Add float utilities, & float formatting with dragonbox

3 files changed, 4429 insertions, 0 deletions

diff --git a/vendor/dragonbox/BUILD.bazel b/vendor/dragonbox/BUILD.bazel
new file mode 100644
index 0000000..7a531d4
--- /dev/null
+++ b/vendor/dragonbox/BUILD.bazel
@@ -0,0 +1,6 @@
+cc_library(

+    name = "dragonbox",

+    hdrs = ["dragonbox.h"],

+    includes = ["."],

+    visibility = ["//:__subpackages__"],

+)

diff --git a/vendor/dragonbox/LICENSE.txt b/vendor/dragonbox/LICENSE.txt
new file mode 100644
index 0000000..bd8b243
--- /dev/null
+++ b/vendor/dragonbox/LICENSE.txt
@@ -0,0 +1,218 @@
+                                 Apache License
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+                        http://www.apache.org/licenses/
+
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diff --git a/vendor/dragonbox/dragonbox.h b/vendor/dragonbox/dragonbox.h
new file mode 100644
index 0000000..7cd8d19
--- /dev/null
+++ b/vendor/dragonbox/dragonbox.h
@@ -0,0 +1,4205 @@
+// Copyright 2020-2024 Junekey Jeon
+//
+// The contents of this file may be used under the terms of
+// the Apache License v2.0 with LLVM Exceptions.
+//
+//    (See accompanying file LICENSE-Apache or copy at
+//     https://llvm.org/foundation/relicensing/LICENSE.txt)
+//
+// Alternatively, the contents of this file may be used under the terms of
+// the Boost Software License, Version 1.0.
+//    (See accompanying file LICENSE-Boost or copy at
+//     https://www.boost.org/LICENSE_1_0.txt)
+//
+// Unless required by applicable law or agreed to in writing, this software
+// is distributed on an "AS IS" BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY
+// KIND, either express or implied.
+
+
+#ifndef JKJ_HEADER_DRAGONBOX
+#define JKJ_HEADER_DRAGONBOX
+
+// Attribute for storing static data into a dedicated place, e.g. flash memory. Every ODR-used
+// static data declaration will be decorated with this macro. The users may define this macro,
+// before including the library headers, into whatever they want.
+#ifndef JKJ_STATIC_DATA_SECTION
+    #define JKJ_STATIC_DATA_SECTION
+#else
+    #define JKJ_STATIC_DATA_SECTION_DEFINED 1
+#endif
+
+// To use the library with toolchains without standard C++ headers, the users may define this macro
+// into their custom namespace which contains the definitions of all the standard C++ library
+// features used in this header. (The list can be found below.)
+#ifndef JKJ_STD_REPLACEMENT_NAMESPACE
+    #define JKJ_STD_REPLACEMENT_NAMESPACE std
+    #include <cassert>
+    #include <cstdint>
+    #include <cstring>
+    #include <limits>
+    #include <type_traits>
+
+    #ifdef __has_include
+        #if __has_include(<version>)
+            #include <version>
+        #endif
+    #endif
+#else
+    #define JKJ_STD_REPLACEMENT_NAMESPACE_DEFINED 1
+#endif
+
+////////////////////////////////////////////////////////////////////////////////////////
+// Language feature detections.
