half.h

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///////////////////////////////////////////////////////////////////////////
//
// Copyright (c) 2002, Industrial Light & Magic, a division of Lucas
// Digital Ltd. LLC
// 
// All rights reserved.
// 
// Redistribution and use in source and binary forms, with or without
// modification, are permitted provided that the following conditions are
// met:
// *       Redistributions of source code must retain the above copyright
// notice, this list of conditions and the following disclaimer.
// *       Redistributions in binary form must reproduce the above
// copyright notice, this list of conditions and the following disclaimer
// in the documentation and/or other materials provided with the
// distribution.
// *       Neither the name of Industrial Light & Magic nor the names of
// its contributors may be used to endorse or promote products derived
// from this software without specific prior written permission. 
// 
// THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS
// "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT
// LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR
// A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT
// OWNER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,
// SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT
// LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE,
// DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY
// THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
// (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE
// OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
//
///////////////////////////////////////////////////////////////////////////

// Primary authors:
//     Florian Kainz <kainz@ilm.com>
//     Rod Bogart <rgb@ilm.com>

//---------------------------------------------------------------------------
//
//      half -- a 16-bit floating point number class:
//
//      Type half can represent positive and negative numbers whose
//      magnitude is between roughly 6.1e-5 and 6.5e+4 with a relative
//      error of 9.8e-4; numbers smaller than 6.1e-5 can be represented
//      with an absolute error of 6.0e-8.  All integers from -2048 to
//      +2048 can be represented exactly.
//
//      Type half behaves (almost) like the built-in C++ floating point
//      types.  In arithmetic expressions, half, float and double can be
//      mixed freely.  Here are a few examples:
//
//          half a (3.5);
//          float b (a + sqrt (a));
//          a += b;
//          b += a;
//          b = a + 7;
//
//      Conversions from half to float are lossless; all half numbers
//      are exactly representable as floats.
//
//      Conversions from float to half may not preserve a float's value
//      exactly.  If a float is not representable as a half, then the
//      float value is rounded to the nearest representable half.  If a
//      float value is exactly in the middle between the two closest
//      representable half values, then the float value is rounded to
//      the closest half whose least significant bit is zero.
//
//      Overflows during float-to-half conversions cause arithmetic
//      exceptions.  An overflow occurs when the float value to be
//      converted is too large to be represented as a half, or if the
//      float value is an infinity or a NAN.
//
//      The implementation of type half makes the following assumptions
//      about the implementation of the built-in C++ types:
//
//          float is an IEEE 754 single-precision number
//          sizeof (float) == 4
//          sizeof (unsigned int) == sizeof (float)
//          alignof (unsigned int) == alignof (float)
//          sizeof (unsigned short) == 2
//
//---------------------------------------------------------------------------

#ifndef _HALF_H_
#define _HALF_H_

#include "halfExport.h"    // for definition of HALF_EXPORT
#include <iostream>

class half
{
  public:

    //-------------
    // Constructors
    //-------------

    half () = default;                  // no initialization
    half (float f);
    // rule of 5
    ~half () = default;
    half (const half &) = default;
    half (half &&) noexcept = default;

    //--------------------
    // Conversion to float
    //--------------------

    operator            float () const;


    //------------
    // Unary minus
    //------------

    half                operator - () const;


    //-----------
    // Assignment
    //-----------

    half &              operator = (const half  &h) = default;
    half &              operator = (half  &&h) noexcept = default;
    half &              operator = (float f);

    half &              operator += (half  h);
    half &              operator += (float f);

    half &              operator -= (half  h);
    half &              operator -= (float f);

    half &              operator *= (half  h);
    half &              operator *= (float f);

    half &              operator /= (half  h);
    half &              operator /= (float f);


    //---------------------------------------------------------
    // Round to n-bit precision (n should be between 0 and 10).
    // After rounding, the significand's 10-n least significant
    // bits will be zero.
    //---------------------------------------------------------

    half                round (unsigned int n) const;


