src/ieee754.c

/*==============================================================================
 ieee754.c -- floating-point conversion between half, double & single-precision

 Copyright (c) 2018-2020, Laurence Lundblade. All rights reserved.

 SPDX-License-Identifier: BSD-3-Clause

 See BSD-3-Clause license in README.md

 Created on 7/23/18
 =============================================================================*/

#include "ieee754.h"
#include <string.h> // For memcpy()


/*
 This code is written for clarity and verifiability, not for size, on
 the assumption that the optimizer will do a good job. The LLVM
 optimizer, -Os, does seem to do the job and the resulting object code
 is smaller from combining code for the many different cases (normal,
 subnormal, infinity, zero...) for the conversions. GCC is no where near
 as good.

 This code has really long lines and is much easier to read because of
 them. Some coding guidelines prefer 80 column lines (can they not afford
 big displays?). It would make this code much worse even to wrap at 120
 columns.

 Dead stripping is also really helpful to get code size down when
 floating-point encoding is not needed. (If this is put in a library
 and linking is against the library, then dead stripping is automatic).

 This code works solely using shifts and masks and thus has no
 dependency on any math libraries. It can even work if the CPU doesn't
 have any floating-point support, though that isn't the most useful
 thing to do.

 The memcpy() dependency is only for CopyFloatToUint32() and friends
 which only is needed to avoid type punning when converting the actual
 float bits to an unsigned value so the bit shifts and masks can work.
 */

/*
 The references used to write this code:

 - IEEE 754-2008, particularly section 3.6 and 6.2.1

 - https://en.wikipedia.org/wiki/IEEE_754 and subordinate pages

 - https://stackoverflow.com/questions/19800415/why-does-ieee-754-reserve-so-many-nan-values

 - https://stackoverflow.com/questions/46073295/implicit-type-promotion-rules

 - https://stackoverflow.com/questions/589575/what-does-the-c-standard-state-the-size-of-int-long-type-to-be
 */


// ----- Half Precsion -----------
#define HALF_NUM_SIGNIFICAND_BITS (10)
#define HALF_NUM_EXPONENT_BITS    (5)
#define HALF_NUM_SIGN_BITS        (1)

#define HALF_SIGNIFICAND_SHIFT    (0)
#define HALF_EXPONENT_SHIFT       (HALF_NUM_SIGNIFICAND_BITS)
#define HALF_SIGN_SHIFT           (HALF_NUM_SIGNIFICAND_BITS + HALF_NUM_EXPONENT_BITS)

#define HALF_SIGNIFICAND_MASK     (0x3ff) // The lower 10 bits  // 0x03ff
#define HALF_EXPONENT_MASK        (0x1f << HALF_EXPONENT_SHIFT) // 0x7c00 5 bits of exponent
#define HALF_SIGN_MASK            (0x01 << HALF_SIGN_SHIFT) //  // 0x80001 bit of sign
#define HALF_QUIET_NAN_BIT        (0x01 << (HALF_NUM_SIGNIFICAND_BITS-1)) // 0x0200

/* Biased    Biased    Unbiased   Use
    0x00       0        -15       0 and subnormal
    0x01       1        -14       Smallest normal exponent
    0x1e      30         15       Largest normal exponent
    0x1F      31         16       NaN and Infinity  */
#define HALF_EXPONENT_BIAS        (15)
#define HALF_EXPONENT_MAX         (HALF_EXPONENT_BIAS)    //  15 Unbiased
#define HALF_EXPONENT_MIN         (-HALF_EXPONENT_BIAS+1) // -14 Unbiased
#define HALF_EXPONENT_ZERO        (-HALF_EXPONENT_BIAS)   // -15 Unbiased
#define HALF_EXPONENT_INF_OR_NAN  (HALF_EXPONENT_BIAS+1)  //  16 Unbiased


// ------ Single-Precision --------
#define SINGLE_NUM_SIGNIFICAND_BITS (23)
#define SINGLE_NUM_EXPONENT_BITS    (8)
#define SINGLE_NUM_SIGN_BITS        (1)

