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Diffstat (limited to 'include/dragonbox')
| -rw-r--r-- | include/dragonbox/dragonbox.h | 4220 | ||||
| -rw-r--r-- | include/dragonbox/dragonbox_revision.txt | 8 |
2 files changed, 4228 insertions, 0 deletions
diff --git a/include/dragonbox/dragonbox.h b/include/dragonbox/dragonbox.h new file mode 100644 index 0000000..6a3671a --- /dev/null +++ b/include/dragonbox/dragonbox.h @@ -0,0 +1,4220 @@ +// Copyright 2020-2024 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. + +/* + * Modifications copyright 2024 Dominick Allen + * + * Modifications to the contents of this file may be used under the terms of + * the Apache License v2.0 with LLVM Exceptions. + * + * http://www.apache.org/licenses/LICENSE-2.0 + * + * Unless required by applicable law or agreed to in writing, software + * distributed under the License is distributed on an "AS IS" BASIS, + * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. + * See the License for the specific language governing permissions and + * limitations under the License. + */ + + +#ifndef JKJ_HEADER_DRAGONBOX +#define JKJ_HEADER_DRAGONBOX + +// Attribute for storing static data into a dedicated place, e.g. flash memory. Every ODR-used +// static data declaration will be decorated with this macro. The users may define this macro, +// before including the library headers, into whatever they want. +#ifndef JKJ_STATIC_DATA_SECTION + #define JKJ_STATIC_DATA_SECTION +#else + #define JKJ_STATIC_DATA_SECTION_DEFINED 1 +#endif + +// To use the library with toolchains without standard C++ headers, the users may define this macro +// into their custom namespace which contains the definitions of all the standard C++ library +// features used in this header. (The list can be found below.) +#ifndef JKJ_STD_REPLACEMENT_NAMESPACE + #define JKJ_STD_REPLACEMENT_NAMESPACE std + #include <cassert> + #include <cstdint> + #include <cstring> + #include <limits> + #include <type_traits> + + #ifdef __has_include + #if __has_include(<version>) + #include <version> + #endif + #endif +#else + #define JKJ_STD_REPLACEMENT_NAMESPACE_DEFINED 1 +#endif + +//////////////////////////////////////////////////////////////////////////////////////// +// Language feature detections. +//////////////////////////////////////////////////////////////////////////////////////// + +// C++14 constexpr +#if defined(__cpp_constexpr) && __cpp_constexpr >= 201304L + #define JKJ_HAS_CONSTEXPR14 1 +#elif __cplusplus >= 201402L + #define JKJ_HAS_CONSTEXPR14 1 +#elif defined(_MSC_VER) && _MSC_VER >= 1910 && _MSVC_LANG >= 201402L + #define JKJ_HAS_CONSTEXPR14 1 +#else + #define JKJ_HAS_CONSTEXPR14 0 +#endif + +#if JKJ_HAS_CONSTEXPR14 + #define JKJ_CONSTEXPR14 constexpr +#else + #define JKJ_CONSTEXPR14 +#endif + +// C++17 constexpr lambdas +#if defined(__cpp_constexpr) && __cpp_constexpr >= 201603L + #define JKJ_HAS_CONSTEXPR17 1 +#elif __cplusplus >= 201703L + #define JKJ_HAS_CONSTEXPR17 1 +#elif defined(_MSC_VER) && _MSC_VER >= 1911 && _MSVC_LANG >= 201703L + #define JKJ_HAS_CONSTEXPR17 1 +#else + #define JKJ_HAS_CONSTEXPR17 0 +#endif + +// C++17 inline variables +#if defined(__cpp_inline_variables) && __cpp_inline_variables >= 201606L + #define JKJ_HAS_INLINE_VARIABLE 1 +#elif __cplusplus >= 201703L + #define JKJ_HAS_INLINE_VARIABLE 1 +#elif defined(_MSC_VER) && _MSC_VER >= 1912 && _MSVC_LANG >= 201703L + #define JKJ_HAS_INLINE_VARIABLE 1 +#else + #define JKJ_HAS_INLINE_VARIABLE 0 +#endif + +#if JKJ_HAS_INLINE_VARIABLE + #define JKJ_INLINE_VARIABLE inline constexpr +#else + #define JKJ_INLINE_VARIABLE static constexpr +#endif + +// C++17 if constexpr +#if defined(__cpp_if_constexpr) && __cpp_if_constexpr >= 201606L + #define JKJ_HAS_IF_CONSTEXPR 1 +#elif __cplusplus >= 201703L + #define JKJ_HAS_IF_CONSTEXPR 1 +#elif defined(_MSC_VER) && _MSC_VER >= 1911 && _MSVC_LANG >= 201703L + #define JKJ_HAS_IF_CONSTEXPR 1 +#else + #define JKJ_HAS_IF_CONSTEXPR 0 +#endif + +#if JKJ_HAS_IF_CONSTEXPR + #define JKJ_IF_CONSTEXPR if constexpr +#else + #define JKJ_IF_CONSTEXPR if +#endif + +// C++20 std::bit_cast +#if JKJ_STD_REPLACEMENT_NAMESPACE_DEFINED + #if JKJ_STD_REPLACEMENT_HAS_BIT_CAST + #define JKJ_HAS_BIT_CAST 1 + #else + #define JKJ_HAS_BIT_CAST 0 + #endif +#elif defined(__cpp_lib_bit_cast) && __cpp_lib_bit_cast >= 201806L + #include <bit> + #define JKJ_HAS_BIT_CAST 1 +#else + #define JKJ_HAS_BIT_CAST 0 +#endif + +// C++23 if consteval or C++20 std::is_constant_evaluated +#if defined(__cpp_if_consteval) && __cpp_is_consteval >= 202106L + #define JKJ_IF_CONSTEVAL if consteval + #define JKJ_IF_NOT_CONSTEVAL if !consteval + #define JKJ_CAN_BRANCH_ON_CONSTEVAL 1 + #define JKJ_USE_IS_CONSTANT_EVALUATED 0 +#elif JKJ_STD_REPLACEMENT_NAMESPACE_DEFINED + #if JKJ_STD_REPLACEMENT_HAS_IS_CONSTANT_EVALUATED + #define JKJ_IF_CONSTEVAL if (stdr::is_constant_evaluated()) + #define JKJ_IF_NOT_CONSTEVAL if (!stdr::is_constant_evaluated()) + #define JKJ_CAN_BRANCH_ON_CONSTEVAL 1 + #define JKJ_USE_IS_CONSTANT_EVALUATED 1 + #elif JKJ_HAS_IF_CONSTEXPR + #define JKJ_IF_CONSTEVAL if constexpr (false) + #define JKJ_IF_NOT_CONSTEVAL if constexpr (true) + #define JKJ_CAN_BRANCH_ON_CONSTEVAL 0 + #define JKJ_USE_IS_CONSTANT_EVALUATED 0 + #else + #define JKJ_IF_CONSTEVAL if (false) + #define JKJ_IF_NOT_CONSTEVAL if (true) + #define JKJ_CAN_BRANCH_ON_CONSTEVAL 0 + #define JKJ_USE_IS_CONSTANT_EVALUATED 0 + #endif +#else + #if defined(__cpp_lib_is_constant_evaluated) && __cpp_lib_is_constant_evaluated >= 201811L + #define JKJ_IF_CONSTEVAL if (stdr::is_constant_evaluated()) + #define JKJ_IF_NOT_CONSTEVAL if (!stdr::is_constant_evaluated()) + #define JKJ_CAN_BRANCH_ON_CONSTEVAL 1 + #define JKJ_USE_IS_CONSTANT_EVALUATED 1 + #elif JKJ_HAS_IF_CONSTEXPR + #define JKJ_IF_CONSTEVAL if constexpr (false) + #define JKJ_IF_NOT_CONSTEVAL if constexpr (true) + #define JKJ_CAN_BRANCH_ON_CONSTEVAL 0 + #define JKJ_USE_IS_CONSTANT_EVALUATED 0 + #else + #define JKJ_IF_CONSTEVAL if (false) + #define JKJ_IF_NOT_CONSTEVAL if (true) + #define JKJ_CAN_BRANCH_ON_CONSTEVAL 0 + #define JKJ_USE_IS_CONSTANT_EVALUATED 0 + #endif +#endif + +#if JKJ_CAN_BRANCH_ON_CONSTEVAL && JKJ_HAS_BIT_CAST + #define JKJ_CONSTEXPR20 constexpr +#else + #define JKJ_CONSTEXPR20 +#endif + +// 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_HAS_BUILTIN(x) __has_builtin(x) +#else + #define JKJ_HAS_BUILTIN(x) false +#endif + +#if defined(_MSC_VER) + #include <intrin.h> +#elif defined(__INTEL_COMPILER) + #include <immintrin.h> +#endif + +namespace jkj { + namespace dragonbox { + //////////////////////////////////////////////////////////////////////////////////////// + // The Compatibility layer for toolchains without standard C++ headers. + //////////////////////////////////////////////////////////////////////////////////////// + namespace detail { + namespace stdr { + // <bit> +#if JKJ_HAS_BIT_CAST + using JKJ_STD_REPLACEMENT_NAMESPACE::bit_cast; +#endif + + // <cassert> + // We need assert() macro, but it is not namespaced anyway, so nothing to do here. + + // <cstdint> + using JKJ_STD_REPLACEMENT_NAMESPACE::int_least8_t; + using JKJ_STD_REPLACEMENT_NAMESPACE::int_least16_t; + using JKJ_STD_REPLACEMENT_NAMESPACE::int_least32_t; + using JKJ_STD_REPLACEMENT_NAMESPACE::int_fast8_t; + using JKJ_STD_REPLACEMENT_NAMESPACE::int_fast16_t; + using JKJ_STD_REPLACEMENT_NAMESPACE::int_fast32_t; + using JKJ_STD_REPLACEMENT_NAMESPACE::uint_least8_t; + using JKJ_STD_REPLACEMENT_NAMESPACE::uint_least16_t; + using JKJ_STD_REPLACEMENT_NAMESPACE::uint_least32_t; + using JKJ_STD_REPLACEMENT_NAMESPACE::uint_least64_t; + using JKJ_STD_REPLACEMENT_NAMESPACE::uint_fast8_t; + using JKJ_STD_REPLACEMENT_NAMESPACE::uint_fast16_t; + using JKJ_STD_REPLACEMENT_NAMESPACE::uint_fast32_t; + // We need INT32_C, UINT32_C and UINT64_C macros too, but again there is nothing to do + // here. + + // <cstring> + using JKJ_STD_REPLACEMENT_NAMESPACE::size_t; + using JKJ_STD_REPLACEMENT_NAMESPACE::memcpy; + + // <limits> + template <class T> + using numeric_limits = JKJ_STD_REPLACEMENT_NAMESPACE::numeric_limits<T>; + + // <type_traits> + template <bool cond, class T = void> + using enable_if = JKJ_STD_REPLACEMENT_NAMESPACE::enable_if<cond, T>; + template <class T> + using add_rvalue_reference = JKJ_STD_REPLACEMENT_NAMESPACE::add_rvalue_reference<T>; + template <bool cond, class T_true, class T_false> + using conditional = JKJ_STD_REPLACEMENT_NAMESPACE::conditional<cond, T_true, T_false>; +#if JKJ_USE_IS_CONSTANT_EVALUATED + using JKJ_STD_REPLACEMENT_NAMESPACE::is_constant_evaluated; +#endif + template <class T1, class T2> + using is_same = JKJ_STD_REPLACEMENT_NAMESPACE::is_same<T1, T2>; +#if !JKJ_HAS_BIT_CAST + template <class T> + using is_trivially_copyable = JKJ_STD_REPLACEMENT_NAMESPACE::is_trivially_copyable<T>; +#endif + template <class T> + using is_integral = JKJ_STD_REPLACEMENT_NAMESPACE::is_integral<T>; + template <class T> + using is_signed = JKJ_STD_REPLACEMENT_NAMESPACE::is_signed<T>; + template <class T> + using is_unsigned = JKJ_STD_REPLACEMENT_NAMESPACE::is_unsigned<T>; + } + } + + + //////////////////////////////////////////////////////////////////////////////////////// + // Some general utilities for C++11-compatibility. + //////////////////////////////////////////////////////////////////////////////////////// + namespace detail { +#if !JKJ_HAS_CONSTEXPR17 + template <stdr::size_t... indices> + struct index_sequence {}; + + template <stdr::size_t current, stdr::size_t total, class Dummy, stdr::size_t... indices> + struct make_index_sequence_impl { + using type = typename make_index_sequence_impl<current + 1, total, Dummy, indices..., + current>::type; + }; + + template <stdr::size_t total, class Dummy, stdr::size_t... indices> + struct make_index_sequence_impl<total, total, Dummy, indices...> { + using type = index_sequence<indices...>; + }; + + template <stdr::size_t N> + using make_index_sequence = typename make_index_sequence_impl<0, N, void>::type; +#endif + + // Available since C++11, but including <utility> just for this is an overkill. + template <class T> + typename stdr::add_rvalue_reference<T>::type declval() noexcept; + + // Similarly, including <array> is an overkill. + template <class T, stdr::size_t N> + struct array { + T data_[N]; + constexpr T operator[](stdr::size_t idx) const noexcept { return data_[idx]; } + JKJ_CONSTEXPR14 T& operator[](stdr::size_t idx) noexcept { return data_[idx]; } + }; + } + + + //////////////////////////////////////////////////////////////////////////////////////// + // Some basic features for encoding/decoding IEEE-754 formats. + //////////////////////////////////////////////////////////////////////////////////////// + namespace detail { + template <class T> + struct physical_bits { + static constexpr stdr::size_t value = + sizeof(T) * stdr::numeric_limits<unsigned char>::digits; + }; + template <class T> + struct value_bits { + static constexpr stdr::size_t value = stdr::numeric_limits< + typename stdr::enable_if<stdr::is_integral<T>::value, T>::type>::digits; + }; + + template <typename To, typename From> + JKJ_CONSTEXPR20 To bit_cast(const From& from) { +#if JKJ_HAS_BIT_CAST + return stdr::bit_cast<To>(from); +#else + static_assert(sizeof(From) == sizeof(To), ""); + static_assert(stdr::is_trivially_copyable<To>::value, ""); + static_assert(stdr::is_trivially_copyable<From>::value, ""); + To to; + stdr::memcpy(&to, &from, sizeof(To)); + return to; +#endif + } + } + + // These classes expose encoding specs of IEEE-754-like floating-point formats. + // Currently available formats are IEEE-754 binary32 & IEEE-754 binary64. + + struct ieee754_binary32 { + static constexpr int total_bits = 32; + 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_significand_digits = 9; + static constexpr int decimal_exponent_digits = 2; + }; + struct ieee754_binary64 { + static constexpr int total_bits = 64; + 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_significand_digits = 17; + static constexpr int decimal_exponent_digits = 3; + }; + + // A floating-point format traits class defines ways to interpret a bit pattern of given size as + // an encoding of floating-point number. This is an implementation of such a traits class, + // supporting ways to interpret IEEE-754 binary floating-point numbers. + template <class Format, class CarrierUInt, class ExponentInt = int> + struct ieee754_binary_traits { + // CarrierUInt needs to have enough size to hold the entire contents of floating-point + // numbers. The actual bits are assumed to be aligned to the LSB, and every other bits are + // assumed to be zeroed. + static_assert(detail::value_bits<CarrierUInt>::value >= Format::total_bits, + "jkj::dragonbox: insufficient number of bits"); + static_assert(detail::stdr::is_unsigned<CarrierUInt>::value, ""); + + // ExponentUInt needs to be large enough to hold (unsigned) exponent bits as well as the + // (signed) actual exponent. + // TODO: static overflow guard against intermediate computations. + static_assert(detail::value_bits<ExponentInt>::value >= Format::exponent_bits + 1, + "jkj::dragonbox: insufficient number of bits"); + static_assert(detail::stdr::is_signed<ExponentInt>::value, ""); + + using format = Format; + using carrier_uint = CarrierUInt; + static constexpr int carrier_bits = int(detail::value_bits<carrier_uint>::value); + using exponent_int = ExponentInt; + + // 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 exponent_int extract_exponent_bits(carrier_uint u) noexcept { + return exponent_int((u >> format::significand_bits) & + ((exponent_int(1) << format::exponent_bits) - 1)); + } + + // 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 { + return carrier_uint(u & ((carrier_uint(1) << format::significand_bits) - 1u)); + } + + // Remove the exponent bits and extract significand bits together with the sign bit. + static constexpr carrier_uint remove_exponent_bits(carrier_uint u) noexcept { + return carrier_uint(u & ~(((carrier_uint(1) << format::exponent_bits) - 1u) + << 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) & + ((((carrier_uint(1) << (Format::total_bits - 1)) - 1u) << 1) | 1u)); + } + + // Obtain the actual value of the binary exponent from the extracted exponent bits. + static constexpr exponent_int binary_exponent(exponent_int exponent_bits) noexcept { + return exponent_int(exponent_bits == 0 ? format::min_exponent + : exponent_bits + format::exponent_bias); + } + + // Obtain the actual value of the binary significand from the extracted significand bits + // and exponent bits. + static constexpr carrier_uint binary_significand(carrier_uint significand_bits, + exponent_int exponent_bits) noexcept { + return carrier_uint( + exponent_bits == 0 + ? significand_bits + : (significand_bits | (carrier_uint(1) << format::significand_bits))); + } + + /* Various boolean observer functions */ + + static constexpr bool is_nonzero(carrier_uint u) noexcept { + return (u & ((carrier_uint(1) << (format::significand_bits + format::exponent_bits)) - + 1u)) != 0; + } + static constexpr bool is_positive(carrier_uint u) noexcept { + return u < (carrier_uint(1) << (format::significand_bits + format::exponent_bits)); + } + static constexpr bool is_negative(carrier_uint u) noexcept { return !is_positive(u); } + static constexpr bool is_finite(exponent_int exponent_bits) noexcept { + return exponent_bits != ((exponent_int(1) << format::exponent_bits) - 1); + } + static constexpr bool has_all_zero_significand_bits(carrier_uint u) noexcept { + return ((u << 1) & + ((((carrier_uint(1) << (Format::total_bits - 1)) - 1u) << 1) | 1u)) == 0; + } + static constexpr bool has_even_significand_bits(carrier_uint u) noexcept { + return u % 2 == 0; + } + }; + + // Convert between bit patterns stored in carrier_uint and instances of an actual + // floating-point type. Depending on format and carrier_uint, this operation might not + // be possible for some specific bit patterns. However, the contract is that u always + // denotes a valid bit pattern, so the functions here are assumed to be noexcept. + // Users might specialize this class to change the behavior for certain types. + // The default provided by the library is to treat the given floating-point type Float as either + // IEEE-754 binary32 or IEEE-754 binary64, depending on the bitwise size of Float. + template <class Float> + struct default_float_bit_carrier_conversion_traits { + // Guards against types that have different internal representations than IEEE-754 + // binary32/64. I don't know if there is a truly reliable way of detecting IEEE-754 binary + // formats. I just did my best here. Note that in some cases + // numeric_limits<Float>::is_iec559 may report false even if the internal representation is + // IEEE-754 compatible. In such a case, the user can specialize this traits template and + // remove this static sanity check in order to make Dragonbox work for Float. + static_assert(detail::stdr::numeric_limits<Float>::is_iec559 && + detail::stdr::numeric_limits<Float>::radix == 2 && + (detail::physical_bits<Float>::value == 32 || + detail::physical_bits<Float>::value == 64), + "jkj::dragonbox: Float may not be of IEEE-754 binary32/binary64"); + + // Specifies the unsigned integer type to hold bitwise value of Float. + using carrier_uint = + typename detail::stdr::conditional<detail::physical_bits<Float>::value == 32, + detail::stdr::uint_least32_t, + detail::stdr::uint_least64_t>::type; + + // Specifies the floating-point format. + using format = typename detail::stdr::conditional<detail::physical_bits<Float>::value == 32, + ieee754_binary32, ieee754_binary64>::type; + + // Converts the floating-point type into the bit-carrier unsigned integer type. + static JKJ_CONSTEXPR20 carrier_uint float_to_carrier(Float x) noexcept { + return detail::bit_cast<carrier_uint>(x); + } + + // Converts the bit-carrier unsigned integer type into the floating-point type. + static JKJ_CONSTEXPR20 Float carrier_to_float(carrier_uint x) noexcept { + return detail::bit_cast<Float>(x); + } + }; + + // 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 FormatTraits> + struct signed_significand_bits { + using format_traits = FormatTraits; + using carrier_uint = typename format_traits::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 format_traits::remove_sign_bit_and_shift(u); + } + + constexpr bool is_positive() const noexcept { return format_traits::is_positive(u); } + constexpr bool is_negative() const noexcept { return format_traits::is_negative(u); } + constexpr bool has_all_zero_significand_bits() const noexcept { + return format_traits::has_all_zero_significand_bits(u); + } + constexpr bool has_even_significand_bits() const noexcept { + return format_traits::has_even_significand_bits(u); + } + }; + + template <class FormatTraits> + struct float_bits { + using format_traits = FormatTraits; + using carrier_uint = typename format_traits::carrier_uint; + using exponent_int = typename format_traits::exponent_int; + + carrier_uint u; + + float_bits() = default; + constexpr explicit float_bits(carrier_uint bit_pattern) noexcept : u{bit_pattern} {} + + // 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 exponent_int extract_exponent_bits() const noexcept { + return format_traits::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 format_traits::extract_significand_bits(u); + } + + // Remove the exponent bits and extract significand bits together with the sign bit. + constexpr signed_significand_bits<format_traits> remove_exponent_bits() const noexcept { + return signed_significand_bits<format_traits>(format_traits::remove_exponent_bits(u)); + } + + // Obtain the actual value of the binary exponent from the extracted exponent bits. + static constexpr exponent_int binary_exponent(exponent_int exponent_bits) noexcept { + return format_traits::binary_exponent(exponent_bits); + } + constexpr exponent_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, + exponent_int exponent_bits) noexcept { + return format_traits::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 format_traits::is_nonzero(u); } + constexpr bool is_positive() const noexcept { return format_traits::is_positive(u); } + constexpr bool is_negative() const noexcept { return format_traits::is_negative(u); } + constexpr bool is_finite(exponent_int exponent_bits) const noexcept { + return format_traits::is_finite(exponent_bits); + } + constexpr bool is_finite() const noexcept { + return format_traits::is_finite(extract_exponent_bits()); + } + constexpr bool has_even_significand_bits() const noexcept { + return format_traits::has_even_significand_bits(u); + } + }; + + template <class Float, + class ConversionTraits = default_float_bit_carrier_conversion_traits<Float>, + class FormatTraits = ieee754_binary_traits<typename ConversionTraits::format, + typename ConversionTraits::carrier_uint>> + JKJ_CONSTEXPR20 float_bits<FormatTraits> make_float_bits(Float x) noexcept { + return float_bits<FormatTraits>(ConversionTraits::float_to_carrier(x)); + } + + namespace detail { + //////////////////////////////////////////////////////////////////////////////////////// + // Bit operation intrinsics. + //////////////////////////////////////////////////////////////////////////////////////// + + namespace bits { + // Most compilers should be able to optimize this into the ROR instruction. + // n is assumed to be at most of bit_width bits. + template <stdr::size_t bit_width, class UInt> + JKJ_CONSTEXPR14 UInt rotr(UInt n, unsigned int r) noexcept { + static_assert(bit_width > 0, "jkj::dragonbox: rotation bit-width must be positive"); + static_assert(bit_width <= value_bits<UInt>::value, + "jkj::dragonbox: rotation bit-width is too large"); + r &= (bit_width - 1); + return (n >> r) | (n << ((bit_width - r) & (bit_width - 1))); + } + } + + //////////////////////////////////////////////////////////////////////////////////////// + // 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; + + stdr::uint_least64_t high_; + stdr::uint_least64_t low_; + + constexpr uint128(stdr::uint_least64_t high, stdr::uint_least64_t low) noexcept + : high_{high}, low_{low} {} + + constexpr stdr::uint_least64_t high() const noexcept { return high_; } + constexpr stdr::uint_least64_t low() const noexcept { return low_; } + + JKJ_CONSTEXPR20 uint128& operator+=(stdr::uint_least64_t n) & noexcept { + auto const generic_impl = [&] { + auto const sum = (low_ + n) & UINT64_C(0xffffffffffffffff); + high_ += (sum < low_ ? 