3308 lines
118 KiB
C++
3308 lines
118 KiB
C++
// Copyright 2020 Junekey Jeon
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//
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// The contents of this file may be used under the terms of
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// the Apache License v2.0 with LLVM Exceptions.
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//
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// (See accompanying file LICENSE-Apache or copy at
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// https://llvm.org/foundation/relicensing/LICENSE.txt)
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//
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// Alternatively, the contents of this file may be used under the terms of
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// the Boost Software License, Version 1.0.
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// (See accompanying file LICENSE-Boost or copy at
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// https://www.boost.org/LICENSE_1_0.txt)
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//
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// Unless required by applicable law or agreed to in writing, this software
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// is distributed on an "AS IS" BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY
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// KIND, either express or implied.
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#ifndef JKJ_DRAGONBOX
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#define JKJ_DRAGONBOX
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#include <cassert>
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#include <cstdint>
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#include <cstring>
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#include <limits>
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#include <type_traits>
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// Suppress additional buffer overrun check
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// I have no idea why MSVC thinks some functions here are vulnerable to the buffer overrun attacks
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// No, they aren't.
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#if defined(__GNUC__) || defined(__clang__)
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#define JKJ_SAFEBUFFERS
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#define JKJ_FORCEINLINE inline __attribute__((always_inline))
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#elif defined(_MSC_VER)
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#define JKJ_SAFEBUFFERS __declspec(safebuffers)
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#define JKJ_FORCEINLINE __forceinline
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#else
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#define JKJ_SAFEBUFFERS
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#define JKJ_FORCEINLINE inline
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#endif
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#if defined(_MSC_VER)
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#include <intrin.h>
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#endif
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namespace jkj::dragonbox {
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namespace detail {
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template <class T>
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constexpr std::size_t physical_bits = sizeof(T) * std::numeric_limits<unsigned char>::digits;
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template <class T>
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constexpr std::size_t value_bits =
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std::numeric_limits<std::enable_if_t<std::is_unsigned_v<T>, T>>::digits;
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}
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enum class ieee754_format {
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binary32,
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binary64
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};
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template <ieee754_format format_>
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struct ieee754_format_info;
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template <>
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struct ieee754_format_info<ieee754_format::binary32> {
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static constexpr auto format = ieee754_format::binary32;
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static constexpr int significand_bits = 23;
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static constexpr int exponent_bits = 8;
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static constexpr int min_exponent = -126;
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static constexpr int max_exponent = 127;
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static constexpr int exponent_bias = -127;
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static constexpr int decimal_digits = 9;
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};
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template <>
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struct ieee754_format_info<ieee754_format::binary64> {
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static constexpr auto format = ieee754_format::binary64;
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static constexpr int significand_bits = 52;
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static constexpr int exponent_bits = 11;
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static constexpr int min_exponent = -1022;
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static constexpr int max_exponent = 1023;
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static constexpr int exponent_bias = -1023;
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static constexpr int decimal_digits = 17;
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};
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// To reduce boilerplates
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template <class T>
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struct default_ieee754_traits {
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static_assert(detail::physical_bits<T> == 32 || detail::physical_bits<T> == 64);
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using type = T;
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static constexpr ieee754_format format =
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detail::physical_bits<T> == 32 ? ieee754_format::binary32 : ieee754_format::binary64;
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using carrier_uint = std::conditional_t<
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detail::physical_bits<T> == 32,
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std::uint32_t,
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std::uint64_t>;
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static_assert(sizeof(carrier_uint) == sizeof(T));
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static constexpr int carrier_bits = int(detail::physical_bits<carrier_uint>);
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static T carrier_to_float(carrier_uint u) noexcept {
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T x;
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std::memcpy(&x, &u, sizeof(carrier_uint));
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return x;
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}
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static carrier_uint float_to_carrier(T x) noexcept {
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carrier_uint u;
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std::memcpy(&u, &x, sizeof(carrier_uint));
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return u;
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}
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static constexpr unsigned int extract_exponent_bits(carrier_uint u) noexcept {
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constexpr int significand_bits = ieee754_format_info<format>::significand_bits;
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constexpr int exponent_bits = ieee754_format_info<format>::exponent_bits;
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static_assert(detail::value_bits<unsigned int> > exponent_bits);
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constexpr auto exponent_bits_mask = (unsigned int)(((unsigned int)(1) << exponent_bits) - 1);
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return (unsigned int)((u >> significand_bits) & exponent_bits_mask);
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}
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static constexpr carrier_uint extract_significand_bits(carrier_uint u) noexcept {
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constexpr int significand_bits = ieee754_format_info<format>::significand_bits;
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constexpr auto significand_bits_mask = carrier_uint((carrier_uint(1) << significand_bits) - 1);
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return carrier_uint(u & significand_bits_mask);
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}
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// Allows positive zero and positive NaN's, but not allow negative zero
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static constexpr bool is_positive(carrier_uint u) noexcept {
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return (u >> (carrier_bits - 1)) == 0;
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}
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// Allows negative zero and negative NaN's, but not allow positive zero
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static constexpr bool is_negative(carrier_uint u) noexcept {
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return (u >> (carrier_bits - 1)) != 0;
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}
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static constexpr int exponent_bias = 1 - (1 << (carrier_bits - ieee754_format_info<format>::significand_bits - 2));
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static constexpr bool is_finite(carrier_uint u) noexcept {
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constexpr int significand_bits = ieee754_format_info<format>::significand_bits;
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constexpr int exponent_bits = ieee754_format_info<format>::exponent_bits;
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constexpr auto exponent_bits_mask =
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carrier_uint(((carrier_uint(1) << exponent_bits) - 1) << significand_bits);
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return (u & exponent_bits_mask) != exponent_bits_mask;
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}
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static constexpr bool is_nonzero(carrier_uint u) noexcept {
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return (u << 1) != 0;
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}
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// Allows positive and negative zeros
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static constexpr bool is_subnormal(carrier_uint u) noexcept {
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constexpr int significand_bits = ieee754_format_info<format>::significand_bits;
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constexpr int exponent_bits = ieee754_format_info<format>::exponent_bits;
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constexpr auto exponent_bits_mask =
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carrier_uint(((carrier_uint(1) << exponent_bits) - 1) << significand_bits);
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return (u & exponent_bits_mask) == 0;
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}
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static constexpr bool is_positive_infinity(carrier_uint u) noexcept {
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constexpr int significand_bits = ieee754_format_info<format>::significand_bits;
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constexpr int exponent_bits = ieee754_format_info<format>::exponent_bits;
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constexpr auto positive_infinity =
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carrier_uint((carrier_uint(1) << exponent_bits) - 1) << significand_bits;
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return u == positive_infinity;
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}
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static constexpr bool is_negative_infinity(carrier_uint u) noexcept {
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constexpr int significand_bits = ieee754_format_info<format>::significand_bits;
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constexpr int exponent_bits = ieee754_format_info<format>::exponent_bits;
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constexpr auto negative_infinity =
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(carrier_uint((carrier_uint(1) << exponent_bits) - 1) << significand_bits)
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| (carrier_uint(1) << (carrier_bits - 1));
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return u == negative_infinity;
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}
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static constexpr bool is_infinity(carrier_uint u) noexcept {
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return is_positive_infinity(u) || is_negative_infinity(u);
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}
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static constexpr bool is_nan(carrier_uint u) noexcept {
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return !is_finite(u) && (extract_significand_bits(u) != 0);
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}
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};
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// Speciailze this class template for possible extensions
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template <class T>
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struct ieee754_traits : default_ieee754_traits<T> {
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// I don't know if there is a truly reliable way of detecting
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// IEEE-754 binary32/binary64 formats; I just did my best here
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static_assert(std::numeric_limits<T>::is_iec559 &&
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std::numeric_limits<T>::radix == 2 &&
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(detail::physical_bits<T> == 32 || detail::physical_bits<T> == 64),
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"default_ieee754_traits only worsk for 32-bits or 64-bits types "
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"supporting binary32 or binary64 formats!");
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};
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// Convenient wrapper for ieee754_traits
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// In order to reduce the argument passing overhead,
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// this class should be as simple as possible
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// (e.g., no inheritance, no private non-static data member, etc.;
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// this is an unfortunate fact about x64 calling convention)
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template <class T>
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struct ieee754_bits {
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using carrier_uint = typename ieee754_traits<T>::carrier_uint;
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carrier_uint u;
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ieee754_bits() = default;
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constexpr explicit ieee754_bits(carrier_uint bit_pattern) noexcept :