+////////////////////////////////////////////////////////////////////////////////////////
+
+// C++14 constexpr
+#if defined(__cpp_constexpr) && __cpp_constexpr >= 201304L
+    #define JKJ_HAS_CONSTEXPR14 1
+#elif __cplusplus >= 201402L
+    #define JKJ_HAS_CONSTEXPR14 1
+#elif defined(_MSC_VER) && _MSC_VER >= 1910 && _MSVC_LANG >= 201402L
+    #define JKJ_HAS_CONSTEXPR14 1
+#else
+    #define JKJ_HAS_CONSTEXPR14 0
+#endif
+
+#if JKJ_HAS_CONSTEXPR14
+    #define JKJ_CONSTEXPR14 constexpr
+#else
+    #define JKJ_CONSTEXPR14
+#endif
+
+// C++17 constexpr lambdas
+#if defined(__cpp_constexpr) && __cpp_constexpr >= 201603L
+    #define JKJ_HAS_CONSTEXPR17 1
+#elif __cplusplus >= 201703L
+    #define JKJ_HAS_CONSTEXPR17 1
+#elif defined(_MSC_VER) && _MSC_VER >= 1911 && _MSVC_LANG >= 201703L
+    #define JKJ_HAS_CONSTEXPR17 1
+#else
+    #define JKJ_HAS_CONSTEXPR17 0
+#endif
+
+// C++17 inline variables
+#if defined(__cpp_inline_variables) && __cpp_inline_variables >= 201606L
+    #define JKJ_HAS_INLINE_VARIABLE 1
+#elif __cplusplus >= 201703L
+    #define JKJ_HAS_INLINE_VARIABLE 1
+#elif defined(_MSC_VER) && _MSC_VER >= 1912 && _MSVC_LANG >= 201703L
+    #define JKJ_HAS_INLINE_VARIABLE 1
+#else
+    #define JKJ_HAS_INLINE_VARIABLE 0
+#endif
+
+#if JKJ_HAS_INLINE_VARIABLE
+    #define JKJ_INLINE_VARIABLE inline constexpr
+#else
+    #define JKJ_INLINE_VARIABLE static constexpr
+#endif
+
+// C++17 if constexpr
+#if defined(__cpp_if_constexpr) && __cpp_if_constexpr >= 201606L
+    #define JKJ_HAS_IF_CONSTEXPR 1
+#elif __cplusplus >= 201703L
+    #define JKJ_HAS_IF_CONSTEXPR 1
+#elif defined(_MSC_VER) && _MSC_VER >= 1911 && _MSVC_LANG >= 201703L
+    #define JKJ_HAS_IF_CONSTEXPR 1
+#else
+    #define JKJ_HAS_IF_CONSTEXPR 0
+#endif
+
+#if JKJ_HAS_IF_CONSTEXPR
+    #define JKJ_IF_CONSTEXPR if constexpr
+#else
+    #define JKJ_IF_CONSTEXPR if
+#endif
+
+// C++20 std::bit_cast
+#if JKJ_STD_REPLACEMENT_NAMESPACE_DEFINED
+    #if JKJ_STD_REPLACEMENT_HAS_BIT_CAST
+        #define JKJ_HAS_BIT_CAST 1
+    #else
+        #define JKJ_HAS_BIT_CAST 0
+    #endif
+#elif defined(__cpp_lib_bit_cast) && __cpp_lib_bit_cast >= 201806L
+    #include <bit>
+    #define JKJ_HAS_BIT_CAST 1
+#else
+    #define JKJ_HAS_BIT_CAST 0
+#endif
+
+// C++23 if consteval or C++20 std::is_constant_evaluated
+#if defined(__cpp_if_consteval) && __cpp_is_consteval >= 202106L
+    #define JKJ_IF_CONSTEVAL if consteval
+    #define JKJ_IF_NOT_CONSTEVAL if !consteval
+    #define JKJ_CAN_BRANCH_ON_CONSTEVAL 1
+    #define JKJ_USE_IS_CONSTANT_EVALUATED 0
+#elif JKJ_STD_REPLACEMENT_NAMESPACE_DEFINED
+    #if JKJ_STD_REPLACEMENT_HAS_IS_CONSTANT_EVALUATED
+        #define JKJ_IF_CONSTEVAL if (stdr::is_constant_evaluated())
+        #define JKJ_IF_NOT_CONSTEVAL if (!stdr::is_constant_evaluated())
+        #define JKJ_CAN_BRANCH_ON_CONSTEVAL 1
+        #define JKJ_USE_IS_CONSTANT_EVALUATED 1
+    #elif JKJ_HAS_IF_CONSTEXPR
+        #define JKJ_IF_CONSTEVAL if constexpr (false)
+        #define JKJ_IF_NOT_CONSTEVAL if constexpr (true)
+        #define JKJ_CAN_BRANCH_ON_CONSTEVAL 0
+        #define JKJ_USE_IS_CONSTANT_EVALUATED 0
+    #else
+        #define JKJ_IF_CONSTEVAL if (false)
+        #define JKJ_IF_NOT_CONSTEVAL if (true)
+        #define JKJ_CAN_BRANCH_ON_CONSTEVAL 0
+        #define JKJ_USE_IS_CONSTANT_EVALUATED 0
+    #endif
+#else
