Page MenuHomePhabricator
Files F5021510

No OneTemporary
Actions

View Options

This file is larger than 256 KB, so syntax highlighting was skipped.
diff --git a/mnn_fp16_merge/CMakeLists.txt b/mnn_fp16_merge/CMakeLists.txt
new file mode 100644
index 0000000..cb4cbce
--- /dev/null
+++ b/mnn_fp16_merge/CMakeLists.txt
@@ -0,0 +1,25 @@
+cmake_minimum_required(VERSION 3.19)
+project(mnn_demo)
+
+set(CMAKE_CXX_STANDARD 14)
+
+find_package(OpenCV REQUIRED)
+set(MNN_DIR /Users/zihaoguo/Desktop/starmaker/MNN)
+include_directories(${MNN_DIR}/include)
+include_directories(${MNN_DIR}/include/MNN)
+include_directories(${MNN_DIR}/express/module)
+include_directories(${MNN_DIR}/tools)
+include_directories(${MNN_DIR}/tools/cpp)
+include_directories(${MNN_DIR}/source)
+include_directories(${MNN_DIR}/source/backend)
+include_directories(${MNN_DIR}/source/core)
+include_directories(${MNN_DIR}/schema/current)
+include_directories(${MNN_DIR}/3rd_party/flatbuffers/include)
+include_directories(/Users/zihaoguo/Desktop/starmaker/jsoncpp/include/json)
+include_directories(${OpenCV_INCLUDE_DIRS})
+
+include_directories(merge)
+
+link_directories(${MNN_DIR}/build)
+add_executable(mnn_demo fuse.cpp merge/merge.cpp)
+target_link_libraries(mnn_demo -lMNN)
\ No newline at end of file
diff --git a/mnn_fp16_merge/ReadME.txt b/mnn_fp16_merge/ReadME.txt
new file mode 100644
index 0000000..175fd24
--- /dev/null
+++ b/mnn_fp16_merge/ReadME.txt
@@ -0,0 +1,11 @@
+MNN fp16模型参数融合用法
+
+1. 半精度类型第三方库下载: https://half.sourceforge.net/, 下载得到的目录放到项目中
+2. 生成flatbuffers解析MNN模型的接口:
+（1）cd MNN_ROOT/3rd_party/3rd_party/flatbuffers
+ mkdir build && cmake .. && make -j8
+ 编译产物包含flatc工具
+（2）./flatc --cpp MNN_ROOT/schema/default/MNN.fbs
+ 生成一个MNN_generated.h头文件包含模型解析接口
+ 将这个文件改名为MNN_schema.h, 放到 MNN_ROOT/3rd_party/3rd_party/flatbuffers/include/flatbuffers/下
+
diff --git a/mnn_fp16_merge/fuse.cpp b/mnn_fp16_merge/fuse.cpp
new file mode 100644
index 0000000..16ad664
--- /dev/null
+++ b/mnn_fp16_merge/fuse.cpp
@@ -0,0 +1,44 @@
+//
+// Created by ZIHAO GUO on 2023/12/15.
+//
+
+#include "merge.h"
+#include <sys/time.h>
+
+int main()
+{
+ const char* model_1_path = "/Users/zihaoguo/Desktop/starmaker/Retrieval-based-Voice-Conversion-WebUI/transfered_models/contentvec_test_v3_fp16.mnn";
+ const char* model_2_path = "/Users/zihaoguo/Desktop/starmaker/Retrieval-based-Voice-Conversion-WebUI/transfered_models/contentvec_test_v3_xf_fp16.mnn";
+ std::vector<const char*> model_paths{model_1_path, model_2_path};
+ std::vector<float> ratios{0.5, 0.5};
+ const char* save_path = "/Users/zihaoguo/Desktop/starmaker/Retrieval-based-Voice-Conversion-WebUI/transfered_models/new_net.mnn";
+
+ std::vector<float> ratios2{0.2, 0.8};
+ const char* save_path2 = "/Users/zihaoguo/Desktop/starmaker/Retrieval-based-Voice-Conversion-WebUI/transfered_models/new_net2.mnn";
+
+ Merge merge;
+// struct timeval start;
+// struct timeval end;
+// gettimeofday(&start, NULL);
+ int err_code = merge.init(model_paths, true);
+ if (err_code == 0)
+ {
+// struct timeval start;
+// struct timeval end;
+// gettimeofday(&start, NULL);
+ err_code = merge.process(save_path, ratios);
+// gettimeofday(&end, NULL);
+// float time = (end.tv_sec - start.tv_sec) * 1000.0 + (end.tv_usec - start.tv_usec) / 1000.0;
+// printf("time: %f \n", time);
+
+// struct timeval start2;
+// struct timeval end2;
+// gettimeofday(&start2, NULL);
+ err_code = merge.process(save_path2, ratios2);
+// gettimeofday(&end2, NULL);
+// float time2 = (end2.tv_sec - start2.tv_sec) * 1000.0 + (end2.tv_usec - start2.tv_usec) / 1000.0;
+// printf("time2: %f \n", time2);
+ }
+// printf("err_code: %d \n", err_code);
+ return 0;
+}
\ No newline at end of file
diff --git a/mnn_fp16_merge/merge/error_code.h b/mnn_fp16_merge/merge/error_code.h
new file mode 100644
index 0000000..634b290
--- /dev/null
+++ b/mnn_fp16_merge/merge/error_code.h
@@ -0,0 +1,17 @@
+//
+// Created by ZIHAO GUO on 2023/12/28.
+//
+
+#ifndef MNN_DEMO_ERROR_CODE_H
+#define MNN_DEMO_ERROR_CODE_H
+
+enum {
+ MERGE_SUCCESS = 0,
+ MERGE_BASE_ERROR = 1000,
+ ERROR_MODEL_NUM_NOT_ENOUGH, // 模型数量不够, 小于2
+ ERROR_RATIO_NUM, // 融合系数的数量与模型数量不一致
+ ERROR_MODEL_PATH_NOT_EXISTS // 模型路径错误
+};
+
+
+#endif //MNN_DEMO_ERROR_CODE_H
diff --git a/mnn_fp16_merge/merge/half-2.2.0/ChangeLog.txt b/mnn_fp16_merge/merge/half-2.2.0/ChangeLog.txt
new file mode 100644
index 0000000..37f3dbf
--- /dev/null
+++ b/mnn_fp16_merge/merge/half-2.2.0/ChangeLog.txt
@@ -0,0 +1,213 @@
+Release Notes {#changelog}
+=============
+
+2.2.0 release (2021-06-12):
+---------------------------
+
+- Added `rsqrt` function for inverse square root.
+- Improved performance of `pow` function.
+- Fixed bug that forgot to include `<immintrin.h>` for F16C intrinsics.
+
+
+2.1.0 release (2019-08-05):
+---------------------------
+
+- Added detection of IEEE floating-point exceptions to operators and functions.
+- Added configuration options for automatic exception handling.
+- Added functions for explicitly managing floating-point exception flags.
+- Improved accuracy of `pow` and `atan2` functions.
+
+
+2.0.0 release (2019-07-23):
+---------------------------
+
+- Made internal implementation independent from built-in floating point
+ facilities for increased reliability and IEEE-conformance.
+- Changed default rounding mode to rounding to nearest.
+- Always round ties to even when rounding to nearest.
+- Extended `constexpr` support to comparison and classification functions.
+- Added support for F16C compiler intrinsics for conversions.
+- Enabled C++11 feature detection for Intel compilers.
+
+
+1.12.0 release (2017-03-06):
+----------------------------
+
+- Changed behaviour of `half_cast` to perform conversions to/from `double`
+ and `long double` directly according to specified rounding mode, without an
+ intermediate `float` conversion.
+- Added `noexcept` specifiers to constructors.
+- Fixed minor portability problem with `logb` and `ilogb`.
+- Tested for *VC++ 2015*.
+
+
+1.11.0 release (2013-11-16):
+----------------------------
+
+- Made tie-breaking behaviour in round to nearest configurable by
+ `HALF_ROUND_TIES_TO_EVEN` macro.
+- Completed support for all C++11 mathematical functions even if single-
+ precision versions from `<cmath>` are unsupported.
+- Fixed inability to disable support for C++11 mathematical functions on
+ *VC++ 2013*.
+
+
+1.10.0 release (2013-11-09):
+----------------------------
+
+- Made default rounding mode configurable by `HALF_ROUND_STYLE` macro.
+- Added support for non-IEEE single-precision implementations.
+- Added `HALF_ENABLE_CPP11_TYPE_TRAITS` preprocessor flag for checking
+ support for C++11 type traits and TMP features.
+- Restricted `half_cast` to support built-in arithmetic types only.
+- Changed behaviour of `half_cast` to respect rounding mode when casting
+ to/from integer types.
+
+
+1.9.2 release (2013-11-01):
+---------------------------
+
+- Tested for *gcc 4.8*.
+- Tested and fixed for *VC++ 2013*.
+- Removed unnecessary warnings in *MSVC*.
+
+
+1.9.1 release (2013-08-08):
+---------------------------
+
+- Fixed problems with older gcc and MSVC versions.
+- Small fix to non-C++11 implementations of `remainder` and `remquo`.
+
+
+1.9.0 release (2013-08-07):
+---------------------------
+
+- Changed behaviour of `nearbyint`, `rint`, `lrint` and `llrint` to use
+ rounding mode of half-precision implementation (which is
+ truncating/indeterminate) instead of single-precision rounding mode.
+- Added support for more C++11 mathematical functions even if single-
+ precision versions from `<cmath>` are unsupported, in particular
+ `remainder`, `remquo` and `cbrt`.
+- Minor implementation changes.
+
+
+1.8.1 release (2013-01-22):
+---------------------------
+
+- Fixed bug resulting in multiple definitions of the `nanh` function due to
+ a missing `inline` specification.
+
+
+1.8.0 release (2013-01-19):
+---------------------------
+
+- Added support for more C++11 mathematical functions even if single-
+ precision versions from `<cmath>` are unsupported, in particular
+ exponential and logarithm functions, hyperbolic area functions and the
+ hypotenuse function.
+- Made `fma` function use default implementation if single-precision version
+ from `<cmath>` is not faster and thus `FP_FAST_FMAH` to be defined always.
+- Fixed overload resolution issues when invoking certain mathematical
+ functions by unqualified calls.
+
+
+1.7.0 release (2012-10-26):
+---------------------------
+
+- Added support for C++11 `noexcept` specifiers.
+- Changed C++11 `long long` to be supported on *VC++ 2003* and up.
+
+
+1.6.1 release (2012-09-13):
+---------------------------
+
+- Made `fma` and `fdim` functions available even if corresponding
+ single-precision functions are not.
+
+
+1.6.0 release (2012-09-12):
+---------------------------
+
+- Added `HALF_ENABLE_CPP11_LONG_LONG` to control support for `long long`
+ integers and corresponding mathematical functions.
+- Fixed C++98 compatibility on non-VC compilers.
+
+
+1.5.1 release (2012-08-17):
+---------------------------
+
+- Recorrected `std::numeric_limits::round_style` to always return
+ `std::round_indeterminate`, due to overflow-handling deviating from
+ correct round-toward-zero behaviour.
+
+
+1.5.0 release (2012-08-16):
+---------------------------
+
+- Added `half_cast` for explicitly casting between half and any type
+ convertible to/from `float` and allowing the explicit specification of
+ the rounding mode to use.
+
+
+1.4.0 release (2012-08-12):
+---------------------------
+
+- Added support for C++11 generalized constant expressions (`constexpr`).
+
+
+1.3.1 release (2012-08-11):
+---------------------------
+
+- Fixed requirement for `std::signbit` and `std::isnan` (even if C++11
+ `<cmath>` functions disabled) on non-VC compilers.
+
+
+1.3.0 release (2012-08-10):
+---------------------------
+
+- Made requirement for `<cstdint>` and `static_assert` optional and thus
+ made the library C++98-compatible.
+- Made support for C++11 features user-overridable through explicit
+ definition of corresponding preprocessor symbols to either 0 or 1.
+- Renamed `HALF_ENABLE_HASH` to `HALF_ENABLE_CPP11_HASH` in correspondence
+ with other C++11 preprocessor symbols.
