything other than binary64
return cache_holder<FloatFormat>::cache[std::size_t(
k - cache_holder<FloatFormat>::min_k)];
}
}
};
}
}
}
namespace policy {
namespace sign {
inline constexpr auto ignore = detail::policy_impl::sign::ignore{};
inline constexpr auto return_sign = detail::policy_impl::sign::return_sign{};
}
namespace trailing_zero {
inline constexpr auto ignore = detail::policy_impl::trailing_zero::ignore{};
inline constexpr auto remove = detail::policy_impl::trailing_zero::remove{};
inline constexpr auto report = detail::policy_impl::trailing_zero::report{};
}
namespace decimal_to_binary_rounding {
inline constexpr auto nearest_to_even =
detail::policy_impl::decimal_to_binary_rounding::nearest_to_even{};
inline constexpr auto nearest_to_odd =
detail::policy_impl::decimal_to_binary_rounding::nearest_to_odd{};
inline constexpr auto nearest_toward_plus_infinity =
detail::policy_impl::decimal_to_binary_rounding::nearest_toward_plus_infinity{};
inline constexpr auto nearest_toward_minus_infinity =
detail::policy_impl::decimal_to_binary_rounding::nearest_toward_minus_infinity{};
inline constexpr auto nearest_toward_zero =
detail::policy_impl::decimal_to_binary_rounding::nearest_toward_zero{};
inline constexpr auto nearest_away_from_zero =
detail::policy_impl::decimal_to_binary_rounding::nearest_away_from_zero{};
inline constexpr auto nearest_to_even_static_boundary =
detail::policy_impl::decimal_to_binary_rounding::nearest_to_even_static_boundary{};
inline constexpr auto nearest_to_odd_static_boundary =
detail::policy_impl::decimal_to_binary_rounding::nearest_to_odd_static_boundary{};
inline constexpr auto nearest_toward_plus_infinity_static_boundary =
detail::policy_impl::decimal_to_binary_rounding::
nearest_toward_plus_infinity_static_boundary{};
inline constexpr auto nearest_toward_minus_infinity_static_boundary =
detail::policy_impl::decimal_to_binary_rounding::
nearest_toward_minus_infinity_static_boundary{};
inline constexpr auto toward_plus_infinity =
detail::policy_impl::decimal_to_binary_rounding::toward_plus_infinity{};
inline constexpr auto toward_minus_infinity =
detail::policy_impl::decimal_to_binary_rounding::toward_minus_infinity{};
inline constexpr auto toward_zero =
detail::policy_impl::decimal_to_binary_rounding::toward_zero{};
inline constexpr auto away_from_zero =
detail::policy_impl::decimal_to_binary_rounding::away_from_zero{};
}
namespace binary_to_decimal_rounding {
inline constexpr auto do_not_care =
detail::policy_impl::binary_to_decimal_rounding::do_not_care{};
inline constexpr auto to_even =
detail::policy_impl::binary_to_decimal_rounding::to_even{};
inline constexpr auto to_odd =
detail::policy_impl::binary_to_decimal_rounding::to_odd{};
inline constexpr auto away_from_zero =
detail::policy_impl::binary_to_decimal_rounding::away_from_zero{};
inline constexpr auto toward_zero =
detail::policy_impl::binary_to_decimal_rounding::toward_zero{};
}
namespace cache {
inline constexpr auto full = detail::policy_impl::cache::full{};
inline constexpr auto compact = detail::policy_impl::cache::compact{};
}
}
namespace detail {
////////////////////////////////////////////////////////////////////////////////////////
// The main algorithm.
////////////////////////////////////////////////////////////////////////////////////////
template <class Float, class FloatTraits>
struct impl : private FloatTraits, private FloatTraits::format {
using format = typename FloatTraits::format;
using carrier_uint = typename FloatTraits::carrier_uint;
using FloatTraits::carrier_bits;
using format::significand_bits;
using format::min_exponent;
using format::max_exponent;
using format::exponent_bias;
using format::decimal_digits;
static constexpr int kappa = std::is_same_v<format, ieee754_binary32> ? 1 : 2;
static_assert(kappa >= 1);
static_assert(carrier_bits >= significand_bits + 2 + log::floor_log2_pow10(kappa + 1));
static constexpr int min_k = [] {
constexpr auto a = -log::floor_log10_pow2_minus_log10_4_over_3(
int(max_exponent - significand_bits));
constexpr auto b =
-log::floor_log10_pow2(int(max_exponent - significand_bits)) + kappa;
return a < b ? a : b;
}();
static_assert(min_k >= cache_holder<format>::min_k);
static constexpr int max_k = [] {
// We do invoke shorter_interval_case for exponent == min_exponent case,
// so we should not add 1 here.