+    #if defined(__cpp_lib_is_constant_evaluated) && __cpp_lib_is_constant_evaluated >= 201811L
+        #define JKJ_IF_CONSTEVAL if (stdr::is_constant_evaluated())
+        #define JKJ_IF_NOT_CONSTEVAL if (!stdr::is_constant_evaluated())
+        #define JKJ_CAN_BRANCH_ON_CONSTEVAL 1
+        #define JKJ_USE_IS_CONSTANT_EVALUATED 1
+    #elif JKJ_HAS_IF_CONSTEXPR
+        #define JKJ_IF_CONSTEVAL if constexpr (false)
+        #define JKJ_IF_NOT_CONSTEVAL if constexpr (true)
+        #define JKJ_CAN_BRANCH_ON_CONSTEVAL 0
+        #define JKJ_USE_IS_CONSTANT_EVALUATED 0
+    #else
+        #define JKJ_IF_CONSTEVAL if (false)
+        #define JKJ_IF_NOT_CONSTEVAL if (true)
+        #define JKJ_CAN_BRANCH_ON_CONSTEVAL 0
+        #define JKJ_USE_IS_CONSTANT_EVALUATED 0
+    #endif
+#endif
+
+#if JKJ_CAN_BRANCH_ON_CONSTEVAL && JKJ_HAS_BIT_CAST
+    #define JKJ_CONSTEXPR20 constexpr
+#else
+    #define JKJ_CONSTEXPR20
+#endif
+
+// Suppress additional buffer overrun check.
+// I have no idea why MSVC thinks some functions here are vulnerable to the buffer overrun
+// attacks. No, they aren't.
+#if defined(__GNUC__) || defined(__clang__)
+    #define JKJ_SAFEBUFFERS
+    #define JKJ_FORCEINLINE inline __attribute__((always_inline))
+#elif defined(_MSC_VER)
+    #define JKJ_SAFEBUFFERS __declspec(safebuffers)
+    #define JKJ_FORCEINLINE __forceinline
+#else
+    #define JKJ_SAFEBUFFERS
+    #define JKJ_FORCEINLINE inline
+#endif
+
+#if defined(__has_builtin)
+    #define JKJ_HAS_BUILTIN(x) __has_builtin(x)
+#else
+    #define JKJ_HAS_BUILTIN(x) false
+#endif
+
+#if defined(_MSC_VER)
+    #include <intrin.h>
+#elif defined(__INTEL_COMPILER)
+    #include <immintrin.h>
+#endif
+
+namespace jkj {
+    namespace dragonbox {
+        ////////////////////////////////////////////////////////////////////////////////////////
+        // The Compatibility layer for toolchains without standard C++ headers.
+        ////////////////////////////////////////////////////////////////////////////////////////
+        namespace detail {
+            namespace stdr {
+                // <bit>
+#if JKJ_HAS_BIT_CAST
+                using JKJ_STD_REPLACEMENT_NAMESPACE::bit_cast;
+#endif
+
+                // <cassert>
+                // We need assert() macro, but it is not namespaced anyway, so nothing to do here.
+
+                // <cstdint>
+                using JKJ_STD_REPLACEMENT_NAMESPACE::int_least8_t;
+                using JKJ_STD_REPLACEMENT_NAMESPACE::int_least16_t;
+                using JKJ_STD_REPLACEMENT_NAMESPACE::int_least32_t;
+                using JKJ_STD_REPLACEMENT_NAMESPACE::int_fast8_t;
+                using JKJ_STD_REPLACEMENT_NAMESPACE::int_fast16_t;
+                using JKJ_STD_REPLACEMENT_NAMESPACE::int_fast32_t;
+                using JKJ_STD_REPLACEMENT_NAMESPACE::uint_least8_t;
+                using JKJ_STD_REPLACEMENT_NAMESPACE::uint_least16_t;
+                using JKJ_STD_REPLACEMENT_NAMESPACE::uint_least32_t;
+                using JKJ_STD_REPLACEMENT_NAMESPACE::uint_least64_t;
+                using JKJ_STD_REPLACEMENT_NAMESPACE::uint_fast8_t;
+                using JKJ_STD_REPLACEMENT_NAMESPACE::uint_fast16_t;
+                using JKJ_STD_REPLACEMENT_NAMESPACE::uint_fast32_t;
+                // We need INT32_C, UINT32_C and UINT64_C macros too, but again there is nothing to do
+                // here.