#define SINGLE_SIGNIFICAND_SHIFT    (0)
#define SINGLE_EXPONENT_SHIFT       (SINGLE_NUM_SIGNIFICAND_BITS)
#define SINGLE_SIGN_SHIFT           (SINGLE_NUM_SIGNIFICAND_BITS + SINGLE_NUM_EXPONENT_BITS)

#define SINGLE_SIGNIFICAND_MASK     (0x7fffffUL) // The lower 23 bits
#define SINGLE_EXPONENT_MASK        (0xffUL << SINGLE_EXPONENT_SHIFT) // 8 bits of exponent
#define SINGLE_SIGN_MASK            (0x01UL << SINGLE_SIGN_SHIFT) // 1 bit of sign
#define SINGLE_QUIET_NAN_BIT        (0x01UL << (SINGLE_NUM_SIGNIFICAND_BITS-1))

/* Biased  Biased   Unbiased  Use
    0x0000     0     -127      0 and subnormal
    0x0001     1     -126      Smallest normal exponent
    0x7f     127        0      1
    0xfe     254      127      Largest normal exponent
    0xff     255      128      NaN and Infinity  */
#define SINGLE_EXPONENT_BIAS        (127)
#define SINGLE_EXPONENT_MAX         (SINGLE_EXPONENT_BIAS)    //  127 unbiased
#define SINGLE_EXPONENT_MIN         (-SINGLE_EXPONENT_BIAS+1) // -126 unbiased
#define SINGLE_EXPONENT_ZERO        (-SINGLE_EXPONENT_BIAS)   // -127 unbiased
#define SINGLE_EXPONENT_INF_OR_NAN  (SINGLE_EXPONENT_BIAS+1)  //  128 unbiased


// --------- Double-Precision ----------
#define DOUBLE_NUM_SIGNIFICAND_BITS (52)
#define DOUBLE_NUM_EXPONENT_BITS    (11)
#define DOUBLE_NUM_SIGN_BITS        (1)

#define DOUBLE_SIGNIFICAND_SHIFT    (0)
#define DOUBLE_EXPONENT_SHIFT       (DOUBLE_NUM_SIGNIFICAND_BITS)
#define DOUBLE_SIGN_SHIFT           (DOUBLE_NUM_SIGNIFICAND_BITS + DOUBLE_NUM_EXPONENT_BITS)

#define DOUBLE_SIGNIFICAND_MASK     (0xfffffffffffffULL) // The lower 52 bits
#define DOUBLE_EXPONENT_MASK        (0x7ffULL << DOUBLE_EXPONENT_SHIFT) // 11 bits of exponent
#define DOUBLE_SIGN_MASK            (0x01ULL << DOUBLE_SIGN_SHIFT) // 1 bit of sign
#define DOUBLE_QUIET_NAN_BIT        (0x01ULL << (DOUBLE_NUM_SIGNIFICAND_BITS-1))


/* Biased      Biased   Unbiased  Use
   0x00000000     0     -1023     0 and subnormal
   0x00000001     1     -1022     Smallest normal exponent
   0x000007fe  2046      1023     Largest normal exponent
   0x000007ff  2047      1024     NaN and Infinity  */
#define DOUBLE_EXPONENT_BIAS        (1023)
#define DOUBLE_EXPONENT_MAX         (DOUBLE_EXPONENT_BIAS)    // unbiased
#define DOUBLE_EXPONENT_MIN         (-DOUBLE_EXPONENT_BIAS+1) // unbiased
#define DOUBLE_EXPONENT_ZERO        (-DOUBLE_EXPONENT_BIAS)   // unbiased
#define DOUBLE_EXPONENT_INF_OR_NAN  (DOUBLE_EXPONENT_BIAS+1)  // unbiased



/*
 Convenient functions to avoid type punning, compiler warnings and
 such. The optimizer reduces them to a simple assignment.  This is a
 crusty corner of C. It shouldn't be this hard.

