// half - IEEE 754-based half-precision floating-point library. // // Copyright (c) 2012-2021 Christian Rau // Copyright (c) 2023, ARM Limited. // // Permission is hereby granted, free of charge, to any person obtaining a copy of this software and associated documentation // files (the "Software"), to deal in the Software without restriction, including without limitation the rights to use, copy, // modify, merge, publish, distribute, sublicense, and/or sell copies of the Software, and to permit persons to whom the // Software is furnished to do so, subject to the following conditions: // // The above copyright notice and this permission notice shall be included in all copies or substantial portions of the Software. // // THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE // WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR // COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, // ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE. // Version 2.2.0 /// \file /// Main header file for half-precision functionality. #ifndef HALF_HALF_HPP #define HALF_HALF_HPP #define HALF_GCC_VERSION (__GNUC__*100+__GNUC_MINOR__) #if defined(__INTEL_COMPILER) #define HALF_ICC_VERSION __INTEL_COMPILER #elif defined(__ICC) #define HALF_ICC_VERSION __ICC #elif defined(__ICL) #define HALF_ICC_VERSION __ICL #else #define HALF_ICC_VERSION 0 #endif // check C++11 language features #if defined(__clang__) // clang #if __has_feature(cxx_static_assert) && !defined(HALF_ENABLE_CPP11_STATIC_ASSERT) #define HALF_ENABLE_CPP11_STATIC_ASSERT 1 #endif #if __has_feature(cxx_constexpr) && !defined(HALF_ENABLE_CPP11_CONSTEXPR) #define HALF_ENABLE_CPP11_CONSTEXPR 1 #endif #if __has_feature(cxx_noexcept) && !defined(HALF_ENABLE_CPP11_NOEXCEPT) #define HALF_ENABLE_CPP11_NOEXCEPT 1 #endif #if __has_feature(cxx_user_literals) && !defined(HALF_ENABLE_CPP11_USER_LITERALS) #define HALF_ENABLE_CPP11_USER_LITERALS 1 #endif #if __has_feature(cxx_thread_local) && !defined(HALF_ENABLE_CPP11_THREAD_LOCAL) #define HALF_ENABLE_CPP11_THREAD_LOCAL 1 #endif #if (defined(__GXX_EXPERIMENTAL_CXX0X__) || __cplusplus >= 201103L) && !defined(HALF_ENABLE_CPP11_LONG_LONG) #define HALF_ENABLE_CPP11_LONG_LONG 1 #endif #elif HALF_ICC_VERSION && defined(__INTEL_CXX11_MODE__) // Intel C++ #if HALF_ICC_VERSION >= 1500 && !defined(HALF_ENABLE_CPP11_THREAD_LOCAL) #define HALF_ENABLE_CPP11_THREAD_LOCAL 1 #endif #if HALF_ICC_VERSION >= 1500 && !defined(HALF_ENABLE_CPP11_USER_LITERALS) #define HALF_ENABLE_CPP11_USER_LITERALS 1 #endif #if HALF_ICC_VERSION >= 1400 && !defined(HALF_ENABLE_CPP11_CONSTEXPR) #define HALF_ENABLE_CPP11_CONSTEXPR 1 #endif #if HALF_ICC_VERSION >= 1400 && !defined(HALF_ENABLE_CPP11_NOEXCEPT) #define HALF_ENABLE_CPP11_NOEXCEPT 1 #endif #if HALF_ICC_VERSION >= 1110 && !defined(HALF_ENABLE_CPP11_STATIC_ASSERT) #define HALF_ENABLE_CPP11_STATIC_ASSERT 1 #endif #if HALF_ICC_VERSION >= 1110 && !defined(HALF_ENABLE_CPP11_LONG_LONG) #define HALF_ENABLE_CPP11_LONG_LONG 1 #endif #elif defined(__GNUC__) // gcc #if defined(__GXX_EXPERIMENTAL_CXX0X__) || __cplusplus >= 201103L #if HALF_GCC_VERSION >= 408 && !defined(HALF_ENABLE_CPP11_THREAD_LOCAL) #define HALF_ENABLE_CPP11_THREAD_LOCAL 1 #endif #if HALF_GCC_VERSION >= 407 && !defined(HALF_ENABLE_CPP11_USER_LITERALS) #define HALF_ENABLE_CPP11_USER_LITERALS 1 #endif #if HALF_GCC_VERSION >= 406 && !defined(HALF_ENABLE_CPP11_CONSTEXPR) #define HALF_ENABLE_CPP11_CONSTEXPR 1 #endif #if HALF_GCC_VERSION >= 406 && !defined(HALF_ENABLE_CPP11_NOEXCEPT) #define HALF_ENABLE_CPP11_NOEXCEPT 1 #endif #if HALF_GCC_VERSION >= 403 && !defined(HALF_ENABLE_CPP11_STATIC_ASSERT) #define HALF_ENABLE_CPP11_STATIC_ASSERT 1 #endif #if !defined(HALF_ENABLE_CPP11_LONG_LONG) #define HALF_ENABLE_CPP11_LONG_LONG 1 #endif #endif #define HALF_TWOS_COMPLEMENT_INT 1 #elif defined(_MSC_VER) // Visual C++ #if _MSC_VER >= 1900 && !defined(HALF_ENABLE_CPP11_THREAD_LOCAL) #define HALF_ENABLE_CPP11_THREAD_LOCAL 1 #endif #if _MSC_VER >= 1900 && !defined(HALF_ENABLE_CPP11_USER_LITERALS) #define HALF_ENABLE_CPP11_USER_LITERALS 1 #endif #if _MSC_VER >= 1900 && !defined(HALF_ENABLE_CPP11_CONSTEXPR) #define HALF_ENABLE_CPP11_CONSTEXPR 1 #endif #if _MSC_VER >= 1900 && !defined(HALF_ENABLE_CPP11_NOEXCEPT) #define HALF_ENABLE_CPP11_NOEXCEPT 1 #endif #if _MSC_VER >= 1600 && !defined(HALF_ENABLE_CPP11_STATIC_ASSERT) #define HALF_ENABLE_CPP11_STATIC_ASSERT 1 #endif #if _MSC_VER >= 1310 && !defined(HALF_ENABLE_CPP11_LONG_LONG) #define HALF_ENABLE_CPP11_LONG_LONG 1 #endif #define HALF_TWOS_COMPLEMENT_INT 1 #define HALF_POP_WARNINGS 1 #pragma warning(push) #pragma warning(disable : 4099 4127 4146) //struct vs class, constant in if, negative unsigned #endif // check C++11 library features #include #if defined(_LIBCPP_VERSION) // libc++ #if defined(__GXX_EXPERIMENTAL_CXX0X__) || __cplusplus >= 201103 #ifndef HALF_ENABLE_CPP11_TYPE_TRAITS #define HALF_ENABLE_CPP11_TYPE_TRAITS 1 #endif #ifndef HALF_ENABLE_CPP11_CSTDINT #define HALF_ENABLE_CPP11_CSTDINT 1 #endif #ifndef HALF_ENABLE_CPP11_CMATH #define HALF_ENABLE_CPP11_CMATH 1 #endif #ifndef HALF_ENABLE_CPP11_HASH #define HALF_ENABLE_CPP11_HASH 1 #endif #ifndef HALF_ENABLE_CPP11_CFENV #define HALF_ENABLE_CPP11_CFENV 1 #endif #endif #elif defined(__GLIBCXX__) // libstdc++ #if defined(__GXX_EXPERIMENTAL_CXX0X__) || __cplusplus >= 201103 #ifdef __clang__ #if __GLIBCXX__ >= 20080606 && !defined(HALF_ENABLE_CPP11_TYPE_TRAITS) #define HALF_ENABLE_CPP11_TYPE_TRAITS 1 #endif #if __GLIBCXX__ >= 20080606 && !defined(HALF_ENABLE_CPP11_CSTDINT) #define HALF_ENABLE_CPP11_CSTDINT 1 #endif #if __GLIBCXX__ >= 20080606 && !defined(HALF_ENABLE_CPP11_CMATH) #define HALF_ENABLE_CPP11_CMATH 1 #endif #if __GLIBCXX__ >= 20080606 && !defined(HALF_ENABLE_CPP11_HASH) #define HALF_ENABLE_CPP11_HASH 1 #endif #if __GLIBCXX__ >= 20080606 && !defined(HALF_ENABLE_CPP11_CFENV) #define HALF_ENABLE_CPP11_CFENV 1 #endif #else #if HALF_GCC_VERSION >= 403 && !defined(HALF_ENABLE_CPP11_TYPE_TRAITS) #define HALF_ENABLE_CPP11_TYPE_TRAITS 1 #endif #if HALF_GCC_VERSION >= 403 && !defined(HALF_ENABLE_CPP11_CSTDINT) #define HALF_ENABLE_CPP11_CSTDINT 1 #endif #if HALF_GCC_VERSION >= 403 && !defined(HALF_ENABLE_CPP11_CMATH) #define HALF_ENABLE_CPP11_CMATH 1 #endif #if HALF_GCC_VERSION >= 403 && !defined(HALF_ENABLE_CPP11_HASH) #define HALF_ENABLE_CPP11_HASH 1 #endif #if HALF_GCC_VERSION >= 403 && !defined(HALF_ENABLE_CPP11_CFENV) #define HALF_ENABLE_CPP11_CFENV 1 #endif #endif #endif #elif defined(_CPPLIB_VER) // Dinkumware/Visual C++ #if _CPPLIB_VER >= 520 && !defined(HALF_ENABLE_CPP11_TYPE_TRAITS) #define HALF_ENABLE_CPP11_TYPE_TRAITS 1 #endif #if _CPPLIB_VER >= 520 && !defined(HALF_ENABLE_CPP11_CSTDINT) #define HALF_ENABLE_CPP11_CSTDINT 1 #endif #if _CPPLIB_VER >= 520 && !defined(HALF_ENABLE_CPP11_HASH) #define HALF_ENABLE_CPP11_HASH 1 #endif #if _CPPLIB_VER >= 610 && !defined(HALF_ENABLE_CPP11_CMATH) #define HALF_ENABLE_CPP11_CMATH 1 #endif #if _CPPLIB_VER >= 610 && !defined(HALF_ENABLE_CPP11_CFENV) #define HALF_ENABLE_CPP11_CFENV 1 #endif #endif #undef HALF_GCC_VERSION #undef HALF_ICC_VERSION // any error throwing C++ exceptions? #if defined(HALF_ERRHANDLING_THROW_INVALID) || defined(HALF_ERRHANDLING_THROW_DIVBYZERO) || defined(HALF_ERRHANDLING_THROW_OVERFLOW) || defined(HALF_ERRHANDLING_THROW_UNDERFLOW) || defined(HALF_ERRHANDLING_THROW_INEXACT) #define HALF_ERRHANDLING_THROWS 1 #endif // any error handling enabled? #define HALF_ERRHANDLING (HALF_ERRHANDLING_FLAGS||HALF_ERRHANDLING_ERRNO||HALF_ERRHANDLING_FENV||HALF_ERRHANDLING_THROWS) #if HALF_ERRHANDLING #define HALF_UNUSED_NOERR(name) name #else #define HALF_UNUSED_NOERR(name) #endif // support constexpr #if HALF_ENABLE_CPP11_CONSTEXPR #define HALF_CONSTEXPR constexpr #define HALF_CONSTEXPR_CONST constexpr #if HALF_ERRHANDLING #define HALF_CONSTEXPR_NOERR #else #define HALF_CONSTEXPR_NOERR constexpr #endif #else #define HALF_CONSTEXPR #define HALF_CONSTEXPR_CONST const #define HALF_CONSTEXPR_NOERR #endif // support noexcept #if HALF_ENABLE_CPP11_NOEXCEPT #define HALF_NOEXCEPT noexcept #define HALF_NOTHROW noexcept #else #define HALF_NOEXCEPT #define HALF_NOTHROW throw() #endif // support thread storage #if HALF_ENABLE_CPP11_THREAD_LOCAL #define HALF_THREAD_LOCAL thread_local #else #define HALF_THREAD_LOCAL static #endif #include #include #include #include #include #include #include #include #include #include #if HALF_ENABLE_CPP11_TYPE_TRAITS #include #endif #if HALF_ENABLE_CPP11_CSTDINT #include #endif #if HALF_ERRHANDLING_ERRNO #include #endif #if HALF_ENABLE_CPP11_CFENV #include #endif #if HALF_ENABLE_CPP11_HASH #include #endif #ifndef HALF_ENABLE_F16C_INTRINSICS /// Enable F16C intruction set intrinsics. /// Defining this to 1 enables the use of [F16C compiler intrinsics](https://en.wikipedia.org/wiki/F16C) for converting between /// half-precision and single-precision values which may result in improved performance. This will not perform additional checks /// for support of the F16C instruction set, so an appropriate target platform is required when enabling this feature. /// /// Unless predefined it will be enabled automatically when the `__F16C__` symbol is defined, which some compilers do on supporting platforms. #define HALF_ENABLE_F16C_INTRINSICS __F16C__ #endif #if HALF_ENABLE_F16C_INTRINSICS #include #endif #ifdef HALF_DOXYGEN_ONLY /// Type for internal floating-point computations. /// This can be predefined to a built-in floating-point type (`float`, `double` or `long double`) to override the internal /// half-precision implementation to use this type for computing arithmetic operations and mathematical function (if available). /// This can result in improved performance for arithmetic operators and mathematical functions but might cause results to /// deviate from the specified half-precision rounding mode and inhibits proper detection of half-precision exceptions. #define HALF_ARITHMETIC_TYPE (undefined) /// Enable internal exception flags. /// Defining this to 1 causes operations on half-precision values to raise internal floating-point exception flags according to /// the IEEE 754 standard. These can then be cleared and checked with clearexcept(), testexcept(). #define HALF_ERRHANDLING_FLAGS 0 /// Enable exception propagation to `errno`. /// Defining this to 1 causes operations on half-precision values to propagate floating-point exceptions to /// [errno](https://en.cppreference.com/w/cpp/error/errno) from ``. Specifically this will propagate domain errors as /// [EDOM](https://en.cppreference.com/w/cpp/error/errno_macros) and pole, overflow and underflow errors as /// [ERANGE](https://en.cppreference.com/w/cpp/error/errno_macros). Inexact errors won't be propagated. #define HALF_ERRHANDLING_ERRNO 0 /// Enable exception propagation to built-in floating-point platform. /// Defining this to 1 causes operations on half-precision values to propagate floating-point exceptions to the built-in /// single- and double-precision implementation's exception flags using the /// [C++11 floating-point environment control](https://en.cppreference.com/w/cpp/numeric/fenv) from ``. However, this /// does not work in reverse and single- or double-precision exceptions will not raise the corresponding half-precision /// exception flags, nor will explicitly clearing flags clear the corresponding built-in flags. #define HALF_ERRHANDLING_FENV 0 /// Throw C++ exception on domain errors. /// Defining this to a string literal causes operations on half-precision values to throw a /// [std::domain_error](https://en.cppreference.com/w/cpp/error/domain_error) with the specified message on domain errors. #define HALF_ERRHANDLING_THROW_INVALID (undefined) /// Throw C++ exception on pole errors. /// Defining this to a string literal causes operations on half-precision values to throw a /// [std::domain_error](https://en.cppreference.com/w/cpp/error/domain_error) with the specified message on pole errors. #define HALF_ERRHANDLING_THROW_DIVBYZERO (undefined) /// Throw C++ exception on overflow errors. /// Defining this to a string literal causes operations on half-precision values to throw a /// [std::overflow_error](https://en.cppreference.com/w/cpp/error/overflow_error) with the specified message on overflows. #define HALF_ERRHANDLING_THROW_OVERFLOW (undefined) /// Throw C++ exception on underflow errors. /// Defining this to a string literal causes operations on half-precision values to throw a /// [std::underflow_error](https://en.cppreference.com/w/cpp/error/underflow_error) with the specified message on underflows. #define HALF_ERRHANDLING_THROW_UNDERFLOW (undefined) /// Throw C++ exception on rounding errors. /// Defining this to 1 causes operations on half-precision values to throw a /// [std::range_error](https://en.cppreference.com/w/cpp/error/range_error) with the specified message on general rounding errors. #define HALF_ERRHANDLING_THROW_INEXACT (undefined) #endif #ifndef HALF_ERRHANDLING_OVERFLOW_TO_INEXACT /// Raise INEXACT exception on overflow. /// Defining this to 1 (default) causes overflow errors to automatically raise inexact exceptions in addition. /// These will be raised after any possible handling of the underflow exception. #define HALF_ERRHANDLING_OVERFLOW_TO_INEXACT 1 #endif #ifndef HALF_ERRHANDLING_UNDERFLOW_TO_INEXACT /// Raise INEXACT exception on underflow. /// Defining this to 1 (default) causes underflow errors to automatically raise inexact exceptions in addition. /// These will be raised after any possible handling of the underflow exception. /// /// **Note:** This will actually cause underflow (and the accompanying inexact) exceptions to be raised *only* when the result /// is inexact, while if disabled bare underflow errors will be raised for *any* (possibly exact) subnormal result. #define HALF_ERRHANDLING_UNDERFLOW_TO_INEXACT 1 #endif /// Default rounding mode. /// This specifies the rounding mode used for all conversions between [half](\ref half_float::half)s and more precise types /// (unless using half_cast() and specifying the rounding mode directly) as well as in arithmetic operations and mathematical /// functions. It can be redefined (before including half.hpp) to one of the standard rounding modes using their respective /// constants or the equivalent values of /// [std::float_round_style](https://en.cppreference.com/w/cpp/types/numeric_limits/float_round_style): /// /// `std::float_round_style` | value | rounding /// ---------------------------------|-------|------------------------- /// `std::round_indeterminate` | -1 | fastest /// `std::round_toward_zero` | 0 | toward zero /// `std::round_to_nearest` | 1 | to nearest (default) /// `std::round_toward_infinity` | 2 | toward positive infinity /// `std::round_toward_neg_infinity` | 3 | toward negative infinity /// /// By default this is set to `1` (`std::round_to_nearest`), which rounds results to the nearest representable value. It can even /// be set to [std::numeric_limits::round_style](https://en.cppreference.com/w/cpp/types/numeric_limits/round_style) to synchronize /// the rounding mode with that of the built-in single-precision implementation (which is likely `std::round_to_nearest`, though). #ifndef HALF_ROUND_STYLE #define HALF_ROUND_STYLE 1 // = std::round_to_nearest #endif /// Value signaling overflow. /// In correspondence with `HUGE_VAL[F|L]` from `` this symbol expands to a positive value signaling the overflow of an /// operation, in particular it just evaluates to positive infinity. /// /// **See also:** Documentation for [HUGE_VAL](https://en.cppreference.com/w/cpp/numeric/math/HUGE_VAL) #define HUGE_VALH std::numeric_limits::infinity() /// Fast half-precision fma function. /// This symbol is defined if the fma() function generally executes as fast as, or faster than, a separate /// half-precision multiplication followed by an addition, which is always the case. /// /// **See also:** Documentation for [FP_FAST_FMA](https://en.cppreference.com/w/cpp/numeric/math/fma) #define FP_FAST_FMAH 1 /// Half rounding mode. /// In correspondence with `FLT_ROUNDS` from `` this symbol expands to the rounding mode used for /// half-precision operations. It is an alias for [HALF_ROUND_STYLE](\ref HALF_ROUND_STYLE). /// /// **See also:** Documentation for [FLT_ROUNDS](https://en.cppreference.com/w/cpp/types/climits/FLT_ROUNDS) #define HLF_ROUNDS HALF_ROUND_STYLE #ifndef FP_ILOGB0 #define FP_ILOGB0 INT_MIN #endif #ifndef FP_ILOGBNAN #define FP_ILOGBNAN INT_MAX #endif #ifndef FP_SUBNORMAL #define FP_SUBNORMAL 0 #endif #ifndef FP_ZERO #define FP_ZERO 1 #endif #ifndef FP_NAN #define FP_NAN 2 #endif #ifndef FP_INFINITE #define FP_INFINITE 3 #endif #ifndef FP_NORMAL #define FP_NORMAL 4 #endif #if !HALF_ENABLE_CPP11_CFENV && !defined(FE_ALL_EXCEPT) #define FE_INVALID 0x10 #define FE_DIVBYZERO 0x08 #define FE_OVERFLOW 0x04 #define FE_UNDERFLOW 0x02 #define FE_INEXACT 0x01 #define FE_ALL_EXCEPT (FE_INVALID|FE_DIVBYZERO|FE_OVERFLOW|FE_UNDERFLOW|FE_INEXACT) #endif /// Main namespace for half-precision functionality. /// This namespace contains all the functionality provided by the library. namespace half_float { class half; #if HALF_ENABLE_CPP11_USER_LITERALS /// Library-defined half-precision literals. /// Import this namespace to enable half-precision floating-point literals: /// ~~~~{.cpp} /// using namespace half_float::literal; /// half_float::half = 4.2_h; /// ~~~~ namespace literal { half operator "" _h(long double); } #endif /// \internal /// \brief Implementation details. namespace detail { #if HALF_ENABLE_CPP11_TYPE_TRAITS /// Conditional type. template struct conditional : std::conditional {}; /// Helper for tag dispatching. template struct bool_type : std::integral_constant {}; using std::true_type; using std::false_type; /// Type traits for floating-point types. template struct is_float : std::is_floating_point {}; #else /// Conditional type. template struct conditional { typedef T type; }; template struct conditional { typedef F type; }; /// Helper for tag dispatching. template struct bool_type {}; typedef bool_type true_type; typedef bool_type false_type; /// Type traits for floating-point types. template struct is_float : false_type {}; template struct is_float : is_float {}; template struct is_float : is_float {}; template struct is_float : is_float {}; template<> struct is_float : true_type {}; template<> struct is_float : true_type {}; template<> struct is_float : true_type {}; #endif /// Type traits for floating-point bits. template struct bits { typedef unsigned char type; }; template struct bits : bits {}; template struct bits : bits {}; template struct bits : bits {}; #if HALF_ENABLE_CPP11_CSTDINT /// Unsigned integer of (at least) 16 bits width. typedef std::uint_least16_t uint16; /// Fastest unsigned integer of (at least) 32 bits width. typedef std::uint_fast32_t uint32; /// Fastest signed integer of (at least) 32 bits width. typedef std::int_fast32_t int32; /// Unsigned integer of (at least) 32 bits width. template<> struct bits { typedef std::uint_least32_t type; }; /// Unsigned integer of (at least) 64 bits width. template<> struct bits { typedef std::uint_least64_t type; }; #else /// Unsigned integer of (at least) 16 bits width. typedef unsigned short uint16; /// Fastest unsigned integer of (at least) 32 bits width. typedef unsigned long uint32; /// Fastest unsigned integer of (at least) 32 bits width. typedef long int32; /// Unsigned integer of (at least) 32 bits width. template<> struct bits : conditional::digits>=32,unsigned int,unsigned long> {}; #if HALF_ENABLE_CPP11_LONG_LONG /// Unsigned integer of (at least) 64 bits width. template<> struct bits : conditional::digits>=64,unsigned long,unsigned long long> {}; #else /// Unsigned integer of (at least) 64 bits width. template<> struct bits { typedef unsigned long type; }; #endif #endif #ifdef HALF_ARITHMETIC_TYPE /// Type to use for arithmetic computations and mathematic functions internally. typedef HALF_ARITHMETIC_TYPE internal_t; #endif /// Tag type for binary construction. struct binary_t {}; /// Tag for binary construction. HALF_CONSTEXPR_CONST binary_t binary = binary_t(); /// \name Implementation defined classification and arithmetic /// \{ /// Check for infinity. /// \tparam T argument type (builtin floating-point type) /// \param arg value to query /// \retval true if infinity /// \retval false else template bool builtin_isinf(T arg) { #if HALF_ENABLE_CPP11_CMATH return std::isinf(arg); #elif defined(_MSC_VER) return !::_finite(static_cast(arg)) && !::_isnan(static_cast(arg)); #else return arg == std::numeric_limits::infinity() || arg == -std::numeric_limits::infinity(); #endif } /// Check for NaN. /// \tparam T argument type (builtin floating-point type) /// \param arg value to query /// \retval true if not a number /// \retval false else template bool builtin_isnan(T arg) { #if HALF_ENABLE_CPP11_CMATH return std::isnan(arg); #elif defined(_MSC_VER) return ::_isnan(static_cast(arg)) != 0; #else return arg != arg; #endif } /// Check sign. /// \tparam T argument type (builtin floating-point type) /// \param arg value to query /// \retval true if signbit set /// \retval false else template bool builtin_signbit(T arg) { #if HALF_ENABLE_CPP11_CMATH return std::signbit(arg); #else return arg < T() || (arg == T() && T(1)/arg < T()); #endif } /// Platform-independent sign mask. /// \param arg integer value in two's complement /// \retval -1 if \a arg negative /// \retval 0 if \a arg positive inline uint32 sign_mask(uint32 arg) { static const int N = std::numeric_limits::digits - 1; #if HALF_TWOS_COMPLEMENT_INT return static_cast(arg) >> N; #else return -((arg>>N)&1); #endif } /// Platform-independent arithmetic right shift. /// \param arg integer value in two's complement /// \param i shift amount (at most 31) /// \return \a arg right shifted for \a i bits with possible sign extension inline uint32 arithmetic_shift(uint32 arg, int i) { #if HALF_TWOS_COMPLEMENT_INT return static_cast(arg) >> i; #else return static_cast(arg)/(static_cast(1)<>(std::numeric_limits::digits-1))&1); #endif } /// \} /// \name Error handling /// \{ /// Internal exception flags. /// \return reference to global exception flags inline int& errflags() { HALF_THREAD_LOCAL int flags = 0; return flags; } /// Raise floating-point exception. /// \param flags exceptions to raise /// \param cond condition to raise exceptions for inline void raise(int HALF_UNUSED_NOERR(flags), bool HALF_UNUSED_NOERR(cond) = true) { #if HALF_ERRHANDLING if(!cond) return; #if HALF_ERRHANDLING_FLAGS errflags() |= flags; #endif #if HALF_ERRHANDLING_ERRNO if(flags & FE_INVALID) errno = EDOM; else if(flags & (FE_DIVBYZERO|FE_OVERFLOW|FE_UNDERFLOW)) errno = ERANGE; #endif #if HALF_ERRHANDLING_FENV && HALF_ENABLE_CPP11_CFENV std::feraiseexcept(flags); #endif #ifdef HALF_ERRHANDLING_THROW_INVALID if(flags & FE_INVALID) throw std::domain_error(HALF_ERRHANDLING_THROW_INVALID); #endif #ifdef HALF_ERRHANDLING_THROW_DIVBYZERO if(flags & FE_DIVBYZERO) throw std::domain_error(HALF_ERRHANDLING_THROW_DIVBYZERO); #endif #ifdef HALF_ERRHANDLING_THROW_OVERFLOW if(flags & FE_OVERFLOW) throw std::overflow_error(HALF_ERRHANDLING_THROW_OVERFLOW); #endif #ifdef HALF_ERRHANDLING_THROW_UNDERFLOW if(flags & FE_UNDERFLOW) throw std::underflow_error(HALF_ERRHANDLING_THROW_UNDERFLOW); #endif #ifdef HALF_ERRHANDLING_THROW_INEXACT if(flags & FE_INEXACT) throw std::range_error(HALF_ERRHANDLING_THROW_INEXACT); #endif #if HALF_ERRHANDLING_UNDERFLOW_TO_INEXACT if((flags & FE_UNDERFLOW) && !