constexpr auto a = -log::floor_log10_pow2_minus_log10_4_over_3(
int(min_exponent - significand_bits /*+ 1*/));
constexpr auto b =
-log::floor_log10_pow2(int(min_exponent - significand_bits)) + kappa;
return a > b ? a : b;
}();
static_assert(max_k <= cache_holder<format>::max_k);
using cache_entry_type = typename cache_holder<format>::cache_entry_type;
static constexpr auto cache_bits = cache_holder<format>::cache_bits;
static constexpr int case_shorter_interval_left_endpoint_lower_threshold = 2;
static constexpr int case_shorter_interval_left_endpoint_upper_threshold =
2 +
log::floor_log2(
compute_power<
count_factors<5>((carrier_uint(1) << (significand_bits + 2)) - 1) + 1>(10) /
3);
static constexpr int case_shorter_interval_right_endpoint_lower_threshold = 0;
static constexpr int case_shorter_interval_right_endpoint_upper_threshold =
2 +
log::floor_log2(
compute_power<
count_factors<5>((carrier_uint(1) << (significand_bits + 1)) + 1) + 1>(10) /
3);
static constexpr int shorter_interval_tie_lower_threshold =
-log::floor_log5_pow2_minus_log5_3(significand_bits + 4) - 2 - significand_bits;
static constexpr int shorter_interval_tie_upper_threshold =
-log::floor_log5_pow2(significand_bits + 2) - 2 - significand_bits;
struct compute_mul_result {
carrier_uint result;
bool is_integer;
};
struct compute_mul_parity_result {
bool parity;
bool is_integer;
};
//// The main algorithm assumes the input is a normal/subnormal finite number
template <class ReturnType, class IntervalType, class TrailingZeroPolicy,
class BinaryToDecimalRoundingPolicy, class CachePolicy,
class... AdditionalArgs>
JKJ_SAFEBUFFERS static ReturnType
compute_nearest_normal(carrier_uint const two_fc, int const exponent,
AdditionalArgs... additional_args) noexcept {
//////////////////////////////////////////////////////////////////////
// Step 1: Schubfach multiplier calculation
//////////////////////////////////////////////////////////////////////
ReturnType ret_value;
IntervalType interval_type{additional_args...};
// Compute k and beta.
int const minus_k = log::floor_log10_pow2(exponent) - kappa;
auto const cache = CachePolicy::template get_cache<format>(-minus_k);
int const beta = exponent + log::floor_log2_pow10(-minus_k);
// Compute zi and deltai.
// 10^kappa <= deltai < 10^(kappa + 1)
auto const deltai = compute_delta(cache, beta);
// For the case of binary32, the result of integer check is not correct for
// 29711844 * 2^-82
// = 6.1442653300000000008655037797566933477355632930994033813476... * 10^-18
// and 29711844 * 2^-81
// = 1.2288530660000000001731007559513386695471126586198806762695... * 10^-17,
// and they are the unique counterexamples. However, since 29711844 is even,
// this does not cause any problem for the endpoints calculations; it can only
// cause a problem when we need to perform integer check for the center.
// Fortunately, with these inputs, that branch is never executed, so we are fine.
auto const [zi, is_z_integer] = compute_mul((two_fc | 1) << beta, cache);
//////////////////////////////////////////////////////////////////////
// Step 2: Try larger divisor; remove trailing zeros if necessary
//////////////////////////////////////////////////////////////////////
constexpr auto big_divisor = compute_power<kappa + 1>(std::uint32_t(10));
constexpr auto small_divisor = compute_power<kappa>(std::uint32_t(10));
// Using an upper bound on zi, we might be able to optimize the division
// better than the compiler; we are computing zi / big_divisor here.
ret_value.significand =
div::divide_by_pow10<kappa + 1, carrier_uint,
(carrier_uint(1) << (significand_bits + 1)) * big_divisor -
1>(zi);
auto r = std::uint32_t(zi - big_divisor * ret_value.significand);
if (r < deltai) {
// Exclude the right endpoint if necessary.
if (r == 0 && (is_z_integer & !interval_type.include_right_endpoint())) {
if constexpr (BinaryToDecimalRoundingPolicy::tag ==
policy_impl::binary_to_decimal_rounding::tag_t::do_not_care) {
ret_value.significand *= 10;
ret_value.exponent = minus_k + kappa;
--ret_value.significand;
TrailingZeroPolicy::template no_trailing_zeros<impl>(ret_value);
return ret_value;
}
else {
--ret_value.significand;
r = big_divisor;
goto small_divisor_case_label;
}
}
}
else if (r > deltai) {
goto small_divisor_case_label;
}
else {
// r == deltai; compare fractional parts.
auto const [xi_parity, x_is_integer] =
compute_mul_parity(two_fc - 1, cache, beta);
if (!(xi_parity | (x_is_integer & interval_type.include_left_endpoint()))) {
goto small_divisor_case_label;
}
}
ret_value.exponent = minus_k + kappa + 1;
// We may need to remove trailing zeros.