+
+                // <cstring>
+                using JKJ_STD_REPLACEMENT_NAMESPACE::size_t;
+                using JKJ_STD_REPLACEMENT_NAMESPACE::memcpy;
+
+                // <limits>
+                template <class T>
+                using numeric_limits = JKJ_STD_REPLACEMENT_NAMESPACE::numeric_limits<T>;
+
+                // <type_traits>
+                template <bool cond, class T = void>
+                using enable_if = JKJ_STD_REPLACEMENT_NAMESPACE::enable_if<cond, T>;
+                template <class T>
+                using add_rvalue_reference = JKJ_STD_REPLACEMENT_NAMESPACE::add_rvalue_reference<T>;
+                template <bool cond, class T_true, class T_false>
+                using conditional = JKJ_STD_REPLACEMENT_NAMESPACE::conditional<cond, T_true, T_false>;
+#if JKJ_USE_IS_CONSTANT_EVALUATED
+                using JKJ_STD_REPLACEMENT_NAMESPACE::is_constant_evaluated;
+#endif
+                template <class T1, class T2>
+                using is_same = JKJ_STD_REPLACEMENT_NAMESPACE::is_same<T1, T2>;
+#if !JKJ_HAS_BIT_CAST
+                template <class T>
+                using is_trivially_copyable = JKJ_STD_REPLACEMENT_NAMESPACE::is_trivially_copyable<T>;
+#endif
+                template <class T>
+                using is_integral = JKJ_STD_REPLACEMENT_NAMESPACE::is_integral<T>;
+                template <class T>
+                using is_signed = JKJ_STD_REPLACEMENT_NAMESPACE::is_signed<T>;
+                template <class T>
+                using is_unsigned = JKJ_STD_REPLACEMENT_NAMESPACE::is_unsigned<T>;
+            }
+        }
+
+
+        ////////////////////////////////////////////////////////////////////////////////////////
+        // Some general utilities for C++11-compatibility.
+        ////////////////////////////////////////////////////////////////////////////////////////
+        namespace detail {
+#if !JKJ_HAS_CONSTEXPR17
+            template <stdr::size_t... indices>
+            struct index_sequence {};
+
+            template <stdr::size_t current, stdr::size_t total, class Dummy, stdr::size_t... indices>
+            struct make_index_sequence_impl {
+                using type = typename make_index_sequence_impl<current + 1, total, Dummy, indices...,
+                                                               current>::type;
+            };
+
+            template <stdr::size_t total, class Dummy, stdr::size_t... indices>
+            struct make_index_sequence_impl<total, total, Dummy, indices...> {
+                using type = index_sequence<indices...>;
+            };
+
+            template <stdr::size_t N>
+            using make_index_sequence = typename make_index_sequence_impl<0, N, void>::type;
+#endif
+
+            // Available since C++11, but including <utility> just for this is an overkill.
+            template <class T>
+            typename stdr::add_rvalue_reference<T>::type declval() noexcept;
+
+            // Similarly, including <array> is an overkill.
+            template <class T, stdr::size_t N>
+            struct array {
+                T data_[N];
+                constexpr T operator[](stdr::size_t idx) const noexcept { return data_[idx]; }
+                JKJ_CONSTEXPR14 T& operator[](stdr::size_t idx) noexcept { return data_[idx]; }
+            };
+        }
+
+
+        ////////////////////////////////////////////////////////////////////////////////////////
+        // Some basic features for encoding/decoding IEEE-754 formats.