(flags & FE_INEXACT)) raise(FE_INEXACT); #endif #if HALF_ERRHANDLING_OVERFLOW_TO_INEXACT if((flags & FE_OVERFLOW) && !(flags & FE_INEXACT)) raise(FE_INEXACT); #endif #endif } /// Check and signal for any NaN. /// \param x first half-precision value to check /// \param y second half-precision value to check /// \retval true if either \a x or \a y is NaN /// \retval false else /// \exception FE_INVALID if \a x or \a y is NaN inline HALF_CONSTEXPR_NOERR bool compsignal(unsigned int x, unsigned int y) { #if HALF_ERRHANDLING raise(FE_INVALID, (x&0x7FFF)>0x7C00 || (y&0x7FFF)>0x7C00); #endif return (x&0x7FFF) > 0x7C00 || (y&0x7FFF) > 0x7C00; } /// Signal and silence signaling NaN. /// \param nan half-precision NaN value /// \return quiet NaN /// \exception FE_INVALID if \a nan is signaling NaN inline HALF_CONSTEXPR_NOERR unsigned int signal(unsigned int nan) { #if HALF_ERRHANDLING raise(FE_INVALID, !(nan&0x200)); #endif return nan | 0x200; } /// Signal and silence signaling NaNs. /// \param x first half-precision value to check /// \param y second half-precision value to check /// \return quiet NaN /// \exception FE_INVALID if \a x or \a y is signaling NaN inline HALF_CONSTEXPR_NOERR unsigned int signal(unsigned int x, unsigned int y) { #if HALF_ERRHANDLING raise(FE_INVALID, ((x&0x7FFF)>0x7C00 && !(x&0x200)) || ((y&0x7FFF)>0x7C00 && !(y&0x200))); #endif return ((x&0x7FFF)>0x7C00) ? (x|0x200) : (y|0x200); } /// Signal and silence signaling NaNs. /// \param x first half-precision value to check /// \param y second half-precision value to check /// \param z third half-precision value to check /// \return quiet NaN /// \exception FE_INVALID if \a x, \a y or \a z is signaling NaN inline HALF_CONSTEXPR_NOERR unsigned int signal(unsigned int x, unsigned int y, unsigned int z) { #if HALF_ERRHANDLING raise(FE_INVALID, ((x&0x7FFF)>0x7C00 && !(x&0x200)) || ((y&0x7FFF)>0x7C00 && !(y&0x200)) || ((z&0x7FFF)>0x7C00 && !(z&0x200))); #endif return ((x&0x7FFF)>0x7C00) ? (x|0x200) : ((y&0x7FFF)>0x7C00) ? (y|0x200) : (z|0x200); } /// Select value or signaling NaN. /// \param x preferred half-precision value /// \param y ignored half-precision value except for signaling NaN /// \return \a y if signaling NaN, \a x otherwise /// \exception FE_INVALID if \a y is signaling NaN inline HALF_CONSTEXPR_NOERR unsigned int select(unsigned int x, unsigned int HALF_UNUSED_NOERR(y)) { #if HALF_ERRHANDLING return (((y&0x7FFF)>0x7C00) && !(y&0x200)) ? signal(y) : x; #else return x; #endif } /// Raise domain error and return NaN. /// return quiet NaN /// \exception FE_INVALID inline HALF_CONSTEXPR_NOERR unsigned int invalid() { #if HALF_ERRHANDLING raise(FE_INVALID); #endif return 0x7FFF; } /// Raise pole error and return infinity. /// \param sign half-precision value with sign bit only /// \return half-precision infinity with sign of \a sign /// \exception FE_DIVBYZERO inline HALF_CONSTEXPR_NOERR unsigned int pole(unsigned int sign = 0) { #if HALF_ERRHANDLING raise(FE_DIVBYZERO); #endif return sign | 0x7C00; } /// Check value for underflow. /// \param arg non-zero half-precision value to check /// \return \a arg /// \exception FE_UNDERFLOW if arg is subnormal inline HALF_CONSTEXPR_NOERR unsigned int check_underflow(unsigned int arg) { #if HALF_ERRHANDLING && !HALF_ERRHANDLING_UNDERFLOW_TO_INEXACT raise(FE_UNDERFLOW, !(arg&0x7C00)); #endif return arg; } /// \} /// \name Conversion and rounding /// \{ /// Half-precision overflow. /// \tparam R rounding mode to use /// \param sign half-precision value with sign bit only /// \return rounded overflowing half-precision value /// \exception FE_OVERFLOW template HALF_CONSTEXPR_NOERR unsigned int overflow(unsigned int sign = 0) { #if HALF_ERRHANDLING raise(FE_OVERFLOW); #endif return (R==std::round_toward_infinity) ? (sign+0x7C00-(sign>>15)) : (R==std::round_toward_neg_infinity) ? (sign+0x7BFF+(sign>>15)) : (R==std::round_toward_zero) ? (sign|0x7BFF) : (sign|0x7C00); } /// Half-precision underflow. /// \tparam R rounding mode to use /// \param sign half-precision value with sign bit only /// \return rounded underflowing half-precision value /// \exception FE_UNDERFLOW template HALF_CONSTEXPR_NOERR unsigned int underflow(unsigned int sign = 0) { #if HALF_ERRHANDLING raise(FE_UNDERFLOW); #endif return (R==std::round_toward_infinity) ? (sign+1-(sign>>15)) : (R==std::round_toward_neg_infinity) ? (sign+(sign>>15)) : sign; } /// Round half-precision number. /// \tparam R rounding mode to use /// \tparam I `true` to always raise INEXACT exception, `false` to raise only for rounded results /// \param value finite half-precision number to round /// \param g guard bit (most significant discarded bit) /// \param s sticky bit (or of all but the most significant discarded bits) /// \return rounded half-precision value /// \exception FE_OVERFLOW on overflows /// \exception FE_UNDERFLOW on underflows /// \exception FE_INEXACT if value had to be rounded or \a I is `true` template HALF_CONSTEXPR_NOERR unsigned int rounded(unsigned int value, int g, int s) { #if HALF_ERRHANDLING value += (R==std::round_to_nearest) ? (g&(s|value)) : (R==std::round_toward_infinity) ? (~(value>>15)&(g|s)) : (R==std::round_toward_neg_infinity) ? ((value>>15)&(g|s)) : 0; if((value&0x7C00) == 0x7C00) raise(FE_OVERFLOW); else if(value & 0x7C00) raise(FE_INEXACT, I || (g|s)!=0); else raise(FE_UNDERFLOW, !(HALF_ERRHANDLING_UNDERFLOW_TO_INEXACT) || I || (g|s)!=0); return value; #else return (R==std::round_to_nearest) ? (value+(g&(s|value))) : (R==std::round_toward_infinity) ? (value+(~(value>>15)&(g|s))) : (R==std::round_toward_neg_infinity) ? (value+((value>>15)&(g|s))) : value; #endif } /// Round half-precision number to nearest integer value. /// \tparam R rounding mode to use /// \tparam E `true` for round to even, `false` for round away from zero /// \tparam I `true` to raise INEXACT exception (if inexact), `false` to never raise it /// \param value half-precision value to round /// \return half-precision bits for nearest integral value /// \exception FE_INVALID for signaling NaN /// \exception FE_INEXACT if value had to be rounded and \a I is `true` template unsigned int integral(unsigned int value) { unsigned int abs = value & 0x7FFF; if(abs < 0x3C00) { raise(FE_INEXACT, I); return ((R==std::round_to_nearest) ? (0x3C00&-static_cast(abs>=(0x3800+E))) : (R==std::round_toward_infinity) ? (0x3C00&-(~(value>>15)&(abs!=0))) : (R==std::round_toward_neg_infinity) ? (0x3C00&-static_cast(value>0x8000)) : 0) | (value&0x8000); } if(abs >= 0x6400) return (abs>0x7C00) ? signal(value) : value; unsigned int exp = 25 - (abs>>10), mask = (1<>exp)&E)) : (R==std::round_toward_infinity) ? (mask&((value>>15)-1)) : (R==std::round_toward_neg_infinity) ? (mask&-(value>>15)) : 0) + value) & ~mask; } /// Convert fixed point to half-precision floating-point. /// \tparam R rounding mode to use /// \tparam F number of fractional bits in [11,31] /// \tparam S `true` for signed, `false` for unsigned /// \tparam N `true` for additional normalization step, `false` if already normalized to 1.F /// \tparam I `true` to always raise INEXACT exception, `false` to raise only for rounded results /// \param m mantissa in Q1.F fixed point format /// \param exp biased exponent - 1 /// \param sign half-precision value with sign bit only /// \param s sticky bit (or of all but the most significant already discarded bits) /// \return value converted to half-precision /// \exception FE_OVERFLOW on overflows /// \exception FE_UNDERFLOW on underflows /// \exception FE_INEXACT if value had to be rounded or \a I is `true` template unsigned int fixed2half(uint32 m, int exp = 14, unsigned int sign = 0, int s = 0) { if(S) { uint32 msign = sign_mask(m); m = (m^msign) - msign; sign = msign & 0x8000; } if(N) for(; m<(static_cast(1)<(sign+(m>>(F-10-exp)), (m>>(F-11-exp))&1, s|((m&((static_cast(1)<<(F-11-exp))-1))!=0)); return rounded(sign+(exp<<10)+(m>>(F-10)), (m>>(F-11))&1, s|((m&((static_cast(1)<<(F-11))-1))!=0)); } /// Convert IEEE single-precision to half-precision. /// Credit for this goes to [Jeroen van der Zijp](ftp://ftp.fox-toolkit.org/pub/fasthalffloatconversion.pdf). /// \tparam R rounding mode to use /// \param value single-precision value to convert /// \return rounded half-precision value /// \exception FE_OVERFLOW on overflows /// \exception FE_UNDERFLOW on underflows /// \exception FE_INEXACT if value had to be rounded template unsigned int float2half_impl(float value, true_type) { #if HALF_ENABLE_F16C_INTRINSICS return _mm_cvtsi128_si32(_mm_cvtps_ph(_mm_set_ss(value), (R==std::round_to_nearest) ? _MM_FROUND_TO_NEAREST_INT : (R==std::round_toward_zero) ? _MM_FROUND_TO_ZERO : (R==std::round_toward_infinity) ? _MM_FROUND_TO_POS_INF : (R==std::round_toward_neg_infinity) ? _MM_FROUND_TO_NEG_INF : _MM_FROUND_CUR_DIRECTION)); #else bits::type fbits; std::memcpy(&fbits, &value, sizeof(float)); #if 1 unsigned int sign = (fbits>>16) & 0x8000; fbits &= 0x7FFFFFFF; if(fbits >= 0x7F800000) return sign | 0x7C00 | ((fbits>0x7F800000) ? (0x200|((fbits>>13)&0x3FF)) : 0); if(fbits >= 0x47800000) return overflow(sign); if(fbits >= 0x38800000) return rounded(sign|(((fbits>>23)-112)<<10)|((fbits>>13)&0x3FF), (fbits>>12)&1, (fbits&0xFFF)!=0); if(fbits >= 0x33000000) { int i = 125 - (fbits>>23); fbits = (fbits&0x7FFFFF) | 0x800000; return rounded(sign|(fbits>>(i+1)), (fbits>>i)&1, (fbits&((static_cast(1)<(sign); return sign; #else static const uint16 base_table[512] = { 0x0000, 0x0000, 0x0000, 0x0000, 0x0000, 0x0000, 0x0000, 0x0000, 0x0000, 0x0000, 0x0000, 0x0000, 0x0000, 0x0000, 0x0000, 0x0000, 0x0000, 0x0000, 0x0000, 0x0000, 0x0000, 0x0000, 0x0000, 0x0000, 0x0000, 0x0000, 0x0000, 0x0000, 0x0000, 0x0000, 0x0000, 0x0000, 0x0000, 0x0000, 0x0000, 0x0000, 0x0000, 0x0000, 0x0000, 0x0000, 0x0000, 0x0000, 0x0000, 0x0000, 0x0000, 0x0000, 0x0000, 0x0000, 0x0000, 0x0000, 0x0000, 0x0000, 0x0000, 0x0000, 0x0000, 0x0000, 0x0000, 0x0000, 0x0000, 0x0000, 0x0000, 0x0000, 0x0000, 0x0000, 0x0000, 0x0000, 0x0000, 0x0000, 0x0000, 0x0000, 0x0000, 0x0000, 0x0000, 0x0000, 0x0000, 0x0000, 0x0000, 0x0000, 0x0000, 0x0000, 0x0000, 0x0000, 0x0000, 0x0000, 0x0000, 0x0000, 0x0000, 0x0000, 0x0000, 0x0000, 0x0000, 0x0000, 0x0000, 0x0000, 0x0000, 0x0000, 0x0000, 0x0000, 0x0000, 0x0000, 0x0000, 0x0000, 0x0000, 0x0001, 0x0002, 0x0004, 0x0008, 0x0010, 0x0020, 0x0040, 0x0080, 0x0100, 0x0200, 0x0400, 0x0800, 0x0C00, 0x1000, 0x1400, 0x1800, 0x1C00, 0x2000, 0x2400, 0x2800, 0x2C00, 0x3000, 0x3400, 0x3800, 0x3C00, 0x4000, 0x4400, 0x4800, 0x4C00, 0x5000, 0x5400, 0x5800, 0x5C00, 0x6000, 0x6400, 0x6800, 0x6C00, 0x7000, 0x7400, 0x7800, 0x7BFF, 0x7BFF, 0x7BFF, 0x7BFF, 0x7BFF, 0x7BFF, 0x7BFF, 0x7BFF, 0x7BFF, 0x7BFF, 0x7BFF, 0x7BFF, 0x7BFF, 0x7BFF, 0x7BFF, 0x7BFF, 0x7BFF, 0x7BFF, 0x7BFF, 0x7BFF, 0x7BFF, 0x7BFF, 0x7BFF, 0x7BFF, 0x7BFF, 0x7BFF, 0x7BFF, 0x7BFF, 0x7BFF, 0x7BFF, 0x7BFF, 0x7BFF, 0x7BFF, 0x7BFF, 0x7BFF, 0x7BFF, 0x7BFF, 0x7BFF, 0x7BFF, 0x7BFF, 0x7BFF, 0x7BFF, 0x7BFF, 0x7BFF, 0x7BFF, 0x7BFF, 0x7BFF, 0x7BFF, 0x7BFF, 0x7BFF, 0x7BFF, 0x7BFF, 0x7BFF, 0x7BFF, 0x7BFF, 0x7BFF, 0x7BFF, 0x7BFF, 0x7BFF, 0x7BFF, 0x7BFF, 0x7BFF, 0x7BFF, 0x7BFF, 0x7BFF, 0x7BFF, 0x7BFF, 0x7BFF, 0x7BFF, 0x7BFF, 0x7BFF, 0x7BFF, 0x7BFF, 0x7BFF, 0x7BFF, 0x7BFF, 0x7BFF, 0x7BFF, 0x7BFF, 0x7BFF, 0x7BFF, 0x7BFF, 0x7BFF, 0x7BFF, 0x7BFF, 0x7BFF, 0x7BFF, 0x7BFF, 0x7BFF, 0x7BFF, 0x7BFF, 0x7BFF, 0x7BFF, 0x7BFF, 0x7BFF, 0x7BFF, 0x7BFF, 0x7BFF, 0x7BFF, 0x7BFF, 0x7BFF, 0x7BFF, 0x7BFF, 0x7BFF, 0x7BFF, 0x7BFF, 0x7BFF, 0x7BFF, 0x7BFF, 0x7BFF, 0x7BFF, 0x7BFF, 0x7C00, 0x8000, 0x8000, 0x8000, 0x8000, 0x8000, 0x8000, 0x8000, 0x8000, 0x8000, 0x8000, 0x8000, 0x8000, 0x8000, 0x8000, 0x8000, 0x8000, 0x8000, 0x8000, 0x8000, 0x8000, 0x8000, 0x8000, 0x8000, 0x8000, 0x8000, 0x8000, 0x8000, 0x8000, 0x8000, 0x8000, 0x8000, 0x8000, 0x8000, 0x8000, 0x8000, 0x8000, 0x8000, 0x8000, 0x8000, 0x8000, 0x8000, 0x8000, 0x8000, 0x8000, 0x8000, 0x8000, 0x8000, 0x8000, 0x8000, 0x8000, 0x8000, 0x8000, 0x8000, 0x8000, 0x8000, 0x8000, 0x8000, 0x8000, 0x8000, 0x8000, 0x8000, 0x8000, 0x8000, 0x8000, 0x8000, 0x8000, 0x8000, 0x8000, 0x8000, 0x8000, 0x8000, 0x8000, 0x8000, 0x8000, 0x8000, 0x8000, 0x8000, 0x8000, 0x8000, 0x8000, 0x8000, 0x8000, 0x8000, 0x8000, 0x8000, 0x8000, 0x8000, 0x8000, 0x8000, 0x8000, 0x8000, 0x8000, 0x8000, 0x8000, 0x8000, 0x8000, 0x8000, 0x8000, 0x8000, 0x8000, 0x8000, 0x8000, 0x8000, 0x8001, 0x8002, 0x8004, 0x8008, 0x8010, 0x8020, 0x8040, 0x8080, 0x8100, 0x8200, 0x8400, 0x8800, 0x8C00, 0x9000, 0x9400, 0x9800, 0x9C00, 0xA000, 0xA400, 0xA800, 0xAC00, 0xB000, 0xB400, 0xB800, 0xBC00, 0xC000, 0xC400, 0xC800, 0xCC00, 0xD000, 0xD400, 0xD800, 0xDC00, 0xE000, 0xE400, 0xE800, 0xEC00, 0xF000, 0xF400, 0xF800, 0xFBFF, 0xFBFF, 0xFBFF, 0xFBFF, 0xFBFF, 0xFBFF, 0xFBFF, 0xFBFF, 0xFBFF, 0xFBFF, 0xFBFF, 0xFBFF, 0xFBFF, 0xFBFF, 0xFBFF, 0xFBFF, 0xFBFF, 0xFBFF, 0xFBFF, 0xFBFF, 0xFBFF, 0xFBFF, 0xFBFF, 0xFBFF, 0xFBFF, 0xFBFF, 0xFBFF, 0xFBFF, 0xFBFF, 0xFBFF, 0xFBFF, 0xFBFF, 0xFBFF, 0xFBFF, 0xFBFF, 0xFBFF, 0xFBFF, 0xFBFF, 0xFBFF, 0xFBFF, 0xFBFF, 0xFBFF, 0xFBFF, 0xFBFF, 0xFBFF, 0xFBFF, 0xFBFF, 0xFBFF, 0xFBFF, 0xFBFF, 0xFBFF, 0xFBFF, 0xFBFF, 0xFBFF, 0xFBFF, 0xFBFF, 0xFBFF, 0xFBFF, 0xFBFF, 0xFBFF, 0xFBFF, 0xFBFF, 0xFBFF, 0xFBFF, 0xFBFF, 0xFBFF, 0xFBFF, 0xFBFF, 0xFBFF, 0xFBFF, 0xFBFF, 0xFBFF, 0xFBFF, 0xFBFF, 0xFBFF, 0xFBFF, 0xFBFF, 0xFBFF, 0xFBFF, 0xFBFF, 0xFBFF, 0xFBFF, 0xFBFF, 0xFBFF, 0xFBFF, 0xFBFF, 0xFBFF, 0xFBFF, 0xFBFF, 0xFBFF, 0xFBFF, 0xFBFF, 0xFBFF, 0xFBFF, 0xFBFF, 0xFBFF, 0xFBFF, 0xFBFF, 0xFBFF, 0xFBFF, 0xFBFF, 0xFBFF, 0xFBFF, 0xFBFF, 0xFBFF, 0xFBFF, 0xFBFF, 0xFBFF, 0xFBFF, 0xFBFF, 0xFBFF, 0xFBFF, 0xFC00 }; static const unsigned char shift_table[256] = { 24, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 13 }; int sexp = fbits >> 23, exp = sexp & 0xFF, i = shift_table[exp]; fbits &= 0x7FFFFF; uint32 m = (fbits|((exp!=0)<<23)) & -static_cast(exp!=0xFF); return rounded(base_table[sexp]+(fbits>>i), (m>>(i-1))&1, (((static_cast(1)<<(i-1))-1)&m)!=0); #endif #endif } /// Convert IEEE double-precision to half-precision. /// \tparam R rounding mode to use /// \param value double-precision value to convert /// \return rounded half-precision value /// \exception FE_OVERFLOW on overflows /// \exception FE_UNDERFLOW on underflows /// \exception FE_INEXACT if value had to be rounded template unsigned int float2half_impl(double value, true_type) { #if HALF_ENABLE_F16C_INTRINSICS if(R == std::round_indeterminate) return _mm_cvtsi128_si32(_mm_cvtps_ph(_mm_cvtpd_ps(_mm_set_sd(value)), _MM_FROUND_CUR_DIRECTION)); #endif bits::type dbits; std::memcpy(&dbits, &value, sizeof(double)); uint32 hi = dbits >> 32, lo = dbits & 0xFFFFFFFF; unsigned int sign = (hi>>16) & 0x8000; hi &= 0x7FFFFFFF; if(hi >= 0x7FF00000) return sign | 0x7C00 | ((dbits&0xFFFFFFFFFFFFF) ? (0x200|((hi>>10)&0x3FF)) : 0); if(hi >= 0x40F00000) return overflow(sign); if(hi >= 0x3F100000) return rounded(sign|(((hi>>20)-1008)<<10)|((hi>>10)&0x3FF), (hi>>9)&1, ((hi&0x1FF)|lo)!=0); if(hi >= 0x3E600000) { int i = 1018 - (hi>>20); hi = (hi&0xFFFFF) | 0x100000; return rounded(sign|(hi>>(i+1)), (hi>>i)&1, ((hi&((static_cast(1)<(sign); return sign; } /// Convert non-IEEE floating-point to half-precision. /// \tparam R rounding mode to use /// \tparam T source type (builtin floating-point type) /// \param value floating-point value to convert /// \return rounded half-precision value /// \exception FE_OVERFLOW on overflows /// \exception FE_UNDERFLOW on underflows /// \exception FE_INEXACT if value had to be rounded template unsigned int float2half_impl(T value, ...) { unsigned int hbits = static_cast(builtin_signbit(value)) << 15; if(value == T()) return hbits; if(builtin_isnan(value)) return hbits | 0x7FFF; if(builtin_isinf(value)) return hbits | 0x7C00; int exp; std::frexp(value, &exp); if(exp > 16) return overflow(hbits); if(exp < -13) value = std::ldexp(value, 25); else { value = std::ldexp(value, 12-exp); hbits |= ((exp+13)<<10); } T ival, frac = std::modf(value, &ival); int m = std::abs(static_cast(ival)); return rounded(hbits+(m>>1), m&1, frac!=T()); } /// Convert floating-point to half-precision. /// \tparam R rounding mode to use /// \tparam T source type (builtin floating-point type) /// \param value floating-point value to convert /// \return rounded half-precision value /// \exception FE_OVERFLOW on overflows /// \exception FE_UNDERFLOW on underflows /// \exception FE_INEXACT if value had to be rounded template unsigned int float2half(T value) { return float2half_impl(value, bool_type::is_iec559&&sizeof(typename bits::type)==sizeof(T)>()); } /// Convert integer to half-precision floating-point. /// \tparam R rounding mode to use /// \tparam T type to convert (builtin integer type) /// \param value integral value to convert /// \return rounded half-precision value /// \exception FE_OVERFLOW on overflows /// \exception FE_INEXACT if value had to be rounded template unsigned int int2half(T value) { unsigned int bits = static_cast(value<0) << 15; if(!value) return bits; if(bits) value = -value; if(value > 0xFFFF) return overflow(bits); unsigned int m = static_cast(value), exp = 24; for(; m<0x400; m<<=1,--exp) ; for(; m>0x7FF; m>>=1,++exp) ; bits |= (exp<<10) + m; return (exp>24) ? rounded(bits, (value>>(exp-25))&1, (((1<<(exp-25))-1)&value)!=0) : bits; } /// Convert half-precision to IEEE single-precision. /// Credit for this goes to [Jeroen van der Zijp](ftp://ftp.fox-toolkit.org/pub/fasthalffloatconversion.pdf). /// \param value half-precision value to convert /// \return single-precision value inline float half2float_impl(unsigned int value, float, true_type) { #if HALF_ENABLE_F16C_INTRINSICS return _mm_cvtss_f32(_mm_cvtph_ps(_mm_cvtsi32_si128(value))); #else #if 0 bits::type fbits = static_cast::type>(value&0x8000) << 16; int abs = value & 0x7FFF; if(abs) { fbits |= 0x38000000 << static_cast(abs>=0x7C00); for(; abs<0x400; abs<<=1,fbits-=0x800000) ; fbits += static_cast::type>(abs) << 13; } #else static const bits::type mantissa_table[2048] = { 0x00000000, 0x33800000, 0x34000000, 0x34400000, 0x34800000, 0x34A00000, 0x34C00000, 0x34E00000, 0x35000000, 0x35100000, 0x35200000, 0x35300000, 0x35400000, 0x35500000, 0x35600000, 0x35700000, 0x35800000, 0x35880000, 0x35900000, 0x35980000, 0x35A00000, 0x35A80000, 0x35B00000, 0x35B80000, 0x35C00000, 0x35C80000, 0x35D00000, 0x35D80000, 0x35E00000, 0x35E80000, 0x35F00000, 0x35F80000, 0x36000000, 0x36040000, 0x36080000, 0x360C0000, 0x36100000, 0x36140000, 0x36180000, 0x361C0000, 0x36200000, 0x36240000, 0x36280000, 0x362C0000, 0x36300000, 0x36340000, 0x36380000, 0x363C0000, 0x36400000, 0x36440000, 0x36480000, 0x364C0000, 0x36500000, 0x36540000, 0x36580000, 0x365C0000, 0x36600000, 0x36640000, 0x36680000, 0x366C0000, 0x36700000, 0x36740000, 0x36780000, 0x367C0000, 0x36800000, 0x36820000, 0x36840000, 0x36860000, 0x36880000, 0x368A0000, 0x368C0000, 0x368E0000, 0x36900000, 0x36920000, 0x36940000, 0x36960000, 0x36980000, 0x369A0000, 0x369C0000, 0x369E0000, 0x36A00000, 0x36A20000, 0x36A40000, 0x36A60000, 0x36A80000, 0x36AA0000, 0x36AC0000, 0x36AE0000, 0x36B00000, 0x36B20000, 0x36B40000, 0x36B60000, 0x36B80000, 0x36BA0000, 0x36BC0000, 0x36BE0000, 0x36C00000, 0x36C20000, 0x36C40000, 0x36C60000, 0x36C80000, 0x36CA0000, 0x36CC0000, 0x36CE0000, 0x36D00000, 0x36D20000, 0x36D40000, 0x36D60000, 0x36D80000, 0x36DA0000, 0x36DC0000, 0x36DE0000, 0x36E00000, 0x36E20000, 0x36E40000, 0x36E60000, 0x36E80000, 0x36EA0000, 0x36EC0000, 0x36EE0000, 0x36F00000, 0x36F20000, 0x36F40000, 0x36F60000, 0x36F80000, 0x36FA0000, 0x36FC0000, 0x36FE0000, 0x37000000, 0x37010000, 0x37020000, 0x37030000, 0x37040000, 0x37050000, 0x37060000, 0x37070000, 0x37080000, 0x37090000, 0x370A0000, 0x370B0000, 0x370C0000, 0x370D0000, 0x370E0000, 0x370F0000, 0x37100000, 0x37110000, 0x37120000, 0x37130000, 0x37140000, 0x37150000, 0x37160000, 0x37170000, 0x37180000, 0x37190000, 0x371A0000, 0x371B0000, 0x371C0000, 0x371D0000, 0x371E0000, 0x371F0000, 0x37200000, 0x37210000, 0x37220000, 0x37230000, 0x37240000, 0x37250000, 0x37260000, 0x37270000, 0x37280000, 0x37290000, 0x372A0000, 0x372B0000, 0x372C0000, 0x372D0000, 0x372E0000, 0x372F0000, 0x37300000, 0x37310000, 0x37320000, 0x37330000, 0x37340000, 0x37350000, 0x37360000, 0x37370000, 0x37380000, 0x37390000, 0x373A0000, 0x373B0000, 0x373C0000, 0x373D0000, 0x373E0000, 0x373F0000, 0x37400000, 0x37410000, 0x37420000, 0x37430000, 0x37440000, 0x37450000, 0x37460000, 0x37470000, 0x37480000, 0x37490000, 0x374A0000, 0x374B0000, 0x374C0000, 0x374D0000, 0x374E0000, 0x374F0000, 0x37500000, 0x37510000, 0x37520000, 0x37530000, 0x37540000, 0x37550000, 0x37560000, 0x37570000, 0x37580000, 0x37590000, 0x375A0000, 0x375B0000, 0x375C0000, 0x375D0000, 0x375E0000, 0x375F0000, 0x37600000, 0x37610000, 0x37620000, 0x37630000, 0x37640000, 0x37650000, 0x37660000, 0x37670000, 0x37680000, 0x37690000, 0x376A0000, 0x376B0000, 0x376C0000, 0x376D0000, 0x376E0000, 0x376F0000, 0x37700000, 0x37710000, 0x37720000, 0x37730000, 0x37740000, 0x37750000, 0x37760000, 0x37770000, 0x37780000, 0x37790000, 0x377A0000, 0x377B0000, 0x377C0000, 0x377D0000, 0x377E0000, 0x377F0000, 0x37800000, 0x37808000, 0x37810000, 0x37818000, 0x37820000, 0x37828000, 0x37830000, 0x37838000, 0x37840000, 0x37848000, 0x37850000, 0x37858000, 0x37860000, 0x37868000, 0x37870000, 0x37878000, 0x37880000, 0x37888000, 0x37890000, 0x37898000, 0x378A0000, 0x378A8000, 0x378B0000, 0x378B8000, 0x378C0000, 0x378C8000, 0x378D0000, 0x378D8000, 0x378E0000, 0x378E8000, 0x378F0000, 0x378F8000, 0x37900000, 0x37908000, 0x37910000, 0x37918000, 0x37920000, 0x37928000, 0x37930000, 0x37938000, 0x37940000, 0x37948000, 0x37950000, 0x37958000, 0x37960000, 0x37968000, 0x37970000, 0x37978000, 0x37980000, 0x37988000, 0x37990000, 0x37998000, 0x379A0000, 0x379A8000, 0x379B0000, 0x379B8000, 0x379C0000, 0x379C8000, 0x379D0000, 0x379D8000, 0x379E0000, 0x379E8000, 0x379F0000, 0x379F8000, 0x37A00000, 0x37A08000, 0x37A10000, 0x37A18000, 0x37A20000, 0x37A28000, 0x37A30000, 0x37A38000, 0x37A40000, 0x37A48000, 0x37A50000, 0x37A58000, 0x37A60000, 0x37A68000, 0x37A70000, 0x37A78000, 0x37A80000, 0x37A88000, 0x37A90000, 0x37A98000, 0x37AA0000, 0x37AA8000, 0x37AB0000, 0x37AB8000, 0x37AC0000, 0x37AC8000, 0x37AD0000, 0x37AD8000, 0x37AE0000, 0x37AE8000, 0x37AF0000, 0x37AF8000, 0x37B00000, 0x37B08000, 0x37B10000, 0x37B18000, 0x37B20000, 0x37B28000, 0x37B30000, 0x37B38000, 0x37B40000, 0x37B48000, 0x37B50000, 0x37B58000, 0x37B60000, 0x37B68000, 0x37B70000, 0x37B78000, 0x37B80000, 0x37B88000, 0x37B90000, 0x37B98000, 0x37BA0000, 0x37BA8000, 0x37BB0000, 0x37BB8000, 0x37BC0000, 0x37BC8000, 0x37BD0000, 0x37BD8000, 0x37BE0000, 0x37BE8000, 0x37BF0000, 0x37BF8000, 0x37C00000, 0x37C08000, 0x37C10000, 0x37C18000, 0x37C20000, 0x37C28000, 0x37C30000, 0x37C38000, 0x37C40000, 0x37C48000, 0x37C50000, 0x37C58000, 0x37C60000, 0x37C68000, 0x37C70000, 0x37C78000, 0x37C80000, 0x37C88000, 0x37C90000, 0x37C98000, 0x37CA0000, 0x37CA8000, 0x37CB0000, 0x37CB8000, 0x37CC0000, 0x37CC8000, 0x37CD0000, 0x37CD8000, 0x37CE0000, 0x37CE8000, 0x37CF0000, 0x37CF8000, 0x37D00000, 0x37D08000, 0x37D10000, 0x37D18000, 0x37D20000, 0x37D28000, 0x37D30000, 0x37D38000, 0x37D40000, 0x37D48000, 0x37D50000, 0x37D58000, 0x37D60000, 0x37D68000, 0x37D70000, 0x37D78000, 