TrailingZeroPolicy::template on_trailing_zeros<impl>(ret_value);
return ret_value;
//////////////////////////////////////////////////////////////////////
// Step 3: Find the significand with the smaller divisor
//////////////////////////////////////////////////////////////////////
small_divisor_case_label:
TrailingZeroPolicy::template no_trailing_zeros<impl>(ret_value);
ret_value.significand *= 10;
ret_value.exponent = minus_k + kappa;
if constexpr (BinaryToDecimalRoundingPolicy::tag ==
policy_impl::binary_to_decimal_rounding::tag_t::do_not_care) {
// Normally, we want to compute
// ret_value.significand += r / small_divisor
// and return, but we need to take care of the case that the resulting
// value is exactly the right endpoint, while that is not included in the
// interval.
if (!interval_type.include_right_endpoint()) {
// Is r divisible by 10^kappa?
if (is_z_integer && div::check_divisibility_and_divide_by_pow10<kappa>(r)) {
// This should be in the interval.
ret_value.significand += r - 1;
}
else {
ret_value.significand += r;
}
}
else {
ret_value.significand += div::small_division_by_pow10<kappa>(r);
}
}
else {
auto dist = r - (deltai / 2) + (small_divisor / 2);
bool const approx_y_parity = ((dist ^ (small_divisor / 2)) & 1) != 0;
// Is dist divisible by 10^kappa?
bool const divisible_by_small_divisor =
div::check_divisibility_and_divide_by_pow10<kappa>(dist);
// Add dist / 10^kappa to the significand.
ret_value.significand += dist;
if (divisible_by_small_divisor) {
// Check z^(f) >= epsilon^(f).
// We have either yi == zi - epsiloni or yi == (zi - epsiloni) - 1,
// where yi == zi - epsiloni if and only if z^(f) >= epsilon^(f).
// Since there are only 2 possibilities, we only need to care about the
// parity. Also, zi and r should have the same parity since the divisor is
// an even number.
auto const [yi_parity, is_y_integer] =
compute_mul_parity(two_fc, cache, beta);
if (yi_parity != approx_y_parity) {
--ret_value.significand;
}
else {
// If z^(f) >= epsilon^(f), we might have a tie
// when z^(f) == epsilon^(f), or equivalently, when y is an integer.
// For tie-to-up case, we can just choose the upper one.
if (BinaryToDecimalRoundingPolicy::prefer_round_down(ret_value) &
is_y_integer) {
--ret_value.significand;
}
}
}
}
return ret_value;
}
template <class ReturnType, class IntervalType, class TrailingZeroPolicy,
class BinaryToDecimalRoundingPolicy, class CachePolicy,
class... AdditionalArgs>
JKJ_SAFEBUFFERS static ReturnType
compute_nearest_shorter(int const exponent,
AdditionalArgs... additional_args) noexcept {
ReturnType ret_value;
IntervalType interval_type{additional_args...};
// Compute k and beta.
int const minus_k = log::floor_log10_pow2_minus_log10_4_over_3(exponent);
int const beta = exponent + log::floor_log2_pow10(-minus_k);
// Compute xi and zi.
auto const cache = CachePolicy::template get_cache<format>(-minus_k);
auto xi = compute_left_endpoint_for_shorter_interval_case(cache, beta);
auto zi = compute_right_endpoint_for_shorter_interval_case(cache, beta);
// If we don't accept the right endpoint and
// if the right endpoint is an integer, decrease it.
if (!interval_type.include_right_endpoint() &&
is_right_endpoint_integer_shorter_interval(exponent)) {
--zi;
}
// If we don't accept the left endpoint or
// if the left endpoint is not an integer, increase it.
if (!interval_type.include_left_endpoint() ||
!is_left_endpoint_integer_shorter_interval(exponent)) {
++xi;
}
// Try bigger divisor.
ret_value.significand = zi / 10;
// If succeed, remove trailing zeros if necessary and return.
if (ret_value.significand * 10 >= xi) {
ret_value.exponent = minus_k + 1;
TrailingZeroPolicy::template on_trailing_zeros<impl>(ret_value);
return ret_value;
}
// Otherwise, compute the round-up of y.
TrailingZeroPolicy::template no_trailing_zeros<impl>(ret_value);
ret_value.significand = compute_round_up_for_shorter_interval_case(cache, beta);
ret_value.exponent = minus_k;
// When tie occurs, choose one of them according to the rule.
if (BinaryToDecimalRoundingPolicy::prefer_round_down(ret_value) &&
exponent >= shorter_interval_tie_lower_threshold &&
exponent <= shorter_interval_tie_upper_threshold) {
--ret_value.significand;
}
else if (ret_value.significand < xi) {
++ret_value.significand;
}
return ret_value;
}
template <class ReturnType, class TrailingZeroPolicy, class CachePolicy>
JKJ_SAFEBUFFERS static ReturnType
compute_left_closed_directed(carrier_uint const two_fc, int exponent) noexcept {
//////////////////////////////////////////////////////////////////////
// Step 1: Schubfach multiplier calculation
//////////////////////////////////////////////////////////////////////
ReturnType ret_value;
// Compute k and beta.
int const minus_k = log::floor_log10_pow2(exponent) - kappa;
auto const cache = CachePolicy::template get_cache<format>(-minus_k);
int const beta = exponent + log::floor_log2_pow10(-minus_k);
// Compute xi and deltai.