+        ////////////////////////////////////////////////////////////////////////////////////////
+        namespace detail {
+            template <class T>
+            struct physical_bits {
+                static constexpr stdr::size_t value =
+                    sizeof(T) * stdr::numeric_limits<unsigned char>::digits;
+            };
+            template <class T>
+            struct value_bits {
+                static constexpr stdr::size_t value = stdr::numeric_limits<
+                    typename stdr::enable_if<stdr::is_integral<T>::value, T>::type>::digits;
+            };
+
+            template <typename To, typename From>
+            JKJ_CONSTEXPR20 To bit_cast(const From& from) {
+#if JKJ_HAS_BIT_CAST
+                return stdr::bit_cast<To>(from);
+#else
+                static_assert(sizeof(From) == sizeof(To), "");
+                static_assert(stdr::is_trivially_copyable<To>::value, "");
+                static_assert(stdr::is_trivially_copyable<From>::value, "");
+                To to;
+                stdr::memcpy(&to, &from, sizeof(To));
+                return to;
+#endif
+            }
+        }
+
+        // These classes expose encoding specs of IEEE-754-like floating-point formats.
+        // Currently available formats are IEEE-754 binary32 & IEEE-754 binary64.
+
+        struct ieee754_binary32 {
+            static constexpr int total_bits = 32;
+            static constexpr int significand_bits = 23;
+            static constexpr int exponent_bits = 8;
+            static constexpr int min_exponent = -126;
+            static constexpr int max_exponent = 127;
+            static constexpr int exponent_bias = -127;
+            static constexpr int decimal_significand_digits = 9;
+            static constexpr int decimal_exponent_digits = 2;
+        };
+        struct ieee754_binary64 {
+            static constexpr int total_bits = 64;
+            static constexpr int significand_bits = 52;
+            static constexpr int exponent_bits = 11;
+            static constexpr int min_exponent = -1022;
+            static constexpr int max_exponent = 1023;
+            static constexpr int exponent_bias = -1023;
+            static constexpr int decimal_significand_digits = 17;
+            static constexpr int decimal_exponent_digits = 3;
+        };
+
+        // A floating-point format traits class defines ways to interpret a bit pattern of given size as
+        // an encoding of floating-point number. This is an implementation of such a traits class,
+        // supporting ways to interpret IEEE-754 binary floating-point numbers.
+        template <class Format, class CarrierUInt, class ExponentInt = int>
+        struct ieee754_binary_traits {
+            // CarrierUInt needs to have enough size to hold the entire contents of floating-point
+            // numbers. The actual bits are assumed to be aligned to the LSB, and every other bits are
+            // assumed to be zeroed.
+            static_assert(detail::value_bits<CarrierUInt>::value >= Format::total_bits,
+                          "jkj::dragonbox: insufficient number of bits");
+            static_assert(detail::stdr::is_unsigned<CarrierUInt>::value, "");
+
+            // ExponentUInt needs to be large enough to hold (unsigned) exponent bits as well as the
+            // (signed) actual exponent.
+            // TODO: static overflow guard against intermediate computations.
+            static_assert(detail::value_bits<ExponentInt>::value >= Format::exponent_bits + 1,
+                          "jkj::dragonbox: insufficient number of bits");
+            static_assert(detail::stdr::is_signed<ExponentInt>::value, "");
+
+            using format = Format;
+            using carrier_uint = CarrierUInt;
+            static constexpr int carrier_bits = int(detail::value_bits<carrier_uint>::value);
+            using exponent_int = ExponentInt;
+
+            // Extract exponent bits from a bit pattern.
+            // The result must be aligned to the LSB so that there is no additional zero paddings
+            // on the right. This function does not do bias adjustment.
+            static constexpr exponent_int extract_exponent_bits(carrier_uint u) noexcept {
+                return exponent_int((u >> format::significand_bits) &
+                                    ((exponent_int(1) << format::exponent_bits) - 1));
+            }
+
+            // Extract significand bits from a bit pattern.
+            // The result must be aligned to the LSB so that there is no additional zero paddings
+            // on the right. The result does not contain the implicit bit.