0x37D80000, 0x37D88000, 0x37D90000, 0x37D98000, 0x37DA0000, 0x37DA8000, 0x37DB0000, 0x37DB8000, 0x37DC0000, 0x37DC8000, 0x37DD0000, 0x37DD8000, 0x37DE0000, 0x37DE8000, 0x37DF0000, 0x37DF8000, 0x37E00000, 0x37E08000, 0x37E10000, 0x37E18000, 0x37E20000, 0x37E28000, 0x37E30000, 0x37E38000, 0x37E40000, 0x37E48000, 0x37E50000, 0x37E58000, 0x37E60000, 0x37E68000, 0x37E70000, 0x37E78000, 0x37E80000, 0x37E88000, 0x37E90000, 0x37E98000, 0x37EA0000, 0x37EA8000, 0x37EB0000, 0x37EB8000, 0x37EC0000, 0x37EC8000, 0x37ED0000, 0x37ED8000, 0x37EE0000, 0x37EE8000, 0x37EF0000, 0x37EF8000, 0x37F00000, 0x37F08000, 0x37F10000, 0x37F18000, 0x37F20000, 0x37F28000, 0x37F30000, 0x37F38000, 0x37F40000, 0x37F48000, 0x37F50000, 0x37F58000, 0x37F60000, 0x37F68000, 0x37F70000, 0x37F78000, 0x37F80000, 0x37F88000, 0x37F90000, 0x37F98000, 0x37FA0000, 0x37FA8000, 0x37FB0000, 0x37FB8000, 0x37FC0000, 0x37FC8000, 0x37FD0000, 0x37FD8000, 0x37FE0000, 0x37FE8000, 0x37FF0000, 0x37FF8000, 0x38000000, 0x38004000, 0x38008000, 0x3800C000, 0x38010000, 0x38014000, 0x38018000, 0x3801C000, 0x38020000, 0x38024000, 0x38028000, 0x3802C000, 0x38030000, 0x38034000, 0x38038000, 0x3803C000, 0x38040000, 0x38044000, 0x38048000, 0x3804C000, 0x38050000, 0x38054000, 0x38058000, 0x3805C000, 0x38060000, 0x38064000, 0x38068000, 0x3806C000, 0x38070000, 0x38074000, 0x38078000, 0x3807C000, 0x38080000, 0x38084000, 0x38088000, 0x3808C000, 0x38090000, 0x38094000, 0x38098000, 0x3809C000, 0x380A0000, 0x380A4000, 0x380A8000, 0x380AC000, 0x380B0000, 0x380B4000, 0x380B8000, 0x380BC000, 0x380C0000, 0x380C4000, 0x380C8000, 0x380CC000, 0x380D0000, 0x380D4000, 0x380D8000, 0x380DC000, 0x380E0000, 0x380E4000, 0x380E8000, 0x380EC000, 0x380F0000, 0x380F4000, 0x380F8000, 0x380FC000, 0x38100000, 0x38104000, 0x38108000, 0x3810C000, 0x38110000, 0x38114000, 0x38118000, 0x3811C000, 0x38120000, 0x38124000, 0x38128000, 0x3812C000, 0x38130000, 0x38134000, 0x38138000, 0x3813C000, 0x38140000, 0x38144000, 0x38148000, 0x3814C000, 0x38150000, 0x38154000, 0x38158000, 0x3815C000, 0x38160000, 0x38164000, 0x38168000, 0x3816C000, 0x38170000, 0x38174000, 0x38178000, 0x3817C000, 0x38180000, 0x38184000, 0x38188000, 0x3818C000, 0x38190000, 0x38194000, 0x38198000, 0x3819C000, 0x381A0000, 0x381A4000, 0x381A8000, 0x381AC000, 0x381B0000, 0x381B4000, 0x381B8000, 0x381BC000, 0x381C0000, 0x381C4000, 0x381C8000, 0x381CC000, 0x381D0000, 0x381D4000, 0x381D8000, 0x381DC000, 0x381E0000, 0x381E4000, 0x381E8000, 0x381EC000, 0x381F0000, 0x381F4000, 0x381F8000, 0x381FC000, 0x38200000, 0x38204000, 0x38208000, 0x3820C000, 0x38210000, 0x38214000, 0x38218000, 0x3821C000, 0x38220000, 0x38224000, 0x38228000, 0x3822C000, 0x38230000, 0x38234000, 0x38238000, 0x3823C000, 0x38240000, 0x38244000, 0x38248000, 0x3824C000, 0x38250000, 0x38254000, 0x38258000, 0x3825C000, 0x38260000, 0x38264000, 0x38268000, 0x3826C000, 0x38270000, 0x38274000, 0x38278000, 0x3827C000, 0x38280000, 0x38284000, 0x38288000, 0x3828C000, 0x38290000, 0x38294000, 0x38298000, 0x3829C000, 0x382A0000, 0x382A4000, 0x382A8000, 0x382AC000, 0x382B0000, 0x382B4000, 0x382B8000, 0x382BC000, 0x382C0000, 0x382C4000, 0x382C8000, 0x382CC000, 0x382D0000, 0x382D4000, 0x382D8000, 0x382DC000, 0x382E0000, 0x382E4000, 0x382E8000, 0x382EC000, 0x382F0000, 0x382F4000, 0x382F8000, 0x382FC000, 0x38300000, 0x38304000, 0x38308000, 0x3830C000, 0x38310000, 0x38314000, 0x38318000, 0x3831C000, 0x38320000, 0x38324000, 0x38328000, 0x3832C000, 0x38330000, 0x38334000, 0x38338000, 0x3833C000, 0x38340000, 0x38344000, 0x38348000, 0x3834C000, 0x38350000, 0x38354000, 0x38358000, 0x3835C000, 0x38360000, 0x38364000, 0x38368000, 0x3836C000, 0x38370000, 0x38374000, 0x38378000, 0x3837C000, 0x38380000, 0x38384000, 0x38388000, 0x3838C000, 0x38390000, 0x38394000, 0x38398000, 0x3839C000, 0x383A0000, 0x383A4000, 0x383A8000, 0x383AC000, 0x383B0000, 0x383B4000, 0x383B8000, 0x383BC000, 0x383C0000, 0x383C4000, 0x383C8000, 0x383CC000, 0x383D0000, 0x383D4000, 0x383D8000, 0x383DC000, 0x383E0000, 0x383E4000, 0x383E8000, 0x383EC000, 0x383F0000, 0x383F4000, 0x383F8000, 0x383FC000, 0x38400000, 0x38404000, 0x38408000, 0x3840C000, 0x38410000, 0x38414000, 0x38418000, 0x3841C000, 0x38420000, 0x38424000, 0x38428000, 0x3842C000, 0x38430000, 0x38434000, 0x38438000, 0x3843C000, 0x38440000, 0x38444000, 0x38448000, 0x3844C000, 0x38450000, 0x38454000, 0x38458000, 0x3845C000, 0x38460000, 0x38464000, 0x38468000, 0x3846C000, 0x38470000, 0x38474000, 0x38478000, 0x3847C000, 0x38480000, 0x38484000, 0x38488000, 0x3848C000, 0x38490000, 0x38494000, 0x38498000, 0x3849C000, 0x384A0000, 0x384A4000, 0x384A8000, 0x384AC000, 0x384B0000, 0x384B4000, 0x384B8000, 0x384BC000, 0x384C0000, 0x384C4000, 0x384C8000, 0x384CC000, 0x384D0000, 0x384D4000, 0x384D8000, 0x384DC000, 0x384E0000, 0x384E4000, 0x384E8000, 0x384EC000, 0x384F0000, 0x384F4000, 0x384F8000, 0x384FC000, 0x38500000, 0x38504000, 0x38508000, 0x3850C000, 0x38510000, 0x38514000, 0x38518000, 0x3851C000, 0x38520000, 0x38524000, 0x38528000, 0x3852C000, 0x38530000, 0x38534000, 0x38538000, 0x3853C000, 0x38540000, 0x38544000, 0x38548000, 0x3854C000, 0x38550000, 0x38554000, 0x38558000, 0x3855C000, 0x38560000, 0x38564000, 0x38568000, 0x3856C000, 0x38570000, 0x38574000, 0x38578000, 0x3857C000, 0x38580000, 0x38584000, 0x38588000, 0x3858C000, 0x38590000, 0x38594000, 0x38598000, 0x3859C000, 0x385A0000, 0x385A4000, 0x385A8000, 0x385AC000, 0x385B0000, 0x385B4000, 0x385B8000, 0x385BC000, 0x385C0000, 0x385C4000, 0x385C8000, 0x385CC000, 0x385D0000, 0x385D4000, 0x385D8000, 0x385DC000, 0x385E0000, 0x385E4000, 0x385E8000, 0x385EC000, 0x385F0000, 0x385F4000, 0x385F8000, 0x385FC000, 0x38600000, 0x38604000, 0x38608000, 0x3860C000, 0x38610000, 0x38614000, 0x38618000, 0x3861C000, 0x38620000, 0x38624000, 0x38628000, 0x3862C000, 0x38630000, 0x38634000, 0x38638000, 0x3863C000, 0x38640000, 0x38644000, 0x38648000, 0x3864C000, 0x38650000, 0x38654000, 0x38658000, 0x3865C000, 0x38660000, 0x38664000, 0x38668000, 0x3866C000, 0x38670000, 0x38674000, 0x38678000, 0x3867C000, 0x38680000, 0x38684000, 0x38688000, 0x3868C000, 0x38690000, 0x38694000, 0x38698000, 0x3869C000, 0x386A0000, 0x386A4000, 0x386A8000, 0x386AC000, 0x386B0000, 0x386B4000, 0x386B8000, 0x386BC000, 0x386C0000, 0x386C4000, 0x386C8000, 0x386CC000, 0x386D0000, 0x386D4000, 0x386D8000, 0x386DC000, 0x386E0000, 0x386E4000, 0x386E8000, 0x386EC000, 0x386F0000, 0x386F4000, 0x386F8000, 0x386FC000, 0x38700000, 0x38704000, 0x38708000, 0x3870C000, 0x38710000, 0x38714000, 0x38718000, 0x3871C000, 0x38720000, 0x38724000, 0x38728000, 0x3872C000, 0x38730000, 0x38734000, 0x38738000, 0x3873C000, 0x38740000, 0x38744000, 0x38748000, 0x3874C000, 0x38750000, 0x38754000, 0x38758000, 0x3875C000, 0x38760000, 0x38764000, 0x38768000, 0x3876C000, 0x38770000, 0x38774000, 0x38778000, 0x3877C000, 0x38780000, 0x38784000, 0x38788000, 0x3878C000, 0x38790000, 0x38794000, 0x38798000, 0x3879C000, 0x387A0000, 0x387A4000, 0x387A8000, 0x387AC000, 0x387B0000, 0x387B4000, 0x387B8000, 0x387BC000, 0x387C0000, 0x387C4000, 0x387C8000, 0x387CC000, 0x387D0000, 0x387D4000, 0x387D8000, 0x387DC000, 0x387E0000, 0x387E4000, 0x387E8000, 0x387EC000, 0x387F0000, 0x387F4000, 0x387F8000, 0x387FC000, 0x38000000, 0x38002000, 0x38004000, 0x38006000, 0x38008000, 0x3800A000, 0x3800C000, 0x3800E000, 0x38010000, 0x38012000, 0x38014000, 0x38016000, 0x38018000, 0x3801A000, 0x3801C000, 0x3801E000, 0x38020000, 0x38022000, 0x38024000, 0x38026000, 0x38028000, 0x3802A000, 0x3802C000, 0x3802E000, 0x38030000, 0x38032000, 0x38034000, 0x38036000, 0x38038000, 0x3803A000, 0x3803C000, 0x3803E000, 0x38040000, 0x38042000, 0x38044000, 0x38046000, 0x38048000, 0x3804A000, 0x3804C000, 0x3804E000, 0x38050000, 0x38052000, 0x38054000, 0x38056000, 0x38058000, 0x3805A000, 0x3805C000, 0x3805E000, 0x38060000, 0x38062000, 0x38064000, 0x38066000, 0x38068000, 0x3806A000, 0x3806C000, 0x3806E000, 0x38070000, 0x38072000, 0x38074000, 0x38076000, 0x38078000, 0x3807A000, 0x3807C000, 0x3807E000, 0x38080000, 0x38082000, 0x38084000, 0x38086000, 0x38088000, 0x3808A000, 0x3808C000, 0x3808E000, 0x38090000, 0x38092000, 0x38094000, 0x38096000, 0x38098000, 0x3809A000, 0x3809C000, 0x3809E000, 0x380A0000, 0x380A2000, 0x380A4000, 0x380A6000, 0x380A8000, 0x380AA000, 0x380AC000, 0x380AE000, 0x380B0000, 0x380B2000, 0x380B4000, 0x380B6000, 0x380B8000, 0x380BA000, 0x380BC000, 0x380BE000, 0x380C0000, 0x380C2000, 0x380C4000, 0x380C6000, 0x380C8000, 0x380CA000, 0x380CC000, 0x380CE000, 0x380D0000, 0x380D2000, 0x380D4000, 0x380D6000, 0x380D8000, 0x380DA000, 0x380DC000, 0x380DE000, 0x380E0000, 0x380E2000, 0x380E4000, 0x380E6000, 0x380E8000, 0x380EA000, 0x380EC000, 0x380EE000, 0x380F0000, 0x380F2000, 0x380F4000, 0x380F6000, 0x380F8000, 0x380FA000, 0x380FC000, 0x380FE000, 0x38100000, 0x38102000, 0x38104000, 0x38106000, 0x38108000, 0x3810A000, 0x3810C000, 0x3810E000, 0x38110000, 0x38112000, 0x38114000, 0x38116000, 0x38118000, 0x3811A000, 0x3811C000, 0x3811E000, 0x38120000, 0x38122000, 0x38124000, 0x38126000, 0x38128000, 0x3812A000, 0x3812C000, 0x3812E000, 0x38130000, 0x38132000, 0x38134000, 0x38136000, 0x38138000, 0x3813A000, 0x3813C000, 0x3813E000, 0x38140000, 0x38142000, 0x38144000, 0x38146000, 0x38148000, 0x3814A000, 0x3814C000, 0x3814E000, 0x38150000, 0x38152000, 0x38154000, 0x38156000, 0x38158000, 0x3815A000, 0x3815C000, 0x3815E000, 0x38160000, 0x38162000, 0x38164000, 0x38166000, 0x38168000, 0x3816A000, 0x3816C000, 0x3816E000, 0x38170000, 0x38172000, 0x38174000, 0x38176000, 0x38178000, 0x3817A000, 0x3817C000, 0x3817E000, 0x38180000, 0x38182000, 0x38184000, 0x38186000, 0x38188000, 0x3818A000, 0x3818C000, 0x3818E000, 0x38190000, 0x38192000, 0x38194000, 0x38196000, 0x38198000, 0x3819A000, 0x3819C000, 0x3819E000, 0x381A0000, 0x381A2000, 0x381A4000, 0x381A6000, 0x381A8000, 0x381AA000, 0x381AC000, 0x381AE000, 0x381B0000, 0x381B2000, 0x381B4000, 0x381B6000, 0x381B8000, 0x381BA000, 0x381BC000, 0x381BE000, 0x381C0000, 0x381C2000, 0x381C4000, 0x381C6000, 0x381C8000, 0x381CA000, 0x381CC000, 0x381CE000, 0x381D0000, 0x381D2000, 0x381D4000, 0x381D6000, 0x381D8000, 0x381DA000, 0x381DC000, 0x381DE000, 0x381E0000, 0x381E2000, 0x381E4000, 0x381E6000, 0x381E8000, 0x381EA000, 0x381EC000, 0x381EE000, 0x381F0000, 0x381F2000, 0x381F4000, 0x381F6000, 0x381F8000, 0x381FA000, 0x381FC000, 0x381FE000, 0x38200000, 0x38202000, 0x38204000, 0x38206000, 0x38208000, 0x3820A000, 0x3820C000, 0x3820E000, 0x38210000, 0x38212000, 0x38214000, 0x38216000, 0x38218000, 0x3821A000, 0x3821C000, 0x3821E000, 0x38220000, 0x38222000, 0x38224000, 0x38226000, 0x38228000, 0x3822A000, 0x3822C000, 0x3822E000, 0x38230000, 0x38232000, 0x38234000, 0x38236000, 0x38238000, 0x3823A000, 0x3823C000, 0x3823E000, 0x38240000, 0x38242000, 0x38244000, 0x38246000, 0x38248000, 0x3824A000, 0x3824C000, 0x3824E000, 0x38250000, 0x38252000, 0x38254000, 0x38256000, 0x38258000, 0x3825A000, 0x3825C000, 0x3825E000, 0x38260000, 0x38262000, 0x38264000, 0x38266000, 0x38268000, 0x3826A000, 0x3826C000, 0x3826E000, 0x38270000, 0x38272000, 0x38274000, 0x38276000, 0x38278000, 0x3827A000, 0x3827C000, 0x3827E000, 0x38280000, 0x38282000, 0x38284000, 0x38286000, 0x38288000, 0x3828A000, 0x3828C000, 0x3828E000, 0x38290000, 0x38292000, 0x38294000, 0x38296000, 0x38298000, 0x3829A000, 0x3829C000, 0x3829E000, 0x382A0000, 0x382A2000, 0x382A4000, 0x382A6000, 0x382A8000, 0x382AA000, 0x382AC000, 0x382AE000, 0x382B0000, 0x382B2000, 0x382B4000, 0x382B6000, 0x382B8000, 0x382BA000, 0x382BC000, 0x382BE000, 0x382C0000, 0x382C2000, 0x382C4000, 0x382C6000, 0x382C8000, 0x382CA000, 0x382CC000, 0x382CE000, 0x382D0000, 0x382D2000, 0x382D4000, 0x382D6000, 0x382D8000, 0x382DA000, 0x382DC000, 0x382DE000, 0x382E0000, 0x382E2000, 0x382E4000, 0x382E6000, 0x382E8000, 0x382EA000, 0x382EC000, 0x382EE000, 0x382F0000, 0x382F2000, 0x382F4000, 0x382F6000, 0x382F8000, 0x382FA000, 0x382FC000, 0x382FE000, 0x38300000, 0x38302000, 0x38304000, 0x38306000, 0x38308000, 0x3830A000, 0x3830C000, 0x3830E000, 0x38310000, 0x38312000, 0x38314000, 0x38316000, 0x38318000, 0x3831A000, 0x3831C000, 0x3831E000, 0x38320000, 0x38322000, 0x38324000, 0x38326000, 0x38328000, 0x3832A000, 0x3832C000, 0x3832E000, 0x38330000, 0x38332000, 0x38334000, 0x38336000, 0x38338000, 0x3833A000, 0x3833C000, 0x3833E000, 0x38340000, 0x38342000, 0x38344000, 0x38346000, 0x38348000, 0x3834A000, 0x3834C000, 0x3834E000, 0x38350000, 0x38352000, 0x38354000, 0x38356000, 0x38358000, 0x3835A000, 0x3835C000, 0x3835E000, 0x38360000, 0x38362000, 0x38364000, 0x38366000, 0x38368000, 0x3836A000, 0x3836C000, 0x3836E000, 0x38370000, 0x38372000, 0x38374000, 0x38376000, 0x38378000, 0x3837A000, 0x3837C000, 0x3837E000, 0x38380000, 0x38382000, 0x38384000, 0x38386000, 0x38388000, 0x3838A000, 0x3838C000, 0x3838E000, 0x38390000, 0x38392000, 0x38394000, 0x38396000, 0x38398000, 0x3839A000, 0x3839C000, 0x3839E000, 0x383A0000, 0x383A2000, 0x383A4000, 0x383A6000, 0x383A8000, 0x383AA000, 0x383AC000, 0x383AE000, 0x383B0000, 0x383B2000, 0x383B4000, 0x383B6000, 0x383B8000, 0x383BA000, 0x383BC000, 0x383BE000, 0x383C0000, 0x383C2000, 0x383C4000, 0x383C6000, 0x383C8000, 0x383CA000, 0x383CC000, 0x383CE000, 0x383D0000, 0x383D2000, 0x383D4000, 0x383D6000, 0x383D8000, 0x383DA000, 0x383DC000, 0x383DE000, 0x383E0000, 0x383E2000, 0x383E4000, 0x383E6000, 0x383E8000, 0x383EA000, 0x383EC000, 0x383EE000, 0x383F0000, 0x383F2000, 0x383F4000, 0x383F6000, 0x383F8000, 0x383FA000, 0x383FC000, 0x383FE000, 0x38400000, 0x38402000, 0x38404000, 0x38406000, 0x38408000, 0x3840A000, 0x3840C000, 0x3840E000, 0x38410000, 0x38412000, 0x38414000, 0x38416000, 0x38418000, 0x3841A000, 0x3841C000, 0x3841E000, 0x38420000, 0x38422000, 0x38424000, 0x38426000, 0x38428000, 0x3842A000, 0x3842C000, 0x3842E000, 0x38430000, 0x38432000, 0x38434000, 0x38436000, 0x38438000, 0x3843A000, 0x3843C000, 0x3843E000, 0x38440000, 0x38442000, 0x38444000, 0x38446000, 0x38448000, 0x3844A000, 0x3844C000, 0x3844E000, 0x38450000, 0x38452000, 0x38454000, 0x38456000, 0x38458000, 0x3845A000, 0x3845C000, 0x3845E000, 0x38460000, 0x38462000, 0x38464000, 0x38466000, 0x38468000, 0x3846A000, 0x3846C000, 0x3846E000, 0x38470000, 0x38472000, 0x38474000, 0x38476000, 0x38478000, 0x3847A000, 0x3847C000, 0x3847E000, 0x38480000, 0x38482000, 0x38484000, 0x38486000, 0x38488000, 0x3848A000, 0x3848C000, 0x3848E000, 0x38490000, 0x38492000, 0x38494000, 0x38496000, 0x38498000, 0x3849A000, 0x3849C000, 0x3849E000, 0x384A0000, 0x384A2000, 0x384A4000, 0x384A6000, 0x384A8000, 0x384AA000, 0x384AC000, 0x384AE000, 0x384B0000, 0x384B2000, 0x384B4000, 0x384B6000, 0x384B8000, 0x384BA000, 0x384BC000, 0x384BE000, 0x384C0000, 0x384C2000, 0x384C4000, 0x384C6000, 0x384C8000, 0x384CA000, 0x384CC000, 0x384CE000, 0x384D0000, 0x384D2000, 0x384D4000, 0x384D6000, 0x384D8000, 0x384DA000, 0x384DC000, 0x384DE000, 0x384E0000, 0x384E2000, 0x384E4000, 0x384E6000, 0x384E8000, 0x384EA000, 0x384EC000, 0x384EE000, 0x384F0000, 0x384F2000, 0x384F4000, 0x384F6000, 0x384F8000, 0x384FA000, 0x384FC000, 0x384FE000, 0x38500000, 0x38502000, 0x38504000, 0x38506000, 0x38508000, 0x3850A000, 0x3850C000, 0x3850E000, 0x38510000, 0x38512000, 0x38514000, 0x38516000, 0x38518000, 0x3851A000, 0x3851C000, 0x3851E000, 0x38520000, 0x38522000, 0x38524000, 0x38526000, 0x38528000, 0x3852A000, 0x3852C000, 0x3852E000, 0x38530000, 0x38532000, 0x38534000, 0x38536000, 0x38538000, 0x3853A000, 0x3853C000, 0x3853E000, 0x38540000, 0x38542000, 0x38544000, 0x38546000, 0x38548000, 0x3854A000, 0x3854C000, 0x3854E000, 0x38550000, 0x38552000, 0x38554000, 0x38556000, 0x38558000, 0x3855A000, 0x3855C000, 0x3855E000, 0x38560000, 0x38562000, 0x38564000, 0x38566000, 0x38568000, 0x3856A000, 0x3856C000, 0x3856E000, 0x38570000, 0x38572000, 0x38574000, 0x38576000, 0x38578000, 0x3857A000, 0x3857C000, 0x3857E000, 0x38580000, 0x38582000, 0x38584000, 0x38586000, 0x38588000, 0x3858A000, 0x3858C000, 0x3858E000, 0x38590000, 0x38592000, 0x38594000, 0x38596000, 0x38598000, 0x3859A000, 0x3859C000, 0x3859E000, 0x385A0000, 0x385A2000, 0x385A4000, 0x385A6000, 0x385A8000, 0x385AA000, 0x385AC000, 0x385AE000, 0x385B0000, 0x385B2000, 0x385B4000, 0x385B6000, 0x385B8000, 0x385BA000, 0x385BC000, 0x385BE000, 0x385C0000, 0x385C2000, 0x385C4000, 0x385C6000, 0x385C8000, 0x385CA000, 0x385CC000, 0x385CE000, 0x385D0000, 0x385D2000, 0x385D4000, 0x385D6000, 0x385D8000, 0x385DA000, 0x385DC000, 0x385DE000, 0x385E0000, 0x385E2000, 0x385E4000, 0x385E6000, 0x385E8000, 0x385EA000, 0x385EC000, 0x385EE000, 0x385F0000, 0x385F2000, 0x385F4000, 0x385F6000, 0x385F8000, 0x385FA000, 0x385FC000, 0x385FE000, 0x38600000, 0x38602000, 0x38604000, 0x38606000, 0x38608000, 0x3860A000, 0x3860C000, 0x3860E000, 0x38610000, 0x38612000, 0x38614000, 0x38616000, 0x38618000, 0x3861A000, 0x3861C000, 0x3861E000, 0x38620000, 0x38622000, 0x38624000, 0x38626000, 0x38628000, 0x3862A000, 0x3862C000, 0x3862E000, 0x38630000, 0x38632000, 0x38634000, 0x38636000, 0x38638000, 0x3863A000, 0x3863C000, 0x3863E000, 0x38640000, 0x38642000, 0x38644000, 0x38646000, 0x38648000, 0x3864A000, 0x3864C000, 0x3864E000, 0x38650000, 0x38652000, 0x38654000, 0x38656000, 0x38658000, 0x3865A000, 0x3865C000, 0x3865E000, 0x38660000, 0x38662000, 0x38664000, 0x38666000, 0x38668000, 0x3866A000, 0x3866C000, 0x3866E000, 0x38670000, 0x38672000, 0x38674000, 0x38676000, 0x38678000, 0x3867A000, 0x3867C000, 0x3867E000, 0x38680000, 0x38682000, 0x38684000, 0x38686000, 0x38688000, 0x3868A000, 0x3868C000, 0x3868E000, 0x38690000, 0x38692000, 0x38694000, 0x38696000, 0x38698000, 0x3869A000, 0x3869C000, 0x3869E000, 0x386A0000, 0x386A2000, 0x386A4000, 0x386A6000, 0x386A8000, 0x386AA000, 0x386AC000, 0x386AE000, 0x386B0000, 0x386B2000, 0x386B4000, 0x386B6000, 0x386B8000, 0x386BA000, 0x386BC000, 0x386BE000, 0x386C0000, 0x386C2000, 0x386C4000, 0x386C6000, 0x386C8000, 0x386CA000, 0x386CC000, 0x386CE000, 0x386D0000, 0x386D2000, 0x386D4000, 0x386D6000, 0x386D8000, 0x386DA000, 0x386DC000, 0x386DE000, 0x386E0000, 0x386E2000, 0x386E4000, 0x386E6000, 0x386E8000, 0x386EA000, 0x386EC000, 0x386EE000, 0x386F0000, 0x386F2000, 0x386F4000, 0x386F6000, 0x386F8000, 0x386FA000, 0x386FC000, 0x386FE000, 0x38700000, 0x38702000, 0x38704000, 0x38706000, 0x38708000, 0x3870A000, 0x3870C000, 0x3870E000, 0x38710000, 0x38712000, 0x38714000, 0x38716000, 0x38718000, 0x3871A000, 0x3871C000, 0x3871E000, 0x38720000, 0x38722000, 0x38724000, 0x38726000, 0x38728000, 0x3872A000, 0x3872C000, 0x3872E000, 0x38730000, 0x38732000, 0x38734000, 0x38736000, 0x38738000, 0x3873A000, 0x3873C000, 0x3873E000, 0x38740000, 0x38742000, 0x38744000, 0x38746000, 0x38748000, 0x3874A000, 0x3874C000, 0x3874E000, 0x38750000, 0x38752000, 0x38754000, 0x38756000, 0x38758000, 0x3875A000, 0x3875C000, 0x3875E000, 0x38760000, 0x38762000, 0x38764000, 0x38766000, 0x38768000, 0x3876A000, 0x3876C000, 0x3876E000, 0x38770000, 0x38772000, 0x38774000, 0x38776000, 0x38778000, 0x3877A000, 0x3877C000, 0x3877E000, 0x38780000, 0x38782000, 0x38784000, 0x38786000, 0x38788000, 0x3878A000, 0x3878C000, 0x3878E000, 0x38790000, 0x38792000, 0x38794000, 0x38796000, 0x38798000, 0x3879A000, 0x3879C000, 0x3879E000, 0x387A0000, 0x387A2000, 0x387A4000, 0x387A6000, 0x387A8000, 0x387AA000, 0x387AC000, 0x387AE000, 0x387B0000, 0x387B2000, 0x387B4000, 0x387B6000, 0x387B8000, 0x387BA000, 0x387BC000, 0x387BE000, 0x387C0000, 0x387C2000, 0x387C4000, 0x387C6000, 0x387C8000, 0x387CA000, 0x387CC000, 0x387CE000, 0x387D0000, 0x387D2000, 0x387D4000, 0x387D6000, 0x387D8000, 0x387DA000, 0x387DC000, 0x387DE000, 0x387E0000, 0x387E2000, 0x387E4000, 0x387E6000, 0x387E8000, 0x387EA000, 0x387EC000, 0x387EE000, 0x387F0000, 0x387F2000, 0x387F4000, 0x387F6000, 0x387F8000, 0x387FA000, 0x387FC000, 0x387FE000 }; static const bits::type exponent_table[64] = { 0x00000000, 0x00800000, 0x01000000, 0x01800000, 0x02000000, 0x02800000, 0x03000000, 0x03800000, 0x04000000, 0x04800000, 0x05000000, 0x05800000, 0x06000000, 0x06800000, 0x07000000, 0x07800000, 0x08000000, 0x08800000, 0x09000000, 0x09800000, 0x0A000000, 0x0A800000, 0x0B000000, 0x0B800000, 0x0C000000, 0x0C800000, 0x0D000000, 0x0D800000, 0x0E000000, 0x0E800000, 0x0F000000, 0x47800000, 0x80000000, 0x80800000, 0x81000000, 0x81800000, 0x82000000, 0x82800000, 0x83000000, 0x83800000, 0x84000000, 0x84800000, 0x85000000, 0x85800000, 0x86000000, 0x86800000, 0x87000000, 0x87800000, 0x88000000, 0x88800000, 0x89000000, 0x89800000, 0x8A000000, 0x8A800000, 0x8B000000, 0x8B800000, 0x8C000000, 0x8C800000, 0x8D000000, 0x8D800000, 0x8E000000, 0x8E800000, 0x8F000000, 0xC7800000 }; static const unsigned short offset_table[64] = { 0, 1024, 1024, 1024, 1024, 1024, 1024, 1024, 1024, 1024, 1024, 1024, 1024, 1024, 1024, 1024, 1024, 1024, 1024, 1024, 1024, 1024, 1024, 1024, 1024, 1024, 1024, 1024, 1024, 1024, 1024, 1024, 0, 1024, 1024, 1024, 1024, 1024, 1024, 1024, 1024, 1024, 1024, 1024, 1024, 1024, 1024, 1024, 1024, 1024, 1024, 1024, 1024, 1024, 1024, 1024, 1024, 1024, 1024, 1024, 1024, 1024, 1024, 1024 }; bits::type fbits = mantissa_table[offset_table[value>>10]+(value&0x3FF)] + exponent_table[value>>10]; #endif float out; std::memcpy(&out, &fbits, sizeof(float)); return out; #endif } /// Convert half-precision to IEEE double-precision. /// \param value half-precision value to convert /// \return double-precision value inline double half2float_impl(unsigned int value, double, true_type) { #if HALF_ENABLE_F16C_INTRINSICS return _mm_cvtsd_f64(_mm_cvtps_pd(_mm_cvtph_ps(_mm_cvtsi32_si128(value)))); #else uint32 hi = static_cast(value&0x8000) << 16; unsigned int abs = value & 0x7FFF; if(abs) { hi |= 0x3F000000 << static_cast(abs>=0x7C00); for(; abs<0x400; abs<<=1,hi-=0x100000) ; hi += static_cast(abs) << 10; } bits::type dbits = static_cast::type>(hi) << 32; double out; std::memcpy(&out, &dbits, sizeof(double)); return out; #endif } /// Convert half-precision to non-IEEE floating-point. /// \tparam T type to convert to (builtin integer type) /// \param value half-precision value to convert /// \return floating-point value template T half2float_impl(unsigned int value, T, ...) { T out; unsigned int abs = value & 0x7FFF; if(abs > 0x7C00) out = (std::numeric_limits::has_signaling_NaN && !(abs&0x200)) ? std::numeric_limits::signaling_NaN() : std::numeric_limits::has_quiet_NaN ? std::numeric_limits::quiet_NaN() : T(); else if(abs == 0x7C00) out = std::numeric_limits::has_infinity ? std::numeric_limits::infinity() : std::numeric_limits::max(); else if(abs > 0x3FF) out = std::ldexp(static_cast((abs&0x3FF)|0x400), (abs>>10)-25); else out = std::ldexp(static_cast(abs), -24); return (value&0x8000) ? -out : out; } /// Convert half-precision to floating-point. /// \tparam T type to convert to (builtin integer type) /// \param value half-precision value to convert /// \return floating-point value template T half2float(unsigned int value) { return half2float_impl(value, T(), bool_type::is_iec559&&sizeof(typename bits::type)==sizeof(T)>()); } /// Convert half-precision floating-point to integer. /// \tparam R rounding mode to use /// \tparam E `true` for round to even, `false` for round away from zero /// \tparam I `true` to raise INEXACT exception (if inexact), `false` to never raise it /// \tparam T type to convert to (buitlin integer type with at least 16 bits precision, excluding any implicit sign bits) /// \param value half-precision value to convert /// \return rounded integer value /// \exception FE_INVALID if value is not representable in type \a T /// \exception FE_INEXACT if value had to be rounded and \a I is `true` template T half2int(unsigned int value) { unsigned int abs = value & 0x7FFF; if(abs >= 0x7C00) { raise(FE_INVALID); return (value&0x8000) ? std::numeric_limits::min() : std::numeric_limits::max(); } if(abs < 0x3800) { raise(FE_INEXACT, I); return (R==std::round_toward_infinity) ? T(~(value>>15)&(abs!