// 10^kappa <= deltai < 10^(kappa + 1)
auto const deltai = compute_delta(cache, beta);
auto [xi, is_x_integer] = compute_mul(two_fc << beta, cache);
// Deal with the unique exceptional cases
// 29711844 * 2^-82
// = 6.1442653300000000008655037797566933477355632930994033813476... * 10^-18
// and 29711844 * 2^-81
// = 1.2288530660000000001731007559513386695471126586198806762695... * 10^-17
// for binary32.
if constexpr (std::is_same_v<format, ieee754_binary32>) {
if (exponent <= -80) {
is_x_integer = false;
}
}
if (!is_x_integer) {
++xi;
}
//////////////////////////////////////////////////////////////////////
// Step 2: Try larger divisor; remove trailing zeros if necessary
//////////////////////////////////////////////////////////////////////
constexpr auto big_divisor = compute_power<kappa + 1>(std::uint32_t(10));
// Using an upper bound on xi, we might be able to optimize the division
// better than the compiler; we are computing xi / big_divisor here.
ret_value.significand =
div::divide_by_pow10<kappa + 1, carrier_uint,
(carrier_uint(1) << (significand_bits + 1)) * big_divisor -
1>(xi);
auto r = std::uint32_t(xi - big_divisor * ret_value.significand);
if (r != 0) {
++ret_value.significand;
r = big_divisor - r;
}
if (r > deltai) {
goto small_divisor_case_label;
}
else if (r == deltai) {
// Compare the fractional parts.
// This branch is never taken for the exceptional cases
// 2f_c = 29711482, e = -81
// (6.1442649164096937243516663440523473127541365101933479309082... * 10^-18)
// and 2f_c = 29711482, e = -80
// (1.2288529832819387448703332688104694625508273020386695861816... * 10^-17).
auto const [zi_parity, is_z_integer] =
compute_mul_parity(two_fc + 2, cache, beta);
if (zi_parity || is_z_integer) {
goto small_divisor_case_label;
}
}
// The ceiling is inside, so we are done.
ret_value.exponent = minus_k + kappa + 1;
TrailingZeroPolicy::template on_trailing_zeros<impl>(ret_value);
return ret_value;
//////////////////////////////////////////////////////////////////////
// Step 3: Find the significand with the smaller divisor
//////////////////////////////////////////////////////////////////////
small_divisor_case_label:
ret_value.significand *= 10;
ret_value.significand -= div::small_division_by_pow10<kappa>(r);
ret_value.exponent = minus_k + kappa;
TrailingZeroPolicy::template no_trailing_zeros<impl>(ret_value);
return ret_value;
}
template <class ReturnType, class TrailingZeroPolicy, class CachePolicy>
JKJ_SAFEBUFFERS static ReturnType
compute_right_closed_directed(carrier_uint const two_fc, int const exponent,
bool shorter_interval) noexcept {
//////////////////////////////////////////////////////////////////////
// Step 1: Schubfach multiplier calculation
//////////////////////////////////////////////////////////////////////
ReturnType ret_value;
// Compute k and beta.
int const minus_k =
log::floor_log10_pow2(exponent - (shorter_interval ? 1 : 0)) - kappa;
auto const cache = CachePolicy::template get_cache<format>(-minus_k);
int const beta = exponent + log::floor_log2_pow10(-minus_k);
// Compute zi and deltai.
// 10^kappa <= deltai < 10^(kappa + 1)
auto const deltai =
shorter_interval ? compute_delta(cache, beta - 1) : compute_delta(cache, beta);
carrier_uint const zi = compute_mul(two_fc << beta, cache).result;
//////////////////////////////////////////////////////////////////////
// Step 2: Try larger divisor; remove trailing zeros if necessary
//////////////////////////////////////////////////////////////////////
constexpr auto big_divisor = compute_power<kappa + 1>(std::uint32_t(10));
// Using an upper bound on zi, we might be able to optimize the division better than
// the compiler; we are computing zi / big_divisor here.
ret_value.significand =
div::divide_by_pow10<kappa + 1, carrier_uint,
(carrier_uint(1) << (significand_bits + 1)) * big_divisor -
1>(zi);
auto const r = std::uint32_t(zi - big_divisor * ret_value.significand);
if (r > deltai) {
goto small_divisor_case_label;
}
else if (r == deltai) {
// Compare the fractional parts.
if (!compute_mul_parity(two_fc - (shorter_interval ? 1 : 2), cache, beta)
.parity) {
goto small_divisor_case_label;
}
}
// The floor is inside, so we are done.
ret_value.exponent = minus_k + kappa + 1;
TrailingZeroPolicy::template on_trailing_zeros<impl>(ret_value);
return ret_value;
//////////////////////////////////////////////////////////////////////
// Step 3: Find the significand with the small divisor
//////////////////////////////////////////////////////////////////////
small_divisor_case_label:
ret_value.significand *= 10;
ret_value.significand += div::small_division_by_pow10<kappa>(r);
ret_value.exponent = minus_k + kappa;
TrailingZeroPolicy::template no_trailing_zeros<impl>(ret_value);
return ret_value;
}
// Remove trailing zeros from n and return the number of zeros removed.