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u{ bit_pattern } {}
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constexpr explicit ieee754_bits(T float_value) noexcept :
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u{ ieee754_traits<T>::float_to_carrier(float_value) } {}
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constexpr T to_float() const noexcept {
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return ieee754_traits<T>::carrier_to_float(u);
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}
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constexpr carrier_uint extract_significand_bits() const noexcept {
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return ieee754_traits<T>::extract_significand_bits(u);
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}
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constexpr unsigned int extract_exponent_bits() const noexcept {
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return ieee754_traits<T>::extract_exponent_bits(u);
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}
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constexpr carrier_uint binary_significand() const noexcept {
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using format_info = ieee754_format_info<ieee754_traits<T>::format>;
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auto s = extract_significand_bits();
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if (extract_exponent_bits() == 0) {
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return s;
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}
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else {
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return s | (carrier_uint(1) << format_info::significand_bits);
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}
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}
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constexpr int binary_exponent() const noexcept {
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using format_info = ieee754_format_info<ieee754_traits<T>::format>;
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auto e = extract_exponent_bits();
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if (e == 0) {
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return format_info::min_exponent;
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}
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else {
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return e + format_info::exponent_bias;
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}
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}
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constexpr bool is_finite() const noexcept {
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return ieee754_traits<T>::is_finite(u);
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}
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constexpr bool is_nonzero() const noexcept {
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return ieee754_traits<T>::is_nonzero(u);
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}
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// Allows positive and negative zeros
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constexpr bool is_subnormal() const noexcept {
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return ieee754_traits<T>::is_subnormal(u);
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}
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// Allows positive zero and positive NaN's, but not allow negative zero
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constexpr bool is_positive() const noexcept {
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return ieee754_traits<T>::is_positive(u);
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}
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// Allows negative zero and negative NaN's, but not allow positive zero
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constexpr bool is_negative() const noexcept {
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return ieee754_traits<T>::is_negative(u);
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}
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constexpr bool is_positive_infinity() const noexcept {
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return ieee754_traits<T>::is_positive_infinity(u);
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}
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constexpr bool is_negative_infinity() const noexcept {
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return ieee754_traits<T>::is_negative_infinity(u);
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}
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// Allows both plus and minus infinities
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constexpr bool is_infinity() const noexcept {
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return ieee754_traits<T>::is_infinity(u);
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}
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constexpr bool is_nan() const noexcept {
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return ieee754_traits<T>::is_nan(u);
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}
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};
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namespace detail {
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////////////////////////////////////////////////////////////////////////////////////////
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// Bit operation intrinsics
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////////////////////////////////////////////////////////////////////////////////////////
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namespace bits {
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template <class UInt>
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inline int countr_zero(UInt n) noexcept {
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static_assert(std::is_unsigned_v<UInt> && value_bits<UInt> <= 64);
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#if defined(__GNUC__) || defined(__clang__)
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#define JKJ_HAS_COUNTR_ZERO_INTRINSIC 1
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if constexpr (std::is_same_v<UInt, unsigned long>) {
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return __builtin_ctzl(n);
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}
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else if constexpr (std::is_same_v<UInt, unsigned long long>) {
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return __builtin_ctzll(n);
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}
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else {
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static_assert(sizeof(UInt) <= sizeof(unsigned int));
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return __builtin_ctz((unsigned int)n);
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}
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#elif defined(_MSC_VER)
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#define JKJ_HAS_COUNTR_ZERO_INTRINSIC 1
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if constexpr (std::is_same_v<UInt, unsigned __int64>) {
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#if defined(_M_X64)
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return int(_tzcnt_u64(n));
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#else
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return ((unsigned int)(n) == 0) ?
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(32 + (_tzcnt_u32((unsigned int)(n >> 32)))) :
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(_tzcnt_u32((unsigned int)n));
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#endif
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}
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else {
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static_assert(sizeof(UInt) <= sizeof(unsigned int));
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return int(_tzcnt_u32((unsigned int)n));
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}
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#else
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#define JKJ_HAS_COUNTR_ZERO_INTRINSIC 0
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int count;
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auto n32 = std::uint32_t(n);
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if constexpr (value_bits<UInt> > 32) {
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if (n32 != 0) {
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count = 31;
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}
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else {
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n32 = std::uint32_t(n >> 32);
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if constexpr (value_bits<UInt> == 64) {
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if (n32 != 0) {
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count = 63;
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}
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else {
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return 64;
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}
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}
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else {
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count = value_bits<UInt>;
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}
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}
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}
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else {
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if constexpr (value_bits<UInt> == 32) {
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if (n32 != 0) {
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count = 31;
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}
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else {
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return 32;
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}
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}
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else {
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count = value_bits<UInt>;
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}
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}
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n32 &= (0 - n32);
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if constexpr (value_bits<UInt> > 16) {
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if ((n32 & 0x0000ffff) != 0) count -= 16;
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}
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if constexpr (value_bits<UInt> > 8) {
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if ((n32 & 0x00ff00ff) != 0) count -= 8;
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}
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if ((n32 & 0x0f0f0f0f) != 0) count -= 4;
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if ((n32 & 0x33333333) != 0) count -= 2;
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if ((n32 & 0x55555555) != 0) count -= 1;
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return count;
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#endif
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}
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}
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////////////////////////////////////////////////////////////////////////////////////////
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// Utilities for wide unsigned integer arithmetic
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////////////////////////////////////////////////////////////////////////////////////////
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namespace wuint {
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struct uint128 {
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uint128() = default;
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#if defined(__SIZEOF_INT128__)
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unsigned __int128 internal_;
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constexpr uint128(std::uint64_t high, std::uint64_t low) noexcept :
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internal_{ ((unsigned __int128)low) | (((unsigned __int128)high) << 64) } {}
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constexpr uint128(unsigned __int128 u) noexcept : internal_{ u } {}
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constexpr std::uint64_t high() const noexcept {
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return std::uint64_t(internal_ >> 64);
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}
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constexpr std::uint64_t low() const noexcept {
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return std::uint64_t(internal_);
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}
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uint128& operator+=(std::uint64_t n) & noexcept {
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internal_ += n;
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return *this;
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}
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#else
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std::uint64_t high_;
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std::uint64_t low_;
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constexpr uint128(std::uint64_t high, std::uint64_t low) noexcept :
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high_{ high }, low_{ low } {}
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constexpr std::uint64_t high() const noexcept {
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return high_;
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}
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constexpr std::uint64_t low() const noexcept {
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return low_;
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}
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uint128& operator+=(std::uint64_t n) & noexcept {
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#if defined(_MSC_VER) && defined(_M_X64)
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auto carry = _addcarry_u64(0, low_, n, &low_);
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_addcarry_u64(carry, high_, 0, &high_);
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return *this;
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#else
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auto sum = low_ + n;
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high_ += (sum < low_ ? 1 : 0);
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low_ = sum;
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return *this;
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#endif
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}
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#endif
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};
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static inline std::uint64_t umul64(std::uint32_t x, std::uint32_t y) noexcept {
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#if defined(_MSC_VER) && defined(_M_IX86)
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return __emulu(x, y);
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#else
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return x * std::uint64_t(y);
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#endif
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}
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// Get 128-bit result of multiplication of two 64-bit unsigned integers
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JKJ_SAFEBUFFERS inline uint128 umul128(std::uint64_t x, std::uint64_t y) noexcept {
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#if defined(__SIZEOF_INT128__)
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return (unsigned __int128)(x) * (unsigned __int128)(y);
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#elif defined(_MSC_VER) && defined(_M_X64)
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uint128 result;