=0)) : (R==std::round_toward_neg_infinity) ? -T(value>0x8000) : T(); } int exp = 25 - (abs>>10); unsigned int m = (value&0x3FF) | 0x400; int32 i = static_cast((exp<=0) ? (m<<-exp) : ((m+( (R==std::round_to_nearest) ? ((1<<(exp-1))-(~(m>>exp)&E)) : (R==std::round_toward_infinity) ? (((1<>15)-1)) : (R==std::round_toward_neg_infinity) ? (((1<>15)) : 0))>>exp)); if((!std::numeric_limits::is_signed && (value&0x8000)) || (std::numeric_limits::digits<16 && ((value&0x8000) ? (-i::min()) : (i>std::numeric_limits::max())))) raise(FE_INVALID); else if(I && exp > 0 && (m&((1<((value&0x8000) ? -i : i); } /// \} /// \name Mathematics /// \{ /// upper part of 64-bit multiplication. /// \tparam R rounding mode to use /// \param x first factor /// \param y second factor /// \return upper 32 bit of \a x * \a y template uint32 mulhi(uint32 x, uint32 y) { uint32 xy = (x>>16) * (y&0xFFFF), yx = (x&0xFFFF) * (y>>16), c = (xy&0xFFFF) + (yx&0xFFFF) + (((x&0xFFFF)*(y&0xFFFF))>>16); return (x>>16)*(y>>16) + (xy>>16) + (yx>>16) + (c>>16) + ((R==std::round_to_nearest) ? ((c>>15)&1) : (R==std::round_toward_infinity) ? ((c&0xFFFF)!=0) : 0); } /// 64-bit multiplication. /// \param x first factor /// \param y second factor /// \return upper 32 bit of \a x * \a y rounded to nearest inline uint32 multiply64(uint32 x, uint32 y) { #if HALF_ENABLE_CPP11_LONG_LONG return static_cast((static_cast(x)*static_cast(y)+0x80000000)>>32); #else return mulhi(x, y); #endif } /// 64-bit division. /// \param x upper 32 bit of dividend /// \param y divisor /// \param s variable to store sticky bit for rounding /// \return (\a x << 32) / \a y inline uint32 divide64(uint32 x, uint32 y, int &s) { #if HALF_ENABLE_CPP11_LONG_LONG unsigned long long xx = static_cast(x) << 32; return s = (xx%y!=0), static_cast(xx/y); #else y >>= 1; uint32 rem = x, div = 0; for(unsigned int i=0; i<32; ++i) { div <<= 1; if(rem >= y) { rem -= y; div |= 1; } rem <<= 1; } return s = rem > 1, div; #endif } /// Half precision positive modulus. /// \tparam Q `true` to compute full quotient, `false` else /// \tparam R `true` to compute signed remainder, `false` for positive remainder /// \param x first operand as positive finite half-precision value /// \param y second operand as positive finite half-precision value /// \param quo adress to store quotient at, `nullptr` if \a Q `false` /// \return modulus of \a x / \a y template unsigned int mod(unsigned int x, unsigned int y, int *quo = NULL) { unsigned int q = 0; if(x > y) { int absx = x, absy = y, expx = 0, expy = 0; for(; absx<0x400; absx<<=1,--expx) ; for(; absy<0x400; absy<<=1,--expy) ; expx += absx >> 10; expy += absy >> 10; int mx = (absx&0x3FF) | 0x400, my = (absy&0x3FF) | 0x400; for(int d=expx-expy; d; --d) { if(!Q && mx == my) return 0; if(mx >= my) { mx -= my; q += Q; } mx <<= 1; q <<= static_cast(Q); } if(!Q && mx == my) return 0; if(mx >= my) { mx -= my; ++q; } if(Q) { q &= (1<<(std::numeric_limits::digits-1)) - 1; if(!mx) return *quo = q, 0; } for(; mx<0x400; mx<<=1,--expy) ; x = (expy>0) ? ((expy<<10)|(mx&0x3FF)) : (mx>>(1-expy)); } if(R) { unsigned int a, b; if(y < 0x800) { a = (x<0x400) ? (x<<1) : (x+0x400); b = y; } else { a = x; b = y - 0x400; } if(a > b || (a == b && (q&1))) { int exp = (y>>10) + (y<=0x3FF), d = exp - (x>>10) - (x<=0x3FF); int m = (((y&0x3FF)|((y>0x3FF)<<10))<<1) - (((x&0x3FF)|((x>0x3FF)<<10))<<(1-d)); for(; m<0x800 && exp>1; m<<=1,--exp) ; x = 0x8000 + ((exp-1)<<10) + (m>>1); q += Q; } } if(Q) *quo = q; return x; } /// Fixed point square root. /// \tparam F number of fractional bits /// \param r radicand in Q1.F fixed point format /// \param exp exponent /// \return square root as Q1.F/2 template uint32 sqrt(uint32 &r, int &exp) { int i = exp & 1; r <<= i; exp = (exp-i) / 2; uint32 m = 0; for(uint32 bit=static_cast(1)<>=2) { if(r < m+bit) m >>= 1; else { r -= m + bit; m = (m>>1) + bit; } } return m; } /// Fixed point binary exponential. /// This uses the BKM algorithm in E-mode. /// \param m exponent in [0,1) as Q0.31 /// \param n number of iterations (at most 32) /// \return 2 ^ \a m as Q1.31 inline uint32 exp2(uint32 m, unsigned int n = 32) { static const uint32 logs[] = { 0x80000000, 0x4AE00D1D, 0x2934F098, 0x15C01A3A, 0x0B31FB7D, 0x05AEB4DD, 0x02DCF2D1, 0x016FE50B, 0x00B84E23, 0x005C3E10, 0x002E24CA, 0x001713D6, 0x000B8A47, 0x0005C53B, 0x0002E2A3, 0x00017153, 0x0000B8AA, 0x00005C55, 0x00002E2B, 0x00001715, 0x00000B8B, 0x000005C5, 0x000002E3, 0x00000171, 0x000000B9, 0x0000005C, 0x0000002E, 0x00000017, 0x0000000C, 0x00000006, 0x00000003, 0x00000001 }; if(!m) return 0x80000000; uint32 mx = 0x80000000, my = 0; for(unsigned int i=1; i> i; } } return mx; } /// Fixed point binary logarithm. /// This uses the BKM algorithm in L-mode. /// \param m mantissa in [1,2) as Q1.30 /// \param n number of iterations (at most 32) /// \return log2(\a m) as Q0.31 inline uint32 log2(uint32 m, unsigned int n = 32) { static const uint32 logs[] = { 0x80000000, 0x4AE00D1D, 0x2934F098, 0x15C01A3A, 0x0B31FB7D, 0x05AEB4DD, 0x02DCF2D1, 0x016FE50B, 0x00B84E23, 0x005C3E10, 0x002E24CA, 0x001713D6, 0x000B8A47, 0x0005C53B, 0x0002E2A3, 0x00017153, 0x0000B8AA, 0x00005C55, 0x00002E2B, 0x00001715, 0x00000B8B, 0x000005C5, 0x000002E3, 0x00000171, 0x000000B9, 0x0000005C, 0x0000002E, 0x00000017, 0x0000000C, 0x00000006, 0x00000003, 0x00000001 }; if(m == 0x40000000) return 0; uint32 mx = 0x40000000, my = 0; for(unsigned int i=1; i>i); if(mz <= m) { mx = mz; my += logs[i]; } } return my; } /// Fixed point sine and cosine. /// This uses the CORDIC algorithm in rotation mode. /// \param mz angle in [-pi/2,pi/2] as Q1.30 /// \param n number of iterations (at most 31) /// \return sine and cosine of \a mz as Q1.30 inline std::pair sincos(uint32 mz, unsigned int n = 31) { static const uint32 angles[] = { 0x3243F6A9, 0x1DAC6705, 0x0FADBAFD, 0x07F56EA7, 0x03FEAB77, 0x01FFD55C, 0x00FFFAAB, 0x007FFF55, 0x003FFFEB, 0x001FFFFD, 0x00100000, 0x00080000, 0x00040000, 0x00020000, 0x00010000, 0x00008000, 0x00004000, 0x00002000, 0x00001000, 0x00000800, 0x00000400, 0x00000200, 0x00000100, 0x00000080, 0x00000040, 0x00000020, 0x00000010, 0x00000008, 0x00000004, 0x00000002, 0x00000001 }; uint32 mx = 0x26DD3B6A, my = 0; for(unsigned int i=0; i0x3FF)<<10); int exp = (abs>>10) + (abs<=0x3FF) - 15; if(abs < 0x3A48) return k = 0, m << (exp+20); #if HALF_ENABLE_CPP11_LONG_LONG unsigned long long y = m * 0xA2F9836E4E442, mask = (1ULL<<(62-exp)) - 1, yi = (y+(mask>>1)) & ~mask, f = y - yi; uint32 sign = -static_cast(f>>63); k = static_cast(yi>>(62-exp)); return (multiply64(static_cast((sign ? -f : f)>>(31-exp)), 0xC90FDAA2)^sign) - sign; #else uint32 yh = m*0xA2F98 + mulhi(m, 0x36E4E442), yl = (m*0x36E4E442) & 0xFFFFFFFF; uint32 mask = (static_cast(1)<<(30-exp)) - 1, yi = (yh+(mask>>1)) & ~mask, sign = -static_cast(yi>yh); k = static_cast(yi>>(30-exp)); uint32 fh = (yh^sign) + (yi^~sign) - ~sign, fl = (yl^sign) - sign; return (multiply64((exp>-1) ? (((fh<<(1+exp))&0xFFFFFFFF)|((fl&0xFFFFFFFF)>>(31-exp))) : fh, 0xC90FDAA2)^sign) - sign; #endif } /// Get arguments for atan2 function. /// \param abs half-precision floating-point value /// \return \a abs and sqrt(1 - \a abs^2) as Q0.30 inline std::pair atan2_args(unsigned int abs) { int exp = -15; for(; abs<0x400; abs<<=1,--exp) ; exp += abs >> 10; uint32 my = ((abs&0x3FF)|0x400) << 5, r = my * my; int rexp = 2 * exp; r = 0x40000000 - ((rexp>-31) ? ((r>>-rexp)|((r&((static_cast(1)<<-rexp)-1))!=0)) : 1); for(rexp=0; r<0x40000000; r<<=1,--rexp) ; uint32 mx = sqrt<30>(r, rexp); int d = exp - rexp; if(d < 0) return std::make_pair((d<-14) ? ((my>>(-d-14))+((my>>(-d-15))&1)) : (my<<(14+d)), (mx<<14)+(r<<13)/mx); if(d > 0) return std::make_pair(my<<14, (d>14) ? ((mx>>(d-14))+((mx>>(d-15))&1)) : ((d==14) ? mx : ((mx<<(14-d))+(r<<(13-d))/mx))); return std::make_pair(my<<13, (mx<<13)+(r<<12)/mx); } /// Get exponentials for hyperbolic computation /// \param abs half-precision floating-point value /// \param exp variable to take unbiased exponent of larger result /// \param n number of BKM iterations (at most 32) /// \return exp(abs) and exp(-\a abs) as Q1.31 with same exponent inline std::pair hyperbolic_args(unsigned int abs, int &exp, unsigned int n = 32) { uint32 mx = detail::multiply64(static_cast((abs&0x3FF)+((abs>0x3FF)<<10))<<21, 0xB8AA3B29), my; int e = (abs>>10) + (abs<=0x3FF); if(e < 14) { exp = 0; mx >>= 14 - e; } else { exp = mx >> (45-e); mx = (mx<<(e-14)) & 0x7FFFFFFF; } mx = exp2(mx, n); int d = exp << 1, s; if(mx > 0x80000000) { my = divide64(0x80000000, mx, s); my |= s; ++d; } else my = mx; return std::make_pair(mx, (d<31) ? ((my>>d)|((my&((static_cast(1)< unsigned int exp2_post(uint32 m, int exp, bool esign, unsigned int sign = 0, unsigned int n = 32) { if(esign) { exp = -exp - (m!=0); if(exp < -25) return underflow(sign); else if(exp == -25) return rounded(sign, 1, m!=0); } else if(exp > 15) return overflow(sign); if(!m) return sign | (((exp+=15)>0) ? (exp<<10) : check_underflow(0x200>>-exp)); m = exp2(m, n); int s = 0; if(esign) m = divide64(0x80000000, m, s); return fixed2half(m, exp+14, sign, s); } /// Postprocessing for binary logarithm. /// \tparam R rounding mode to use /// \tparam L logarithm for base transformation as Q1.31 /// \param m fractional part of logarithm as Q0.31 /// \param ilog signed integer part of logarithm /// \param exp biased exponent of result /// \param sign sign bit of result /// \return value base-transformed and converted to half-precision /// \exception FE_OVERFLOW on overflows /// \exception FE_UNDERFLOW on underflows /// \exception FE_INEXACT if no other exception occurred template unsigned int log2_post(uint32 m, int ilog, int exp, unsigned int sign = 0) { uint32 msign = sign_mask(ilog); m = (((static_cast(ilog)<<27)+(m>>4))^msign) - msign; if(!m) return 0; for(; m<0x80000000; m<<=1,--exp) ; int i = m >= L, s; exp += i; m >>= 1 + i; sign ^= msign & 0x8000; if(exp < -11) return underflow(sign); m = divide64(m, L, s); return fixed2half(m, exp, sign, 1); } /// Hypotenuse square root and postprocessing. /// \tparam R rounding mode to use /// \param r mantissa as Q2.30 /// \param exp biased exponent /// \return square root converted to half-precision /// \exception FE_OVERFLOW on overflows /// \exception FE_UNDERFLOW on underflows /// \exception FE_INEXACT if value had to be rounded template unsigned int hypot_post(uint32 r, int exp) { int i = r >> 31; if((exp+=i) > 46) return overflow(); if(exp < -34) return underflow(); r = (r>>i) | (r&i); uint32 m = sqrt<30>(r, exp+=15); return fixed2half(m, exp-1, 0, r!=0); } /// Division and postprocessing for tangents. /// \tparam R rounding mode to use /// \param my dividend as Q1.31 /// \param mx divisor as Q1.31 /// \param exp biased exponent of result /// \param sign sign bit of result /// \return quotient converted to half-precision /// \exception FE_OVERFLOW on overflows /// \exception FE_UNDERFLOW on underflows /// \exception FE_INEXACT if no other exception occurred template unsigned int tangent_post(uint32 my, uint32 mx, int exp, unsigned int sign = 0) { int i = my >= mx, s; exp += i; if(exp > 29) return overflow(sign); if(exp < -11) return underflow(sign); uint32 m = divide64(my>>(i+1), mx, s); return fixed2half(m, exp, sign, s); } /// Area function and postprocessing. /// This computes the value directly in Q2.30 using the representation `asinh|acosh(x) = log(x+sqrt(x^2+|-1))`. /// \tparam R rounding mode to use /// \tparam S `true` for asinh, `false` for acosh /// \param arg half-precision argument /// \return asinh|acosh(\a arg) converted to half-precision /// \exception FE_OVERFLOW on overflows /// \exception FE_UNDERFLOW on underflows /// \exception FE_INEXACT if no other exception occurred template unsigned int area(unsigned int arg) { int abs = arg & 0x7FFF, expx = (abs>>10) + (abs<=0x3FF) - 15, expy = -15, ilog, i; uint32 mx = static_cast((abs&0x3FF)|((abs>0x3FF)<<10)) << 20, my, r; for(; abs<0x400; abs<<=1,--expy) ; expy += abs >> 10; r = ((abs&0x3FF)|0x400) << 5; r *= r; i = r >> 31; expy = 2*expy + i; r >>= i; if(S) { if(expy < 0) { r = 0x40000000 + ((expy>-30) ? ((r>>-expy)|((r&((static_cast(1)<<-expy)-1))!=0)) : 1); expy = 0; } else { r += 0x40000000 >> expy; i = r >> 31; r = (r>>i) | (r&i); expy += i; } } else { r -= 0x40000000 >> expy; for(; r<0x40000000; r<<=1,--expy) ; } my = sqrt<30>(r, expy); my = (my<<15) + (r<<14)/my; if(S) { mx >>= expy - expx; ilog = expy; } else { my >>= expx - expy; ilog = expx; } my += mx; i = my >> 31; static const int G = S && (R==std::round_to_nearest); return log2_post(log2(my>>i, 26+S+G)+(G<<3), ilog+i, 17, arg&(static_cast(S)<<15)); } /// Class for 1.31 unsigned floating-point computation struct f31 { /// Constructor. /// \param mant mantissa as 1.31 /// \param e exponent HALF_CONSTEXPR f31(uint32 mant, int e) : m(mant), exp(e) {} /// Constructor. /// \param abs unsigned half-precision value f31(unsigned int abs) : exp(-15) { for(; abs<0x400; abs<<=1,--exp) ; m = static_cast((abs&0x3FF)|0x400) << 21; exp += (abs>>10); } /// Addition operator. /// \param a first operand /// \param b second operand /// \return \a a + \a b friend f31 operator+(f31 a, f31 b) { if(b.exp > a.exp) std::swap(a, b); int d = a.exp - b.exp; uint32 m = a.m + ((d<32) ? (b.m>>d) : 0); int i = (m&0xFFFFFFFF) < a.m; return f31(((m+i)>>i)|0x80000000, a.exp+i); } /// Subtraction operator. /// \param a first operand /// \param b second operand /// \return \a a - \a b friend f31 operator-(f31 a, f31 b) { int d = a.exp - b.exp, exp = a.exp; uint32 m = a.m - ((d<32) ? (b.m>>d) : 0); if(!m) return f31(0, -32); for(; m<0x80000000; m<<=1,--exp) ; return f31(m, exp); } /// Multiplication operator. /// \param a first operand /// \param b second operand /// \return \a a * \a b friend f31 operator*(f31 a, f31 b) { uint32 m = multiply64(a.m, b.m); int i = m >> 31; return f31(m<<(1-i), a.exp + b.exp + i); } /// Division operator. /// \param a first operand /// \param b second operand /// \return \a a / \a b friend f31 operator/(f31 a, f31 b) { int i = a.m >= b.m, s; uint32 m = divide64((a.m+i)>>i, b.m, s); return f31(m, a.exp - b.exp + i - 1); } uint32 m; ///< mantissa as 1.31. int exp; ///< exponent. }; /// Error function and postprocessing. /// This computes the value directly in Q1.31 using the approximations given /// [here](https://en.wikipedia.org/wiki/Error_function#Approximation_with_elementary_functions). /// \tparam R rounding mode to use /// \tparam C `true` for comlementary error function, `false` else /// \param arg half-precision function argument /// \return approximated value of error function in half-precision /// \exception FE_OVERFLOW on overflows /// \exception FE_UNDERFLOW on underflows /// \exception FE_INEXACT if no other exception occurred template unsigned int erf(unsigned int arg) { unsigned int abs = arg & 0x7FFF, sign = arg & 0x8000; f31 x(abs), x2 = x * x * f31(0xB8AA3B29, 0), t = f31(0x80000000, 0) / (f31(0x80000000, 0)+f31(0xA7BA054A, -2)*x), t2 = t * t; f31 e = ((f31(0x87DC2213, 0)*t2+f31(0xB5F0E2AE, 0))*t2+f31(0x82790637, -2)-(f31(0xBA00E2B8, 0)*t2+f31(0x91A98E62, -2))*t) * t / ((x2.exp<0) ? f31(exp2((x2.exp>-32) ? (x2.m>>-x2.exp) : 0, 30), 0) : f31(exp2((x2.m<>(31-x2.exp))); return (!C || sign) ? fixed2half(0x80000000-(e.m>>(C-e.exp)), 14+C, sign&(C-1U)) : (e.exp<-25) ? underflow() : fixed2half(e.m>>1, e.exp+14, 0, e.m&1); } /// Gamma function and postprocessing. /// This approximates the value of either the gamma function or its logarithm directly in Q1.31. /// \tparam R rounding mode to use /// \tparam L `true` for lograithm of gamma function, `false` for gamma function /// \param arg half-precision floating-point value /// \return lgamma/tgamma(\a arg) in half-precision /// \exception FE_OVERFLOW on overflows /// \exception FE_UNDERFLOW on underflows /// \exception FE_INEXACT if \a arg is not a positive integer template unsigned int gamma(unsigned int arg) { /* static const double p[] ={ 2.50662827563479526904, 225.525584619175212544, -268.295973841304927459, 80.9030806934622512966, -5.00757863970517583837, 0.0114684895434781459556 }; double t = arg + 4.65, s = p[0]; for(unsigned int i=0; i<5; ++i) s += p[i+1] / (arg+i); return std::log(s) + (arg-0.5)*std::log(t) - t; */ static const f31 pi(0xC90FDAA2, 1), lbe(0xB8AA3B29, 0); unsigned int abs = arg & 0x7FFF, sign = arg & 0x8000; bool bsign = sign != 0; f31 z(abs), x = sign ? (z+f31(0x80000000, 0)) : z, t = x + f31(0x94CCCCCD, 2), s = f31(0xA06C9901, 1) + f31(0xBBE654E2, -7)/(x+f31(0x80000000, 2)) + f31(0xA1CE6098, 6)/(x+f31(0x80000000, 1)) + f31(0xE1868CB7, 7)/x - f31(0x8625E279, 8)/(x+f31(0x80000000, 0)) - f31(0xA03E158F, 2)/(x+f31(0xC0000000, 1)); int i = (s.exp>=2) + (s.exp>=4) + (s.exp>=8) + (s.exp>=16); s = f31((static_cast(s.exp)<<(31-i))+(log2(s.m>>1, 28)>>i), i) / lbe; if(x.exp != -1 || x.m != 0x80000000) { i = (t.exp>=2) + (t.exp>=4) + (t.exp>=8); f31 l = f31((static_cast(t.exp)<<(31-i))+(log2(t.m>>1, 30)>>i), i) / lbe; s = (x.exp<-1) ? (s-(f31(0x80000000, -1)-x)*l) : (s+(x-f31(0x80000000, -1))*l); } s = x.exp ? (s-t) : (t-s); if(bsign) { if(z.exp >= 0) { sign &= (L|((z.m>>(31-z.exp))&1)) - 1; for(z=f31((z.m<<(1+z.exp))&0xFFFFFFFF, -1); z.m<0x80000000; z.m<<=1,--z.exp) ; } if(z.exp == -1) z = f31(0x80000000, 0) - z; if(z.exp < -1) { z = z * pi; z.m = sincos(z.m>>(1-z.exp), 30).first; for(z.exp=1; z.m<0x80000000; z.m<<=1,--z.exp) ; } else z = f31(0x80000000, 0); } if(L) { if(bsign) { f31 l(0x92868247, 0); if(z.exp < 0) { uint32 m = log2((z.m+1)>>1, 27); z = f31(-((static_cast(z.exp)<<26)+(m>>5)), 5); for(; z.m<0x80000000; z.m<<=1,--z.exp) ; l = l + z / lbe; } sign = static_cast(x.exp&&(l.exp(x.exp==0) << 15; if(s.exp < -24) return underflow(sign); if(s.exp > 15) return overflow(sign); } } else { s = s * lbe; uint32 m; if(s.exp < 0) { m = s.m >> -s.exp; s.exp = 0; } else { m = (s.m<>(31-s.exp)); } s.m = exp2(m, 27); if(!x.exp) s = f31(0x80000000, 0) / s; if(bsign) { if(z.exp < 0) s = s * z; s = pi / s; if(s.exp < -24) return underflow(sign); } else if(z.exp > 0 && !(z.m&((1<<(31-z.exp))-1))) return ((s.exp+14)<<10) + (s.m>>21); if(s.exp > 15) return overflow(sign); } return fixed2half(s.m, s.exp+14, sign); } /// \} template struct half_caster; } /// Half-precision floating-point type. /// This class implements an IEEE-conformant half-precision floating-point type with the usual arithmetic /// operators and conversions. It is implicitly convertible to single-precision floating-point, which makes artihmetic /// expressions and functions with mixed-type operands to be of the most precise operand type. /// /// According to the C++98/03 definition, the half type is not a POD type. But according to C++11's less strict and /// extended definitions it is both a standard layout type and a trivially copyable type (even if not a POD type), which /// means it can be standard-conformantly copied using raw binary copies. But in this context some more words about the /// actual size of the type. Although the half is representing an IEEE 16-bit type, it does not neccessarily have to be of /// exactly 16-bits size. But on any reasonable implementation the actual binary representation of this type will most /// probably not ivolve any additional "magic" or padding beyond the simple binary representation of the underlying 16-bit /// IEEE number, even if not strictly guaranteed by the standard. But even then it only has an actual size of 16 bits if /// your C++ implementation supports an unsigned integer type of exactly 16 bits width. But this should be the case on /// nearly any reasonable platform. /// /// So if your C++ implementation is not totally exotic or imposes special alignment requirements, it is a reasonable /// assumption that the data of a half is just comprised of the 2 bytes of the underlying IEEE representation. class half { public: /// \name Construction and assignment /// \{ /// Default constructor. /// This initializes the half to 0. Although this does not match the builtin types' default-initialization semantics /// and may be less efficient than no initialization, it is needed to provide proper value-initialization semantics. HALF_CONSTEXPR half() HALF_NOEXCEPT : data_() {} /// Conversion constructor. /// \param rhs float to convert /// \exception FE_OVERFLOW, ...UNDERFLOW, ...INEXACT according to rounding explicit half(float rhs) : data_(static_cast(detail::float2half(rhs))) {} /// Conversion to single-precision. /// \return single precision value representing expression value operator float() const { return detail::half2float(data_); } /// Assignment operator. /// \param rhs single-precision value to copy from /// \return reference to this half /// \exception FE_OVERFLOW, ...UNDERFLOW, ...INEXACT according to rounding half& operator=(float rhs) { data_ = static_cast(detail::float2half(rhs)); return *this; } /// \} /// \name Arithmetic updates /// \{ /// Arithmetic assignment. /// \tparam T type of concrete half expression /// \param rhs half expression to add /// \return reference to this half /// \exception FE_... according to operator+(half,half) half& operator+=(half rhs) { return *this = *this + rhs; } /// Arithmetic assignment. /// \tparam T type of concrete half expression /// \param rhs half expression to subtract /// \return reference to this half /// \exception FE_... according to operator-(half,half) half& operator-=(half rhs) { return *this = *this - rhs; } /// Arithmetic assignment. /// \tparam T type of concrete half expression /// \param rhs half expression to multiply with /// \return reference to this half /// \exception FE_... according to operator*(half,half) half& operator*=(half rhs) { return *this = *this * rhs; } /// Arithmetic assignment. /// \tparam T type of concrete half expression /// \param rhs half expression to divide by /// \return reference to this half /// \exception FE_... according to operator/(half,half) half& operator/=(half rhs) { return *this = *this / rhs; } /// Arithmetic assignment. /// \param rhs single-precision value to add /// \return reference to this half /// \exception FE_... according to operator=() half& operator+=(float rhs) { return *this = *this + rhs; } /// Arithmetic assignment. /// \param rhs single-precision value to subtract /// \return reference to this half /// \exception FE_... according to operator=() half& operator-=(float rhs) { return *this = *this - rhs; } /// Arithmetic assignment. /// \param rhs single-precision value to multiply with /// \return reference to this half /// \exception FE_... according to operator=() half& operator*=(float rhs) { return *this = *this * rhs; } /// Arithmetic assignment. /// \param rhs single-precision value to divide by /// \return reference to this half /// \exception FE_... according to operator=() half& operator/=(float rhs) { return *this = *this / rhs; } /// \} /// \name Increment and decrement /// \{ /// Prefix increment. /// \return incremented half value /// \exception FE_... according to operator+(half,half) half& operator++() { return *this = *this + half(detail::binary, 0x3C00); } /// Prefix decrement. /// \return decremented half value /// \exception FE_... according to operator-(half,half) half& operator--() { return *this = *this + half(detail::binary, 0xBC00); } /// Postfix increment. /// \return non-incremented half value /// \exception FE_... according to operator+(half,half) half operator++(int) { half out(*this); ++*this; return out; } /// Postfix decrement. /// \return non-decremented half value /// \exception FE_... according to operator-(half,half) half operator--(int) { half out(*this); --*this; return out; } /// \} private: /// Rounding mode to use static const std::float_round_style round_style = (std::float_round_style)(HALF_ROUND_STYLE); /// Constructor. /// \param bits binary representation to set half to HALF_CONSTEXPR half(detail::binary_t, unsigned int bits) HALF_NOEXCEPT : data_(static_cast(bits)) {} /// Internal binary representation detail::uint16 data_; #ifndef HALF_DOXYGEN_ONLY friend HALF_CONSTEXPR_NOERR bool operator==(half, half); friend HALF_CONSTEXPR_NOERR bool operator!