    //--------------------------------------------------------------------
    // Classification:
    //
    //  h.isFinite()            returns true if h is a normalized number,
    //                          a denormalized number or zero
    //
    //  h.isNormalized()        returns true if h is a normalized number
    //
    //  h.isDenormalized()      returns true if h is a denormalized number
    //
    //  h.isZero()              returns true if h is zero
    //
    //  h.isNan()               returns true if h is a NAN
    //
    //  h.isInfinity()          returns true if h is a positive
    //                          or a negative infinity
    //
    //  h.isNegative()          returns true if the sign bit of h
    //                          is set (negative)
    //--------------------------------------------------------------------

    bool                isFinite () const;
    bool                isNormalized () const;
    bool                isDenormalized () const;
    bool                isZero () const;
    bool                isNan () const;
    bool                isInfinity () const;
    bool                isNegative () const;


    //--------------------------------------------
    // Special values
    //
    //  posInf()        returns +infinity
    //
    //  negInf()        returns -infinity
    //
    //  qNan()          returns a NAN with the bit
    //                  pattern 0111111111111111
    //
    //  sNan()          returns a NAN with the bit
    //                  pattern 0111110111111111
    //--------------------------------------------

    static half         posInf ();
    static half         negInf ();
    static half         qNan ();
    static half         sNan ();


    //--------------------------------------
    // Access to the internal representation
    //--------------------------------------

    HALF_EXPORT unsigned short  bits () const;
    HALF_EXPORT void            setBits (unsigned short bits);


  public:

    union uif
    {
        unsigned int    i;
        float           f;
    };

  private:

    HALF_EXPORT static short                  convert (int i);
    HALF_EXPORT static float                  overflow ();

    unsigned short                            _h;

    HALF_EXPORT static const uif              _toFloat[1 << 16];
    HALF_EXPORT static const unsigned short   _eLut[1 << 9];
};



//-----------
// Stream I/O
//-----------

HALF_EXPORT std::ostream &      operator << (std::ostream &os, half  h);
HALF_EXPORT std::istream &      operator >> (std::istream &is, half &h);


//----------
// Debugging
//----------

HALF_EXPORT void        printBits   (std::ostream &os, half  h);
HALF_EXPORT void        printBits   (std::ostream &os, float f);
HALF_EXPORT void        printBits   (char  c[19], half  h);
HALF_EXPORT void        printBits   (char  c[35], float f);


//-------------------------------------------------------------------------
// Limits
//
// Visual C++ will complain if HALF_MIN, HALF_NRM_MIN etc. are not float
// constants, but at least one other compiler (gcc 2.96) produces incorrect
// results if they are.
//-------------------------------------------------------------------------

#if (defined _WIN32 || defined _WIN64) && defined _MSC_VER

  #define HALF_MIN      5.96046448e-08f // Smallest positive half

  #define HALF_NRM_MIN  6.10351562e-05f // Smallest positive normalized half

  #define HALF_MAX      65504.0f        // Largest positive half

  #define HALF_EPSILON  0.00097656f     // Smallest positive e for which
                                        // half (1.0 + e) != half (1.0)
#else

  #define HALF_MIN      5.96046448e-08  // Smallest positive half

  #define HALF_NRM_MIN  6.10351562e-05  // Smallest positive normalized half

  #define HALF_MAX      65504.0         // Largest positive half

  #define HALF_EPSILON  0.00097656      // Smallest positive e for which
                                        // half (1.0 + e) != half (1.0)
#endif


#define HALF_MANT_DIG   11              // Number of digits in mantissa
                                        // (significand + hidden leading 1)

// 
// floor( (HALF_MANT_DIG - 1) * log10(2) ) => 3.01... -> 3
#define HALF_DIG        3               // Number of base 10 digits that
                                        // can be represented without change

// ceil(HALF_MANT_DIG * log10(2) + 1) => 4.31... -> 5
#define HALF_DECIMAL_DIG        5       // Number of base-10 digits that are
                                        // necessary to uniquely represent all
                                        // distinct values

#define HALF_RADIX      2               // Base of the exponent

#define HALF_MIN_EXP    -13             // Minimum negative integer such that
                                        // HALF_RADIX raised to the power of
                                        // one less than that integer is a
                                        // normalized half

#define HALF_MAX_EXP    16              // Maximum positive integer such that
                                        // HALF_RADIX raised to the power of
                                        // one less than that integer is a
                                        // normalized half

#define HALF_MIN_10_EXP -4              // Minimum positive integer such
                                        // that 10 raised to that power is
                                        // a normalized half

#define HALF_MAX_10_EXP 4               // Maximum positive integer such
                                        // that 10 raised to that power is
                                        // a normalized half