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result.low_ = _umul128(x, y, &result.high_);
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return result;
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#else
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auto a = std::uint32_t(x >> 32);
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auto b = std::uint32_t(x);
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auto c = std::uint32_t(y >> 32);
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auto d = std::uint32_t(y);
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auto ac = umul64(a, c);
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auto bc = umul64(b, c);
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auto ad = umul64(a, d);
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auto bd = umul64(b, d);
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auto intermediate = (bd >> 32) + std::uint32_t(ad) + std::uint32_t(bc);
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return{ ac + (intermediate >> 32) + (ad >> 32) + (bc >> 32),
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(intermediate << 32) + std::uint32_t(bd) };
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#endif
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}
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JKJ_SAFEBUFFERS inline std::uint64_t umul128_upper64(std::uint64_t x, std::uint64_t y) noexcept {
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#if defined(__SIZEOF_INT128__)
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auto p = (unsigned __int128)(x) * (unsigned __int128)(y);
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return std::uint64_t(p >> 64);
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#elif defined(_MSC_VER) && defined(_M_X64)
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return __umulh(x, y);
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#else
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auto a = std::uint32_t(x >> 32);
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auto b = std::uint32_t(x);
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auto c = std::uint32_t(y >> 32);
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auto d = std::uint32_t(y);
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auto ac = umul64(a, c);
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auto bc = umul64(b, c);
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auto ad = umul64(a, d);
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auto bd = umul64(b, d);
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auto intermediate = (bd >> 32) + std::uint32_t(ad) + std::uint32_t(bc);
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return ac + (intermediate >> 32) + (ad >> 32) + (bc >> 32);
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#endif
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}
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// Get upper 64-bits of multiplication of a 64-bit unsigned integer and a 128-bit unsigned integer
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JKJ_SAFEBUFFERS inline std::uint64_t umul192_upper64(std::uint64_t x, uint128 y) noexcept {
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auto g0 = umul128(x, y.high());
|
|
g0 += umul128_upper64(x, y.low());
|
|
return g0.high();
|
|
}
|
|
|
|
// Get upper 32-bits of multiplication of a 32-bit unsigned integer and a 64-bit unsigned integer
|
|
inline std::uint32_t umul96_upper32(std::uint32_t x, std::uint64_t y) noexcept {
|
|
#if defined(__SIZEOF_INT128__) || (defined(_MSC_VER) && defined(_M_X64))
|
|
return std::uint32_t(umul128_upper64(x, y));
|
|
#else
|
|
//std::uint32_t a = 0;
|
|
auto b = x;
|
|
auto c = std::uint32_t(y >> 32);
|
|
auto d = std::uint32_t(y);
|
|
|
|
//std::uint64_t ac = 0;
|
|
auto bc = umul64(b, c);
|
|
//std::uint64_t ad = 0;
|
|
auto bd = umul64(b, d);
|
|
|
|
auto intermediate = (bd >> 32) + bc;
|
|
return std::uint32_t(intermediate >> 32);
|
|
#endif
|
|
}
|
|
|
|
// Get middle 64-bits of multiplication of a 64-bit unsigned integer and a 128-bit unsigned integer
|
|
JKJ_SAFEBUFFERS inline std::uint64_t umul192_middle64(std::uint64_t x, uint128 y) noexcept {
|
|
auto g01 = x * y.high();
|
|
auto g10 = umul128_upper64(x, y.low());
|
|
return g01 + g10;
|
|
}
|
|
|
|
// Get middle 32-bits of multiplication of a 32-bit unsigned integer and a 64-bit unsigned integer
|
|
inline std::uint64_t umul96_lower64(std::uint32_t x, std::uint64_t y) noexcept {
|
|
return x * y;
|
|
}
|
|
}
|
|
|
|
template <int k, class Int>
|
|
constexpr Int compute_power(Int a) noexcept {
|
|
static_assert(k >= 0);
|
|
Int p = 1;
|
|
for (int i = 0; i < k; ++i) {
|
|
p *= a;
|
|
}
|
|
return p;
|
|
}
|
|
|
|
template <int a, class UInt>
|
|
constexpr int count_factors(UInt n) noexcept {
|
|
static_assert(a > 1);
|
|
int c = 0;
|
|
while (n % a == 0) {
|
|
n /= a;
|
|
++c;
|
|
}
|
|
return c;
|
|
}
|
|
|
|
////////////////////////////////////////////////////////////////////////////////////////
|
|
// Utilities for fast/constexpr log computation
|
|
////////////////////////////////////////////////////////////////////////////////////////
|
|
|
|
namespace log {
|
|
constexpr std::int32_t floor_shift(
|
|
std::uint32_t integer_part,
|
|
std::uint64_t fractional_digits,
|
|
std::size_t shift_amount) noexcept
|
|
{
|
|
assert(shift_amount < 32);
|
|
// Ensure no overflow
|
|
assert(shift_amount == 0 || integer_part < (std::uint32_t(1) << (32 - shift_amount)));
|
|
|
|
return shift_amount == 0 ? std::int32_t(integer_part) :
|
|
std::int32_t(
|
|
(integer_part << shift_amount) |
|
|
(fractional_digits >> (64 - shift_amount)));
|
|
}
|
|
|
|
// Compute floor(e * c - s)
|
|
template <
|
|
std::uint32_t c_integer_part,
|
|
std::uint64_t c_fractional_digits,
|
|
std::size_t shift_amount,
|
|
std::int32_t max_exponent,
|
|
std::uint32_t s_integer_part = 0,
|
|
std::uint64_t s_fractional_digits = 0
|
|
>
|
|
constexpr int compute(int e) noexcept {
|
|
assert(e <= max_exponent && e >= -max_exponent);
|
|
constexpr auto c = floor_shift(c_integer_part, c_fractional_digits, shift_amount);
|
|
constexpr auto s = floor_shift(s_integer_part, s_fractional_digits, shift_amount);
|
|
return int((std::int32_t(e) * c - s) >> shift_amount);
|
|
}
|
|
|
|
inline constexpr std::uint64_t log10_2_fractional_digits{ 0x4d10'4d42'7de7'fbcc };
|
|
inline constexpr std::uint64_t log10_4_over_3_fractional_digits{ 0x1ffb'fc2b'bc78'0375 };
|
|
inline constexpr std::size_t floor_log10_pow2_shift_amount = 22;
|
|
inline constexpr int floor_log10_pow2_input_limit = 1700;
|
|
inline constexpr int floor_log10_pow2_minus_log10_4_over_3_input_limit = 1700;
|
|
|
|
inline constexpr std::uint64_t log2_10_fractional_digits{ 0x5269'e12f'346e'2bf9 };
|
|
inline constexpr std::size_t floor_log2_pow10_shift_amount = 19;
|
|
inline constexpr int floor_log2_pow10_input_limit = 1233;
|
|
|
|
inline constexpr std::uint64_t log5_2_fractional_digits{ 0x6e40'd1a4'143d'cb94 };
|
|
inline constexpr std::uint64_t log5_3_fractional_digits{ 0xaebf'4791'5d44'3b24 };
|
|
inline constexpr std::size_t floor_log5_pow2_shift_amount = 20;
|
|
inline constexpr int floor_log5_pow2_input_limit = 1492;
|
|
inline constexpr int floor_log5_pow2_minus_log5_3_input_limit = 2427;
|
|
|
|
// For constexpr computation
|
|
// Returns -1 when n = 0
|
|
template <class UInt>
|
|
constexpr int floor_log2(UInt n) noexcept {
|
|
int count = -1;
|
|
while (n != 0) {
|
|
++count;
|
|
n >>= 1;
|
|
}
|
|
return count;
|
|
}
|
|
|
|
constexpr int floor_log10_pow2(int e) noexcept {
|
|
using namespace log;
|
|
return compute<
|
|
0, log10_2_fractional_digits,
|
|
floor_log10_pow2_shift_amount,
|
|
floor_log10_pow2_input_limit>(e);
|
|
}
|
|
|
|
constexpr int floor_log2_pow10(int e) noexcept {
|
|
using namespace log;
|
|
return compute<
|
|
3, log2_10_fractional_digits,
|
|
floor_log2_pow10_shift_amount,
|
|
floor_log2_pow10_input_limit>(e);
|
|
}
|
|
|
|
constexpr int floor_log5_pow2(int e) noexcept {
|
|
using namespace log;
|
|
return compute<
|
|
0, log5_2_fractional_digits,
|
|
floor_log5_pow2_shift_amount,
|
|
floor_log5_pow2_input_limit>(e);
|
|
}
|
|
|
|
constexpr int floor_log5_pow2_minus_log5_3(int e) noexcept {
|
|
using namespace log;
|
|
return compute<
|
|
0, log5_2_fractional_digits,
|
|
floor_log5_pow2_shift_amount,
|
|
floor_log5_pow2_minus_log5_3_input_limit,
|
|
0, log5_3_fractional_digits>(e);
|
|
}
|
|
|
|
constexpr int floor_log10_pow2_minus_log10_4_over_3(int e) noexcept {
|
|
using namespace log;
|
|
return compute<
|
|
0, log10_2_fractional_digits,
|
|
floor_log10_pow2_shift_amount,
|
|
floor_log10_pow2_minus_log10_4_over_3_input_limit,
|
|
0, log10_4_over_3_fractional_digits>(e);
|
|
}
|
|
}
|
|
|
|
////////////////////////////////////////////////////////////////////////////////////////
|
|
// Utilities for fast divisibility test
|
|
////////////////////////////////////////////////////////////////////////////////////////
|
|
|
|
namespace div {
|
|
template <class UInt, UInt a>
|
|
constexpr UInt modular_inverse(unsigned int bit_width = unsigned(value_bits<UInt>)) noexcept {
|
|
// By Euler's theorem, a^phi(2^n) == 1 (mod 2^n),
|
|
// where phi(2^n) = 2^(n-1), so the modular inverse of a is
|
|
// a^(2^(n-1) - 1) = a^(1 + 2 + 2^2 + ... + 2^(n-2))
|
|
std::common_type_t<UInt, unsigned int> mod_inverse = 1;
|
|
for (unsigned int i = 1; i < bit_width; ++i) {
|
|
mod_inverse = mod_inverse * mod_inverse * a;
|
|
}
|
|
if (bit_width < value_bits<UInt>) {
|
|
auto mask = UInt((UInt(1) << bit_width) - 1);
|
|
return UInt(mod_inverse & mask);
|
|
}
|
|
else {
|
|
return UInt(mod_inverse);
|
|
}
|
|
}
|
|
|
|
template <class UInt, UInt a, std::size_t N>
|
|
struct table_t {
|
|
static_assert(std::is_unsigned_v<UInt>);
|
|
static_assert(a % 2 != 0);
|
|
static_assert(N > 0);
|
|
|
|
static constexpr std::size_t size = N;
|
|
UInt mod_inv[N];
|
|
UInt max_quotients[N];
|
|
};
|
|
|
|
template <class UInt, UInt a, std::size_t N>
|
|
struct table_holder {
|
|
static constexpr table_t<UInt, a, N> table = [] {
|
|
constexpr auto mod_inverse = modular_inverse<UInt, a>();
|
|
table_t<UInt, a, N> table{};
|
|
std::common_type_t<UInt, unsigned int> pow_of_mod_inverse = 1;
|
|
UInt pow_of_a = 1;
|
|
for (std::size_t i = 0; i < N; ++i) {
|
|
table.mod_inv[i] = UInt(pow_of_mod_inverse);
|
|
table.max_quotients[i] = UInt(std::numeric_limits<UInt>::max() / pow_of_a);
|
|
|
|
pow_of_mod_inverse *= mod_inverse;
|
|
pow_of_a *= a;
|
|
}
|
|
|
|
return table;
|
|
}();
|
|
};
|
|
|
|
template <std::size_t table_size, class UInt>
|
|
constexpr bool divisible_by_power_of_5(UInt x, unsigned int exp) noexcept {
|
|
auto const& table = table_holder<UInt, 5, table_size>::table;
|
|
assert(exp < table.size);
|
|
return (x * table.mod_inv[exp]) <= table.max_quotients[exp];
|
|
}
|
|
|
|
template <class UInt>
|
|
constexpr bool divisible_by_power_of_2(UInt x, unsigned int exp) noexcept {
|
|
assert(exp >= 1);
|
|
assert(x != 0);
|
|
#if JKJ_HAS_COUNTR_ZERO_INTRINSIC
|
|
return bits::countr_zero(x) >= int(exp);
|
|
#else
|
|
if (exp >= value_bits<UInt>) {
|
|
return false;
|
|
}
|
|
auto mask = UInt((UInt(1) << exp) - 1);
|
|
return (x & mask) == 0;
|
|
#endif
|
|
}
|
|
|
|
// Replace n by floor(n / 5^N)
|
|
// Returns true if and only if n is divisible by 5^N
|
|
// Precondition: n <= 2 * 5^(N+1)
|
|
template <int N>
|
|
struct check_divisibility_and_divide_by_pow5_info;
|
|
|
|
template <>
|
|
struct check_divisibility_and_divide_by_pow5_info<1> {
|
|
static constexpr std::uint32_t magic_number = 0xcccd;
|
|
static constexpr int bits_for_comparison = 16;
|
|
static constexpr std::uint32_t threshold = 0x3333;
|
|
static constexpr int shift_amount = 18;
|
|
};
|
|
|
|
template <>
|
|
struct check_divisibility_and_divide_by_pow5_info<2> {
|
|
static constexpr std::uint32_t magic_number = 0xa429;
|
|
static constexpr int bits_for_comparison = 8;
|
|
static constexpr std::uint32_t threshold = 0x0a;
|
|
static constexpr int shift_amount = 20;
|
|
};
|
|
|
|
template <int N>
|
|
constexpr bool check_divisibility_and_divide_by_pow5(std::uint32_t& n) noexcept
|
|
{
|
|
// Make sure the computation for max_n does not overflow
|
|
static_assert(N + 1 <= log::floor_log5_pow2(31));
|
|
assert(n <= compute_power<N + 1>(std::uint32_t(5)) * 2);
|
|
|
|
using info = check_divisibility_and_divide_by_pow5_info<N>;
|
|
n *= info::magic_number;
|
|
constexpr std::uint32_t comparison_mask =
|
|
info::bits_for_comparison >= 32 ? std::numeric_limits<std::uint32_t>::max() :
|
|
std::uint32_t((std::uint32_t(1) << info::bits_for_comparison) - 1);
|
|
|
|
if ((n & comparison_mask) <= info::threshold) {
|
|
n >>= info::shift_amount;
|
|
return true;
|
|
}
|
|
else {
|
|
n >>= info::shift_amount;
|
|
return false;
|
|
}
|
|
}
|
|
|
|
// Compute floor(n / 10^N) for small n and N
|
|
// Precondition: n <= 10^(N+1)
|
|
template <int N>
|
|
struct small_division_by_pow10_info;
|
|
|
|
template <>
|
|
struct small_division_by_pow10_info<1> {
|
|
static constexpr std::uint32_t magic_number = 0xcccd;
|
|
static constexpr int shift_amount = 19;
|
|
};
|
|
|
|
template <>
|
|
struct small_division_by_pow10_info<2> {
|
|
static constexpr std::uint32_t magic_number = 0xa3d8;
|
|
static constexpr int shift_amount = 22;
|
|
};
|
|
|
|
template <int N>
|
|
constexpr std::uint32_t small_division_by_pow10(std::uint32_t n) noexcept
|
|
{
|
|
assert(n <= compute_power<N + 1>(std::uint32_t(10)));
|
|
return (n * small_division_by_pow10_info<N>::magic_number)
|
|
>> small_division_by_pow10_info<N>::shift_amount;
|
|
}
|
|
|
|
// Compute floor(n / 10^N) for small N
|
|
// Precondition: n <= 2^a * 5^b (a = max_pow2, b = max_pow5)
|
|
template <int N, int max_pow2, int max_pow5, class UInt>
|
|
constexpr UInt divide_by_pow10(UInt n) noexcept
|
|
{
|
|
static_assert(N >= 0);
|
|
|
|
// Ensure no overflow
|
|
static_assert(max_pow2 + (log::floor_log2_pow10(max_pow5) - max_pow5) < value_bits<UInt>);
|
|
|
|
// Specialize for 64bit division by 1000
|
|
// Ensure that the correctness condition is met
|
|
if constexpr (std::is_same_v<UInt, std::uint64_t> && N == 3 &&
|
|
max_pow2 + (log::floor_log2_pow10(N + max_pow5) - (N + max_pow5)) < 70)
|
|
{
|
|
return wuint::umul128_upper64(n, 0x8312'6e97'8d4f'df3c) >> 9;
|
|
}
|
|
else {
|
|
constexpr auto divisor = compute_power<N>(UInt(10));
|
|
return n / divisor;
|
|
}
|
|
}
|
|
}
|
|
}
|
|
|
|
////////////////////////////////////////////////////////////////////////////////////////
|
|
// DIY floating-point data type
|
|
////////////////////////////////////////////////////////////////////////////////////////
|
|
|
|
template <class Float, bool is_signed, bool trailing_zero_flag>
|
|
struct fp_t;
|
|
|
|
template <class Float>
|
|
struct fp_t<Float, false, false> {
|
|
using float_type = Float;
|
|
using carrier_uint = typename ieee754_traits<Float>::carrier_uint;
|
|
|
|
carrier_uint significand;
|
|
int exponent;
|
|
};
|
|
|
|
template <class Float>
|
|
struct fp_t<Float, true, false> {
|
|
using float_type = Float;
|
|
using carrier_uint = typename ieee754_traits<Float>::carrier_uint;
|
|
|
|
carrier_uint significand;
|
|
int exponent;
|
|
bool is_negative;
|
|
};
|
|
|
|
template <class Float>
|
|
struct fp_t<Float, false, true> {
|
|
using float_type = Float;
|
|
using carrier_uint = typename ieee754_traits<Float>::carrier_uint;
|
|
|
|
carrier_uint significand;
|
|
int exponent;
|
|
bool may_have_trailing_zeros;
|
|
};
|
|
|
|
template <class Float>
|
|
struct fp_t<Float, true, true> {
|
|
using float_type = Float;
|
|
using carrier_uint = typename ieee754_traits<Float>::carrier_uint;
|
|
|
|
carrier_uint significand;
|
|
int exponent;
|
|
bool is_negative;
|
|
bool may_have_trailing_zeros;
|
|
};
|
|
|
|
template <class Float>
|
|
using unsigned_fp_t = fp_t<Float, false, false>;
|
|
|
|
template <class Float>
|
|
using signed_fp_t = fp_t<Float, true, false>;
|
|
|
|
|
|
////////////////////////////////////////////////////////////////////////////////////////
|
|
// Computed cache entries
|
|
////////////////////////////////////////////////////////////////////////////////////////
|
|
|
|
namespace detail {
|
|
template <ieee754_format format>
|
|
struct cache_holder;
|
|
|
|
template <>
|
|
struct cache_holder<ieee754_format::binary32> {
|
|
using cache_entry_type = std::uint64_t;
|
|
static constexpr int cache_bits = 64;
|
|
static constexpr int min_k = -31;
|
|
static constexpr int max_k = 46;
|
|
static constexpr cache_entry_type cache[] = {
|
|
0x81ceb32c4b43fcf5,
|
|
0xa2425ff75e14fc32,
|
|
0xcad2f7f5359a3b3f,
|
|
0xfd87b5f28300ca0e,
|
|
0x9e74d1b791e07e49,
|
|
0xc612062576589ddb,
|
|
0xf79687aed3eec552,
|
|
0x9abe14cd44753b53,
|
|
0xc16d9a0095928a28,
|
|
0xf1c90080baf72cb2,
|
|
0x971da05074da7bef,
|
|
0xbce5086492111aeb,
|
|
0xec1e4a7db69561a6,
|
|
0x9392ee8e921d5d08,
|
|
0xb877aa3236a4b44a,
|
|