=(half, half); friend HALF_CONSTEXPR_NOERR bool operator<(half, half); friend HALF_CONSTEXPR_NOERR bool operator>(half, half); friend HALF_CONSTEXPR_NOERR bool operator<=(half, half); friend HALF_CONSTEXPR_NOERR bool operator>=(half, half); friend HALF_CONSTEXPR half operator-(half); friend half operator+(half, half); friend half operator-(half, half); friend half operator*(half, half); friend half operator/(half, half); template friend std::basic_ostream& operator<<(std::basic_ostream&, half); template friend std::basic_istream& operator>>(std::basic_istream&, half&); friend HALF_CONSTEXPR half fabs(half); friend half fmod(half, half); friend half remainder(half, half); friend half remquo(half, half, int*); friend half fma(half, half, half); friend HALF_CONSTEXPR_NOERR half fmax(half, half); friend HALF_CONSTEXPR_NOERR half fmin(half, half); friend half fdim(half, half); friend half nanh(const char*); friend half exp(half); friend half exp2(half); friend half expm1(half); friend half log(half); friend half log10(half); friend half log2(half); friend half log1p(half); friend half sqrt(half); friend half rsqrt(half); friend half cbrt(half); friend half hypot(half, half); friend half hypot(half, half, half); friend half pow(half, half); friend void sincos(half, half*, half*); friend half sin(half); friend half cos(half); friend half tan(half); friend half asin(half); friend half acos(half); friend half atan(half); friend half atan2(half, half); friend half sinh(half); friend half cosh(half); friend half tanh(half); friend half asinh(half); friend half acosh(half); friend half atanh(half); friend half erf(half); friend half erfc(half); friend half lgamma(half); friend half tgamma(half); friend half ceil(half); friend half floor(half); friend half trunc(half); friend half round(half); friend long lround(half); friend half rint(half); friend long lrint(half); friend half nearbyint(half); #ifdef HALF_ENABLE_CPP11_LONG_LONG friend long long llround(half); friend long long llrint(half); #endif friend half frexp(half, int*); friend half scalbln(half, long); friend half modf(half, half*); friend int ilogb(half); friend half logb(half); friend half nextafter(half, half); friend half nexttoward(half, long double); friend HALF_CONSTEXPR half copysign(half, half); friend HALF_CONSTEXPR int fpclassify(half); friend HALF_CONSTEXPR bool isfinite(half); friend HALF_CONSTEXPR bool isinf(half); friend HALF_CONSTEXPR bool isnan(half); friend HALF_CONSTEXPR bool isnormal(half); friend HALF_CONSTEXPR bool signbit(half); friend HALF_CONSTEXPR bool isgreater(half, half); friend HALF_CONSTEXPR bool isgreaterequal(half, half); friend HALF_CONSTEXPR bool isless(half, half); friend HALF_CONSTEXPR bool islessequal(half, half); friend HALF_CONSTEXPR bool islessgreater(half, half); template friend struct detail::half_caster; friend class std::numeric_limits; #if HALF_ENABLE_CPP11_HASH friend struct std::hash; #endif #if HALF_ENABLE_CPP11_USER_LITERALS friend half literal::operator "" _h(long double); #endif #endif }; #if HALF_ENABLE_CPP11_USER_LITERALS namespace literal { /// Half literal. /// While this returns a properly rounded half-precision value, half literals can unfortunately not be constant /// expressions due to rather involved conversions. So don't expect this to be a literal literal without involving /// conversion operations at runtime. It is a convenience feature, not a performance optimization. /// \param value literal value /// \return half with of given value (possibly rounded) /// \exception FE_OVERFLOW, ...UNDERFLOW, ...INEXACT according to rounding inline half operator "" _h(long double value) { return half(detail::binary, detail::float2half(value)); } } #endif namespace detail { /// Helper class for half casts. /// This class template has to be specialized for all valid cast arguments to define an appropriate static /// `cast` member function and a corresponding `type` member denoting its return type. /// \tparam T destination type /// \tparam U source type /// \tparam R rounding mode to use template struct half_caster {}; template struct half_caster { #if HALF_ENABLE_CPP11_STATIC_ASSERT && HALF_ENABLE_CPP11_TYPE_TRAITS static_assert(std::is_arithmetic::value, "half_cast from non-arithmetic type unsupported"); #endif static half cast(U arg) { return cast_impl(arg, is_float()); }; private: static half cast_impl(U arg, true_type) { return half(binary, float2half(arg)); } static half cast_impl(U arg, false_type) { return half(binary, int2half(arg)); } }; template struct half_caster { #if HALF_ENABLE_CPP11_STATIC_ASSERT && HALF_ENABLE_CPP11_TYPE_TRAITS static_assert(std::is_arithmetic::value, "half_cast to non-arithmetic type unsupported"); #endif static T cast(half arg) { return cast_impl(arg, is_float()); } private: static T cast_impl(half arg, true_type) { return half2float(arg.data_); } static T cast_impl(half arg, false_type) { return half2int(arg.data_); } }; template struct half_caster { static half cast(half arg) { return arg; } }; } } /// Extensions to the C++ standard library. namespace std { /// Numeric limits for half-precision floats. /// **See also:** Documentation for [std::numeric_limits](https://en.cppreference.com/w/cpp/types/numeric_limits) template<> class numeric_limits { public: /// Is template specialization. static HALF_CONSTEXPR_CONST bool is_specialized = true; /// Supports signed values. static HALF_CONSTEXPR_CONST bool is_signed = true; /// Is not an integer type. static HALF_CONSTEXPR_CONST bool is_integer = false; /// Is not exact. static HALF_CONSTEXPR_CONST bool is_exact = false; /// Doesn't provide modulo arithmetic. static HALF_CONSTEXPR_CONST bool is_modulo = false; /// Has a finite set of values. static HALF_CONSTEXPR_CONST bool is_bounded = true; /// IEEE conformant. static HALF_CONSTEXPR_CONST bool is_iec559 = true; /// Supports infinity. static HALF_CONSTEXPR_CONST bool has_infinity = true; /// Supports quiet NaNs. static HALF_CONSTEXPR_CONST bool has_quiet_NaN = true; /// Supports signaling NaNs. static HALF_CONSTEXPR_CONST bool has_signaling_NaN = true; /// Supports subnormal values. static HALF_CONSTEXPR_CONST float_denorm_style has_denorm = denorm_present; /// Supports no denormalization detection. static HALF_CONSTEXPR_CONST bool has_denorm_loss = false; #if HALF_ERRHANDLING_THROWS static HALF_CONSTEXPR_CONST bool traps = true; #else /// Traps only if [HALF_ERRHANDLING_THROW_...](\ref HALF_ERRHANDLING_THROW_INVALID) is acitvated. static HALF_CONSTEXPR_CONST bool traps = false; #endif /// Does not support no pre-rounding underflow detection. static HALF_CONSTEXPR_CONST bool tinyness_before = false; /// Rounding mode. static HALF_CONSTEXPR_CONST float_round_style round_style = half_float::half::round_style; /// Significant digits. static HALF_CONSTEXPR_CONST int digits = 11; /// Significant decimal digits. static HALF_CONSTEXPR_CONST int digits10 = 3; /// Required decimal digits to represent all possible values. static HALF_CONSTEXPR_CONST int max_digits10 = 5; /// Number base. static HALF_CONSTEXPR_CONST int radix = 2; /// One more than smallest exponent. static HALF_CONSTEXPR_CONST int min_exponent = -13; /// Smallest normalized representable power of 10. static HALF_CONSTEXPR_CONST int min_exponent10 = -4; /// One more than largest exponent static HALF_CONSTEXPR_CONST int max_exponent = 16; /// Largest finitely representable power of 10. static HALF_CONSTEXPR_CONST int max_exponent10 = 4; /// Smallest positive normal value. static HALF_CONSTEXPR half_float::half min() HALF_NOTHROW { return half_float::half(half_float::detail::binary, 0x0400); } /// Smallest finite value. static HALF_CONSTEXPR half_float::half lowest() HALF_NOTHROW { return half_float::half(half_float::detail::binary, 0xFBFF); } /// Largest finite value. static HALF_CONSTEXPR half_float::half max() HALF_NOTHROW { return half_float::half(half_float::detail::binary, 0x7BFF); } /// Difference between 1 and next representable value. static HALF_CONSTEXPR half_float::half epsilon() HALF_NOTHROW { return half_float::half(half_float::detail::binary, 0x1400); } /// Maximum rounding error in ULP (units in the last place). static HALF_CONSTEXPR half_float::half round_error() HALF_NOTHROW { return half_float::half(half_float::detail::binary, (round_style==std::round_to_nearest) ? 0x3800 : 0x3C00); } /// Positive infinity. static HALF_CONSTEXPR half_float::half infinity() HALF_NOTHROW { return half_float::half(half_float::detail::binary, 0x7C00); } /// Quiet NaN. static HALF_CONSTEXPR half_float::half quiet_NaN() HALF_NOTHROW { return half_float::half(half_float::detail::binary, 0x7FFF); } /// Signaling NaN. static HALF_CONSTEXPR half_float::half signaling_NaN() HALF_NOTHROW { return half_float::half(half_float::detail::binary, 0x7DFF); } /// Smallest positive subnormal value. static HALF_CONSTEXPR half_float::half denorm_min() HALF_NOTHROW { return half_float::half(half_float::detail::binary, 0x0001); } }; #if HALF_ENABLE_CPP11_HASH /// Hash function for half-precision floats. /// This is only defined if C++11 `std::hash` is supported and enabled. /// /// **See also:** Documentation for [std::hash](https://en.cppreference.com/w/cpp/utility/hash) template<> struct hash { /// Type of function argument. typedef half_float::half argument_type; /// Function return type. typedef size_t result_type; /// Compute hash function. /// \param arg half to hash /// \return hash value result_type operator()(argument_type arg) const { return hash()(arg.data_&-static_cast(arg.data_!=0x8000)); } }; #endif } namespace half_float { /// \anchor compop /// \name Comparison operators /// \{ /// Comparison for equality. /// \param x first operand /// \param y second operand /// \retval true if operands equal /// \retval false else /// \exception FE_INVALID if \a x or \a y is NaN inline HALF_CONSTEXPR_NOERR bool operator==(half x, half y) { return !detail::compsignal(x.data_, y.data_) && (x.data_==y.data_ || !((x.data_|y.data_)&0x7FFF)); } /// Comparison for inequality. /// \param x first operand /// \param y second operand /// \retval true if operands not equal /// \retval false else /// \exception FE_INVALID if \a x or \a y is NaN inline HALF_CONSTEXPR_NOERR bool operator!=(half x, half y) { return detail::compsignal(x.data_, y.data_) || (x.data_!=y.data_ && ((x.data_|y.data_)&0x7FFF)); } /// Comparison for less than. /// \param x first operand /// \param y second operand /// \retval true if \a x less than \a y /// \retval false else /// \exception FE_INVALID if \a x or \a y is NaN inline HALF_CONSTEXPR_NOERR bool operator<(half x, half y) { return !detail::compsignal(x.data_, y.data_) && ((x.data_^(0x8000|(0x8000-(x.data_>>15))))+(x.data_>>15)) < ((y.data_^(0x8000|(0x8000-(y.data_>>15))))+(y.data_>>15)); } /// Comparison for greater than. /// \param x first operand /// \param y second operand /// \retval true if \a x greater than \a y /// \retval false else /// \exception FE_INVALID if \a x or \a y is NaN inline HALF_CONSTEXPR_NOERR bool operator>(half x, half y) { return !detail::compsignal(x.data_, y.data_) && ((x.data_^(0x8000|(0x8000-(x.data_>>15))))+(x.data_>>15)) > ((y.data_^(0x8000|(0x8000-(y.data_>>15))))+(y.data_>>15)); } /// Comparison for less equal. /// \param x first operand /// \param y second operand /// \retval true if \a x less equal \a y /// \retval false else /// \exception FE_INVALID if \a x or \a y is NaN inline HALF_CONSTEXPR_NOERR bool operator<=(half x, half y) { return !detail::compsignal(x.data_, y.data_) && ((x.data_^(0x8000|(0x8000-(x.data_>>15))))+(x.data_>>15)) <= ((y.data_^(0x8000|(0x8000-(y.data_>>15))))+(y.data_>>15)); } /// Comparison for greater equal. /// \param x first operand /// \param y second operand /// \retval true if \a x greater equal \a y /// \retval false else /// \exception FE_INVALID if \a x or \a y is NaN inline HALF_CONSTEXPR_NOERR bool operator>=(half x, half y) { return !detail::compsignal(x.data_, y.data_) && ((x.data_^(0x8000|(0x8000-(x.data_>>15))))+(x.data_>>15)) >= ((y.data_^(0x8000|(0x8000-(y.data_>>15))))+(y.data_>>15)); } /// \} /// \anchor arithmetics /// \name Arithmetic operators /// \{ /// Identity. /// \param arg operand /// \return unchanged operand inline HALF_CONSTEXPR half operator+(half arg) { return arg; } /// Negation. /// \param arg operand /// \return negated operand inline HALF_CONSTEXPR half operator-(half arg) { return half(detail::binary, arg.data_^0x8000); } /// Addition. /// This operation is exact to rounding for all rounding modes. /// \param x left operand /// \param y right operand /// \return sum of half expressions /// \exception FE_INVALID if \a x and \a y are infinities with different signs or signaling NaNs /// \exception FE_OVERFLOW, ...UNDERFLOW, ...INEXACT according to rounding inline half operator+(half x, half y) { #ifdef HALF_ARITHMETIC_TYPE return half(detail::binary, detail::float2half(detail::half2float(x.data_)+detail::half2float(y.data_))); #else int absx = x.data_ & 0x7FFF, absy = y.data_ & 0x7FFF; bool sub = ((x.data_^y.data_)&0x8000) != 0; if(absx >= 0x7C00 || absy >= 0x7C00) return half(detail::binary, (absx>0x7C00 || absy>0x7C00) ? detail::signal(x.data_, y.data_) : (absy!=0x7C00) ? x.data_ : (sub && absx==0x7C00) ? detail::invalid() : y.data_); if(!absx) return absy ? y : half(detail::binary, (half::round_style==std::round_toward_neg_infinity) ? (x.data_|y.data_) : (x.data_&y.data_)); if(!absy) return x; unsigned int sign = ((sub && absy>absx) ? y.data_ : x.data_) & 0x8000; if(absy > absx) std::swap(absx, absy); int exp = (absx>>10) + (absx<=0x3FF), d = exp - (absy>>10) - (absy<=0x3FF), mx = ((absx&0x3FF)|((absx>0x3FF)<<10)) << 3, my; if(d < 13) { my = ((absy&0x3FF)|((absy>0x3FF)<<10)) << 3; my = (my>>d) | ((my&((1<(half::round_style==std::round_toward_neg_infinity)<<15); for(; mx<0x2000 && exp>1; mx<<=1,--exp) ; } else { mx += my; int i = mx >> 14; if((exp+=i) > 30) return half(detail::binary, detail::overflow(sign)); mx = (mx>>i) | (mx&i); } return half(detail::binary, detail::rounded(sign+((exp-1)<<10)+(mx>>3), (mx>>2)&1, (mx&0x3)!=0)); #endif } /// Subtraction. /// This operation is exact to rounding for all rounding modes. /// \param x left operand /// \param y right operand /// \return difference of half expressions /// \exception FE_INVALID if \a x and \a y are infinities with equal signs or signaling NaNs /// \exception FE_OVERFLOW, ...UNDERFLOW, ...INEXACT according to rounding inline half operator-(half x, half y) { #ifdef HALF_ARITHMETIC_TYPE return half(detail::binary, detail::float2half(detail::half2float(x.data_)-detail::half2float(y.data_))); #else return x + -y; #endif } /// Multiplication. /// This operation is exact to rounding for all rounding modes. /// \param x left operand /// \param y right operand /// \return product of half expressions /// \exception FE_INVALID if multiplying 0 with infinity or if \a x or \a y is signaling NaN /// \exception FE_OVERFLOW, ...UNDERFLOW, ...INEXACT according to rounding inline half operator*(half x, half y) { #ifdef HALF_ARITHMETIC_TYPE return half(detail::binary, detail::float2half(detail::half2float(x.data_)*detail::half2float(y.data_))); #else int absx = x.data_ & 0x7FFF, absy = y.data_ & 0x7FFF, exp = -16; unsigned int sign = (x.data_^y.data_) & 0x8000; if(absx >= 0x7C00 || absy >= 0x7C00) return half(detail::binary, (absx>0x7C00 || absy>0x7C00) ? detail::signal(x.data_, y.data_) : ((absx==0x7C00 && !absy)||(absy==0x7C00 && !absx)) ? detail::invalid() : (sign|0x7C00)); if(!absx || !absy) return half(detail::binary, sign); for(; absx<0x400; absx<<=1,--exp) ; for(; absy<0x400; absy<<=1,--exp) ; detail::uint32 m = static_cast((absx&0x3FF)|0x400) * static_cast((absy&0x3FF)|0x400); int i = m >> 21, s = m & i; exp += (absx>>10) + (absy>>10) + i; if(exp > 29) return half(detail::binary, detail::overflow(sign)); else if(exp < -11) return half(detail::binary, detail::underflow(sign)); return half(detail::binary, detail::fixed2half(m>>i, exp, sign, s)); #endif } /// Division. /// This operation is exact to rounding for all rounding modes. /// \param x left operand /// \param y right operand /// \return quotient of half expressions /// \exception FE_INVALID if dividing 0s or infinities with each other or if \a x or \a y is signaling NaN /// \exception FE_DIVBYZERO if dividing finite value by 0 /// \exception FE_OVERFLOW, ...UNDERFLOW, ...INEXACT according to rounding inline half operator/(half x, half y) { #ifdef HALF_ARITHMETIC_TYPE return half(detail::binary, detail::float2half(detail::half2float(x.data_)/detail::half2float(y.data_))); #else int absx = x.data_ & 0x7FFF, absy = y.data_ & 0x7FFF, exp = 14; unsigned int sign = (x.data_^y.data_) & 0x8000; if(absx >= 0x7C00 || absy >= 0x7C00) return half(detail::binary, (absx>0x7C00 || absy>0x7C00) ? detail::signal(x.data_, y.data_) : (absx==absy) ? detail::invalid() : (sign|((absx==0x7C00) ? 0x7C00 : 0))); if(!absx) return half(detail::binary, absy ? sign : detail::invalid()); if(!absy) return half(detail::binary, detail::pole(sign)); for(; absx<0x400; absx<<=1,--exp) ; for(; absy<0x400; absy<<=1,++exp) ; detail::uint32 mx = (absx&0x3FF) | 0x400, my = (absy&0x3FF) | 0x400; int i = mx < my; exp += (absx>>10) - (absy>>10) - i; if(exp > 29) return half(detail::binary, detail::overflow(sign)); else if(exp < -11) return half(detail::binary, detail::underflow(sign)); mx <<= 12 + i; my <<= 1; return half(detail::binary, detail::fixed2half(mx/my, exp, sign, mx%my!=0)); #endif } /// \} /// \anchor streaming /// \name Input and output /// \{ /// Output operator. /// This uses the built-in functionality for streaming out floating-point numbers. /// \param out output stream to write into /// \param arg half expression to write /// \return reference to output stream template std::basic_ostream& operator<<(std::basic_ostream &out, half arg) { #ifdef HALF_ARITHMETIC_TYPE return out << detail::half2float(arg.data_); #else return out << detail::half2float(arg.data_); #endif } /// Input operator. /// This uses the built-in functionality for streaming in floating-point numbers, specifically double precision floating /// point numbers (unless overridden with [HALF_ARITHMETIC_TYPE](\ref HALF_ARITHMETIC_TYPE)). So the input string is first /// rounded to double precision using the underlying platform's current floating-point rounding mode before being rounded /// to half-precision using the library's half-precision rounding mode. /// \param in input stream to read from /// \param arg half to read into /// \return reference to input stream /// \exception FE_OVERFLOW, ...UNDERFLOW, ...INEXACT according to rounding template std::basic_istream& operator>>(std::basic_istream &in, half &arg) { #ifdef HALF_ARITHMETIC_TYPE detail::internal_t f; #else double f; #endif if(in >> f) arg.data_ = detail::float2half(f); return in; } /// \} /// \anchor basic /// \name Basic mathematical operations /// \{ /// Absolute value. /// **See also:** Documentation for [std::fabs](https://en.cppreference.com/w/cpp/numeric/math/fabs). /// \param arg operand /// \return absolute value of \a arg inline HALF_CONSTEXPR half fabs(half arg) { return half(detail::binary, arg.data_&0x7FFF); } /// Absolute value. /// **See also:** Documentation for [std::abs](https://en.cppreference.com/w/cpp/numeric/math/fabs). /// \param arg operand /// \return absolute value of \a arg inline HALF_CONSTEXPR half abs(half arg) { return fabs(arg); } /// Remainder of division. /// **See also:** Documentation for [std::fmod](https://en.cppreference.com/w/cpp/numeric/math/fmod). /// \param x first operand /// \param y second operand /// \return remainder of floating-point division. /// \exception FE_INVALID if \a x is infinite or \a y is 0 or if \a x or \a y is signaling NaN inline half fmod(half x, half y) { unsigned int absx = x.data_ & 0x7FFF, absy = y.data_ & 0x7FFF, sign = x.data_ & 0x8000; if(absx >= 0x7C00 || absy >= 0x7C00) return half(detail::binary, (absx>0x7C00 || absy>0x7C00) ? detail::signal(x.data_, y.data_) : (absx==0x7C00) ? detail::invalid() : x.data_); if(!absy) return half(detail::binary, detail::invalid()); if(!absx) return x; if(absx == absy) return half(detail::binary, sign); return half(detail::binary, sign|detail::mod(absx, absy)); } /// Remainder of division. /// **See also:** Documentation for [std::remainder](https://en.cppreference.com/w/cpp/numeric/math/remainder). /// \param x first operand /// \param y second operand /// \return remainder of floating-point division. /// \exception FE_INVALID if \a x is infinite or \a y is 0 or if \a x or \a y is signaling NaN inline half remainder(half x, half y) { unsigned int absx = x.data_ & 0x7FFF, absy = y.data_ & 0x7FFF, sign = x.data_ & 0x8000; if(absx >= 0x7C00 || absy >= 0x7C00) return half(detail::binary, (absx>0x7C00 || absy>0x7C00) ? detail::signal(x.data_, y.data_) : (absx==0x7C00) ? detail::invalid() : x.data_); if(!absy) return half(detail::binary, detail::invalid()); if(absx == absy) return half(detail::binary, sign); return half(detail::binary, sign^detail::mod(absx, absy)); } /// Remainder of division. /// **See also:** Documentation for [std::remquo](https://en.cppreference.com/w/cpp/numeric/math/remquo). /// \param x first operand /// \param y second operand /// \param quo address to store some bits of quotient at /// \return remainder of floating-point division. /// \exception FE_INVALID if \a x is infinite or \a y is 0 or if \a x or \a y is signaling NaN inline half remquo(half x, half y, int *quo) { unsigned int absx = x.data_ & 0x7FFF, absy = y.data_ & 0x7FFF, value = x.data_ & 0x8000; if(absx >= 0x7C00 || absy >= 0x7C00) return half(detail::binary, (absx>0x7C00 || absy>0x7C00) ? detail::signal(x.data_, y.data_) : (absx==0x7C00) ? detail::invalid() : (*quo = 0, x.data_)); if(!absy) return half(detail::binary, detail::invalid()); bool qsign = ((value^y.data_)&0x8000) != 0; int q = 1; if(absx != absy) value ^= detail::mod(absx, absy, &q); return *quo = qsign ? -q : q, half(detail::binary, value); } /// Fused multiply add. /// This function is exact to rounding for all rounding modes. /// /// **See also:** Documentation for [std::fma](https://en.cppreference.com/w/cpp/numeric/math/fma). /// \param x first operand /// \param y second operand /// \param z third operand /// \return ( \a x * \a y ) + \a z rounded as one operation. /// \exception FE_INVALID according to