1 : 0); + low_ = sum; + }; + // To suppress warning. + static_cast<void>(generic_impl); + + JKJ_IF_CONSTEXPR(value_bits<stdr::uint_least64_t>::value > 64) { + generic_impl(); + return *this; + } + + JKJ_IF_CONSTEVAL { + generic_impl(); + return *this; + } + + // See https://github.com/fmtlib/fmt/pull/2985. +#if JKJ_HAS_BUILTIN(__builtin_addcll) && !defined(__ibmxl__) + JKJ_IF_CONSTEXPR( + stdr::is_same<stdr::uint_least64_t, unsigned long long>::value) { + unsigned long long carry{}; + low_ = stdr::uint_least64_t(__builtin_addcll(low_, n, 0, &carry)); + high_ = stdr::uint_least64_t(__builtin_addcll(high_, 0, carry, &carry)); + return *this; + } +#endif +#if JKJ_HAS_BUILTIN(__builtin_addcl) && !defined(__ibmxl__) + JKJ_IF_CONSTEXPR(stdr::is_same<stdr::uint_least64_t, unsigned long>::value) { + unsigned long carry{}; + low_ = stdr::uint_least64_t( + __builtin_addcl(static_cast<unsigned long>(low_), + static_cast<unsigned long>(n), 0, &carry)); + high_ = stdr::uint_least64_t( + __builtin_addcl(static_cast<unsigned long>(high_), 0, carry, &carry)); + return *this; + } +#endif +#if JKJ_HAS_BUILTIN(__builtin_addc) && !defined(__ibmxl__) + JKJ_IF_CONSTEXPR(stdr::is_same<stdr::uint_least64_t, unsigned int>::value) { + unsigned int carry{}; + low_ = stdr::uint_least64_t(__builtin_addc(static_cast<unsigned int>(low_), + static_cast<unsigned int>(n), 0, + &carry)); + high_ = stdr::uint_least64_t( + __builtin_addc(static_cast<unsigned int>(high_), 0, carry, &carry)); + return *this; + } +#endif + +#if JKJ_HAS_BUILTIN(__builtin_ia32_addcarry_u64) + // __builtin_ia32_addcarry_u64 is not documented, but it seems it takes unsigned + // long long arguments. + unsigned long long result{}; + auto const carry = __builtin_ia32_addcarry_u64(0, low_, n, &result); + low_ = stdr::uint_least64_t(result); + __builtin_ia32_addcarry_u64(carry, high_, 0, &result); + high_ = stdr::uint_least64_t(result); +#elif defined(_MSC_VER) && defined(_M_X64) + // On MSVC, uint_least64_t and __int64 must be unsigned long long; see + // https://learn.microsoft.com/en-us/cpp/c-runtime-library/standard-types + // and https://learn.microsoft.com/en-us/cpp/cpp/int8-int16-int32-int64. + static_assert(stdr::is_same<unsigned long long, stdr::uint_least64_t>::value, + ""); + auto const carry = _addcarry_u64(0, low_, n, &low_); + _addcarry_u64(carry, high_, 0, &high_); +#elif defined(__INTEL_COMPILER) && (defined(_M_X64) || defined(__x86_64)) + // Cannot find any documentation on how things are defined, but hopefully this + // is always true... + static_assert(stdr::is_same<unsigned __int64, stdr::uint_least64_t>::value, ""); + auto const carry = _addcarry_u64(0, low_, n, &low_); + _addcarry_u64(carry, high_, 0, &high_); +#else + generic_impl(); +#endif + return *this; + } + }; + + inline JKJ_CONSTEXPR20 stdr::uint_least64_t umul64(stdr::uint_least32_t x, + stdr::uint_least32_t y) noexcept { +#if defined(_MSC_VER) && defined(_M_IX86) + JKJ_IF_NOT_CONSTEVAL { return __emulu(x, y); } +#endif + return x * stdr::uint_least64_t(y); + } + + // Get 128-bit result of multiplication of two 64-bit unsigned integers. + JKJ_SAFEBUFFERS inline JKJ_CONSTEXPR20 uint128 + umul128(stdr::uint_least64_t x, stdr::uint_least64_t y) noexcept { + auto const generic_impl = [=]() -> uint128 { + auto const a = stdr::uint_least32_t(x >> 32); + auto const b = stdr::uint_least32_t(x); + auto const c = stdr::uint_least32_t(y >> 32); + auto const d = stdr::uint_least32_t(y); + + auto const ac = umul64(a, c); + auto const bc = umul64(b, c); + auto const ad = umul64(a, d); + auto const bd = umul64(b, d); + + auto const intermediate = + (bd >> 32) + stdr::uint_least32_t(ad) + stdr::uint_least32_t(bc); + + return {ac + (intermediate >> 32) + (ad >> 32) + (bc >> 32), + (intermediate << 32) + stdr::uint_least32_t(bd)}; + }; + // To silence warning. + static_cast<void>(generic_impl); + +#if defined(__SIZEOF_INT128__) + auto const result = builtin_uint128_t(x) * builtin_uint128_t(y); + return {stdr::uint_least64_t(result >> 64), stdr::uint_least64_t(result)}; +#elif defined(_MSC_VER) && defined(_M_X64) + JKJ_IF_CONSTEVAL { + // This redundant variable is to workaround MSVC's codegen bug caused by the + // interaction of NRVO and intrinsics. + auto const result = generic_impl(); + return result; + } + uint128 result; + #if defined(__AVX2__) + result.low_ = _mulx_u64(x, y, &result.high_); + #else + result.low_ = _umul128(x, y, &result.high_); + #endif + return result; +#else + return generic_impl(); +#endif + } + + // Get high half of the 128-bit result of multiplication of two 64-bit unsigned + // integers. + JKJ_SAFEBUFFERS inline JKJ_CONSTEXPR20 stdr::uint_least64_t + umul128_upper64(stdr::uint_least64_t x, stdr::uint_least64_t y) noexcept { + auto const generic_impl = [=]() -> stdr::uint_least64_t { + auto const a = stdr::uint_least32_t(x >> 32); + auto const b = stdr::uint_least32_t(x); + auto const c = stdr::uint_least32_t(y >> 32); + auto const d = stdr::uint_least32_t(y); + + auto const ac = umul64(a, c); + auto const bc = umul64(b, c); + auto const ad = umul64(a, d); + auto const bd = umul64(b, d); + + auto const intermediate = + (bd >> 32) + stdr::uint_least32_t(ad) + stdr::uint_least32_t(bc); + + return ac + (intermediate >> 32) + (ad >> 32) + (bc >> 32); + }; + // To silence warning. + static_cast<void>(generic_impl); + +#if defined(__SIZEOF_INT128__) + auto const result = builtin_uint128_t(x) * builtin_uint128_t(y); + return stdr::uint_least64_t(result >> 64); +#elif defined(_MSC_VER) && defined(_M_X64) + JKJ_IF_CONSTEVAL { + // This redundant variable is to workaround MSVC's codegen bug caused by the + // interaction of NRVO and intrinsics. + auto const result = generic_impl(); + return result; + } + stdr::uint_least64_t result; + #if defined(__AVX2__) + _mulx_u64(x, y, &result); + #else + result = __umulh(x, y); + #endif + return result; +#else + return generic_impl(); +#endif + } + + // Get upper 128-bits of multiplication of a 64-bit unsigned integer and a 128-bit + // unsigned integer. + JKJ_SAFEBUFFERS inline JKJ_CONSTEXPR20 uint128 umul192_upper128(stdr::uint_least64_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 JKJ_CONSTEXPR20 stdr::uint_least64_t + umul96_upper64(stdr::uint_least32_t x, stdr::uint_least64_t y) noexcept { +#if defined(__SIZEOF_INT128__) || (defined(_MSC_VER) && defined(_M_X64)) + return umul128_upper64(stdr::uint_least64_t(x) << 32, y); +#else + auto const yh = stdr::uint_least32_t(y >> 32); + auto const yl = stdr::uint_least32_t(y); + + auto const xyh = umul64(x, yh); + auto const 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 JKJ_CONSTEXPR20 uint128 umul192_lower128(stdr::uint_least64_t x, + uint128 y) noexcept { + auto const high = x * y.high(); + auto const high_low = umul128(x, y.low()); + return {(high + high_low.high()) & UINT64_C(0xffffffffffffffff), high_low.low()}; + } + + // Get lower 64-bits of multiplication of a 32-bit unsigned integer and a 64-bit + // unsigned integer. + constexpr stdr::uint_least64_t umul96_lower64(stdr::uint_least32_t x, + stdr::uint_least64_t y) noexcept { + return (x * y) & UINT64_C(0xffffffffffffffff); + } + } + + //////////////////////////////////////////////////////////////////////////////////////// + // Some simple utilities for constexpr computation. + //////////////////////////////////////////////////////////////////////////////////////// + + template <int k, class Int> + constexpr Int compute_power(Int a) noexcept { + static_assert(k >= 0, ""); +#if JKJ_HAS_CONSTEXPR14 + Int p = 1; + for (int i = 0; i < k; ++i) { + p *= a; + } + return p; +#else + return k == 0 ? 1 + : k % 2 == 0 ? compute_power<k / 2, Int>(a * a) + : a * compute_power<k / 2, Int>(a * a); +#endif + } + + template <int a, class UInt> + constexpr int count_factors(UInt n) noexcept { + static_assert(a > 1, ""); +#if JKJ_HAS_CONSTEXPR14 + int c = 0; + while (n % a == 0) { + n /= a; + ++c; + } + return c; +#else + return n % a == 0 ? count_factors<a, UInt>(n / a) + 1 : 0; +#endif + } + + //////////////////////////////////////////////////////////////////////////////////////// + // Utilities for fast/constexpr log computation. + //////////////////////////////////////////////////////////////////////////////////////// + + namespace log { + static_assert((stdr::int_fast32_t(-1) >> 1) == stdr::int_fast32_t(-1) && + (stdr::int_fast16_t(-1) >> 1) == stdr::int_fast16_t(-1), + "jkj::dragonbox: right-shift for signed integers must be arithmetic"); + + // For constexpr computation. + // Returns -1 when n = 0. + template <class UInt> + constexpr int floor_log2(UInt n) noexcept { +#if JKJ_HAS_CONSTEXPR14 + int count = -1; + while (n != 0) { + ++count; + n >>= 1; + } + return count; +#else + return n == 0 ? -1 : floor_log2<UInt>(n / 2) + 1; +#endif + } + + template <template <stdr::size_t> class Info, stdr::int_least32_t min_exponent, + stdr::int_least32_t max_exponent, stdr::size_t current_tier, + stdr::int_least32_t supported_min_exponent = Info<current_tier>::min_exponent, + stdr::int_least32_t supported_max_exponent = Info<current_tier>::max_exponent> + constexpr bool is_in_range(int) noexcept { + return min_exponent >= supported_min_exponent && + max_exponent <= supported_max_exponent; + } + template <template <stdr::size_t> class Info, stdr::int_least32_t min_exponent, + stdr::int_least32_t max_exponent, stdr::size_t current_tier> + constexpr bool is_in_range(...) noexcept { + // Supposed to be always false, but formally dependent on the template parameters. + static_assert(min_exponent > max_exponent, + "jkj::dragonbox: exponent range is too wide"); + return false; + } + + template <template <stdr::size_t> class Info, stdr::int_least32_t min_exponent, + stdr::int_least32_t max_exponent, stdr::size_t current_tier = 0, + bool = is_in_range<Info, min_exponent, max_exponent, current_tier>(0)> + struct compute_impl; + + template <template <stdr::size_t> class Info, stdr::int_least32_t min_exponent, + stdr::int_least32_t max_exponent, stdr::size_t current_tier> + struct compute_impl<Info, min_exponent, max_exponent, current_tier, true> { + using info = Info<current_tier>; + using default_return_type = typename info::default_return_type; + template <class ReturnType, class Int> + static constexpr ReturnType compute(Int e) noexcept { +#if JKJ_HAS_CONSTEXPR14 + assert(min_exponent <= e && e <= max_exponent); +#endif + // The sign is irrelevant for the mathematical validity of the formula, but + // assuming positivity makes the overflow analysis simpler. + static_assert(info::multiply >= 0 && info::subtract >= 0, ""); + return static_cast<ReturnType>((e * info::multiply - info::subtract) >> + info::shift); + } + }; + + template <template <stdr::size_t> class Info, stdr::int_least32_t min_exponent, + stdr::int_least32_t max_exponent, stdr::size_t current_tier> + struct compute_impl<Info, min_exponent, max_exponent, current_tier, false> { + using next_tier = compute_impl<Info, min_exponent, max_exponent, current_tier + 1>; + using default_return_type = typename next_tier::default_return_type; + template <class ReturnType, class Int> + static constexpr ReturnType compute(Int e) noexcept { + return next_tier::template compute<ReturnType>(e); + } + }; + + template <stdr::size_t tier> + struct floor_log10_pow2_info; + template <> + struct floor_log10_pow2_info<0> { + using default_return_type = stdr::int_fast8_t; + static constexpr stdr::int_fast16_t multiply = 77; + static constexpr stdr::int_fast16_t subtract = 0; + static constexpr stdr::size_t shift = 8; + static constexpr stdr::int_least32_t min_exponent = -102; + static constexpr stdr::int_least32_t max_exponent = 102; + }; + template <> + struct floor_log10_pow2_info<1> { + using default_return_type = stdr::int_fast8_t; + // 24-bits are enough in fact. + static constexpr stdr::int_fast32_t multiply = 1233; + static constexpr stdr::int_fast32_t subtract = 0; + static constexpr stdr::size_t shift = 12; + // Formula itself holds on [-680,680]; [-425,425] is to ensure that the output is + // within [-127,127]. + static constexpr stdr::int_least32_t min_exponent = -425; + static constexpr stdr::int_least32_t max_exponent = 425; + }; + template <> + struct floor_log10_pow2_info<2> { + using default_return_type = stdr::int_fast16_t; + static constexpr stdr::int_fast32_t multiply = INT32_C(315653); + static constexpr stdr::int_fast32_t subtract = 0; + static constexpr stdr::size_t shift = 20; + static constexpr stdr::int_least32_t min_exponent = -2620; + static constexpr stdr::int_least32_t max_exponent = 2620; + }; + template <stdr::int_least32_t min_exponent = -2620, + stdr::int_least32_t max_exponent = 2620, + class ReturnType = typename compute_impl<floor_log10_pow2_info, min_exponent, + max_exponent>::default_return_type, + class Int> + constexpr ReturnType floor_log10_pow2(Int e) noexcept { + return compute_impl<floor_log10_pow2_info, min_exponent, + max_exponent>::template compute<ReturnType>(e); + } + + template <stdr::size_t tier> + struct floor_log2_pow10_info; + template <> + struct floor_log2_pow10_info<0> { + using default_return_type = stdr::int_fast8_t; + static constexpr stdr::int_fast16_t multiply = 53; + static constexpr stdr::int_fast16_t subtract = 0; + static constexpr stdr::size_t shift = 4; + static constexpr stdr::int_least32_t min_exponent = -15; + static constexpr stdr::int_least32_t max_exponent = 18; + }; + template <> + struct floor_log2_pow10_info<1> { + using default_return_type = stdr::int_fast16_t; + // 24-bits are enough in fact. + static constexpr stdr::int_fast32_t multiply = 1701; + static constexpr stdr::int_fast32_t subtract = 0; + static constexpr stdr::size_t shift = 9; + static constexpr stdr::int_least32_t min_exponent = -58; + static constexpr stdr::int_least32_t max_exponent = 58; + }; + template <> + struct floor_log2_pow10_info<2> { + using default_return_type = stdr::int_fast16_t; + static constexpr stdr::int_fast32_t multiply = INT32_C(1741647); + static constexpr stdr::int_fast32_t subtract = 0; + static constexpr stdr::size_t shift = 19; + // Formula itself holds on [-4003,4003]; [-1233,1233] is to ensure no overflow. + static constexpr stdr::int_least32_t min_exponent = -1233; + static constexpr stdr::int_least32_t max_exponent = 1233; + }; + template <stdr::int_least32_t min_exponent = -1233, + stdr::int_least32_t max_exponent = 1233, + class ReturnType = typename compute_impl<floor_log2_pow10_info, min_exponent, + max_exponent>::default_return_type, + class Int> + constexpr ReturnType floor_log2_pow10(Int e) noexcept { + return compute_impl<floor_log2_pow10_info, min_exponent, + max_exponent>::template compute<ReturnType>(e); + } + + template <stdr::size_t tier> + struct floor_log10_pow2_minus_log10_4_over_3_info; + template <> + struct floor_log10_pow2_minus_log10_4_over_3_info<0> { + using default_return_type = stdr::int_fast8_t; + static constexpr stdr::int_fast16_t multiply = 77; + static constexpr stdr::int_fast16_t subtract = 31; + static constexpr stdr::size_t shift = 8; + static constexpr stdr::int_least32_t min_exponent = -75; + static constexpr stdr::int_least32_t max_exponent = 129; + }; + template <> + struct floor_log10_pow2_minus_log10_4_over_3_info<1> { + using default_return_type = stdr::int_fast8_t; + // 24-bits are enough in fact. + static constexpr stdr::int_fast32_t multiply = 19728; + static constexpr stdr::int_fast32_t subtract = 8241; + static constexpr stdr::size_t shift = 16; + // Formula itself holds on [-849,315]; [-424,315] is to ensure that the output is + // within [-127,127]. + static constexpr stdr::int_least32_t min_exponent = -424; + static constexpr stdr::int_least32_t max_exponent = 315; + }; + template <> + struct floor_log10_pow2_minus_log10_4_over_3_info<2> { + using default_return_type = stdr::int_fast16_t; + static constexpr stdr::int_fast32_t multiply = INT32_C(631305); + static constexpr stdr::int_fast32_t subtract = INT32_C(261663); + static constexpr stdr::size_t shift = 21; + static constexpr stdr::int_least32_t min_exponent = -2985; + static constexpr stdr::int_least32_t max_exponent = 2936; + }; + template <stdr::int_least32_t min_exponent = -2985, + stdr::int_least32_t max_exponent = 2936, + class ReturnType = + typename compute_impl<floor_log10_pow2_minus_log10_4_over_3_info, + min_exponent, max_exponent>::default_return_type, + class Int> + constexpr ReturnType floor_log10_pow2_minus_log10_4_over_3(Int e) noexcept { + return compute_impl<floor_log10_pow2_minus_log10_4_over_3_info, min_exponent, + max_exponent>::template compute<ReturnType>(e); + } + + template <stdr::size_t tier> + struct floor_log5_pow2_info; + template <> + struct floor_log5_pow2_info<0> { + using default_return_type = stdr::int_fast32_t; + static constexpr stdr::int_fast32_t multiply = INT32_C(225799); + static constexpr stdr::int_fast32_t subtract = 0; + static constexpr stdr::size_t shift = 19; + static constexpr stdr::int_least32_t min_exponent = -1831; + static constexpr stdr::int_least32_t max_exponent = 1831; + }; + template <stdr::int_least32_t min_exponent = -1831, + stdr::int_least32_t max_exponent = 1831, + class ReturnType = typename compute_impl<floor_log5_pow2_info, min_exponent, + max_exponent>::default_return_type, + class Int> + constexpr ReturnType floor_log5_pow2(Int e) noexcept { + return compute_impl<floor_log5_pow2_info, min_exponent, + max_exponent>::template compute<ReturnType>(e); + } + + template <stdr::size_t tier> + struct floor_log5_pow2_minus_log5_3_info; + template <> + struct floor_log5_pow2_minus_log5_3_info<0> { + using default_return_type = stdr::int_fast32_t; + static constexpr stdr::int_fast32_t multiply = INT32_C(451597); + static constexpr stdr::int_fast32_t subtract = INT32_C(715764); + static constexpr stdr::size_t shift = 20; + static constexpr stdr::int_least32_t min_exponent = -3543; + static constexpr stdr::int_least32_t max_exponent = 2427; + }; + template <stdr::int_least32_t min_exponent = -3543, + stdr::int_least32_t max_exponent = 2427, + class ReturnType = + typename compute_impl<floor_log5_pow2_minus_log5_3_info, min_exponent, + max_exponent>::default_return_type, + class Int> + constexpr ReturnType floor_log5_pow2_minus_log5_3(Int e) noexcept { + return compute_impl<floor_log5_pow2_minus_log5_3_info, min_exponent, + max_exponent>::template compute<ReturnType>(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, class UInt> + struct divide_by_pow10_info; + + template <class UInt> + struct divide_by_pow10_info<1, UInt> { + static constexpr stdr::uint_fast32_t magic_number = 6554; + static constexpr int shift_amount = 16; + }; + + template <> + struct divide_by_pow10_info<1, stdr::uint_least8_t> { + static constexpr stdr::uint_fast16_t magic_number = 103; + static constexpr int shift_amount = 10; + }; + + template <> + struct divide_by_pow10_info<1, stdr::uint_least16_t> { + static constexpr stdr::uint_fast16_t magic_number = 103; + static constexpr int shift_amount = 10; + }; + + template <class UInt> + struct divide_by_pow10_info<2, UInt> { + static constexpr stdr::uint_fast32_t magic_number = 656; + static constexpr int shift_amount = 16; + }; + + template <> + struct divide_by_pow10_info<2, stdr::uint_least16_t> { + static constexpr stdr::uint_fast32_t magic_number = 41; + static constexpr int shift_amount = 12; + }; + + template <int N, class UInt> + JKJ_CONSTEXPR14 bool check_divisibility_and_divide_by_pow10(UInt& n) noexcept { + // Make sure the computation for max_n does not overflow. + static_assert(N + 1 <= log::floor_log10_pow2(int(value_bits<UInt>::value)), ""); + assert(n <= compute_power<N + 1>(UInt(10))); + + using info = divide_by_pow10_info<N, UInt>; + using intermediate_type = decltype(info::magic_number); + auto const prod = intermediate_type(n * info::magic_number); + + constexpr auto mask = + intermediate_type((intermediate_type(1) << info::shift_amount) - 1); + bool const result = ((prod & mask) < info::magic_number); + + n = UInt(prod >> info::shift_amount); + return result; + } + + // Compute floor(n / 10^N) for small n and N. + // Precondition: n <= 10^(N+1) + template <int N, class UInt> + JKJ_CONSTEXPR14 UInt small_division_by_pow10(UInt n) noexcept { + // Make sure the computation for max_n does not overflow. + static_assert(N + 1 <= log::floor_log10_pow2(int(value_bits<UInt>::value)), ""); + assert(n <= compute_power<N + 1>(UInt(10))); + + return UInt((n * divide_by_pow10_info<N, UInt>::magic_number) >> + divide_by_pow10_info<N, UInt>::shift_amount); + } + + // Compute floor(n / 10^N) for small N. + // Precondition: n <= n_max + template <int N, class UInt, UInt n_max> + JKJ_CONSTEXPR20 UInt divide_by_pow10(UInt n) noexcept { + static_assert(N >= 0, ""); + + // Specialize for 32-bit division by 10. + // Without the bound on n_max (which compilers these days never leverage), the + // minimum needed amount of shift is larger than 32. Hence, this may generate better + // code for 32-bit or smaller architectures. Even for 64-bit architectures, it seems + // compilers tend to generate mov + mul instead of a single imul for an unknown + // reason if we just write n / 10. + JKJ_IF_CONSTEXPR(stdr::is_same<UInt, stdr::uint_least32_t>::value && N == 1 && + n_max <= UINT32_C(1073741828)) { + return UInt(wuint::umul64(n, UINT32_C(429496730)) >> 32); + } + // Specialize for 64-bit division by 10. + // Without the bound on n_max (which compilers these days never leverage), the + // minimum needed amount of shift is larger than 64. + else JKJ_IF_CONSTEXPR(stdr::is_same<UInt, stdr::uint_least64_t>::value && N == 1 && + n_max <= UINT64_C(4611686018427387908)) { + return UInt(wuint::umul128_upper64(n, UINT64_C(1844674407370955162))); + } + // Specialize for 32-bit division by 100. + // It seems compilers tend to generate mov + mul instead of a single imul for an + // unknown reason if we just write n / 100. + else JKJ_IF_CONSTEXPR(stdr::is_same<UInt, stdr::uint_least32_t>::value && N == 2) { + return UInt(wuint::umul64(n, UINT32_C(1374389535)) >> 37); + } + // Specialize for 64-bit division by 1000. + // Without the bound on n_max (which compilers these days never leverage), the + // smallest magic number for this computation does not fit into 64-bits. + else JKJ_IF_CONSTEXPR(stdr::is_same<UInt, stdr::uint_least64_t>::value && N == 3 && + n_max <= UINT64_C(15534100272597517998)) { + return UInt(wuint::umul128_upper64(n, UINT64_C(4722366482869645214)) >> 8); + } + else { + constexpr auto divisor = compute_power<N>(UInt(10)); + return n / divisor; + } + } + } + } + + //////////////////////////////////////////////////////////////////////////////////////// + // Return types for the main interface function. + //////////////////////////////////////////////////////////////////////////////////////// + + template <class SignificandType, class ExponentType, bool is_signed, bool trailing_zero_flag> + struct decimal_fp; + + template <class SignificandType, class ExponentType> + struct decimal_fp<SignificandType, ExponentType, false, false> { + SignificandType significand; + ExponentType exponent; + }; + + template <class SignificandType, class ExponentType> + struct decimal_fp<SignificandType, ExponentType, true, false> { + SignificandType significand; + ExponentType exponent; + bool is_negative; + }; + + template <class SignificandType, class ExponentType> + struct decimal_fp<SignificandType, ExponentType, false, true> { + SignificandType significand; + ExponentType exponent; + bool may_have_trailing_zeros; + }; + + template <class SignificandType, class ExponentType> + struct decimal_fp<SignificandType, ExponentType, true, true> { + SignificandType significand; + ExponentType exponent; + bool may_have_trailing_zeros; + bool is_negative; + }; + + template <class SignificandType, class ExponentType, bool trailing_zero_flag = false> + using unsigned_decimal_fp = + decimal_fp<SignificandType, ExponentType, false, trailing_zero_flag>; + + template <class SignificandType, class ExponentType, bool trailing_zero_flag = false> + using signed_decimal_fp = decimal_fp<SignificandType, ExponentType, true, trailing_zero_flag>; + + template <class SignificandType, class ExponentType> + constexpr signed_decimal_fp<SignificandType, ExponentType, false> + add_sign_to_unsigned_decimal_fp( + bool is_negative, unsigned_decimal_fp<SignificandType, ExponentType, false> r) noexcept { + return {r.significand, r.exponent, is_negative}; + } + + template <class SignificandType, class ExponentType> + constexpr signed_decimal_fp<SignificandType, ExponentType, true> + add_sign_to_unsigned_decimal_fp( + bool is_negative, unsigned_decimal_fp<SignificandType, ExponentType, true> r) noexcept { + return {r.significand, r.exponent, r.may_have_trailing_zeros, is_negative}; + } + + namespace detail { + template <class UnsignedDecimalFp> + struct unsigned_decimal_fp_to_signed; + + template <class SignificandType, class ExponentType, bool trailing_zero_flag> + struct unsigned_decimal_fp_to_signed< + unsigned_decimal_fp<SignificandType, ExponentType, trailing_zero_flag>> { + using type = signed_decimal_fp<SignificandType, ExponentType, trailing_zero_flag>; + }; + + template <class UnsignedDecimalFp> + using unsigned_decimal_fp_to_signed_t = + typename unsigned_decimal_fp_to_signed<UnsignedDecimalFp>::type; + } + + + //////////////////////////////////////////////////////////////////////////////////////// + // Computed cache entries. + //////////////////////////////////////////////////////////////////////////////////////// + + template <class FloatFormat, class Dummy = void> + struct cache_holder; + + template <class Dummy> + struct cache_holder<ieee754_binary32, Dummy> { + using cache_entry_type = detail::stdr::uint_least64_t; + static constexpr int cache_bits = 64; + static constexpr int min_k = -31; + static constexpr int max_k = 46; + static constexpr detail::array<cache_entry_type, detail::stdr::size_t(max_k - min_k + 1)> + cache JKJ_STATIC_DATA_SECTION = { + {UINT64_C(0x81ceb32c4b43fcf5), UINT64_C(0xa2425ff75e14fc32), + UINT64_C(0xcad2f7f5359a3b3f), UINT64_C(0xfd87b5f28300ca0e), + UINT64_C(0x9e74d1b791e07e49), UINT64_C(0xc612062576589ddb), + UINT64_C(0xf79687aed3eec552), UINT64_C(0x9abe14cd44753b53), + UINT64_C(0xc16d9a0095928a28), UINT64_C(0xf1c90080baf72cb2), + UINT64_C(0x971da05074da7bef), UINT64_C(0xbce5086492111aeb), + UINT64_C(0xec1e4a7db69561a6), UINT64_C(0x9392ee8e921d5d08), + UINT64_C(0xb877aa3236a4b44a), UINT64_C(0xe69594bec44de15c), + UINT64_C(0x901d7cf73ab0acda), UINT64_C(0xb424dc35095cd810), + UINT64_C(0xe12e13424bb40e14), UINT64_C(0x8cbccc096f5088cc), + UINT64_C(0xafebff0bcb24aaff), UINT64_C(0xdbe6fecebdedd5bf), + UINT64_C(0x89705f4136b4a598), UINT64_C(0xabcc77118461cefd), + UINT64_C(0xd6bf94d5e57a42bd), UINT64_C(0x8637bd05af6c69b6), + UINT64_C(0xa7c5ac471b478424), UINT64_C(0xd1b71758e219652c), + UINT64_C(0x83126e978d4fdf3c), UINT64_C(0xa3d70a3d70a3d70b), + UINT64_C(0xcccccccccccccccd), UINT64_C(0x8000000000000000), + UINT64_C(0xa000000000000000), UINT64_C(0xc800000000000000), + UINT64_C(0xfa00000000000000), UINT64_C(0x9c40000000000000), + UINT64_C(0xc350000000000000), UINT64_C(0xf424000000000000), + UINT64_C(0x9896800000000000), UINT64_C(0xbebc200000000000), + UINT64_C(0xee6b280000000000), UINT64_C(0x9502f90000000000), + UINT64_C(0xba43b74000000000), UINT64_C(0xe8d4a51000000000), + UINT64_C(0x9184e72a00000000), UINT64_C(0xb5e620f480000000), + UINT64_C(0xe35fa931a0000000), UINT64_C(0x8e1bc9bf04000000), + UINT64_C(0xb1a2bc2ec5000000), UINT64_C(0xde0b6b3a76400000), + UINT64_C(0x8ac7230489e80000), UINT64_C(0xad78ebc5ac620000), + UINT64_C(0xd8d726b7177a8000), UINT64_C(0x878678326eac9000), + UINT64_C(0xa968163f0a57b400), UINT64_C(0xd3c21bcecceda100), + UINT64_C(0x84595161401484a0), UINT64_C(0xa56fa5b99019a5c8), + UINT64_C(0xcecb8f27f4200f3a), UINT64_C(0x813f3978f8940985), + UINT64_C(0xa18f07d736b90be6), UINT64_C(0xc9f2c9cd04674edf), + UINT64_C(0xfc6f7c4045812297), UINT64_C(0x9dc5ada82b70b59e), + UINT64_C(0xc5371912364ce306), UINT64_C(0xf684df56c3e01bc7), + UINT64_C(0x9a130b963a6c115d), UINT64_C(0xc097ce7bc90715b4), + UINT64_C(0xf0bdc21abb48db21), UINT64_C(0x96769950b50d88f5), + UINT64_C(0xbc143fa4e250eb32), UINT64_C(0xeb194f8e1ae525fe), + UINT64_C(0x92efd1b8d0cf37bf), UINT64_C(0xb7abc627050305ae), + UINT64_C(0xe596b7b0c643c71a), UINT64_C(0x8f7e32ce7bea5c70), + UINT64_C(0xb35dbf821ae4f38c), UINT64_C(0xe0352f62a19e306f)}}; + }; +#if !JKJ_HAS_INLINE_VARIABLE + // decltype(...) should not depend on Dummy; see + // https://stackoverflow.com/questions/76438400/decltype-on-static-variable-in-template-class. + template <class Dummy> + constexpr decltype(cache_holder<ieee754_binary32>::cache) + cache_holder<ieee754_binary32, Dummy>::cache; +#endif + + template <class Dummy> + struct cache_holder<ieee754_binary64, Dummy> { + using cache_entry_type = detail::wuint::uint128; + static constexpr int cache_bits = 128; + static constexpr int min_k = -292; + static constexpr int max_k = 326; + static constexpr detail::array<cache_entry_type, detail::stdr::size_t(max_k - min_k + 1)> + cache JKJ_STATIC_DATA_SECTION = { + {{UINT64_C(0xff77b1fcbebcdc4f), UINT64_C(0x25e8e89c13bb0f7b)}, + {UINT64_C(0x9faacf3df73609b1), UINT64_C(0x77b191618c54e9ad)}, + {UINT64_C(0xc795830d75038c1d), UINT64_C(0xd59df5b9ef6a2418)}, + {UINT64_C(0xf97ae3d0d2446f25), UINT64_C(0x4b0573286b44ad1e)}, + {UINT64_C(0x9becce62836ac577), UINT64_C(0x4ee367f9430aec33)}, + {UINT64_C(0xc2e801fb244576d5), UINT64_C(0x229c41f793cda740)}, + {UINT64_C(0xf3a20279ed56d48a), UINT64_C(0x6b43527578c11110)}, + {UINT64_C(0x9845418c345644d6), UINT64_C(0x830a13896b78aaaa)}, + {UINT64_C(0xbe5691ef416bd60c), UINT64_C(0x23cc986bc656d554)}, + {UINT64_C(0xedec366b11c6cb8f), UINT64_C(0x2cbfbe86b7ec8aa9)}, + {UINT64_C(0x94b3a202eb1c3f39), UINT64_C(0x7bf7d71432f3d6aa)}, + {UINT64_C(0xb9e08a83a5e34f07), UINT64_C(0xdaf5ccd93fb0cc54)}, + {UINT64_C(0xe858ad248f5c22c9), UINT64_C(0xd1b3400f8f9cff69)}, + {UINT64_C(0x91376c36d99995be), UINT64_C(0x23100809b9c21fa2)}, + {UINT64_C(0xb58547448ffffb2d), UINT64_C(0xabd40a0c2832a78b)}, + {UINT64_C(0xe2e69915b3fff9f9), UINT64_C(0x16c90c8f323f516d)}, + {UINT64_C(0x8dd01fad907ffc3b), UINT64_C(0xae3da7d97f6792e4)}, + {UINT64_C(0xb1442798f49ffb4a), UINT64_C(0x99cd11cfdf41779d)}, + {UINT64_C(0xdd95317f31c7fa1d), UINT64_C(0x40405643d711d584)}, + {UINT64_C(0x8a7d3eef7f1cfc52), UINT64_C(0x482835ea666b2573)}, + {UINT64_C(0xad1c8eab5ee43b66), UINT64_C(0xda3243650005eed0)}, + {UINT64_C(0xd863b256369d4a40), UINT64_C(0x90bed43e40076a83)}, + {UINT64_C(0x873e4f75e2224e68), UINT64_C(0x5a7744a6e804a292)}, + {UINT64_C(0xa90de3535aaae202), UINT64_C(0x711515d0a205cb37)}, + {UINT64_C(0xd3515c2831559a83), UINT64_C(0x0d5a5b44ca873e04)}, + {UINT64_C(0x8412d9991ed58091), UINT64_C(0xe858790afe9486c3)}, + {UINT64_C(0xa5178fff668ae0b6), UINT64_C(0x626e974dbe39a873)}, + {UINT64_C(0xce5d73ff402d98e3), UINT64_C(0xfb0a3d212dc81290)}, + {UINT64_C(0x80fa687f881c7f8e), UINT64_C(0x7ce66634bc9d0b9a)}, + {UINT64_C(0xa139029f6a239f72), UINT64_C(0x1c1fffc1ebc44e81)}, + {UINT64_C(0xc987434744ac874e), UINT64_C(0xa327ffb266b56221)}, + {UINT64_C(0xfbe9141915d7a922), UINT64_C(0x4bf1ff9f0062baa9)}, + {UINT64_C(0x9d71ac8fada6c9b5), UINT64_C(0x6f773fc3603db4aa)}, + {UINT64_C(0xc4ce17b399107c22), UINT64_C(0xcb550fb4384d21d4)}, + {UINT64_C(0xf6019da07f549b2b), UINT64_C(0x7e2a53a146606a49)}, + {UINT64_C(0x99c102844f94e0fb), UINT64_C(0x2eda7444cbfc426e)}, + {UINT64_C(0xc0314325637a1939), UINT64_C(0xfa911155fefb5309)}, + {UINT64_C(0xf03d93eebc589f88), UINT64_C(0x793555ab7eba27cb)}, + {UINT64_C(0x96267c7535b763b5), UINT64_C(0x4bc1558b2f3458df)}, + {UINT64_C(0xbbb01b9283253ca2), UINT64_C(0x9eb1aaedfb016f17)}, + {UINT64_C(0xea9c227723ee8bcb), UINT64_C(0x465e15a979c1cadd)}, + {UINT64_C(0x92a1958a7675175f), UINT64_C(0x0bfacd89ec191eca)}, + {UINT64_C(0xb749faed14125d36), UINT64_C(0xcef980ec671f667c)}, + {UINT64_C(0xe51c79a85916f484), UINT64_C(0x82b7e12780e7401b)}, + {UINT64_C(0x8f31cc0937ae58d2), UINT64_C(0xd1b2ecb8b0908811)}, + {UINT64_C(0xb2fe3f0b8599ef07), UINT64_C(0x861fa7e6dcb4aa16)}, + {UINT64_C(0xdfbdcece67006ac9), UINT64_C(0x67a791e093e1d49b)}, + {UINT64_C(0x8bd6a141006042bd), UINT64_C(0xe0c8bb2c5c6d24e1)}, + {UINT64_C(0xaecc49914078536d), UINT64_C(0x58fae9f773886e19)}, + {UINT64_C(0xda7f5bf590966848), UINT64_C(0xaf39a475506a899f)}, + {UINT64_C(0x888f99797a5e012d), UINT64_C(0x6d8406c952429604)}, + {UINT64_C(0xaab37fd7d8f58178), UINT64_C(0xc8e5087ba6d33b84)}, + {UINT64_C(0xd5605fcdcf32e1d6), UINT64_C(0xfb1e4a9a90880a65)}, + {UINT64_C(0x855c3be0a17fcd26), UINT64_C(0x5cf2eea09a550680)}, + {UINT64_C(0xa6b34ad8c9dfc06f), UINT64_C(0xf42faa48c0ea481f)}, + {UINT64_C(0xd0601d8efc57b08b), UINT64_C(0xf13b94daf124da27)}, + {UINT64_C(0x823c12795db6ce57), UINT64_C(0x76c53d08d6b70859)}, + {UINT64_C(0xa2cb1717b52481ed), UINT64_C(0x54768c4b0c64ca6f)}, + {UINT64_C(0xcb7ddcdda26da268), UINT64_C(0xa9942f5dcf7dfd0a)}, + {UINT64_C(0xfe5d54150b090b02), UINT64_C(0xd3f93b35435d7c4d)}, + {UINT64_C(0x9efa548d26e5a6e1), UINT64_C(0xc47bc5014a1a6db0)}, + {UINT64_C(0xc6b8e9b0709f109a), UINT64_C(0x359ab6419ca1091c)}, + {UINT64_C(0xf867241c8cc6d4c0), UINT64_C(0xc30163d203c94b63)}, + {UINT64_C(0x9b407691d7fc44f8), UINT64_C(0x79e0de63425dcf1e)}, + {UINT64_C(0xc21094364dfb5636), UINT64_C(0x985915fc12f542e5)}, + {UINT64_C(0xf294b943e17a2bc4), UINT64_C(0x3e6f5b7b17b2939e)}, + {UINT64_C(0x979cf3ca6cec5b5a), UINT64_C(0xa705992ceecf9c43)}, + {UINT64_C(0xbd8430bd08277231), UINT64_C(0x50c6ff782a838354)}, + {UINT64_C(0xece53cec4a314ebd), UINT64_C(0xa4f8bf5635246429)}, + {UINT64_C(0x940f4613ae5ed136), UINT64_C(0x871b7795e136be9a)}, + {UINT64_C(0xb913179899f68584), UINT64_C(0x28e2557b59846e40)}, + {UINT64_C(0xe757dd7ec07426e5), UINT64_C(0x331aeada2fe589d0)}, + {UINT64_C(0x9096ea6f3848984f), UINT64_C(0x3ff0d2c85def7622)}, + {UINT64_C(0xb4bca50b065abe63), UINT64_C(0x0fed077a756b53aa)}, + {UINT64_C(0xe1ebce4dc7f16dfb), UINT64_C(0xd3e8495912c62895)}, + {UINT64_C(0x8d3360f09cf6e4bd), UINT64_C(0x64712dd7abbbd95d)}, + {UINT64_C(0xb080392cc4349dec), UINT64_C(0xbd8d794d96aacfb4)}, + {UINT64_C(0xdca04777f541c567), UINT64_C(0xecf0d7a0fc5583a1)}, + {UINT64_C(0x89e42caaf9491b60), UINT64_C(0xf41686c49db57245)}, + {UINT64_C(0xac5d37d5b79b6239), UINT64_C(0x311c2875c522ced6)}, + {UINT64_C(0xd77485cb25823ac7), UINT64_C(0x7d633293366b828c)}, + {UINT64_C(0x86a8d39ef77164bc), UINT64_C(0xae5dff9c02033198)}, + {UINT64_C(0xa8530886b54dbdeb), UINT64_C(0xd9f57f830283fdfd)}, + {UINT64_C(0xd267caa862a12d66), UINT64_C(0xd072df63c324fd7c)}, + {UINT64_C(0x8380dea93da4bc60), UINT64_C(0x4247cb9e59f71e6e)}, + {UINT64_C(0xa46116538d0deb78), UINT64_C(0x52d9be85f074e609)}, + {UINT64_C(0xcd795be870516656), UINT64_C(0x67902e276c921f8c)}, + {UINT64_C(0x806bd9714632dff6), UINT64_C(0x00ba1cd8a3db53b7)}, + {UINT64_C(0xa086cfcd97bf97f3), UINT64_C(0x80e8a40eccd228a5)}, + {UINT64_C(0xc8a883c0fdaf7df0), UINT64_C(0x6122cd128006b2ce)}, + {UINT64_C(0xfad2a4b13d1b5d6c), UINT64_C(0x796b805720085f82)}, + {UINT64_C(0x9cc3a6eec6311a63), UINT64_C(0xcbe3303674053bb1)}, + {UINT64_C(0xc3f490aa77bd60fc), UINT64_C(0xbedbfc4411068a9d)}, + {UINT64_C(0xf4f1b4d515acb93b), UINT64_C(0xee92fb5515482d45)}, + {UINT64_C(0x991711052d8bf3c5), UINT64_C(0x751bdd152d4d1c4b)}, + {UINT64_C(0xbf5cd54678eef0b6), UINT64_C(0xd262d45a78a0635e)}, + {UINT64_C(0xef340a98172aace4), UINT64_C(0x86fb897116c87c35)}, + {UINT64_C(0x9580869f0e7aac0e), UINT64_C(0xd45d35e6ae3d4da1)}, + {UINT64_C(0xbae0a846d2195712), UINT64_C(0x8974836059cca10a)}, + {UINT64_C(0xe998d258869facd7), UINT64_C(0x2bd1a438703fc94c)}, + {UINT64_C(0x91ff83775423cc06), UINT64_C(0x7b6306a34627ddd0)}, + {UINT64_C(0xb67f6455292cbf08), UINT64_C(0x1a3bc84c17b1d543)}, + {UINT64_C(0xe41f3d6a7377eeca), UINT64_C(0x20caba5f1d9e4a94)}, + {UINT64_C(0x8e938662882af53e), UINT64_C(0x547eb47b7282ee9d)}, + {UINT64_C(0xb23867fb2a35b28d), UINT64_C(0xe99e619a4f23aa44)}, + {UINT64_C(0xdec681f9f4c31f31), UINT64_C(0x6405fa00e2ec94d5)}, + {UINT64_C(0x8b3c113c38f9f37e), UINT64_C(0xde83bc408dd3dd05)}, + {UINT64_C(0xae0b158b4738705e), UINT64_C(0x9624ab50b148d446)}, + {UINT64_C(0xd98ddaee19068c76), UINT64_C(0x3badd624dd9b0958)}, + {UINT64_C(0x87f8a8d4cfa417c9), UINT64_C(0xe54ca5d70a80e5d7)}, + {UINT64_C(0xa9f6d30a038d1dbc), UINT64_C(0x5e9fcf4ccd211f4d)}, + {UINT64_C(0xd47487cc8470652b), UINT64_C(0x7647c32000696720)}, + {UINT64_C(0x84c8d4dfd2c63f3b), UINT64_C(0x29ecd9f40041e074)}, + {UINT64_C(0xa5fb0a17c777cf09), UINT64_C(0xf468107100525891)}, + {UINT64_C(0xcf79cc9db955c2cc), UINT64_C(0x7182148d4066eeb5)}, + {UINT64_C(0x81ac1fe293d599bf), UINT64_C(0xc6f14cd848405531)}, + {UINT64_C(0xa21727db38cb002f), UINT64_C(0xb8ada00e5a506a7d)}, + {UINT64_C(0xca9cf1d206fdc03b), UINT64_C(0xa6d90811f0e4851d)}, + {UINT64_C(0xfd442e4688bd304a), UINT64_C(0x908f4a166d1da664)}, + {UINT64_C(0x9e4a9cec15763e2e), UINT64_C(0x9a598e4e043287ff)}, + {UINT64_C(0xc5dd44271ad3cdba), UINT64_C(0x40eff1e1853f29fe)}, + {UINT64_C(0xf7549530e188c128), UINT64_C(0xd12bee59e68ef47d)}, + {UINT64_C(0x9a94dd3e8cf578b9), UINT64_C(0x82bb74f8301958cf)}, + {UINT64_C(0xc13a148e3032d6e7), UINT64_C(0xe36a52363c1faf02)}, + {UINT64_C(0xf18899b1bc3f8ca1), UINT64_C(0xdc44e6c3cb279ac2)}, + {UINT64_C(0x96f5600f15a7b7e5), UINT64_C(0x29ab103a5ef8c0ba)}, + {UINT64_C(0xbcb2b812db11a5de), UINT64_C(0x7415d448f6b6f0e8)}, + {UINT64_C(0xebdf661791d60f56), UINT64_C(0x111b495b3464ad22)}, + {UINT64_C(0x936b9fcebb25c995), UINT64_C(0xcab10dd900beec35)}, + {UINT64_C(0xb84687c269ef3bfb), UINT64_C(0x3d5d514f40eea743)}, + {UINT64_C(0xe65829b3046b0afa), UINT64_C(0x0cb4a5a3112a5113)}, + {UINT64_C(0x8ff71a0fe2c2e6dc), UINT64_C(0x47f0e785eaba72ac)}, + {UINT64_C(0xb3f4e093db73a093), UINT64_C(0x59ed216765690f57)}, + {UINT64_C(0xe0f218b8d25088b8), UINT64_C(0x306869c13ec3532d)}, + {UINT64_C(0x8c974f7383725573), UINT64_C(0x1e414218c73a13fc)}, + {UINT64_C(0xafbd2350644eeacf), UINT64_C(0xe5d1929ef90898fb)}, + {UINT64_C(0xdbac6c247d62a583), UINT64_C(0xdf45f746b74abf3a)}, + {UINT64_C(0x894bc396ce5da772), UINT64_C(0x6b8bba8c328eb784)}, + {UINT64_C(0xab9eb47c81f5114f), UINT64_C(0x066ea92f3f326565)}, + {UINT64_C(0xd686619ba27255a2), UINT64_C(0xc80a537b0efefebe)}, + {UINT64_C(0x8613fd0145877585), UINT64_C(0xbd06742ce95f5f37)}, + {UINT64_C(0xa798fc4196e952e7), UINT64_C(0x2c48113823b73705)}, + {UINT64_C(0xd17f3b51fca3a7a0), UINT64_C(0xf75a15862ca504c6)}, + {UINT64_C(0x82ef85133de648c4), UINT64_C(0x9a984d73dbe722fc)}, + {UINT64_C(0xa3ab66580d5fdaf5), UINT64_C(0xc13e60d0d2e0ebbb)}, + {UINT64_C(0xcc963fee10b7d1b3), UINT64_C(0x318df905079926a9)}, + {UINT64_C(0xffbbcfe994e5c61f), UINT64_C(0xfdf17746497f7053)}, + {UINT64_C(0x9fd561f1fd0f9bd3), UINT64_C(0xfeb6ea8bedefa634)}, + {UINT64_C(0xc7caba6e7c5382c8), UINT64_C(0xfe64a52ee96b8fc1)}, + {UINT64_C(0xf9bd690a1b68637b), UINT64_C(0x3dfdce7aa3c673b1)}, + {UINT64_C(0x9c1661a651213e2d), UINT64_C(0x06bea10ca65c084f)}, + {UINT64_C(0xc31bfa0fe5698db8), UINT64_C(0x486e494fcff30a63)}, + {UINT64_C(0xf3e2f893dec3f126), UINT64_C(0x5a89dba3c3efccfb)}, + {UINT64_C(0x986ddb5c6b3a76b7), UINT64_C(0xf89629465a75e01d)}, + {UINT64_C(0xbe89523386091465), UINT64_C(0xf6bbb397f1135824)}, + {UINT64_C(0xee2ba6c0678b597f), UINT64_C(0x746aa07ded582e2d)}, + {UINT64_C(0x94db483840b717ef), UINT64_C(0xa8c2a44eb4571cdd)}, + {UINT64_C(0xba121a4650e4ddeb), UINT64_C(0x92f34d62616ce414)}, + {UINT64_C(0xe896a0d7e51e1566), UINT64_C(0x77b020baf9c81d18)}, + {UINT64_C(0x915e2486ef32cd60), UINT64_C(0x0ace1474dc1d122f)}, + {UINT64_C(0xb5b5ada8aaff80b8), UINT64_C(0x0d819992132456bb)}, + {UINT64_C(0xe3231912d5bf60e6), UINT64_C(0x10e1fff697ed6c6a)}, + {UINT64_C(0x8df5efabc5979c8f), UINT64_C(0xca8d3ffa1ef463c2)}, + {UINT64_C(0xb1736b96b6fd83b3), UINT64_C(0xbd308ff8a6b17cb3)}, + {UINT64_C(0xddd0467c64bce4a0), UINT64_C(0xac7cb3f6d05ddbdf)}, + {UINT64_C(0x8aa22c0dbef60ee4), UINT64_C(0x6bcdf07a423aa96c)}, + {UINT64_C(0xad4ab7112eb3929d), UINT64_C(0x86c16c98d2c953c7)}, + {UINT64_C(0xd89d64d57a607744), UINT64_C(0xe871c7bf077ba8b8)}, + {UINT64_C(0x87625f056c7c4a8b), UINT64_C(0x11471cd764ad4973)}, + {UINT64_C(0xa93af6c6c79b5d2d), UINT64_C(0xd598e40d3dd89bd0)}, + {UINT64_C(0xd389b47879823479), UINT64_C(0x4aff1d108d4ec2c4)}, + {UINT64_C(0x843610cb4bf160cb), UINT64_C(0xcedf722a585139bb)}, + {UINT64_C(0xa54394fe1eedb8fe), UINT64_C(0xc2974eb4ee658829)}, + {UINT64_C(0xce947a3da6a9273e), UINT64_C(0x733d226229feea33)}, + {UINT64_C(0x811ccc668829b887), UINT64_C(0x0806357d5a3f5260)}, + {UINT64_C(0xa163ff802a3426a8), UINT64_C(0xca07c2dcb0cf26f8)}, + {UINT64_C(0xc9bcff6034c13052), UINT64_C(0xfc89b393dd02f0b6)}, + {UINT64_C(0xfc2c3f3841f17c67), UINT64_C(0xbbac2078d443ace3)}, + {UINT64_C(0x9d9ba7832936edc0), UINT64_C(0xd54b944b84aa4c0e)}, + {UINT64_C(0xc5029163f384a931), UINT64_C(0x0a9e795e65d4df12)}, + {UINT64_C(0xf64335bcf065d37d), UINT64_C(0x4d4617b5ff4a16d6)}, + {UINT64_C(0x99ea0196163fa42e), UINT64_C(0x504bced1bf8e4e46)}, + {UINT64_C(0xc06481fb9bcf8d39), UINT64_C(0xe45ec2862f71e1d7)}, + {UINT64_C(0xf07da27a82c37088), UINT64_C(0x5d767327bb4e5a4d)}, + {UINT64_C(0x964e858c91ba2655), UINT64_C(0x3a6a07f8d510f870)}, + {UINT64_C(0xbbe226efb628afea), UINT64_C(0x890489f70a55368c)}, + {UINT64_C(0xeadab0aba3b2dbe5), UINT64_C(0x2b45ac74ccea842f)}, + {UINT64_C(0x92c8ae6b464fc96f), UINT64_C(0x3b0b8bc90012929e)}, + {UINT64_C(0xb77ada0617e3bbcb), UINT64_