+
+
+1.2.0 release (2012-08-07):
+---------------------------
+
+- Added proper preprocessor definitions for `HUGE_VALH` and `FP_FAST_FMAH`
+ in correspondence with their single-precision counterparts from `<cmath>`.
+- Fixed internal preprocessor macros to be properly undefined after use.
+
+
+1.1.2 release (2012-08-07):
+---------------------------
+
+- Revised `std::numeric_limits::round_style` to return
+ `std::round_toward_zero` if the `float` version also does and
+ `std::round_indeterminate` otherwise.
+- Fixed `std::numeric_limits::round_error` to reflect worst-case round
+ toward zero behaviour.
+
+
+1.1.1 release (2012-08-06):
+---------------------------
+
+- Fixed `std::numeric_limits::min` to return smallest positive normal
+ number, instead of subnormal number.
+- Fixed `std::numeric_limits::round_style` to return
+ `std::round_indeterminate` due to mixture of separately rounded
+ single-precision arithmetics with truncating single-to-half conversions.
+
+
+1.1.0 release (2012-08-06):
+---------------------------
+
+- Added half-precision literals.
+
+
+1.0.0 release (2012-08-05):
+---------------------------
+
+- First release.
diff --git a/mnn_fp16_merge/merge/half-2.2.0/LICENSE.txt b/mnn_fp16_merge/merge/half-2.2.0/LICENSE.txt
new file mode 100644
index 0000000..45f55db
--- /dev/null
+++ b/mnn_fp16_merge/merge/half-2.2.0/LICENSE.txt
@@ -0,0 +1,21 @@
+The MIT License
+
+Copyright (c) 2012-2021 Christian Rau
+
+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.
diff --git a/mnn_fp16_merge/merge/half-2.2.0/README.txt b/mnn_fp16_merge/merge/half-2.2.0/README.txt
new file mode 100644
index 0000000..3dd0d1c
--- /dev/null
+++ b/mnn_fp16_merge/merge/half-2.2.0/README.txt
@@ -0,0 +1,317 @@
+HALF-PRECISION FLOATING-POINT LIBRARY (Version 2.2.0)
+-----------------------------------------------------
+
+This is a C++ header-only library to provide an IEEE 754 conformant 16-bit
+half-precision floating-point type along with corresponding arithmetic
+operators, type conversions and common mathematical functions. It aims for both
+efficiency and ease of use, trying to accurately mimic the behaviour of the
+built-in floating-point types at the best performance possible.
+
+
+INSTALLATION AND REQUIREMENTS
+-----------------------------
+
+Conveniently, the library consists of just a single header file containing all
+the functionality, which can be directly included by your projects, without the
+neccessity to build anything or link to anything.
+
+Whereas this library is fully C++98-compatible, it can profit from certain
+C++11 features. Support for those features is checked automatically at compile
+(or rather preprocessing) time, but can be explicitly enabled or disabled by
+predefining the corresponding preprocessor symbols to either 1 or 0 yourself
+before including half.hpp. This is useful when the automatic detection fails
+(for more exotic implementations) or when a feature should be explicitly
+disabled:
+
+ - 'long long' integer type for mathematical functions returning 'long long'
+ results (enabled for VC++ 2003 and icc 11.1 and newer, gcc and clang,
+ overridable with 'HALF_ENABLE_CPP11_LONG_LONG').
+
+ - Static assertions for extended compile-time checks (enabled for VC++ 2010,
+ gcc 4.3, clang 2.9, icc 11.1 and newer, overridable with
+ 'HALF_ENABLE_CPP11_STATIC_ASSERT').
+
+ - Generalized constant expressions (enabled for VC++ 2015, gcc 4.6, clang 3.1,
+ icc 14.0 and newer, overridable with 'HALF_ENABLE_CPP11_CONSTEXPR').
+
+ - noexcept exception specifications (enabled for VC++ 2015, gcc 4.6,
+ clang 3.0, icc 14.0 and newer, overridable with 'HALF_ENABLE_CPP11_NOEXCEPT').
+
+ - User-defined literals for half-precision literals to work (enabled for
+ VC++ 2015, gcc 4.7, clang 3.1, icc 15.0 and newer, overridable with
+ 'HALF_ENABLE_CPP11_USER_LITERALS').
+
+ - Thread-local storage for per-thread floating-point exception flags (enabled
+ for VC++ 2015, gcc 4.8, clang 3.3, icc 15.0 and newer, overridable with
+ 'HALF_ENABLE_CPP11_THREAD_LOCAL').
+
+ - Type traits and template meta-programming features from <type_traits>
+ (enabled for VC++ 2010, libstdc++ 4.3, libc++ and newer, overridable with
+ 'HALF_ENABLE_CPP11_TYPE_TRAITS').
+
+ - Special integer types from <cstdint> (enabled for VC++ 2010, libstdc++ 4.3,
+ libc++ and newer, overridable with 'HALF_ENABLE_CPP11_CSTDINT').
+
+ - Certain C++11 single-precision mathematical functions from <cmath> for
+ floating-point classification during conversions from higher precision types
+ (enabled for VC++ 2013, libstdc++ 4.3, libc++ and newer, overridable with
+ 'HALF_ENABLE_CPP11_CMATH').
+
+ - Floating-point environment control from <cfenv> for possible exception
+ propagation to the built-in floating-point platform (enabled for VC++ 2013,
+ libstdc++ 4.3, libc++ and newer, overridable with 'HALF_ENABLE_CPP11_CFENV').
+
+ - Hash functor 'std::hash' from <functional> (enabled for VC++ 2010,
+ libstdc++ 4.3, libc++ and newer, overridable with 'HALF_ENABLE_CPP11_HASH').
+
+The library has been tested successfully with Visual C++ 2005-2015, gcc 4-8
+and clang 3-8 on 32- and 64-bit x86 systems. Please contact me if you have any
+problems, suggestions or even just success testing it on other platforms.
+
+
+DOCUMENTATION
+-------------
+
+What follows are some general words about the usage of the library and its
+implementation. For a complete documentation of its interface consult the
+corresponding website http://half.sourceforge.net. You may also generate the
+complete developer documentation from the library's only include file's doxygen
+comments, but this is more relevant to developers rather than mere users.
+
+BASIC USAGE
+
+To make use of the library just include its only header file half.hpp, which
+defines all half-precision functionality inside the 'half_float' namespace. The
+actual 16-bit half-precision data type is represented by the 'half' type, which
+uses the standard IEEE representation with 1 sign bit, 5 exponent bits and 11
+mantissa bits (including the hidden bit) and supports all types of special
+values, like subnormal values, infinity and NaNs. This type behaves like the
+built-in floating-point types as much as possible, supporting the usual
+arithmetic, comparison and streaming operators, which makes its use pretty
+straight-forward:
+
+ using half_float::half;
+ half a(3.4), b(5);
+ half c = a * b;
+ c += 3;
+ if(c > a)
+ std::cout << c << std::endl;
+
+Additionally the 'half_float' namespace also defines half-precision versions
+for all mathematical functions of the C++ standard library, which can be used
+directly through ADL:
+
+ half a(-3.14159);
+ half s = sin(abs(a));
+ long l = lround(s);
+
+You may also specify explicit half-precision literals, since the library
+provides a user-defined literal inside the 'half_float::literal' namespace,
+which you just need to import (assuming support for C++11 user-defined literals):
+
+ using namespace half_float::literal;
+ half x = 1.0_h;
+
+Furthermore the library provides proper specializations for
+'std::numeric_limits', defining various implementation properties, and
+'std::hash' for hashing half-precision numbers (assuming support for C++11
+'std::hash'). Similar to the corresponding preprocessor symbols from <cmath>
+the library also defines the 'HUGE_VALH' constant and maybe the 'FP_FAST_FMAH'
+symbol.
+
+CONVERSIONS AND ROUNDING
+
+The half is explicitly constructible/convertible from a single-precision float
+argument. Thus it is also explicitly constructible/convertible from any type
+implicitly convertible to float, but constructing it from types like double or
+int will involve the usual warnings arising when implicitly converting those to
+float because of the lost precision. On the one hand those warnings are
+intentional, because converting those types to half neccessarily also reduces
+precision. But on the other hand they are raised for explicit conversions from
+those types, when the user knows what he is doing. So if those warnings keep
+bugging you, then you won't get around first explicitly converting to float
+before converting to half, or use the 'half_cast' described below. In addition
+you can also directly assign float values to halfs.
+
+In contrast to the float-to-half conversion, which reduces precision, the
+conversion from half to float (and thus to any other type implicitly
+convertible from float) is implicit, because all values represetable with
+half-precision are also representable with single-precision. This way the
+half-to-float conversion behaves similar to the builtin float-to-double
+conversion and all arithmetic expressions involving both half-precision and
+single-precision arguments will be of single-precision type. This way you can
+also directly use the mathematical functions of the C++ standard library,
+though in this case you will invoke the single-precision versions which will
+also return single-precision values, which is (even if maybe performing the
+exact same computation, see below) not as conceptually clean when working in a
+half-precision environment.
+
+The default rounding mode for conversions between half and more precise types
+as well as for rounding results of arithmetic operations and mathematical
+functions rounds to the nearest representable value. But by predefining the
+'HALF_ROUND_STYLE' preprocessor symbol this default can be overridden with one
+of the other standard rounding modes using their respective constants or the
+equivalent values of 'std::float_round_style' (it can even be synchronized with
+the built-in single-precision implementation by defining it to
+'std::numeric_limits<float>::round_style'):
+
+ - 'std::round_indeterminate' (-1) for the fastest rounding.
+
+ - 'std::round_toward_zero' (0) for rounding toward zero.
+
+ - 'std::round_to_nearest' (1) for rounding to the nearest value (default).
+
+ - 'std::round_toward_infinity' (2) for rounding toward positive infinity.
+
+ - 'std::round_toward_neg_infinity' (3) for rounding toward negative infinity.
+
+In addition to changing the overall default rounding mode one can also use the
+'half_cast'. This converts between half and any built-in arithmetic type using
+a configurable rounding mode (or the default rounding mode if none is
+specified). In addition to a configurable rounding mode, 'half_cast' has
+another big difference to a mere 'static_cast': Any conversions are performed
+directly using the given rounding mode, without any intermediate conversion
+to/from 'float'. This is especially relevant for conversions to integer types,
+which don't necessarily truncate anymore. But also for conversions from
+'double' or 'long double' this may produce more precise results than a
+pre-conversion to 'float' using the single-precision implementation's current
+rounding mode would.
+
+ half a = half_cast<half>(4.2);
+ half b = half_cast<half,std::numeric_limits<float>::round_style>(4.2f);
+ assert( half_cast<int, std::round_to_nearest>( 0.7_h ) == 1 );
+ assert( half_cast<half,std::round_toward_zero>( 4097 ) == 4096.0_h );
+ assert( half_cast<half,std::round_toward_infinity>( 4097 ) == 4100.0_h );
+ assert( half_cast<half,std::round_toward_infinity>( std::numeric_limits<double>::min() ) > 0.0_h );
+
+ACCURACY AND PERFORMANCE
+
+From version 2.0 onward the library is implemented without employing the
+underlying floating-point implementation of the system (except for conversions,
+of course), providing an entirely self-contained half-precision implementation
+with results independent from the system's existing single- or double-precision
+implementation and its rounding behaviour.
+
+As to accuracy, many of the operators and functions provided by this library
+are exact to rounding for all rounding modes, i.e. the error to the exact
+result is at most 0.5 ULP (unit in the last place) for rounding to nearest and
+less than 1 ULP for all other rounding modes. This holds for all the operations
+required by the IEEE 754 standard and many more. Specifically the following
+functions might exhibit a deviation from the correctly rounded exact result by
+1 ULP for a select few input values: 'expm1', 'log1p', 'pow', 'atan2', 'erf',
+'erfc', 'lgamma', 'tgamma' (for more details see the documentation of the
+individual functions). All other functions and operators are always exact to
+rounding or independent of the rounding mode altogether.
+
+The increased IEEE-conformance and cleanliness of this implementation comes
+with a certain performance cost compared to doing computations and mathematical
+functions in hardware-accelerated single-precision. On average and depending on
+the platform, the arithemtic operators are about 75% as fast and the
+mathematical functions about 33-50% as fast as performing the corresponding
+operations in single-precision and converting between the inputs and outputs.