operator*() and operator+() unless any argument is a quiet NaN and no argument is a signaling NaN /// \exception FE_OVERFLOW, ...UNDERFLOW, ...INEXACT according to rounding the final addition inline half fma(half x, half y, half z) { #ifdef HALF_ARITHMETIC_TYPE detail::internal_t fx = detail::half2float(x.data_), fy = detail::half2float(y.data_), fz = detail::half2float(z.data_); #if HALF_ENABLE_CPP11_CMATH && FP_FAST_FMA return half(detail::binary, detail::float2half(std::fma(fx, fy, fz))); #else return half(detail::binary, detail::float2half(fx*fy+fz)); #endif #else int absx = x.data_ & 0x7FFF, absy = y.data_ & 0x7FFF, absz = z.data_ & 0x7FFF, exp = -15; unsigned int sign = (x.data_^y.data_) & 0x8000; bool sub = ((sign^z.data_)&0x8000) != 0; if(absx >= 0x7C00 || absy >= 0x7C00 || absz >= 0x7C00) return (absx>0x7C00 || absy>0x7C00 || absz>0x7C00) ? half(detail::binary, detail::signal(x.data_, y.data_, z.data_)) : (absx==0x7C00) ? half(detail::binary, (!absy || (sub && absz==0x7C00)) ? detail::invalid() : (sign|0x7C00)) : (absy==0x7C00) ? half(detail::binary, (!absx || (sub && absz==0x7C00)) ? detail::invalid() : (sign|0x7C00)) : z; if(!absx || !absy) return absz ? z : half(detail::binary, (half::round_style==std::round_toward_neg_infinity) ? (z.data_|sign) : (z.data_&sign)); for(; absx<0x400; absx<<=1,--exp) ; for(; absy<0x400; absy<<=1,--exp) ; detail::uint32 m = static_cast((absx&0x3FF)|0x400) * static_cast((absy&0x3FF)|0x400); int i = m >> 21; exp += (absx>>10) + (absy>>10) + i; m <<= 3 - i; if(absz) { int expz = 0; for(; absz<0x400; absz<<=1,--expz) ; expz += absz >> 10; detail::uint32 mz = static_cast((absz&0x3FF)|0x400) << 13; if(expz > exp || (expz == exp && mz > m)) { std::swap(m, mz); std::swap(exp, expz); if(sub) sign = z.data_ & 0x8000; } int d = exp - expz; mz = (d<23) ? ((mz>>d)|((mz&((static_cast(1)<(half::round_style==std::round_toward_neg_infinity)<<15); for(; m<0x800000; m<<=1,--exp) ; } else { m += mz; i = m >> 24; m = (m>>i) | (m&i); exp += i; } } if(exp > 30) return half(detail::binary, detail::overflow(sign)); else if(exp < -10) return half(detail::binary, detail::underflow(sign)); return half(detail::binary, detail::fixed2half(m, exp-1, sign)); #endif } /// Maximum of half expressions. /// **See also:** Documentation for [std::fmax](https://en.cppreference.com/w/cpp/numeric/math/fmax). /// \param x first operand /// \param y second operand /// \return maximum of operands, ignoring quiet NaNs /// \exception FE_INVALID if \a x or \a y is signaling NaN inline HALF_CONSTEXPR_NOERR half fmax(half x, half y) { return half(detail::binary, (!isnan(y) && (isnan(x) || (x.data_^(0x8000|(0x8000-(x.data_>>15)))) < (y.data_^(0x8000|(0x8000-(y.data_>>15)))))) ? detail::select(y.data_, x.data_) : detail::select(x.data_, y.data_)); } /// Minimum of half expressions. /// **See also:** Documentation for [std::fmin](https://en.cppreference.com/w/cpp/numeric/math/fmin). /// \param x first operand /// \param y second operand /// \return minimum of operands, ignoring quiet NaNs /// \exception FE_INVALID if \a x or \a y is signaling NaN inline HALF_CONSTEXPR_NOERR half fmin(half x, half y) { return half(detail::binary, (!isnan(y) && (isnan(x) || (x.data_^(0x8000|(0x8000-(x.data_>>15)))) > (y.data_^(0x8000|(0x8000-(y.data_>>15)))))) ? detail::select(y.data_, x.data_) : detail::select(x.data_, y.data_)); } /// Positive difference. /// This function is exact to rounding for all rounding modes. /// /// **See also:** Documentation for [std::fdim](https://en.cppreference.com/w/cpp/numeric/math/fdim). /// \param x first operand /// \param y second operand /// \return \a x - \a y or 0 if difference negative /// \exception FE_... according to operator-(half,half) inline half fdim(half x, half y) { if(isnan(x) || isnan(y)) return half(detail::binary, detail::signal(x.data_, y.data_)); return (x.data_^(0x8000|(0x8000-(x.data_>>15)))) <= (y.data_^(0x8000|(0x8000-(y.data_>>15)))) ? half(detail::binary, 0) : (x-y); } /// Get NaN value. /// **See also:** Documentation for [std::nan](https://en.cppreference.com/w/cpp/numeric/math/nan). /// \param arg string code /// \return quiet NaN inline half nanh(const char *arg) { unsigned int value = 0x7FFF; while(*arg) value ^= static_cast(*arg++) & 0xFF; return half(detail::binary, value); } /// \} /// \anchor exponential /// \name Exponential functions /// \{ /// Exponential function. /// This function is exact to rounding for all rounding modes. /// /// **See also:** Documentation for [std::exp](https://en.cppreference.com/w/cpp/numeric/math/exp). /// \param arg function argument /// \return e raised to \a arg /// \exception FE_INVALID for signaling NaN /// \exception FE_OVERFLOW, ...UNDERFLOW, ...INEXACT according to rounding inline half exp(half arg) { #ifdef HALF_ARITHMETIC_TYPE return half(detail::binary, detail::float2half(std::exp(detail::half2float(arg.data_)))); #else int abs = arg.data_ & 0x7FFF, e = (abs>>10) + (abs<=0x3FF), exp; if(!abs) return half(detail::binary, 0x3C00); if(abs >= 0x7C00) return half(detail::binary, (abs==0x7C00) ? (0x7C00&((arg.data_>>15)-1U)) : detail::signal(arg.data_)); if(abs >= 0x4C80) return half(detail::binary, (arg.data_&0x8000) ? detail::underflow() : detail::overflow()); detail::uint32 m = detail::multiply64(static_cast((abs&0x3FF)+((abs>0x3FF)<<10))<<21, 0xB8AA3B29); if(e < 14) { exp = 0; m >>= 14 - e; } else { exp = m >> (45-e); m = (m<<(e-14)) & 0x7FFFFFFF; } return half(detail::binary, detail::exp2_post(m, exp, (arg.data_&0x8000)!=0, 0, 26)); #endif } /// Binary exponential. /// This function is exact to rounding for all rounding modes. /// /// **See also:** Documentation for [std::exp2](https://en.cppreference.com/w/cpp/numeric/math/exp2). /// \param arg function argument /// \return 2 raised to \a arg /// \exception FE_INVALID for signaling NaN /// \exception FE_OVERFLOW, ...UNDERFLOW, ...INEXACT according to rounding inline half exp2(half arg) { #if defined(HALF_ARITHMETIC_TYPE) && HALF_ENABLE_CPP11_CMATH return half(detail::binary, detail::float2half(std::exp2(detail::half2float(arg.data_)))); #else int abs = arg.data_ & 0x7FFF, e = (abs>>10) + (abs<=0x3FF), exp = (abs&0x3FF) + ((abs>0x3FF)<<10); if(!abs) return half(detail::binary, 0x3C00); if(abs >= 0x7C00) return half(detail::binary, (abs==0x7C00) ? (0x7C00&((arg.data_>>15)-1U)) : detail::signal(arg.data_)); if(abs >= 0x4E40) return half(detail::binary, (arg.data_&0x8000) ? detail::underflow() : detail::overflow()); return half(detail::binary, detail::exp2_post( (static_cast(exp)<<(6+e))&0x7FFFFFFF, exp>>(25-e), (arg.data_&0x8000)!=0, 0, 28)); #endif } /// Exponential minus one. /// This function may be 1 ULP off the correctly rounded exact result in <0.05% of inputs for `std::round_to_nearest` /// and in <1% of inputs for any other rounding mode. /// /// **See also:** Documentation for [std::expm1](https://en.cppreference.com/w/cpp/numeric/math/expm1). /// \param arg function argument /// \return e raised to \a arg and subtracted by 1 /// \exception FE_INVALID for signaling NaN /// \exception FE_OVERFLOW, ...UNDERFLOW, ...INEXACT according to rounding inline half expm1(half arg) { #if defined(HALF_ARITHMETIC_TYPE) && HALF_ENABLE_CPP11_CMATH return half(detail::binary, detail::float2half(std::expm1(detail::half2float(arg.data_)))); #else unsigned int abs = arg.data_ & 0x7FFF, sign = arg.data_ & 0x8000, e = (abs>>10) + (abs<=0x3FF), exp; if(!abs) return arg; if(abs >= 0x7C00) return half(detail::binary, (abs==0x7C00) ? (0x7C00+(sign>>1)) : detail::signal(arg.data_)); if(abs >= 0x4A00) return half(detail::binary, (arg.data_&0x8000) ? detail::rounded(0xBBFF, 1, 1) : detail::overflow()); detail::uint32 m = detail::multiply64(static_cast((abs&0x3FF)+((abs>0x3FF)<<10))<<21, 0xB8AA3B29); if(e < 14) { exp = 0; m >>= 14 - e; } else { exp = m >> (45-e); m = (m<<(e-14)) & 0x7FFFFFFF; } m = detail::exp2(m); if(sign) { int s = 0; if(m > 0x80000000) { ++exp; m = detail::divide64(0x80000000, m, s); } m = 0x80000000 - ((m>>exp)|((m&((static_cast(1)<>exp) : 1; for(exp+=14; m<0x80000000 && exp; m<<=1,--exp) ; if(exp > 29) return half(detail::binary, detail::overflow()); return half(detail::binary, detail::rounded(sign+(exp<<10)+(m>>21), (m>>20)&1, (m&0xFFFFF)!=0)); #endif } /// Natural logarithm. /// This function is exact to rounding for all rounding modes. /// /// **See also:** Documentation for [std::log](https://en.cppreference.com/w/cpp/numeric/math/log). /// \param arg function argument /// \return logarithm of \a arg to base e /// \exception FE_INVALID for signaling NaN or negative argument /// \exception FE_DIVBYZERO for 0 /// \exception FE_OVERFLOW, ...UNDERFLOW, ...INEXACT according to rounding inline half log(half arg) { #ifdef HALF_ARITHMETIC_TYPE return half(detail::binary, detail::float2half(std::log(detail::half2float(arg.data_)))); #else int abs = arg.data_ & 0x7FFF, exp = -15; if(!abs) return half(detail::binary, detail::pole(0x8000)); if(arg.data_ & 0x8000) return half(detail::binary, (arg.data_<=0xFC00) ? detail::invalid() : detail::signal(arg.data_)); if(abs >= 0x7C00) return (abs==0x7C00) ? arg : half(detail::binary, detail::signal(arg.data_)); for(; abs<0x400; abs<<=1,--exp) ; exp += abs >> 10; return half(detail::binary, detail::log2_post( detail::log2(static_cast((abs&0x3FF)|0x400)<<20, 27)+8, exp, 17)); #endif } /// Common logarithm. /// This function is exact to rounding for all rounding modes. /// /// **See also:** Documentation for [std::log10](https://en.cppreference.com/w/cpp/numeric/math/log10). /// \param arg function argument /// \return logarithm of \a arg to base 10 /// \exception FE_INVALID for signaling NaN or negative argument /// \exception FE_DIVBYZERO for 0 /// \exception FE_OVERFLOW, ...UNDERFLOW, ...INEXACT according to rounding inline half log10(half arg) { #ifdef HALF_ARITHMETIC_TYPE return half(detail::binary, detail::float2half(std::log10(detail::half2float(arg.data_)))); #else int abs = arg.data_ & 0x7FFF, exp = -15; if(!abs) return half(detail::binary, detail::pole(0x8000)); if(arg.data_ & 0x8000) return half(detail::binary, (arg.data_<=0xFC00) ? detail::invalid() : detail::signal(arg.data_)); if(abs >= 0x7C00) return (abs==0x7C00) ? arg : half(detail::binary, detail::signal(arg.data_)); switch(abs) { case 0x4900: return half(detail::binary, 0x3C00); case 0x5640: return half(detail::binary, 0x4000); case 0x63D0: return half(detail::binary, 0x4200); case 0x70E2: return half(detail::binary, 0x4400); } for(; abs<0x400; abs<<=1,--exp) ; exp += abs >> 10; return half(detail::binary, detail::log2_post( detail::log2(static_cast((abs&0x3FF)|0x400)<<20, 27)+8, exp, 16)); #endif } /// Binary logarithm. /// This function is exact to rounding for all rounding modes. /// /// **See also:** Documentation for [std::log2](https://en.cppreference.com/w/cpp/numeric/math/log2). /// \param arg function argument /// \return logarithm of \a arg to base 2 /// \exception FE_INVALID for signaling NaN or negative argument /// \exception FE_DIVBYZERO for 0 /// \exception FE_OVERFLOW, ...UNDERFLOW, ...INEXACT according to rounding inline half log2(half arg) { #if defined(HALF_ARITHMETIC_TYPE) && HALF_ENABLE_CPP11_CMATH return half(detail::binary, detail::float2half(std::log2(detail::half2float(arg.data_)))); #else int abs = arg.data_ & 0x7FFF, exp = -15, s = 0; if(!abs) return half(detail::binary, detail::pole(0x8000)); if(arg.data_ & 0x8000) return half(detail::binary, (arg.data_<=0xFC00) ? detail::invalid() : detail::signal(arg.data_)); if(abs >= 0x7C00) return (abs==0x7C00) ? arg : half(detail::binary, detail::signal(arg.data_)); if(abs == 0x3C00) return half(detail::binary, 0); for(; abs<0x400; abs<<=1,--exp) ; exp += (abs>>10); if(!(abs&0x3FF)) { unsigned int value = static_cast(exp<0) << 15, m = std::abs(exp) << 6; for(exp=18; m<0x400; m<<=1,--exp) ; return half(detail::binary, value+(exp<<10)+m); } detail::uint32 ilog = exp, sign = detail::sign_mask(ilog), m = (((ilog<<27)+(detail::log2(static_cast((abs&0x3FF)|0x400)<<20, 28)>>4))^sign) - sign; if(!m) return half(detail::binary, 0); for(exp=14; m<0x8000000 && exp; m<<=1,--exp) ; for(; m>0xFFFFFFF; m>>=1,++exp) s |= m & 1; return half(detail::binary, detail::fixed2half(m, exp, sign&0x8000, s)); #endif } /// Natural logarithm plus one. /// This function may be 1 ULP off the correctly rounded exact result in <0.05% of inputs for `std::round_to_nearest` /// and in ~1% of inputs for any other rounding mode. /// /// **See also:** Documentation for [std::log1p](https://en.cppreference.com/w/cpp/numeric/math/log1p). /// \param arg function argument /// \return logarithm of \a arg plus 1 to base e /// \exception FE_INVALID for signaling NaN or argument <-1 /// \exception FE_DIVBYZERO for -1 /// \exception FE_OVERFLOW, ...UNDERFLOW, ...INEXACT according to rounding inline half log1p(half arg) { #if defined(HALF_ARITHMETIC_TYPE) && HALF_ENABLE_CPP11_CMATH return half(detail::binary, detail::float2half(std::log1p(detail::half2float(arg.data_)))); #else if(arg.data_ >= 0xBC00) return half(detail::binary, (arg.data_==0xBC00) ? detail::pole(0x8000) : (arg.data_<=0xFC00) ? detail::invalid() : detail::signal(arg.data_)); int abs = arg.data_ & 0x7FFF, exp = -15; if(!abs || abs >= 0x7C00) return (abs>0x7C00) ? half(detail::binary, detail::signal(arg.data_)) : arg; for(; abs<0x400; abs<<=1,--exp) ; exp += abs >> 10; detail::uint32 m = static_cast((abs&0x3FF)|0x400) << 20; if(arg.data_ & 0x8000) { m = 0x40000000 - (m>>-exp); for(exp=0; m<0x40000000; m<<=1,--exp) ; } else { if(exp < 0) { m = 0x40000000 + (m>>-exp); exp = 0; } else { m += 0x40000000 >> exp; int i = m >> 31; m >>= i; exp += i; } } return half(detail::binary, detail::log2_post(detail::log2(m), exp, 17)); #endif } /// \} /// \anchor power /// \name Power functions /// \{ /// Square root. /// This function is exact to rounding for all rounding modes. /// /// **See also:** Documentation for [std::sqrt](https://en.cppreference.com/w/cpp/numeric/math/sqrt). /// \param arg function argument /// \return square root of \a arg /// \exception FE_INVALID for signaling NaN and negative arguments /// \exception FE_INEXACT according to rounding inline half sqrt(half arg) { #ifdef HALF_ARITHMETIC_TYPE return half(detail::binary, detail::float2half(std::sqrt(detail::half2float(arg.data_)))); #else int abs = arg.data_ & 0x7FFF, exp = 15; if(!abs || arg.data_ >= 0x7C00) return half(detail::binary, (abs>0x7C00) ? detail::signal(arg.data_) : (arg.data_>0x8000) ? detail::invalid() : arg.data_); for(; abs<0x400; abs<<=1,--exp) ; detail::uint32 r = static_cast((abs&0x3FF)|0x400) << 10, m = detail::sqrt<20>(r, exp+=abs>>10); return half(detail::binary, detail::rounded((exp<<10)+(m&0x3FF), r>m, r!=0)); #endif } /// Inverse square root. /// This function is exact to rounding for all rounding modes and thus generally more accurate than directly computing /// 1 / sqrt(\a arg) in half-precision, in addition to also being faster. /// \param arg function argument /// \return reciprocal of square root of \a arg /// \exception FE_INVALID for signaling NaN and negative arguments /// \exception FE_INEXACT according to rounding inline half rsqrt(half arg) { #ifdef HALF_ARITHMETIC_TYPE return half(detail::binary, detail::float2half(detail::internal_t(1)/std::sqrt(detail::half2float(arg.data_)))); #else unsigned int abs = arg.data_ & 0x7FFF, bias = 0x4000; if(!abs || arg.data_ >= 0x7C00) return half(detail::binary, (abs>0x7C00) ? detail::signal(arg.data_) : (arg.data_>0x8000) ? detail::invalid() : !abs ? detail::pole(arg.data_&0x8000) : 0); for(; abs<0x400; abs<<=1,bias-=0x400) ; unsigned int frac = (abs+=bias) & 0x7FF; if(frac == 0x400) return half(detail::binary, 0x7A00-(abs>>1)); if((half::round_style == std::round_to_nearest && (frac == 0x3FE || frac == 0x76C)) || (half::round_style != std::round_to_nearest && (frac == 0x15A || frac == 0x3FC || frac == 0x401 || frac == 0x402 || frac == 0x67B))) return pow(arg, half(detail::binary, 0xB800)); detail::uint32 f = 0x17376 - abs, mx = (abs&0x3FF) | 0x400, my = ((f>>1)&0x3FF) | 0x400, mz = my * my; int expy = (f>>11) - 31, expx = 32 - (abs>>10), i = mz >> 21; for(mz=0x60000000-(((mz>>i)*mx)>>(expx-2*expy-i)); mz<0x40000000; mz<<=1,--expy) ; i = (my*=mz>>10) >> 31; expy += i; my = (my>>(20+i)) + 1; i = (mz=my*my) >> 21; for(mz=0x60000000-(((mz>>i)*mx)>>(expx-2*expy-i)); mz<0x40000000; mz<<=1,--expy) ; i = (my*=(mz>>10)+1) >> 31; return half(detail::binary, detail::fixed2half(my>>i, expy+i+14)); #endif } /// Cubic root. /// This function is exact to rounding for all rounding modes. /// /// **See also:** Documentation for [std::cbrt](https://en.cppreference.com/w/cpp/numeric/math/cbrt). /// \param arg function argument /// \return cubic root of \a arg /// \exception FE_INVALID for signaling NaN /// \exception FE_INEXACT according to rounding inline half cbrt(half arg) { #if defined(HALF_ARITHMETIC_TYPE) && HALF_ENABLE_CPP11_CMATH return half(detail::binary, detail::float2half(std::cbrt(detail::half2float(arg.data_)))); #else int abs = arg.data_ & 0x7FFF, exp = -15; if(!abs || abs == 0x3C00 || abs >= 0x7C00) return (abs>0x7C00) ? half(detail::binary, detail::signal(arg.data_)) : arg; for(; abs<0x400; abs<<=1, --exp); detail::uint32 ilog = exp + (abs>>10), sign = detail::sign_mask(ilog), f, m = (((ilog<<27)+(detail::log2(static_cast((abs&0x3FF)|0x400)<<20, 24)>>4))^sign) - sign; for(exp=2; m<0x80000000; m<<=1,--exp) ; m = detail::multiply64(m, 0xAAAAAAAB); int i = m >> 31, s; exp += i; m <<= 1 - i; if(exp < 0) { f = m >> -exp; exp = 0; } else { f = (m<> (31-exp); } m = detail::exp2(f, (half::round_style==std::round_to_nearest) ? 29 : 26); if(sign) { if(m > 0x80000000) { m = detail::divide64(0x80000000, m, s); ++exp; } exp = -exp; } return half(detail::binary, (half::round_style==std::round_to_nearest) ? detail::fixed2half(m, exp+14, arg.data_&0x8000) : detail::fixed2half((m+0x80)>>8, exp+14, arg.data_&0x8000)); #endif } /// Hypotenuse function. /// This function is exact to rounding for all rounding modes. /// /// **See also:** Documentation for [std::hypot](https://en.cppreference.com/w/cpp/numeric/math/hypot). /// \param x first argument /// \param y second argument /// \return square root of sum of squares without internal over- or underflows /// \exception FE_INVALID if \a x or \a y is signaling NaN /// \exception FE_OVERFLOW, ...UNDERFLOW, ...INEXACT according to rounding of the final square root inline half hypot(half x, half y) { #ifdef HALF_ARITHMETIC_TYPE detail::internal_t fx = detail::half2float(x.data_), fy = detail::half2float(y.data_); #if HALF_ENABLE_CPP11_CMATH return half(detail::binary, detail::float2half(std::hypot(fx, fy))); #else return half(detail::binary, detail::float2half(std::sqrt(fx*fx+fy*fy))); #endif #else int absx = x.data_ & 0x7FFF, absy = y.data_ & 0x7FFF, expx = 0, expy = 0; if(absx >= 0x7C00 || absy >= 0x7C00) return half(detail::binary, (absx==0x7C00) ? detail::select(0x7C00, y.data_) : (absy==0x7C00) ? detail::select(0x7C00, x.data_) : detail::signal(x.data_, y.data_)); if(!absx) return half(detail::binary, absy ? detail::check_underflow(absy) : 0); if(!absy) return half(detail::binary, detail::check_underflow(absx)); if(absy > absx) std::swap(absx, absy); for(; absx<0x400; absx<<=1,--expx) ; for(; absy<0x400; absy<<=1,--expy) ; detail::uint32 mx = (absx&0x3FF) | 0x400, my = (absy&0x3FF) | 0x400; mx *= mx; my *= my; int ix = mx >> 21, iy = my >> 21; expx = 2*(expx+(absx>>10)) - 15 + ix; expy = 2*(expy+(absy>>10)) - 15 + iy; mx <<= 10 - ix; my <<= 10 - iy; int d = expx - expy; my = (d<30) ? ((my>>d)|((my&((static_cast(1)<(mx+my, expx)); #endif } /// Hypotenuse function. /// This function is exact to rounding for all rounding modes. /// /// **See also:** Documentation for [std::hypot](https://en.cppreference.com/w/cpp/numeric/math/hypot). /// \param x first argument /// \param y second argument /// \param z third argument /// \return square root of sum of squares without internal over- or underflows /// \exception FE_INVALID if \a x, \a y or \a z is signaling NaN /// \exception FE_OVERFLOW, ...UNDERFLOW, ...INEXACT according to rounding of the final square root inline half hypot(half x, half y, half z) { #ifdef HALF_ARITHMETIC_TYPE detail::internal_t fx = detail::half2float(x.data_), fy = detail::half2float(y.data_), fz = detail::half2float(z.data_); return half(detail::binary, detail::float2half(std::sqrt(fx*fx+fy*fy+fz*fz))); #else int absx = x.data_ & 0x7FFF, absy = y.data_ & 0x7FFF, absz = z.data_ & 0x7FFF, expx = 0, expy = 0, expz = 0; if(!absx) return hypot(y, z); if(!absy) return hypot(x, z); if(!absz) return hypot(x, y); if(absx >= 0x7C00 || absy >= 0x7C00 || absz >= 0x7C00) return half(detail::binary, (absx==0x7C00) ? detail::select(0x7C00, detail::select(y.data_, z.data_)) : (absy==0x7C00) ? detail::select(0x7C00, detail::select(x.data_, z.data_)) : (absz==0x7C00) ? detail::select(0x7C00, detail::select(x.data_, y.data_)) : detail::signal(x.data_, y.data_, z.data_)); if(absz > absy) std::swap(absy, absz); if(absy > absx) std::swap(absx, absy); if(absz > absy) std::swap(absy, absz); for(; absx<0x400; absx<<=1,--expx) ; for(; absy<0x400; absy<<=1,--expy) ; for(; absz<0x400; absz<<=1,--expz) ; detail::uint32 mx = (absx&0x3FF) | 0x400, my = (absy&0x3FF) | 0x400, mz = (absz&0x3FF) | 0x400; mx *= mx; my *= my; mz *= mz; int ix = mx >> 21, iy = my >> 21, iz = mz >> 21; expx = 2*(expx+(absx>>10)) - 15 + ix; expy = 2*(expy+(absy>>10)) - 15 + iy; expz = 2*(expz+(absz>>10)) - 15 + iz; mx <<= 10 - ix; my <<= 10 - iy; mz <<= 10 - iz; int d = expy - expz; mz = (d<30) ? ((mz>>d)|((mz&((static_cast(1)<>1) | (my&1); if(++expy > expx) { std::swap(mx, my); std::swap(expx, expy); } } d = expx - expy; my = (d<30) ? ((my>>d)|((my&((static_cast(1)<(mx+my, expx)); #endif } /// Power function. /// This function may be 1 ULP off the correctly rounded exact result for any rounding mode in ~0.00025% of inputs. /// /// **See also:** Documentation for [std::pow](https://en.cppreference.com/w/cpp/numeric/math/pow). /// \param x base /// \param y exponent /// \return \a x raised to \a y /// \exception FE_INVALID if \a x or \a y is signaling NaN or if \a x is finite an negative and \a y is finite and not integral /// \exception FE_DIVBYZERO if \a x is 0 and \a y is negative /// \exception FE_OVERFLOW, ...UNDERFLOW, ...INEXACT according to rounding inline half pow(half x, half y) { #ifdef HALF_ARITHMETIC_TYPE return half(detail::binary, detail::float2half(std::pow(detail::half2float(x.data_), detail::half2float(y.data_)))); #else int absx = x.data_ & 0x7FFF, absy = y.data_ & 0x7FFF, exp = -15; if(!absy || x.data_ == 0x3C00) return half(detail::binary, detail::select(0x3C00, (x.data_==0x3C00) ? y.data_ : x.data_)); bool is_int = absy >= 0x6400 || (absy>=0x3C00 && !(absy&((1<<(25-(absy>>10)))-1))); unsigned int sign = x.data_ & (static_cast((absy<0x6800)&&is_int&&((absy>>(25-(absy>>10)))&1))<<15); if(absx >= 0x7C00 || absy >= 0x7C00) return half(detail::binary, (absx>0x7C00 || absy>0x7C00) ? detail::signal(x.data_, y.data_) : (absy==0x7C00) ? ((absx==0x3C00) ? 