0xe69594bec44de15c,
|
|
0x901d7cf73ab0acda,
|
|
0xb424dc35095cd810,
|
|
0xe12e13424bb40e14,
|
|
0x8cbccc096f5088cc,
|
|
0xafebff0bcb24aaff,
|
|
0xdbe6fecebdedd5bf,
|
|
0x89705f4136b4a598,
|
|
0xabcc77118461cefd,
|
|
0xd6bf94d5e57a42bd,
|
|
0x8637bd05af6c69b6,
|
|
0xa7c5ac471b478424,
|
|
0xd1b71758e219652c,
|
|
0x83126e978d4fdf3c,
|
|
0xa3d70a3d70a3d70b,
|
|
0xcccccccccccccccd,
|
|
0x8000000000000000,
|
|
0xa000000000000000,
|
|
0xc800000000000000,
|
|
0xfa00000000000000,
|
|
0x9c40000000000000,
|
|
0xc350000000000000,
|
|
0xf424000000000000,
|
|
0x9896800000000000,
|
|
0xbebc200000000000,
|
|
0xee6b280000000000,
|
|
0x9502f90000000000,
|
|
0xba43b74000000000,
|
|
0xe8d4a51000000000,
|
|
0x9184e72a00000000,
|
|
0xb5e620f480000000,
|
|
0xe35fa931a0000000,
|
|
0x8e1bc9bf04000000,
|
|
0xb1a2bc2ec5000000,
|
|
0xde0b6b3a76400000,
|
|
0x8ac7230489e80000,
|
|
0xad78ebc5ac620000,
|
|
0xd8d726b7177a8000,
|
|
0x878678326eac9000,
|
|
0xa968163f0a57b400,
|
|
0xd3c21bcecceda100,
|
|
0x84595161401484a0,
|
|
0xa56fa5b99019a5c8,
|
|
0xcecb8f27f4200f3a,
|
|
0x813f3978f8940984,
|
|
0xa18f07d736b90be5,
|
|
0xc9f2c9cd04674ede,
|
|
0xfc6f7c4045812296,
|
|
0x9dc5ada82b70b59d,
|
|
0xc5371912364ce305,
|
|
0xf684df56c3e01bc6,
|
|
0x9a130b963a6c115c,
|
|
0xc097ce7bc90715b3,
|
|
0xf0bdc21abb48db20,
|
|
0x96769950b50d88f4,
|
|
0xbc143fa4e250eb31,
|
|
0xeb194f8e1ae525fd,
|
|
0x92efd1b8d0cf37be,
|
|
0xb7abc627050305ad,
|
|
0xe596b7b0c643c719,
|
|
0x8f7e32ce7bea5c6f,
|
|
0xb35dbf821ae4f38b,
|
|
0xe0352f62a19e306e
|
|
};
|
|
};
|
|
|
|
template <>
|
|
struct cache_holder<ieee754_format::binary64> {
|
|
using cache_entry_type = wuint::uint128;
|
|
static constexpr int cache_bits = 128;
|
|
static constexpr int min_k = -292;
|
|
static constexpr int max_k = 326;
|
|
static constexpr cache_entry_type cache[] = {
|
|
{ 0xff77b1fcbebcdc4f, 0x25e8e89c13bb0f7b },
|
|
{ 0x9faacf3df73609b1, 0x77b191618c54e9ad },
|
|
{ 0xc795830d75038c1d, 0xd59df5b9ef6a2418 },
|
|
{ 0xf97ae3d0d2446f25, 0x4b0573286b44ad1e },
|
|
{ 0x9becce62836ac577, 0x4ee367f9430aec33 },
|
|
{ 0xc2e801fb244576d5, 0x229c41f793cda740 },
|
|
{ 0xf3a20279ed56d48a, 0x6b43527578c11110 },
|
|
{ 0x9845418c345644d6, 0x830a13896b78aaaa },
|
|
{ 0xbe5691ef416bd60c, 0x23cc986bc656d554 },
|
|
{ 0xedec366b11c6cb8f, 0x2cbfbe86b7ec8aa9 },
|
|
{ 0x94b3a202eb1c3f39, 0x7bf7d71432f3d6aa },
|
|
{ 0xb9e08a83a5e34f07, 0xdaf5ccd93fb0cc54 },
|
|
{ 0xe858ad248f5c22c9, 0xd1b3400f8f9cff69 },
|
|
{ 0x91376c36d99995be, 0x23100809b9c21fa2 },
|
|
{ 0xb58547448ffffb2d, 0xabd40a0c2832a78b },
|
|
{ 0xe2e69915b3fff9f9, 0x16c90c8f323f516d },
|
|
{ 0x8dd01fad907ffc3b, 0xae3da7d97f6792e4 },
|
|
{ 0xb1442798f49ffb4a, 0x99cd11cfdf41779d },
|
|
{ 0xdd95317f31c7fa1d, 0x40405643d711d584 },
|
|
{ 0x8a7d3eef7f1cfc52, 0x482835ea666b2573 },
|
|
{ 0xad1c8eab5ee43b66, 0xda3243650005eed0 },
|
|
{ 0xd863b256369d4a40, 0x90bed43e40076a83 },
|
|
{ 0x873e4f75e2224e68, 0x5a7744a6e804a292 },
|
|
{ 0xa90de3535aaae202, 0x711515d0a205cb37 },
|
|
{ 0xd3515c2831559a83, 0x0d5a5b44ca873e04 },
|
|
{ 0x8412d9991ed58091, 0xe858790afe9486c3 },
|
|
{ 0xa5178fff668ae0b6, 0x626e974dbe39a873 },
|
|
{ 0xce5d73ff402d98e3, 0xfb0a3d212dc81290 },
|
|
{ 0x80fa687f881c7f8e, 0x7ce66634bc9d0b9a },
|
|
{ 0xa139029f6a239f72, 0x1c1fffc1ebc44e81 },
|
|
{ 0xc987434744ac874e, 0xa327ffb266b56221 },
|
|
{ 0xfbe9141915d7a922, 0x4bf1ff9f0062baa9 },
|
|
{ 0x9d71ac8fada6c9b5, 0x6f773fc3603db4aa },
|
|
{ 0xc4ce17b399107c22, 0xcb550fb4384d21d4 },
|
|
{ 0xf6019da07f549b2b, 0x7e2a53a146606a49 },
|
|
{ 0x99c102844f94e0fb, 0x2eda7444cbfc426e },
|
|
{ 0xc0314325637a1939, 0xfa911155fefb5309 },
|
|
{ 0xf03d93eebc589f88, 0x793555ab7eba27cb },
|
|
{ 0x96267c7535b763b5, 0x4bc1558b2f3458df },
|
|
{ 0xbbb01b9283253ca2, 0x9eb1aaedfb016f17 },
|
|
{ 0xea9c227723ee8bcb, 0x465e15a979c1cadd },
|
|
{ 0x92a1958a7675175f, 0x0bfacd89ec191eca },
|
|
{ 0xb749faed14125d36, 0xcef980ec671f667c },
|
|
{ 0xe51c79a85916f484, 0x82b7e12780e7401b },
|
|
{ 0x8f31cc0937ae58d2, 0xd1b2ecb8b0908811 },
|
|
{ 0xb2fe3f0b8599ef07, 0x861fa7e6dcb4aa16 },
|
|
{ 0xdfbdcece67006ac9, 0x67a791e093e1d49b },
|
|
{ 0x8bd6a141006042bd, 0xe0c8bb2c5c6d24e1 },
|
|
{ 0xaecc49914078536d, 0x58fae9f773886e19 },
|
|
{ 0xda7f5bf590966848, 0xaf39a475506a899f },
|
|
{ 0x888f99797a5e012d, 0x6d8406c952429604 },
|
|
{ 0xaab37fd7d8f58178, 0xc8e5087ba6d33b84 },
|
|
{ 0xd5605fcdcf32e1d6, 0xfb1e4a9a90880a65 },
|
|
{ 0x855c3be0a17fcd26, 0x5cf2eea09a550680 },
|
|
{ 0xa6b34ad8c9dfc06f, 0xf42faa48c0ea481f },
|
|
{ 0xd0601d8efc57b08b, 0xf13b94daf124da27 },
|
|
{ 0x823c12795db6ce57, 0x76c53d08d6b70859 },
|
|
{ 0xa2cb1717b52481ed, 0x54768c4b0c64ca6f },
|
|
{ 0xcb7ddcdda26da268, 0xa9942f5dcf7dfd0a },
|
|
{ 0xfe5d54150b090b02, 0xd3f93b35435d7c4d },
|
|
{ 0x9efa548d26e5a6e1, 0xc47bc5014a1a6db0 },
|
|
{ 0xc6b8e9b0709f109a, 0x359ab6419ca1091c },
|
|
{ 0xf867241c8cc6d4c0, 0xc30163d203c94b63 },
|
|
{ 0x9b407691d7fc44f8, 0x79e0de63425dcf1e },
|
|
{ 0xc21094364dfb5636, 0x985915fc12f542e5 },
|
|
{ 0xf294b943e17a2bc4, 0x3e6f5b7b17b2939e },
|
|
{ 0x979cf3ca6cec5b5a, 0xa705992ceecf9c43 },
|
|
{ 0xbd8430bd08277231, 0x50c6ff782a838354 },
|
|
{ 0xece53cec4a314ebd, 0xa4f8bf5635246429 },
|
|
{ 0x940f4613ae5ed136, 0x871b7795e136be9a },
|
|
{ 0xb913179899f68584, 0x28e2557b59846e40 },
|
|
{ 0xe757dd7ec07426e5, 0x331aeada2fe589d0 },
|
|
{ 0x9096ea6f3848984f, 0x3ff0d2c85def7622 },
|
|
{ 0xb4bca50b065abe63, 0x0fed077a756b53aa },
|
|
{ 0xe1ebce4dc7f16dfb, 0xd3e8495912c62895 },
|
|
{ 0x8d3360f09cf6e4bd, 0x64712dd7abbbd95d },
|
|
{ 0xb080392cc4349dec, 0xbd8d794d96aacfb4 },
|
|
{ 0xdca04777f541c567, 0xecf0d7a0fc5583a1 },
|
|
{ 0x89e42caaf9491b60, 0xf41686c49db57245 },
|
|
{ 0xac5d37d5b79b6239, 0x311c2875c522ced6 },
|
|
{ 0xd77485cb25823ac7, 0x7d633293366b828c },
|
|
{ 0x86a8d39ef77164bc, 0xae5dff9c02033198 },
|
|
{ 0xa8530886b54dbdeb, 0xd9f57f830283fdfd },
|
|
{ 0xd267caa862a12d66, 0xd072df63c324fd7c },
|
|
{ 0x8380dea93da4bc60, 0x4247cb9e59f71e6e },
|
|
{ 0xa46116538d0deb78, 0x52d9be85f074e609 },
|
|
{ 0xcd795be870516656, 0x67902e276c921f8c },
|
|
{ 0x806bd9714632dff6, 0x00ba1cd8a3db53b7 },
|
|
{ 0xa086cfcd97bf97f3, 0x80e8a40eccd228a5 },
|
|
{ 0xc8a883c0fdaf7df0, 0x6122cd128006b2ce },
|
|
{ 0xfad2a4b13d1b5d6c, 0x796b805720085f82 },
|
|
{ 0x9cc3a6eec6311a63, 0xcbe3303674053bb1 },
|
|
{ 0xc3f490aa77bd60fc, 0xbedbfc4411068a9d },
|
|
{ 0xf4f1b4d515acb93b, 0xee92fb5515482d45 },
|
|
{ 0x991711052d8bf3c5, 0x751bdd152d4d1c4b },
|
|
{ 0xbf5cd54678eef0b6, 0xd262d45a78a0635e },
|
|
{ 0xef340a98172aace4, 0x86fb897116c87c35 },
|
|
{ 0x9580869f0e7aac0e, 0xd45d35e6ae3d4da1 },
|
|
{ 0xbae0a846d2195712, 0x8974836059cca10a },
|
|
{ 0xe998d258869facd7, 0x2bd1a438703fc94c },
|
|
{ 0x91ff83775423cc06, 0x7b6306a34627ddd0 },
|
|
{ 0xb67f6455292cbf08, 0x1a3bc84c17b1d543 },
|
|
{ 0xe41f3d6a7377eeca, 0x20caba5f1d9e4a94 },
|
|
{ 0x8e938662882af53e, 0x547eb47b7282ee9d },
|
|
{ 0xb23867fb2a35b28d, 0xe99e619a4f23aa44 },
|
|
{ 0xdec681f9f4c31f31, 0x6405fa00e2ec94d5 },
|
|
{ 0x8b3c113c38f9f37e, 0xde83bc408dd3dd05 },
|
|
{ 0xae0b158b4738705e, 0x9624ab50b148d446 },
|
|
{ 0xd98ddaee19068c76, 0x3badd624dd9b0958 },
|
|
{ 0x87f8a8d4cfa417c9, 0xe54ca5d70a80e5d7 },
|
|
{ 0xa9f6d30a038d1dbc, 0x5e9fcf4ccd211f4d },
|
|
{ 0xd47487cc8470652b, 0x7647c32000696720 },
|
|
{ 0x84c8d4dfd2c63f3b, 0x29ecd9f40041e074 },
|
|
{ 0xa5fb0a17c777cf09, 0xf468107100525891 },
|
|
{ 0xcf79cc9db955c2cc, 0x7182148d4066eeb5 },
|
|
{ 0x81ac1fe293d599bf, 0xc6f14cd848405531 },
|
|
{ 0xa21727db38cb002f, 0xb8ada00e5a506a7d },
|
|
{ 0xca9cf1d206fdc03b, 0xa6d90811f0e4851d },
|
|
{ 0xfd442e4688bd304a, 0x908f4a166d1da664 },
|
|
{ 0x9e4a9cec15763e2e, 0x9a598e4e043287ff },
|
|
{ 0xc5dd44271ad3cdba, 0x40eff1e1853f29fe },
|
|
{ 0xf7549530e188c128, 0xd12bee59e68ef47d },
|
|
{ 0x9a94dd3e8cf578b9, 0x82bb74f8301958cf },
|
|
{ 0xc13a148e3032d6e7, 0xe36a52363c1faf02 },
|
|
{ 0xf18899b1bc3f8ca1, 0xdc44e6c3cb279ac2 },
|
|
{ 0x96f5600f15a7b7e5, 0x29ab103a5ef8c0ba },
|
|
{ 0xbcb2b812db11a5de, 0x7415d448f6b6f0e8 },
|
|
{ 0xebdf661791d60f56, 0x111b495b3464ad22 },
|
|
{ 0x936b9fcebb25c995, 0xcab10dd900beec35 },
|
|
{ 0xb84687c269ef3bfb, 0x3d5d514f40eea743 },
|
|
{ 0xe65829b3046b0afa, 0x0cb4a5a3112a5113 },
|
|
{ 0x8ff71a0fe2c2e6dc, 0x47f0e785eaba72ac },
|
|
{ 0xb3f4e093db73a093, 0x59ed216765690f57 },
|
|
{ 0xe0f218b8d25088b8, 0x306869c13ec3532d },
|
|
{ 0x8c974f7383725573, 0x1e414218c73a13fc },
|
|
{ 0xafbd2350644eeacf, 0xe5d1929ef90898fb },
|
|
{ 0xdbac6c247d62a583, 0xdf45f746b74abf3a },
|
|
{ 0x894bc396ce5da772, 0x6b8bba8c328eb784 },
|
|
{ 0xab9eb47c81f5114f, 0x066ea92f3f326565 },
|
|
{ 0xd686619ba27255a2, 0xc80a537b0efefebe },
|
|
{ 0x8613fd0145877585, 0xbd06742ce95f5f37 },
|
|
{ 0xa798fc4196e952e7, 0x2c48113823b73705 },
|
|
{ 0xd17f3b51fca3a7a0, 0xf75a15862ca504c6 },
|
|
{ 0x82ef85133de648c4, 0x9a984d73dbe722fc },
|
|
{ 0xa3ab66580d5fdaf5, 0xc13e60d0d2e0ebbb },
|
|
{ 0xcc963fee10b7d1b3, 0x318df905079926a9 },
|
|
{ 0xffbbcfe994e5c61f, 0xfdf17746497f7053 },
|
|
{ 0x9fd561f1fd0f9bd3, 0xfeb6ea8bedefa634 },
|
|
{ 0xc7caba6e7c5382c8, 0xfe64a52ee96b8fc1 },
|
|
{ 0xf9bd690a1b68637b, 0x3dfdce7aa3c673b1 },
|
|
{ 0x9c1661a651213e2d, 0x06bea10ca65c084f },
|
|
{ 0xc31bfa0fe5698db8, 0x486e494fcff30a63 },
|
|
{ 0xf3e2f893dec3f126, 0x5a89dba3c3efccfb },
|
|
{ 0x986ddb5c6b3a76b7, 0xf89629465a75e01d },
|
|
{ 0xbe89523386091465, 0xf6bbb397f1135824 },
|
|
{ 0xee2ba6c0678b597f, 0x746aa07ded582e2d },
|
|
{ 0x94db483840b717ef, 0xa8c2a44eb4571cdd },
|
|
{ 0xba121a4650e4ddeb, 0x92f34d62616ce414 },
|
|
{ 0xe896a0d7e51e1566, 0x77b020baf9c81d18 },
|
|
{ 0x915e2486ef32cd60, 0x0ace1474dc1d122f },
|
|
{ 0xb5b5ada8aaff80b8, 0x0d819992132456bb },
|
|
{ 0xe3231912d5bf60e6, 0x10e1fff697ed6c6a },
|
|
{ 0x8df5efabc5979c8f, 0xca8d3ffa1ef463c2 },
|
|
{ 0xb1736b96b6fd83b3, 0xbd308ff8a6b17cb3 },
|
|
{ 0xddd0467c64bce4a0, 0xac7cb3f6d05ddbdf },
|
|
{ 0x8aa22c0dbef60ee4, 0x6bcdf07a423aa96c },
|
|
{ 0xad4ab7112eb3929d, 0x86c16c98d2c953c7 },
|
|
{ 0xd89d64d57a607744, 0xe871c7bf077ba8b8 },
|
|
{ 0x87625f056c7c4a8b, 0x11471cd764ad4973 },
|
|
{ 0xa93af6c6c79b5d2d, 0xd598e40d3dd89bd0 },
|
|
{ 0xd389b47879823479, 0x4aff1d108d4ec2c4 },
|
|
{ 0x843610cb4bf160cb, 0xcedf722a585139bb },
|
|
{ 0xa54394fe1eedb8fe, 0xc2974eb4ee658829 },
|
|
{ 0xce947a3da6a9273e, 0x733d226229feea33 },
|
|
{ 0x811ccc668829b887, 0x0806357d5a3f5260 },
|
|
{ 0xa163ff802a3426a8, 0xca07c2dcb0cf26f8 },
|
|
{ 0xc9bcff6034c13052, 0xfc89b393dd02f0b6 },
|
|
{ 0xfc2c3f3841f17c67, 0xbbac2078d443ace3 },
|
|
{ 0x9d9ba7832936edc0, 0xd54b944b84aa4c0e },
|
|
{ 0xc5029163f384a931, 0x0a9e795e65d4df12 },
|
|
{ 0xf64335bcf065d37d, 0x4d4617b5ff4a16d6 },
|
|
{ 0x99ea0196163fa42e, 0x504bced1bf8e4e46 },
|
|
{ 0xc06481fb9bcf8d39, 0xe45ec2862f71e1d7 },
|
|
{ 0xf07da27a82c37088, 0x5d767327bb4e5a4d },
|
|
{ 0x964e858c91ba2655, 0x3a6a07f8d510f870 },
|
|
{ 0xbbe226efb628afea, 0x890489f70a55368c },
|
|
{ 0xeadab0aba3b2dbe5, 0x2b45ac74ccea842f },
|
|
{ 0x92c8ae6b464fc96f, 0x3b0b8bc90012929e },
|
|
{ 0xb77ada0617e3bbcb, 0x09ce6ebb40173745 },
|
|
{ 0xe55990879ddcaabd, 0xcc420a6a101d0516 },
|
|
{ 0x8f57fa54c2a9eab6, 0x9fa946824a12232e },
|
|
{ 0xb32df8e9f3546564, 0x47939822dc96abfa },
|
|
{ 0xdff9772470297ebd, 0x59787e2b93bc56f8 },
|
|
{ 0x8bfbea76c619ef36, 0x57eb4edb3c55b65b },
|
|
{ 0xaefae51477a06b03, 0xede622920b6b23f2 },
|
|
{ 0xdab99e59958885c4, 0xe95fab368e45ecee },
|
|
{ 0x88b402f7fd75539b, 0x11dbcb0218ebb415 },
|
|
{ 0xaae103b5fcd2a881, 0xd652bdc29f26a11a },
|
|
{ 0xd59944a37c0752a2, 0x4be76d3346f04960 },
|
|
{ 0x857fcae62d8493a5, 0x6f70a4400c562ddc },
|
|
{ 0xa6dfbd9fb8e5b88e, 0xcb4ccd500f6bb953 },
|
|
{ 0xd097ad07a71f26b2, 0x7e2000a41346a7a8 },
|
|
{ 0x825ecc24c873782f, 0x8ed400668c0c28c9 },
|
|
{ 0xa2f67f2dfa90563b, 0x728900802f0f32fb },
|
|
{ 0xcbb41ef979346bca, 0x4f2b40a03ad2ffba },
|
|
{ 0xfea126b7d78186bc, 0xe2f610c84987bfa9 },
|
|
{ 0x9f24b832e6b0f436, 0x0dd9ca7d2df4d7ca },
|
|
{ 0xc6ede63fa05d3143, 0x91503d1c79720dbc },
|
|
{ 0xf8a95fcf88747d94, 0x75a44c6397ce912b },
|
|
{ 0x9b69dbe1b548ce7c, 0xc986afbe3ee11abb },
|
|
{ 0xc24452da229b021b, 0xfbe85badce996169 },
|
|
{ 0xf2d56790ab41c2a2, 0xfae27299423fb9c4 },
|
|
{ 0x97c560ba6b0919a5, 0xdccd879fc967d41b },
|
|
{ 0xbdb6b8e905cb600f, 0x5400e987bbc1c921 },
|
|
{ 0xed246723473e3813, 0x290123e9aab23b69 },
|
|
{ 0x9436c0760c86e30b, 0xf9a0b6720aaf6522 },
|
|
{ 0xb94470938fa89bce, 0xf808e40e8d5b3e6a },
|
|
{ 0xe7958cb87392c2c2, 0xb60b1d1230b20e05 },
|
|
{ 0x90bd77f3483bb9b9, 0xb1c6f22b5e6f48c3 },
|
|
{ 0xb4ecd5f01a4aa828, 0x1e38aeb6360b1af4 },
|
|
{ 0xe2280b6c20dd5232, 0x25c6da63c38de1b1 },
|
|
{ 0x8d590723948a535f, 0x579c487e5a38ad0f },
|
|
{ 0xb0af48ec79ace837, 0x2d835a9df0c6d852 },
|
|
{ 0xdcdb1b2798182244, 0xf8e431456cf88e66 },
|
|
{ 0x8a08f0f8bf0f156b, 0x1b8e9ecb641b5900 },
|
|
{ 0xac8b2d36eed2dac5, 0xe272467e3d222f40 },
|
|
{ 0xd7adf884aa879177, 0x5b0ed81dcc6abb10 },
|
|
{ 0x86ccbb52ea94baea, 0x98e947129fc2b4ea },
|
|
{ 0xa87fea27a539e9a5, 0x3f2398d747b36225 },
|
|
{ 0xd29fe4b18e88640e, 0x8eec7f0d19a03aae },
|
|
{ 0x83a3eeeef9153e89, 0x1953cf68300424ad },
|
|
{ 0xa48ceaaab75a8e2b, 0x5fa8c3423c052dd8 },
|
|
{ 0xcdb02555653131b6, 0x3792f412cb06794e },
|
|
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|
|
{ 0xdb2e51bfe9d0696a, 0x06997b05fcc0319e },
|
|
{ 0x88fcf317f22241e2, 0x441fece3bdf81f03 },
|
|
{ 0xab3c2fddeeaad25a, 0xd527e81cad7626c3 },
|
|
{ 0xd60b3bd56a5586f1, 0x8a71e223d8d3b074 },
|
|
{ 0x85c7056562757456, 0xf6872d5667844e49 },
|
|
{ 0xa738c6bebb12d16c, 0xb428f8ac016561db },
|
|
{ 0xd106f86e69d785c7, 0xe13336d701beba52 },
|
|
{ 0x82a45b450226b39c, 0xecc0024661173473 },
|
|
{ 0xa34d721642b06084, 0x27f002d7f95d0190 },
|
|
{ 0xcc20ce9bd35c78a5, 0x31ec038df7b441f4 },
|
|
{ 0xff290242c83396ce, 0x7e67047175a15271 },
|
|
{ 0x9f79a169bd203e41, 0x0f0062c6e984d386 },
|
|
{ 0xc75809c42c684dd1, 0x52c07b78a3e60868 },
|
|
{ 0xf92e0c3537826145, 0xa7709a56ccdf8a82 },
|
|
{ 0x9bbcc7a142b17ccb, 0x88a66076400bb691 },
|
|
{ 0xc2abf989935ddbfe, 0x6acff893d00ea435 },
|
|
{ 0xf356f7ebf83552fe, 0x0583f6b8c4124d43 },
|
|
{ 0x98165af37b2153de, 0xc3727a337a8b704a },
|
|
{ 0xbe1bf1b059e9a8d6, 0x744f18c0592e4c5c },
|
|
{ 0xeda2ee1c7064130c, 0x1162def06f79df73 },
|
|
{ 0x9485d4d1c63e8be7, 0x8addcb5645ac2ba8 },
|
|
{ 0xb9a74a0637ce2ee1, 0x6d953e2bd7173692 },
|
|
{ 0xe8111c87c5c1ba99, 0xc8fa8db6ccdd0437 },
|
|
{ 0x910ab1d4db9914a0, 0x1d9c9892400a22a2 },
|
|
{ 0xb54d5e4a127f59c8, 0x2503beb6d00cab4b },
|
|
{ 0xe2a0b5dc971f303a, 0x2e44ae64840fd61d },
|
|
{ 0x8da471a9de737e24, 0x5ceaecfed289e5d2 },
|
|
{ 0xb10d8e1456105dad, 0x7425a83e872c5f47 },
|
|
{ 0xdd50f1996b947518, 0xd12f124e28f77719 },
|
|
{ 0x8a5296ffe33cc92f, 0x82bd6b70d99aaa6f },
|
|
{ 0xace73cbfdc0bfb7b, 0x636cc64d1001550b },
|
|
{ 0xd8210befd30efa5a, 0x3c47f7e05401aa4e },
|
|
{ 0x8714a775e3e95c78, 0x65acfaec34810a71 },
|
|
{ 0xa8d9d1535ce3b396, 0x7f1839a741a14d0d },
|
|
{ 0xd31045a8341ca07c, 0x1ede48111209a050 },
|
|
{ 0x83ea2b892091e44d, 0x934aed0aab460432 },
|
|
{ 0xa4e4b66b68b65d60, 0xf81da84d5617853f },
|
|
{ 0xce1de40642e3f4b9, 0x36251260ab9d668e },
|
|
{ 0x80d2ae83e9ce78f3, 0xc1d72b7c6b426019 },
|
|
{ 0xa1075a24e4421730, 0xb24cf65b8612f81f },
|
|
{ 0xc94930ae1d529cfc, 0xdee033f26797b627 },
|
|
{ 0xfb9b7cd9a4a7443c, 0x169840ef017da3b1 },
|
|
{ 0x9d412e0806e88aa5, 0x8e1f289560ee864e },
|
|
{ 0xc491798a08a2ad4e, 0xf1a6f2bab92a27e2 },
|
|
{ 0xf5b5d7ec8acb58a2, 0xae10af696774b1db },
|
|
{ 0x9991a6f3d6bf1765, 0xacca6da1e0a8ef29 },
|
|
{ 0xbff610b0cc6edd3f, 0x17fd090a58d32af3 },
|
|
{ 0xeff394dcff8a948e, 0xddfc4b4cef07f5b0 },
|
|
{ 0x95f83d0a1fb69cd9, 0x4abdaf101564f98e },
|
|
{ 0xbb764c4ca7a4440f, 0x9d6d1ad41abe37f1 },
|
|
{ 0xea53df5fd18d5513, 0x84c86189216dc5ed },
|
|
{ 0x92746b9be2f8552c, 0x32fd3cf5b4e49bb4 },
|
|
{ 0xb7118682dbb66a77, 0x3fbc8c33221dc2a1 },
|
|
{ 0xe4d5e82392a40515, 0x0fabaf3feaa5334a },
|
|
{ 0x8f05b1163ba6832d, 0x29cb4d87f2a7400e },
|
|
{ 0xb2c71d5bca9023f8, 0x743e20e9ef511012 },
|
|
{ 0xdf78e4b2bd342cf6, 0x914da9246b255416 },
|
|
{ 0x8bab8eefb6409c1a, 0x1ad089b6c2f7548e },
|
|
{ 0xae9672aba3d0c320, 0xa184ac2473b529b1 },
|
|
{ 0xda3c0f568cc4f3e8, 0xc9e5d72d90a2741e },
|
|
{ 0x8865899617fb1871, 0x7e2fa67c7a658892 },
|
|
{ 0xaa7eebfb9df9de8d, 0xddbb901b98feeab7 },
|
|
{ 0xd51ea6fa85785631, 0x552a74227f3ea565 },
|
|
{ 0x8533285c936b35de, 0xd53a88958f87275f },
|
|
{ 0xa67ff273b8460356, 0x8a892abaf368f137 },
|
|
{ 0xd01fef10a657842c, 0x2d2b7569b0432d85 },
|
|
{ 0x8213f56a67f6b29b, 0x9c3b29620e29fc73 },
|
|
{ 0xa298f2c501f45f42, 0x8349f3ba91b47b8f },
|
|
{ 0xcb3f2f7642717713, 0x241c70a936219a73 },
|
|
{ 0xfe0efb53d30dd4d7, 0xed238cd383aa0110 },
|
|
{ 0x9ec95d1463e8a506, 0xf4363804324a40aa },
|
|
{ 0xc67bb4597ce2ce48, 0xb143c6053edcd0d5 },
|
|
{ 0xf81aa16fdc1b81da, 0xdd94b7868e94050a },
|
|
{ 0x9b10a4e5e9913128, 0xca7cf2b4191c8326 },
|
|
{ 0xc1d4ce1f63f57d72, 0xfd1c2f611f63a3f0 },
|
|
{ 0xf24a01a73cf2dccf, 0xbc633b39673c8cec },
|
|
{ 0x976e41088617ca01, 0xd5be0503e085d813 },
|
|
{ 0xbd49d14aa79dbc82, 0x4b2d8644d8a74e18 },
|
|
{ 0xec9c459d51852ba2, 0xddf8e7d60ed1219e },
|
|
{ 0x93e1ab8252f33b45, 0xcabb90e5c942b503 },
|
|