JKJ_FORCEINLINE static int remove_trailing_zeros(carrier_uint& n) noexcept {
assert(n != 0);
if constexpr (std::is_same_v<format, ieee754_binary32>) {
constexpr auto mod_inv_5 = std::uint32_t(0xcccc'cccd);
constexpr auto mod_inv_25 = mod_inv_5 * mod_inv_5;
int s = 0;
while (true) {
auto q = bits::rotr(n * mod_inv_25, 2);
if (q <= std::numeric_limits<std::uint32_t>::max() / 100) {
n = q;
s += 2;
}
else {
break;
}
}
auto q = bits::rotr(n * mod_inv_5, 1);
if (q <= std::numeric_limits<std::uint32_t>::max() / 10) {
n = q;
s |= 1;
}
return s;
}
else {
static_assert(std::is_same_v<format, ieee754_binary64>);
// Divide by 10^8 and reduce to 32-bits if divisible.
// Since ret_value.significand <= (2^53 * 1000 - 1) / 1000 < 10^16,
// n is at most of 16 digits.
// This magic number is ceil(2^90 / 10^8).
constexpr auto magic_number = std::uint64_t(12379400392853802749ull);
auto nm = wuint::umul128(n, magic_number);
// Is n is divisible by 10^8?
if ((nm.high() & ((std::uint64_t(1) << (90 - 64)) - 1)) == 0 &&
nm.low() < magic_number) {
// If yes, work with the quotient.
auto n32 = std::uint32_t(nm.high() >> (90 - 64));
constexpr auto mod_inv_5 = std::uint32_t(0xcccc'cccd);
constexpr auto mod_inv_25 = mod_inv_5 * mod_inv_5;
int s = 8;
while (true) {
auto q = bits::rotr(n32 * mod_inv_25, 2);
if (q <= std::numeric_limits<std::uint32_t>::max() / 100) {
n32 = q;
s += 2;
}
else {
break;
}
}
auto q = bits::rotr(n32 * mod_inv_5, 1);
if (q <= std::numeric_limits<std::uint32_t>::max() / 10) {
n32 = q;
s |= 1;
}
n = n32;
return s;
}
// If n is not divisible by 10^8, work with n itself.
constexpr auto mod_inv_5 = std::uint64_t(0xcccc'cccc'cccc'cccd);
constexpr auto mod_inv_25 = mod_inv_5 * mod_inv_5;
int s = 0;
while (true) {
auto q = bits::rotr(n * mod_inv_25, 2);
if (q <= std::numeric_limits<std::uint64_t>::max() / 100) {
n = q;
s += 2;
}
else {
break;
}
}
auto q = bits::rotr(n * mod_inv_5, 1);
if (q <= std::numeric_limits<std::uint64_t>::max() / 10) {
n = q;
s |= 1;
}
return s;
}
}
static compute_mul_result compute_mul(carrier_uint u,
cache_entry_type const& cache) noexcept {
if constexpr (std::is_same_v<format, ieee754_binary32>) {
auto r = wuint::umul96_upper64(u, cache);
return {carrier_uint(r >> 32), carrier_uint(r) == 0};
}
else {
static_assert(std::is_same_v<format, ieee754_binary64>);
auto r = wuint::umul192_upper128(u, cache);
return {r.high(), r.low() == 0};
}
}
static constexpr std::uint32_t compute_delta(cache_entry_type const& cache,
int beta) noexcept {
if constexpr (std::is_same_v<format, ieee754_binary32>) {
return std::uint32_t(cache >> (cache_bits - 1 - beta));
}
else {
static_assert(std::is_same_v<format, ieee754_binary64>);
return std::uint32_t(cache.high() >> (carrier_bits - 1 - beta));
}
}
static compute_mul_parity_result compute_mul_parity(carrier_uint two_f,
cache_entry_type const& cache,
int beta) noexcept {
assert(beta >= 1);
assert(beta < 64);
if constexpr (std::is_same_v<format, ieee754_binary32>) {
auto r = wuint::umul96_lower64(two_f, cache);
return {((r >> (64 - beta)) & 1) != 0, std::uint32_t(r >> (32 - beta)) == 0};
}
else {
static_assert(std::is_same_v<format, ieee754_binary64>);
auto r = wuint::umul192_lower128(two_f, cache);
return {((r.high() >> (64 - beta)) & 1) != 0,
((r.high() << beta) | (r.low() >> (64 - beta))) == 0};
}
}
static constexpr carrier_uint
compute_left_endpoint_for_shorter_interval_case(cache_entry_type const& cache,
int beta) noexcept {
if constexpr (std::is_same_v<format, ieee754_binary32>) {
return carrier_uint((cache - (cache >> (significand_bits + 2))) >>
(cache_bits - significand_bits - 1 - beta));
}
else {
static_assert(std::is_same_v<format, ieee754_binary64>);
return (cache.high() - (cache.high() >> (significand_bits + 2))) >>
(carrier_bits - significand_bits - 1 - beta);
}
}
static constexpr carrier_uint