+            static constexpr carrier_uint extract_significand_bits(carrier_uint u) noexcept {
+                return carrier_uint(u & ((carrier_uint(1) << format::significand_bits) - 1u));
+            }
+
+            // Remove the exponent bits and extract significand bits together with the sign bit.
+            static constexpr carrier_uint remove_exponent_bits(carrier_uint u) noexcept {
+                return carrier_uint(u & ~(((carrier_uint(1) << format::exponent_bits) - 1u)
+                                          << format::significand_bits));
+            }
+
+            // Shift the obtained signed significand bits to the left by 1 to remove the sign bit.
+            static constexpr carrier_uint remove_sign_bit_and_shift(carrier_uint u) noexcept {
+                return carrier_uint((carrier_uint(u) << 1) &
+                                    ((((carrier_uint(1) << (Format::total_bits - 1)) - 1u) << 1) | 1u));
+            }
+
+            // Obtain the actual value of the binary exponent from the extracted exponent bits.
+            static constexpr exponent_int binary_exponent(exponent_int exponent_bits) noexcept {
+                return exponent_int(exponent_bits == 0 ? format::min_exponent
+                                                       : exponent_bits + format::exponent_bias);
+            }
+
+            // Obtain the actual value of the binary significand from the extracted significand bits
+            // and exponent bits.
+            static constexpr carrier_uint binary_significand(carrier_uint significand_bits,
+                                                             exponent_int exponent_bits) noexcept {
+                return carrier_uint(
+                    exponent_bits == 0
+                        ? significand_bits
+                        : (significand_bits | (carrier_uint(1) << format::significand_bits)));
+            }
+
+            /* Various boolean observer functions */
+
+            static constexpr bool is_nonzero(carrier_uint u) noexcept {
+                return (u & ((carrier_uint(1) << (format::significand_bits + format::exponent_bits)) -
+                             1u)) != 0;
+            }
+            static constexpr bool is_positive(carrier_uint u) noexcept {
+                return u < (carrier_uint(1) << (format::significand_bits + format::exponent_bits));
+            }
+            static constexpr bool is_negative(carrier_uint u) noexcept { return !is_positive(u); }
+            static constexpr bool is_finite(exponent_int exponent_bits) noexcept {
+                return exponent_bits != ((exponent_int(1) << format::exponent_bits) - 1);
+            }
+            static constexpr bool has_all_zero_significand_bits(carrier_uint u) noexcept {
+                return ((u << 1) &
+                        ((((carrier_uint(1) << (Format::total_bits - 1)) - 1u) << 1) | 1u)) == 0;
+            }
+            static constexpr bool has_even_significand_bits(carrier_uint u) noexcept {
+                return u % 2 == 0;
+            }
+        };
+
+        // Convert between bit patterns stored in carrier_uint and instances of an actual
+        // floating-point type. Depending on format and carrier_uint, this operation might not
+        // be possible for some specific bit patterns. However, the contract is that u always
+        // denotes a valid bit pattern, so the functions here are assumed to be noexcept.
+        // Users might specialize this class to change the behavior for certain types.
+        // The default provided by the library is to treat the given floating-point type Float as either
+        // IEEE-754 binary32 or IEEE-754 binary64, depending on the bitwise size of Float.
+        template <class Float>
+        struct default_float_bit_carrier_conversion_traits {
+            // Guards against types that have different internal representations than IEEE-754
+            // binary32/64. I don't know if there is a truly reliable way of detecting IEEE-754 binary
+            // formats. I just did my best here. Note that in some cases
+            // numeric_limits<Float>::is_iec559 may report false even if the internal representation is
+            // IEEE-754 compatible. In such a case, the user can specialize this traits template and
+            // remove this static sanity check in order to make Dragonbox work for Float.
+            static_assert(detail::stdr::numeric_limits<Float>::is_iec559 &&
+                              detail::stdr::numeric_limits<Float>::radix == 2 &&
+                              (detail::physical_bits<Float>::value == 32 ||
+                               detail::physical_bits<Float>::value == 64),
+                          "jkj::dragonbox: Float may not be of IEEE-754 binary32/binary64");
+
+            // Specifies the unsigned integer type to hold bitwise value of Float.