 These are also in UsefulBuf.h under a different name. They are copied
 here to avoid a dependency on UsefulBuf.h. There is no object code
 size impact because these always optimze down to a simple assignment.
 */
static inline uint32_t CopyFloatToUint32(float f)
{
    uint32_t u32;
    memcpy(&u32, &f, sizeof(uint32_t));
    return u32;
}

static inline uint64_t CopyDoubleToUint64(double d)
{
    uint64_t u64;
    memcpy(&u64, &d, sizeof(uint64_t));
    return u64;
}

static inline float CopyUint32ToFloat(uint32_t u32)
{
    float f;
    memcpy(&f, &u32, sizeof(uint32_t));
    return f;
}

static inline double CopyUint64ToDouble(uint64_t u64)
{
    double d;
    memcpy(&d, &u64, sizeof(uint64_t));
    return d;
}


// Public function; see ieee754.h
uint16_t IEEE754_FloatToHalf(float f)
{
    // Pull the three parts out of the single-precision float
    const uint32_t uSingle = CopyFloatToUint32(f);
    const int32_t  nSingleUnbiasedExponent = (int32_t)((uSingle & SINGLE_EXPONENT_MASK) >> SINGLE_EXPONENT_SHIFT) - SINGLE_EXPONENT_BIAS;
    const uint32_t uSingleSign             = (uSingle & SINGLE_SIGN_MASK) >> SINGLE_SIGN_SHIFT;
    const uint32_t uSingleSignificand      = uSingle & SINGLE_SIGNIFICAND_MASK;


    // Now convert the three parts to half-precision.

    // All works is done on uint32_t with conversion to uint16_t at the end.
    // This avoids integer promotions that static analyzers complain about and
    // reduces code size.
    uint32_t uHalfSign, uHalfSignificand, uHalfBiasedExponent;

    if(nSingleUnbiasedExponent == SINGLE_EXPONENT_INF_OR_NAN) {
        // +/- Infinity and NaNs -- single biased exponent is 0xff
        uHalfBiasedExponent = HALF_EXPONENT_INF_OR_NAN + HALF_EXPONENT_BIAS;
        if(!uSingleSignificand) {
            // Infinity
            uHalfSignificand = 0;
        } else {
            // Copy the LSBs of the NaN payload that will fit from the single to the half
            uHalfSignificand = uSingleSignificand & (HALF_SIGNIFICAND_MASK & ~HALF_QUIET_NAN_BIT);
            if(uSingleSignificand & SINGLE_QUIET_NAN_BIT) {
                // It's a qNaN; copy the qNaN bit
                uHalfSignificand |= HALF_QUIET_NAN_BIT;
            } else {
                // It's an sNaN; make sure the significand is not zero so it stays a NaN
                // This is needed because not all significand bits are copied from single
                if(!uHalfSignificand) {
                    // Set the LSB. This is what wikipedia shows for sNAN.
                    uHalfSignificand |= 0x01;
                }
            }
        }
    } else if(nSingleUnbiasedExponent == SINGLE_EXPONENT_ZERO) {
        // 0 or a subnormal number  -- singled biased exponent is 0
        uHalfBiasedExponent = 0;
        uHalfSignificand    = 0; // Any subnormal single will be too small to express as a half precision
    } else if(nSingleUnbiasedExponent > HALF_EXPONENT_MAX) {
        // Exponent is too large to express in half-precision; round up to infinity
        uHalfBiasedExponent = HALF_EXPONENT_INF_OR_NAN + HALF_EXPONENT_BIAS;
        uHalfSignificand    = 0;
    } else if(nSingleUnbiasedExponent < HALF_EXPONENT_MIN) {
        // Exponent is too small to express in half-precision normal; make it a half-precision subnormal
        uHalfBiasedExponent = HALF_EXPONENT_ZERO + HALF_EXPONENT_BIAS;
        // Difference between single normal exponent and the base exponent of a half subnormal
        const uint32_t uExpDiff = (uint32_t)-(nSingleUnbiasedExponent - HALF_EXPONENT_MIN);
        // Also have to shift the significand by the difference in number of bits between a single and a half significand
        const uint32_t uSignificandBitsDiff = SINGLE_NUM_SIGNIFICAND_BITS - HALF_NUM_SIGNIFICAND_BITS;
        // Add in the 1 that is implied in the significand of a normal number; it needs to be present in a subnormal
        const uint32_t uSingleSignificandSubnormal = uSingleSignificand + (0x01UL << SINGLE_NUM_SIGNIFICAND_BITS);
        uHalfSignificand = uSingleSignificandSubnormal >> (uExpDiff + uSignificandBitsDiff);
    } else {
        // The normal case, exponent is in range for half-precision
        uHalfBiasedExponent = (uint32_t)(nSingleUnbiasedExponent + HALF_EXPONENT_BIAS);
        uHalfSignificand    = uSingleSignificand >> (SINGLE_NUM_SIGNIFICAND_BITS - HALF_NUM_SIGNIFICAND_BITS);
    }
    uHalfSign = uSingleSign;