compute_right_endpoint_for_shorter_interval_case(cache_entry_type const& cache,
int beta) noexcept {
if constexpr (std::is_same_v<format, ieee754_binary32>) {
return carrier_uint((cache + (cache >> (significand_bits + 1))) >>
(cache_bits - significand_bits - 1 - beta));
}
else {
static_assert(std::is_same_v<format, ieee754_binary64>);
return (cache.high() + (cache.high() >> (significand_bits + 1))) >>
(carrier_bits - significand_bits - 1 - beta);
}
}
static constexpr carrier_uint
compute_round_up_for_shorter_interval_case(cache_entry_type const& cache,
int beta) noexcept {
if constexpr (std::is_same_v<format, ieee754_binary32>) {
return (carrier_uint(cache >> (cache_bits - significand_bits - 2 - beta)) + 1) /
2;
}
else {
static_assert(std::is_same_v<format, ieee754_binary64>);
return ((cache.high() >> (carrier_bits - significand_bits - 2 - beta)) + 1) / 2;
}
}
static constexpr bool
is_right_endpoint_integer_shorter_interval(int exponent) noexcept {
return exponent >= case_shorter_interval_right_endpoint_lower_threshold &&
exponent <= case_shorter_interval_right_endpoint_upper_threshold;
}
static constexpr bool is_left_endpoint_integer_shorter_interval(int exponent) noexcept {
return exponent >= case_shorter_interval_left_endpoint_lower_threshold &&
exponent <= case_shorter_interval_left_endpoint_upper_threshold;
}
};
////////////////////////////////////////////////////////////////////////////////////////
// Policy holder.
////////////////////////////////////////////////////////////////////////////////////////
namespace policy_impl {
// The library will specify a list of accepted kinds of policies and their defaults, and
// the user will pass a list of policies. The aim of helper classes/functions here is to
// do the following:
// 1. Check if the policy parameters given by the user are all valid; that means,
// each of them should be of the kinds specified by the library.
// If that's not the case, then the compilation fails.
// 2. Check if multiple policy parameters for the same kind is specified by the user.
// If that's the case, then the compilation fails.
// 3. Build a class deriving from all policies the user have given, and also from
// the default policies if the user did not specify one for some kinds.
// A policy belongs to a certain kind if it is deriving from a base class.
// For a given kind, find a policy belonging to that kind.
// Check if there are more than one such policies.
enum class policy_found_info { not_found, unique, repeated };
template <class Policy, policy_found_info info>
struct found_policy_pair {
using policy = Policy;
static constexpr auto found_info = info;
};
template <class Base, class DefaultPolicy>
struct base_default_pair {
using base = Base;
template <class FoundPolicyInfo>
static constexpr FoundPolicyInfo get_policy_impl(FoundPolicyInfo) {
return {};
}
template <class FoundPolicyInfo, class FirstPolicy, class... RemainingPolicies>
static constexpr auto get_policy_impl(FoundPolicyInfo, FirstPolicy,
RemainingPolicies... remainings) {
if constexpr (std::is_base_of_v<Base, FirstPolicy>) {
if constexpr (FoundPolicyInfo::found_info == policy_found_info::not_found) {
return get_policy_impl(
found_policy_pair<FirstPolicy, policy_found_info::unique>{},
remainings...);
}
else {
return get_policy_impl(
found_policy_pair<FirstPolicy, policy_found_info::repeated>{},
remainings...);
}
}
else {
return get_policy_impl(FoundPolicyInfo{}, remainings...);
}
}
template <class... Policies>
static constexpr auto get_policy(Policies... policies) {
return get_policy_impl(
found_policy_pair<DefaultPolicy, policy_found_info::not_found>{},
policies...);
}
};
template <class... BaseDefaultPairs>
struct base_default_pair_list {};
// Check if a given policy belongs to one of the kinds specified by the library.
