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+HALF-PRECISION FLOATING POINT LIBRARY (Version 1.12.0)
+------------------------------------------------------
+
+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
+builtin floating point types at the best performance possible.
+
+
+INSTALLATION AND REQUIREMENTS
+-----------------------------
+
+Comfortably enough, 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
+defining the corresponding preprocessor symbols to either 1 or 0 yourself. 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 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 and newer, overridable with 'HALF_ENABLE_CPP11_STATIC_ASSERT').
+
+ - Generalized constant expressions (enabled for VC++ 2015, gcc 4.6, clang 3.1
+ and newer, overridable with 'HALF_ENABLE_CPP11_CONSTEXPR').
+
+ - noexcept exception specifications (enabled for VC++ 2015, gcc 4.6, clang 3.0
+ 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 and newer, overridable with
+ 'HALF_ENABLE_CPP11_USER_LITERALS').
+
+ - 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
+ an improved implementation of their half-precision counterparts to work
+ (enabled for VC++ 2013, libstdc++ 4.3, libc++ and newer, overridable with
+ 'HALF_ENABLE_CPP11_CMATH').
+
+ - 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 4.4-4.8
+and clang 3.1. Please contact me if you have any problems, suggestions or even
+just success testing it on other platforms.
+
+
+DOCUMENTATION
+-------------
+
+Here follow some general words about the usage of the library and its
+implementation. For a complete documentation of its iterface look at 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 (for
+reasons described below).
+
+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. This
+type behaves like the builtin 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 from float to half uses truncation
+(round toward zero, but mapping overflows to infinity) for rounding values not
+representable exactly in half-precision. This is the fastest rounding possible
+and is usually sufficient. But by redefining the 'HALF_ROUND_STYLE'
+preprocessor symbol (before including half.hpp) 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 underlying single-precision implementation by defining it
+to 'std::numeric_limits<float>::round_style'):
+
+ - 'std::round_indeterminate' or -1 for the fastest rounding (default).
+
+ - 'std::round_toward_zero' or 0 for rounding toward zero.
+
+ - std::round_to_nearest' or 1 for rounding to the nearest value.
+
+ - std::round_toward_infinity' or 2 for rounding toward positive infinity.
+
+ - std::round_toward_neg_infinity' or 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 );
+
+When using round to nearest (either as default or through 'half_cast') ties are
+by default resolved by rounding them away from zero (and thus equal to the
+behaviour of the 'round' function). But by redefining the
+'HALF_ROUND_TIES_TO_EVEN' preprocessor symbol to 1 (before including half.hpp)
+this default can be changed to the slightly slower but less biased and more
+IEEE-conformant behaviour of rounding half-way cases to the nearest even value.
+
+ #define HALF_ROUND_TIES_TO_EVEN 1
+ #include <half.hpp>
+ ...
+ assert( half_cast<int,std::round_to_nearest>(3.5_h)
+ == half_cast<int,std::round_to_nearest>(4.5_h) );
+
+IMPLEMENTATION
+
+For performance reasons (and ease of implementation) many of the mathematical
+functions provided by the library as well as all arithmetic operations are
+actually carried out in single-precision under the hood, calling to the C++
+standard library implementations of those functions whenever appropriate,
+meaning the arguments are converted to floats and the result back to half. But
+to reduce the conversion overhead as much as possible any temporary values
+inside of lengthy expressions are kept in single-precision as long as possible,
+while still maintaining a strong half-precision type to the outside world. Only
+when finally assigning the value to a half or calling a function that works
+directly on halfs is the actual conversion done (or never, when further
+converting the result to float.
+
+This approach has two implications. First of all you have to treat the
+library's documentation at http://half.sourceforge.net as a simplified version,
+describing the behaviour of the library as if implemented this way. The actual
+argument and return types of functions and operators may involve other internal
+types (feel free to generate the exact developer documentation from the Doxygen
+comments in the library's header file if you really need to). But nevertheless
+the behaviour is exactly like specified in the documentation. The other
+implication is, that in the presence of rounding errors or over-/underflows
+arithmetic expressions may produce different results when compared to
+converting to half-precision after each individual operation:
+
+ half a = std::numeric_limits<half>::max() * 2.0_h / 2.0_h; // a = MAX
+ half b = half(std::numeric_limits<half>::max() * 2.0_h) / 2.0_h; // b = INF
+ assert( a != b );
+
+But this should only be a problem in very few cases. One last word has to be
+said when talking about performance. Even with its efforts in reducing
+conversion overhead as much as possible, the software half-precision
+implementation can most probably not beat the direct use of single-precision
+computations. Usually using actual float values for all computations and
+temproraries and using halfs only for storage is the recommended way. On the
+one hand this somehow makes the provided mathematical functions obsolete
+(especially in light of the implicit conversion from half to float), but
+nevertheless the goal of this library was to provide a complete and
+conceptually clean half-precision implementation, to which the standard
+mathematical functions belong, even if usually not needed.
+
+IEEE CONFORMANCE
+
+The half type uses the standard IEEE representation with 1 sign bit, 5 exponent
+bits and 10 mantissa bits (11 when counting the hidden bit). It supports all
+types of special values, like subnormal values, infinity and NaNs. But there
+are some limitations to the complete conformance to the IEEE 754 standard:
+
+ - The implementation does not differentiate between signalling and quiet
+ NaNs, this means operations on halfs are not specified to trap on
+ signalling NaNs (though they may, see last point).
+
+ - Though arithmetic operations are internally rounded to single-precision
+ using the underlying single-precision implementation's current rounding
+ mode, those values are then converted to half-precision using the default
+ half-precision rounding mode (changed by defining 'HALF_ROUND_STYLE'
+ accordingly). This mixture of rounding modes is also the reason why
+ 'std::numeric_limits<half>::round_style' may actually return
+ 'std::round_indeterminate' when half- and single-precision rounding modes
+ don't match.
+
+ - Because of internal truncation it may also be that certain single-precision
+ NaNs will be wrongly converted to half-precision infinity, though this is
+ very unlikely to happen, since most single-precision implementations don't
+ tend to only set the lowest bits of a NaN mantissa.
+
+ - The implementation does not provide any floating point exceptions, thus
+ arithmetic operations or mathematical functions are not specified to invoke
+ proper floating point exceptions. But due to many functions implemented in
+ single-precision, those may still invoke floating point exceptions of the
+ underlying single-precision implementation.
+
+Some of those points could have been circumvented by controlling the floating
+point environment using <cfenv> or implementing a similar exception mechanism.
+But this would have required excessive runtime checks giving two high an impact
+on performance for something that is rarely ever needed. If you really need to
+rely on proper floating point exceptions, it is recommended to explicitly
+perform computations using the built-in floating point types to be on the safe
+side. In the same way, if you really need to rely on a particular rounding
+behaviour, it is recommended to either use single-precision computations and
+explicitly convert the result to half-precision using 'half_cast' and
+specifying the desired rounding mode, or synchronize the default half-precision
+rounding mode to the rounding mode of the single-precision implementation (most
+likely 'HALF_ROUND_STYLE=1', 'HALF_ROUND_TIES_TO_EVEN=1'). But this is really
+considered an expert-scenario that should be used only when necessary, since
+actually working with half-precision usually comes with a certain
+tolerance/ignorance of exactness considerations and proper rounding comes with
+a certain performance cost.
+
+
+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.