//---------------------------------------------------------------------------
//
// Implementation --
//
// Representation of a float:
//
//      We assume that a float, f, is an IEEE 754 single-precision
//      floating point number, whose bits are arranged as follows:
//
//          31 (msb)
//          | 
//          | 30     23
//          | |      | 
//          | |      | 22                    0 (lsb)
//          | |      | |                     |
//          X XXXXXXXX XXXXXXXXXXXXXXXXXXXXXXX
//
//          s e        m
//
//      S is the sign-bit, e is the exponent and m is the significand.
//
//      If e is between 1 and 254, f is a normalized number:
//
//                  s    e-127
//          f = (-1)  * 2      * 1.m
//
//      If e is 0, and m is not zero, f is a denormalized number:
//
//                  s    -126
//          f = (-1)  * 2      * 0.m
//
//      If e and m are both zero, f is zero:
//
//          f = 0.0
//
//      If e is 255, f is an "infinity" or "not a number" (NAN),
//      depending on whether m is zero or not.
//
//      Examples:
//
//          0 00000000 00000000000000000000000 = 0.0
//          0 01111110 00000000000000000000000 = 0.5
//          0 01111111 00000000000000000000000 = 1.0
//          0 10000000 00000000000000000000000 = 2.0
//          0 10000000 10000000000000000000000 = 3.0
//          1 10000101 11110000010000000000000 = -124.0625
//          0 11111111 00000000000000000000000 = +infinity
//          1 11111111 00000000000000000000000 = -infinity
//          0 11111111 10000000000000000000000 = NAN
//          1 11111111 11111111111111111111111 = NAN
//
// Representation of a half:
//
//      Here is the bit-layout for a half number, h:
//
//          15 (msb)
//          | 
//          | 14  10
//          | |   |
//          | |   | 9        0 (lsb)
//          | |   | |        |
//          X XXXXX XXXXXXXXXX
//
//          s e     m
//
//      S is the sign-bit, e is the exponent and m is the significand.
//
//      If e is between 1 and 30, h is a normalized number:
//
//                  s    e-15
//          h = (-1)  * 2     * 1.m
//
//      If e is 0, and m is not zero, h is a denormalized number:
//
//                  S    -14
//          h = (-1)  * 2     * 0.m
//
//      If e and m are both zero, h is zero:
//
//          h = 0.0
//
//      If e is 31, h is an "infinity" or "not a number" (NAN),
//      depending on whether m is zero or not.
//
//      Examples:
//
//          0 00000 0000000000 = 0.0
//          0 01110 0000000000 = 0.5
//          0 01111 0000000000 = 1.0
//          0 10000 0000000000 = 2.0
//          0 10000 1000000000 = 3.0
//          1 10101 1111000001 = -124.0625
//          0 11111 0000000000 = +infinity
//          1 11111 0000000000 = -infinity
//          0 11111 1000000000 = NAN
//          1 11111 1111111111 = NAN
//
// Conversion:
//
//      Converting from a float to a half requires some non-trivial bit
//      manipulations.  In some cases, this makes conversion relatively
//      slow, but the most common case is accelerated via table lookups.
//
//      Converting back from a half to a float is easier because we don't
//      have to do any rounding.  In addition, there are only 65536
//      different half numbers; we can convert each of those numbers once
//      and store the results in a table.  Later, all conversions can be
//      done using only simple table lookups.
//
//---------------------------------------------------------------------------


//----------------------------
// Half-from-float constructor
//----------------------------

inline
half::half (float f)
{
    uif x;

    x.f = f;

    if (f == 0)
    {
        //
        // Common special case - zero.
        // Preserve the zero's sign bit.
        //

        _h = (x.i >> 16);
    }
    else
    {
        //
        // We extract the combined sign and exponent, e, from our
        // floating-point number, f.  Then we convert e to the sign
        // and exponent of the half number via a table lookup.
        //
        // For the most common case, where a normalized half is produced,
        // the table lookup returns a non-zero value; in this case, all
        // we have to do is round f's significand to 10 bits and combine
        // the result with e.
        //
        // For all other cases (overflow, zeroes, denormalized numbers
        // resulting from underflow, infinities and NANs), the table
        // lookup returns zero, and we call a longer, non-inline function
        // to do the float-to-half conversion.
        //

        int e = (x.i >> 23) & 0x000001ff;

        e = _eLut[e];

        if (e)
        {
            //
            // Simple case - round the significand, m, to 10
            // bits and combine it with the sign and exponent.
            //

            int m = x.i & 0x007fffff;
            _h = e + ((m + 0x00000fff + ((m >> 13) & 1)) >> 13);
        }
        else
        {
            //
            // Difficult case - call a function.
            //