{ 0xb8da1662e7b00a17, 0x3d6a751f3b936243 },
|
|
{ 0xe7109bfba19c0c9d, 0x0cc512670a783ad4 },
|
|
{ 0x906a617d450187e2, 0x27fb2b80668b24c5 },
|
|
{ 0xb484f9dc9641e9da, 0xb1f9f660802dedf6 },
|
|
{ 0xe1a63853bbd26451, 0x5e7873f8a0396973 },
|
|
{ 0x8d07e33455637eb2, 0xdb0b487b6423e1e8 },
|
|
{ 0xb049dc016abc5e5f, 0x91ce1a9a3d2cda62 },
|
|
{ 0xdc5c5301c56b75f7, 0x7641a140cc7810fb },
|
|
{ 0x89b9b3e11b6329ba, 0xa9e904c87fcb0a9d },
|
|
{ 0xac2820d9623bf429, 0x546345fa9fbdcd44 },
|
|
{ 0xd732290fbacaf133, 0xa97c177947ad4095 },
|
|
{ 0x867f59a9d4bed6c0, 0x49ed8eabcccc485d },
|
|
{ 0xa81f301449ee8c70, 0x5c68f256bfff5a74 },
|
|
{ 0xd226fc195c6a2f8c, 0x73832eec6fff3111 },
|
|
{ 0x83585d8fd9c25db7, 0xc831fd53c5ff7eab },
|
|
{ 0xa42e74f3d032f525, 0xba3e7ca8b77f5e55 },
|
|
{ 0xcd3a1230c43fb26f, 0x28ce1bd2e55f35eb },
|
|
{ 0x80444b5e7aa7cf85, 0x7980d163cf5b81b3 },
|
|
{ 0xa0555e361951c366, 0xd7e105bcc332621f },
|
|
{ 0xc86ab5c39fa63440, 0x8dd9472bf3fefaa7 },
|
|
{ 0xfa856334878fc150, 0xb14f98f6f0feb951 },
|
|
{ 0x9c935e00d4b9d8d2, 0x6ed1bf9a569f33d3 },
|
|
{ 0xc3b8358109e84f07, 0x0a862f80ec4700c8 },
|
|
{ 0xf4a642e14c6262c8, 0xcd27bb612758c0fa },
|
|
{ 0x98e7e9cccfbd7dbd, 0x8038d51cb897789c },
|
|
{ 0xbf21e44003acdd2c, 0xe0470a63e6bd56c3 },
|
|
{ 0xeeea5d5004981478, 0x1858ccfce06cac74 },
|
|
{ 0x95527a5202df0ccb, 0x0f37801e0c43ebc8 },
|
|
{ 0xbaa718e68396cffd, 0xd30560258f54e6ba },
|
|
{ 0xe950df20247c83fd, 0x47c6b82ef32a2069 },
|
|
{ 0x91d28b7416cdd27e, 0x4cdc331d57fa5441 },
|
|
{ 0xb6472e511c81471d, 0xe0133fe4adf8e952 },
|
|
{ 0xe3d8f9e563a198e5, 0x58180fddd97723a6 },
|
|
{ 0x8e679c2f5e44ff8f, 0x570f09eaa7ea7648 },
|
|
{ 0xb201833b35d63f73, 0x2cd2cc6551e513da },
|
|
{ 0xde81e40a034bcf4f, 0xf8077f7ea65e58d1 },
|
|
{ 0x8b112e86420f6191, 0xfb04afaf27faf782 },
|
|
{ 0xadd57a27d29339f6, 0x79c5db9af1f9b563 },
|
|
{ 0xd94ad8b1c7380874, 0x18375281ae7822bc },
|
|
{ 0x87cec76f1c830548, 0x8f2293910d0b15b5 },
|
|
{ 0xa9c2794ae3a3c69a, 0xb2eb3875504ddb22 },
|
|
{ 0xd433179d9c8cb841, 0x5fa60692a46151eb },
|
|
{ 0x849feec281d7f328, 0xdbc7c41ba6bcd333 },
|
|
{ 0xa5c7ea73224deff3, 0x12b9b522906c0800 },
|
|
{ 0xcf39e50feae16bef, 0xd768226b34870a00 },
|
|
{ 0x81842f29f2cce375, 0xe6a1158300d46640 },
|
|
{ 0xa1e53af46f801c53, 0x60495ae3c1097fd0 },
|
|
{ 0xca5e89b18b602368, 0x385bb19cb14bdfc4 },
|
|
{ 0xfcf62c1dee382c42, 0x46729e03dd9ed7b5 },
|
|
{ 0x9e19db92b4e31ba9, 0x6c07a2c26a8346d1 },
|
|
{ 0xc5a05277621be293, 0xc7098b7305241885 },
|
|
{ 0xf70867153aa2db38, 0xb8cbee4fc66d1ea7 }
|
|
};
|
|
};
|
|
|
|
// Compressed cache for double
|
|
struct compressed_cache_detail {
|
|
static constexpr int compression_ratio = 27;
|
|
static constexpr std::size_t compressed_table_size =
|
|
(cache_holder<ieee754_format::binary64>::max_k -
|
|
cache_holder<ieee754_format::binary64>::min_k + compression_ratio) / compression_ratio;
|
|
|
|
struct cache_holder_t {
|
|
wuint::uint128 table[compressed_table_size];
|
|
};
|
|
static constexpr cache_holder_t cache = [] {
|
|
cache_holder_t res{};
|
|
for (std::size_t i = 0; i < compressed_table_size; ++i) {
|
|
res.table[i] = cache_holder<ieee754_format::binary64>::cache[i * compression_ratio];
|
|
}
|
|
return res;
|
|
}();
|
|
|
|
struct pow5_holder_t {
|
|
std::uint64_t table[compression_ratio];
|
|
};
|
|
static constexpr pow5_holder_t pow5 = [] {
|
|
pow5_holder_t res{};
|
|
std::uint64_t p = 1;
|
|
for (std::size_t i = 0; i < compression_ratio; ++i) {
|
|
res.table[i] = p;
|
|
p *= 5;
|
|
}
|
|
return res;
|
|
}();
|
|
|
|
static constexpr std::uint32_t errors[] = {
|
|
0x50001400, 0x54044100, 0x54014555, 0x55954415, 0x54115555,
|
|
0x00000001, 0x50000000, 0x00104000, 0x54010004, 0x05004001,
|
|
0x55555544, 0x41545555, 0x54040551, 0x15445545, 0x51555514,
|
|
0x10000015, 0x00101100, 0x01100015, 0x00000000, 0x00000000,
|
|
0x00000000, 0x00000000, 0x04450514, 0x45414110, 0x55555145,
|
|
0x50544050, 0x15040155, 0x11054140, 0x50111514, 0x11451454,
|
|
0x00400541, 0x00000000, 0x55555450, 0x10056551, 0x10054011,
|
|
0x55551014, 0x69514555, 0x05151109, 0x00155555
|
|
};
|
|
};
|
|
}
|
|
|
|
|
|
////////////////////////////////////////////////////////////////////////////////////////
|
|
// Policies
|
|
////////////////////////////////////////////////////////////////////////////////////////
|
|
|
|
namespace detail {
|
|
// Forward declare the implementation class
|
|
template <class Float>
|
|
struct impl;
|
|
|
|
namespace policy_impl {
|
|
// Sign policy
|
|
namespace sign {
|
|
struct base {};
|
|
|
|
struct ignore : base {
|
|
using sign_policy = ignore;
|
|
static constexpr bool return_has_sign = false;
|
|
|
|
template <class Float, class Fp>
|
|
static constexpr void handle_sign(ieee754_bits<Float>, Fp&) noexcept {}
|
|
};
|
|
|
|
struct return_sign : base {
|
|
using sign_policy = return_sign;
|
|
static constexpr bool return_has_sign = true;
|
|
|
|
template <class Float, class Fp>
|
|
static constexpr void handle_sign(ieee754_bits<Float> br, Fp& fp) noexcept {
|
|
fp.is_negative = br.is_negative();
|
|
}
|
|
};
|
|
}
|
|
|
|
// Trailing zero policy
|
|
namespace trailing_zero {
|
|
struct base {};
|
|
|
|
struct ignore : base {
|
|
using trailing_zero_policy = ignore;
|
|
static constexpr bool report_trailing_zeros = false;
|
|
|
|
template <class Fp>
|
|
static constexpr void on_trailing_zeros(Fp&) noexcept {}
|
|
|
|
template <class Fp>
|
|
static constexpr void no_trailing_zeros(Fp&) noexcept {}
|
|
};
|
|
|
|
struct remove : base {
|
|
using trailing_zero_policy = remove;
|
|
static constexpr bool report_trailing_zeros = false;
|
|
|
|
template <class Fp>
|
|
static constexpr void on_trailing_zeros(Fp& fp) noexcept {
|
|
fp.exponent +=
|
|
impl<typename Fp::float_type>::remove_trailing_zeros(fp.significand);
|
|
}
|
|
|
|
template <class Fp>
|
|
static constexpr void no_trailing_zeros(Fp&) noexcept {}
|
|
};
|
|
|
|
struct report : base {
|
|
using trailing_zero_policy = report;
|
|
static constexpr bool report_trailing_zeros = true;
|
|
|
|
template <class Fp>
|
|
static constexpr void on_trailing_zeros(Fp& fp) noexcept {
|
|
fp.may_have_trailing_zeros = true;
|
|
}
|
|
|
|
template <class Fp>
|
|
static constexpr void no_trailing_zeros(Fp& fp) noexcept {
|
|
fp.may_have_trailing_zeros = false;
|
|
}
|
|
};
|
|
}
|
|
|
|
// Rounding mode policy
|
|
namespace rounding_mode {
|
|
struct base {};
|
|
|
|
enum class tag_t {
|
|
to_nearest,
|
|
left_closed_directed,
|
|
right_closed_directed
|
|
};
|
|
namespace interval_type {
|
|
struct symmetric_boundary {
|
|
static constexpr bool is_symmetric = true;
|
|
bool is_closed;
|
|
constexpr bool include_left_endpoint() const noexcept {
|
|
return is_closed;
|
|
}
|
|
constexpr bool include_right_endpoint() const noexcept {
|
|
return is_closed;
|
|
}
|
|
};
|
|
struct asymmetric_boundary {
|
|
static constexpr bool is_symmetric = false;
|
|
bool is_left_closed;
|
|
constexpr bool include_left_endpoint() const noexcept {
|
|
return is_left_closed;
|
|
}
|
|
constexpr bool include_right_endpoint() const noexcept {
|
|
return !is_left_closed;
|
|
}
|
|
};
|
|
struct closed {
|
|
static constexpr bool is_symmetric = true;
|
|
static constexpr bool include_left_endpoint() noexcept {
|
|
return true;
|
|
}
|
|
static constexpr bool include_right_endpoint() noexcept {
|
|
return true;
|
|
}
|
|
};
|
|
struct open {
|
|
static constexpr bool is_symmetric = true;
|
|
static constexpr bool include_left_endpoint() noexcept {
|
|
return false;
|
|
}
|
|
static constexpr bool include_right_endpoint() noexcept {
|
|
return false;
|
|
}
|
|
};
|
|
struct left_closed_right_open {
|
|
static constexpr bool is_symmetric = false;
|
|
static constexpr bool include_left_endpoint() noexcept {
|
|
return true;
|
|
}
|
|
static constexpr bool include_right_endpoint() noexcept {
|
|
return false;
|
|
}
|
|
};
|
|
struct right_closed_left_open {
|
|
static constexpr bool is_symmetric = false;
|
|
static constexpr bool include_left_endpoint() noexcept {
|
|
return false;
|
|
}
|
|
static constexpr bool include_right_endpoint() noexcept {
|
|
return true;
|
|
}
|
|
};
|
|
}
|
|
|
|
struct nearest_to_even : base {
|
|
using rounding_mode_policy = nearest_to_even;
|
|
static constexpr auto tag = tag_t::to_nearest;
|
|
|
|
template <class Float, class Func>
|
|
static auto delegate(ieee754_bits<Float>, Func&& f) noexcept {
|
|
return f(nearest_to_even{});
|
|
}
|
|
|
|
template <class Float>
|
|
static constexpr interval_type::symmetric_boundary
|
|
interval_type_normal(ieee754_bits<Float> br) noexcept
|
|
{
|
|
return{ br.u % 2 == 0 };
|
|
}
|
|
template <class Float>
|
|
static constexpr interval_type::closed
|
|
interval_type_shorter(ieee754_bits<Float>) noexcept
|
|
{
|
|
return{};
|
|
}
|
|
};
|
|
struct nearest_to_odd : base {
|
|
using rounding_mode_policy = nearest_to_odd;
|
|
static constexpr auto tag = tag_t::to_nearest;
|
|
|
|
template <class Float, class Func>
|
|
static auto delegate(ieee754_bits<Float>, Func&& f) noexcept {
|
|
return f(nearest_to_odd{});
|
|
}
|
|
|
|
template <class Float>
|
|
static constexpr interval_type::symmetric_boundary
|
|
interval_type_normal(ieee754_bits<Float> br) noexcept
|
|
{
|
|
return{ br.u % 2 != 0 };
|
|
}
|
|
template <class Float>
|
|
static constexpr interval_type::closed
|
|
interval_type_shorter(ieee754_bits<Float>) noexcept
|
|
{
|
|
return{};
|
|
}
|
|
};
|
|
struct nearest_toward_plus_infinity : base {
|
|
using rounding_mode_policy = nearest_toward_plus_infinity;
|
|
static constexpr auto tag = tag_t::to_nearest;
|
|
|
|
template <class Float, class Func>
|
|
static auto delegate(ieee754_bits<Float>, Func&& f) noexcept {
|
|
return f(nearest_toward_plus_infinity{});
|
|
}
|
|
|
|
template <class Float>
|
|
static constexpr interval_type::asymmetric_boundary
|
|
interval_type_normal(ieee754_bits<Float> br) noexcept
|
|
{
|
|
return{ !br.is_negative() };
|
|
}
|
|
template <class Float>
|
|
static constexpr interval_type::asymmetric_boundary
|
|
interval_type_shorter(ieee754_bits<Float> br) noexcept
|
|
{
|
|
return{ !br.is_negative() };
|
|
}
|
|
};
|
|
struct nearest_toward_minus_infinity : base {
|
|
using rounding_mode_policy = nearest_toward_minus_infinity;
|
|
static constexpr auto tag = tag_t::to_nearest;
|
|
|
|
template <class Float, class Func>
|
|
static auto delegate(ieee754_bits<Float>, Func&& f) noexcept {
|
|
return f(nearest_toward_minus_infinity{});
|
|
}
|
|
|
|
template <class Float>
|
|
static constexpr interval_type::asymmetric_boundary
|
|
interval_type_normal(ieee754_bits<Float> br) noexcept
|
|
{
|
|
return{ br.is_negative() };
|
|
}
|
|
template <class Float>
|
|
static constexpr interval_type::asymmetric_boundary
|
|
interval_type_shorter(ieee754_bits<Float> br) noexcept
|
|
{
|
|
return{ br.is_negative() };
|
|
}
|
|
};
|
|
struct nearest_toward_zero : base {
|
|
using rounding_mode_policy = nearest_toward_zero;
|
|
static constexpr auto tag = tag_t::to_nearest;
|
|
|
|
template <class Float, class Func>
|
|
static auto delegate(ieee754_bits<Float>, Func&& f) noexcept {
|
|
return f(nearest_toward_zero{});
|
|
}
|
|
template <class Float>
|
|
static constexpr interval_type::right_closed_left_open
|
|
interval_type_normal(ieee754_bits<Float>) noexcept
|
|
{
|
|
return{};
|
|
}
|
|
template <class Float>
|
|
static constexpr interval_type::right_closed_left_open
|
|
interval_type_shorter(ieee754_bits<Float>) noexcept
|
|
{
|
|
return{};
|
|
}
|
|
};
|
|
struct nearest_away_from_zero : base {
|
|
using rounding_mode_policy = nearest_away_from_zero;
|
|
static constexpr auto tag = tag_t::to_nearest;
|
|
|
|
template <class Float, class Func>
|
|
static auto delegate(ieee754_bits<Float>, Func&& f) noexcept {
|
|
return f(nearest_away_from_zero{});
|
|
}
|
|
template <class Float>
|
|
static constexpr interval_type::left_closed_right_open
|
|
interval_type_normal(ieee754_bits<Float>) noexcept
|
|
{
|
|
return{};
|
|
}
|
|
template <class Float>
|
|
static constexpr interval_type::left_closed_right_open
|
|
interval_type_shorter(ieee754_bits<Float>) noexcept
|
|
{
|
|
return{};
|
|
}
|
|
};
|
|
|
|
namespace detail {
|
|
struct nearest_always_closed {
|
|
static constexpr auto tag = tag_t::to_nearest;
|
|
|
|
template <class Float>
|
|
static constexpr interval_type::closed
|
|
interval_type_normal(ieee754_bits<Float>) noexcept
|
|
{
|
|
return{};
|
|
}
|
|
template <class Float>
|
|
static constexpr interval_type::closed
|
|
interval_type_shorter(ieee754_bits<Float>) noexcept
|
|
{
|
|
return{};
|
|
}
|
|
};
|
|
struct nearest_always_open {
|
|
static constexpr auto tag = tag_t::to_nearest;
|
|
|
|
template <class Float>
|
|
static constexpr interval_type::open
|
|
interval_type_normal(ieee754_bits<Float>) noexcept
|
|
{
|
|
return{};
|
|
}
|
|
template <class Float>
|
|
static constexpr interval_type::open
|
|
interval_type_shorter(ieee754_bits<Float>) noexcept
|
|
{
|
|
return{};
|
|
}
|
|
};
|
|
}
|
|
|
|
struct nearest_to_even_static_boundary : base {
|
|
using rounding_mode_policy = nearest_to_even_static_boundary;
|
|
template <class Float, class Func>
|
|
static auto delegate(ieee754_bits<Float> br, Func&& f) noexcept {
|
|
if (br.u % 2 == 0) {
|
|
return f(detail::nearest_always_closed{});
|
|
}
|
|
else {
|
|
return f(detail::nearest_always_open{});
|
|
}
|
|
}
|
|
};
|
|
struct nearest_to_odd_static_boundary : base {
|
|
using rounding_mode_policy = nearest_to_odd_static_boundary;
|
|
template <class Float, class Func>
|
|
static auto delegate(ieee754_bits<Float> br, Func&& f) noexcept {
|
|
if (br.u % 2 == 0) {
|
|
return f(detail::nearest_always_open{});
|
|
}
|
|
else {
|
|
return f(detail::nearest_always_closed{});
|
|
}
|
|
}
|
|
};
|
|
struct nearest_toward_plus_infinity_static_boundary : base {
|
|
using rounding_mode_policy = nearest_toward_plus_infinity_static_boundary;
|
|
template <class Float, class Func>
|
|
static auto delegate(ieee754_bits<Float> br, Func&& f) noexcept {
|
|
if (br.is_negative()) {
|
|
return f(nearest_toward_zero{});
|
|
}
|
|
else {
|
|
return f(nearest_away_from_zero{});
|
|
}
|
|
}
|
|
};
|
|
struct nearest_toward_minus_infinity_static_boundary : base {
|
|
using rounding_mode_policy = nearest_toward_minus_infinity_static_boundary;
|
|
template <class Float, class Func>
|
|
static auto delegate(ieee754_bits<Float> br, Func&& f) noexcept {
|
|
if (br.is_negative()) {
|
|
return f(nearest_away_from_zero{});
|
|
}
|
|
else {
|
|
return f(nearest_toward_zero{});
|
|
}
|
|
}
|
|
};
|
|
|
|
namespace detail {
|
|
struct left_closed_directed {
|
|
static constexpr auto tag = tag_t::left_closed_directed;
|
|
|
|
template <class Float>
|
|
static constexpr interval_type::left_closed_right_open
|
|
interval_type_normal(ieee754_bits<Float>) noexcept
|
|
{
|
|
return{};
|
|
}
|
|
};
|
|
struct right_closed_directed {
|
|
static constexpr auto tag = tag_t::right_closed_directed;
|
|
|
|
template <class Float>
|
|
static constexpr interval_type::right_closed_left_open
|
|
interval_type_normal(ieee754_bits<Float>) noexcept
|
|
{
|
|
return{};
|
|
}
|
|
};
|
|
}
|
|
|
|
struct toward_plus_infinity : base {
|
|
using rounding_mode_policy = toward_plus_infinity;
|
|
template <class Float, class Func>
|
|
static auto delegate(ieee754_bits<Float> br, Func&& f) noexcept {
|
|
if (br.is_negative()) {
|
|
return f(detail::left_closed_directed{});
|
|
}
|
|
else {
|
|
return f(detail::right_closed_directed{});
|
|
}
|
|
}
|
|
};
|
|
struct toward_minus_infinity : base {
|
|
using rounding_mode_policy = toward_minus_infinity;
|
|
template <class Float, class Func>
|
|
static auto delegate(ieee754_bits<Float> br, Func&& f) noexcept {
|
|
if (br.is_negative()) {
|
|
return f(detail::right_closed_directed{});
|
|
}
|
|
else {
|
|
return f(detail::left_closed_directed{});
|
|
}
|
|
}
|
|
};
|
|
struct toward_zero : base {
|
|
using rounding_mode_policy = toward_zero;
|
|
template <class Float, class Func>
|
|
static auto delegate(ieee754_bits<Float>, Func&& f) noexcept {
|
|
return f(detail::left_closed_directed{});
|
|
}
|
|
};
|
|
struct away_from_zero : base {
|
|
using rounding_mode_policy = away_from_zero;
|
|
template <class Float, class Func>
|
|
static auto delegate(ieee754_bits<Float>, Func&& f) noexcept {
|
|
return f(detail::right_closed_directed{});
|
|
}
|
|
};
|
|
}
|
|
|
|
// Correct rounding policy
|
|
namespace correct_rounding {
|
|
struct base {};
|
|
|
|
enum class tag_t {
|
|
do_not_care,
|
|
to_even,
|
|
to_odd,
|
|
away_from_zero,
|
|
toward_zero
|
|
};
|
|
|
|
struct do_not_care : base {
|
|
using correct_rounding_policy = do_not_care;
|
|
static constexpr auto tag = tag_t::do_not_care;
|
|
|
|
template <class Fp>
|
|
static constexpr void break_rounding_tie(Fp&) noexcept {}
|
|
};
|
|
|
|
struct to_even : base {
|
|
using correct_rounding_policy = to_even;
|
|
static constexpr auto tag = tag_t::to_even;
|
|
|
|
template <class Fp>
|
|
static constexpr void break_rounding_tie(Fp& fp) noexcept
|
|
{
|
|
fp.significand = fp.significand % 2 == 0 ?