+However, directly computing with half-precision values is a rather rare
+use-case and usually using actual 'float' values for all computations and
+temproraries and using 'half's only for storage is the recommended way. But
+nevertheless the goal of this library was to provide a complete and
+conceptually clean IEEE-confromant half-precision implementation and in the few
+cases when you do need to compute directly in half-precision you do so for a
+reason and want accurate results.
+
+If necessary, this internal implementation can be overridden by predefining the
+'HALF_ARITHMETIC_TYPE' preprocessor symbol to one of the built-in
+floating-point types ('float', 'double' or 'long double'), which will cause the
+library to use this type for computing arithmetic operations and mathematical
+functions (if available). However, due to using the platform's floating-point
+implementation (and its rounding behaviour) internally, this might cause
+results to deviate from the specified half-precision rounding mode. It will of
+course also inhibit the automatic exception detection described below.
+
+The conversion operations between half-precision and single-precision types can
+also make use of the F16C extension for x86 processors by using the
+corresponding compiler intrinsics from <immintrin.h>. Support for this is
+checked at compile-time by looking for the '__F16C__' macro which at least gcc
+and clang define based on the target platform. It can also be enabled manually
+by predefining the 'HALF_ENABLE_F16C_INTRINSICS' preprocessor symbol to 1, or 0
+for explicitly disabling it. However, this will directly use the corresponding
+intrinsics for conversion without checking if they are available at runtime
+(possibly crashing if they are not), so make sure they are supported on the
+target platform before enabling this.
+
+EXCEPTION HANDLING
+
+The half-precision implementation supports all 5 required floating-point
+exceptions from the IEEE standard to indicate erroneous inputs or inexact
+results during operations. These are represented by exception flags which
+actually use the same values as the corresponding 'FE_...' flags defined in
+C++11's <cfenv> header if supported, specifically:
+
+ - 'FE_INVALID' for invalid inputs to an operation.
+ - 'FE_DIVBYZERO' for finite inputs producing infinite results.
+ - 'FE_OVERFLOW' if a result is too large to represent finitely.
+ - 'FE_UNDERFLOW' for a subnormal or zero result after rounding.
+ - 'FE_INEXACT' if a result needed rounding to be representable.
+ - 'FE_ALL_EXCEPT' as a convenient OR of all possible exception flags.
+
+The internal exception flag state will start with all flags cleared and is
+maintained per thread if C++11 thread-local storage is supported, otherwise it
+will be maintained globally and will theoretically NOT be thread-safe (while
+practically being as thread-safe as a simple integer variable can be). These
+flags can be managed explicitly using the library's error handling functions,
+which again try to mimic the built-in functions for handling floating-point
+exceptions from <cfenv>. You can clear them with 'feclearexcept' (which is the
+only way a flag can be cleared), test them with 'fetestexcept', explicitly
+raise errors with 'feraiseexcept' and save and restore their state using
+'fegetexceptflag' and 'fesetexceptflag'. You can also throw corresponding C++
+exceptions based on the current flag state using 'fethrowexcept'.
+
+However, any automatic exception detection and handling during half-precision
+operations and functions is DISABLED by default, since it comes with a minor
+performance overhead due to runtime checks, and reacting to IEEE floating-point
+exceptions is rarely ever needed in application code. But the library fully
+supports IEEE-conformant detection of floating-point exceptions and various
+ways for handling them, which can be enabled by pre-defining the corresponding
+preprocessor symbols to 1. They can be enabled individually or all at once and
+they will be processed in the order they are listed here:
+
+ - 'HALF_ERRHANDLING_FLAGS' sets the internal exception flags described above
+ whenever the corresponding exception occurs.
+ - 'HALF_ERRHANDLING_ERRNO' sets the value of 'errno' from <cerrno> similar to
+ the behaviour of the built-in floating-point types when 'MATH_ERRNO' is used.
+ - 'HALF_ERRHANDLING_FENV' will propagate exceptions to the built-in
+ floating-point implementation using 'std::feraiseexcept' if support for
+ C++11 floating-point control is enabled. However, this does not synchronize
+ exceptions: neither will clearing propagate nor will it work in reverse.
+ - 'HALF_ERRHANDLING_THROW_...' can be defined to a string literal which will
+ be used as description message for a C++ exception that is thrown whenever
+ a 'FE_...' exception occurs, similar to the behaviour of 'fethrowexcept'.
+
+If any of the above error handling is activated, non-quiet operations on
+half-precision values will also raise a 'FE_INVALID' exception whenever
+they encounter a signaling NaN value, in addition to transforming the value
+into a quiet NaN. If error handling is disabled, signaling NaNs will be
+treated like quiet NaNs (while still getting explicitly quieted if propagated
+to the result). There can also be additional treatment of overflow and
+underflow errors after they have been processed as above, which is ENABLED by
+default (but of course only takes effect if any other exception handling is
+activated) unless overridden by pre-defining the corresponding preprocessor
+symbol to 0:
+
+ - 'HALF_ERRHANDLING_OVERFLOW_TO_INEXACT' will cause overflow errors to also
+ raise a 'FE_INEXACT' exception.
+ - 'HALF_ERRHANDLING_UNDERFLOW_TO_INEXACT' will cause underflow errors to also
+ raise a 'FE_INEXACT' exception. This will also slightly change the
+ behaviour of the underflow exception, which will ONLY be raised if the
+ result is actually inexact due to underflow. If this is disabled, underflow
+ exceptions will be raised for ANY (possibly exact) subnormal result.
+
+
+CREDITS AND CONTACT
+-------------------
+
+This library is developed by CHRISTIAN RAU and released under the MIT License
+(see LICENSE.txt). If you have any questions or problems with it, feel free to
+contact me at rauy@users.sourceforge.net.
+
+Additional credit goes to JEROEN VAN DER ZIJP for his paper on "Fast Half Float
+Conversions", whose algorithms have been used in the library for converting
+between half-precision and single-precision values.
diff --git a/mnn_fp16_merge/merge/half-2.2.0/include/half.hpp b/mnn_fp16_merge/merge/half-2.2.0/include/half.hpp
new file mode 100644
index 0000000..f4d8614
--- /dev/null
+++ b/mnn_fp16_merge/merge/half-2.2.0/include/half.hpp
@@ -0,0 +1,4601 @@
+// half - IEEE 754-based half-precision floating-point library.
+//
+// Copyright (c) 2012-2021 Christian Rau <rauy@users.sourceforge.net>
+//
+// 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 <utility>
+#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 <utility>
+#include <algorithm>
+#include <istream>
+#include <ostream>
+#include <limits>
+#include <stdexcept>
+#include <climits>
+#include <cmath>
+#include <cstring>
+#include <cstdlib>
+#if HALF_ENABLE_CPP11_TYPE_TRAITS
+ #include <type_traits>
+#endif
+#if HALF_ENABLE_CPP11_CSTDINT
+ #include <cstdint>
+#endif
+#if HALF_ERRHANDLING_ERRNO
+ #include <cerrno>
+#endif
+#if HALF_ENABLE_CPP11_CFENV
+ #include <cfenv>
+#endif
+#if HALF_ENABLE_CPP11_HASH
+ #include <functional>
+#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 <immintrin.h>
+#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 `<cerrno>`. 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 `<cfenv>`. 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<float>::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 `<cmath>` 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<half_float::half>::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 `<cfloat>` 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<bool B,typename T,typename F> struct conditional : std::conditional<B,T,F> {};
+
+ /// Helper for tag dispatching.
+ template<bool B> struct bool_type : std::integral_constant<bool,B> {};
+ using std::true_type;
+ using std::false_type;
+
+ /// Type traits for floating-point types.
+ template<typename T> struct is_float : std::is_floating_point<T> {};
+ #else
+ /// Conditional type.
+ template<bool,typename T,typename> struct conditional { typedef T type; };
+ template<typename T,typename F> struct conditional<false,T,F> { typedef F type; };
+
+ /// Helper for tag dispatching.
+ template<bool> struct bool_type {};
+ typedef bool_type<true> true_type;
+ typedef bool_type<false> false_type;
+
+ /// Type traits for floating-point types.
+ template<typename> struct is_float : false_type {};
+ template<typename T> struct is_float<const T> : is_float<T> {};
+ template<typename T> struct is_float<volatile T> : is_float<T> {};
+ template<typename T> struct is_float<const volatile T> : is_float<T> {};
+ template<> struct is_float<float> : true_type {};
+ template<> struct is_float<double> : true_type {};
+ template<> struct is_float<long double> : true_type {};
+ #endif
+
+ /// Type traits for floating-point bits.
+ template<typename T> struct bits { typedef unsigned char type; };
+ template<typename T> struct bits<const T> : bits<T> {};
+ template<typename T> struct bits<volatile T> : bits<T> {};
+ template<typename T> struct bits<const volatile T> : bits<T> {};
+
+ #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<float> { typedef std::uint_least32_t type; };
+
+ /// Unsigned integer of (at least) 64 bits width.
+ template<> struct bits<double> { 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<float> : conditional<std::numeric_limits<unsigned int>::digits>=32,unsigned int,unsigned long> {};
+
+ #if HALF_ENABLE_CPP11_LONG_LONG
+ /// Unsigned integer of (at least) 64 bits width.
+ template<> struct bits<double> : conditional<std::numeric_limits<unsigned long>::digits>=64,unsigned long,unsigned long long> {};
+ #else
+ /// Unsigned integer of (at least) 64 bits width.
+ template<> struct bits<double> { 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<typename T> bool builtin_isinf(T arg)
+ {
+ #if HALF_ENABLE_CPP11_CMATH
+ return std::isinf(arg);
+ #elif defined(_MSC_VER)
+ return !::_finite(static_cast<double>(arg)) && !::_isnan(static_cast<double>(arg));
+ #else
+ return arg == std::numeric_limits<T>::infinity() || arg == -std::numeric_limits<T>::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<typename T> bool builtin_isnan(T arg)
+ {
+ #if HALF_ENABLE_CPP11_CMATH
+ return std::isnan(arg);
+ #elif defined(_MSC_VER)
+ return ::_isnan(static_cast<double>(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<typename T> 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<uint32>::digits - 1;
+ #if HALF_TWOS_COMPLEMENT_INT
+ return static_cast<int32>(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<int32>(arg) >> i;
+ #else
+ return static_cast<int32>(arg)/(static_cast<int32>(1)<<i) - ((arg>>(std::numeric_limits<uint32>::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<std::float_round_style R> 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<std::float_round_style R> 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<std::float_round_style R,bool I> 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<std::float_round_style R,bool E,bool I> 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<unsigned>(abs>=(0x3800+E))) :
+ (R==std::round_toward_infinity) ? (0x3C00&-(~(value>>15)&(abs!=0))) :
+ (R==std::round_toward_neg_infinity) ? (0x3C00&-static_cast<unsigned>(value>0x8000)) :
+ 0) | (value&0x8000);
+ }
+ if(abs >= 0x6400)
+ return (abs>0x7C00) ? signal(value) : value;
+ unsigned int exp = 25 - (abs>>10), mask = (1<<exp) - 1;
+ raise(FE_INEXACT, I && (value&mask));
+ return (( (R==std::round_to_nearest) ? ((1<<(exp-1))-(~(value>>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<std::float_round_style R,unsigned int F,bool S,bool N,bool I> 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<uint32>(1)<<F) && exp; m<<=1,--exp) ;
+ else if(exp < 0)
+ return rounded<R,I>(sign+(m>>(F-10-exp)), (m>>(F-11-exp))&1, s|((m&((static_cast<uint32>(1)<<(F-11-exp))-1))!=0));
+ return rounded<R,I>(sign+(exp<<10)+(m>>(F-10)), (m>>(F-11))&1, s|((m&((static_cast<uint32>(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<std::float_round_style R> 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<float>::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<R>(sign);
+ if(fbits >= 0x38800000)
+ return rounded<R,false>(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<R,false>(sign|(fbits>>(i+1)), (fbits>>i)&1, (fbits&((static_cast<uint32>(1)<<i)-1))!=0);
+ }
+ if(fbits != 0)
+ return underflow<R>(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<uint32>(exp!=0xFF);
+ return rounded<R,false>(base_table[sexp]+(fbits>>i), (m>>(i-1))&1, (((static_cast<uint32>(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<std::float_round_style R> 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<double>::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<R>(sign);
+ if(hi >= 0x3F100000)
+ return rounded<R,false>(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<R,false>(sign|(hi>>(i+1)), (hi>>i)&1, ((hi&((static_cast<uint32>(1)<<i)-1))|lo)!=0);
+ }
+ if((hi|lo) != 0)
+ return underflow<R>(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<std::float_round_style R,typename T> unsigned int float2half_impl(T value, ...)