C(0x09ce6ebb40173745)}, + {UINT64_C(0xe55990879ddcaabd), UINT64_C(0xcc420a6a101d0516)}, + {UINT64_C(0x8f57fa54c2a9eab6), UINT64_C(0x9fa946824a12232e)}, + {UINT64_C(0xb32df8e9f3546564), UINT64_C(0x47939822dc96abfa)}, + {UINT64_C(0xdff9772470297ebd), UINT64_C(0x59787e2b93bc56f8)}, + {UINT64_C(0x8bfbea76c619ef36), UINT64_C(0x57eb4edb3c55b65b)}, + {UINT64_C(0xaefae51477a06b03), UINT64_C(0xede622920b6b23f2)}, + {UINT64_C(0xdab99e59958885c4), UINT64_C(0xe95fab368e45ecee)}, + {UINT64_C(0x88b402f7fd75539b), UINT64_C(0x11dbcb0218ebb415)}, + {UINT64_C(0xaae103b5fcd2a881), UINT64_C(0xd652bdc29f26a11a)}, + {UINT64_C(0xd59944a37c0752a2), UINT64_C(0x4be76d3346f04960)}, + {UINT64_C(0x857fcae62d8493a5), UINT64_C(0x6f70a4400c562ddc)}, + {UINT64_C(0xa6dfbd9fb8e5b88e), UINT64_C(0xcb4ccd500f6bb953)}, + {UINT64_C(0xd097ad07a71f26b2), UINT64_C(0x7e2000a41346a7a8)}, + {UINT64_C(0x825ecc24c873782f), UINT64_C(0x8ed400668c0c28c9)}, + {UINT64_C(0xa2f67f2dfa90563b), UINT64_C(0x728900802f0f32fb)}, + {UINT64_C(0xcbb41ef979346bca), UINT64_C(0x4f2b40a03ad2ffba)}, + {UINT64_C(0xfea126b7d78186bc), UINT64_C(0xe2f610c84987bfa9)}, + {UINT64_C(0x9f24b832e6b0f436), UINT64_C(0x0dd9ca7d2df4d7ca)}, + {UINT64_C(0xc6ede63fa05d3143), UINT64_C(0x91503d1c79720dbc)}, + {UINT64_C(0xf8a95fcf88747d94), UINT64_C(0x75a44c6397ce912b)}, + {UINT64_C(0x9b69dbe1b548ce7c), UINT64_C(0xc986afbe3ee11abb)}, + {UINT64_C(0xc24452da229b021b), UINT64_C(0xfbe85badce996169)}, + {UINT64_C(0xf2d56790ab41c2a2), UINT64_C(0xfae27299423fb9c4)}, + {UINT64_C(0x97c560ba6b0919a5), UINT64_C(0xdccd879fc967d41b)}, + {UINT64_C(0xbdb6b8e905cb600f), UINT64_C(0x5400e987bbc1c921)}, + {UINT64_C(0xed246723473e3813), UINT64_C(0x290123e9aab23b69)}, + {UINT64_C(0x9436c0760c86e30b), UINT64_C(0xf9a0b6720aaf6522)}, + {UINT64_C(0xb94470938fa89bce), UINT64_C(0xf808e40e8d5b3e6a)}, + {UINT64_C(0xe7958cb87392c2c2), UINT64_C(0xb60b1d1230b20e05)}, + {UINT64_C(0x90bd77f3483bb9b9), UINT64_C(0xb1c6f22b5e6f48c3)}, + {UINT64_C(0xb4ecd5f01a4aa828), UINT64_C(0x1e38aeb6360b1af4)}, + {UINT64_C(0xe2280b6c20dd5232), UINT64_C(0x25c6da63c38de1b1)}, + {UINT64_C(0x8d590723948a535f), UINT64_C(0x579c487e5a38ad0f)}, + {UINT64_C(0xb0af48ec79ace837), UINT64_C(0x2d835a9df0c6d852)}, + {UINT64_C(0xdcdb1b2798182244), UINT64_C(0xf8e431456cf88e66)}, + {UINT64_C(0x8a08f0f8bf0f156b), UINT64_C(0x1b8e9ecb641b5900)}, + {UINT64_C(0xac8b2d36eed2dac5), UINT64_C(0xe272467e3d222f40)}, + {UINT64_C(0xd7adf884aa879177), UINT64_C(0x5b0ed81dcc6abb10)}, + {UINT64_C(0x86ccbb52ea94baea), UINT64_C(0x98e947129fc2b4ea)}, + {UINT64_C(0xa87fea27a539e9a5), UINT64_C(0x3f2398d747b36225)}, + {UINT64_C(0xd29fe4b18e88640e), UINT64_C(0x8eec7f0d19a03aae)}, + {UINT64_C(0x83a3eeeef9153e89), UINT64_C(0x1953cf68300424ad)}, + {UINT64_C(0xa48ceaaab75a8e2b), UINT64_C(0x5fa8c3423c052dd8)}, + {UINT64_C(0xcdb02555653131b6), UINT64_C(0x3792f412cb06794e)}, + {UINT64_C(0x808e17555f3ebf11), UINT64_C(0xe2bbd88bbee40bd1)}, + {UINT64_C(0xa0b19d2ab70e6ed6), UINT64_C(0x5b6aceaeae9d0ec5)}, + {UINT64_C(0xc8de047564d20a8b), UINT64_C(0xf245825a5a445276)}, + {UINT64_C(0xfb158592be068d2e), UINT64_C(0xeed6e2f0f0d56713)}, + {UINT64_C(0x9ced737bb6c4183d), UINT64_C(0x55464dd69685606c)}, + {UINT64_C(0xc428d05aa4751e4c), UINT64_C(0xaa97e14c3c26b887)}, + {UINT64_C(0xf53304714d9265df), UINT64_C(0xd53dd99f4b3066a9)}, + {UINT64_C(0x993fe2c6d07b7fab), UINT64_C(0xe546a8038efe402a)}, + {UINT64_C(0xbf8fdb78849a5f96), UINT64_C(0xde98520472bdd034)}, + {UINT64_C(0xef73d256a5c0f77c), UINT64_C(0x963e66858f6d4441)}, + {UINT64_C(0x95a8637627989aad), UINT64_C(0xdde7001379a44aa9)}, + {UINT64_C(0xbb127c53b17ec159), UINT64_C(0x5560c018580d5d53)}, + {UINT64_C(0xe9d71b689dde71af), UINT64_C(0xaab8f01e6e10b4a7)}, + {UINT64_C(0x9226712162ab070d), UINT64_C(0xcab3961304ca70e9)}, + {UINT64_C(0xb6b00d69bb55c8d1), UINT64_C(0x3d607b97c5fd0d23)}, + {UINT64_C(0xe45c10c42a2b3b05), UINT64_C(0x8cb89a7db77c506b)}, + {UINT64_C(0x8eb98a7a9a5b04e3), UINT64_C(0x77f3608e92adb243)}, + {UINT64_C(0xb267ed1940f1c61c), UINT64_C(0x55f038b237591ed4)}, + {UINT64_C(0xdf01e85f912e37a3), UINT64_C(0x6b6c46dec52f6689)}, + {UINT64_C(0x8b61313bbabce2c6), UINT64_C(0x2323ac4b3b3da016)}, + {UINT64_C(0xae397d8aa96c1b77), UINT64_C(0xabec975e0a0d081b)}, + {UINT64_C(0xd9c7dced53c72255), UINT64_C(0x96e7bd358c904a22)}, + {UINT64_C(0x881cea14545c7575), UINT64_C(0x7e50d64177da2e55)}, + {UINT64_C(0xaa242499697392d2), UINT64_C(0xdde50bd1d5d0b9ea)}, + {UINT64_C(0xd4ad2dbfc3d07787), UINT64_C(0x955e4ec64b44e865)}, + {UINT64_C(0x84ec3c97da624ab4), UINT64_C(0xbd5af13bef0b113f)}, + {UINT64_C(0xa6274bbdd0fadd61), UINT64_C(0xecb1ad8aeacdd58f)}, + {UINT64_C(0xcfb11ead453994ba), UINT64_C(0x67de18eda5814af3)}, + {UINT64_C(0x81ceb32c4b43fcf4), UINT64_C(0x80eacf948770ced8)}, + {UINT64_C(0xa2425ff75e14fc31), UINT64_C(0xa1258379a94d028e)}, + {UINT64_C(0xcad2f7f5359a3b3e), UINT64_C(0x096ee45813a04331)}, + {UINT64_C(0xfd87b5f28300ca0d), UINT64_C(0x8bca9d6e188853fd)}, + {UINT64_C(0x9e74d1b791e07e48), UINT64_C(0x775ea264cf55347e)}, + {UINT64_C(0xc612062576589dda), UINT64_C(0x95364afe032a819e)}, + {UINT64_C(0xf79687aed3eec551), UINT64_C(0x3a83ddbd83f52205)}, + {UINT64_C(0x9abe14cd44753b52), UINT64_C(0xc4926a9672793543)}, + {UINT64_C(0xc16d9a0095928a27), UINT64_C(0x75b7053c0f178294)}, + {UINT64_C(0xf1c90080baf72cb1), UINT64_C(0x5324c68b12dd6339)}, + {UINT64_C(0x971da05074da7bee), UINT64_C(0xd3f6fc16ebca5e04)}, + {UINT64_C(0xbce5086492111aea), UINT64_C(0x88f4bb1ca6bcf585)}, + {UINT64_C(0xec1e4a7db69561a5), UINT64_C(0x2b31e9e3d06c32e6)}, + {UINT64_C(0x9392ee8e921d5d07), UINT64_C(0x3aff322e62439fd0)}, + {UINT64_C(0xb877aa3236a4b449), UINT64_C(0x09befeb9fad487c3)}, + {UINT64_C(0xe69594bec44de15b), 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{UINT64_C(0x8000000000000000), UINT64_C(0x0000000000000000)}, + {UINT64_C(0xa000000000000000), UINT64_C(0x0000000000000000)}, + {UINT64_C(0xc800000000000000), UINT64_C(0x0000000000000000)}, + {UINT64_C(0xfa00000000000000), UINT64_C(0x0000000000000000)}, + {UINT64_C(0x9c40000000000000), UINT64_C(0x0000000000000000)}, + {UINT64_C(0xc350000000000000), UINT64_C(0x0000000000000000)}, + {UINT64_C(0xf424000000000000), UINT64_C(0x0000000000000000)}, + {UINT64_C(0x9896800000000000), UINT64_C(0x0000000000000000)}, + {UINT64_C(0xbebc200000000000), UINT64_C(0x0000000000000000)}, + {UINT64_C(0xee6b280000000000), UINT64_C(0x0000000000000000)}, + {UINT64_C(0x9502f90000000000), UINT64_C(0x0000000000000000)}, + {UINT64_C(0xba43b74000000000), UINT64_C(0x0000000000000000)}, + {UINT64_C(0xe8d4a51000000000), UINT64_C(0x0000000000000000)}, + {UINT64_C(0x9184e72a00000000), UINT64_C(0x0000000000000000)}, + {UINT64_C(0xb5e620f480000000), UINT64_C(0x0000000000000000)}, + {UINT64_C(0xe35fa931a0000000), UINT64_C(0x0000000000000000)}, + {UINT64_C(0x8e1bc9bf04000000), UINT64_C(0x0000000000000000)}, + {UINT64_C(0xb1a2bc2ec5000000), UINT64_C(0x0000000000000000)}, + {UINT64_C(0xde0b6b3a76400000), UINT64_C(0x0000000000000000)}, + {UINT64_C(0x8ac7230489e80000), UINT64_C(0x0000000000000000)}, + {UINT64_C(0xad78ebc5ac620000), UINT64_C(0x0000000000000000)}, + {UINT64_C(0xd8d726b7177a8000), UINT64_C(0x0000000000000000)}, + {UINT64_C(0x878678326eac9000), UINT64_C(0x0000000000000000)}, + {UINT64_C(0xa968163f0a57b400), UINT64_C(0x0000000000000000)}, + {UINT64_C(0xd3c21bcecceda100), UINT64_C(0x0000000000000000)}, + {UINT64_C(0x84595161401484a0), UINT64_C(0x0000000000000000)}, + {UINT64_C(0xa56fa5b99019a5c8), UINT64_C(0x0000000000000000)}, + {UINT64_C(0xcecb8f27f4200f3a), UINT64_C(0x0000000000000000)}, + {UINT64_C(0x813f3978f8940984), UINT64_C(0x4000000000000000)}, + {UINT64_C(0xa18f07d736b90be5), UINT64_C(0x5000000000000000)}, + {UINT64_C(0xc9f2c9cd04674ede), UINT64_C(0xa400000000000000)}, + {UINT64_C(0xfc6f7c4045812296), UINT64_C(0x4d00000000000000)}, + {UINT64_C(0x9dc5ada82b70b59d), UINT64_C(0xf020000000000000)}, + {UINT64_C(0xc5371912364ce305), UINT64_C(0x6c28000000000000)}, + {UINT64_C(0xf684df56c3e01bc6), UINT64_C(0xc732000000000000)}, + {UINT64_C(0x9a130b963a6c115c), UINT64_C(0x3c7f400000000000)}, + {UINT64_C(0xc097ce7bc90715b3), UINT64_C(0x4b9f100000000000)}, + {UINT64_C(0xf0bdc21abb48db20), UINT64_C(0x1e86d40000000000)}, + {UINT64_C(0x96769950b50d88f4), UINT64_C(0x1314448000000000)}, + {UINT64_C(0xbc143fa4e250eb31), UINT64_C(0x17d955a000000000)}, + {UINT64_C(0xeb194f8e1ae525fd), UINT64_C(0x5dcfab0800000000)}, + {UINT64_C(0x92efd1b8d0cf37be), UINT64_C(0x5aa1cae500000000)}, + {UINT64_C(0xb7abc627050305ad), UINT64_C(0xf14a3d9e40000000)}, + {UINT64_C(0xe596b7b0c643c719), UINT64_C(0x6d9ccd05d0000000)}, + {UINT64_C(0x8f7e32ce7bea5c6f), UINT64_C(0xe4820023a2000000)}, + {UINT64_C(0xb35dbf821ae4f38b), UINT64_C(0xdda2802c8a800000)}, + {UINT64_C(0xe0352f62a19e306e), UINT64_C(0xd50b2037ad200000)}, + {UINT64_C(0x8c213d9da502de45), UINT64_C(0x4526f422cc340000)}, + {UINT64_C(0xaf298d050e4395d6), UINT64_C(0x9670b12b7f410000)}, + {UINT64_C(0xdaf3f04651d47b4c), UINT64_C(0x3c0cdd765f114000)}, + {UINT64_C(0x88d8762bf324cd0f), UINT64_C(0xa5880a69fb6ac800)}, + {UINT64_C(0xab0e93b6efee0053), UINT64_C(0x8eea0d047a457a00)}, + {UINT64_C(0xd5d238a4abe98068), UINT64_C(0x72a4904598d6d880)}, + {UINT64_C(0x85a36366eb71f041), UINT64_C(0x47a6da2b7f864750)}, + {UINT64_C(0xa70c3c40a64e6c51), UINT64_C(0x999090b65f67d924)}, + {UINT64_C(0xd0cf4b50cfe20765), UINT64_C(0xfff4b4e3f741cf6d)}, + {UINT64_C(0x82818f1281ed449f), UINT64_C(0xbff8f10e7a8921a5)}, + {UINT64_C(0xa321f2d7226895c7), UINT64_C(0xaff72d52192b6a0e)}, + {UINT64_C(0xcbea6f8ceb02bb39), UINT64_C(0x9bf4f8a69f764491)}, + {UINT64_C(0xfee50b7025c36a08), UINT64_C(0x02f236d04753d5b5)}, + {UINT64_C(0x9f4f2726179a2245), UINT64_C(0x01d762422c946591)}, + {UINT64_C(0xc722f0ef9d80aad6), UINT64_C(0x424d3ad2b7b97ef6)}, + {UINT64_C(0xf8ebad2b84e0d58b), UINT64_C(0xd2e0898765a7deb3)}, + {UINT64_C(0x9b934c3b330c8577), UINT64_C(0x63cc55f49f88eb30)}, + {UINT64_C(0xc2781f49ffcfa6d5), UINT64_C(0x3cbf6b71c76b25fc)}, + {UINT64_C(0xf316271c7fc3908a), UINT64_C(0x8bef464e3945ef7b)}, + {UINT64_C(0x97edd871cfda3a56), UINT64_C(0x97758bf0e3cbb5ad)}, + {UINT64_C(0xbde94e8e43d0c8ec), UINT64_C(0x3d52eeed1cbea318)}, + {UINT64_C(0xed63a231d4c4fb27), UINT64_C(0x4ca7aaa863ee4bde)}, + {UINT64_C(0x945e455f24fb1cf8), UINT64_C(0x8fe8caa93e74ef6b)}, + {UINT64_C(0xb975d6b6ee39e436), UINT64_C(0xb3e2fd538e122b45)}, + {UINT64_C(0xe7d34c64a9c85d44), UINT64_C(0x60dbbca87196b617)}, + {UINT64_C(0x90e40fbeea1d3a4a), UINT64_C(0xbc8955e946fe31ce)}, + {UINT64_C(0xb51d13aea4a488dd), UINT64_C(0x6babab6398bdbe42)}, + {UINT64_C(0xe264589a4dcdab14), UINT64_C(0xc696963c7eed2dd2)}, + {UINT64_C(0x8d7eb76070a08aec), UINT64_C(0xfc1e1de5cf543ca3)}, + {UINT64_C(0xb0de65388cc8ada8), UINT64_C(0x3b25a55f43294bcc)}, + {UINT64_C(0xdd15fe86affad912), UINT64_C(0x49ef0eb713f39ebf)}, + {UINT64_C(0x8a2dbf142dfcc7ab), UINT64_C(0x6e3569326c784338)}, + {UINT64_C(0xacb92ed9397bf996), UINT64_C(0x49c2c37f07965405)}, + {UINT64_C(0xd7e77a8f87daf7fb), UINT64_C(0xdc33745ec97be907)}, + {UINT64_C(0x86f0ac99b4e8dafd), UINT64_C(0x69a028bb3ded71a4)}, + {UINT64_C(0xa8acd7c0222311bc), UINT64_C(0xc40832ea0d68ce0d)}, + {UINT64_C(0xd2d80db02aabd62b), UINT64_C(0xf50a3fa490c30191)}, + {UINT64_C(0x83c7088e1aab65db), UINT64_C(0x792667c6da79e0fb)}, + {UINT64_C(0xa4b8cab1a1563f52), UINT64_C(0x577001b891185939)}, + {UINT64_C(0xcde6fd5e09abcf26), UINT64_C(0xed4c0226b55e6f87)}, + {UINT64_C(0x80b05e5ac60b6178), UINT64_C(0x544f8158315b05b5)}, + {UINT64_C(0xa0dc75f1778e39d6), UINT64_C(0x696361ae3db1c722)}, + {UINT64_C(0xc913936dd571c84c), UINT64_C(0x03bc3a19cd1e38ea)}, + {UINT64_C(0xfb5878494ace3a5f), UINT64_C(0x04ab48a04065c724)}, + {UINT64_C(0x9d174b2dcec0e47b), UINT64_C(0x62eb0d64283f9c77)}, + {UINT64_C(0xc45d1df942711d9a), UINT64_C(0x3ba5d0bd324f8395)}, + {UINT64_C(0xf5746577930d6500), UINT64_C(0xca8f44ec7ee3647a)}, + {UINT64_C(0x9968bf6abbe85f20), UINT64_C(0x7e998b13cf4e1ecc)}, + {UINT64_C(0xbfc2ef456ae276e8), UINT64_C(0x9e3fedd8c321a67f)}, + {UINT64_C(0xefb3ab16c59b14a2), UINT64_C(0xc5cfe94ef3ea101f)}, + {UINT64_C(0x95d04aee3b80ece5), UINT64_C(0xbba1f1d158724a13)}, + {UINT64_C(0xbb445da9ca61281f), UINT64_C(0x2a8a6e45ae8edc98)}, + {UINT64_C(0xea1575143cf97226), UINT64_C(0xf52d09d71a3293be)}, + {UINT64_C(0x924d692ca61be758), UINT64_C(0x593c2626705f9c57)}, + {UINT64_C(0xb6e0c377cfa2e12e), UINT64_C(0x6f8b2fb00c77836d)}, + {UINT64_C(0xe498f455c38b997a), UINT64_C(0x0b6dfb9c0f956448)}, + {UINT64_C(0x8edf98b59a373fec), UINT64_C(0x4724bd4189bd5ead)}, + {UINT64_C(0xb2977ee300c50fe7), UINT64_C(0x58edec91ec2cb658)}, + {UINT64_C(0xdf3d5e9bc0f653e1), UINT64_C(0x2f2967b66737e3ee)}, + {UINT64_C(0x8b865b215899f46c), UINT64_C(0xbd79e0d20082ee75)}, + {UINT64_C(0xae67f1e9aec07187), UINT64_C(0xecd8590680a3aa12)}, + {UINT64_C(0xda01ee641a708de9), UINT64_C(0xe80e6f4820cc9496)}, + {UINT64_C(0x884134fe908658b2), UINT64_C(0x3109058d147fdcde)}, + {UINT64_C(0xaa51823e34a7eede), UINT64_C(0xbd4b46f0599fd416)}, + {UINT64_C(0xd4e5e2cdc1d1ea96), UINT64_C(0x6c9e18ac7007c91b)}, + {UINT64_C(0x850fadc09923329e), UINT64_C(0x03e2cf6bc604ddb1)}, + {UINT64_C(0xa6539930bf6bff45), UINT64_C(0x84db8346b786151d)}, + {UINT64_C(0xcfe87f7cef46ff16), UINT64_C(0xe612641865679a64)}, + {UINT64_C(0x81f14fae158c5f6e), UINT64_C(0x4fcb7e8f3f60c07f)}, + {UINT64_C(0xa26da3999aef7749), UINT64_C(0xe3be5e330f38f09e)}, + {UINT64_C(0xcb090c8001ab551c), UINT64_C(0x5cadf5bfd3072cc6)}, + {UINT64_C(0xfdcb4fa002162a63), UINT64_C(0x73d9732fc7c8f7f7)}, + {UINT64_C(0x9e9f11c4014dda7e), UINT64_C(0x2867e7fddcdd9afb)}, + {UINT64_C(0xc646d63501a1511d), UINT64_C(0xb281e1fd541501b9)}, + {UINT64_C(0xf7d88bc24209a565), UINT64_C(0x1f225a7ca91a4227)}, + {UINT64_C(0x9ae757596946075f), UINT64_C(0x3375788de9b06959)}, + {UINT64_C(0xc1a12d2fc3978937), UINT64_C(0x0052d6b1641c83af)}, + {UINT64_C(0xf209787bb47d6b84), UINT64_C(0xc0678c5dbd23a49b)}, + {UINT64_C(0x9745eb4d50ce6332), UINT64_C(0xf840b7ba963646e1)}, + {UINT64_C(0xbd176620a501fbff), UINT64_C(0xb650e5a93bc3d899)}, + {UINT64_C(0xec5d3fa8ce427aff), UINT64_C(0xa3e51f138ab4cebf)}, + {UINT64_C(0x93ba47c980e98cdf), UINT64_C(0xc66f336c36b10138)}, + {UINT64_C(0xb8a8d9bbe123f017), UINT64_C(0xb80b0047445d4185)}, + {UINT64_C(0xe6d3102ad96cec1d), UINT64_C(0xa60dc059157491e6)}, + {UINT64_C(0x9043ea1ac7e41392), UINT64_C(0x87c89837ad68db30)}, + {UINT64_C(0xb454e4a179dd1877), UINT64_C(0x29babe4598c311fc)}, + {UINT64_C(0xe16a1dc9d8545e94), UINT64_C(0xf4296dd6fef3d67b)}, + {UINT64_C(0x8ce2529e2734bb1d), UINT64_C(0x1899e4a65f58660d)}, + {UINT64_C(0xb01ae745b101e9e4), UINT64_C(0x5ec05dcff72e7f90)}, + {UINT64_C(0xdc21a1171d42645d), UINT64_C(0x76707543f4fa1f74)}, + {UINT64_C(0x899504ae72497eba), UINT64_C(0x6a06494a791c53a9)}, + {UINT64_C(0xabfa45da0edbde69), UINT64_C(0x0487db9d17636893)}, + {UINT64_C(0xd6f8d7509292d603), UINT64_C(0x45a9d2845d3c42b7)}, + {UINT64_C(0x865b86925b9bc5c2), UINT64_C(0x0b8a2392ba45a9b3)}, + {UINT64_C(0xa7f26836f282b732), UINT64_C(0x8e6cac7768d7141f)}, + {UINT64_C(0xd1ef0244af2364ff), UINT64_C(0x3207d795430cd927)}, + {UINT64_C(0x8335616aed761f1f), UINT64_C(0x7f44e6bd49e807b9)}, + {UINT64_C(0xa402b9c5a8d3a6e7), UINT64_C(0x5f16206c9c6209a7)}, + {UINT64_C(0xcd036837130890a1), UINT64_C(0x36dba887c37a8c10)}, + {UINT64_C(0x802221226be55a64), UINT64_C(0xc2494954da2c978a)}, + {UINT64_C(0xa02aa96b06deb0fd), UINT64_C(0xf2db9baa10b7bd6d)}, + {UINT64_C(0xc83553c5c8965d3d), UINT64_C(0x6f92829494e5acc8)}, + {UINT64_C(0xfa42a8b73abbf48c), UINT64_C(0xcb772339ba1f17fa)}, + {UINT64_C(0x9c69a97284b578d7), UINT64_C(0xff2a760414536efc)}, + {UINT64_C(0xc38413cf25e2d70d), UINT64_C(0xfef5138519684abb)}, + {UINT64_C(0xf46518c2ef5b8cd1), UINT64_C(0x7eb258665fc25d6a)}, + {UINT64_C(0x98bf2f79d5993802), UINT64_C(0xef2f773ffbd97a62)}, + {UINT64_C(0xbeeefb584aff8603), UINT64_C(0xaafb550ffacfd8fb)}, + {UINT64_C(0xeeaaba2e5dbf6784), UINT64_C(0x95ba2a53f983cf39)}, + {UINT64_C(0x952ab45cfa97a0b2), UINT64_C(0xdd945a747bf26184)}, + {UINT64_C(0xba756174393d88df), UINT64_C(0x94f971119aeef9e5)}, + {UINT64_C(0xe912b9d1478ceb17), UINT64_C(0x7a37cd5601aab85e)}, + {UINT64_C(0x91abb422ccb812ee), UINT64_C(0xac62e055c10ab33b)}, + {UINT64_C(0xb616a12b7fe617aa), UINT64_C(0x577b986b314d600a)}, + {UINT64_C(0xe39c49765fdf9d94), UINT64_C(0xed5a7e85fda0b80c)}, + {UINT64_C(0x8e41ade9fbebc27d), UINT64_C(0x14588f13be847308)}, + {UINT64_C(0xb1d219647ae6b31c), UINT64_C(0x596eb2d8ae258fc9)}, + {UINT64_C(0xde469fbd99a05fe3), UINT64_C(0x6fca5f8ed9aef3bc)}, + {UINT64_C(0x8aec23d680043bee), UINT64_C(0x25de7bb9480d5855)}, + {UINT64_C(0xada72ccc20054ae9), UINT64_C(0xaf561aa79a10ae6b)}, + {UINT64_C(0xd910f7ff28069da4), UINT64_C(0x1b2ba1518094da05)}, + {UINT64_C(0x87aa9aff79042286), UINT64_C(0x90fb44d2f05d0843)}, + {UINT64_C(0xa99541bf57452b28), UINT64_C(0x353a1607ac744a54)}, + {UINT64_C(0xd3fa922f2d1675f2), UINT64_C(0x42889b8997915ce9)}, + {UINT64_C(0x847c9b5d7c2e09b7), UINT64_C(0x69956135febada12)}, + {UINT64_C(0xa59bc234db398c25), UINT64_C(0x43fab9837e699096)}, + {UINT64_C(0xcf02b2c21207ef2e), UINT64_C(0x94f967e45e03f4bc)}, + {UINT64_C(0x8161afb94b44f57d), UINT64_C(0x1d1be0eebac278f6)}, + {UINT64_C(0xa1ba1ba79e1632dc), UINT64_C(0x6462d92a69731733)}, + {UINT64_C(0xca28a291859bbf93), UINT64_C(0x7d7b8f7503cfdcff)}, + {UINT64_C(0xfcb2cb35e702af78), UINT64_C(0x5cda735244c3d43f)}, + {UINT64_C(0x9defbf01b061adab), UINT64_C(0x3a0888136afa64a8)}, + {UINT64_C(0xc56baec21c7a1916), UINT64_C(0x088aaa1845b8fdd1)}, + {UINT64_C(0xf6c69a72a3989f5b), UINT64_C(0x8aad549e57273d46)}, + {UINT64_C(0x9a3c2087a63f6399), UINT64_C(0x36ac54e2f678864c)}, + {UINT64_C(0xc0cb28a98fcf3c7f), UINT64_C(0x84576a1bb416a7de)}, + {UINT64_C(0xf0fdf2d3f3c30b9f), UINT64_C(0x656d44a2a11c51d6)}, + {UINT64_C(0x969eb7c47859e743), UINT64_C(0x9f644ae5a4b1b326)}, + {UINT64_C(0xbc4665b596706114), UINT64_C(0x873d5d9f0dde1fef)}, + {UINT64_C(0xeb57ff22fc0c7959), UINT64_C(0xa90cb506d155a7eb)}, + {UINT64_C(0x9316ff75dd87cbd8), UINT64_C(0x09a7f12442d588f3)}, + {UINT64_C(0xb7dcbf5354e9bece), UINT64_C(0x0c11ed6d538aeb30)}, + {UINT64_C(0xe5d3ef282a242e81), UINT64_C(0x8f1668c8a86da5fb)}, + {UINT64_C(0x8fa475791a569d10), UINT64_C(0xf96e017d694487bd)}, + {UINT64_C(0xb38d92d760ec4455), UINT64_C(0x37c981dcc395a9ad)}, + {UINT64_C(0xe070f78d3927556a), UINT64_C(0x85bbe253f47b1418)}, + {UINT64_C(0x8c469ab843b89562), UINT64_C(0x93956d7478ccec8f)}, + {UINT64_C(0xaf58416654a6babb), UINT64_C(0x387ac8d1970027b3)}, + {UINT64_C(0xdb2e51bfe9d0696a), UINT64_C(0x06997b05fcc0319f)}, + {UINT64_C(0x88fcf317f22241e2), UINT64_C(0x441fece3bdf81f04)}, + {UINT64_C(0xab3c2fddeeaad25a), UINT64_C(0xd527e81cad7626c4)}, + {UINT64_C(0xd60b3bd56a5586f1), UINT64_C(0x8a71e223d8d3b075)}, + {UINT64_C(0x85c7056562757456), UINT64_C(0xf6872d5667844e4a)}, + {UINT64_C(0xa738c6bebb12d16c), UINT64_C(0xb428f8ac016561dc)}, + {UINT64_C(0xd106f86e69d785c7), UINT64_C(0xe13336d701beba53)}, + {UINT64_C(0x82a45b450226b39c), UINT64_C(0xecc0024661173474)}, + {UINT64_C(0xa34d721642b06084), UINT64_C(0x27f002d7f95d0191)}, + {UINT64_C(0xcc20ce9bd35c78a5), UINT64_C(0x31ec038df7b441f5)}, + {UINT64_C(0xff290242c83396ce), UINT64_C(0x7e67047175a15272)}, + {UINT64_C(0x9f79a169bd203e41), UINT64_C(0x0f0062c6e984d387)}, + {UINT64_C(0xc75809c42c684dd1), UINT64_C(0x52c07b78a3e60869)}, + {UINT64_C(0xf92e0c3537826145), UINT64_C(0xa7709a56ccdf8a83)}, + {UINT64_C(0x9bbcc7a142b17ccb), UINT64_C(0x88a66076400bb692)}, + {UINT64_C(0xc2abf989935ddbfe), UINT64_C(0x6acff893d00ea436)}, + {UINT64_C(0xf356f7ebf83552fe), UINT64_C(0x0583f6b8c4124d44)}, + {UINT64_C(0x98165af37b2153de), UINT64_C(0xc3727a337a8b704b)}, + {UINT64_C(0xbe1bf1b059e9a8d6), UINT64_C(0x744f18c0592e4c5d)}, + {UINT64_C(0xeda2ee1c7064130c), UINT64_C(0x1162def06f79df74)}, + {UINT64_C(0x9485d4d1c63e8be7), UINT64_C(0x8addcb5645ac2ba9)}, + {UINT64_C(0xb9a74a0637ce2ee1), UINT64_C(0x6d953e2bd7173693)}, + {UINT64_C(0xe8111c87c5c1ba99), UINT64_C(0xc8fa8db6ccdd0438)}, + {UINT64_C(0x910ab1d4db9914a0), UINT64_C(0x1d9c9892400a22a3)}, + {UINT64_C(0xb54d5e4a127f59c8), UINT64_C(0x2503beb6d00cab4c)}, + {UINT64_C(0xe2a0b5dc971f303a), UINT64_C(0x2e44ae64840fd61e)}, + {UINT64_C(0x8da471a9de737e24), UINT64_C(0x5ceaecfed289e5d3)}, + {UINT64_C(0xb10d8e1456105dad), UINT64_C(0x7425a83e872c5f48)}, + {UINT64_C(0xdd50f1996b947518), UINT64_C(0xd12f124e28f7771a)}, + {UINT64_C(0x8a5296ffe33cc92f), UINT64_C(0x82bd6b70d99aaa70)}, + {UINT64_C(0xace73cbfdc0bfb7b), UINT64_C(0x636cc64d1001550c)}, + {UINT64_C(0xd8210befd30efa5a), UINT64_C(0x3c47f7e05401aa4f)}, + {UINT64_C(0x8714a775e3e95c78), UINT64_C(0x65acfaec34810a72)}, + {UINT64_C(0xa8d9d1535ce3b396), UINT64_C(0x7f1839a741a14d0e)}, + {UINT64_C(0xd31045a8341ca07c), UINT64_C(0x1ede48111209a051)}, + {UINT64_C(0x83ea2b892091e44d), UINT64_C(0x934aed0aab460433)}, + {UINT64_C(0xa4e4b66b68b65d60), UINT64_C(0xf81da84d56178540)}, + {UINT64_C(0xce1de40642e3f4b9), UINT64_C(0x36251260ab9d668f)}, + {UINT64_C(0x80d2ae83e9ce78f3), UINT64_C(0xc1d72b7c6b42601a)}, + {UINT64_C(0xa1075a24e4421730), UINT64_C(0xb24cf65b8612f820)}, + {UINT64_C(0xc94930ae1d529cfc), UINT64_C(0xdee033f26797b628)}, + {UINT64_C(0xfb9b7cd9a4a7443c), UINT64_C(0x169840ef017da3b2)}, + {UINT64_C(0x9d412e0806e88aa5), UINT64_C(0x8e1f289560ee864f)}, + {UINT64_C(0xc491798a08a2ad4e), UINT64_C(0xf1a6f2bab92a27e3)}, + {UINT64_C(0xf5b5d7ec8acb58a2), UINT64_C(0xae10af696774b1dc)}, + {UINT64_C(0x9991a6f3d6bf1765), UINT64_C(0xacca6da1e0a8ef2a)}, + {UINT64_C(0xbff610b0cc6edd3f), UINT64_C(0x17fd090a58d32af4)}, + {UINT64_C(0xeff394dcff8a948e), UINT64_C(0xddfc4b4cef07f5b1)}, + {UINT64_C(0x95f83d0a1fb69cd9), UINT64_C(0x4abdaf101564f98f)}, + {UINT64_C(0xbb764c4ca7a4440f), UINT64_C(0x9d6d1ad41abe37f2)}, + {UINT64_C(0xea53df5fd18d5513), UINT64_C(0x84c86189216dc5ee)}, + {UINT64_C(0x92746b9be2f8552c), UINT64_C(0x32fd3cf5b4e49bb5)}, + {UINT64_C(0xb7118682dbb66a77), UINT64_C(0x3fbc8c33221dc2a2)}, + {UINT64_C(0xe4d5e82392a40515), UINT64_C(0x0fabaf3feaa5334b)}, + {UINT64_C(0x8f05b1163ba6832d), UINT64_C(0x29cb4d87f2a7400f)}, + {UINT64_C(0xb2c71d5bca9023f8), UINT64_C(0x743e20e9ef511013)}, + {UINT64_C(0xdf78e4b2bd342cf6), UINT64_C(0x914da9246b255417)}, + {UINT64_C(0x8bab8eefb6409c1a), UINT64_C(0x1ad089b6c2f7548f)}, + {UINT64_C(0xae9672aba3d0c320), UINT64_C(0xa184ac2473b529b2)}, + {UINT64_C(0xda3c0f568cc4f3e8), UINT64_C(0xc9e5d72d90a2741f)}, + {UINT64_C(0x8865899617fb1871), UINT64_C(0x7e2fa67c7a658893)}, + {UINT64_C(0xaa7eebfb9df9de8d), UINT64_C(0xddbb901b98feeab8)}, + {UINT64_C(0xd51ea6fa85785631), UINT64_C(0x552a74227f3ea566)}, + {UINT64_C(0x8533285c936b35de), UINT64_C(0xd53a88958f872760)}, + {UINT64_C(0xa67ff273b8460356), UINT64_C(0x8a892abaf368f138)}, + {UINT64_C(0xd01fef10a657842c), UINT64_C(0x2d2b7569b0432d86)}, + {UINT64_C(0x8213f56a67f6b29b), UINT64_C(0x9c3b29620e29fc74)}, + {UINT64_C(0xa298f2c501f45f42), UINT64_C(0x8349f3ba91b47b90)}, + {UINT64_C(0xcb3f2f7642717713), UINT64_C(0x241c70a936219a74)}, + {UINT64_C(0xfe0efb53d30dd4d7), UINT64_C(0xed238cd383aa0111)}, + {UINT64_C(0x9ec95d1463e8a506), UINT64_C(0xf4363804324a40ab)}, + {UINT64_C(0xc67bb4597ce2ce48), UINT64_C(0xb143c6053edcd0d6)}, + {UINT64_C(0xf81aa16fdc1b81da), UINT64_C(0xdd94b7868e94050b)}, + {UINT64_C(0x9b10a4e5e9913128), UINT64_C(0xca7cf2b4191c8327)}, + {UINT64_C(0xc1d4ce1f63f57d72), UINT64_C(0xfd1c2f611f63a3f1)}, + {UINT64_C(0xf24a01a73cf2dccf), UINT64_C(0xbc633b39673c8ced)}, + {UINT64_C(0x976e41088617ca01), UINT64_C(0xd5be0503e085d814)}, + {UINT64_C(0xbd49d14aa79dbc82), UINT64_C(0x4b2d8644d8a74e19)}, + {UINT64_C(0xec9c459d51852ba2), UINT64_C(0xddf8e7d60ed1219f)}, + {UINT64_C(0x93e1ab8252f33b45), UINT64_C(0xcabb90e5c942b504)}, + {UINT64_C(0xb8da1662e7b00a17), UINT64_C(0x3d6a751f3b936244)}, + {UINT64_C(0xe7109bfba19c0c9d), UINT64_C(0x0cc512670a783ad5)}, + {UINT64_C(0x906a617d450187e2), UINT64_C(0x27fb2b80668b24c6)}, + {UINT64_C(0xb484f9dc9641e9da), UINT64_C(0xb1f9f660802dedf7)}, + {UINT64_C(0xe1a63853bbd26451), UINT64_C(0x5e7873f8a0396974)}, + {UINT64_C(0x8d07e33455637eb2), UINT64_C(0xdb0b487b6423e1e9)}, + {UINT64_C(0xb049dc016abc5e5f), UINT64_C(0x91ce1a9a3d2cda63)}, + {UINT64_C(0xdc5c5301c56b75f7), UINT64_C(0x7641a140cc7810fc)}, + {UINT64_C(0x89b9b3e11b6329ba), UINT64_C(0xa9e904c87fcb0a9e)}, + {UINT64_C(0xac2820d9623bf429), UINT64_C(0x546345fa9fbdcd45)}, + {UINT64_C(0xd732290fbacaf133), UINT64_C(0xa97c177947ad4096)}, + {UINT64_C(0x867f59a9d4bed6c0), UINT64_C(0x49ed8eabcccc485e)}, + {UINT64_C(0xa81f301449ee8c70), UINT64_C(0x5c68f256bfff5a75)}, + {UINT64_C(0xd226fc195c6a2f8c), UINT64_C(0x73832eec6fff3112)}, + {UINT64_C(0x83585d8fd9c25db7), UINT64_C(0xc831fd53c5ff7eac)}, + {UINT64_C(0xa42e74f3d032f525), UINT64_C(0xba3e7ca8b77f5e56)}, + {UINT64_C(0xcd3a1230c43fb26f), UINT64_C(0x28ce1bd2e55f35ec)}, + {UINT64_C(0x80444b5e7aa7cf85), UINT64_C(0x7980d163cf5b81b4)}, + {UINT64_C(0xa0555e361951c366), UINT64_C(0xd7e105bcc3326220)}, + {UINT64_C(0xc86ab5c39fa63440), UINT64_C(0x8dd9472bf3fefaa8)}, + {UINT64_C(0xfa856334878fc150), UINT64_C(0xb14f98f6f0feb952)}, + {UINT64_C(0x9c935e00d4b9d8d2), UINT64_C(0x6ed1bf9a569f33d4)}, + {UINT64_C(0xc3b8358109e84f07), UINT64_C(0x0a862f80ec4700c9)}, + {UINT64_C(0xf4a642e14c6262c8), UINT64_C(0xcd27bb612758c0fb)}, + {UINT64_C(0x98e7e9cccfbd7dbd), UINT64_C(0x8038d51cb897789d)}, + {UINT64_C(0xbf21e44003acdd2c), UINT64_C(0xe0470a63e6bd56c4)}, + {UINT64_C(0xeeea5d5004981478), UINT64_C(0x1858ccfce06cac75)}, + {UINT64_C(0x95527a5202df0ccb), UINT64_C(0x0f37801e0c43ebc9)}, + {UINT64_C(0xbaa718e68396cffd), UINT64_C(0xd30560258f54e6bb)}, + {UINT64_C(0xe950df20247c83fd), UINT64_C(0x47c6b82ef32a206a)}, + {UINT64_C(0x91d28b7416cdd27e), UINT64_C(0x4cdc331d57fa5442)}, + {UINT64_C(0xb6472e511c81471d), UINT64_C(0xe0133fe4adf8e953)}, + {UINT64_C(0xe3d8f9e563a198e5), UINT64_C(0x58180fddd97723a7)}, + {UINT64_C(0x8e679c2f5e44ff8f), UINT64_C(0x570f09eaa7ea7649)}, + {UINT64_C(0xb201833b35d63f73), UINT64_C(0x2cd2cc6551e513db)}, + {UINT64_C(0xde81e40a034bcf4f), UINT64_C(0xf8077f7ea65e58d2)}, + {UINT64_C(0x8b112e86420f6191), UINT64_C(0xfb04afaf27faf783)}, + {UINT64_C(0xadd57a27d29339f6), UINT64_C(0x79c5db9af1f9b564)}, + {UINT64_C(0xd94ad8b1c7380874), UINT64_C(0x18375281ae7822bd)}, + {UINT64_C(0x87cec76f1c830548), UINT64_C(0x8f2293910d0b15b6)}, + {UINT64_C(0xa9c2794ae3a3c69a), UINT64_C(0xb2eb3875504ddb23)}, + {UINT64_C(0xd433179d9c8cb841), UINT64_C(0x5fa60692a46151ec)}, + {UINT64_C(0x849feec281d7f328), UINT64_C(0xdbc7c41ba6bcd334)}, + {UINT64_C(0xa5c7ea73224deff3), UINT64_C(0x12b9b522906c0801)}, + {UINT64_C(0xcf39e50feae16bef), UINT64_C(0xd768226b34870a01)}, + {UINT64_C(0x81842f29f2cce375), UINT64_C(0xe6a1158300d46641)}, + {UINT64_C(0xa1e53af46f801c53), UINT64_C(0x60495ae3c1097fd1)}, + {UINT64_C(0xca5e89b18b602368), UINT64_C(0x385bb19cb14bdfc5)}, + {UINT64_C(0xfcf62c1dee382c42), UINT64_C(0x46729e03dd9ed7b6)}, + {UINT64_C(0x9e19db92b4e31ba9), UINT64_C(0x6c07a2c26a8346d2)}, + {UINT64_C(0xc5a05277621be293), UINT64_C(0xc7098b7305241886)}, + {UINT64_C(0xf70867153aa2db38), UINT64_C(0xb8cbee4fc66d1ea8)}}}; + }; +#if !JKJ_HAS_INLINE_VARIABLE + // decltype(...) should not depend on Dummy; see + // https://stackoverflow.com/questions/76438400/decltype-on-static-variable-in-template-class. + template <class Dummy> + constexpr decltype(cache_holder<ieee754_binary64>::cache) + cache_holder<ieee754_binary64, Dummy>::cache; +#endif + + // Compressed cache. + template <class FloatFormat, class Dummy = void> + struct compressed_cache_holder { + using cache_entry_type = typename cache_holder<FloatFormat>::cache_entry_type; + static constexpr int cache_bits = cache_holder<FloatFormat>::cache_bits; + static constexpr int min_k = cache_holder<FloatFormat>::min_k; + static constexpr int max_k = cache_holder<FloatFormat>::max_k; + + template <class ShiftAmountType, class DecimalExponentType> + static constexpr cache_entry_type get_cache(DecimalExponentType k) noexcept { + return cache_holder<FloatFormat>::cache[k - min_k]; + } + }; + + template <class Dummy> + struct compressed_cache_holder<ieee754_binary32, Dummy> { + using cache_entry_type = cache_holder<ieee754_binary32>::cache_entry_type; + static constexpr int cache_bits = cache_holder<ieee754_binary32>::cache_bits; + static constexpr int min_k = cache_holder<ieee754_binary32>::min_k; + static constexpr int max_k = cache_holder<ieee754_binary32>::max_k; + static constexpr int compression_ratio = 13; + static constexpr detail::stdr::size_t compressed_table_size = + detail::stdr::size_t((max_k - min_k + compression_ratio) / compression_ratio); + static constexpr detail::stdr::size_t pow5_table_size = + detail::stdr::size_t((compression_ratio + 1) / 2); + + using cache_holder_t = detail::array<cache_entry_type, compressed_table_size>; + using pow5_holder_t = detail::array<detail::stdr::uint_least16_t, pow5_table_size>; + +#if JKJ_HAS_CONSTEXPR17 + static constexpr cache_holder_t cache JKJ_STATIC_DATA_SECTION = [] { + cache_holder_t res{}; + for (detail::stdr::size_t i = 0; i < compressed_table_size; ++i) { + res[i] = cache_holder<ieee754_binary32>::cache[i * compression_ratio]; + } + return res; + }(); + static constexpr pow5_holder_t pow5_table JKJ_STATIC_DATA_SECTION = [] { + pow5_holder_t res{}; + detail::stdr::uint_least16_t p = 1; + for (detail::stdr::size_t i = 0; i < pow5_table_size; ++i) { + res[i] = p; + p *= 5; + } + return res; + }(); +#else + template <detail::stdr::size_t... indices> + static constexpr cache_holder_t make_cache(detail::index_sequence<indices...>) { + return {cache_holder<ieee754_binary32>::cache[indices * compression_ratio]...}; + } + static constexpr cache_holder_t cache JKJ_STATIC_DATA_SECTION = + make_cache(detail::make_index_sequence<compressed_table_size>{}); + + template <detail::stdr::size_t... indices> + static constexpr pow5_holder_t make_pow5_table(detail::index_sequence<indices...>) { + return {detail::compute_power<indices>(detail::stdr::uint_least16_t(5))...}; + } + static constexpr pow5_holder_t pow5_table JKJ_STATIC_DATA_SECTION = + make_pow5_table(detail::make_index_sequence<pow5_table_size>{}); +#endif + + template <class ShiftAmountType, class DecimalExponentType> + static JKJ_CONSTEXPR20 cache_entry_type get_cache(DecimalExponentType k) noexcept { + // Compute the base index. + // Supposed to compute (k - min_k) / compression_ratio. + static_assert(max_k - min_k <= 89 && compression_ratio == 13, ""); + static_assert(max_k - min_k <= detail::stdr::numeric_limits<DecimalExponentType>::max(), + ""); + auto const cache_index = + DecimalExponentType(detail::stdr::uint_fast16_t(DecimalExponentType(k - min_k) * + detail::stdr::int_fast16_t(79)) >> + 10); + auto const kb = DecimalExponentType(cache_index * compression_ratio + min_k); + auto const offset = DecimalExponentType(k - kb); + + // Get the base cache. + auto const base_cache = cache[cache_index]; + + if (offset == 0) { + return base_cache; + } + else { + // Compute the required amount of bit-shift. + auto const alpha = + ShiftAmountType(detail::log::floor_log2_pow10<min_k, max_k>(k) - + detail::log::floor_log2_pow10<min_k, max_k>(kb) - offset); + assert(alpha > 0 && alpha < 64); + + // Try to recover the real cache. + auto const pow5 = + offset >= 7 + ? detail::stdr::uint_fast32_t(detail::stdr::uint_fast32_t(pow5_table[6]) * + pow5_table[offset - 6]) + : detail::stdr::uint_fast32_t(pow5_table[offset]); + auto mul_result = detail::wuint::umul128(base_cache, pow5); + auto const recovered_cache = + cache_entry_type((((mul_result.high() << ShiftAmountType(64 - alpha)) | + (mul_result.low() >> alpha)) + + 1) & + UINT64_C(0xffffffffffffffff)); + assert(recovered_cache != 0); + + return recovered_cache; + } + } + }; +#if !JKJ_HAS_INLINE_VARIABLE + template <class Dummy> + constexpr typename compressed_cache_holder<ieee754_binary32, Dummy>::cache_holder_t + compressed_cache_holder<ieee754_binary32, Dummy>::cache; + template <class Dummy> + constexpr typename compressed_cache_holder<ieee754_binary32, Dummy>::pow5_holder_t + compressed_cache_holder<ieee754_binary32, Dummy>::pow5_table; +#endif + + template <class Dummy> + struct compressed_cache_holder<ieee754_binary64, Dummy> { + using cache_entry_type = cache_holder<ieee754_binary64>::cache_entry_type; + static constexpr int cache_bits = cache_holder<ieee754_binary64>::cache_bits; + static constexpr int min_k = cache_holder<ieee754_binary64>::min_k; + static constexpr int max_k = cache_holder<ieee754_binary64>::max_k; + static constexpr int compression_ratio = 27; + static constexpr detail::stdr::size_t compressed_table_size = + detail::stdr::size_t((max_k - min_k + compression_ratio) / compression_ratio); + static constexpr detail::stdr::size_t pow5_table_size = + detail::stdr::size_t(compression_ratio); + + using cache_holder_t = detail::array<cache_entry_type, compressed_table_size>; + using pow5_holder_t = detail::array<detail::stdr::uint_least64_t, pow5_table_size>; + +#if JKJ_HAS_CONSTEXPR17 + static constexpr cache_holder_t cache JKJ_STATIC_DATA_SECTION = [] { + cache_holder_t res{}; + for (detail::stdr::size_t i = 0; i < compressed_table_size; ++i) { + res[i] = cache_holder<ieee754_binary64>::cache[i * compression_ratio]; + } + return res; + }(); + static constexpr pow5_holder_t pow5_table JKJ_STATIC_DATA_SECTION = [] { + pow5_holder_t res{}; + detail::stdr::uint_least64_t p = 1; + for (detail::stdr::size_t i = 0; i < pow5_table_size; ++i) { + res[i] = p; + p *= 5; + } + return res; + }(); +#else + template <detail::stdr::size_t... indices> + static constexpr cache_holder_t make_cache(detail::index_sequence<indices...>) { + return {cache_holder<ieee754_binary64>::cache[indices * compression_ratio]...}; + } + static constexpr cache_holder_t cache JKJ_STATIC_DATA_SECTION = + make_cache(detail::make_index_sequence<compressed_table_size>{}); + + template <detail::stdr::size_t... indices> + static constexpr pow5_holder_t make_pow5_table(detail::index_sequence<indices...>) { + return {detail::compute_power<indices>(detail::stdr::uint_least64_t(5))...}; + } + static constexpr pow5_holder_t pow5_table JKJ_STATIC_DATA_SECTION = + make_pow5_table(detail::make_index_sequence<pow5_table_size>{}); +#endif + + template <class ShiftAmountType, class DecimalExponentType> + static JKJ_CONSTEXPR20 cache_entry_type get_cache(DecimalExponentType k) noexcept { + // Compute the base index. + // Supposed to compute (k - min_k) / compression_ratio. + static_assert(max_k - min_k <= 619 && compression_ratio == 27, ""); + static_assert(max_k - min_k <= detail::stdr::numeric_limits<DecimalExponentType>::max(), + ""); + auto const cache_index = + DecimalExponentType(detail::stdr::uint_fast32_t(DecimalExponentType(k - min_k) * + detail::stdr::int_fast32_t(607)) >> + 14); + auto const kb = DecimalExponentType(cache_index * compression_ratio + min_k); + auto const offset = DecimalExponentType(k - kb); + + // Get the base cache. + auto const base_cache = cache[cache_index]; + + if (offset == 0) { + return base_cache; + } + else { + // Compute the required amount of bit-shift. + auto const alpha = + ShiftAmountType(detail::log::floor_log2_pow10<min_k, max_k>(k) - + detail::log::floor_log2_pow10<min_k, max_k>(kb) - offset); + assert(alpha > 0 && alpha < 64); + + // Try to recover the real cache. + auto const pow5 = pow5_table[offset]; + auto recovered_cache = detail::wuint::umul128(base_cache.high(), pow5); + auto const middle_low = detail::wuint::umul128(base_cache.low(), pow5); + + recovered_cache += middle_low.high(); + + auto const high_to_middle = detail::stdr::uint_least64_t( + (recovered_cache.high() << ShiftAmountType(64 - alpha)) & + UINT64_C(0xffffffffffffffff)); + auto const middle_to_low = detail::stdr::uint_least64_t( + (recovered_cache.low() << ShiftAmountType(64 - alpha)) & + UINT64_C(0xffffffffffffffff)); + + recovered_cache = {(recovered_cache.low() >> alpha) | high_to_middle, + ((middle_low.low() >> alpha) | middle_to_low)}; + + assert(recovered_cache.low() != UINT64_C(0xffffffffffffffff)); + recovered_cache = {recovered_cache.high(), + detail::stdr::uint_least64_t(recovered_cache.low() + 1)}; + + return recovered_cache; + } + } + }; +#if !JKJ_HAS_INLINE_VARIABLE + template <class Dummy> + constexpr typename compressed_cache_holder<ieee754_binary64, Dummy>::cache_holder_t + compressed_cache_holder<ieee754_binary64, Dummy>::cache; + template <class Dummy> + constexpr typename compressed_cache_holder<ieee754_binary64, Dummy>::pow5_holder_t + compressed_cache_holder<ieee754_binary64, Dummy>::pow5_table; +#endif + + //////////////////////////////////////////////////////////////////////////////////////// + // Forward declarations of user-specializable templates used in the main algorithm. + //////////////////////////////////////////////////////////////////////////////////////// + + // Remove trailing zeros from significand and add the number of removed zeros into + // exponent. + template <class TrailingZeroPolicy, class Format, class DecimalSignificand, + class DecimalExponentType> + struct remove_trailing_zeros_traits; + + // Users can specialize this traits class to make Dragonbox work with their own formats. + // However, this requires detailed knowledge on how the algorithm works, so it is recommended to + // read through the paper. + template <class FormatTraits, class CacheEntryType, detail::stdr::size_t cache_bits_> + struct multiplication_traits; + + // A collection of some common definitions to reduce boilerplate. + template <class FormatTraits, class CacheEntryType, detail::stdr::size_t cache_bits_> + struct multiplication_traits_base { + using format = typename FormatTraits::format; + static constexpr int significand_bits = format::significand_bits; + static constexpr int total_bits = format::total_bits; + using carrier_uint = typename FormatTraits::carrier_uint; + using cache_entry_type = CacheEntryType; + static constexpr int cache_bits = int(cache_bits_); + + struct compute_mul_result { + carrier_uint integer_part; + bool is_integer; + }; + struct compute_mul_parity_result { + bool parity; + bool is_integer; + }; + }; + + + //////////////////////////////////////////////////////////////////////////////////////// + // Policies. + //////////////////////////////////////////////////////////////////////////////////////// + + namespace detail { + template <class T> + struct dummy {}; + } + + namespace policy { + namespace sign { + JKJ_INLINE_VARIABLE struct ignore_t { + using sign_policy = ignore_t; + static constexpr bool return_has_sign = false; + +#if defined(_MSC_VER) && !defined(__clang__) + // See + // https://developercommunity.visualstudio.com/t/Failure-to-optimize-intrinsics/10628226 + template <class SignedSignificandBits, class DecimalSignificand, + class DecimalExponentType> + static constexpr decimal_fp<DecimalSignificand, DecimalExponentType, false, false> + handle_sign( + SignedSignificandBits, + decimal_fp<DecimalSignificand, DecimalExponentType, false, false> r) noexcept { + return {r.significand, r.exponent}; + } + template <class SignedSignificandBits, class DecimalSignificand, + class DecimalExponentType> + static constexpr decimal_fp<DecimalSignificand, DecimalExponentType, false, true> + handle_sign( + SignedSignificandBits, + decimal_fp<DecimalSignificand, DecimalExponentType, false, true> r) noexcept { + return {r.significand, r.exponent, r.may_have_trailing_zeros}; + } +#else + template <class SignedSignificandBits, class UnsignedDecimalFp> + static constexpr UnsignedDecimalFp handle_sign(SignedSignificandBits, + UnsignedDecimalFp r) noexcept { + return r; + } +#endif + } ignore = {}; + + JKJ_INLINE_VARIABLE struct return_sign_t { + using sign_policy = return_sign_t; + static constexpr bool return_has_sign = true; + + template <class SignedSignificandBits, class UnsignedDecimalFp> + static constexpr detail::unsigned_decimal_fp_to_signed_t<UnsignedDecimalFp> + handle_sign(SignedSignificandBits s, UnsignedDecimalFp r) noexcept { + return add_sign_to_unsigned_decimal_fp(s.is_negative(), r); + } + } return_sign = {}; + } + + namespace trailing_zero { + JKJ_INLINE_VARIABLE struct ignore_t { + using trailing_zero_policy = ignore_t; + static constexpr bool report_trailing_zeros = false; + + template <class Format, class DecimalSignificand, class DecimalExponentType> + static constexpr unsigned_decimal_fp<DecimalSignificand, DecimalExponentType, false> + on_trailing_zeros(DecimalSignificand significand, + DecimalExponentType exponent) noexcept { + return {significand, exponent}; + } + + template <class Format, class DecimalSignificand, class DecimalExponentType> + static constexpr unsigned_decimal_fp<DecimalSignificand, DecimalExponentType, false> + no_trailing_zeros(DecimalSignificand significand, + DecimalExponentType exponent) noexcept { + return {significand, exponent}; + } + } ignore = {}; + + JKJ_INLINE_VARIABLE struct remove_t { + using trailing_zero_policy = remove_t; + static constexpr bool report_trailing_zeros = false; + + template <class Format, class DecimalSignificand, class DecimalExponentType> + JKJ_FORCEINLINE static JKJ_CONSTEXPR14 + unsigned_decimal_fp<DecimalSignificand, DecimalExponentType, false> + on_trailing_zeros(DecimalSignificand significand, + DecimalExponentType exponent) noexcept { + remove_trailing_zeros_traits< + remove_t, Format, DecimalSignificand, + DecimalExponentType>::remove_trailing_zeros(significand, exponent); + return {significand, exponent}; + } + + template <class Format, class DecimalSignificand, class DecimalExponentType> + static constexpr unsigned_decimal_fp<DecimalSignificand, DecimalExponentType, false> + no_trailing_zeros(DecimalSignificand significand, + DecimalExponentType exponent) noexcept { + return {significand, exponent}; + } + } remove = {}; + + JKJ_INLINE_VARIABLE struct remove_compact_t { + using trailing_zero_policy = remove_compact_t; + static constexpr bool report_trailing_zeros = false; + + template <class Format, class DecimalSignificand, class DecimalExponentType> + JKJ_FORCEINLINE static JKJ_CONSTEXPR14 + unsigned_decimal_fp<DecimalSignificand, DecimalExponentType, false> + on_trailing_zeros(DecimalSignificand significand, + DecimalExponentType exponent) noexcept { + remove_trailing_zeros_traits< + remove_compact_t, Format, DecimalSignificand, + DecimalExponentType>::remove_trailing_zeros(significand, exponent); + return {significand, exponent}; + } + + template <class Format, class DecimalSignificand, class DecimalExponentType> + static constexpr unsigned_decimal_fp<DecimalSignificand, DecimalExponentType, false> + no_trailing_zeros(DecimalSignificand significand, + DecimalExponentType exponent) noexcept { + return {significand, exponent}; + } + } remove_compact = {}; + + JKJ_INLINE_VARIABLE struct report_t { + using trailing_zero_policy = report_t; + static constexpr bool report_trailing_zeros = true; + + template <class Format, class