|
|
fp.significand : fp.significand - 1;
|
|
}
|
|
};
|
|
|
|
struct to_odd : base {
|
|
using correct_rounding_policy = to_odd;
|
|
static constexpr auto tag = tag_t::to_odd;
|
|
|
|
template <class Fp>
|
|
static constexpr void break_rounding_tie(Fp& fp) noexcept
|
|
{
|
|
fp.significand = fp.significand % 2 != 0 ?
|
|
fp.significand : fp.significand - 1;
|
|
}
|
|
};
|
|
|
|
struct away_from_zero : base {
|
|
using correct_rounding_policy = away_from_zero;
|
|
static constexpr auto tag = tag_t::away_from_zero;
|
|
|
|
template <class Fp>
|
|
static constexpr void break_rounding_tie(Fp& /*fp*/) noexcept {}
|
|
};
|
|
|
|
struct toward_zero : base {
|
|
using correct_rounding_policy = toward_zero;
|
|
static constexpr auto tag = tag_t::toward_zero;
|
|
|
|
template <class Fp>
|
|
static constexpr void break_rounding_tie(Fp& fp) noexcept
|
|
{
|
|
--fp.significand;
|
|
}
|
|
};
|
|
}
|
|
|
|
namespace cache {
|
|
struct base {};
|
|
|
|
struct normal : base {
|
|
using cache_policy = normal;
|
|
template <ieee754_format format>
|
|
static constexpr typename cache_holder<format>::cache_entry_type get_cache(int k) noexcept {
|
|
assert(k >= cache_holder<format>::min_k && k <= cache_holder<format>::max_k);
|
|
return cache_holder<format>::cache[std::size_t(k - cache_holder<format>::min_k)];
|
|
}
|
|
};
|
|
|
|
struct compressed : base {
|
|
using cache_policy = compressed;
|
|
template <ieee754_format format>
|
|
static constexpr typename cache_holder<format>::cache_entry_type get_cache(int k) noexcept {
|
|
assert(k >= cache_holder<format>::min_k && k <= cache_holder<format>::max_k);
|
|
|
|
if constexpr (format == ieee754_format::binary64)
|
|
{
|
|
// Compute base index
|
|
auto cache_index = (k - cache_holder<format>::min_k) /
|
|
compressed_cache_detail::compression_ratio;
|
|
auto kb = cache_index * compressed_cache_detail::compression_ratio
|
|
+ cache_holder<format>::min_k;
|
|
auto offset = k - kb;
|
|
|
|
// Get base cache
|
|
auto base_cache = compressed_cache_detail::cache.table[cache_index];
|
|
|
|
if (offset == 0) {
|
|
return base_cache;
|
|
}
|
|
else {
|
|
// Compute the required amount of bit-shift
|
|
auto alpha = log::floor_log2_pow10(kb + offset) - log::floor_log2_pow10(kb) - offset;
|
|
assert(alpha > 0 && alpha < 64);
|
|
|
|
// Try to recover the real cache
|
|
auto pow5 = compressed_cache_detail::pow5.table[offset];
|
|
auto recovered_cache = wuint::umul128(base_cache.high(), pow5);
|
|
auto middle_low = wuint::umul128(base_cache.low() - (kb < 0 ? 1 : 0), pow5);
|
|
|
|
recovered_cache += middle_low.high();
|
|
|
|
auto high_to_middle = recovered_cache.high() << (64 - alpha);
|
|
auto middle_to_low = recovered_cache.low() << (64 - alpha);
|
|
|
|
recovered_cache = wuint::uint128{
|
|
(recovered_cache.low() >> alpha) | high_to_middle,
|
|
((middle_low.low() >> alpha) | middle_to_low)
|
|
};
|
|
|
|
if (kb < 0) {
|
|
recovered_cache += 1;
|
|
}
|
|
|
|
// Get error
|
|
auto error_idx = (k - cache_holder<format>::min_k) / 16;
|
|
auto error = (compressed_cache_detail::errors[error_idx] >>
|
|
((k - cache_holder<format>::min_k) % 16) * 2) & 0x3;
|
|
|
|
// Add the error back
|
|
assert(recovered_cache.low() + error >= recovered_cache.low());
|
|
recovered_cache = {
|
|
recovered_cache.high(),
|
|
recovered_cache.low() + error
|
|
};
|
|
|
|
return recovered_cache;
|
|
}
|
|
}
|
|
else
|
|
{
|
|
return cache_holder<format>::cache[std::size_t(k - cache_holder<format>::min_k)];
|
|
}
|
|
}
|
|
};
|
|
}
|
|
|
|
namespace input_validation {
|
|
struct base {};
|
|
|
|
struct assert_finite : base {
|
|
using input_validation_policy = assert_finite;
|
|
template <class Float>
|
|
static void validate_input([[maybe_unused]] ieee754_bits<Float> br) noexcept
|
|
{
|
|
assert(br.is_finite());
|
|
}
|
|
};
|
|
|
|
struct do_nothing : base {
|
|
using input_validation_policy = do_nothing;
|
|
template <class Float>
|
|
static void validate_input(ieee754_bits<Float>) noexcept {}
|
|
};
|
|
}
|
|
}
|
|
}
|
|
|
|
namespace policy {
|
|
namespace sign {
|
|
inline constexpr auto ignore = detail::policy_impl::sign::ignore{};
|
|
inline constexpr auto return_sign = detail::policy_impl::sign::return_sign{};
|
|
}
|
|
|
|
namespace trailing_zero {
|
|
inline constexpr auto ignore = detail::policy_impl::trailing_zero::ignore{};
|
|
inline constexpr auto remove = detail::policy_impl::trailing_zero::remove{};
|
|
inline constexpr auto report = detail::policy_impl::trailing_zero::report{};
|
|
}
|
|
|
|
namespace rounding_mode {
|
|
inline constexpr auto nearest_to_even =
|
|
detail::policy_impl::rounding_mode::nearest_to_even{};
|
|
inline constexpr auto nearest_to_odd =
|
|
detail::policy_impl::rounding_mode::nearest_to_odd{};
|
|
inline constexpr auto nearest_toward_plus_infinity =
|
|
detail::policy_impl::rounding_mode::nearest_toward_plus_infinity{};
|
|
inline constexpr auto nearest_toward_minus_infinity =
|
|
detail::policy_impl::rounding_mode::nearest_toward_minus_infinity{};
|
|
inline constexpr auto nearest_toward_zero =
|
|
detail::policy_impl::rounding_mode::nearest_toward_zero{};
|
|
inline constexpr auto nearest_away_from_zero =
|
|
detail::policy_impl::rounding_mode::nearest_away_from_zero{};
|
|
|
|
inline constexpr auto nearest_to_even_static_boundary =
|
|
detail::policy_impl::rounding_mode::nearest_to_even_static_boundary{};
|
|
inline constexpr auto nearest_to_odd_static_boundary =
|
|
detail::policy_impl::rounding_mode::nearest_to_odd_static_boundary{};
|
|
inline constexpr auto nearest_toward_plus_infinity_static_boundary =
|
|
detail::policy_impl::rounding_mode::nearest_toward_plus_infinity_static_boundary{};
|
|
inline constexpr auto nearest_toward_minus_infinity_static_boundary =
|
|
detail::policy_impl::rounding_mode::nearest_toward_minus_infinity_static_boundary{};
|
|
|
|
inline constexpr auto toward_plus_infinity =
|
|
detail::policy_impl::rounding_mode::toward_plus_infinity{};
|
|
inline constexpr auto toward_minus_infinity =
|
|
detail::policy_impl::rounding_mode::toward_minus_infinity{};
|
|
inline constexpr auto toward_zero =
|
|
detail::policy_impl::rounding_mode::toward_zero{};
|
|
inline constexpr auto away_from_zero =
|
|
detail::policy_impl::rounding_mode::away_from_zero{};
|
|
}
|
|
|
|
namespace correct_rounding {
|
|
inline constexpr auto do_not_care = detail::policy_impl::correct_rounding::do_not_care{};
|
|
inline constexpr auto to_even = detail::policy_impl::correct_rounding::to_even{};
|
|
inline constexpr auto to_odd = detail::policy_impl::correct_rounding::to_odd{};
|
|
inline constexpr auto away_from_zero = detail::policy_impl::correct_rounding::away_from_zero{};
|
|
inline constexpr auto toward_zero = detail::policy_impl::correct_rounding::toward_zero{};
|
|
}
|
|
|
|
namespace cache {
|
|
inline constexpr auto normal = detail::policy_impl::cache::normal{};
|
|
inline constexpr auto compressed = detail::policy_impl::cache::compressed{};
|
|
}
|
|
|
|
namespace input_validation {
|
|
inline constexpr auto assert_finite = detail::policy_impl::input_validation::assert_finite{};
|
|
inline constexpr auto do_nothing = detail::policy_impl::input_validation::do_nothing{};
|
|
}
|
|
}
|
|
|
|
namespace detail {
|
|
////////////////////////////////////////////////////////////////////////////////////////
|
|
// The main algorithm
|
|
////////////////////////////////////////////////////////////////////////////////////////
|
|
|
|
// Get sign/decimal significand/decimal exponent from
|
|
// the bit representation of a floating-point number
|
|
template <class Float>
|
|
struct impl : private ieee754_traits<Float>,
|
|
private ieee754_format_info<ieee754_traits<Float>::format>
|
|
{
|
|
using carrier_uint = typename ieee754_traits<Float>::carrier_uint;
|
|
|
|
using ieee754_traits<Float>::format;
|
|
using ieee754_traits<Float>::carrier_bits;
|
|
using ieee754_format_info<format>::significand_bits;
|
|
using ieee754_format_info<format>::min_exponent;
|
|
using ieee754_format_info<format>::max_exponent;
|
|
using ieee754_format_info<format>::exponent_bias;
|
|
using ieee754_format_info<format>::decimal_digits;
|
|
|
|
static constexpr int kappa = format == ieee754_format::binary32 ? 1 : 2;
|
|
static_assert(kappa >= 1);
|
|
static_assert(carrier_bits >= significand_bits + 2 + log::floor_log2_pow10(kappa + 1));
|
|
|
|
static constexpr int min_k = [] {
|
|
constexpr auto a = -log::floor_log10_pow2_minus_log10_4_over_3(
|
|
int(max_exponent - significand_bits));
|
|
constexpr auto b = -log::floor_log10_pow2(
|
|
int(max_exponent - significand_bits)) + kappa;
|
|
return a < b ? a : b;
|
|
}();
|
|
static_assert(min_k >= cache_holder<format>::min_k);
|
|
|
|
static constexpr int max_k = [] {
|
|
constexpr auto a = -log::floor_log10_pow2_minus_log10_4_over_3(
|
|
int(min_exponent - significand_bits + 1));
|
|
constexpr auto b = -log::floor_log10_pow2(
|
|
int(min_exponent - significand_bits)) + kappa;
|
|
return a > b ? a : b;
|
|
}();
|
|
static_assert(max_k <= cache_holder<format>::max_k);
|
|
|
|
using cache_entry_type =
|
|
typename cache_holder<format>::cache_entry_type;
|
|
static constexpr auto cache_bits =
|
|
cache_holder<format>::cache_bits;
|
|
|
|
static constexpr int max_power_of_factor_of_5 = log::floor_log5_pow2(int(significand_bits + 2));
|
|
static constexpr int divisibility_check_by_5_threshold =
|
|
log::floor_log2_pow10(max_power_of_factor_of_5 + kappa + 1);
|
|
|
|
static constexpr int case_fc_pm_half_lower_threshold = -kappa - log::floor_log5_pow2(kappa);
|
|
static constexpr int case_fc_pm_half_upper_threshold = log::floor_log2_pow10(kappa + 1);
|
|
|
|
static constexpr int case_fc_lower_threshold = -kappa - 1 - log::floor_log5_pow2(kappa + 1);
|
|
static constexpr int case_fc_upper_threshold = log::floor_log2_pow10(kappa + 1);
|
|
|
|
static constexpr int case_shorter_interval_left_endpoint_lower_threshold = 2;
|
|
static constexpr int case_shorter_interval_left_endpoint_upper_threshold = 2 +
|
|
log::floor_log2(compute_power<
|
|
count_factors<5>((carrier_uint(1) << (significand_bits + 2)) - 1) + 1
|
|
>(10) / 3);
|
|
|
|
static constexpr int case_shorter_interval_right_endpoint_lower_threshold = 0;
|
|
static constexpr int case_shorter_interval_right_endpoint_upper_threshold = 2 +
|
|
log::floor_log2(compute_power<
|
|
count_factors<5>((carrier_uint(1) << (significand_bits + 1)) + 1) + 1
|
|
>(10) / 3);
|
|
|
|
static constexpr int shorter_interval_tie_lower_threshold =
|
|
-log::floor_log5_pow2_minus_log5_3(significand_bits + 4) - 2 - significand_bits;
|
|
static constexpr int shorter_interval_tie_upper_threshold =
|
|
-log::floor_log5_pow2(significand_bits + 2) - 2 - significand_bits;
|
|
|
|
//// The main algorithm assumes the input is a normal/subnormal finite number
|
|
|
|
template <class ReturnType, class IntervalTypeProvider, class SignPolicy,
|
|
class TrailingZeroPolicy, class CorrectRoundingPolicy, class CachePolicy>
|
|
JKJ_SAFEBUFFERS static ReturnType compute_nearest(ieee754_bits<Float> const br) noexcept
|
|
{
|
|
//////////////////////////////////////////////////////////////////////
|
|
// Step 1: integer promotion & Schubfach multiplier calculation
|
|
//////////////////////////////////////////////////////////////////////
|
|
|
|
ReturnType ret_value;
|
|
|
|
SignPolicy::handle_sign(br, ret_value);
|
|
|
|
auto significand = br.extract_significand_bits();
|
|
auto exponent = int(br.extract_exponent_bits());
|
|
|
|
// Deal with normal/subnormal dichotomy
|
|
if (exponent != 0) {
|
|
exponent += exponent_bias - significand_bits;
|
|
|
|
// Shorter interval case; proceed like Schubfach
|
|
if (significand == 0) {
|
|
shorter_interval_case<TrailingZeroPolicy, CorrectRoundingPolicy, CachePolicy>(
|
|
ret_value, exponent,
|
|
IntervalTypeProvider::interval_type_shorter(br));
|
|
return ret_value;
|
|
}
|
|
|
|
significand |= (carrier_uint(1) << significand_bits);
|
|
}
|
|
// Subnormal case; interval is always regular
|
|
else {
|
|
exponent = min_exponent - significand_bits;
|
|
}
|
|
|
|
auto const interval_type = IntervalTypeProvider::interval_type_normal(br);
|
|
|
|
// Compute k and beta
|
|
int const minus_k = log::floor_log10_pow2(exponent) - kappa;
|
|
auto const cache = CachePolicy::template get_cache<format>(-minus_k);
|
|
int const beta_minus_1 = exponent + log::floor_log2_pow10(-minus_k);
|
|
|
|
// Compute zi and deltai
|
|
// 10^kappa <= deltai < 10^(kappa + 1)
|
|
auto const deltai = compute_delta(cache, beta_minus_1);
|
|
carrier_uint const two_fc = significand << 1;
|
|
carrier_uint const two_fr = two_fc | 1;
|
|
carrier_uint const zi = compute_mul(two_fr << beta_minus_1, cache);
|
|
|
|
|
|
//////////////////////////////////////////////////////////////////////
|
|
// Step 2: Try larger divisor; remove trailing zeros if necessary
|
|
//////////////////////////////////////////////////////////////////////
|
|
|
|
constexpr auto big_divisor = compute_power<kappa + 1>(std::uint32_t(10));
|
|
constexpr auto small_divisor = compute_power<kappa>(std::uint32_t(10));
|
|
|
|
// Using an upper bound on zi, we might be able to optimize the division
|
|
// better than the compiler; we are computing zi / big_divisor here
|
|
ret_value.significand = div::divide_by_pow10<kappa + 1,
|
|
significand_bits + kappa + 2, kappa + 1>(zi);
|
|
auto r = std::uint32_t(zi - big_divisor * ret_value.significand);
|
|
|
|
if (r > deltai) {
|
|
goto small_divisor_case_label;
|
|
}
|
|
else if (r < deltai) {
|
|
// Exclude the right endpoint if necessary
|
|
if (r == 0 && !interval_type.include_right_endpoint() &&
|
|
is_product_integer<integer_check_case_id::fc_pm_half>(two_fr, exponent, minus_k))
|
|
{
|
|
if constexpr (CorrectRoundingPolicy::tag ==
|
|
policy_impl::correct_rounding::tag_t::do_not_care)
|
|
{
|
|
ret_value.significand *= 10;
|
|
ret_value.exponent = minus_k + kappa;
|
|
--ret_value.significand;
|
|
return ret_value;
|
|
}
|
|
else {
|
|
--ret_value.significand;
|
|
r = big_divisor;
|
|
goto small_divisor_case_label;
|
|
}
|
|
}
|
|
}
|
|
else {
|
|
// r == deltai; compare fractional parts
|
|
// Check conditions in the order different from the paper
|
|
// to take advantage of short-circuiting
|
|
auto const two_fl = two_fc - 1;
|
|
if ((!interval_type.include_left_endpoint() ||
|
|
!is_product_integer<integer_check_case_id::fc_pm_half>(
|
|
two_fl, exponent, minus_k)) &&
|
|
!compute_mul_parity(two_fl, cache, beta_minus_1))
|
|
{
|
|
goto small_divisor_case_label;
|
|
}
|
|
}
|
|
ret_value.exponent = minus_k + kappa + 1;
|
|
|
|
// We may need to remove trailing zeros
|
|
TrailingZeroPolicy::on_trailing_zeros(ret_value);
|
|
return ret_value;
|
|
|
|
|
|
//////////////////////////////////////////////////////////////////////
|
|
// Step 3: Find the significand with the smaller divisor
|
|
//////////////////////////////////////////////////////////////////////
|
|
|
|
small_divisor_case_label:
|
|
TrailingZeroPolicy::no_trailing_zeros(ret_value);
|
|
ret_value.significand *= 10;
|
|
ret_value.exponent = minus_k + kappa;
|
|
|
|
constexpr auto mask = (std::uint32_t(1) << kappa) - 1;
|
|
|
|
if constexpr (CorrectRoundingPolicy::tag ==
|
|
policy_impl::correct_rounding::tag_t::do_not_care)
|
|
{
|
|
// Normally, we want to compute
|
|
// ret_value.significand += r / small_divisor
|
|
// and return, but we need to take care of the case that the resulting
|
|
// value is exactly the right endpoint, while that is not included in the interval
|
|
if (!interval_type.include_right_endpoint()) {
|
|
// Is r divisible by 2^kappa?