+ {
+ unsigned int hbits = static_cast<unsigned>(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<R>(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<int>(ival));
+ return rounded<R,false>(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<std::float_round_style R,typename T> unsigned int float2half(T value)
+ {
+ return float2half_impl<R>(value, bool_type<std::numeric_limits<T>::is_iec559&&sizeof(typename bits<T>::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<std::float_round_style R,typename T> unsigned int int2half(T value)
+ {
+ unsigned int bits = static_cast<unsigned>(value<0) << 15;
+ if(!value)
+ return bits;
+ if(bits)
+ value = -value;
+ if(value > 0xFFFF)
+ return overflow<R>(bits);
+ unsigned int m = static_cast<unsigned int>(value), exp = 24;
+ for(; m<0x400; m<<=1,--exp) ;
+ for(; m>0x7FF; m>>=1,++exp) ;
+ bits |= (exp<<10) + m;
+ return (exp>24) ? rounded<R,false>(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<float>::type fbits = static_cast<bits<float>::type>(value&0x8000) << 16;
+ int abs = value & 0x7FFF;
+ if(abs)
+ {
+ fbits |= 0x38000000 << static_cast<unsigned>(abs>=0x7C00);
+ for(; abs<0x400; abs<<=1,fbits-=0x800000) ;
+ fbits += static_cast<bits<float>::type>(abs) << 13;
+ }
+ #else
+ static const bits<float>::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<float>::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<float>::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<uint32>(value&0x8000) << 16;
+ unsigned int abs = value & 0x7FFF;
+ if(abs)
+ {
+ hi |= 0x3F000000 << static_cast<unsigned>(abs>=0x7C00);
+ for(; abs<0x400; abs<<=1,hi-=0x100000) ;
+ hi += static_cast<uint32>(abs) << 10;
+ }
+ bits<double>::type dbits = static_cast<bits<double>::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<typename T> T half2float_impl(unsigned int value, T, ...)
+ {
+ T out;
+ unsigned int abs = value & 0x7FFF;
+ if(abs > 0x7C00)
+ out = (std::numeric_limits<T>::has_signaling_NaN && !(abs&0x200)) ? std::numeric_limits<T>::signaling_NaN() :
+ std::numeric_limits<T>::has_quiet_NaN ? std::numeric_limits<T>::quiet_NaN() : T();
+ else if(abs == 0x7C00)
+ out = std::numeric_limits<T>::has_infinity ? std::numeric_limits<T>::infinity() : std::numeric_limits<T>::max();
+ else if(abs > 0x3FF)
+ out = std::ldexp(static_cast<T>((abs&0x3FF)|0x400), (abs>>10)-25);
+ else
+ out = std::ldexp(static_cast<T>(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<typename T> T half2float(unsigned int value)
+ {
+ return half2float_impl(value, T(), bool_type<std::numeric_limits<T>::is_iec559&&sizeof(typename bits<T>::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<std::float_round_style R,bool E,bool I,typename T> T half2int(unsigned int value)
+ {
+ unsigned int abs = value & 0x7FFF;
+ if(abs >= 0x7C00)
+ {
+ raise(FE_INVALID);
+ return (value&0x8000) ? std::numeric_limits<T>::min() : std::numeric_limits<T>::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<int32>((exp<=0) ? (m<<-exp) : ((m+(
+ (R==std::round_to_nearest) ? ((1<<(exp-1))-(~(m>>exp)&E)) :
+ (R==std::round_toward_infinity) ? (((1<<exp)-1)&((value>>15)-1)) :
+ (R==std::round_toward_neg_infinity) ? (((1<<exp)-1)&-(value>>15)) : 0))>>exp));
+ if((!std::numeric_limits<T>::is_signed && (value&0x8000)) || (std::numeric_limits<T>::digits<16 &&
+ ((value&0x8000) ? (-i<std::numeric_limits<T>::min()) : (i>std::numeric_limits<T>::max()))))
+ raise(FE_INVALID);
+ else if(I && exp > 0 && (m&((1<<exp)-1)))
+ raise(FE_INEXACT);
+ return static_cast<T>((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<std::float_round_style R> 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<uint32>((static_cast<unsigned long long>(x)*static_cast<unsigned long long>(y)+0x80000000)>>32);
+ #else
+ return mulhi<std::round_to_nearest>(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<unsigned long long>(x) << 32;
+ return s = (xx%y!=0), static_cast<uint32>(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<bool Q,bool R> 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<int>(Q);
+ }
+ if(!Q && mx == my)
+ return 0;
+ if(mx >= my)
+ {
+ mx -= my;
+ ++q;
+ }
+ if(Q)
+ {
+ q &= (1<<(std::numeric_limits<int>::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<unsigned int F> uint32 sqrt(uint32 &r, int &exp)
+ {
+ int i = exp & 1;
+ r <<= i;
+ exp = (exp-i) / 2;
+ uint32 m = 0;
+ for(uint32 bit=static_cast<uint32>(1)<<F; bit; bit>>=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<n; ++i)
+ {
+ uint32 mz = my + logs[i];
+ if(mz <= m)
+ {
+ my = mz;
+ mx += mx >> 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<n; ++i)
+ {
+ uint32 mz = mx + (mx>>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<uint32,uint32> 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; i<n; ++i)
+ {
+ uint32 sign = sign_mask(mz);
+ uint32 tx = mx - (arithmetic_shift(my, i)^sign) + sign;
+ uint32 ty = my + (arithmetic_shift(mx, i)^sign) - sign;
+ mx = tx; my = ty; mz -= (angles[i]^sign) - sign;
+ }
+ return std::make_pair(my, mx);
+ }
+
+ /// Fixed point arc tangent.
+ /// This uses the CORDIC algorithm in vectoring mode.
+ /// \param my y coordinate as Q0.30
+ /// \param mx x coordinate as Q0.30
+ /// \param n number of iterations (at most 31)
+ /// \return arc tangent of \a my / \a mx as Q1.30
+ inline uint32 atan2(uint32 my, uint32 mx, 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 mz = 0;
+ for(unsigned int i=0; i<n; ++i)
+ {
+ uint32 sign = sign_mask(my);
+ uint32 tx = mx + (arithmetic_shift(my, i)^sign) - sign;
+ uint32 ty = my - (arithmetic_shift(mx, i)^sign) + sign;
+ mx = tx; my = ty; mz += (angles[i]^sign) - sign;
+ }
+ return mz;
+ }
+
+ /// Reduce argument for trigonometric functions.
+ /// \param abs half-precision floating-point value
+ /// \param k value to take quarter period
+ /// \return \a abs reduced to [-pi/4,pi/4] as Q0.30
+ inline uint32 angle_arg(unsigned int abs, int &k)
+ {
+ uint32 m = (abs&0x3FF) | ((abs>0x3FF)<<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<uint32>(f>>63);
+ k = static_cast<int>(yi>>(62-exp));
+ return (multiply64(static_cast<uint32>((sign ? -f : f)>>(31-exp)), 0xC90FDAA2)^sign) - sign;
+ #else
+ uint32 yh = m*0xA2F98 + mulhi<std::round_toward_zero>(m, 0x36E4E442), yl = (m*0x36E4E442) & 0xFFFFFFFF;
+ uint32 mask = (static_cast<uint32>(1)<<(30-exp)) - 1, yi = (yh+(mask>>1)) & ~mask, sign = -static_cast<uint32>(yi>yh);
+ k = static_cast<int>(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<uint32,uint32> 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<uint32>(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<uint32,uint32> hyperbolic_args(unsigned int abs, int &exp, unsigned int n = 32)
+ {
+ uint32 mx = detail::multiply64(static_cast<uint32>((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<uint32>(1)<<d)-1))!=0)) : 1);
+ }
+
+ /// Postprocessing for binary exponential.
+ /// \tparam R rounding mode to use
+ /// \param m fractional part of as Q0.31
+ /// \param exp absolute value of unbiased exponent
+ /// \param esign sign of actual exponent
+ /// \param sign sign bit of result
+ /// \param n number of BKM iterations (at most 32)
+ /// \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<std::float_round_style R> 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<R>(sign);
+ else if(exp == -25)
+ return rounded<R,false>(sign, 1, m!=0);
+ }
+ else if(exp > 15)
+ return overflow<R>(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<R,31,false,false,true>(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<std::float_round_style R,uint32 L> unsigned int log2_post(uint32 m, int ilog, int exp, unsigned int sign = 0)
+ {
+ uint32 msign = sign_mask(ilog);
+ m = (((static_cast<uint32>(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<R>(sign);
+ m = divide64(m, L, s);
+ return fixed2half<R,30,false,false,true>(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<std::float_round_style R> unsigned int hypot_post(uint32 r, int exp)
+ {
+ int i = r >> 31;
+ if((exp+=i) > 46)
+ return overflow<R>();
+ if(exp < -34)
+ return underflow<R>();
+ r = (r>>i) | (r&i);
+ uint32 m = sqrt<30>(r, exp+=15);
+ return fixed2half<R,15,false,false,false>(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<std::float_round_style R> 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<R>(sign);
+ if(exp < -11)
+ return underflow<R>(sign);
+ uint32 m = divide64(my>>(i+1), mx, s);
+ return fixed2half<R,30,false,false,true>(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<std::float_round_style R,bool S> unsigned int area(unsigned int arg)
+ {
+ int abs = arg & 0x7FFF, expx = (abs>>10) + (abs<=0x3FF) - 15, expy = -15, ilog, i;
+ uint32 mx = static_cast<uint32>((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<uint32>(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<R,0xB8AA3B2A>(log2(my>>i, 26+S+G)+(G<<3), ilog+i, 17, arg&(static_cast<unsigned>(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<uint32>((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<std::float_round_style R,bool C> 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<<x2.exp)&0x7FFFFFFF, 22), x2.m>>(31-x2.exp)));
+ return (!C || sign) ? fixed2half<R,31,false,true,true>(0x80000000-(e.m>>(C-e.exp)), 14+C, sign&(C-1U)) :
+ (e.exp<-25) ? underflow<R>() : fixed2half<R,30,false,false,true>(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<std::float_round_style R,bool L> 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<uint32>(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<uint32>(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<uint32>(z.exp)<<26)+(m>>5)), 5);
+ for(; z.m<0x80000000; z.m<<=1,--z.exp) ;
+ l = l + z / lbe;
+ }
+ sign = static_cast<unsigned>(x.exp&&(l.exp<s.exp||(l.exp==s.exp&&l.m<s.m))) << 15;
+ s = sign ? (s-l) : x.exp ? (l-s) : (l+s);
+ }
+ else
+ {
+ sign = static_cast<unsigned>(x.exp==0) << 15;
+ if(s.exp < -24)
+ return underflow<R>(sign);
+ if(s.exp > 15)
+ return overflow<R>(sign);
+ }
+ }
+ else
+ {
+ s = s * lbe;
+ uint32 m;
+ if(s.exp < 0)
+ {
+ m = s.m >> -s.exp;
+ s.exp = 0;
+ }
+ else
+ {
+ m = (s.m<<s.exp) & 0x7FFFFFFF;
+ s.exp = (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<R>(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<R>(sign);
+ }
+ return fixed2half<R,31,false,false,true>(s.m, s.exp+14, sign);
+ }
+ /// \}
+
+ template<typename,typename,std::float_round_style> 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::uint16>(detail::float2half<round_style>(rhs))) {}
+
+ /// Conversion to single-precision.