template <class Policy>
constexpr bool check_policy_validity(Policy, base_default_pair_list<>) {
return false;
}
template <class Policy, class FirstBaseDefaultPair, class... RemainingBaseDefaultPairs>
constexpr bool check_policy_validity(
Policy,
base_default_pair_list<FirstBaseDefaultPair, RemainingBaseDefaultPairs...>) {
return std::is_base_of_v<typename FirstBaseDefaultPair::base, Policy> ||
check_policy_validity(
Policy{}, base_default_pair_list<RemainingBaseDefaultPairs...>{});
}
template <class BaseDefaultPairList>
constexpr bool check_policy_list_validity(BaseDefaultPairList) {
return true;
}
template <class BaseDefaultPairList, class FirstPolicy, class... RemainingPolicies>
constexpr bool check_policy_list_validity(BaseDefaultPairList, FirstPolicy,
RemainingPolicies... remaining_policies) {
return check_policy_validity(FirstPolicy{}, BaseDefaultPairList{}) &&
check_policy_list_validity(BaseDefaultPairList{}, remaining_policies...);
}
// Build policy_holder.
template <bool repeated_, class... FoundPolicyPairs>
struct found_policy_pair_list {
static constexpr bool repeated = repeated_;
};
template <class... Policies>
struct policy_holder : Policies... {};
template <bool repeated, class... FoundPolicyPairs, class... Policies>
constexpr auto
make_policy_holder_impl(base_default_pair_list<>,
found_policy_pair_list<repeated, FoundPolicyPairs...>,
Policies...) {
return found_policy_pair_list<repeated, FoundPolicyPairs...>{};
}
template <class FirstBaseDefaultPair, class... RemainingBaseDefaultPairs, bool repeated,
class... FoundPolicyPairs, class... Policies>
constexpr auto make_policy_holder_impl(
base_default_pair_list<FirstBaseDefaultPair, RemainingBaseDefaultPairs...>,
found_policy_pair_list<repeated, FoundPolicyPairs...>, Policies... policies) {
using new_found_policy_pair =
decltype(FirstBaseDefaultPair::get_policy(policies...));
return make_policy_holder_impl(
base_default_pair_list<RemainingBaseDefaultPairs...>{},
found_policy_pair_list < repeated ||
new_found_policy_pair::found_info == policy_found_info::repeated,
new_found_policy_pair, FoundPolicyPairs... > {}, policies...);
}
template <bool repeated, class... RawPolicies>
constexpr auto convert_to_policy_holder(found_policy_pair_list<repeated>,
RawPolicies...) {
return policy_holder<RawPolicies...>{};
}
template <bool repeated, class FirstFoundPolicyPair, class... RemainingFoundPolicyPairs,
class... RawPolicies>
constexpr auto
convert_to_policy_holder(found_policy_pair_list<repeated, FirstFoundPolicyPair,
RemainingFoundPolicyPairs...>,
RawPolicies... policies) {
return convert_to_policy_holder(
found_policy_pair_list<repeated, RemainingFoundPolicyPairs...>{},
typename FirstFoundPolicyPair::policy{}, policies...);
}
template <class BaseDefaultPairList, class... Policies>
constexpr auto make_policy_holder(BaseDefaultPairList, Policies... policies) {
static_assert(check_policy_list_validity(BaseDefaultPairList{}, Policies{}...),
"jkj::dragonbox: an invalid policy is specified");
using policy_pair_list = decltype(make_policy_holder_impl(
BaseDefaultPairList{}, found_policy_pair_list<false>{}, policies...));
static_assert(!policy_pair_list::repeated,
"jkj::dragonbox: each policy should be specified at most once");
return convert_to_policy_holder(policy_pair_list{});
}
}
}
////////////////////////////////////////////////////////////////////////////////////////
// The interface function.
////////////////////////////////////////////////////////////////////////////////////////
template <class Float, class FloatTraits = default_float_traits<Float>, class... Policies>
JKJ_FORCEINLINE JKJ_SAFEBUFFERS auto
to_decimal(signed_significand_bits<Float, FloatTraits> signed_significand_bits,
unsigned int exponent_bits, Policies... policies) noexcept {
// Build policy holder type.
using namespace detail::policy_impl;
using policy_holder = decltype(make_policy_holder(
base_default_pair_list<base_default_pair<sign::base, sign::return_sign>,
base_default_pair<trailing_zero::base, trailing_zero::remove>,
base_default_pair<decimal_to_binary_rounding::base,
decimal_to_binary_rounding::nearest_to_even>,
base_default_pair<binary_to_decimal_rounding::base,
binary_to_decimal_rounding::to_even>,
base_default_pair<cache::base, cache::full>>{},
policies...));
using return_type =
decimal_fp<typename FloatTraits::carrier_uint, policy_holder::return_has_sign,
policy_holder::report_trailing_zeros>;
return_type ret = policy_holder::delegate(
signed_significand_bits,
[exponent_bits, signed_significand_bits](auto interval_type_provider) {
using format = typename FloatTraits::format;
constexpr auto tag = decltype(interval_type_provider)::tag;
auto two_fc = signed_significand_bits.remove_sign_bit_and_shift();
auto exponent = int(exponent_bits);
if constexpr (tag == decimal_to_binary_rounding::tag_t::to_nearest) {
// Is the input a normal number?
if (exponent != 0) {
exponent += format::exponent_bias - format::significand_bits;
// Shorter interval case; proceed like Schubfach.