+            using carrier_uint =
+                typename detail::stdr::conditional<detail::physical_bits<Float>::value == 32,
+                                                   detail::stdr::uint_least32_t,
+                                                   detail::stdr::uint_least64_t>::type;
+
+            // Specifies the floating-point format.
+            using format = typename detail::stdr::conditional<detail::physical_bits<Float>::value == 32,
+                                                              ieee754_binary32, ieee754_binary64>::type;
+
+            // Converts the floating-point type into the bit-carrier unsigned integer type.
+            static JKJ_CONSTEXPR20 carrier_uint float_to_carrier(Float x) noexcept {
+                return detail::bit_cast<carrier_uint>(x);
+            }
+
+            // Converts the bit-carrier unsigned integer type into the floating-point type.
+            static JKJ_CONSTEXPR20 Float carrier_to_float(carrier_uint x) noexcept {
+                return detail::bit_cast<Float>(x);
+            }
+        };
+
+        // Convenient wrappers for floating-point traits classes.
+        // In order to reduce the argument passing overhead, these classes should be as simple as
+        // possible (e.g., no inheritance, no private non-static data member, etc.; this is an
+        // unfortunate fact about common ABI convention).
+
+        template <class FormatTraits>
+        struct signed_significand_bits {
+            using format_traits = FormatTraits;
+            using carrier_uint = typename format_traits::carrier_uint;
+
+            carrier_uint u;
+
+            signed_significand_bits() = default;
+            constexpr explicit signed_significand_bits(carrier_uint bit_pattern) noexcept
+                : u{bit_pattern} {}
+
+            // Shift the obtained signed significand bits to the left by 1 to remove the sign bit.
+            constexpr carrier_uint remove_sign_bit_and_shift() const noexcept {
+                return format_traits::remove_sign_bit_and_shift(u);
+            }
+
+            constexpr bool is_positive() const noexcept { return format_traits::is_positive(u); }
+            constexpr bool is_negative() const noexcept { return format_traits::is_negative(u); }
+            constexpr bool has_all_zero_significand_bits() const noexcept {
+                return format_traits::has_all_zero_significand_bits(u);
+            }
+            constexpr bool has_even_significand_bits() const noexcept {
+                return format_traits::has_even_significand_bits(u);
+            }
+        };
+
+        template <class FormatTraits>
+        struct float_bits {
+            using format_traits = FormatTraits;
+            using carrier_uint = typename format_traits::carrier_uint;
+            using exponent_int = typename format_traits::exponent_int;
+
+            carrier_uint u;
+
+            float_bits() = default;
+            constexpr explicit float_bits(carrier_uint bit_pattern) noexcept : u{bit_pattern} {}
+
+            // Extract exponent bits from a bit pattern.
+            // The result must be aligned to the LSB so that there is no additional zero paddings
+            // on the right. This function does not do bias adjustment.
+            constexpr exponent_int extract_exponent_bits() const noexcept {
+                return format_traits::extract_exponent_bits(u);
+            }
+
+            // Extract significand bits from a bit pattern.
+            // The result must be aligned to the LSB so that there is no additional zero paddings
+            // on the right. The result does not contain the implicit bit.
+            constexpr carrier_uint extract_significand_bits() const noexcept {
+                return format_traits::extract_significand_bits(u);
+            }
+
+            // Remove the exponent bits and extract significand bits together with the sign bit.
+            constexpr signed_significand_bits<format_traits> remove_exponent_bits() const noexcept {
+                return signed_significand_bits<format_traits>(format_traits::remove_exponent_bits(u));
+            }
+
+            // Obtain the actual value of the binary exponent from the extracted exponent bits.
+            static constexpr exponent_int binary_exponent(exponent_int exponent_bits) noexcept {
+                return format_traits::binary_exponent(exponent_bits);
+            }
+            constexpr exponent_int binary_exponent() const noexcept {
+                return binary_exponent(extract_exponent_bits());
+            }
+
+            // Obtain the actual value of the binary exponent from the extracted significand bits
+            // and exponent bits.
+            static constexpr carrier_uint binary_signific