+ /// \return single precision value representing expression value
+ operator float() const { return detail::half2float<float>(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::uint16>(detail::float2half<round_style>(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<detail::uint16>(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<typename charT,typename traits> friend std::basic_ostream<charT,traits>& operator<<(std::basic_ostream<charT,traits>&, half);
+ template<typename charT,typename traits> friend std::basic_istream<charT,traits>& operator>>(std::basic_istream<charT,traits>&, 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<typename,typename,std::float_round_style> friend struct detail::half_caster;
+ friend class std::numeric_limits<half>;
+ #if HALF_ENABLE_CPP11_HASH
+ friend struct std::hash<half>;
+ #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<half::round_style>(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<typename T,typename U,std::float_round_style R=(std::float_round_style)(HALF_ROUND_STYLE)> struct half_caster {};
+ template<typename U,std::float_round_style R> struct half_caster<half,U,R>
+ {
+ #if HALF_ENABLE_CPP11_STATIC_ASSERT && HALF_ENABLE_CPP11_TYPE_TRAITS
+ static_assert(std::is_arithmetic<U>::value, "half_cast from non-arithmetic type unsupported");
+ #endif
+
+ static half cast(U arg) { return cast_impl(arg, is_float<U>()); };
+
+ private:
+ static half cast_impl(U arg, true_type) { return half(binary, float2half<R>(arg)); }
+ static half cast_impl(U arg, false_type) { return half(binary, int2half<R>(arg)); }
+ };
+ template<typename T,std::float_round_style R> struct half_caster<T,half,R>
+ {
+ #if HALF_ENABLE_CPP11_STATIC_ASSERT && HALF_ENABLE_CPP11_TYPE_TRAITS
+ static_assert(std::is_arithmetic<T>::value, "half_cast to non-arithmetic type unsupported");
+ #endif
+
+ static T cast(half arg) { return cast_impl(arg, is_float<T>()); }
+
+ private:
+ static T cast_impl(half arg, true_type) { return half2float<T>(arg.data_); }
+ static T cast_impl(half arg, false_type) { return half2int<R,true,true,T>(arg.data_); }
+ };
+ template<std::float_round_style R> struct half_caster<half,half,R>
+ {
+ 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<half_float::half>
+ {
+ 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<half_float::half>
+ {
+ /// 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<half_float::detail::uint16>()(arg.data_&-static_cast<unsigned>(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<half::round_style>(detail::half2float<detail::internal_t>(x.data_)+detail::half2float<detail::internal_t>(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<<d)-1))!=0);
+ }
+ else
+ my = 1;
+ if(sub)
+ {
+ if(!(mx-=my))
+ return half(detail::binary, static_cast<unsigned>(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<half::round_style>(sign));
+ mx = (mx>>i) | (mx&i);
+ }
+ return half(detail::binary, detail::rounded<half::round_style,false>(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<half::round_style>(detail::half2float<detail::internal_t>(x.data_)-detail::half2float<detail::internal_t>(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<half::round_style>(detail::half2float<detail::internal_t>(x.data_)*detail::half2float<detail::internal_t>(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<detail::uint32>((absx&0x3FF)|0x400) * static_cast<detail::uint32>((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<half::round_style>(sign));
+ else if(exp < -11)
+ return half(detail::binary, detail::underflow<half::round_style>(sign));
+ return half(detail::binary, detail::fixed2half<half::round_style,20,false,false,false>(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<half::round_style>(detail::half2float<detail::internal_t>(x.data_)/detail::half2float<detail::internal_t>(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<half::round_style>(sign));
+ else if(exp < -11)
+ return half(detail::binary, detail::underflow<half::round_style>(sign));
+ mx <<= 12 + i;
+ my <<= 1;
+ return half(detail::binary, detail::fixed2half<half::round_style,11,false,false,false>(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<typename charT,typename traits> std::basic_ostream<charT,traits>& operator<<(std::basic_ostream<charT,traits> &out, half arg)
+ {
+ #ifdef HALF_ARITHMETIC_TYPE
+ return out << detail::half2float<detail::internal_t>(arg.data_);
+ #else
+ return out << detail::half2float<float>(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<typename charT,typename traits> std::basic_istream<charT,traits>& operator>>(std::basic_istream<charT,traits> &in, half &arg)
+ {
+ #ifdef HALF_ARITHMETIC_TYPE
+ detail::internal_t f;
+ #else
+ double f;
+ #endif
+ if(in >> f)
+ arg.data_ = detail::float2half<half::round_style>(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<false,false>(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<false,true>(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<true, true>(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<detail::internal_t>(x.data_), fy = detail::half2float<detail::internal_t>(y.data_), fz = detail::half2float<detail::internal_t>(z.data_);
+ #if HALF_ENABLE_CPP11_CMATH && FP_FAST_FMA
+ return half(detail::binary, detail::float2half<half::round_style>(std::fma(fx, fy, fz)));
+ #else
+ return half(detail::binary, detail::float2half<half::round_style>(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<detail::uint32>((absx&0x3FF)|0x400) * static_cast<detail::uint32>((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<detail::uint32>((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<detail::uint32>(1)<<d)-1))!=0)) : 1;
+ if(sub)
+ {
+ m = m - mz;
+ if(!m)
+ return half(detail::binary, static_cast<unsigned>(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<half::round_style>(sign));
+ else if(exp < -10)
+ return half(detail::binary, detail::underflow<half::round_style>(sign));
+ return half(detail::binary, detail::fixed2half<half::round_style,23,false,false,false>(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<unsigned>(*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<half::round_style>(std::exp(detail::half2float<detail::internal_t>(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<half::round_style>() : detail::overflow<half::round_style>());
+ detail::uint32 m = detail::multiply64(static_cast<detail::uint32>((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<half::round_style>(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<half::round_style>(std::exp2(detail::half2float<detail::internal_t>(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<half::round_style>() : detail::overflow<half::round_style>());
+ return half(detail::binary, detail::exp2_post<half::round_style>(
+ (static_cast<detail::uint32>(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<half::round_style>(std::expm1(detail::half2float<detail::internal_t>(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<half::round_style,true>(0xBBFF, 1, 1) : detail::overflow<half::round_style>());
+ detail::uint32 m = detail::multiply64(static_cast<detail::uint32>((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<detail::uint32>(1)<<exp)-1))!=0)|s);
+ exp = 0;
+ }
+ else
+ m -= (exp<31) ? (0x80000000>>exp) : 1;
+ for(exp+=14; m<0x80000000 && exp; m<<=1,--exp) ;
+ if(exp > 29)
+ return half(detail::binary, detail::overflow<half::round_style>());
+ return half(detail::binary, detail::rounded<half::round_style,true>(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<half::round_style>(std::log(detail::half2float<detail::internal_t>(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<half::round_style,0xB8AA3B2A>(
+ detail::log2(static_cast<detail::uint32>((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<half::round_style>(std::log10(detail::half2float<detail::internal_t>(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<half::round_style,0xD49A784C>(
+ detail::log2(static_cast<detail::uint32>((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<half::round_style>(std::log2(detail::half2float<detail::internal_t>(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<unsigned>(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<detail::uint32>((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<half::round_style,27,false,false,true>(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<half::round_style>(std::log1p(detail::half2float<detail::internal_t>(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<detail::uint32>((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<half::round_style,0xB8AA3B2A>(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<half::round_style>(std::sqrt(detail::half2float<detail::internal_t>(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<detail::uint32>((abs&0x3FF)|0x400) << 10, m = detail::sqrt<20>(r, exp+=abs>>10);
+ return half(detail::binary, detail::rounded<half::round_style,false>((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<half::round_style>(detail::internal_t(1)/std::sqrt(detail::half2float<detail::internal_t>(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<half::round_style,30,false,false,true>(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<half::round_style>(std::cbrt(detail::half2float<detail::internal_t>(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<detail::uint32>((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<<exp) & 0x7FFFFFFF;
+ exp = 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<half::round_style,31,false,false,false>(m, exp+14, arg.data_&0x8000) :
+ detail::fixed2half<half::round_style,23,false,false,false>((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<detail::internal_t>(x.data_), fy = detail::half2float<detail::internal_t>(y.data_);
+ #if HALF_ENABLE_CPP11_CMATH
+ return half(detail::binary, detail::float2half<half::round_style>(std::hypot(fx, fy)));
+ #else
+ return half(detail::binary, detail::float2half<half::round_style>(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<detail::uint32>(1)<<d)-1))!=0)) : 1;
+ return half(detail::binary, detail::hypot_post<half::round_style>(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<detail::internal_t>(x.data_), fy = detail::half2float<detail::internal_t>(y.data_), fz = detail::half2float<detail::internal_t>(z.data_);
+ return half(detail::binary, detail::float2half<half::round_style>(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<detail::uint32>(1)<<d)-1))!=0)) : 1;
+ my += mz;
+ if(my & 0x80000000)
+ {
+ my = (my>>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<detail::uint32>(1)<<d)-1))!=0)) : 1;
+ return half(detail::binary, detail::hypot_post<half::round_style>(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<half::round_style>(std::pow(detail::half2float<detail::internal_t>(x.data_), detail::half2float<detail::internal_t>(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<unsigned>((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<detail::uint32>((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<detail::uint32>((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<<exp) & 0x7FFFFFFF;
+ exp = m >> (31-exp);
+ }
+ return half(detail::binary, detail::exp2_post<half::round_style>(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<detail::internal_t>(arg.data_);
+ *sin = half(detail::binary, detail::float2half<half::round_style>(std::sin(f)));
+ *cos = half(detail::binary, detail::float2half<half::round_style>(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<half::round_style,true>(arg.data_-1, 1, 1));
+ *cos = half(detail::binary, detail::rounded<half::round_style,true>(0x3BFF, 1, 1));
+ }
+ else
+ {
+ if(half::round_style != std::round_to_nearest)
+ {
+ switch(abs)
+ {
+ case 0x48B7:
+ *sin = half(detail::binary, detail::rounded<half::round_style,true>((~arg.data_&0x8000)|0x1D07, 1, 1));
+ *cos = half(detail::binary, detail::rounded<half::round_style,true>(0xBBFF, 1, 1));
+ return;
+ case 0x598C:
+ *sin = half(detail::binary, detail::rounded<half::round_style,true>((arg.data_&0x8000)|0x3BFF, 1, 1));
+ *cos = half(detail::binary, detail::rounded<half::round_style,true>(0x80FC, 1, 1));
+ return;
+ case 0x6A64:
+ *sin = half(detail::binary, detail::rounded<half::round_style,true>((~arg.data_&0x8000)|0x3BFE, 1, 1));
+ *cos = half(detail::binary, detail::rounded<half::round_style,true>(0x27FF, 1, 1));
+ return;
+ case 0x6D8C:
+ *sin = half(detail::binary, detail::rounded<half::round_style,true>((arg.data_&0x8000)|0x0FE6, 1, 1));
+ *cos = half(detail::binary, detail::rounded<half::round_style,true>(0x3BFF, 1, 1));
+ return;
+ }
+ }
+ std::pair<detail::uint32,detail::uint32> 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<half::round_style,30,true,true,true>((sc.first^-static_cast<detail::uint32>(sign))+sign));
+ *cos = half(detail::binary, detail::fixed2half<half::round_style,30,true,true,true>(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<half::round_style>(std::sin(detail::half2float<detail::internal_t>(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<half::round_style,true>(arg.data_-1, 1, 1));
+ if(half::round_style != std::round_to_nearest)
+ switch(abs)
+ {
+ case 0x48B7: return half(detail::binary, detail::rounded<half::round_style,true>((~arg.data_&0x8000)|0x1D07, 1, 1));