DecimalSignificand, class DecimalExponentType> + static constexpr unsigned_decimal_fp<DecimalSignificand, DecimalExponentType, true> + on_trailing_zeros(DecimalSignificand significand, + DecimalExponentType exponent) noexcept { + return {significand, exponent, true}; + } + + template <class Format, class DecimalSignificand, class DecimalExponentType> + static constexpr unsigned_decimal_fp<DecimalSignificand, DecimalExponentType, true> + no_trailing_zeros(DecimalSignificand significand, + DecimalExponentType exponent) noexcept { + return {significand, exponent, false}; + } + } report = {}; + } + + namespace decimal_to_binary_rounding { + 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; } + }; + } + + JKJ_INLINE_VARIABLE struct nearest_to_even_t { + using decimal_to_binary_rounding_policy = nearest_to_even_t; + using interval_type_provider = nearest_to_even_t; + static constexpr auto tag = tag_t::to_nearest; + + template <class SignedSignificandBits, class Func, class... Args> + JKJ_FORCEINLINE JKJ_SAFEBUFFERS static constexpr decltype(Func{}( + dragonbox::detail::declval<nearest_to_even_t>(), Args{}...)) + delegate(SignedSignificandBits, Func f, Args... args) noexcept { + return f(nearest_to_even_t{}, args...); + } + + template <class SignedSignificandBits> + static constexpr interval_type::symmetric_boundary + normal_interval(SignedSignificandBits s) noexcept { + return {s.has_even_significand_bits()}; + } + + template <class SignedSignificandBits> + static constexpr interval_type::closed + shorter_interval(SignedSignificandBits) noexcept { + return {}; + } + } nearest_to_even = {}; + + JKJ_INLINE_VARIABLE struct nearest_to_odd_t { + using decimal_to_binary_rounding_policy = nearest_to_odd_t; + using interval_type_provider = nearest_to_odd_t; + static constexpr auto tag = tag_t::to_nearest; + + template <class SignedSignificandBits, class Func, class... Args> + JKJ_FORCEINLINE JKJ_SAFEBUFFERS static constexpr decltype(Func{}( + dragonbox::detail::declval<nearest_to_odd_t>(), Args{}...)) + delegate(SignedSignificandBits, Func f, Args... args) noexcept { + return f(nearest_to_odd_t{}, args...); + } + + template <class SignedSignificandBits> + static constexpr interval_type::symmetric_boundary + normal_interval(SignedSignificandBits s) noexcept { + return {!s.has_even_significand_bits()}; + } + template <class SignedSignificandBits> + static constexpr interval_type::open + shorter_interval(SignedSignificandBits) noexcept { + return {}; + } + } nearest_to_odd = {}; + + JKJ_INLINE_VARIABLE struct nearest_toward_plus_infinity_t { + using decimal_to_binary_rounding_policy = nearest_toward_plus_infinity_t; + using interval_type_provider = nearest_toward_plus_infinity_t; + static constexpr auto tag = tag_t::to_nearest; + + template <class SignedSignificandBits, class Func, class... Args> + JKJ_FORCEINLINE JKJ_SAFEBUFFERS static constexpr decltype(Func{}( + dragonbox::detail::declval<nearest_toward_plus_infinity_t>(), Args{}...)) + delegate(SignedSignificandBits, Func f, Args... args) noexcept { + return f(nearest_toward_plus_infinity_t{}, args...); + } + + template <class SignedSignificandBits> + static constexpr interval_type::asymmetric_boundary + normal_interval(SignedSignificandBits s) noexcept { + return {!s.is_negative()}; + } + template <class SignedSignificandBits> + static constexpr interval_type::asymmetric_boundary + shorter_interval(SignedSignificandBits s) noexcept { + return {!s.is_negative()}; + } + } nearest_toward_plus_infinity = {}; + + JKJ_INLINE_VARIABLE struct nearest_toward_minus_infinity_t { + using decimal_to_binary_rounding_policy = nearest_toward_minus_infinity_t; + using interval_type_provider = nearest_toward_minus_infinity_t; + static constexpr auto tag = tag_t::to_nearest; + + template <class SignedSignificandBits, class Func, class... Args> + JKJ_FORCEINLINE JKJ_SAFEBUFFERS static constexpr decltype(Func{}( + dragonbox::detail::declval<nearest_toward_minus_infinity_t>(), Args{}...)) + delegate(SignedSignificandBits, Func f, Args... args) noexcept { + return f(nearest_toward_minus_infinity_t{}, args...); + } + + template <class SignedSignificandBits> + static constexpr interval_type::asymmetric_boundary + normal_interval(SignedSignificandBits s) noexcept { + return {s.is_negative()}; + } + template <class SignedSignificandBits> + static constexpr interval_type::asymmetric_boundary + shorter_interval(SignedSignificandBits s) noexcept { + return {s.is_negative()}; + } + } nearest_toward_minus_infinity = {}; + + JKJ_INLINE_VARIABLE struct nearest_toward_zero_t { + using decimal_to_binary_rounding_policy = nearest_toward_zero_t; + using interval_type_provider = nearest_toward_zero_t; + static constexpr auto tag = tag_t::to_nearest; + + template <class SignedSignificandBits, class Func, class... Args> + JKJ_FORCEINLINE JKJ_SAFEBUFFERS static constexpr decltype(Func{}( + dragonbox::detail::declval<nearest_toward_zero_t>(), Args{}...)) + delegate(SignedSignificandBits, Func f, Args... args) noexcept { + return f(nearest_toward_zero_t{}, args...); + } + + template <class SignedSignificandBits> + static constexpr interval_type::right_closed_left_open + normal_interval(SignedSignificandBits) noexcept { + return {}; + } + template <class SignedSignificandBits> + static constexpr interval_type::right_closed_left_open + shorter_interval(SignedSignificandBits) noexcept { + return {}; + } + } nearest_toward_zero = {}; + + JKJ_INLINE_VARIABLE struct nearest_away_from_zero_t { + using decimal_to_binary_rounding_policy = nearest_away_from_zero_t; + using interval_type_provider = nearest_away_from_zero_t; + static constexpr auto tag = tag_t::to_nearest; + + template <class SignedSignificandBits, class Func, class... Args> + JKJ_FORCEINLINE JKJ_SAFEBUFFERS static constexpr decltype(Func{}( + dragonbox::detail::declval<nearest_away_from_zero_t>(), Args{}...)) + delegate(SignedSignificandBits, Func f, Args... args) noexcept { + return f(nearest_away_from_zero_t{}, args...); + } + + template <class SignedSignificandBits> + static constexpr interval_type::left_closed_right_open + normal_interval(SignedSignificandBits) noexcept { + return {}; + } + template <class SignedSignificandBits> + static constexpr interval_type::left_closed_right_open + shorter_interval(SignedSignificandBits) noexcept { + return {}; + } + } nearest_away_from_zero = {}; + + namespace detail { + struct nearest_always_closed_t { + using interval_type_provider = nearest_always_closed_t; + static constexpr auto tag = tag_t::to_nearest; + + template <class SignedSignificandBits> + static constexpr interval_type::closed + normal_interval(SignedSignificandBits) noexcept { + return {}; + } + template <class SignedSignificandBits> + static constexpr interval_type::closed + shorter_interval(SignedSignificandBits) noexcept { + return {}; + } + }; + struct nearest_always_open_t { + using interval_type_provider = nearest_always_open_t; + static constexpr auto tag = tag_t::to_nearest; + + template <class SignedSignificandBits> + static constexpr interval_type::open + normal_interval(SignedSignificandBits) noexcept { + return {}; + } + template <class SignedSignificandBits> + static constexpr interval_type::open + shorter_interval(SignedSignificandBits) noexcept { + return {}; + } + }; + } + + JKJ_INLINE_VARIABLE struct nearest_to_even_static_boundary_t { + using decimal_to_binary_rounding_policy = nearest_to_even_static_boundary_t; + + template <class SignedSignificandBits, class Func, class... Args> + JKJ_FORCEINLINE JKJ_SAFEBUFFERS static constexpr decltype(Func{}( + detail::nearest_always_closed_t{}, Args{}...)) + delegate(SignedSignificandBits s, Func f, Args... args) noexcept { + return s.has_even_significand_bits() + ? f(detail::nearest_always_closed_t{}, args...) + : f(detail::nearest_always_open_t{}, args...); + } + } nearest_to_even_static_boundary = {}; + + JKJ_INLINE_VARIABLE struct nearest_to_odd_static_boundary_t { + using decimal_to_binary_rounding_policy = nearest_to_odd_static_boundary_t; + + template <class SignedSignificandBits, class Func, class... Args> + JKJ_FORCEINLINE JKJ_SAFEBUFFERS static constexpr decltype(Func{}( + detail::nearest_always_closed_t{}, Args{}...)) + delegate(SignedSignificandBits s, Func f, Args... args) noexcept { + return s.has_even_significand_bits() + ? f(detail::nearest_always_open_t{}, args...) + : f(detail::nearest_always_closed_t{}, args...); + } + } nearest_to_odd_static_boundary = {}; + + JKJ_INLINE_VARIABLE struct nearest_toward_plus_infinity_static_boundary_t { + using decimal_to_binary_rounding_policy = + nearest_toward_plus_infinity_static_boundary_t; + + template <class SignedSignificandBits, class Func, class... Args> + JKJ_FORCEINLINE + JKJ_SAFEBUFFERS static constexpr decltype(Func{}(nearest_toward_zero, + Args{}...)) + delegate(SignedSignificandBits s, Func f, Args... args) noexcept { + return s.is_negative() ? f(nearest_toward_zero, args...) + : f(nearest_away_from_zero, args...); + } + } nearest_toward_plus_infinity_static_boundary = {}; + + JKJ_INLINE_VARIABLE struct nearest_toward_minus_infinity_static_boundary_t { + using decimal_to_binary_rounding_policy = + nearest_toward_minus_infinity_static_boundary_t; + + template <class SignedSignificandBits, class Func, class... Args> + JKJ_FORCEINLINE + JKJ_SAFEBUFFERS static constexpr decltype(Func{}(nearest_toward_zero, + Args{}...)) + delegate(SignedSignificandBits s, Func f, Args... args) noexcept { + return s.is_negative() ? f(nearest_away_from_zero, args...) + : f(nearest_toward_zero, args...); + } + } nearest_toward_minus_infinity_static_boundary = {}; + + namespace detail { + struct left_closed_directed_t { + static constexpr auto tag = tag_t::left_closed_directed; + }; + struct right_closed_directed_t { + static constexpr auto tag = tag_t::right_closed_directed; + }; + } + + JKJ_INLINE_VARIABLE struct toward_plus_infinity_t { + using decimal_to_binary_rounding_policy = toward_plus_infinity_t; + + template <class SignedSignificandBits, class Func, class... Args> + JKJ_FORCEINLINE JKJ_SAFEBUFFERS static constexpr decltype(Func{}( + detail::left_closed_directed_t{}, Args{}...)) + delegate(SignedSignificandBits s, Func f, Args... args) noexcept { + return s.is_negative() ? f(detail::left_closed_directed_t{}, args...) + : f(detail::right_closed_directed_t{}, args...); + } + } toward_plus_infinity = {}; + + JKJ_INLINE_VARIABLE struct toward_minus_infinity_t { + using decimal_to_binary_rounding_policy = toward_minus_infinity_t; + + template <class SignedSignificandBits, class Func, class... Args> + JKJ_FORCEINLINE JKJ_SAFEBUFFERS static constexpr decltype(Func{}( + detail::left_closed_directed_t{}, Args{}...)) + delegate(SignedSignificandBits s, Func f, Args... args) noexcept { + return s.is_negative() ? f(detail::right_closed_directed_t{}, args...) + : f(detail::left_closed_directed_t{}, args...); + } + } toward_minus_infinity = {}; + + JKJ_INLINE_VARIABLE struct toward_zero_t { + using decimal_to_binary_rounding_policy = toward_zero_t; + + template <class SignedSignificandBits, class Func, class... Args> + JKJ_FORCEINLINE JKJ_SAFEBUFFERS static constexpr decltype(Func{}( + detail::left_closed_directed_t{}, Args{}...)) + delegate(SignedSignificandBits, Func f, Args... args) noexcept { + return f(detail::left_closed_directed_t{}, args...); + } + } toward_zero = {}; + + JKJ_INLINE_VARIABLE struct away_from_zero_t { + using decimal_to_binary_rounding_policy = away_from_zero_t; + + template <class SignedSignificandBits, class Func, class... Args> + JKJ_FORCEINLINE JKJ_SAFEBUFFERS static constexpr decltype(Func{}( + detail::right_closed_directed_t{}, Args{}...)) + delegate(SignedSignificandBits, Func f, Args... args) noexcept { + return f(detail::right_closed_directed_t{}, args...); + } + } away_from_zero = {}; + } + + namespace binary_to_decimal_rounding { + // (Always assumes nearest rounding modes, as there can be no tie for other rounding + // modes.) + enum class tag_t { do_not_care, to_even, to_odd, away_from_zero, toward_zero }; + + // The parameter significand corresponds to 10\tilde{s}+t in the paper. + + JKJ_INLINE_VARIABLE struct do_not_care_t { + using binary_to_decimal_rounding_policy = do_not_care_t; + static constexpr auto tag = tag_t::do_not_care; + + template <class CarrierUInt> + static constexpr bool prefer_round_down(CarrierUInt) noexcept { + return false; + } + } do_not_care = {}; + + JKJ_INLINE_VARIABLE struct to_even_t { + using binary_to_decimal_rounding_policy = to_even_t; + static constexpr auto tag = tag_t::to_even; + + template <class CarrierUInt> + static constexpr bool prefer_round_down(CarrierUInt significand) noexcept { + return significand % 2 != 0; + } + } to_even = {}; + + JKJ_INLINE_VARIABLE struct to_odd_t { + using binary_to_decimal_rounding_policy = to_odd_t; + static constexpr auto tag = tag_t::to_odd; + + template <class CarrierUInt> + static constexpr bool prefer_round_down(CarrierUInt significand) noexcept { + return significand % 2 == 0; + } + } to_odd = {}; + + JKJ_INLINE_VARIABLE struct away_from_zero_t { + using binary_to_decimal_rounding_policy = away_from_zero_t; + static constexpr auto tag = tag_t::away_from_zero; + + template <class CarrierUInt> + static constexpr bool prefer_round_down(CarrierUInt) noexcept { + return false; + } + } away_from_zero = {}; + + JKJ_INLINE_VARIABLE struct toward_zero_t { + using binary_to_decimal_rounding_policy = toward_zero_t; + static constexpr auto tag = tag_t::toward_zero; + + template <class CarrierUInt> + static constexpr bool prefer_round_down(CarrierUInt) noexcept { + return true; + } + } toward_zero = {}; + } + + namespace cache { + JKJ_INLINE_VARIABLE struct full_t { + using cache_policy = full_t; + template <class FloatFormat> + using cache_holder_type = cache_holder<FloatFormat>; + + template <class FloatFormat, class ShiftAmountType, class DecimalExponentType> + static constexpr typename cache_holder_type<FloatFormat>::cache_entry_type + get_cache(DecimalExponentType k) noexcept { +#if JKJ_HAS_CONSTEXPR14 + assert(k >= cache_holder_type<FloatFormat>::min_k && + k <= cache_holder_type<FloatFormat>::max_k); +#endif + return cache_holder_type<FloatFormat>::cache[detail::stdr::size_t( + k - cache_holder_type<FloatFormat>::min_k)]; + } + } full = {}; + + JKJ_INLINE_VARIABLE struct compact_t { + using cache_policy = compact_t; + template <class FloatFormat> + using cache_holder_type = compressed_cache_holder<FloatFormat>; + + template <class FloatFormat, class ShiftAmountType, class DecimalExponentType> + static JKJ_CONSTEXPR20 typename cache_holder<FloatFormat>::cache_entry_type + get_cache(DecimalExponentType k) noexcept { + assert(k >= cache_holder<FloatFormat>::min_k && + k <= cache_holder<FloatFormat>::max_k); + + return cache_holder_type<FloatFormat>::template get_cache<ShiftAmountType>(k); + } + } compact = {}; + } + + namespace preferred_integer_types { + JKJ_INLINE_VARIABLE struct match_t { + using preferred_integer_types_policy = match_t; + + template <class FormatTraits, detail::stdr::uint_least64_t upper_bound> + using remainder_type = typename FormatTraits::carrier_uint; + + template <class FormatTraits, detail::stdr::int_least32_t lower_bound, + detail::stdr::uint_least32_t upper_bound> + using decimal_exponent_type = typename FormatTraits::exponent_int; + + template <class FormatTraits> + using shift_amount_type = typename FormatTraits::exponent_int; + } match; + + JKJ_INLINE_VARIABLE struct prefer_32_t { + using preferred_integer_types_policy = prefer_32_t; + + template <class FormatTraits, detail::stdr::uint_least64_t upper_bound> + using remainder_type = typename detail::stdr::conditional< + upper_bound <= + detail::stdr::numeric_limits<detail::stdr::uint_least32_t>::max(), + detail::stdr::uint_least32_t, typename FormatTraits::carrier_uint>::type; + + template <class FormatTraits, detail::stdr::int_least32_t lower_bound, + detail::stdr::uint_least32_t upper_bound> + using decimal_exponent_type = typename detail::stdr::conditional< + FormatTraits::format::exponent_bits <= + detail::value_bits<detail::stdr::int_least32_t>::value, + detail::stdr::int_least32_t, typename FormatTraits::exponent_int>::type; + + template <class FormatTraits> + using shift_amount_type = detail::stdr::int_least32_t; + } prefer_32; + + JKJ_INLINE_VARIABLE struct minimal_t { + using preferred_integer_types_policy = minimal_t; + + template <class FormatTraits, detail::stdr::uint_least64_t upper_bound> + using remainder_type = typename detail::stdr::conditional< + upper_bound <= detail::stdr::numeric_limits<detail::stdr::uint_least8_t>::max(), + detail::stdr::uint_least8_t, + typename detail::stdr::conditional< + upper_bound <= + detail::stdr::numeric_limits<detail::stdr::uint_least16_t>::max(), + detail::stdr::uint_least16_t, + typename detail::stdr::conditional< + upper_bound <= + detail::stdr::numeric_limits<detail::stdr::uint_least32_t>::max(), + detail::stdr::uint_least32_t, + typename detail::stdr::conditional< + upper_bound <= detail::stdr::numeric_limits< + detail::stdr::uint_least64_t>::max(), + detail::stdr::uint_least64_t, + typename FormatTraits::carrier_uint>::type>::type>::type>::type; + + template <class FormatTraits, detail::stdr::int_least32_t lower_bound, + detail::stdr::uint_least32_t upper_bound> + using decimal_exponent_type = typename detail::stdr::conditional< + lower_bound >= + detail::stdr::numeric_limits<detail::stdr::int_least8_t>::min() && + upper_bound <= + detail::stdr::numeric_limits<detail::stdr::int_least8_t>::max(), + detail::stdr::int_least8_t, + typename detail::stdr::conditional< + lower_bound >= + detail::stdr::numeric_limits<detail::stdr::int_least16_t>::min() && + upper_bound <= + detail::stdr::numeric_limits<detail::stdr::int_least16_t>::max(), + detail::stdr::int_least16_t, + typename detail::stdr::conditional< + lower_bound >= detail::stdr::numeric_limits< + detail::stdr::int_least32_t>::min() && + upper_bound <= detail::stdr::numeric_limits< + detail::stdr::int_least32_t>::max(), + detail::stdr::int_least32_t, + typename FormatTraits::exponent_int>::type>::type>::type; + + template <class FormatTraits> + using shift_amount_type = detail::stdr::int_least8_t; + } minimal; + } + } + + //////////////////////////////////////////////////////////////////////////////////////// + // Specializations of user-specializable templates used in the main algorithm. + //////////////////////////////////////////////////////////////////////////////////////// + + template <class DecimalExponentType> + struct remove_trailing_zeros_traits<policy::trailing_zero::remove_t, ieee754_binary32, + detail::stdr::uint_least32_t, DecimalExponentType> { + JKJ_FORCEINLINE static JKJ_CONSTEXPR14 void + remove_trailing_zeros(detail::stdr::uint_least32_t& significand, + DecimalExponentType& exponent) noexcept { + // See https://github.com/jk-jeon/rtz_benchmark. + // The idea of branchless search below is by reddit users r/pigeon768 and + // r/TheoreticalDumbass. + + auto r = detail::bits::rotr<32>( + detail::stdr::uint_least32_t(significand * UINT32_C(184254097)), 4); + auto b = r < UINT32_C(429497); + auto s = detail::stdr::size_t(b); + significand = b ? r : significand; + + r = detail::bits::rotr<32>( + detail::stdr::uint_least32_t(significand * UINT32_C(42949673)), 2); + b = r < UINT32_C(42949673); + s = s * 2 + b; + significand = b ? r : significand; + + r = detail::bits::rotr<32>( + detail::stdr::uint_least32_t(significand * UINT32_C(1288490189)), 1); + b = r < UINT32_C(429496730); + s = s * 2 + b; + significand = b ? r : significand; + + assert(s < detail::stdr::numeric_limits<DecimalExponentType>::max()); + auto sAddend = static_cast<DecimalExponentType>(s); + assert(detail::stdr::numeric_limits<DecimalExponentType>::max() - sAddend > exponent); + exponent += sAddend; + } + }; + + template <class DecimalExponentType> + struct remove_trailing_zeros_traits<policy::trailing_zero::remove_t, ieee754_binary64, + detail::stdr::uint_least64_t, DecimalExponentType> { + JKJ_FORCEINLINE static JKJ_CONSTEXPR14 void + remove_trailing_zeros(detail::stdr::uint_least64_t& significand, + DecimalExponentType& exponent) noexcept { + // See https://github.com/jk-jeon/rtz_benchmark. + // The idea of branchless search below is by reddit users r/pigeon768 and + // r/TheoreticalDumbass. + + auto r = detail::bits::rotr<64>( + detail::stdr::uint_least64_t(significand * UINT64_C(28999941890838049)), 8); + auto b = r < UINT64_C(184467440738); + auto s = detail::stdr::size_t(b); + significand = b ? r : significand; + + r = detail::bits::rotr<64>( + detail::stdr::uint_least64_t(significand * UINT64_C(182622766329724561)), 4); + b = r < UINT64_C(1844674407370956); + s = s * 2 + b; + significand = b ? r : significand; + + r = detail::bits::rotr<64>( + detail::stdr::uint_least64_t(significand * UINT64_C(10330176681277348905)), 2); + b = r < UINT64_C(184467440737095517); + s = s * 2 + b; + significand = b ? r : significand; + + r = detail::bits::rotr<64>( + detail::stdr::uint_least64_t(significand * UINT64_C(14757395258967641293)), 1); + b = r < UINT64_C(1844674407370955162); + s = s * 2 + b; + significand = b ? r : significand; + + assert(s < detail::stdr::numeric_limits<DecimalExponentType>::max()); + auto sAddend = static_cast<DecimalExponentType>(s); + assert(detail::stdr::numeric_limits<DecimalExponentType>::max() - sAddend > exponent); + exponent += sAddend; + } + }; + + template <class DecimalExponentType> + struct remove_trailing_zeros_traits<policy::trailing_zero::remove_compact_t, ieee754_binary32, + detail::stdr::uint_least32_t, DecimalExponentType> { + JKJ_FORCEINLINE static JKJ_CONSTEXPR14 void + remove_trailing_zeros(detail::stdr::uint_least32_t& significand, + DecimalExponentType& exponent) noexcept { + // See https://github.com/jk-jeon/rtz_benchmark. + while (true) { + auto const r = detail::stdr::uint_least32_t(significand * UINT32_C(1288490189)); + if (r < UINT32_C(429496731)) { + significand = detail::stdr::uint_least32_t(r >> 1); + exponent += 1; + } + else { + break; + } + } + } + }; + + template <class DecimalExponentType> + struct remove_trailing_zeros_traits<policy::trailing_zero::remove_compact_t, ieee754_binary64, + detail::stdr::uint_least64_t, DecimalExponentType> { + JKJ_FORCEINLINE static JKJ_CONSTEXPR14 void + remove_trailing_zeros(detail::stdr::uint_least64_t& significand, + DecimalExponentType& exponent) noexcept { + // See https://github.com/jk-jeon/rtz_benchmark. + while (true) { + auto const r = + detail::stdr::uint_least64_t(significand * UINT64_C(5534023222112865485)); + if (r < UINT64_C(1844674407370955163)) { + significand = detail::stdr::uint_least64_t(r >> 1); + exponent += 1; + } + else { + break; + } + } + } + }; + + template <class ExponentInt> + struct multiplication_traits< + ieee754_binary_traits<ieee754_binary32, detail::stdr::uint_least32_t, ExponentInt>, + detail::stdr::uint_least64_t, 64> + : public multiplication_traits_base< + ieee754_binary_traits<ieee754_binary32, detail::stdr::uint_least32_t>, + detail::stdr::uint_least64_t, 64> { + static JKJ_CONSTEXPR20 compute_mul_result + compute_mul(carrier_uint u, cache_entry_type const& cache) noexcept { + auto const r = detail::wuint::umul96_upper64(u, cache); + return {carrier_uint(r >> 32), carrier_uint(r) == 0}; + } + + template <class ShiftAmountType> + static constexpr detail::stdr::uint_least64_t compute_delta(cache_entry_type const& cache, + ShiftAmountType beta) noexcept { + return detail::stdr::uint_least64_t(cache >> ShiftAmountType(cache_bits - 1 - beta)); + } + + template <class ShiftAmountType> + static JKJ_CONSTEXPR20 compute_mul_parity_result compute_mul_parity( + carrier_uint two_f, cache_entry_type const& cache, ShiftAmountType beta) noexcept { + assert(beta >= 1); + assert(beta <= 32); + + auto const r = detail::wuint::umul96_lower64(two_f, cache); + return {((r >> ShiftAmountType(64 - beta)) & 1) != 0, + (UINT32_C(0xffffffff) & (r >> ShiftAmountType(32 - beta))) == 0}; + } + + template <class ShiftAmountType> + static constexpr carrier_uint + compute_left_endpoint_for_shorter_interval_case(cache_entry_type const& cache, + ShiftAmountType beta) noexcept { + return carrier_uint((cache - (cache >> (significand_bits + 2))) >> + ShiftAmountType(cache_bits - significand_bits - 1 - beta)); + } + + template <class ShiftAmountType> + static constexpr carrier_uint + compute_right_endpoint_for_shorter_interval_case(cache_entry_type const& cache, + ShiftAmountType beta) noexcept { + return carrier_uint((cache + (cache >> (significand_bits + 1))) >> + ShiftAmountType(cache_bits - significand_bits - 1 - beta)); + } + + template <class ShiftAmountType> + static constexpr carrier_uint + compute_round_up_for_shorter_interval_case(cache_entry_type const& cache, + ShiftAmountType beta) noexcept { + return (carrier_uint(cache >> + ShiftAmountType(cache_bits - significand_bits - 2 - beta)) + + 1) / + 2; + } + }; + + template <class ExponentInt> + struct multiplication_traits< + ieee754_binary_traits<ieee754_binary64, detail::stdr::uint_least64_t, ExponentInt>, + detail::wuint::uint128, 128> + : public multiplication_traits_base< + ieee754_binary_traits<ieee754_binary64, detail::stdr::uint_least64_t>, + detail::wuint::uint128, 128> { + static JKJ_CONSTEXPR20 compute_mul_result + compute_mul(carrier_uint u, cache_entry_type const& cache) noexcept { + auto const r = detail::wuint::umul192_upper128(u, cache); + return {r.high(), r.low() == 0}; + } + + template <class ShiftAmountType> + static constexpr detail::stdr::uint_least64_t compute_delta(cache_entry_type const& cache, + ShiftAmountType beta) noexcept { + return detail::stdr::uint_least64_t(cache.high() >> + ShiftAmountType(total_bits - 1 - beta)); + } + + template <class ShiftAmountType> + static JKJ_CONSTEXPR20 compute_mul_parity_result compute_mul_parity( + carrier_uint two_f, cache_entry_type const& cache, ShiftAmountType beta) noexcept { + assert(beta >= 1); + assert(beta < 64); + + auto const r = detail::wuint::umul192_lower128(two_f, cache); + return {((r.high() >> ShiftAmountType(64 - beta)) & 1) != 0, + (((r.high() << beta) & UINT64_C(0xffffffffffffffff)) | + (r.low() >> ShiftAmountType(64 - beta))) == 0}; + } + + template <class ShiftAmountType> + static constexpr carrier_uint + compute_left_endpoint_for_shorter_interval_case(cache_entry_type const& cache, + ShiftAmountType beta) noexcept { + return (cache.high() - (cache.high() >> (significand_bits + 2))) >> + ShiftAmountType(total_bits - significand_bits - 1 - beta); + } + + template <class ShiftAmountType> + static constexpr carrier_uint + compute_right_endpoint_for_shorter_interval_case(cache_entry_type const& cache, + ShiftAmountType beta) noexcept { + return (cache.high() + (cache.high() >> (significand_bits + 1))) >> + ShiftAmountType(total_bits - significand_bits - 1 - beta); + } + + template <class ShiftAmountType> + static constexpr carrier_uint + compute_round_up_for_shorter_interval_case(cache_entry_type const& cache, + ShiftAmountType beta) noexcept { + return ((cache.high() >> ShiftAmountType(total_bits - significand_bits - 2 - beta)) + + 1) / + 2; + } + }; + + namespace detail { + //////////////////////////////////////////////////////////////////////////////////////// + // The main algorithm. + //////////////////////////////////////////////////////////////////////////////////////// + + template <class FormatTraits> + struct impl : private FormatTraits::format { + using format = typename FormatTraits::format; + using carrier_uint = typename FormatTraits::carrier_uint; + static constexpr int carrier_bits = FormatTraits::carrier_bits; + using exponent_int = typename FormatTraits::exponent_int; + + using format::significand_bits; + using format::min_exponent; + using format::max_exponent; + using format::exponent_bias; + + static constexpr int kappa = + log::floor_log10_pow2(carrier_bits - significand_bits - 2) - 1; + static_assert(kappa >= 1, ""); + static_assert(carrier_bits >= significand_bits + 2 + log::floor_log2_pow10(kappa + 1), + ""); + + static constexpr int min(int x, int y) noexcept { return x < y ? x : y; } + static constexpr int max(int x, int y) noexcept { return x > y ? x : y; } + + static constexpr int min_k = + min(-log::floor_log10_pow2_minus_log10_4_over_3(max_exponent - significand_bits), + -log::floor_log10_pow2(max_exponent - significand_bits) + kappa); + + // We do invoke shorter_interval_case for exponent == min_exponent case, + // so we should not add 1 here. + static constexpr int max_k = + max(-log::floor_log10_pow2_minus_log10_4_over_3(min_exponent - + significand_bits /*+ 1*/), + -log::floor_log10_pow2(min_exponent - significand_bits) + kappa); + + 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; + + template <class PreferredIntegerTypesPolicy> + using remainder_type = typename PreferredIntegerTypesPolicy::template remainder_type< + FormatTraits, compute_power<kappa + 1>(detail::stdr::uint_least64_t(10))>; + + template <class PreferredIntegerTypesPolicy> + using decimal_exponent_type = + typename PreferredIntegerTypesPolicy::template decimal_exponent_type< + FormatTraits, detail::stdr::int_least32_t(min(-max_k, min_k)), + detail::stdr::int_least32_t(max(max_k, -min_k + kappa + 1))>; + + template <class SignPolicy, class TrailingZeroPolicy, class PreferredIntegerTypesPolicy> + using return_type = + decimal_fp<carrier_uint, decimal_exponent_type<PreferredIntegerTypesPolicy>, + SignPolicy::return_has_sign, TrailingZeroPolicy::report_trailing_zeros>; + + //// The main algorithm assumes the input is a normal/subnormal finite number. + + template <class SignPolicy, class TrailingZeroPolicy, class IntervalTypeProvider, + class BinaryToDecimalRoundingPolicy, class CachePolicy, + class PreferredIntegerTypesPolicy> + JKJ_SAFEBUFFERS static JKJ_CONSTEXPR20 + return_type<SignPolicy, TrailingZeroPolicy, PreferredIntegerTypesPolicy> + compute_nearest(signed_significand_bits<FormatTraits> s, + exponent_int exponent_bits) noexcept { + using cache_holder_type = typename CachePolicy::template cache_holder_type<format>; + static_assert( + min_k >= cache_holder_type::min_k && max_k <= cache_holder_type::max_k, ""); + + using remainder_type_ = remainder_type<PreferredIntegerTypesPolicy>; + using decimal_exponent_type_ = decimal_exponent_type<PreferredIntegerTypesPolicy>; + using shift_amount_type = + typename PreferredIntegerTypesPolicy::template shift_amount_type<FormatTraits>; + + using multiplication_traits_ = + multiplication_traits<FormatTraits, + typename cache_holder_type::cache_entry_type, + cache_holder_type::cache_bits>; + + auto two_fc = s.remove_sign_bit_and_shift(); + auto binary_exponent = exponent_bits; + + // Is the input a normal number? + if (binary_exponent != 0) { + binary_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(stdr::is_same<format, ieee754_binary32>::value || + stdr::is_same<format, ieee754_binary64>::value, + ""); + + // Shorter interval case. + if (two_fc == 0) { + auto interval_type = IntervalTypeProvider::shorter_interval(s); + + // Compute k and beta. + auto const minus_k = log::floor_log10_pow2_minus_log10_4_over_3< + min_exponent - format::significand_bits, + max_exponent - format::significand_bits, decimal_exponent_type_>( + binary_exponent); + auto const beta = shift_amount_type( + binary_exponent + + log::floor_log2_pow10<min_k, max_k>(decimal_exponent_type_(-minus_k))); + + // Compute xi and zi. + auto const cache = + CachePolicy::template get_cache<format, shift_amount_type>( + decimal_exponent_type_(-minus_k)); + + auto xi = + multiplication_traits_::compute_left_endpoint_for_shorter_interval_case( + cache, beta); + auto zi = multiplication_traits_:: + 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(binary_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(binary_exponent)) { + ++xi; + } + + // Try bigger divisor. + // zi is at most floor((f_c + 1/2) * 2^e * 10^k0). + // Substituting f_c = 2^p and k0 = -floor(log10(3 * 2^(e-2))), we get + // zi <= floor((2^(p+1) + 1) * 20/3) <= ceil((2^(p+1) + 1)/3) * 20. + // This computation does not overflow for any of the formats I care about. + carrier_uint decimal_significand = div::divide_by_pow10< + 1, carrier_uint, + carrier_uint( + ((((carrier_uint(1) << (significand_bits + 1)) + 1) / 3) + 1) * + 20)>(zi); + + // If succeed, remove trailing zeros if necessary and return. + if (decimal_significand * 10 >= xi) { + return SignPolicy::handle_sign( + s, TrailingZeroPolicy::template on_trailing_zeros<format>( + decimal_significand, decimal_exponent_type_(minus_k + 1))); + } + + // Otherwise, compute the round-up of y. + decimal_significand = + multiplication_traits_::compute_round_up_for_shorter_interval_case( + cache, beta); + + // When tie occurs, choose one of them according to the rule. + if (BinaryToDecimalRoundingPolicy::prefer_round_down(decimal_significand) && + binary_exponent >= shorter_interval_tie_lower_threshold && + binary_exponent <= shorter_interval_tie_upper_threshold) { + --decimal_significand; + } + else if (decimal_significand < xi) { + ++decimal_significand; + } + return SignPolicy::handle_sign( + s, TrailingZeroPolicy::template no_trailing_zeros<format>( + decimal_significand, decimal_exponent_type_(minus_k))); + } + + // Normal interval case. + two_fc |= (carrier_uint(1) << (format::significand_bits + 1)); + } + // Is the input a subnormal number? + else { + // Normal interval case. + binary_exponent = format::min_exponent - format::significand_bits; + } + + ////////////////////////////////////////////////////////////////////// + // Step 1: Schubfach multiplier calculation. + ////////////////////////////////////////////////////////////////////// + + auto interval_type = IntervalTypeProvider::normal_interval(s); + + // Compute k and beta. + auto const minus_k = decimal_exponent_type_( + log::floor_log10_pow2<min_exponent - format::significand_bits, + max_exponent - format::significand_bits, + decimal_exponent_type_>(binary_exponent) - + kappa); + auto const cache = CachePolicy::template get_cache<format, shift_amount_type>( + decimal_exponent_type_(-minus_k)); + auto const beta = + shift_amount_type(binary_exponent + log::floor_log2_pow10<min_k, max_k>( + decimal_exponent_type_(-minus_k))); + + // Compute zi and deltai. + // 10^kappa <= deltai < 10^(kappa + 1) + auto const deltai = static_cast<remainder_type_>( + multiplication_traits_::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 z_result = + multiplication_traits_::compute_mul(carrier_uint((two_fc | 1) << beta), cache); + + + ////////////////////////////////////////////////////////////////////// + // Step 2: Try larger divisor; remove trailing zeros if necessary. + ////////////////////////////////////////////////////////////////////// + + constexpr auto big_divisor = compute_power<kappa + 1>(remainder_type_(10)); + constexpr auto small_divisor = compute_power<kappa>(remainder_type_(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. + carrier_uint decimal_significand = div::divide_by_pow10< + kappa + 1, carrier_uint, + carrier_uint((carrier_uint(1) << (significand_bits + 1)) * big_divisor - 1)>( + z_result.integer_part); + auto r = remainder_type_(z_result.integer_part - big_divisor * decimal_significand); + + do { + if (r < deltai) { + // Exclude the right endpoint if necessary. + if ((r | remainder_type_(!z_result.is_integer) | + remainder_type_(interval_type.include_right_endpoint())) == 0) { + JKJ_IF_CONSTEXPR( + BinaryToDecimalRoundingPolicy::tag == + policy::binary_to_decimal_rounding::tag_t::do_not_care) { + decimal_significand *= 10; + --decimal_significand; + return SignPolicy::handle_sign( + s, TrailingZeroPolicy::template no_trailing_zeros<format>( + decimal_significand, + decimal_exponent_type_(minus_k + kappa))); + } + else { + --decimal_significand; + r = big_divisor; + break; + } + } + } + else if (r > deltai) { + break; + } + else { + // r == deltai; compare fractional parts. + auto const x_result = multiplication_traits_::compute_mul_parity( + carrier_uint(two_fc - 1), cache, beta); + + if (!(x_result.parity | + (x_result.is_integer & interval_type.include_left_endpoint()))) { + break; + } + } + + // We may need to remove trailing zeros. + return SignPolicy::handle_sign( + s, TrailingZeroPolicy::template on_trailing_zeros<format>( + decimal_significand, decimal_exponent_type_(minus_k + kappa + 1))); + } while (false); + + + ////////////////////////////////////////////////////////////////////// + // Step 3: Find the significand with the smaller divisor. + ////////////////////////////////////////////////////////////////////// + + decimal_significand *= 10; + + JKJ_IF_CONSTEXPR(BinaryToDecimalRoundingPolicy::tag == + policy::binary_to_decimal_rounding::tag_t::do_not_care) { + // Normally, we want to compute + // 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 (div::check_divisibility_and_divide_by_pow10<kappa>(r) && + z_result.is_integer) { + // This should be in the interval. + decimal_significand += r - 1; + } + else { + decimal_significand += r; + } + } + else { + decimal_significand += div::small_division_by_pow10<kappa>(r); + } + } + else { + // delta is equal to 10^(kappa + elog10(2) - floor(elog10(2))), so dist cannot + // be larger than r. + auto dist = remainder_type_(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. + decimal_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 y_result = + multiplication_traits_::compute_mul_parity(two_fc, cache, beta); + if (y_result.parity != approx_y_parity) { + --decimal_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( + decimal_significand) & + y_result.is_integer) { + --decimal_significand; + } + } + } + } + return SignPolicy::handle_sign( + s, TrailingZeroPolicy::template no_trailing_zeros<format>( + decimal_significand, decimal_exponent_type_(minus_k + kappa))); + } + + template <class SignPolicy, class TrailingZeroPolicy, class CachePolicy, + class PreferredIntegerTypesPolicy> + JKJ_FORCEINLINE JKJ_SAFEBUFFERS static JKJ_CONSTEXPR20 + return_type<SignPolicy, TrailingZeroPolicy, PreferredIntegerTypesPolicy> + compute_left_closed_directed(signed_significand_bits<FormatTraits> s, + exponent_int exponent_bits) noexcept { + using cache_holder_type = typename CachePolicy::template cache_holder_type<format>; + static_assert( + min_k >= cache_holder_type::min_k && max_k <= cache_holder_type::max_k, ""); + + using remainder_type_ = remainder_type<PreferredIntegerTypesPolicy>; + using decimal_exponent_type_ = decimal_exponent_type<PreferredIntegerTypesPolicy>; + using shift_amount_type = + typename PreferredIntegerTypesPolicy::template shift_amount_type<FormatTraits>; + + using multiplication_traits_ = + multiplication_traits<FormatTraits, + typename cache_holder_type::cache_entry_type, + cache_holder_type::cache_bits>; + + auto two_fc = s.remove_sign_bit_and_shift(); + auto binary_exponent = exponent_bits; + + // Is the input a normal number? + if (binary_exponent != 0) { + binary_exponent += format::exponent_bias - format::significand_bits; + two_fc |= (carrier_uint(1) << (format::significand_bits + 1)); + } + // Is the input a subnormal number? + else { + binary_exponent = format::min_exponent - format::significand_bits; + } + + ////////////////////////////////////////////////////////////////////// + // Step 1: Schubfach multiplier calculation. + ////////////////////////////////////////////////////////////////////// + + // Compute k and beta. + auto const minus_k = decimal_exponent_type_( + log::floor_log10_pow2<format::min_exponent - format::significand_bits, + format::max_exponent - format::significand_bits, + decimal_exponent_type_>(binary_exponent) - + kappa); + auto const cache = CachePolicy::template get_cache<format, shift_amount_type>( + decimal_exponent_type_(-minus_k)); + auto const beta = + shift_amount_type(binary_exponent + log::floor_log2_pow10<min_k, max_k>( + decimal_exponent_type_(-minus_k))); + + // Compute xi and deltai. + // 10^kappa <= deltai < 10^(kappa + 1) + auto const deltai = static_cast<remainder_type_>( + multiplication_traits_::compute_delta(cache, beta)); + auto x_result = + multiplication_traits_::compute_mul(carrier_uint(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. + JKJ_IF_CONSTEXPR(stdr::is_same<format, ieee754_binary32>::value) { + if (binary_exponent <= -80) { + x_result.is_integer = false; + } + } + + if (!x_result.is_integer) { + ++x_result.integer_part; + } + + ////////////////////////////////////////////////////////////////////// + // Step 2: Try larger divisor; remove trailing zeros if necessary. + ////////////////////////////////////////////////////////////////////// + + constexpr auto big_divisor = compute_power<kappa + 1>(remainder_type_(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. + carrier_uint decimal_significand = div::divide_by_pow10< + kappa + 1, carrier_uint, + carrier_uint((carrier_uint(1) << (significand_bits + 1)) * big_divisor - 1)>( + x_result.integer_part); + auto r = remainder_type_(x_result.integer_part - big_divisor * decimal_significand); + + if (r != 0) { + ++decimal_significand; + r = remainder_type_(big_divisor - r); + } + + do { + if (r > deltai) { + break; + } + 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). + // For the case of compressed cache for binary32, there is another + // exceptional case 2f_c = 33554430, e = -10 (16383.9990234375). In this + // case, the recovered cache is two large to make compute_mul_parity + // mistakenly conclude that z is not an integer, but actually z = 16384 is + // an integer. + JKJ_IF_CONSTEXPR( + stdr::is_same<cache_holder_type, + compressed_cache_holder<ieee754_binary32>>::value) { + if (two_fc == 33554430 && binary_exponent == -10) { + break; + } + } + auto const z_result = multiplication_traits_::compute_mul_parity( + carrier_uint(two_fc + 2), cache, beta); + if (z_result.parity || z_result.is_integer) { + break; + } + } + + // The ceiling is inside, so we are done. + return SignPolicy::handle_sign( + s, TrailingZeroPolicy::template on_trailing_zeros<format>( + decimal_significand, decimal_exponent_type_(minus_k + kappa + 1))); + } while (false); + + + ////////////////////////////////////////////////////////////////////// + // Step 3: Find the significand with the smaller divisor. + ////////////////////////////////////////////////////////////////////// + + decimal_significand *= 10; + decimal_significand -= div::small_division_by_pow10<kappa>(r); + return SignPolicy::handle_sign( + s, TrailingZeroPolicy::template no_trailing_zeros<format>( + decimal_significand, decimal_exponent_type_(minus_k + kappa))); + } + + template <class SignPolicy, class TrailingZeroPolicy, class CachePolicy, + class PreferredIntegerTypesPolicy> + JKJ_FORCEINLINE JKJ_SAFEBUFFERS static JKJ_CONSTEXPR20 + return_type<SignPolicy, TrailingZeroPolicy, PreferredIntegerTypesPolicy> + compute_right_closed_directed(signed_significand_bits<FormatTraits> s, + exponent_int exponent_bits) noexcept { + using cache_holder_type = typename CachePolicy::template cache_holder_type<format>; + static_assert( + min_k >= cache_holder_type::min_k && max_k <= cache_holder_type::max_k, ""); + + using remainder_type_ = remainder_type<PreferredIntegerTypesPolicy>; + using decimal_exponent_type_ = decimal_exponent_type<PreferredIntegerTypesPolicy>; + using shift_amount_type = + typename PreferredIntegerTypesPolicy::template shift_amount_type<FormatTraits>; + + using multiplication_traits_ = + multiplication_traits<FormatTraits, + typename cache_holder_type::cache_entry_type, + cache_holder_type::cache_bits>; + + auto two_fc = s.remove_sign_bit_and_shift(); + auto binary_exponent = exponent_bits; + bool shorter_interval = false; + + // Is the input a normal number? + if (binary_exponent != 0) { + if (two_fc == 0 && binary_exponent != 1) { + shorter_interval = true; + } + binary_exponent += format::exponent_bias - format::significand_bits; + two_fc |= (carrier_uint(1) << (format::significand_bits + 1)); + } + // Is the input a subnormal number? + else { + binary_exponent = format::min_exponent - format::significand_bits; + } + + ////////////////////////////////////////////////////////////////////// + // Step 1: Schubfach multiplier calculation. + ////////////////////////////////////////////////////////////////////// + + // Compute k and beta. + auto const minus_k = decimal_exponent_type_( + log::floor_log10_pow2<format::min_exponent - format::significand_bits, + format::max_exponent - format::significand_bits, + decimal_exponent_type_>( + exponent_int(binary_exponent - (shorter_interval ? 