// One might think this condition is wrong, since when exponent_bits == 1
// and two_fc == 0, the interval is actually regular. However, it turns out
// that this seemingly wrong condition is actually fine, because the end
// result is anyway the same.
//
// [binary32]
// (fc-1/2) * 2^e = 1.175'494'28... * 10^-38
// (fc-1/4) * 2^e = 1.175'494'31... * 10^-38
// fc * 2^e = 1.175'494'35... * 10^-38
// (fc+1/2) * 2^e = 1.175'494'42... * 10^-38
//
// Hence, shorter_interval_case will return 1.175'494'4 * 10^-38.
// 1.175'494'3 * 10^-38 is also a correct shortest representation that will
// be rejected if we assume shorter interval, but 1.175'494'4 * 10^-38 is
// closer to the true value so it doesn't matter.
//
// [binary64]
// (fc-1/2) * 2^e = 2.225'073'858'507'201'13... * 10^-308
// (fc-1/4) * 2^e = 2.225'073'858'507'201'25... * 10^-308
// fc * 2^e = 2.225'073'858'507'201'38... * 10^-308
// (fc+1/2) * 2^e = 2.225'073'858'507'201'63... * 10^-308
//
// Hence, shorter_interval_case will return 2.225'073'858'507'201'4 *
// 10^-308. This is indeed of the shortest length, and it is the unique one
// closest to the true value among valid representations of the same length.
static_assert(std::is_same_v<format, ieee754_binary32> ||
std::is_same_v<format, ieee754_binary64>);
if (two_fc == 0) {
return decltype(interval_type_provider)::invoke_shorter_interval_case(
signed_significand_bits, [exponent](auto... additional_args) {
return detail::impl<Float, FloatTraits>::
template compute_nearest_shorter<
return_type,
typename decltype(interval_type_provider)::
shorter_interval_type,
typename policy_holder::trailing_zero_policy,
typename policy_holder::
binary_to_decimal_rounding_policy,
typename policy_holder::cache_policy>(
exponent, additional_args...);
});
}
two_fc |= (decltype(two_fc)(1) << (format::significand_bits + 1));
}
// Is the input a subnormal number?
else {
exponent = format::min_exponent - format::significand_bits;
}
return decltype(interval_type_provider)::invoke_normal_interval_case(
signed_significand_bits, [two_fc, exponent](auto... additional_args) {
return detail::impl<Float, FloatTraits>::
template compute_nearest_normal<
return_type,
typename decltype(interval_type_provider)::normal_interval_type,
typename policy_holder::trailing_zero_policy,
typename policy_holder::binary_to_decimal_rounding_policy,
typename policy_holder::cache_policy>(two_fc, exponent,
additional_args...);
});
}
else if constexpr (tag == decimal_to_binary_rounding::tag_t::left_closed_directed) {
// Is the input a normal number?
if (exponent != 0) {
exponent += format::exponent_bias - format::significand_bits;
two_fc |= (decltype(two_fc)(1) << (format::significand_bits + 1));
}
// Is the input a subnormal number?
else {
exponent = format::min_exponent - format::significand_bits;
}
return detail::impl<Float>::template compute_left_closed_directed<
return_type, typename policy_holder::trailing_zero_policy,
typename policy_holder::cache_policy>(two_fc, exponent);
}
else {
static_assert(tag == decimal_to_binary_rounding::tag_t::right_closed_directed);
bool shorter_interval = false;
// Is the input a normal number?
if (exponent != 0) {
if (two_fc == 0 && exponent != 1) {
shorter_interval = true;
}
exponent += format::exponent_bias - format::significand_bits;
two_fc |= (decltype(two_fc)(1) << (format::significand_bits + 1));
}
// Is the input a subnormal number?
else {
exponent = format::min_exponent - format::significand_bits;
}
return detail::impl<Float>::template compute_right_closed_directed<
return_type, typename policy_holder::trailing_zero_policy,
typename policy_holder::cache_policy>(two_fc, exponent, shorter_interval);
}
});
policy_holder::handle_sign(signed_significand_bits, ret);
return ret;
}
template <class Float, class FloatTraits = default_float_traits<Float>, class... Policies>
JKJ_FORCEINLINE JKJ_SAFEBUFFERS auto to_decimal(Float x, Policies... policies) noexcept {
auto const br = float_bits<Float, FloatTraits>(x);
auto const exponent_bits = br.extract_exponent_bits();
auto const s = br.remove_exponent_bits(exponent_bits);
assert(br.is_finite());
return to_decimal<Float, FloatTraits>(s, exponent_bits, policies...);
}
}
#undef JKJ_FORCEINLINE
#undef JKJ_SAFEBUFFERS
#undef JKJ_DRAGONBOX_HAS_BUILTIN
#endif
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