+ case 0x6A64: return half(detail::binary, detail::rounded<half::round_style,true>((~arg.data_&0x8000)|0x3BFE, 1, 1));
+ case 0x6D8C: return half(detail::binary, detail::rounded<half::round_style,true>((arg.data_&0x8000)|0x0FE6, 1, 1));
+ }
+ std::pair<detail::uint32,detail::uint32> sc = detail::sincos(detail::angle_arg(abs, k), 28);
+ detail::uint32 sign = -static_cast<detail::uint32>(((k>>1)&1)^(arg.data_>>15));
+ return half(detail::binary, detail::fixed2half<half::round_style,30,true,true,true>((((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<half::round_style>(std::cos(detail::half2float<detail::internal_t>(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<half::round_style,true>(0x3BFF, 1, 1));
+ if(half::round_style != std::round_to_nearest && abs == 0x598C)
+ return half(detail::binary, detail::rounded<half::round_style,true>(0x80FC, 1, 1));
+ std::pair<detail::uint32,detail::uint32> sc = detail::sincos(detail::angle_arg(abs, k), 28);
+ detail::uint32 sign = -static_cast<detail::uint32>(((k>>1)^k)&1);
+ return half(detail::binary, detail::fixed2half<half::round_style,30,true,true,true>((((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<half::round_style>(std::tan(detail::half2float<detail::internal_t>(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<half::round_style,true>(arg.data_, 0, 1));
+ if(half::round_style != std::round_to_nearest)
+ switch(abs)
+ {
+ case 0x658C: return half(detail::binary, detail::rounded<half::round_style,true>((arg.data_&0x8000)|0x07E6, 1, 1));
+ case 0x7330: return half(detail::binary, detail::rounded<half::round_style,true>((~arg.data_&0x8000)|0x4B62, 1, 1));
+ }
+ std::pair<detail::uint32,detail::uint32> 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<half::round_style>(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<half::round_style>(std::asin(detail::half2float<detail::internal_t>(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<half::round_style,true>(sign|0x3E48, 0, 1));
+ if(abs < 0x2900)
+ return half(detail::binary, detail::rounded<half::round_style,true>(arg.data_, 0, 1));
+ if(half::round_style != std::round_to_nearest && (abs == 0x2B44 || abs == 0x2DC3))
+ return half(detail::binary, detail::rounded<half::round_style,true>(arg.data_+1, 1, 1));
+ std::pair<detail::uint32,detail::uint32> 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<half::round_style,30,false,true,true>(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<half::round_style>(std::acos(detail::half2float<detail::internal_t>(arg.data_))));
+ #else
+ unsigned int abs = arg.data_ & 0x7FFF, sign = arg.data_ >> 15;
+ if(!abs)
+ return half(detail::binary, detail::rounded<half::round_style,true>(0x3E48, 0, 1));
+ if(abs >= 0x3C00)
+ return half(detail::binary, (abs>0x7C00) ? detail::signal(arg.data_) : (abs>0x3C00) ? detail::invalid() :
+ sign ? detail::rounded<half::round_style,true>(0x4248, 0, 1) : 0);
+ std::pair<detail::uint32,detail::uint32> cs = detail::atan2_args(abs);
+ detail::uint32 m = detail::atan2(cs.second, cs.first, 28);
+ return half(detail::binary, detail::fixed2half<half::round_style,31,false,true,true>(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<half::round_style>(std::atan(detail::half2float<detail::internal_t>(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<half::round_style,true>(sign|0x3E48, 0, 1) : detail::signal(arg.data_));
+ if(abs <= 0x2700)
+ return half(detail::binary, detail::rounded<half::round_style,true>(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<half::round_style,30,false,true,true>(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<half::round_style>(std::atan2(detail::half2float<detail::internal_t>(y.data_), detail::half2float<detail::internal_t>(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<half::round_style,true>(signy|0x3E48, 0, 1) :
+ signx ? detail::rounded<half::round_style,true>(signy|0x40B6, 0, 1) :
+ detail::rounded<half::round_style,true>(signy|0x3A48, 0, 1));
+ return (x.data_==0x7C00) ? half(detail::binary, signy) : half(detail::binary, detail::rounded<half::round_style,true>(signy|0x4248, 0, 1));
+ }
+ if(!absy)
+ return signx ? half(detail::binary, detail::rounded<half::round_style,true>(signy|0x4248, 0, 1)) : y;
+ if(!absx)
+ return half(detail::binary, detail::rounded<half::round_style,true>(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<half::round_style,true>(signy|0x3E48, 0, 1));
+ if(signx && d < -11)
+ return half(detail::binary, detail::rounded<half::round_style,true>(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<half::round_style>(signy));
+ my <<= 11 + i;
+ return half(detail::binary, detail::fixed2half<half::round_style,11,false,false,true>(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<half::round_style,31,false,true,true>(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<half::round_style>(std::sinh(detail::half2float<detail::internal_t>(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<half::round_style,true>(arg.data_, 0, 1));
+ std::pair<detail::uint32,detail::uint32> 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<half::round_style>(sign));
+ return half(detail::binary, detail::fixed2half<half::round_style,31,false,false,true>(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<half::round_style>(std::cosh(detail::half2float<detail::internal_t>(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<detail::uint32,detail::uint32> 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<half::round_style>());
+ return half(detail::binary, detail::fixed2half<half::round_style,31,false,false,true>(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<half::round_style>(std::tanh(detail::half2float<detail::internal_t>(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<half::round_style,true>((arg.data_&0x8000)|0x3BFF, 1, 1));
+ if(abs < 0x2700)
+ return half(detail::binary, detail::rounded<half::round_style,true>(arg.data_-1, 1, 1));
+ if(half::round_style != std::round_to_nearest && abs == 0x2D3F)
+ return half(detail::binary, detail::rounded<half::round_style,true>(arg.data_-3, 0, 1));
+ std::pair<detail::uint32,detail::uint32> 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<half::round_style>(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<half::round_style>(std::asinh(detail::half2float<detail::internal_t>(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<half::round_style,true>(arg.data_-1, 1, 1));
+ if(half::round_style != std::round_to_nearest)
+ switch(abs)
+ {
+ case 0x32D4: return half(detail::binary, detail::rounded<half::round_style,true>(arg.data_-13, 1, 1));
+ case 0x3B5B: return half(detail::binary, detail::rounded<half::round_style,true>(arg.data_-197, 1, 1));
+ }
+ return half(detail::binary, detail::area<half::round_style,true>(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<half::round_style>(std::acosh(detail::half2float<detail::internal_t>(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<half::round_style,false>(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<half::round_style>(std::atanh(detail::half2float<detail::internal_t>(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<half::round_style,true>(arg.data_, 0, 1));
+ detail::uint32 m = static_cast<detail::uint32>((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<half::round_style,0xB8AA3B2A>(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<half::round_style>(std::erf(detail::half2float<detail::internal_t>(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<half::round_style,true>((arg.data_&0x8000)|0x3BFF, 1, 1));
+ return half(detail::binary, detail::erf<half::round_style,false>(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<half::round_style>(std::erfc(detail::half2float<detail::internal_t>(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<half::round_style,true>((sign>>1)-(sign>>15), sign>>15, 1));
+ return half(detail::binary, detail::erf<half::round_style,true>(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<half::round_style>(std::lgamma(detail::half2float<detail::internal_t>(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<half::round_style,true>(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<half::round_style>(std::tgamma(detail::half2float<detail::internal_t>(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<half::round_style>((1-((abs>>(25-(abs>>10)))&1))<<15));
+ if(arg.data_ <= 0x100 || (arg.data_ >= 0x4900 && arg.data_ < 0x8000))
+ return half(detail::binary, detail::overflow<half::round_style>());
+ if(arg.data_ == 0x3C00)
+ return arg;
+ return half(detail::binary, detail::gamma<half::round_style,false>(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<std::round_toward_infinity,true,true>(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<std::round_toward_neg_infinity,true,true>(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<std::round_toward_zero,true,true>(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<std::round_to_nearest,false,true>(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<std::round_to_nearest,false,false,long>(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<half::round_style,true,true>(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<half::round_style,true,true,long>(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<half::round_style,true,false>(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<std::round_to_nearest,false,false,long long>(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<half::round_style,true,true,long long>(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<half::round_style>(sign));
+ else if(exp < -10)
+ return half(detail::binary, detail::underflow<half::round_style>(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<half::round_style,false>(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<unsigned>(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<unsigned>(
+ (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<long double>(from);
+ if(detail::builtin_isnan(to) || lfrom == to)
+ return half(static_cast<float>(to));
+ if(!fabs)
+ {
+ detail::raise(FE_UNDERFLOW, !HALF_ERRHANDLING_UNDERFLOW_TO_INEXACT);
+ return half(detail::binary, (static_cast<unsigned>(detail::builtin_signbit(to))<<15)+1);
+ }
+ unsigned int out = from.data_ + (((from.data_>>15)^static_cast<unsigned>(lfrom<to))<<1) - 1;
+ detail::raise(FE_OVERFLOW, (out&0x7FFF)==0x7C00);
+ detail::raise(FE_UNDERFLOW, !HALF_ERRHANDLING_UNDERFLOW_TO_INEXACT && (out&0x7FFF)<0x400);
+ return half(detail::binary, out);
+ }
+
+ /// Take sign.
+ /// **See also:** Documentation for [std::copysign](https://en.cppreference.com/w/cpp/numeric/math/copysign).
+ /// \param x value to change sign for
+ /// \param y value to take sign from
+ /// \return value equal to \a x in magnitude and to \a y in sign
+ inline HALF_CONSTEXPR half copysign(half x, half y) { return half(detail::binary, x.data_^((x.data_^y.data_)&0x8000)); }
+
+ /// \}
+ /// \anchor classification
+ /// \name Floating point classification
+ /// \{
+
+ /// Classify floating-point value.
+ /// **See also:** Documentation for [std::fpclassify](https://en.cppreference.com/w/cpp/numeric/math/fpclassify).
+ /// \param arg number to classify
+ /// \retval FP_ZERO for positive and negative zero
+ /// \retval FP_SUBNORMAL for subnormal numbers
+ /// \retval FP_INFINITY for positive and negative infinity
+ /// \retval FP_NAN for NaNs
+ /// \retval FP_NORMAL for all other (normal) values
+ inline HALF_CONSTEXPR int fpclassify(half arg)
+ {
+ return !(arg.data_&0x7FFF) ? FP_ZERO :
+ ((arg.data_&0x7FFF)<0x400) ? FP_SUBNORMAL :
+ ((arg.data_&0x7FFF)<0x7C00) ? FP_NORMAL :
+ ((arg.data_&0x7FFF)==0x7C00) ? FP_INFINITE :
+ FP_NAN;
+ }
+
+ /// Check if finite number.
+ /// **See also:** Documentation for [std::isfinite](https://en.cppreference.com/w/cpp/numeric/math/isfinite).
+ /// \param arg number to check
+ /// \retval true if neither infinity nor NaN
+ /// \retval false else
+ inline HALF_CONSTEXPR bool isfinite(half arg) { return (arg.data_&0x7C00) != 0x7C00; }
+
+ /// Check for infinity.
+ /// **See also:** Documentation for [std::isinf](https://en.cppreference.com/w/cpp/numeric/math/isinf).
+ /// \param arg number to check
+ /// \retval true for positive or negative infinity
+ /// \retval false else
+ inline HALF_CONSTEXPR bool isinf(half arg) { return (arg.data_&0x7FFF) == 0x7C00; }
+
+ /// Check for NaN.
+ /// **See also:** Documentation for [std::isnan](https://en.cppreference.com/w/cpp/numeric/math/isnan).
+ /// \param arg number to check
+ /// \retval true for NaNs
+ /// \retval false else
+ inline HALF_CONSTEXPR bool isnan(half arg) { return (arg.data_&0x7FFF) > 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<typename T,typename U> T half_cast(U arg) { return detail::half_caster<T,U>::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<typename T,std::float_round_style R,typename U> T half_cast(U arg) { return detail::half_caster<T,U,R>::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
+ 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);
+ }
+ /// \}
+}
+
+
+#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
diff --git a/mnn_fp16_merge/merge/merge.cpp b/mnn_fp16_merge/merge/merge.cpp
new file mode 100644
index 0000000..12f1758
--- /dev/null
+++ b/mnn_fp16_merge/merge/merge.cpp
@@ -0,0 +1,528 @@
+//
+// Created by ZIHAO GUO on 2023/12/27.