1 : 0))) - + kappa); + auto const cache = CachePolicy::template get_cache<format, shift_amount_type>( + decimal_exponent_type_(-minus_k)); + auto const beta = shift_amount_type( + binary_exponent + log::floor_log2_pow10(decimal_exponent_type_(-minus_k))); + + // Compute zi and deltai. + // 10^kappa <= deltai < 10^(kappa + 1) + auto const deltai = + static_cast<remainder_type_>(multiplication_traits_::compute_delta( + cache, shift_amount_type(beta - (shorter_interval ? 1 : 0)))); + carrier_uint const zi = + multiplication_traits_::compute_mul(carrier_uint(two_fc << beta), cache) + .integer_part; + + + ////////////////////////////////////////////////////////////////////// + // Step 2: Try larger divisor; remove trailing zeros if necessary. + ////////////////////////////////////////////////////////////////////// + + constexpr auto big_divisor = compute_power<kappa + 1>(remainder_type_(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. + carrier_uint decimal_significand = div::divide_by_pow10< + kappa + 1, carrier_uint, + carrier_uint((carrier_uint(1) << (significand_bits + 1)) * big_divisor - 1)>( + zi); + auto const r = remainder_type_(zi - big_divisor * decimal_significand); + + do { + if (r > deltai) { + break; + } + else if (r == deltai) { + // Compare the fractional parts. + if (!multiplication_traits_::compute_mul_parity( + carrier_uint(two_fc - (shorter_interval ? 1 : 2)), cache, beta) + .parity) { + break; + } + } + + // The floor is inside, so we are done. + return SignPolicy::handle_sign( + s, TrailingZeroPolicy::template on_trailing_zeros<format>( + decimal_significand, decimal_exponent_type_(minus_k + kappa + 1))); + } while (false); + + + ////////////////////////////////////////////////////////////////////// + // Step 3: Find the significand with the small divisor. + ////////////////////////////////////////////////////////////////////// + + decimal_significand *= 10; + decimal_significand += div::small_division_by_pow10<kappa>(r); + return SignPolicy::handle_sign( + s, TrailingZeroPolicy::template no_trailing_zeros<format>( + decimal_significand, decimal_exponent_type_(minus_k + kappa))); + } + + static constexpr bool + is_right_endpoint_integer_shorter_interval(exponent_int binary_exponent) noexcept { + return binary_exponent >= case_shorter_interval_right_endpoint_lower_threshold && + binary_exponent <= case_shorter_interval_right_endpoint_upper_threshold; + } + + static constexpr bool + is_left_endpoint_integer_shorter_interval(exponent_int binary_exponent) noexcept { + return binary_exponent >= case_shorter_interval_left_endpoint_lower_threshold && + binary_exponent <= case_shorter_interval_left_endpoint_upper_threshold; + } + }; + + + //////////////////////////////////////////////////////////////////////////////////////// + // Policy holder. + //////////////////////////////////////////////////////////////////////////////////////// + + // The library will specify a list of accepted kinds of policies and their defaults, + // and the user will pass a list of policies parameters. The policy parameters are + // supposed to be stateless and only convey information through their types. + // The aim of the helper classes/functions given below 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 at least of one 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. + // The library considers a certain policy parameter to belong to a specific kind if and only + // if the parameter's type has a member type with a specific name; for example, it belongs + // to "sign policy" kind if there is a member type sign_policy. + + // 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 { + // Either the policy parameter type given by the user, or the default policy. + using policy = Policy; + static constexpr auto found_info = info; + }; + + template <class KindDetector, class DefaultPolicy> + struct detector_default_pair { + using kind_detector = KindDetector; + + // Iterate through all given policy parameter types and see if there is a policy + // parameter type belonging to the policy kind specified by KindDetector. + // 1. If there is none, get_found_policy_pair returns + // found_policy_pair<DefaultPolicy, policy_found_info::not_found>. + // 2. If there is only one parameter type belonging to the specified kind, then + // get_found_policy_pair returns + // found_policy_pair<Policy, policy_found_info::unique> + // where Policy is the unique parameter type belonging to the specified kind. + // 3. If there are multiple parameter types belonging to the specified kind, then + // get_found_policy_pair returns + // found_policy_pair<FirstPolicy, policy_found_info::repeated> + // where FirstPolicy is the first parameter type belonging to the specified kind. + // The compilation must fail if this happens. + // This is done by first setting FoundPolicyInfo below to + // found_policy_pair<DefaultPolicy, policy_found_info::not_found>, and then iterate + // over Policies, replacing FoundPolicyInfo by the appropriate one if a parameter + // type belonging to the specified kind is found. + + template <class FoundPolicyInfo, class... Policies> + struct get_found_policy_pair_impl; + + template <class FoundPolicyInfo> + struct get_found_policy_pair_impl<FoundPolicyInfo> { + using type = FoundPolicyInfo; + }; + + template <class FoundPolicyInfo, class FirstPolicy, class... RemainingPolicies> + struct get_found_policy_pair_impl<FoundPolicyInfo, FirstPolicy, RemainingPolicies...> { + using type = typename stdr::conditional< + KindDetector{}(dummy<FirstPolicy>{}), + typename stdr::conditional< + FoundPolicyInfo::found_info == policy_found_info::not_found, + typename get_found_policy_pair_impl< + found_policy_pair<FirstPolicy, policy_found_info::unique>, + RemainingPolicies...>::type, + typename get_found_policy_pair_impl< + found_policy_pair<FirstPolicy, policy_found_info::repeated>, + RemainingPolicies...>::type>::type, + typename get_found_policy_pair_impl<FoundPolicyInfo, + RemainingPolicies...>::type>::type; + }; + + template <class... Policies> + using get_found_policy_pair = typename get_found_policy_pair_impl< + found_policy_pair<DefaultPolicy, policy_found_info::not_found>, Policies...>::type; + }; + + // Simple typelist of detector_default_pair's. + template <class... DetectorDefaultPairs> + struct detector_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(dummy<Policy>, detector_default_pair_list<>) noexcept { + return false; + } + template <class Policy, class FirstDetectorDefaultPair, + class... RemainingDetectorDefaultPairs> + constexpr bool check_policy_validity( + dummy<Policy>, detector_default_pair_list<FirstDetectorDefaultPair, + RemainingDetectorDefaultPairs...>) noexcept { + return typename FirstDetectorDefaultPair::kind_detector{}(dummy<Policy>{}) || + check_policy_validity( + dummy<Policy>{}, + detector_default_pair_list<RemainingDetectorDefaultPairs...>{}); + } + + // Check if all of policies belong to some of the kinds specified by the library. + template <class DetectorDefaultPairList> + constexpr bool check_policy_list_validity(DetectorDefaultPairList) noexcept { + return true; + } + template <class DetectorDefaultPairList, class FirstPolicy, class... RemainingPolicies> + constexpr bool check_policy_list_validity(DetectorDefaultPairList, + dummy<FirstPolicy> first_policy, + dummy<RemainingPolicies>... remaining_policies) { + return check_policy_validity(first_policy, DetectorDefaultPairList{}) && + check_policy_list_validity(DetectorDefaultPairList{}, remaining_policies...); + } + + // Actual policy holder class deriving from all specified policy types. + template <class... Policies> + struct policy_holder : Policies... {}; + + // Iterate through the library-specified list of base-default pairs, i.e., the list of + // policy kinds and their defaults. For each base-default pair, call + // base_default_pair::get_found_policy_pair on the list of user-specified list of + // policies to get found_policy_pair, and build the list of them. + + template <bool repeated_, class... FoundPolicyPairs> + struct found_policy_pair_list { + // This will be set to be true if and only if there exists at least one + // found_policy_pair inside FoundPolicyPairs with + // found_info == policy_found_info::repeated, in which case the compilation must + // fail. + static constexpr bool repeated = repeated_; + }; + + // Iterate through DetectorDefaultPairList and augment FoundPolicyPairList by one at each + // iteration. + template <class DetectorDefaultPairList, class FoundPolicyPairList, class... Policies> + struct make_policy_pair_list_impl; + + // When there is no more detector-default pair to iterate, then the current + // found_policy_pair_list is the final result. + template <bool repeated, class... FoundPolicyPairs, class... Policies> + struct make_policy_pair_list_impl<detector_default_pair_list<>, + found_policy_pair_list<repeated, FoundPolicyPairs...>, + Policies...> { + using type = found_policy_pair_list<repeated, FoundPolicyPairs...>; + }; + + // For the first detector-default pair in the remaining list, call + // detector_default_pair::get_found_policy_pair on Policies and add the returned + // found_policy_pair into the current list of found_policy_pair's, and move to the next + // detector-default pair. + template <class FirstDetectorDefaultPair, class... RemainingDetectorDefaultPairs, + bool repeated, class... FoundPolicyPairs, class... Policies> + struct make_policy_pair_list_impl< + detector_default_pair_list<FirstDetectorDefaultPair, RemainingDetectorDefaultPairs...>, + found_policy_pair_list<repeated, FoundPolicyPairs...>, Policies...> { + using new_found_policy_pair = + typename FirstDetectorDefaultPair::template get_found_policy_pair<Policies...>; + + using type = typename make_policy_pair_list_impl< + detector_default_pair_list<RemainingDetectorDefaultPairs...>, + found_policy_pair_list<(repeated || new_found_policy_pair::found_info == + policy_found_info::repeated), + new_found_policy_pair, FoundPolicyPairs...>, + Policies...>::type; + }; + + template <class DetectorDefaultPairList, class... Policies> + using policy_pair_list = + typename make_policy_pair_list_impl<DetectorDefaultPairList, + found_policy_pair_list<false>, Policies...>::type; + + // Unpack FoundPolicyPairList into found_policy_pair's and build the policy_holder type + // from the corresponding typelist of found_policy_pair::policy's. + template <class FoundPolicyPairList, class... RawPolicies> + struct convert_to_policy_holder_impl; + + template <bool repeated, class... RawPolicies> + struct convert_to_policy_holder_impl<found_policy_pair_list<repeated>, RawPolicies...> { + using type = policy_holder<RawPolicies...>; + }; + + template <bool repeated, class FirstFoundPolicyPair, class... RemainingFoundPolicyPairs, + class... RawPolicies> + struct convert_to_policy_holder_impl< + found_policy_pair_list<repeated, FirstFoundPolicyPair, RemainingFoundPolicyPairs...>, + RawPolicies...> { + using type = typename convert_to_policy_holder_impl< + found_policy_pair_list<repeated, RemainingFoundPolicyPairs...>, + typename FirstFoundPolicyPair::policy, RawPolicies...>::type; + }; + + template <class FoundPolicyPairList> + using convert_to_policy_holder = + typename convert_to_policy_holder_impl<FoundPolicyPairList>::type; + + template <class DetectorDefaultPairList, class... Policies> + struct make_policy_holder_impl { + static_assert(check_policy_list_validity(DetectorDefaultPairList{}, + dummy<Policies>{}...), + "jkj::dragonbox: an invalid policy is specified"); + + static_assert( + !policy_pair_list<DetectorDefaultPairList, Policies...>::repeated, + "jkj::dragonbox: at most one policy should be specified for each policy kind"); + + using type = + convert_to_policy_holder<policy_pair_list<DetectorDefaultPairList, Policies...>>; + }; + + template <class DetectorDefaultPairList, class... Policies> + using make_policy_holder = + typename make_policy_holder_impl<DetectorDefaultPairList, Policies...>::type; + + + // Policy kind detectors. + struct is_sign_policy { + constexpr bool operator()(...) noexcept { return false; } + template <class Policy, class = typename Policy::sign_policy> + constexpr bool operator()(dummy<Policy>) noexcept { + return true; + } + }; + struct is_trailing_zero_policy { + constexpr bool operator()(...) noexcept { return false; } + template <class Policy, class = typename Policy::trailing_zero_policy> + constexpr bool operator()(dummy<Policy>) noexcept { + return true; + } + }; + struct is_decimal_to_binary_rounding_policy { + constexpr bool operator()(...) noexcept { return false; } + template <class Policy, class = typename Policy::decimal_to_binary_rounding_policy> + constexpr bool operator()(dummy<Policy>) noexcept { + return true; + } + }; + struct is_binary_to_decimal_rounding_policy { + constexpr bool operator()(...) noexcept { return false; } + template <class Policy, class = typename Policy::binary_to_decimal_rounding_policy> + constexpr bool operator()(dummy<Policy>) noexcept { + return true; + } + }; + struct is_cache_policy { + constexpr bool operator()(...) noexcept { return false; } + template <class Policy, class = typename Policy::cache_policy> + constexpr bool operator()(dummy<Policy>) noexcept { + return true; + } + }; + struct is_preferred_integer_types_policy { + constexpr bool operator()(...) noexcept { return false; } + template <class Policy, class = typename Policy::preferred_integer_types_policy> + constexpr bool operator()(dummy<Policy>) noexcept { + return true; + } + }; + + template <class... Policies> + using to_decimal_policy_holder = make_policy_holder< + detector_default_pair_list< + detector_default_pair<is_sign_policy, policy::sign::return_sign_t>, + detector_default_pair<is_trailing_zero_policy, policy::trailing_zero::remove_t>, + detector_default_pair<is_decimal_to_binary_rounding_policy, + policy::decimal_to_binary_rounding::nearest_to_even_t>, + detector_default_pair<is_binary_to_decimal_rounding_policy, + policy::binary_to_decimal_rounding::to_even_t>, + detector_default_pair<is_cache_policy, policy::cache::full_t>, + detector_default_pair<is_preferred_integer_types_policy, + policy::preferred_integer_types::match_t>>, + Policies...>; + + template <class FormatTraits, class... Policies> + using to_decimal_return_type = typename impl<FormatTraits>::template return_type< + typename to_decimal_policy_holder<Policies...>::sign_policy, + typename to_decimal_policy_holder<Policies...>::trailing_zero_policy, + typename to_decimal_policy_holder<Policies...>::preferred_integer_types_policy>; + + template <class FormatTraits, class PolicyHolder> + struct to_decimal_dispatcher { + using sign_policy = typename PolicyHolder::sign_policy; + using trailing_zero_policy = typename PolicyHolder::trailing_zero_policy; + using binary_to_decimal_rounding_policy = + typename PolicyHolder::binary_to_decimal_rounding_policy; + using cache_policy = typename PolicyHolder::cache_policy; + using preferred_integer_types_policy = + typename PolicyHolder::preferred_integer_types_policy; + using return_type = + typename impl<FormatTraits>::template return_type<sign_policy, trailing_zero_policy, + preferred_integer_types_policy>; + + template <class IntervalTypeProvider> + JKJ_FORCEINLINE JKJ_SAFEBUFFERS JKJ_CONSTEXPR20 return_type + operator()(IntervalTypeProvider, signed_significand_bits<FormatTraits> s, + typename FormatTraits::exponent_int exponent_bits) noexcept { + constexpr auto tag = IntervalTypeProvider::tag; + + JKJ_IF_CONSTEXPR(tag == policy::decimal_to_binary_rounding::tag_t::to_nearest) { + return impl<FormatTraits>::template compute_nearest< + sign_policy, trailing_zero_policy, IntervalTypeProvider, + binary_to_decimal_rounding_policy, cache_policy, + preferred_integer_types_policy>(s, exponent_bits); + } + else JKJ_IF_CONSTEXPR( + tag == policy::decimal_to_binary_rounding::tag_t::left_closed_directed) { + return impl<FormatTraits>::template compute_left_closed_directed< + sign_policy, trailing_zero_policy, cache_policy, + preferred_integer_types_policy>(s, exponent_bits); + } + else { +#if JKJ_HAS_IF_CONSTEXPR + static_assert( + tag == policy::decimal_to_binary_rounding::tag_t::right_closed_directed, + ""); +#endif + return impl<FormatTraits>::template compute_right_closed_directed< + sign_policy, trailing_zero_policy, cache_policy, + preferred_integer_types_policy>(s, exponent_bits); + } + } + }; + } + + + //////////////////////////////////////////////////////////////////////////////////////// + // The interface function. + //////////////////////////////////////////////////////////////////////////////////////// + + template <class FormatTraits, class ExponentBits, class... Policies> + JKJ_FORCEINLINE + JKJ_SAFEBUFFERS JKJ_CONSTEXPR20 detail::to_decimal_return_type<FormatTraits, Policies...> + to_decimal_ex(signed_significand_bits<FormatTraits> s, ExponentBits exponent_bits, + Policies...) noexcept { + // Build policy holder type. + using policy_holder = detail::to_decimal_policy_holder<Policies...>; + + return policy_holder::delegate( + s, detail::to_decimal_dispatcher<FormatTraits, policy_holder>{}, s, exponent_bits); + } + + template <class Float, + class ConversionTraits = default_float_bit_carrier_conversion_traits<Float>, + class FormatTraits = ieee754_binary_traits<typename ConversionTraits::format, + typename ConversionTraits::carrier_uint>, + class... Policies> + JKJ_FORCEINLINE + JKJ_SAFEBUFFERS JKJ_CONSTEXPR20 detail::to_decimal_return_type<FormatTraits, Policies...> + to_decimal(Float x, Policies... policies) noexcept { + auto const br = make_float_bits<Float, ConversionTraits, FormatTraits>(x); + auto const exponent_bits = br.extract_exponent_bits(); + auto const s = br.remove_exponent_bits(); + assert(br.is_finite() && br.is_nonzero()); + + return to_decimal_ex(s, exponent_bits, policies...); + } + } +} + +#undef JKJ_HAS_BUILTIN +#undef JKJ_FORCEINLINE +#undef JKJ_SAFEBUFFERS +#undef JKJ_CONSTEXPR20 +#undef JKJ_USE_IS_CONSTANT_EVALUATED +#undef JKJ_CAN_BRANCH_ON_CONSTEVAL +#undef JKJ_IF_NOT_CONSTEVAL +#undef JKJ_IF_CONSTEVAL +#undef JKJ_HAS_BIT_CAST +#undef JKJ_IF_CONSTEXPR +#undef JKJ_HAS_IF_CONSTEXPR +#undef JKJ_INLINE_VARIABLE +#undef JKJ_HAS_INLINE_VARIABLE +#undef JKJ_HAS_CONSTEXPR17 +#undef JKJ_CONSTEXPR14 +#undef JKJ_HAS_CONSTEXPR14 +#if JKJ_STD_REPLACEMENT_NAMESPACE_DEFINED + #undef JKJ_STD_REPLACEMENT_NAMESPACE_DEFINED +#else + #undef JKJ_STD_REPLACEMENT_NAMESPACE +#endif +#if JKJ_STATIC_DATA_SECTION_DEFINED + #undef JKJ_STATIC_DATA_SECTION_DEFINED +#else + #undef JKJ_STATIC_DATA_SECTION +#endif + +#endif diff --git a/include/dragonbox/dragonbox_revision.txt b/include/dragonbox/dragonbox_revision.txt new file mode 100644 index 0000000..7abbc11 --- /dev/null +++ b/include/dragonbox/dragonbox_revision.txt @@ -0,0 +1,8 @@ +commit bd692342bc182a7ff20ac44562451eefb746401c +Merge: b27a9ab fbb6ae3 +Author: Junekey Jeon <33922675+jk-jeon@users.noreply.github.com> +Date: Tue Oct 22 11:05:18 2024 -0700 + + Merge pull request #69 from SGSSGene/patch-1 + + fix: typo |