0x3C00 : (!absx && y.data_==0xFC00) ? detail::pole() : (0x7C00&-((y.data_>>15)^(absx>0x3C00)))) : (sign|(0x7C00&((y.data_>>15)-1U)))); if(!absx) return half(detail::binary, (y.data_&0x8000) ? detail::pole(sign) : sign); if((x.data_&0x8000) && !is_int) return half(detail::binary, detail::invalid()); if(x.data_ == 0xBC00) return half(detail::binary, sign|0x3C00); switch(y.data_) { case 0x3800: return sqrt(x); case 0x3C00: return half(detail::binary, detail::check_underflow(x.data_)); case 0x4000: return x * x; case 0xBC00: return half(detail::binary, 0x3C00) / x; } for(; absx<0x400; absx<<=1,--exp) ; detail::uint32 ilog = exp + (absx>>10), msign = detail::sign_mask(ilog), f, m = (((ilog<<27)+((detail::log2(static_cast((absx&0x3FF)|0x400)<<20)+8)>>4))^msign) - msign; for(exp=-11; m<0x80000000; m<<=1,--exp) ; for(; absy<0x400; absy<<=1,--exp) ; m = detail::multiply64(m, static_cast((absy&0x3FF)|0x400)<<21); int i = m >> 31; exp += (absy>>10) + i; m <<= 1 - i; if(exp < 0) { f = m >> -exp; exp = 0; } else { f = (m<> (31-exp); } return half(detail::binary, detail::exp2_post(f, exp, ((msign&1)^(y.data_>>15))!=0, sign)); #endif } /// \} /// \anchor trigonometric /// \name Trigonometric functions /// \{ /// Compute sine and cosine simultaneously. /// This returns the same results as sin() and cos() but is faster than calling each function individually. /// /// This function is exact to rounding for all rounding modes. /// \param arg function argument /// \param sin variable to take sine of \a arg /// \param cos variable to take cosine of \a arg /// \exception FE_INVALID for signaling NaN or infinity /// \exception FE_OVERFLOW, ...UNDERFLOW, ...INEXACT according to rounding inline void sincos(half arg, half *sin, half *cos) { #ifdef HALF_ARITHMETIC_TYPE detail::internal_t f = detail::half2float(arg.data_); *sin = half(detail::binary, detail::float2half(std::sin(f))); *cos = half(detail::binary, detail::float2half(std::cos(f))); #else int abs = arg.data_ & 0x7FFF, sign = arg.data_ >> 15, k; if(abs >= 0x7C00) *sin = *cos = half(detail::binary, (abs==0x7C00) ? detail::invalid() : detail::signal(arg.data_)); else if(!abs) { *sin = arg; *cos = half(detail::binary, 0x3C00); } else if(abs < 0x2500) { *sin = half(detail::binary, detail::rounded(arg.data_-1, 1, 1)); *cos = half(detail::binary, detail::rounded(0x3BFF, 1, 1)); } else { if(half::round_style != std::round_to_nearest) { switch(abs) { case 0x48B7: *sin = half(detail::binary, detail::rounded((~arg.data_&0x8000)|0x1D07, 1, 1)); *cos = half(detail::binary, detail::rounded(0xBBFF, 1, 1)); return; case 0x598C: *sin = half(detail::binary, detail::rounded((arg.data_&0x8000)|0x3BFF, 1, 1)); *cos = half(detail::binary, detail::rounded(0x80FC, 1, 1)); return; case 0x6A64: *sin = half(detail::binary, detail::rounded((~arg.data_&0x8000)|0x3BFE, 1, 1)); *cos = half(detail::binary, detail::rounded(0x27FF, 1, 1)); return; case 0x6D8C: *sin = half(detail::binary, detail::rounded((arg.data_&0x8000)|0x0FE6, 1, 1)); *cos = half(detail::binary, detail::rounded(0x3BFF, 1, 1)); return; } } std::pair sc = detail::sincos(detail::angle_arg(abs, k), 28); switch(k & 3) { case 1: sc = std::make_pair(sc.second, -sc.first); break; case 2: sc = std::make_pair(-sc.first, -sc.second); break; case 3: sc = std::make_pair(-sc.second, sc.first); break; } *sin = half(detail::binary, detail::fixed2half((sc.first^-static_cast(sign))+sign)); *cos = half(detail::binary, detail::fixed2half(sc.second)); } #endif } /// Sine function. /// This function is exact to rounding for all rounding modes. /// /// **See also:** Documentation for [std::sin](https://en.cppreference.com/w/cpp/numeric/math/sin). /// \param arg function argument /// \return sine value of \a arg /// \exception FE_INVALID for signaling NaN or infinity /// \exception FE_OVERFLOW, ...UNDERFLOW, ...INEXACT according to rounding inline half sin(half arg) { #ifdef HALF_ARITHMETIC_TYPE return half(detail::binary, detail::float2half(std::sin(detail::half2float(arg.data_)))); #else int abs = arg.data_ & 0x7FFF, k; if(!abs) return arg; if(abs >= 0x7C00) return half(detail::binary, (abs==0x7C00) ? detail::invalid() : detail::signal(arg.data_)); if(abs < 0x2900) return half(detail::binary, detail::rounded(arg.data_-1, 1, 1)); if(half::round_style != std::round_to_nearest) switch(abs) { case 0x48B7: return half(detail::binary, detail::rounded((~arg.data_&0x8000)|0x1D07, 1, 1)); case 0x6A64: return half(detail::binary, detail::rounded((~arg.data_&0x8000)|0x3BFE, 1, 1)); case 0x6D8C: return half(detail::binary, detail::rounded((arg.data_&0x8000)|0x0FE6, 1, 1)); } std::pair sc = detail::sincos(detail::angle_arg(abs, k), 28); detail::uint32 sign = -static_cast(((k>>1)&1)^(arg.data_>>15)); return half(detail::binary, detail::fixed2half((((k&1) ? sc.second : sc.first)^sign) - sign)); #endif } /// Cosine function. /// This function is exact to rounding for all rounding modes. /// /// **See also:** Documentation for [std::cos](https://en.cppreference.com/w/cpp/numeric/math/cos). /// \param arg function argument /// \return cosine value of \a arg /// \exception FE_INVALID for signaling NaN or infinity /// \exception FE_OVERFLOW, ...UNDERFLOW, ...INEXACT according to rounding inline half cos(half arg) { #ifdef HALF_ARITHMETIC_TYPE return half(detail::binary, detail::float2half(std::cos(detail::half2float(arg.data_)))); #else int abs = arg.data_ & 0x7FFF, k; if(!abs) return half(detail::binary, 0x3C00); if(abs >= 0x7C00) return half(detail::binary, (abs==0x7C00) ? detail::invalid() : detail::signal(arg.data_)); if(abs < 0x2500) return half(detail::binary, detail::rounded(0x3BFF, 1, 1)); if(half::round_style != std::round_to_nearest && abs == 0x598C) return half(detail::binary, detail::rounded(0x80FC, 1, 1)); std::pair sc = detail::sincos(detail::angle_arg(abs, k), 28); detail::uint32 sign = -static_cast(((k>>1)^k)&1); return half(detail::binary, detail::fixed2half((((k&1) ? sc.first : sc.second)^sign) - sign)); #endif } /// Tangent function. /// This function is exact to rounding for all rounding modes. /// /// **See also:** Documentation for [std::tan](https://en.cppreference.com/w/cpp/numeric/math/tan). /// \param arg function argument /// \return tangent value of \a arg /// \exception FE_INVALID for signaling NaN or infinity /// \exception FE_OVERFLOW, ...UNDERFLOW, ...INEXACT according to rounding inline half tan(half arg) { #ifdef HALF_ARITHMETIC_TYPE return half(detail::binary, detail::float2half(std::tan(detail::half2float(arg.data_)))); #else int abs = arg.data_ & 0x7FFF, exp = 13, k; if(!abs) return arg; if(abs >= 0x7C00) return half(detail::binary, (abs==0x7C00) ? detail::invalid() : detail::signal(arg.data_)); if(abs < 0x2700) return half(detail::binary, detail::rounded(arg.data_, 0, 1)); if(half::round_style != std::round_to_nearest) switch(abs) { case 0x658C: return half(detail::binary, detail::rounded((arg.data_&0x8000)|0x07E6, 1, 1)); case 0x7330: return half(detail::binary, detail::rounded((~arg.data_&0x8000)|0x4B62, 1, 1)); } std::pair sc = detail::sincos(detail::angle_arg(abs, k), 30); if(k & 1) sc = std::make_pair(-sc.second, sc.first); detail::uint32 signy = detail::sign_mask(sc.first), signx = detail::sign_mask(sc.second); detail::uint32 my = (sc.first^signy) - signy, mx = (sc.second^signx) - signx; for(; my<0x80000000; my<<=1,--exp) ; for(; mx<0x80000000; mx<<=1,++exp) ; return half(detail::binary, detail::tangent_post(my, mx, exp, (signy^signx^arg.data_)&0x8000)); #endif } /// Arc sine. /// This function is exact to rounding for all rounding modes. /// /// **See also:** Documentation for [std::asin](https://en.cppreference.com/w/cpp/numeric/math/asin). /// \param arg function argument /// \return arc sine value of \a arg /// \exception FE_INVALID for signaling NaN or if abs(\a arg) > 1 /// \exception FE_OVERFLOW, ...UNDERFLOW, ...INEXACT according to rounding inline half asin(half arg) { #ifdef HALF_ARITHMETIC_TYPE return half(detail::binary, detail::float2half(std::asin(detail::half2float(arg.data_)))); #else unsigned int abs = arg.data_ & 0x7FFF, sign = arg.data_ & 0x8000; if(!abs) return arg; if(abs >= 0x3C00) return half(detail::binary, (abs>0x7C00) ? detail::signal(arg.data_) : (abs>0x3C00) ? detail::invalid() : detail::rounded(sign|0x3E48, 0, 1)); if(abs < 0x2900) return half(detail::binary, detail::rounded(arg.data_, 0, 1)); if(half::round_style != std::round_to_nearest && (abs == 0x2B44 || abs == 0x2DC3)) return half(detail::binary, detail::rounded(arg.data_+1, 1, 1)); std::pair sc = detail::atan2_args(abs); detail::uint32 m = detail::atan2(sc.first, sc.second, (half::round_style==std::round_to_nearest) ? 27 : 26); return half(detail::binary, detail::fixed2half(m, 14, sign)); #endif } /// Arc cosine function. /// This function is exact to rounding for all rounding modes. /// /// **See also:** Documentation for [std::acos](https://en.cppreference.com/w/cpp/numeric/math/acos). /// \param arg function argument /// \return arc cosine value of \a arg /// \exception FE_INVALID for signaling NaN or if abs(\a arg) > 1 /// \exception FE_OVERFLOW, ...UNDERFLOW, ...INEXACT according to rounding inline half acos(half arg) { #ifdef HALF_ARITHMETIC_TYPE return half(detail::binary, detail::float2half(std::acos(detail::half2float(arg.data_)))); #else unsigned int abs = arg.data_ & 0x7FFF, sign = arg.data_ >> 15; if(!abs) return half(detail::binary, detail::rounded(0x3E48, 0, 1)); if(abs >= 0x3C00) return half(detail::binary, (abs>0x7C00) ? detail::signal(arg.data_) : (abs>0x3C00) ? detail::invalid() : sign ? detail::rounded(0x4248, 0, 1) : 0); std::pair cs = detail::atan2_args(abs); detail::uint32 m = detail::atan2(cs.second, cs.first, 28); return half(detail::binary, detail::fixed2half(sign ? (0xC90FDAA2-m) : m, 15, 0, sign)); #endif } /// Arc tangent function. /// This function is exact to rounding for all rounding modes. /// /// **See also:** Documentation for [std::atan](https://en.cppreference.com/w/cpp/numeric/math/atan). /// \param arg function argument /// \return arc tangent value of \a arg /// \exception FE_INVALID for signaling NaN /// \exception FE_OVERFLOW, ...UNDERFLOW, ...INEXACT according to rounding inline half atan(half arg) { #ifdef HALF_ARITHMETIC_TYPE return half(detail::binary, detail::float2half(std::atan(detail::half2float(arg.data_)))); #else unsigned int abs = arg.data_ & 0x7FFF, sign = arg.data_ & 0x8000; if(!abs) return arg; if(abs >= 0x7C00) return half(detail::binary, (abs==0x7C00) ? detail::rounded(sign|0x3E48, 0, 1) : detail::signal(arg.data_)); if(abs <= 0x2700) return half(detail::binary, detail::rounded(arg.data_-1, 1, 1)); int exp = (abs>>10) + (abs<=0x3FF); detail::uint32 my = (abs&0x3FF) | ((abs>0x3FF)<<10); detail::uint32 m = (exp>15) ? detail::atan2(my<<19, 0x20000000>>(exp-15), (half::round_style==std::round_to_nearest) ? 26 : 24) : detail::atan2(my<<(exp+4), 0x20000000, (half::round_style==std::round_to_nearest) ? 30 : 28); return half(detail::binary, detail::fixed2half(m, 14, sign)); #endif } /// Arc tangent function. /// This function may be 1 ULP off the correctly rounded exact result in ~0.005% of inputs for `std::round_to_nearest`, /// in ~0.1% of inputs for `std::round_toward_zero` and in ~0.02% of inputs for any other rounding mode. /// /// **See also:** Documentation for [std::atan2](https://en.cppreference.com/w/cpp/numeric/math/atan2). /// \param y numerator /// \param x denominator /// \return arc tangent value /// \exception FE_INVALID if \a x or \a y is signaling NaN /// \exception FE_OVERFLOW, ...UNDERFLOW, ...INEXACT according to rounding inline half atan2(half y, half x) { #ifdef HALF_ARITHMETIC_TYPE return half(detail::binary, detail::float2half(std::atan2(detail::half2float(y.data_), detail::half2float(x.data_)))); #else unsigned int absx = x.data_ & 0x7FFF, absy = y.data_ & 0x7FFF, signx = x.data_ >> 15, signy = y.data_ & 0x8000; if(absx >= 0x7C00 || absy >= 0x7C00) { if(absx > 0x7C00 || absy > 0x7C00) return half(detail::binary, detail::signal(x.data_, y.data_)); if(absy == 0x7C00) return half(detail::binary, (absx<0x7C00) ? detail::rounded(signy|0x3E48, 0, 1) : signx ? detail::rounded(signy|0x40B6, 0, 1) : detail::rounded(signy|0x3A48, 0, 1)); return (x.data_==0x7C00) ? half(detail::binary, signy) : half(detail::binary, detail::rounded(signy|0x4248, 0, 1)); } if(!absy) return signx ? half(detail::binary, detail::rounded(signy|0x4248, 0, 1)) : y; if(!absx) return half(detail::binary, detail::rounded(signy|0x3E48, 0, 1)); int d = (absy>>10) + (absy<=0x3FF) - (absx>>10) - (absx<=0x3FF); if(d > (signx ? 18 : 12)) return half(detail::binary, detail::rounded(signy|0x3E48, 0, 1)); if(signx && d < -11) return half(detail::binary, detail::rounded(signy|0x4248, 0, 1)); if(!signx && d < ((half::round_style==std::round_toward_zero) ? -15 : -9)) { for(; absy<0x400; absy<<=1,--d) ; detail::uint32 mx = ((absx<<1)&0x7FF) | 0x800, my = ((absy<<1)&0x7FF) | 0x800; int i = my < mx; d -= i; if(d < -25) return half(detail::binary, detail::underflow(signy)); my <<= 11 + i; return half(detail::binary, detail::fixed2half(my/mx, d+14, signy, my%mx!=0)); } detail::uint32 m = detail::atan2( ((absy&0x3FF)|((absy>0x3FF)<<10))<<(19+((d<0) ? d : (d>0) ? 0 : -1)), ((absx&0x3FF)|((absx>0x3FF)<<10))<<(19-((d>0) ? d : (d<0) ? 0 : 1))); return half(detail::binary, detail::fixed2half(signx ? (0xC90FDAA2-m) : m, 15, signy, signx)); #endif } /// \} /// \anchor hyperbolic /// \name Hyperbolic functions /// \{ /// Hyperbolic sine. /// This function is exact to rounding for all rounding modes. /// /// **See also:** Documentation for [std::sinh](https://en.cppreference.com/w/cpp/numeric/math/sinh). /// \param arg function argument /// \return hyperbolic sine value of \a arg /// \exception FE_INVALID for signaling NaN /// \exception FE_OVERFLOW, ...UNDERFLOW, ...INEXACT according to rounding inline half sinh(half arg) { #ifdef HALF_ARITHMETIC_TYPE return half(detail::binary, detail::float2half(std::sinh(detail::half2float(arg.data_)))); #else int abs = arg.data_ & 0x7FFF, exp; if(!abs || abs >= 0x7C00) return (abs>0x7C00) ? half(detail::binary, detail::signal(arg.data_)) : arg; if(abs <= 0x2900) return half(detail::binary, detail::rounded(arg.data_, 0, 1)); std::pair mm = detail::hyperbolic_args(abs, exp, (half::round_style==std::round_to_nearest) ? 29 : 27); detail::uint32 m = mm.first - mm.second; for(exp+=13; m<0x80000000 && exp; m<<=1,--exp) ; unsigned int sign = arg.data_ & 0x8000; if(exp > 29) return half(detail::binary, detail::overflow(sign)); return half(detail::binary, detail::fixed2half(m, exp, sign)); #endif } /// Hyperbolic cosine. /// This function is exact to rounding for all rounding modes. /// /// **See also:** Documentation for [std::cosh](https://en.cppreference.com/w/cpp/numeric/math/cosh). /// \param arg function argument /// \return hyperbolic cosine value of \a arg /// \exception FE_INVALID for signaling NaN /// \exception FE_OVERFLOW, ...UNDERFLOW, ...INEXACT according to rounding inline half cosh(half arg) { #ifdef HALF_ARITHMETIC_TYPE return half(detail::binary, detail::float2half(std::cosh(detail::half2float(arg.data_)))); #else int abs = arg.data_ & 0x7FFF, exp; if(!abs) return half(detail::binary, 0x3C00); if(abs >= 0x7C00) return half(detail::binary, (abs>0x7C00) ? detail::signal(arg.data_) : 0x7C00); std::pair mm = detail::hyperbolic_args(abs, exp, (half::round_style==std::round_to_nearest) ? 23 : 26); detail::uint32 m = mm.first + mm.second, i = (~m&0xFFFFFFFF) >> 31; m = (m>>i) | (m&i) | 0x80000000; if((exp+=13+i) > 29) return half(detail::binary, detail::overflow()); return half(detail::binary, detail::fixed2half(m, exp)); #endif } /// Hyperbolic tangent. /// This function is exact to rounding for all rounding modes. /// /// **See also:** Documentation for [std::tanh](https://en.cppreference.com/w/cpp/numeric/math/tanh). /// \param arg function argument /// \return hyperbolic tangent value of \a arg /// \exception FE_INVALID for signaling NaN /// \exception FE_OVERFLOW, ...UNDERFLOW, ...INEXACT according to rounding inline half tanh(half arg) { #ifdef HALF_ARITHMETIC_TYPE return half(detail::binary, detail::float2half(std::tanh(detail::half2float(arg.data_)))); #else int abs = arg.data_ & 0x7FFF, exp; if(!abs) return arg; if(abs >= 0x7C00) return half(detail::binary, (abs>0x7C00) ? detail::signal(arg.data_) : (arg.data_-0x4000)); if(abs >= 0x4500) return half(detail::binary, detail::rounded((arg.data_&0x8000)|0x3BFF, 1, 1)); if(abs < 0x2700) return half(detail::binary, detail::rounded(arg.data_-1, 1, 1)); if(half::round_style != std::round_to_nearest && abs == 0x2D3F) return half(detail::binary, detail::rounded(arg.data_-3, 0, 1)); std::pair mm = detail::hyperbolic_args(abs, exp, 27); detail::uint32 my = mm.first - mm.second - (half::round_style!=std::round_to_nearest), mx = mm.first + mm.second, i = (~mx&0xFFFFFFFF) >> 31; for(exp=13; my<0x80000000; my<<=1,--exp) ; mx = (mx>>i) | 0x80000000; return half(detail::binary, detail::tangent_post(my, mx, exp-i, arg.data_&0x8000)); #endif } /// Hyperbolic area sine. /// This function is exact to rounding for all rounding modes. /// /// **See also:** Documentation for [std::asinh](https://en.cppreference.com/w/cpp/numeric/math/asinh). /// \param arg function argument /// \return area sine value of \a arg /// \exception FE_INVALID for signaling NaN /// \exception FE_OVERFLOW, ...UNDERFLOW, ...INEXACT according to rounding inline half asinh(half arg) { #if defined(HALF_ARITHMETIC_TYPE) && HALF_ENABLE_CPP11_CMATH return half(detail::binary, detail::float2half(std::asinh(detail::half2float(arg.data_)))); #else int abs = arg.data_ & 0x7FFF; if(!abs || abs >= 0x7C00) return (abs>0x7C00) ? half(detail::binary, detail::signal(arg.data_)) : arg; if(abs <= 0x2900) return half(detail::binary, detail::rounded(arg.data_-1, 1, 1)); if(half::round_style != std::round_to_nearest) switch(abs) { case 0x32D4: return half(detail::binary, detail::rounded(arg.data_-13, 1, 1)); case 0x3B5B: return half(detail::binary, detail::rounded(arg.data_-197, 1, 1)); } return half(detail::binary, detail::area(arg.data_)); #endif } /// Hyperbolic area cosine. /// This function is exact to rounding for all rounding modes. /// /// **See also:** Documentation for [std::acosh](https://en.cppreference.com/w/cpp/numeric/math/acosh). /// \param arg function argument /// \return area cosine value of \a arg /// \exception FE_INVALID for signaling NaN or arguments <1 /// \exception FE_OVERFLOW, ...UNDERFLOW, ...INEXACT according to rounding inline half acosh(half arg) { #if defined(HALF_ARITHMETIC_TYPE) && HALF_ENABLE_CPP11_CMATH return half(detail::binary, detail::float2half(std::acosh(detail::half2float(arg.data_)))); #else int abs = arg.data_ & 0x7FFF; if((arg.data_&0x8000) || abs < 0x3C00) return half(detail::binary, (abs<=0x7C00) ? detail::invalid() : detail::signal(arg.data_)); if(abs == 0x3C00) return half(detail::binary, 0); if(arg.data_ >= 0x7C00) return (abs>0x7C00) ? half(detail::binary, detail::signal(arg.data_)) : arg; return half(detail::binary, detail::area(arg.data_)); #endif } /// Hyperbolic area tangent. /// This function is exact to rounding for all rounding modes. /// /// **See also:** Documentation for [std::atanh](https://en.cppreference.com/w/cpp/numeric/math/atanh). /// \param arg function argument /// \return area tangent value of \a arg /// \exception FE_INVALID for signaling NaN or if abs(\a arg) > 1 /// \exception FE_DIVBYZERO for +/-1 /// \exception FE_OVERFLOW, ...UNDERFLOW, ...INEXACT according to rounding inline half atanh(half arg) { #if defined(HALF_ARITHMETIC_TYPE) && HALF_ENABLE_CPP11_CMATH return half(detail::binary, detail::float2half(std::atanh(detail::half2float(arg.data_)))); #else int abs = arg.data_ & 0x7FFF, exp = 0; if(!abs) return arg; if(abs >= 0x3C00) return half(detail::binary, (abs==0x3C00) ? detail::pole(arg.data_&0x8000) : (abs<=0x7C00) ? detail::invalid() : detail::signal(arg.data_)); if(abs < 0x2700) return half(detail::binary, detail::rounded(arg.data_, 0, 1)); detail::uint32 m = static_cast((abs&0x3FF)|((abs>0x3FF)<<10)) << ((abs>>10)+(abs<=0x3FF)+6), my = 0x80000000 + m, mx = 0x80000000 - m; for(; mx<0x80000000; mx<<=1,++exp) ; int i = my >= mx, s; return half(detail::binary, detail::log2_post(detail::log2( (detail::divide64(my>>i, mx, s)+1)>>1, 27)+0x10, exp+i-1, 16, arg.data_&0x8000)); #endif } /// \} /// \anchor special /// \name Error and gamma functions /// \{ /// Error function. /// This function may be 1 ULP off the correctly rounded exact result for any rounding mode in <0.5% of inputs. /// /// **See also:** Documentation for [std::erf](https://en.cppreference.com/w/cpp/numeric/math/erf). /// \param arg function argument /// \return error function value of \a arg /// \exception FE_INVALID for signaling NaN /// \exception FE_OVERFLOW, ...UNDERFLOW, ...INEXACT according to rounding inline half erf(half arg) { #if defined(HALF_ARITHMETIC_TYPE) && HALF_ENABLE_CPP11_CMATH return half(detail::binary, detail::float2half(std::erf(detail::half2float(arg.data_)))); #else unsigned int abs = arg.data_ & 0x7FFF; if(!abs || abs >= 0x7C00) return (abs>=0x7C00) ? half(detail::binary, (abs==0x7C00) ? (arg.data_-0x4000) : detail::signal(arg.data_)) : arg; if(abs >= 0x4200) return half(detail::binary, detail::rounded((arg.data_&0x8000)|0x3BFF, 1, 1)); return half(detail::binary, detail::erf(arg.data_)); #endif } /// Complementary error function. /// This function may be 1 ULP off the correctly rounded exact result for any rounding mode in <0.5% of inputs. /// /// **See also:** Documentation for [std::erfc](https://en.cppreference.com/w/cpp/numeric/math/erfc). /// \param arg function argument /// \return 1 minus error function value of \a arg /// \exception FE_INVALID for signaling NaN /// \exception FE_OVERFLOW, ...UNDERFLOW, ...INEXACT according to rounding inline half erfc(half arg) { #if defined(HALF_ARITHMETIC_TYPE) && HALF_ENABLE_CPP11_CMATH return half(detail::binary, detail::float2half(std::erfc(detail::half2float(arg.data_)))); #else unsigned int abs = arg.data_ & 0x7FFF, sign = arg.data_ & 0x8000; if(abs >= 0x7C00) return (abs>=0x7C00) ? half(detail::binary, (abs==0x7C00) ? (sign>>1) : detail::signal(arg.data_)) : arg; if(!abs) return half(detail::binary, 0x3C00); if(abs >= 0x4400) return half(detail::binary, detail::rounded((sign>>1)-(sign>>15), sign>>15, 1)); return half(detail::binary, detail::erf(arg.data_)); #endif } /// Natural logarithm of gamma function. /// This function may be 1 ULP off the correctly rounded exact result for any rounding mode in ~0.025% of inputs. /// /// **See also:** Documentation for [std::lgamma](https://en.cppreference.com/w/cpp/numeric/math/lgamma). /// \param arg function argument /// \return natural logarith of gamma function for \a arg /// \exception FE_INVALID for signaling NaN /// \exception FE_DIVBYZERO for 0 or negative integer arguments /// \exception FE_OVERFLOW, ...UNDERFLOW, ...INEXACT according to rounding inline half lgamma(half arg) { #if defined(HALF_ARITHMETIC_TYPE) && HALF_ENABLE_CPP11_CMATH return half(detail::binary, detail::float2half(std::lgamma(detail::half2float(arg.data_)))); #else int abs = arg.data_ & 0x7FFF; if(abs >= 0x7C00) return half(detail::binary, (abs==0x7C00) ? 0x7C00 : detail::signal(arg.data_)); if(!abs || arg.data_ >= 0xE400 || (arg.data_ >= 0xBC00 && !