+//
+
+#include "merge.h"
+#include "error_code.h"
+#include <unistd.h>
+#include <sys/time.h>
+
+Merge::Merge(){}
+
+Merge::~Merge()
+{
+ uninit();
+}
+
+int Merge::init(std::vector<const char *> model_paths, bool save_float_cache)
+{
+ int err_code = MERGE_SUCCESS;
+ m_net_num = model_paths.size();
+ if (m_net_num < 2)
+ {
+ err_code = ERROR_MODEL_NUM_NOT_ENOUGH;
+ goto exit_init;
+ }
+
+ for (int i = 0; i < m_net_num; i++)
+ {
+ if (access(model_paths[i], F_OK) != 0)
+ {
+ err_code = ERROR_MODEL_PATH_NOT_EXISTS;
+ goto exit_init;
+ }
+ get_nets(model_paths[i]);
+ }
+
+ if (save_float_cache)
+ {
+ float_param_cache();
+ m_save_float_cache = save_float_cache;
+ }
+
+ exit_init:
+ if (err_code != MERGE_SUCCESS)
+ {
+ uninit();
+ }
+ return err_code;
+}
+
+int Merge::process(const char* save_path, std::vector<float> ratios)
+{
+ int err_code = MERGE_SUCCESS;
+ if (m_net_num != (int)ratios.size())
+ {
+ err_code = ERROR_RATIO_NUM;
+ uninit();
+ return err_code;
+ }
+
+ m_ratio_sum = 0.f;
+ m_ratios.clear();
+ for (int i = 0; i < m_net_num; i++)
+ {
+ m_ratio_sum += ratios[i];
+ }
+
+ for (int i = 0; i < m_net_num; i++)
+ {
+ m_ratios.push_back(ratios[i] / m_ratio_sum);
+ }
+
+ if (m_save_float_cache)
+ {
+ merge_from_float_cache(save_path);
+ return err_code;
+ }
+
+ int size = m_nets[0]->oplists()->size();
+ for (int i = 0; i < size; i++)
+ {
+ auto main_type = EnumNameOpParameter(m_nets[0]->oplists()->Get(i)->main_type());
+ if (main_type == "Convolution2D")
+ {
+ for (int j = 0; j < m_net_num; j++)
+ {
+ m_conv_ops.push_back(m_nets[j]->oplists()->Get(i));
+ }
+ conv2d_merge();
+ m_conv_ops.clear();
+ }
+
+ if (main_type == "Blob")
+ {
+ auto blob_op_1 = m_nets[0]->oplists()->Get(i);
+ auto blob_opt_1 = blob_op_1->UnPack();
+ std::vector<uint8_t> quan_param_1 = blob_opt_1->main.AsBlob()->uint8s;
+
+ auto blob_op_2 = m_nets[1]->oplists()->Get(i);
+ auto blob_opt_2 = blob_op_2->UnPack();
+ std::vector<uint8_t> quan_param2 = blob_opt_2->main.AsBlob()->uint8s;
+ if (quan_param_1 != quan_param2)
+ {
+ for (int j = 0; j < m_net_num; j++)
+ {
+ m_blob_ops.push_back(m_nets[j]->oplists()->Get(i));
+ }
+ blob_merge();
+ }
+ m_blob_ops.clear();
+ }
+
+ if (main_type == "LayerNorm")
+ {
+ for (int j = 0; j < m_net_num; j++)
+ {
+ m_layernorm_ops.push_back(m_nets[j]->oplists()->Get(i));
+ }
+ layer_norm_merge();
+ m_layernorm_ops.clear();
+ }
+ }
+ std::ofstream output(save_path, std::ofstream::binary);
+ output.write((const char*)m_buffer[0], m_buffer_len);
+ return err_code;
+}
+
+
+/*****************内部函数**********************/
+void Merge::get_nets(const char *model_path)
+{
+ std::ifstream infile;
+ infile.open(model_path, std::ios::binary | std::ios::in);
+ infile.seekg(0, std::ios::end);
+ int length = infile.tellg();
+ m_buffer_len = length;
+ infile.seekg(0, std::ios::beg);
+ char* buffer_pointer = new char[length];
+ m_buffer.push_back(buffer_pointer);
+ infile.read(buffer_pointer, length);
+ infile.close();
+ auto net = MNN::GetNet(buffer_pointer);
+ m_nets.push_back(net);
+}
+
+void Merge::float_param_cache()
+{
+ m_conv2d_weight_cache.resize(m_net_num);
+ m_conv2d_bias_cache.resize(m_net_num);
+ m_blob_cache.resize(m_net_num);
+ m_layernorm_gamma_cache.resize(m_net_num);
+ m_layernorm_beta_cache.resize(m_net_num);
+ int size = m_nets[0]->oplists()->size(); // 单个模型的op数量
+ for (int i = 0; i < size; i++)
+ {
+ auto main_type = EnumNameOpParameter(m_nets[0]->oplists()->Get(i)->main_type());
+ if (main_type == "Convolution2D")
+ {
+ conv2d_float_cache(i);
+ m_param_idx.push_back(i);
+ }
+ if (main_type == "Blob")
+ {
+ auto blob_op_1 = m_nets[0]->oplists()->Get(i);
+ auto blob_opt_1 = blob_op_1->UnPack();
+ std::vector<uint8_t> quan_param_1 = blob_opt_1->main.AsBlob()->uint8s;
+
+ auto blob_op_2 = m_nets[1]->oplists()->Get(i);
+ auto blob_opt_2 = blob_op_2->UnPack();
+ std::vector<uint8_t> quan_param2 = blob_opt_2->main.AsBlob()->uint8s;
+ if (quan_param_1 != quan_param2)
+ {
+ blob_float_cache(i);
+ m_param_idx.push_back(i);
+ }
+ }
+ if (main_type == "LayerNorm")
+ {
+ layer_norm_float_cache(i);
+ m_param_idx.push_back(i);
+ }
+ }
+
+ for (int i = 0; i < m_net_num - 1; i++)
+ {
+ char* pointer = m_buffer.back();
+ m_buffer.pop_back();
+ delete[] pointer;
+ m_nets.pop_back();
+ }
+}
+
+void Merge::conv2d_float_cache(int i)
+{
+ if (EnumNameOpType(m_nets[0]->oplists()->Get(i)->type()) == "Deconvolution")
+ {
+ int deconv_size = m_nets[0]->oplists()->Get(i)->main_as_Convolution2D()->weight()->size();
+ for (int j = 0; j < m_net_num; j++)
+ {
+ float* data_ptr = const_cast<float*>(m_nets[j]->oplists()->Get(i)->main_as_Convolution2D()->weight()->data());
+ std::vector<float> weight(data_ptr, data_ptr + deconv_size);
+ m_conv2d_weight_cache[j].push_back(weight);
+ }
+ }
+ else
+ {
+ int quan_size = m_nets[0]->oplists()->Get(i)->main_as_Convolution2D()->quanParameter()->buffer()->size();
+ for (int j = 0; j < m_net_num; j++)
+ {
+ int8_t *char_p = const_cast<int8_t *>(m_nets[j]->oplists()->Get(i)->main_as_Convolution2D()->quanParameter()->buffer()->data());
+ half_float::half *half_p = (half_float::half *) char_p;
+ std::vector<float> weight;
+ for (int k = 0; k < quan_size / 2; k++)
+ {
+ float w = (float)(*(half_p + k));
+ w = fmaxf(w, -65504.0f);
+ w = fminf(w, 65504.0f);
+ weight.push_back(w);
+ }
+ m_conv2d_weight_cache[j].push_back(weight);
+ }
+ }
+ int bias_size = m_nets[0]->oplists()->Get(i)->main_as_Convolution2D()->bias()->size();
+ for (int j = 0; j < m_net_num; j++)
+ {
+ float* data_ptr = const_cast<float*>(m_nets[j]->oplists()->Get(i)->main_as_Convolution2D()->bias()->data());
+ std::vector<float> bias(data_ptr, data_ptr + bias_size);
+ m_conv2d_bias_cache[j].push_back(bias);
+ }
+}
+
+void Merge::blob_float_cache(int i)
+{
+ int quan_size = m_nets[0]->oplists()->Get(i)->main_as_Blob()->uint8s()->size();
+ for (int j = 0; j < m_net_num; j++)
+ {
+ uint8_t *uchar_p = const_cast<uint8_t *>(m_nets[j]->oplists()->Get(i)->main_as_Blob()->uint8s()->data());
+ half_float::half *half_p = (half_float::half *) uchar_p;
+ std::vector<float> weight;
+ for (int k = 0; k < quan_size / 2; k++)
+ {
+ float w = (float)(*(half_p + k));
+ w = fmaxf(w, -65504.0f);
+ w = fminf(w, 65504.0f);
+ weight.push_back(w);
+ }
+ m_blob_cache[j].push_back(weight);
+ }
+}
+
+void Merge::layer_norm_float_cache(int i)
+{
+ int gamma_size = m_nets[0]->oplists()->Get(i)->main_as_LayerNorm()->gamma()->size();
+ int beta_size = m_nets[0]->oplists()->Get(i)->main_as_LayerNorm()->beta()->size();
+ for (int j = 0; j < m_net_num; j++)
+ {
+ float* gamma_ptr = const_cast<float*>(m_nets[j]->oplists()->Get(i)->main_as_LayerNorm()->gamma()->data());
+ float* beta_ptr = const_cast<float*>(m_nets[j]->oplists()->Get(i)->main_as_LayerNorm()->beta()->data());
+ std::vector<float> gamma(gamma_ptr, gamma_ptr + gamma_size);
+ std::vector<float> beta(beta_ptr, beta_ptr + beta_size);
+ m_layernorm_gamma_cache[j].push_back(gamma);
+ m_layernorm_beta_cache[j].push_back(beta);
+ }
+}
+
+void Merge::merge_from_float_cache(const char* save_path)
+{
+ int cur_conv2d = 0;
+ int cur_blob = 0;
+ int cur_layernorm = 0;
+ for (int i : m_param_idx)
+ {
+ auto target_op = m_nets[0]->oplists()->Get(i);
+ auto main_type = EnumNameOpParameter(target_op->main_type());
+ if (main_type == "Convolution2D")
+ {
+ merge_conv2d_from_float_cache(cur_conv2d, target_op);
+ cur_conv2d++;
+ }
+
+ if (main_type == "Blob")
+ {
+ merge_blob_from_float_cache(cur_blob, target_op);
+ cur_blob++;
+ }
+
+ if (main_type == "LayerNorm")
+ {
+ merge_layernorm_from_float_cache(cur_layernorm, target_op);
+ cur_layernorm++;
+ }
+ }
+ std::ofstream output(save_path, std::ofstream::binary);
+ output.write((const char*)m_buffer[0], m_buffer_len);
+}
+
+void Merge::merge_conv2d_from_float_cache(int cur_conv2d, const MNN::Op* target_op)
+{
+ if (EnumNameOpType(target_op->type()) == "Deconvolution")
+ {
+ float* target_w_ptr = const_cast<float *>(target_op->main_as_Convolution2D()->weight()->data());
+ for (int i = 0; i < m_conv2d_weight_cache[0][cur_conv2d].size(); i++)
+ {
+ float w = 0.f;
+ for (int j = 0; j < m_net_num; j++)
+ {
+ w += m_ratios[j] * m_conv2d_weight_cache[j][cur_conv2d][i];
+ }
+ *(target_w_ptr + i) = w;
+ }
+ }
+ else
+ {
+ int8_t * target_w_ptr = const_cast<int8_t *>(target_op->main_as_Convolution2D()->quanParameter()->buffer()->data());
+ for (int i = 0; i < m_conv2d_weight_cache[0][cur_conv2d].size(); i++)
+ {
+ float w = 0.f;
+ for (int j = 0; j < m_net_num; j++)
+ {
+ w += m_ratios[j] * m_conv2d_weight_cache[j][cur_conv2d][i];
+ }
+ half_float::half merged_hf = half_float::half(w);
+ half_float::half* merged_ptr = &merged_hf;
+ int8_t * merged_quan = reinterpret_cast<int8_t*>(merged_ptr);
+ target_w_ptr[i * 2] = *merged_quan;
+ target_w_ptr[i * 2 + 1] = *(merged_quan + 1);
+ }
+ }