|
|
if ((r & mask) == 0) {
|
|
r >>= kappa;
|
|
|
|
// Is r divisible by 5^kappa?
|
|
if (div::check_divisibility_and_divide_by_pow5<kappa>(r) &&
|
|
is_product_integer<integer_check_case_id::fc_pm_half>(two_fr, exponent, minus_k))
|
|
{
|
|
// This should be in the interval
|
|
ret_value.significand += r - 1;
|
|
}
|
|
else {
|
|
ret_value.significand += r;
|
|
}
|
|
}
|
|
else {
|
|
ret_value.significand += div::small_division_by_pow10<kappa>(r);
|
|
}
|
|
}
|
|
else {
|
|
ret_value.significand += div::small_division_by_pow10<kappa>(r);
|
|
}
|
|
}
|
|
else
|
|
{
|
|
auto dist = r - (deltai / 2) + (small_divisor / 2);
|
|
|
|
// Is dist divisible by 2^kappa?
|
|
if ((dist & mask) == 0) {
|
|
bool const approx_y_parity = ((dist ^ (small_divisor / 2)) & 1) != 0;
|
|
dist >>= kappa;
|
|
|
|
// Is dist divisible by 5^kappa?
|
|
if (div::check_divisibility_and_divide_by_pow5<kappa>(dist)) {
|
|
ret_value.significand += dist;
|
|
|
|
// Check z^(f) >= epsilon^(f)
|
|
// We have either yi == zi - epsiloni or yi == (zi - epsiloni) - 1,
|
|
// where yi == zi - epsiloni if and only if z^(f) >= epsilon^(f)
|
|
// Since there are only 2 possibilities, we only need to care about the parity
|
|
// Also, zi and r should have the same parity since the divisor
|
|
// is an even number
|
|
if (compute_mul_parity(two_fc, cache, beta_minus_1) != approx_y_parity) {
|
|
--ret_value.significand;
|
|
}
|
|
else {
|
|
// If z^(f) >= epsilon^(f), we might have a tie
|
|
// when z^(f) == epsilon^(f), or equivalently, when y is an integer
|
|
// For tie-to-up case, we can just choose the upper one
|
|
if constexpr (CorrectRoundingPolicy::tag !=
|
|
policy_impl::correct_rounding::tag_t::away_from_zero)
|
|
{
|
|
if (is_product_integer<integer_check_case_id::fc>(
|
|
two_fc, exponent, minus_k))
|
|
{
|
|
CorrectRoundingPolicy::break_rounding_tie(ret_value);
|
|
}
|
|
}
|
|
}
|
|
}
|
|
// Is dist not divisible by 5^kappa?
|
|
else {
|
|
ret_value.significand += dist;
|
|
}
|
|
}
|
|
// Is dist not divisible by 2^kappa?
|
|
else {
|
|
// Since we know dist is small, we might be able to optimize the division
|
|
// better than the compiler; we are computing dist / small_divisor here
|
|
ret_value.significand += div::small_division_by_pow10<kappa>(dist);
|
|
}
|
|
}
|
|
return ret_value;
|
|
}
|
|
|
|
template <class TrailingZeroPolicy, class CorrectRoundingPolicy,
|
|
class CachePolicy, class ReturnType, class IntervalType>
|
|
JKJ_FORCEINLINE JKJ_SAFEBUFFERS static void shorter_interval_case(
|
|
ReturnType& ret_value, int const exponent, IntervalType const interval_type) noexcept
|
|
{
|
|
// Compute k and beta
|
|
int const minus_k = log::floor_log10_pow2_minus_log10_4_over_3(exponent);
|
|
int const beta_minus_1 = exponent + log::floor_log2_pow10(-minus_k);
|
|
|
|
// Compute xi and zi
|
|
auto const cache = CachePolicy::template get_cache<format>(-minus_k);
|
|
|
|
auto xi = compute_left_endpoint_for_shorter_interval_case(cache, beta_minus_1);
|
|
auto zi = compute_right_endpoint_for_shorter_interval_case(cache, beta_minus_1);
|
|
|
|
// If we don't accept the right endpoint and
|
|
// if the right endpoint is an integer, decrease it
|
|
if (!interval_type.include_right_endpoint() &&
|
|
is_right_endpoint_integer_shorter_interval(exponent))
|
|
{
|
|
--zi;
|
|
}
|
|
// If we don't accept the left endpoint or
|
|
// if the left endpoint is not an integer, increase it
|
|
if (!interval_type.include_left_endpoint() ||
|
|
!is_left_endpoint_integer_shorter_interval(exponent))
|
|
{
|
|
++xi;
|
|
}
|
|
|
|
// Try bigger divisor
|
|
ret_value.significand = zi / 10;
|
|
|
|
// If succeed, remove trailing zeros if necessary and return
|
|
if (ret_value.significand * 10 >= xi) {
|
|
ret_value.exponent = minus_k + 1;
|
|
TrailingZeroPolicy::on_trailing_zeros(ret_value);
|
|
return;
|
|
}
|
|
|
|
// Otherwise, compute the round-up of y
|
|
TrailingZeroPolicy::no_trailing_zeros(ret_value);
|
|
ret_value.significand = compute_round_up_for_shorter_interval_case(cache, beta_minus_1);
|
|
ret_value.exponent = minus_k;
|
|
|
|
// When tie occurs, choose one of them according to the rule
|
|
if constexpr (CorrectRoundingPolicy::tag !=
|
|
policy_impl::correct_rounding::tag_t::do_not_care &&
|
|
CorrectRoundingPolicy::tag !=
|
|
policy_impl::correct_rounding::tag_t::away_from_zero)
|
|
{
|
|
if (exponent >= shorter_interval_tie_lower_threshold &&
|
|
exponent <= shorter_interval_tie_upper_threshold)
|
|
{
|
|
CorrectRoundingPolicy::break_rounding_tie(ret_value);
|
|
}
|
|
else if (ret_value.significand < xi) {
|
|
++ret_value.significand;
|
|
}
|
|
}
|
|
else
|
|
{
|
|
if (ret_value.significand < xi) {
|
|
++ret_value.significand;
|
|
}
|
|
}
|
|
}
|
|
|
|
template <class ReturnType, class SignPolicy, class TrailingZeroPolicy, class CachePolicy>
|
|
JKJ_SAFEBUFFERS static ReturnType
|
|
compute_left_closed_directed(ieee754_bits<Float> const br) noexcept
|
|
{
|
|
//////////////////////////////////////////////////////////////////////
|
|
// Step 1: integer promotion & Schubfach multiplier calculation
|
|
//////////////////////////////////////////////////////////////////////
|
|
|
|
ReturnType ret_value;
|
|
|
|
SignPolicy::handle_sign(br, ret_value);
|
|
|
|
auto significand = br.extract_significand_bits();
|
|
auto exponent = int(br.extract_exponent_bits());
|
|
|
|
// Deal with normal/subnormal dichotomy
|
|
if (exponent != 0) {
|
|
exponent += exponent_bias - significand_bits;
|
|
significand |= (carrier_uint(1) << significand_bits);
|
|
}
|
|
// Subnormal case; interval is always regular
|
|
else {
|
|
exponent = min_exponent - significand_bits;
|
|
}
|
|
|
|
// Compute k and beta
|
|
int const minus_k = log::floor_log10_pow2(exponent) - kappa;
|
|
auto const cache = CachePolicy::template get_cache<format>(-minus_k);
|
|
int const beta = exponent + log::floor_log2_pow10(-minus_k) + 1;
|
|
|
|
// Compute xi and deltai
|
|
// 10^kappa <= deltai < 10^(kappa + 1)
|
|
auto const deltai = compute_delta(cache, beta - 1);
|
|
carrier_uint xi = compute_mul(significand << beta, cache);
|
|
|
|
if (!is_product_integer<integer_check_case_id::fc>(significand, exponent + 1, minus_k)) {
|
|
++xi;
|
|
}
|
|
|
|
//////////////////////////////////////////////////////////////////////
|
|
// Step 2: Try larger divisor; remove trailing zeros if necessary
|
|
//////////////////////////////////////////////////////////////////////
|
|
|
|
constexpr auto big_divisor = compute_power<kappa + 1>(std::uint32_t(10));
|
|
constexpr auto small_divisor = compute_power<kappa>(std::uint32_t(10));
|
|
|
|
// Using an upper bound on xi, we might be able to optimize the division
|
|
// better than the compiler; we are computing xi / big_divisor here
|
|
ret_value.significand = div::divide_by_pow10<kappa + 1,
|
|
significand_bits + kappa + 2, kappa + 1>(xi);
|
|
auto r = std::uint32_t(xi - big_divisor * ret_value.significand);
|
|
|
|
if (r != 0) {
|
|
++ret_value.significand;
|
|
r = big_divisor - r;
|
|
}
|
|
|
|
if (r > deltai) {
|
|
goto small_divisor_case_label;
|
|
}
|
|
else if (r == deltai) {
|
|
// Compare the fractional parts
|
|
if (compute_mul_parity(significand + 1, cache, beta) ||
|
|
is_product_integer<integer_check_case_id::fc>(significand + 1, exponent + 1, minus_k))
|
|
{
|
|
goto small_divisor_case_label;
|
|
}
|
|
|
|
}
|
|
|
|
// The ceiling is inside, so we are done
|
|
ret_value.exponent = minus_k + kappa + 1;
|
|
TrailingZeroPolicy::on_trailing_zeros(ret_value);
|
|
return ret_value;
|
|
|
|
|
|
//////////////////////////////////////////////////////////////////////
|
|
// Step 3: Find the significand with the smaller divisor
|
|
//////////////////////////////////////////////////////////////////////
|
|
|
|
small_divisor_case_label:
|
|
ret_value.significand *= 10;
|
|
ret_value.significand -= div::small_division_by_pow10<kappa>(r);
|
|
ret_value.exponent = minus_k + kappa;
|
|
TrailingZeroPolicy::no_trailing_zeros(ret_value);
|
|
return ret_value;
|
|
}
|
|
|
|
template <class ReturnType, class SignPolicy, class TrailingZeroPolicy, class CachePolicy>
|
|
JKJ_SAFEBUFFERS static ReturnType
|
|
compute_right_closed_directed(ieee754_bits<Float> const br) noexcept
|
|
{
|
|
//////////////////////////////////////////////////////////////////////
|
|
// Step 1: integer promotion & Schubfach multiplier calculation
|
|
//////////////////////////////////////////////////////////////////////
|
|
|
|
ReturnType ret_value;
|
|
|
|
SignPolicy::handle_sign(br, ret_value);
|
|
|
|
auto significand = br.extract_significand_bits();
|
|
auto exponent = int(br.extract_exponent_bits());
|
|
|
|
// Deal with normal/subnormal dichotomy
|
|
bool closer_boundary = false;
|
|
if (exponent != 0) {
|
|
exponent += exponent_bias - significand_bits;
|
|
if (significand == 0) {
|
|
closer_boundary = true;
|
|
}
|
|
significand |= (carrier_uint(1) << significand_bits);
|
|
}
|
|
// Subnormal case; interval is always regular
|
|
else {
|
|
exponent = min_exponent - significand_bits;
|
|
}
|
|
|
|
// Compute k and beta
|
|
int const minus_k = log::floor_log10_pow2(exponent - (closer_boundary ? 1 : 0)) - kappa;
|
|
auto const cache = CachePolicy::template get_cache<format>(-minus_k);
|
|
int const beta = exponent + log::floor_log2_pow10(-minus_k) + 1;
|
|
|
|
// Compute zi and deltai
|
|
// 10^kappa <= deltai < 10^(kappa + 1)
|
|
auto const deltai = closer_boundary ?