            _h = convert (x.i);
        }
    }
}


//------------------------------------------
// Half-to-float conversion via table lookup
//------------------------------------------

inline
half::operator float () const
{
    return _toFloat[_h].f;
}


//-------------------------
// Round to n-bit precision
//-------------------------

inline half
half::round (unsigned int n) const
{
    //
    // Parameter check.
    //

    if (n >= 10)
        return *this;

    //
    // Disassemble h into the sign, s,
    // and the combined exponent and significand, e.
    //

    unsigned short s = _h & 0x8000;
    unsigned short e = _h & 0x7fff;

    //
    // Round the exponent and significand to the nearest value
    // where ones occur only in the (10-n) most significant bits.
    // Note that the exponent adjusts automatically if rounding
    // up causes the significand to overflow.
    //

    e >>= 9 - n;
    e  += e & 1;
    e <<= 9 - n;

    //
    // Check for exponent overflow.
    //

    if (e >= 0x7c00)
    {
        //
        // Overflow occurred -- truncate instead of rounding.
        //

        e = _h;
        e >>= 10 - n;
        e <<= 10 - n;
    }

    //
    // Put the original sign bit back.
    //

    half h;
    h._h = s | e;

    return h;
}


//-----------------------
// Other inline functions
//-----------------------

inline half     
half::operator - () const
{
    half h;
    h._h = _h ^ 0x8000;
    return h;
}


inline half &
half::operator = (float f)
{
    *this = half (f);
    return *this;
}


inline half &
half::operator += (half h)
{
    *this = half (float (*this) + float (h));
    return *this;
}


inline half &
half::operator += (float f)
{
    *this = half (float (*this) + f);
    return *this;
}


inline half &
half::operator -= (half h)
{
    *this = half (float (*this) - float (h));
    return *this;
}


inline half &
half::operator -= (float f)
{
    *this = half (float (*this) - f);
    return *this;
}


inline half &
half::operator *= (half h)
{
    *this = half (float (*this) * float (h));
    return *this;
}


inline half &
half::operator *= (float f)
{
    *this = half (float (*this) * f);
    return *this;
}


inline half &
half::operator /= (half h)
{
    *this = half (float (*this) / float (h));
    return *this;
}


inline half &
half::operator /= (float f)
{
    *this = half (float (*this) / f);
    return *this;
}


inline bool     
half::isFinite () const
{
    unsigned short e = (_h >> 10) & 0x001f;
    return e < 31;
}


inline bool
half::isNormalized () const
{
    unsigned short e = (_h >> 10) & 0x001f;
    return e > 0 && e < 31;
}


inline bool
half::isDenormalized () const
{
    unsigned short e = (_h >> 10) & 0x001f;
    unsigned short m =  _h & 0x3ff;
    return e == 0 && m != 0;
}


inline bool
half::isZero () const
{
    return (_h & 0x7fff) == 0;
}


inline bool
half::isNan () const
{
    unsigned short e = (_h >> 10) & 0x001f;
    unsigned short m =  _h & 0x3ff;
    return e == 31 && m != 0;
}


inline bool
half::isInfinity () const
{
    unsigned short e = (_h >> 10) & 0x001f;
    unsigned short m =  _h & 0x3ff;
    return e == 31 && m == 0;
}


inline bool     
half::isNegative () const
{
    return (_h & 0x8000) != 0;
}


inline half
half::posInf ()
{
    half h;
    h._h = 0x7c00;
    return h;
}


inline half
half::negInf ()
{
    half h;
    h._h = 0xfc00;
    return h;
}


inline half
half::qNan ()
{
    half h;
    h._h = 0x7fff;
    return h;
}


inline half
half::sNan ()
{
    half h;
    h._h = 0x7dff;
    return h;
}


inline unsigned short
half::bits () const
{
    return _h;
}


inline void
half::setBits (unsigned short bits)
{
    _h = bits;
}

#endif