+ float* target_b_ptr = const_cast<float *>(target_op->main_as_Convolution2D()->bias()->data());
+ for (int i = 0; i < m_conv2d_bias_cache[0][cur_conv2d].size(); i++)
+ {
+ float b = 0.f;
+ for (int j = 0; j < m_net_num; j++)
+ {
+ b += m_ratios[j] * m_conv2d_bias_cache[j][cur_conv2d][i];
+ }
+ *(target_b_ptr + i) = b;
+ }
+}
+
+void Merge::merge_blob_from_float_cache(int cur_blob, const MNN::Op* target_op)
+{
+ uint8_t * target_blob_ptr = const_cast<uint8_t *>(target_op->main_as_Blob()->uint8s()->data());
+ for (int i = 0; i < m_blob_cache[0][cur_blob].size(); i++)
+ {
+ float w = 0.f;
+ for (int j = 0; j < m_net_num; j++)
+ {
+ w += m_ratios[j] * m_blob_cache[j][cur_blob][i];
+ }
+ half_float::half merged_hf = half_float::half(w);
+ half_float::half* merged_ptr = &merged_hf;
+ uint8_t * merged_quan = reinterpret_cast<uint8_t*>(merged_ptr);
+ target_blob_ptr[i * 2] = *merged_quan;
+ target_blob_ptr[i * 2 + 1] = *(merged_quan + 1);
+ }
+}
+
+void Merge::merge_layernorm_from_float_cache(int cur_layernorm, const MNN::Op* target_op)
+{
+ float* target_gamma_ptr = const_cast<float *>(target_op->main_as_LayerNorm()->gamma()->data());
+ for (int i = 0; i < m_layernorm_gamma_cache[0][cur_layernorm].size(); i++)
+ {
+ float gamma = 0.f;
+ for (int j = 0; j < m_net_num; j++)
+ {
+ gamma += m_ratios[j] * m_layernorm_gamma_cache[j][cur_layernorm][i];
+ }
+ *(target_gamma_ptr + i) = gamma;
+ }
+
+ float* target_beta_ptr = const_cast<float *>(target_op->main_as_LayerNorm()->beta()->data());
+ for (int i = 0; i < m_layernorm_beta_cache[0][cur_layernorm].size(); i++)
+ {
+ float beta = 0.f;
+ for (int j = 0; j < m_net_num; j++)
+ {
+ beta += m_ratios[j] * m_layernorm_beta_cache[j][cur_layernorm][i];
+ }
+ *(target_beta_ptr + i) = beta;
+ }
+}
+
+void Merge::conv2d_merge()
+{
+ if (EnumNameOpType(m_conv_ops[0]->type()) == "Deconvolution")
+ {
+ for (int i = 0; i < m_net_num; i++)
+ {
+ m_float_data.push_back(const_cast<float *>(m_conv_ops[i]->main_as_Convolution2D()->weight()->data()));
+ }
+ int conv_size = m_conv_ops[0]->main_as_Convolution2D()->weight()->size();
+ float_merge(conv_size);
+ }
+ else
+ {
+ for (int i = 0; i < m_net_num; i++)
+ {
+ m_char_data.push_back(const_cast<int8_t *>(m_conv_ops[i]->main_as_Convolution2D()->quanParameter()->buffer()->data()));
+ }
+ int quan_size = m_conv_ops[0]->main_as_Convolution2D()->quanParameter()->buffer()->size();
+ quan_merge(quan_size, 0);
+ }
+
+ for (int i = 0; i < m_net_num; i++)
+ {
+ m_float_data.push_back(const_cast<float *>(m_conv_ops[i]->main_as_Convolution2D()->bias()->data()));
+ }
+ int bias_size = m_conv_ops[0]->main_as_Convolution2D()->bias()->size();
+ float_merge(bias_size);
+}
+
+void Merge::blob_merge()
+{
+ for (int i = 0; i < m_net_num; i++)
+ {
+ m_uchar_data.push_back(const_cast<uint8_t *>(m_blob_ops[i]->main_as_Blob()->uint8s()->data()));
+ }
+ int quan_size = m_blob_ops[0]->main_as_Blob()->uint8s()->size();
+ quan_merge(quan_size, 1);
+}
+
+void Merge::layer_norm_merge()
+{
+ for (int i = 0; i < m_net_num; i++)
+ {
+ m_float_data.push_back(const_cast<float *>(m_layernorm_ops[i]->main_as_LayerNorm()->gamma()->data()));
+ }
+ int gamma_size = m_layernorm_ops[0]->main_as_LayerNorm()->gamma()->size();
+ float_merge(gamma_size);
+
+ for (int i = 0; i < m_net_num; i++)
+ {
+ m_float_data.push_back(const_cast<float *>(m_layernorm_ops[i]->main_as_LayerNorm()->beta()->data()));
+ }
+ int beta_size = m_layernorm_ops[0]->main_as_LayerNorm()->beta()->size();
+ float_merge(beta_size);
+}
+
+void Merge::float_merge(int size)
+{
+ for (int i = 0; i < size; i++)
+ {
+ float merged_value = 0.f;
+ for (int j = 0; j < m_net_num; j++)
+ {
+ merged_value += m_ratios[j] * (*(m_float_data[j] + i));
+ }
+ *(m_float_data[0] + i) = merged_value;
+ }
+ m_float_data.clear();
+}
+
+
+void Merge::quan_merge(int size, int src_data_type)
+{
+ for (int i = 0; i < m_net_num; i++)
+ {
+ if (src_data_type == 0)
+ {
+ m_half_data.push_back((half_float::half*)m_char_data[i]);
+ }
+ else
+ {
+ m_half_data.push_back((half_float::half*)m_uchar_data[i]);
+ }
+ }
+
+ for (int i = 0; i < size / 2; i++)
+ {
+ float merged_data = 0.f;
+ for (int j = 0; j < m_net_num; j++)
+ {
+ float w = (float)(*(m_half_data[j] + i));
+ w = fmaxf(w, -65504.0f);
+ w = fminf(w, 65504.0f);
+ merged_data += m_ratios[j] * w;
+ }
+ half_float::half merged_hf = half_float::half(merged_data);
+ half_float::half* merged_ptr = &merged_hf;
+ if (src_data_type == 0)
+ {
+ int8_t * merged_quan = reinterpret_cast<int8_t*>(merged_ptr);
+ m_char_data[0][i * 2] = *merged_quan;
+ m_char_data[0][i * 2 + 1] = *(merged_quan + 1);
+ }
+ else
+ {
+ uint8_t * merged_quan = reinterpret_cast<uint8_t*>(merged_ptr);
+ m_uchar_data[0][i * 2] = *merged_quan;
+ m_uchar_data[0][i * 2 + 1] = *(merged_quan + 1);
+ }
+ }
+ m_char_data.clear();
+ m_uchar_data.clear();
+ m_half_data.clear();
+}
+
+void Merge::uninit()
+{
+ for (auto & i : m_buffer)
+ {
+ delete[] i;
+ }
+ m_ratios.clear();
+ m_nets.clear();
+ m_buffer.clear();
+ m_conv_ops.clear();
+ m_blob_ops.clear();
+ m_layernorm_ops.clear();
+ m_char_data.clear();
+ m_uchar_data.clear();
+ m_half_data.clear();
+ m_float_data.clear();
+ m_buffer_len = 0;
+ m_net_num = 0;
+ m_ratio_sum = 0.f;
+ if (m_save_float_cache)
+ {
+ m_conv2d_weight_cache.clear();
+ m_conv2d_bias_cache.clear();
+ m_blob_cache.clear();
+ m_layernorm_gamma_cache.clear();
+ m_layernorm_beta_cache.clear();
+ m_param_idx.clear();
+ m_save_float_cache = false;
+ }
+}
diff --git a/mnn_fp16_merge/merge/merge.h b/mnn_fp16_merge/merge/merge.h
new file mode 100644
index 0000000..499295f
--- /dev/null
+++ b/mnn_fp16_merge/merge/merge.h
@@ -0,0 +1,80 @@
+//
+// Created by ZIHAO GUO on 2023/12/27.
+//
+
+#ifndef MNN_DEMO_MERGE_H
+#define MNN_DEMO_MERGE_H
+
+#include <vector>
+#include <iostream>
+#include <fstream>
+#include "MNN_generated.h"
+#include "flatbuffers/MNN_schema.h"
+#include "half-2.2.0/include/half.hpp"
+
+class Merge{
+public:
+ Merge();
+ ~Merge();
+
+public:
+ int init(std::vector<const char*> model_paths, bool save_float_cache);
+ int process(const char* save_path, std::vector<float> ratios);
+
+private:
+ void get_nets(const char* model_path);
+ void conv2d_merge();
+ void blob_merge();
+ void layer_norm_merge();
+ void quan_merge(int size, int src_data_type);
+ void float_merge(int size);
+ void uninit();
+
+private:
+ /*
+ * 选择缓存全部模型float参数时，用到的成员函数
+ */
+ void float_param_cache(); // 缓存参数的总体调度函数, 主要是遍历模型Op, 在init时被调用
+ void conv2d_float_cache(int i); // 遍历当前op为卷积层, 被调用存入缓存
+ void blob_float_cache(int i); // 遍历当前op为blob层, 被调用存入缓存
+ void layer_norm_float_cache(int i); // 遍历当前op为layernorm层, 被调用存入缓存
+ void merge_from_float_cache(const char* save_path); // 通过缓存好的参数做融合的总体调度函数, 主要是遍历含参Op, 在process时被调用
+ void merge_conv2d_from_float_cache(int cur_conv2d, const MNN::Op* target_op); // 遍历含参Op为卷积时, 被调用做融合
+ void merge_blob_from_float_cache(int cur_blob, const MNN::Op* target_op); // 遍历含参Op为blob时, 被调用做融合
+ void merge_layernorm_from_float_cache(int cur_layernorm, const MNN::Op* target_op); // 遍历含参Op为layernorm时, 被调用做融合
+
+private:
+ std::vector<float> m_ratios; // 存放每个模型的融合比例
+ float m_ratio_sum = 0.f; // 融合比例之和
+ std::vector<const MNN::Net*> m_nets; // 存放每个模型
+ std::vector<char*> m_buffer; // 存放每个模型读取时new出来的buffer, 需要释放
+ int m_buffer_len = 0; // 单个模型的buffer长度
+ int m_net_num = 0; // 模型数量
+ std::vector<const MNN::Op*> m_conv_ops; // 所有模型在同一层(若该层为卷积层)的所有卷积Op
+ std::vector<const MNN::Op*> m_blob_ops; // 所有模型在同一层(若该层为blob)的所有blob Op
+ std::vector<const MNN::Op*> m_layernorm_ops; // 所有模型在同一层(若该层为layer norm层)的所有layer norm Op
+ std::vector<float*> m_float_data; // 所有模型在同一层的float类型参数的数据指针
+ std::vector<int8_t *> m_char_data; // 所有模型在同一层的signed char类型参数的数据指针
+ std::vector<uint8_t *> m_uchar_data; // 所有模型在同一层的unsigned char类型参数的数据指针
+ std::vector<half_float::half*> m_half_data; // m_char_data或m_uchar_data中元素解析到半精度后的数据指针
+
+private:
+ /*
+ * 从这里开始是选择缓存全部模型float参数时，需要使用的成员变量, 均是用来缓存模型参数
+ * 所有用来缓存模型参数的成员变量都是3-d结构
+ * dim0 - 模型数量
+ * dim1 - 算子
+ * dim2 - 数据
+ * 示例: 以卷积bias为例, 假设2个模型, 3个卷积, 则缓存卷积bias的结构为
+ * [[[...], [...], [...]], [[...], [...], [...]]]
+ */
+ bool m_save_float_cache = false;
+ std::vector<std::vector<std::vector<float>>> m_conv2d_weight_cache; // 卷积weight参数缓存
+ std::vector<std::vector<std::vector<float>>> m_conv2d_bias_cache; // 卷积bias参数缓存
+ std::vector<std::vector<std::vector<float>>> m_blob_cache; // blob参数缓存
+ std::vector<std::vector<std::vector<float>>> m_layernorm_gamma_cache; // layer norm gamma缓存
+ std::vector<std::vector<std::vector<float>>> m_layernorm_beta_cache; // layer norm beta缓存
+ std::vector<int> m_param_idx; // 含参的Op索引, 融合时遍历该变量即可
+};
+
+#endif //MNN_DEMO_MERGE_H

File Metadata

Mime Type
text/x-diff
Expires
Mon, May 19, 10:40 (1 d, 4 h)
Storage Engine
blob
Storage Format
Raw Data
Storage Handle
1433548
Default Alt Text
(267 KB)

Event Timeline