|
|
compute_delta(cache, beta - 2) :
|
|
compute_delta(cache, beta - 1);
|
|
carrier_uint const zi = compute_mul(significand << beta, cache);
|
|
|
|
|
|
//////////////////////////////////////////////////////////////////////
|
|
// Step 2: Try larger divisor; remove trailing zeros if necessary
|
|
//////////////////////////////////////////////////////////////////////
|
|
|
|
constexpr auto big_divisor = compute_power<kappa + 1>(std::uint32_t(10));
|
|
constexpr auto small_divisor = compute_power<kappa>(std::uint32_t(10));
|
|
|
|
// Using an upper bound on zi, we might be able to optimize the division
|
|
// better than the compiler; we are computing zi / big_divisor here
|
|
ret_value.significand = div::divide_by_pow10<kappa + 1,
|
|
significand_bits + kappa + 2, kappa + 1>(zi);
|
|
auto const r = std::uint32_t(zi - big_divisor * ret_value.significand);
|
|
|
|
if (r > deltai) {
|
|
goto small_divisor_case_label;
|
|
}
|
|
else if (r == deltai) {
|
|
// Compare the fractional parts
|
|
if (closer_boundary) {
|
|
if (!compute_mul_parity((significand * 2) - 1, cache, beta - 1))
|
|
{
|
|
goto small_divisor_case_label;
|
|
}
|
|
}
|
|
else {
|
|
if (!compute_mul_parity(significand - 1, cache, beta))
|
|
{
|
|
goto small_divisor_case_label;
|
|
}
|
|
}
|
|
}
|
|
|
|
// The floor is inside, so we are done
|
|
ret_value.exponent = minus_k + kappa + 1;
|
|
TrailingZeroPolicy::on_trailing_zeros(ret_value);
|
|
return ret_value;
|
|
|
|
|
|
//////////////////////////////////////////////////////////////////////
|
|
// Step 3: Find the significand with the small divisor
|
|
//////////////////////////////////////////////////////////////////////
|
|
|
|
small_divisor_case_label:
|
|
ret_value.significand *= 10;
|
|
ret_value.significand += div::small_division_by_pow10<kappa>(r);
|
|
ret_value.exponent = minus_k + kappa;
|
|
TrailingZeroPolicy::no_trailing_zeros(ret_value);
|
|
return ret_value;
|
|
}
|
|
|
|
// Remove trailing zeros from n and return the number of zeros removed
|
|
JKJ_FORCEINLINE static int remove_trailing_zeros(carrier_uint& n) noexcept {
|
|
constexpr auto max_power = [] {
|
|
auto max_possible_significand =
|
|
std::numeric_limits<carrier_uint>::max() /
|
|
compute_power<kappa + 1>(std::uint32_t(10));
|
|
|
|
int k = 0;
|
|
carrier_uint p = 1;
|
|
while (p < max_possible_significand / 10) {
|
|
p *= 10;
|
|
++k;
|
|
}
|
|
return k;
|
|
}();
|
|
|
|
auto t = bits::countr_zero(n);
|
|
if (t > max_power) {
|
|
t = max_power;
|
|
}
|
|
|
|
if constexpr (format == ieee754_format::binary32) {
|
|
constexpr auto const& divtable =
|
|
div::table_holder<carrier_uint, 5, decimal_digits>::table;
|
|
|
|
int s = 0;
|
|
for (; s < t - 1; s += 2) {
|
|
if (n * divtable.mod_inv[2] > divtable.max_quotients[2]) {
|
|
break;
|
|
}
|
|
n *= divtable.mod_inv[2];
|
|
}
|
|
if (s < t && n * divtable.mod_inv[1] <= divtable.max_quotients[1])
|
|
{
|
|
n *= divtable.mod_inv[1];
|
|
++s;
|
|
}
|
|
n >>= s;
|
|
return s;
|
|
}
|
|
else {
|
|
static_assert(format == ieee754_format::binary64);
|
|
static_assert(kappa >= 2);
|
|
|
|
// Divide by 10^8 and reduce to 32-bits
|
|
// Since ret_value.significand <= (2^64 - 1) / 1000 < 10^17,
|
|
// both of the quotient and the r should fit in 32-bits
|
|
|
|
constexpr auto const& divtable =
|
|
div::table_holder<carrier_uint, 5, decimal_digits>::table;
|
|
|
|
// If the number is divisible by 1'0000'0000, work with the quotient
|
|
if (t >= 8) {
|
|
auto quotient_candidate = n * divtable.mod_inv[8];
|
|
|
|
if (quotient_candidate <= divtable.max_quotients[8]) {
|
|
auto quotient = std::uint32_t(quotient_candidate >> 8);
|
|
|
|
constexpr auto mod_inverse = std::uint32_t(divtable.mod_inv[1]);
|
|
constexpr auto max_quotient =
|
|
std::numeric_limits<std::uint32_t>::max() / 5;
|
|
|
|
int s = 8;
|
|
for (; s < t; ++s) {
|
|
if (quotient * mod_inverse > max_quotient) {
|
|
break;
|
|
}
|
|
quotient *= mod_inverse;
|
|
}
|
|
quotient >>= (s - 8);
|
|
n = quotient;
|
|
return s;
|
|
}
|
|
}
|
|
|
|
// Otherwise, work with the remainder
|
|
auto quotient = std::uint32_t(div::divide_by_pow10<8, 54, 0>(n));
|
|
auto remainder = std::uint32_t(n - 1'0000'0000 * quotient);
|
|
|
|
constexpr auto mod_inverse = std::uint32_t(divtable.mod_inv[1]);
|
|
constexpr auto max_quotient =
|
|
std::numeric_limits<std::uint32_t>::max() / 5;
|
|
|
|
if (t == 0 || remainder * mod_inverse > max_quotient) {
|
|
return 0;
|
|
}
|
|
remainder *= mod_inverse;
|
|
|
|
if (t == 1 || remainder * mod_inverse > max_quotient) {
|
|
n = (remainder >> 1)
|
|
+ quotient * carrier_uint(1000'0000);
|
|
return 1;
|
|
}
|
|
remainder *= mod_inverse;
|
|
|
|
if (t == 2 || remainder * mod_inverse > max_quotient) {
|
|
n = (remainder >> 2)
|
|
+ quotient * carrier_uint(100'0000);
|
|
return 2;
|
|
}
|
|
remainder *= mod_inverse;
|
|
|
|
if (t == 3 || remainder * mod_inverse > max_quotient) {
|
|
n = (remainder >> 3)
|
|
+ quotient * carrier_uint(10'0000);
|
|
return 3;
|
|
}
|
|
remainder *= mod_inverse;
|
|
|
|
if (t == 4 || remainder * mod_inverse > max_quotient) {
|
|
n = (remainder >> 4)
|
|
+ quotient * carrier_uint(1'0000);
|
|
return 4;
|
|
}
|
|
remainder *= mod_inverse;
|
|
|
|
if (t == 5 || remainder * mod_inverse > max_quotient) {
|
|
n = (remainder >> 5)
|
|
+ quotient * carrier_uint(1000);
|
|
return 5;
|
|
}
|
|
remainder *= mod_inverse;
|
|
|
|
if (t == 6 || remainder * mod_inverse > max_quotient) {
|
|
n = (remainder >> 6)
|
|
+ quotient * carrier_uint(100);
|
|
return 6;
|
|
}
|
|
remainder *= mod_inverse;
|
|
|
|
n = (remainder >> 7)
|
|
+ quotient * carrier_uint(10);
|
|
return 7;
|
|
}
|
|
}
|
|
|
|
static carrier_uint compute_mul(carrier_uint u, cache_entry_type const& cache) noexcept
|
|
{
|
|
if constexpr (format == ieee754_format::binary32) {
|
|
return wuint::umul96_upper32(u, cache);
|
|
}
|
|
else {
|
|
return wuint::umul192_upper64(u, cache);
|
|
}
|
|
}
|
|
|
|
static std::uint32_t compute_delta(cache_entry_type const& cache, int beta_minus_1) noexcept
|
|
{
|
|
if constexpr (format == ieee754_format::binary32) {
|
|
return std::uint32_t(cache >> (cache_bits - 1 - beta_minus_1));
|
|
}
|
|
else {
|
|
return std::uint32_t(cache.high() >> (carrier_bits - 1 - beta_minus_1));
|
|
}
|
|
}
|
|
|
|
static bool compute_mul_parity(carrier_uint two_f, cache_entry_type const& cache, int beta_minus_1) noexcept
|
|
{
|
|
assert(beta_minus_1 >= 1);
|
|
assert(beta_minus_1 < 64);
|
|
|
|
if constexpr (format == ieee754_format::binary32) {
|
|
return ((wuint::umul96_lower64(two_f, cache) >>
|
|
(64 - beta_minus_1)) & 1) != 0;
|
|
}
|
|
else {
|
|
return ((wuint::umul192_middle64(two_f, cache) >>
|
|
(64 - beta_minus_1)) & 1) != 0;
|
|
}
|
|
}
|
|
|
|
static carrier_uint compute_left_endpoint_for_shorter_interval_case(
|
|
cache_entry_type const& cache, int beta_minus_1) noexcept
|
|
{
|
|
if constexpr (format == ieee754_format::binary32) {
|
|
return carrier_uint(
|
|
(cache - (cache >> (significand_bits + 2))) >>
|
|
(cache_bits - significand_bits - 1 - beta_minus_1));
|
|
}
|
|
else {
|
|
return (cache.high() - (cache.high() >> (significand_bits + 2))) >>
|
|
(carrier_bits - significand_bits - 1 - beta_minus_1);
|
|
}
|
|
}
|
|
|
|
static carrier_uint compute_right_endpoint_for_shorter_interval_case(
|
|
cache_entry_type const& cache, int beta_minus_1) noexcept
|
|
{
|
|
if constexpr (format == ieee754_format::binary32) {
|
|
return carrier_uint(
|
|
(cache + (cache >> (significand_bits + 1))) >>
|
|
(cache_bits - significand_bits - 1 - beta_minus_1));
|
|
}
|
|
else {
|
|
return (cache.high() + (cache.high() >> (significand_bits + 1))) >>
|
|
(carrier_bits - significand_bits - 1 - beta_minus_1);
|
|
}
|
|
}
|
|
|
|
static carrier_uint compute_round_up_for_shorter_interval_case(
|
|
cache_entry_type const& cache, int beta_minus_1) noexcept
|
|
{
|
|
if constexpr (format == ieee754_format::binary32) {
|
|
return (carrier_uint(cache >> (cache_bits - significand_bits - 2 - beta_minus_1)) + 1) / 2;
|
|
}
|
|
else {
|
|
return ((cache.high() >> (carrier_bits - significand_bits - 2 - beta_minus_1)) + 1) / 2;
|
|
}
|
|
}
|
|
|
|
static bool is_right_endpoint_integer_shorter_interval(int exponent) noexcept {
|
|
return exponent >= case_shorter_interval_right_endpoint_lower_threshold &&
|
|
exponent <= case_shorter_interval_right_endpoint_upper_threshold;
|
|
}
|
|
|
|
static bool is_left_endpoint_integer_shorter_interval(int exponent) noexcept {
|
|
return exponent >= case_shorter_interval_left_endpoint_lower_threshold &&
|
|
exponent <= case_shorter_interval_left_endpoint_upper_threshold;
|
|
}
|
|
|
|
enum class integer_check_case_id {
|
|
fc_pm_half,
|
|
fc
|
|
};
|
|
template <integer_check_case_id case_id>
|
|
static bool is_product_integer(carrier_uint two_f, int exponent, int minus_k) noexcept
|
|
{
|
|
// Case I: f = fc +- 1/2
|
|
if constexpr (case_id == integer_check_case_id::fc_pm_half)
|
|
{
|
|
if (exponent < case_fc_pm_half_lower_threshold) {
|
|
return false;
|
|
}
|
|
// For k >= 0
|
|
else if (exponent <= case_fc_pm_half_upper_threshold) {
|
|
return true;
|
|
}
|
|
// For k < 0
|
|
else if (exponent > divisibility_check_by_5_threshold) {
|
|
return false;
|
|
}
|
|
else {
|
|
return div::divisible_by_power_of_5<max_power_of_factor_of_5 + 1>(two_f, minus_k);
|
|
}
|
|
}
|
|
// Case II: f = fc + 1
|
|
// Case III: f = fc
|
|
else
|
|
{
|
|
// Exponent for 5 is negative
|
|
if (exponent > divisibility_check_by_5_threshold) {
|
|
return false;
|
|
}
|
|
else if (exponent > case_fc_upper_threshold) {
|
|
return div::divisible_by_power_of_5<max_power_of_factor_of_5 + 1>(two_f, minus_k);
|
|
}
|
|
// Both exponents are nonnegative
|
|
else if (exponent >= case_fc_lower_threshold) {
|
|
return true;
|
|
}
|
|
// Exponent for 2 is negative
|
|
else {
|
|
return div::divisible_by_power_of_2(two_f, minus_k - exponent + 1);
|
|
}
|
|
}
|
|
}
|
|
};
|
|
|
|
|
|
////////////////////////////////////////////////////////////////////////////////////////
|
|
// Policy holder
|
|
////////////////////////////////////////////////////////////////////////////////////////
|
|
|
|
namespace policy_impl {
|
|
// The library will specify a list of accepted kinds of policies and their defaults,
|
|
// and the user will pass a list of policies. The aim of helper classes/functions here
|
|
// is to do the following:
|
|
// 1. Check if the policy parameters given by the user are all valid; that means,
|
|
// each of them should be of the kinds specified by the library.
|
|
// If that's not the case, then the compilation fails.
|
|
// 2. Check if multiple policy parameters for the same kind is specified by the user.
|
|
// If that's the case, then the compilation fails.
|
|
// 3. Build a class deriving from all policies the user have given, and also from
|
|
// the default policies if the user did not specify one for some kinds.
|
|
// A policy belongs to a certain kind if it is deriving from a base class.
|
|
|
|
// For a given kind, find a policy belonging to that kind.
|
|
// Check if there are more than one such policies.
|
|
enum class policy_found_info {
|
|
not_found, unique, repeated
|
|
};
|
|
template <class Policy, policy_found_info info>
|
|
struct found_policy_pair {
|
|
using policy = Policy;
|
|
static constexpr auto found_info = info;
|
|
};
|
|
|
|
template <class Base, class DefaultPolicy>
|
|
struct base_default_pair {
|
|
using base = Base;
|
|
|
|
template <class FoundPolicyInfo>
|
|
static constexpr FoundPolicyInfo get_policy_impl(FoundPolicyInfo) {
|
|
return{};
|
|
}
|
|
template <class FoundPolicyInfo, class FirstPolicy, class... RemainingPolicies>
|
|
static constexpr auto get_policy_impl(FoundPolicyInfo, FirstPolicy, RemainingPolicies... remainings) {
|
|
if constexpr (std::is_base_of_v<Base, FirstPolicy>) {
|
|
if constexpr (FoundPolicyInfo::found_info == policy_found_info::not_found) {
|
|
return get_policy_impl(
|
|
found_policy_pair<FirstPolicy, policy_found_info::unique>{},
|
|
remainings...);
|
|
}
|
|
else {
|
|
return get_policy_impl(
|
|
found_policy_pair<FirstPolicy, policy_found_info::repeated>{},
|
|
remainings...);
|
|
}
|
|
}
|
|
else {
|
|
return get_policy_impl(FoundPolicyInfo{},
|
|
remainings...);
|
|
}
|
|
}
|
|
|
|
template <class... Policies>
|
|
static constexpr auto get_policy(Policies... policies) {
|
|
return get_policy_impl(
|
|
found_policy_pair<DefaultPolicy, policy_found_info::not_found>{},
|
|
policies...);
|
|
}
|
|
};
|
|
template <class... BaseDefaultPairs>
|
|
struct base_default_pair_list {};
|
|
|
|
// Check if a given policy belongs to one of the kinds specified by the library
|
|
template <class Policy>
|
|
constexpr bool check_policy_validity(Policy, base_default_pair_list<>)
|
|
{
|
|
return false;
|
|
}
|
|
template <class Policy, class FirstBaseDefaultPair, class... RemainingBaseDefaultPairs>
|
|
constexpr bool check_policy_validity(Policy,
|
|
base_default_pair_list<FirstBaseDefaultPair, RemainingBaseDefaultPairs...>)
|
|
{
|
|
return std::is_base_of_v<typename FirstBaseDefaultPair::base, Policy> ||
|
|
check_policy_validity(Policy{}, base_default_pair_list< RemainingBaseDefaultPairs...>{});
|
|
}
|
|
|
|
template <class BaseDefaultPairList>
|
|
constexpr bool check_policy_list_validity(BaseDefaultPairList) {
|
|
return true;
|
|
}
|
|
|
|
template <class BaseDefaultPairList, class FirstPolicy, class... RemainingPolicies>
|
|
constexpr bool check_policy_list_validity(BaseDefaultPairList,
|
|
FirstPolicy, RemainingPolicies... remaining_policies)
|
|
{
|
|
return check_policy_validity(FirstPolicy{}, BaseDefaultPairList{}) &&
|
|
check_policy_list_validity(BaseDefaultPairList{}, remaining_policies...);
|
|
}
|
|
|
|
// Build policy_holder
|
|
template <bool repeated_, class... FoundPolicyPairs>
|
|
struct found_policy_pair_list {
|
|
static constexpr bool repeated = repeated_;
|
|
};
|
|
|
|
template <class... Policies>
|
|
struct policy_holder : Policies... {};
|
|
|
|
template <bool repeated, class... FoundPolicyPairs, class... Policies>
|
|
constexpr auto make_policy_holder_impl(
|
|
base_default_pair_list<>,
|
|
found_policy_pair_list<repeated, FoundPolicyPairs...>,
|
|
Policies...)
|
|
{
|
|
return found_policy_pair_list<repeated, FoundPolicyPairs...>{};
|
|
}
|
|
|
|
template <class FirstBaseDefaultPair, class... RemainingBaseDefaultPairs,
|
|
bool repeated, class... FoundPolicyPairs, class... Policies>
|
|
constexpr auto make_policy_holder_impl(
|
|
base_default_pair_list<FirstBaseDefaultPair, RemainingBaseDefaultPairs...>,
|
|
found_policy_pair_list<repeated, FoundPolicyPairs...>,
|
|
Policies... policies)
|
|
{
|
|
using new_found_policy_pair = decltype(FirstBaseDefaultPair::get_policy(policies...));
|
|
|
|
return make_policy_holder_impl(
|
|
base_default_pair_list<RemainingBaseDefaultPairs...>{},
|
|
found_policy_pair_list<
|
|
repeated || new_found_policy_pair::found_info == policy_found_info::repeated,
|
|
new_found_policy_pair, FoundPolicyPairs...
|
|
>{}, policies...);
|
|
}
|
|
|
|
template <bool repeated, class... RawPolicies>
|
|
constexpr auto convert_to_policy_holder(found_policy_pair_list<repeated>, RawPolicies...) {
|
|
return policy_holder<RawPolicies...>{};
|
|
}
|
|
|
|
template <bool repeated, class FirstFoundPolicyPair, class... RemainingFoundPolicyPairs, class... RawPolicies>
|
|
constexpr auto convert_to_policy_holder(
|
|
found_policy_pair_list<repeated, FirstFoundPolicyPair, RemainingFoundPolicyPairs...>, RawPolicies... policies)
|
|
{
|
|
return convert_to_policy_holder(found_policy_pair_list<repeated, RemainingFoundPolicyPairs...>{},
|
|
typename FirstFoundPolicyPair::policy{}, policies...);
|
|
}
|
|
|
|
template <class BaseDefaultPairList, class... Policies>
|
|
constexpr auto make_policy_holder(BaseDefaultPairList, Policies... policies) {
|
|
static_assert(check_policy_list_validity(BaseDefaultPairList{}, Policies{}...),
|
|
"jkj::dragonbox: an invalid policy is specified");
|
|
|
|
using policy_pair_list = decltype(make_policy_holder_impl(BaseDefaultPairList{},
|
|
found_policy_pair_list<false>{}, policies...));
|
|
|
|
static_assert(!policy_pair_list::repeated,
|
|
"jkj::dragonbox: each policy should be specified at most once");
|
|
|
|
return convert_to_policy_holder(policy_pair_list{});
|
|
}
|
|
}
|
|
}
|
|
|
|
|
|
////////////////////////////////////////////////////////////////////////////////////////
|
|
// The interface function
|
|
////////////////////////////////////////////////////////////////////////////////////////
|
|
|
|
template <class Float, class... Policies>
|
|
JKJ_SAFEBUFFERS JKJ_FORCEINLINE auto to_decimal(Float x, Policies... policies)
|
|
{
|
|
// Build policy holder type
|
|
using namespace detail::policy_impl;
|
|
using policy_holder = decltype(make_policy_holder(
|
|
base_default_pair_list<
|
|
base_default_pair<sign::base, sign::return_sign>,
|
|
base_default_pair<trailing_zero::base, trailing_zero::remove>,
|
|
base_default_pair<rounding_mode::base, rounding_mode::nearest_to_even>,
|
|
base_default_pair<correct_rounding::base, correct_rounding::to_even>,
|
|
base_default_pair<cache::base, cache::normal>,
|
|
base_default_pair<input_validation::base, input_validation::assert_finite>
|
|
>{}, policies...));
|
|
|
|
using return_type = fp_t<Float,
|
|
policy_holder::return_has_sign,
|
|
policy_holder::report_trailing_zeros>;
|
|
|
|
auto br = ieee754_bits(x);
|
|
policy_holder::validate_input(br);
|
|
|
|
return policy_holder::delegate(br,
|
|
[br](auto interval_type_provider) {
|
|
constexpr auto tag = decltype(interval_type_provider)::tag;
|
|
|
|
if constexpr (tag == rounding_mode::tag_t::to_nearest) {
|
|
return detail::impl<Float>::template
|
|
compute_nearest<return_type, decltype(interval_type_provider),
|
|
typename policy_holder::sign_policy,
|
|
typename policy_holder::trailing_zero_policy,
|
|
typename policy_holder::correct_rounding_policy,
|
|
typename policy_holder::cache_policy
|
|
>(br);
|
|
}
|
|
else if constexpr (tag == rounding_mode::tag_t::left_closed_directed) {
|
|
return detail::impl<Float>::template
|
|
compute_left_closed_directed<return_type,
|
|
typename policy_holder::sign_policy,
|
|
typename policy_holder::trailing_zero_policy,
|
|
typename policy_holder::cache_policy
|
|
>(br);
|
|
}
|
|
else {
|
|
return detail::impl<Float>::template
|
|
compute_right_closed_directed<return_type,
|
|
typename policy_holder::sign_policy,
|
|
typename policy_holder::trailing_zero_policy,
|
|
typename policy_holder::cache_policy
|
|
>(br);
|
|
}
|
|
});
|
|
}
|
|
}
|
|
|
|
#undef JKJ_HAS_COUNTR_ZERO_INTRINSIC
|
|
#undef JKJ_FORCEINLINE
|
|
#undef JKJ_SAFEBUFFERS
|
|
|
|
#endif
|