(abs&((1<<(25-(abs>>10)))-1)))) return half(detail::binary, detail::pole()); if(arg.data_ == 0x3C00 || arg.data_ == 0x4000) return half(detail::binary, 0); return half(detail::binary, detail::gamma(arg.data_)); #endif } /// Gamma function. /// This function may be 1 ULP off the correctly rounded exact result for any rounding mode in <0.25% of inputs. /// /// **See also:** Documentation for [std::tgamma](https://en.cppreference.com/w/cpp/numeric/math/tgamma). /// \param arg function argument /// \return gamma function value of \a arg /// \exception FE_INVALID for signaling NaN, negative infinity or negative integer arguments /// \exception FE_DIVBYZERO for 0 /// \exception FE_OVERFLOW, ...UNDERFLOW, ...INEXACT according to rounding inline half tgamma(half arg) { #if defined(HALF_ARITHMETIC_TYPE) && HALF_ENABLE_CPP11_CMATH return half(detail::binary, detail::float2half(std::tgamma(detail::half2float(arg.data_)))); #else unsigned int abs = arg.data_ & 0x7FFF; if(!abs) return half(detail::binary, detail::pole(arg.data_)); if(abs >= 0x7C00) return (arg.data_==0x7C00) ? arg : half(detail::binary, detail::signal(arg.data_)); if(arg.data_ >= 0xE400 || (arg.data_ >= 0xBC00 && !(abs&((1<<(25-(abs>>10)))-1)))) return half(detail::binary, detail::invalid()); if(arg.data_ >= 0xCA80) return half(detail::binary, detail::underflow((1-((abs>>(25-(abs>>10)))&1))<<15)); if(arg.data_ <= 0x100 || (arg.data_ >= 0x4900 && arg.data_ < 0x8000)) return half(detail::binary, detail::overflow()); if(arg.data_ == 0x3C00) return arg; return half(detail::binary, detail::gamma(arg.data_)); #endif } /// \} /// \anchor rounding /// \name Rounding /// \{ /// Nearest integer not less than half value. /// **See also:** Documentation for [std::ceil](https://en.cppreference.com/w/cpp/numeric/math/ceil). /// \param arg half to round /// \return nearest integer not less than \a arg /// \exception FE_INVALID for signaling NaN /// \exception FE_INEXACT if value had to be rounded inline half ceil(half arg) { return half(detail::binary, detail::integral(arg.data_)); } /// Nearest integer not greater than half value. /// **See also:** Documentation for [std::floor](https://en.cppreference.com/w/cpp/numeric/math/floor). /// \param arg half to round /// \return nearest integer not greater than \a arg /// \exception FE_INVALID for signaling NaN /// \exception FE_INEXACT if value had to be rounded inline half floor(half arg) { return half(detail::binary, detail::integral(arg.data_)); } /// Nearest integer not greater in magnitude than half value. /// **See also:** Documentation for [std::trunc](https://en.cppreference.com/w/cpp/numeric/math/trunc). /// \param arg half to round /// \return nearest integer not greater in magnitude than \a arg /// \exception FE_INVALID for signaling NaN /// \exception FE_INEXACT if value had to be rounded inline half trunc(half arg) { return half(detail::binary, detail::integral(arg.data_)); } /// Nearest integer. /// **See also:** Documentation for [std::round](https://en.cppreference.com/w/cpp/numeric/math/round). /// \param arg half to round /// \return nearest integer, rounded away from zero in half-way cases /// \exception FE_INVALID for signaling NaN /// \exception FE_INEXACT if value had to be rounded inline half round(half arg) { return half(detail::binary, detail::integral(arg.data_)); } /// Nearest integer. /// **See also:** Documentation for [std::lround](https://en.cppreference.com/w/cpp/numeric/math/round). /// \param arg half to round /// \return nearest integer, rounded away from zero in half-way cases /// \exception FE_INVALID if value is not representable as `long` inline long lround(half arg) { return detail::half2int(arg.data_); } /// Nearest integer using half's internal rounding mode. /// **See also:** Documentation for [std::rint](https://en.cppreference.com/w/cpp/numeric/math/rint). /// \param arg half expression to round /// \return nearest integer using default rounding mode /// \exception FE_INVALID for signaling NaN /// \exception FE_INEXACT if value had to be rounded inline half rint(half arg) { return half(detail::binary, detail::integral(arg.data_)); } /// Nearest integer using half's internal rounding mode. /// **See also:** Documentation for [std::lrint](https://en.cppreference.com/w/cpp/numeric/math/rint). /// \param arg half expression to round /// \return nearest integer using default rounding mode /// \exception FE_INVALID if value is not representable as `long` /// \exception FE_INEXACT if value had to be rounded inline long lrint(half arg) { return detail::half2int(arg.data_); } /// Nearest integer using half's internal rounding mode. /// **See also:** Documentation for [std::nearbyint](https://en.cppreference.com/w/cpp/numeric/math/nearbyint). /// \param arg half expression to round /// \return nearest integer using default rounding mode /// \exception FE_INVALID for signaling NaN inline half nearbyint(half arg) { return half(detail::binary, detail::integral(arg.data_)); } #if HALF_ENABLE_CPP11_LONG_LONG /// Nearest integer. /// **See also:** Documentation for [std::llround](https://en.cppreference.com/w/cpp/numeric/math/round). /// \param arg half to round /// \return nearest integer, rounded away from zero in half-way cases /// \exception FE_INVALID if value is not representable as `long long` inline long long llround(half arg) { return detail::half2int(arg.data_); } /// Nearest integer using half's internal rounding mode. /// **See also:** Documentation for [std::llrint](https://en.cppreference.com/w/cpp/numeric/math/rint). /// \param arg half expression to round /// \return nearest integer using default rounding mode /// \exception FE_INVALID if value is not representable as `long long` /// \exception FE_INEXACT if value had to be rounded inline long long llrint(half arg) { return detail::half2int(arg.data_); } #endif /// \} /// \anchor float /// \name Floating point manipulation /// \{ /// Decompress floating-point number. /// **See also:** Documentation for [std::frexp](https://en.cppreference.com/w/cpp/numeric/math/frexp). /// \param arg number to decompress /// \param exp address to store exponent at /// \return significant in range [0.5, 1) /// \exception FE_INVALID for signaling NaN inline half frexp(half arg, int *exp) { *exp = 0; unsigned int abs = arg.data_ & 0x7FFF; if(abs >= 0x7C00 || !abs) return (abs>0x7C00) ? half(detail::binary, detail::signal(arg.data_)) : arg; for(; abs<0x400; abs<<=1,--*exp) ; *exp += (abs>>10) - 14; return half(detail::binary, (arg.data_&0x8000)|0x3800|(abs&0x3FF)); } /// Multiply by power of two. /// This function is exact to rounding for all rounding modes. /// /// **See also:** Documentation for [std::scalbln](https://en.cppreference.com/w/cpp/numeric/math/scalbn). /// \param arg number to modify /// \param exp power of two to multiply with /// \return \a arg multplied by 2 raised to \a exp /// \exception FE_INVALID for signaling NaN /// \exception FE_OVERFLOW, ...UNDERFLOW, ...INEXACT according to rounding inline half scalbln(half arg, long exp) { unsigned int abs = arg.data_ & 0x7FFF, sign = arg.data_ & 0x8000; if(abs >= 0x7C00 || !abs) return (abs>0x7C00) ? half(detail::binary, detail::signal(arg.data_)) : arg; for(; abs<0x400; abs<<=1,--exp) ; exp += abs >> 10; if(exp > 30) return half(detail::binary, detail::overflow(sign)); else if(exp < -10) return half(detail::binary, detail::underflow(sign)); else if(exp > 0) return half(detail::binary, sign|(exp<<10)|(abs&0x3FF)); unsigned int m = (abs&0x3FF) | 0x400; return half(detail::binary, detail::rounded(sign|(m>>(1-exp)), (m>>-exp)&1, (m&((1<<-exp)-1))!=0)); } /// Multiply by power of two. /// This function is exact to rounding for all rounding modes. /// /// **See also:** Documentation for [std::scalbn](https://en.cppreference.com/w/cpp/numeric/math/scalbn). /// \param arg number to modify /// \param exp power of two to multiply with /// \return \a arg multplied by 2 raised to \a exp /// \exception FE_INVALID for signaling NaN /// \exception FE_OVERFLOW, ...UNDERFLOW, ...INEXACT according to rounding inline half scalbn(half arg, int exp) { return scalbln(arg, exp); } /// Multiply by power of two. /// This function is exact to rounding for all rounding modes. /// /// **See also:** Documentation for [std::ldexp](https://en.cppreference.com/w/cpp/numeric/math/ldexp). /// \param arg number to modify /// \param exp power of two to multiply with /// \return \a arg multplied by 2 raised to \a exp /// \exception FE_INVALID for signaling NaN /// \exception FE_OVERFLOW, ...UNDERFLOW, ...INEXACT according to rounding inline half ldexp(half arg, int exp) { return scalbln(arg, exp); } /// Extract integer and fractional parts. /// **See also:** Documentation for [std::modf](https://en.cppreference.com/w/cpp/numeric/math/modf). /// \param arg number to decompress /// \param iptr address to store integer part at /// \return fractional part /// \exception FE_INVALID for signaling NaN inline half modf(half arg, half *iptr) { unsigned int abs = arg.data_ & 0x7FFF; if(abs > 0x7C00) { arg = half(detail::binary, detail::signal(arg.data_)); return *iptr = arg, arg; } if(abs >= 0x6400) return *iptr = arg, half(detail::binary, arg.data_&0x8000); if(abs < 0x3C00) return iptr->data_ = arg.data_ & 0x8000, arg; unsigned int exp = abs >> 10, mask = (1<<(25-exp)) - 1, m = arg.data_ & mask; iptr->data_ = arg.data_ & ~mask; if(!m) return half(detail::binary, arg.data_&0x8000); for(; m<0x400; m<<=1,--exp) ; return half(detail::binary, (arg.data_&0x8000)|(exp<<10)|(m&0x3FF)); } /// Extract exponent. /// **See also:** Documentation for [std::ilogb](https://en.cppreference.com/w/cpp/numeric/math/ilogb). /// \param arg number to query /// \return floating-point exponent /// \retval FP_ILOGB0 for zero /// \retval FP_ILOGBNAN for NaN /// \retval INT_MAX for infinity /// \exception FE_INVALID for 0 or infinite values inline int ilogb(half arg) { int abs = arg.data_ & 0x7FFF, exp; if(!abs || abs >= 0x7C00) { detail::raise(FE_INVALID); return !abs ? FP_ILOGB0 : (abs==0x7C00) ? INT_MAX : FP_ILOGBNAN; } for(exp=(abs>>10)-15; abs<0x200; abs<<=1,--exp) ; return exp; } /// Extract exponent. /// **See also:** Documentation for [std::logb](https://en.cppreference.com/w/cpp/numeric/math/logb). /// \param arg number to query /// \return floating-point exponent /// \exception FE_INVALID for signaling NaN /// \exception FE_DIVBYZERO for 0 inline half logb(half arg) { int abs = arg.data_ & 0x7FFF, exp; if(!abs) return half(detail::binary, detail::pole(0x8000)); if(abs >= 0x7C00) return half(detail::binary, (abs==0x7C00) ? 0x7C00 : detail::signal(arg.data_)); for(exp=(abs>>10)-15; abs<0x200; abs<<=1,--exp) ; unsigned int value = static_cast(exp<0) << 15; if(exp) { unsigned int m = std::abs(exp) << 6; for(exp=18; m<0x400; m<<=1,--exp) ; value |= (exp<<10) + m; } return half(detail::binary, value); } /// Next representable value. /// **See also:** Documentation for [std::nextafter](https://en.cppreference.com/w/cpp/numeric/math/nextafter). /// \param from value to compute next representable value for /// \param to direction towards which to compute next value /// \return next representable value after \a from in direction towards \a to /// \exception FE_INVALID for signaling NaN /// \exception FE_OVERFLOW for infinite result from finite argument /// \exception FE_UNDERFLOW for subnormal result inline half nextafter(half from, half to) { int fabs = from.data_ & 0x7FFF, tabs = to.data_ & 0x7FFF; if(fabs > 0x7C00 || tabs > 0x7C00) return half(detail::binary, detail::signal(from.data_, to.data_)); if(from.data_ == to.data_ || !(fabs|tabs)) return to; if(!fabs) { detail::raise(FE_UNDERFLOW, !HALF_ERRHANDLING_UNDERFLOW_TO_INEXACT); return half(detail::binary, (to.data_&0x8000)+1); } unsigned int out = from.data_ + (((from.data_>>15)^static_cast( (from.data_^(0x8000|(0x8000-(from.data_>>15))))<(to.data_^(0x8000|(0x8000-(to.data_>>15))))))<<1) - 1; detail::raise(FE_OVERFLOW, fabs<0x7C00 && (out&0x7C00)==0x7C00); detail::raise(FE_UNDERFLOW, !HALF_ERRHANDLING_UNDERFLOW_TO_INEXACT && (out&0x7C00)<0x400); return half(detail::binary, out); } /// Next representable value. /// **See also:** Documentation for [std::nexttoward](https://en.cppreference.com/w/cpp/numeric/math/nexttoward). /// \param from value to compute next representable value for /// \param to direction towards which to compute next value /// \return next representable value after \a from in direction towards \a to /// \exception FE_INVALID for signaling NaN /// \exception FE_OVERFLOW for infinite result from finite argument /// \exception FE_UNDERFLOW for subnormal result inline half nexttoward(half from, long double to) { int fabs = from.data_ & 0x7FFF; if(fabs > 0x7C00) return half(detail::binary, detail::signal(from.data_)); long double lfrom = static_cast(from); if(detail::builtin_isnan(to) || lfrom == to) return half(static_cast(to)); if(!fabs) { detail::raise(FE_UNDERFLOW, !HALF_ERRHANDLING_UNDERFLOW_TO_INEXACT); return half(detail::binary, (static_cast(detail::builtin_signbit(to))<<15)+1); } unsigned int out = from.data_ + (((from.data_>>15)^static_cast(lfrom 0x7C00; } /// Check if normal number. /// **See also:** Documentation for [std::isnormal](https://en.cppreference.com/w/cpp/numeric/math/isnormal). /// \param arg number to check /// \retval true if normal number /// \retval false if either subnormal, zero, infinity or NaN inline HALF_CONSTEXPR bool isnormal(half arg) { return ((arg.data_&0x7C00)!=0) & ((arg.data_&0x7C00)!=0x7C00); } /// Check sign. /// **See also:** Documentation for [std::signbit](https://en.cppreference.com/w/cpp/numeric/math/signbit). /// \param arg number to check /// \retval true for negative number /// \retval false for positive number inline HALF_CONSTEXPR bool signbit(half arg) { return (arg.data_&0x8000) != 0; } /// \} /// \anchor compfunc /// \name Comparison /// \{ /// Quiet comparison for greater than. /// **See also:** Documentation for [std::isgreater](https://en.cppreference.com/w/cpp/numeric/math/isgreater). /// \param x first operand /// \param y second operand /// \retval true if \a x greater than \a y /// \retval false else inline HALF_CONSTEXPR bool isgreater(half x, half y) { return ((x.data_^(0x8000|(0x8000-(x.data_>>15))))+(x.data_>>15)) > ((y.data_^(0x8000|(0x8000-(y.data_>>15))))+(y.data_>>15)) && !isnan(x) && !isnan(y); } /// Quiet comparison for greater equal. /// **See also:** Documentation for [std::isgreaterequal](https://en.cppreference.com/w/cpp/numeric/math/isgreaterequal). /// \param x first operand /// \param y second operand /// \retval true if \a x greater equal \a y /// \retval false else inline HALF_CONSTEXPR bool isgreaterequal(half x, half y) { return ((x.data_^(0x8000|(0x8000-(x.data_>>15))))+(x.data_>>15)) >= ((y.data_^(0x8000|(0x8000-(y.data_>>15))))+(y.data_>>15)) && !isnan(x) && !isnan(y); } /// Quiet comparison for less than. /// **See also:** Documentation for [std::isless](https://en.cppreference.com/w/cpp/numeric/math/isless). /// \param x first operand /// \param y second operand /// \retval true if \a x less than \a y /// \retval false else inline HALF_CONSTEXPR bool isless(half x, half y) { return ((x.data_^(0x8000|(0x8000-(x.data_>>15))))+(x.data_>>15)) < ((y.data_^(0x8000|(0x8000-(y.data_>>15))))+(y.data_>>15)) && !isnan(x) && !isnan(y); } /// Quiet comparison for less equal. /// **See also:** Documentation for [std::islessequal](https://en.cppreference.com/w/cpp/numeric/math/islessequal). /// \param x first operand /// \param y second operand /// \retval true if \a x less equal \a y /// \retval false else inline HALF_CONSTEXPR bool islessequal(half x, half y) { return ((x.data_^(0x8000|(0x8000-(x.data_>>15))))+(x.data_>>15)) <= ((y.data_^(0x8000|(0x8000-(y.data_>>15))))+(y.data_>>15)) && !isnan(x) && !isnan(y); } /// Quiet comarison for less or greater. /// **See also:** Documentation for [std::islessgreater](https://en.cppreference.com/w/cpp/numeric/math/islessgreater). /// \param x first operand /// \param y second operand /// \retval true if either less or greater /// \retval false else inline HALF_CONSTEXPR bool islessgreater(half x, half y) { return x.data_!=y.data_ && ((x.data_|y.data_)&0x7FFF) && !isnan(x) && !isnan(y); } /// Quiet check if unordered. /// **See also:** Documentation for [std::isunordered](https://en.cppreference.com/w/cpp/numeric/math/isunordered). /// \param x first operand /// \param y second operand /// \retval true if unordered (one or two NaN operands) /// \retval false else inline HALF_CONSTEXPR bool isunordered(half x, half y) { return isnan(x) || isnan(y); } /// \} /// \anchor casting /// \name Casting /// \{ /// Cast to or from half-precision floating-point number. /// This casts between [half](\ref half_float::half) and any built-in arithmetic type. The values are converted /// directly using the default rounding mode, without any roundtrip over `float` that a `static_cast` would otherwise do. /// /// Using this cast with neither of the two types being a [half](\ref half_float::half) or with any of the two types /// not being a built-in arithmetic type (apart from [half](\ref half_float::half), of course) results in a compiler /// error and casting between [half](\ref half_float::half)s returns the argument unmodified. /// \tparam T destination type (half or built-in arithmetic type) /// \tparam U source type (half or built-in arithmetic type) /// \param arg value to cast /// \return \a arg converted to destination type /// \exception FE_INVALID if \a T is integer type and result is not representable as \a T /// \exception FE_OVERFLOW, ...UNDERFLOW, ...INEXACT according to rounding template T half_cast(U arg) { return detail::half_caster::cast(arg); } /// Cast to or from half-precision floating-point number. /// This casts between [half](\ref half_float::half) and any built-in arithmetic type. The values are converted /// directly using the specified rounding mode, without any roundtrip over `float` that a `static_cast` would otherwise do. /// /// Using this cast with neither of the two types being a [half](\ref half_float::half) or with any of the two types /// not being a built-in arithmetic type (apart from [half](\ref half_float::half), of course) results in a compiler /// error and casting between [half](\ref half_float::half)s returns the argument unmodified. /// \tparam T destination type (half or built-in arithmetic type) /// \tparam R rounding mode to use. /// \tparam U source type (half or built-in arithmetic type) /// \param arg value to cast /// \return \a arg converted to destination type /// \exception FE_INVALID if \a T is integer type and result is not representable as \a T /// \exception FE_OVERFLOW, ...UNDERFLOW, ...INEXACT according to rounding template T half_cast(U arg) { return detail::half_caster::cast(arg); } /// \} /// \} /// \anchor errors /// \name Error handling /// \{ /// Clear exception flags. /// This function works even if [automatic exception flag handling](\ref HALF_ERRHANDLING_FLAGS) is disabled, /// but in that case manual flag management is the only way to raise flags. /// /// **See also:** Documentation for [std::feclearexcept](https://en.cppreference.com/w/cpp/numeric/fenv/feclearexcept). /// \param excepts OR of exceptions to clear /// \retval 0 all selected flags cleared successfully inline int feclearexcept(int excepts) { detail::errflags() &= ~excepts; return 0; } /// Test exception flags. /// This function works even if [automatic exception flag handling](\ref HALF_ERRHANDLING_FLAGS) is disabled, /// but in that case manual flag management is the only way to raise flags. /// /// **See also:** Documentation for [std::fetestexcept](https://en.cppreference.com/w/cpp/numeric/fenv/fetestexcept). /// \param excepts OR of exceptions to test /// \return OR of selected exceptions if raised inline int fetestexcept(int excepts) { return detail::errflags() & excepts; } /// Raise exception flags. /// This raises the specified floating point exceptions and also invokes any additional automatic exception handling as /// configured with the [HALF_ERRHANDLIG_...](\ref HALF_ERRHANDLING_ERRNO) preprocessor symbols. /// This function works even if [automatic exception flag handling](\ref HALF_ERRHANDLING_FLAGS) is disabled, /// but in that case manual flag management is the only way to raise flags. /// /// **See also:** Documentation for [std::feraiseexcept](https://en.cppreference.com/w/cpp/numeric/fenv/feraiseexcept). /// \param excepts OR of exceptions to raise /// \retval 0 all selected exceptions raised successfully inline int feraiseexcept(int excepts) { detail::errflags() |= excepts; detail::raise(excepts); return 0; } /// Save exception flags. /// This function works even if [automatic exception flag handling](\ref HALF_ERRHANDLING_FLAGS) is disabled, /// but in that case manual flag management is the only way to raise flags. /// /// **See also:** Documentation for [std::fegetexceptflag](https://en.cppreference.com/w/cpp/numeric/fenv/feexceptflag). /// \param flagp adress to store flag state at /// \param excepts OR of flags to save /// \retval 0 for success inline int fegetexceptflag(int *flagp, int excepts) { *flagp = detail::errflags() & excepts; return 0; } /// Restore exception flags. /// This only copies the specified exception state (including unset flags) without incurring any additional exception handling. /// This function works even if [automatic exception flag handling](\ref HALF_ERRHANDLING_FLAGS) is disabled, /// but in that case manual flag management is the only way to raise flags. /// /// **See also:** Documentation for [std::fesetexceptflag](https://en.cppreference.com/w/cpp/numeric/fenv/feexceptflag). /// \param flagp adress to take flag state from /// \param excepts OR of flags to restore /// \retval 0 for success inline int fesetexceptflag(const int *flagp, int excepts) { detail::errflags() = (detail::errflags()|(*flagp&excepts)) & (*flagp|~excepts); return 0; } /// Throw C++ exceptions based on set exception flags. /// This function manually throws a corresponding C++ exception if one of the specified flags is set, /// no matter if automatic throwing (via [HALF_ERRHANDLING_THROW_...](\ref HALF_ERRHANDLING_THROW_INVALID)) is enabled or not. /// This function works even if [automatic exception flag handling](\ref HALF_ERRHANDLING_FLAGS) is disabled, /// but in that case manual flag management is the only way to raise flags. /// \param excepts OR of exceptions to test /// \param msg error message to use for exception description /// \throw std::domain_error if `FE_INVALID` or `FE_DIVBYZERO` is selected and set /// \throw std::overflow_error if `FE_OVERFLOW` is selected and set /// \throw std::underflow_error if `FE_UNDERFLOW` is selected and set /// \throw std::range_error if `FE_INEXACT` is selected and set #if not defined HALF_ENABLE_CPP11_NOEXCEPT inline void fethrowexcept(int excepts, const char *msg = "") { excepts &= detail::errflags(); if(excepts & (FE_INVALID|FE_DIVBYZERO)) throw std::domain_error(msg); if(excepts & FE_OVERFLOW) throw std::overflow_error(msg); if(excepts & FE_UNDERFLOW) throw std::underflow_error(msg); if(excepts & FE_INEXACT) throw std::range_error(msg); } #endif //HALF_ENABLE_CPP11_NOEXCEPT /// \} } #undef HALF_UNUSED_NOERR #undef HALF_CONSTEXPR #undef HALF_CONSTEXPR_CONST #undef HALF_CONSTEXPR_NOERR #undef HALF_NOEXCEPT #undef HALF_NOTHROW #undef HALF_THREAD_LOCAL #undef HALF_TWOS_COMPLEMENT_INT #ifdef HALF_POP_WARNINGS #pragma warning(pop) #undef HALF_POP_WARNINGS #endif #endif