    // Put the 3 values in the right place for a half precision
    const uint32_t uHalfPrecision =  uHalfSignificand |
                                    (uHalfBiasedExponent << HALF_EXPONENT_SHIFT) |
                                    (uHalfSign << HALF_SIGN_SHIFT);
    // Cast is safe because all the masks and shifts above work to make
    // a half precision value which is only 16 bits.
    return (uint16_t)uHalfPrecision;
}


// Public function; see ieee754.h
uint16_t IEEE754_DoubleToHalf(double d)
{
    // Pull the three parts out of the double-precision float
    const uint64_t uDouble = CopyDoubleToUint64(d);
    const int64_t  nDoubleUnbiasedExponent = (int64_t)((uDouble & DOUBLE_EXPONENT_MASK) >> DOUBLE_EXPONENT_SHIFT) - DOUBLE_EXPONENT_BIAS;
    const uint64_t uDoubleSign             = (uDouble & DOUBLE_SIGN_MASK) >> DOUBLE_SIGN_SHIFT;
    const uint64_t uDoubleSignificand      = uDouble & DOUBLE_SIGNIFICAND_MASK;

    // Now convert the three parts to half-precision.

    // All works is done on uint64_t with conversion to uint16_t at the end.
    // This avoids integer promotions that static analyzers complain about.
    // Other options are for these to be unsigned int or fast_int16_t. Code
    // size doesn't vary much between all these options for 64-bit LLVM,
    // 64-bit GCC and 32-bit Armv7 LLVM.
    uint64_t uHalfSign, uHalfSignificand, uHalfBiasedExponent;

    if(nDoubleUnbiasedExponent == DOUBLE_EXPONENT_INF_OR_NAN) {
        // +/- Infinity and NaNs -- single biased exponent is 0xff
        uHalfBiasedExponent = HALF_EXPONENT_INF_OR_NAN + HALF_EXPONENT_BIAS;
        if(!uDoubleSignificand) {
            // Infinity
            uHalfSignificand = 0;
        } else {
            // Copy the LSBs of the NaN payload that will fit from the double to the half
            uHalfSignificand = uDoubleSignificand & (HALF_SIGNIFICAND_MASK & ~HALF_QUIET_NAN_BIT);
            if(uDoubleSignificand & DOUBLE_QUIET_NAN_BIT) {
                // It's a qNaN; copy the qNaN bit
                uHalfSignificand |= HALF_QUIET_NAN_BIT;
            } else {
                // It's an sNaN; make sure the significand is not zero so it stays a NaN
                // This is needed because not all significand bits are copied from single
                if(!uHalfSignificand) {
                    // Set the LSB. This is what wikipedia shows for sNAN.
                    uHalfSignificand |= 0x01;
                }
            }
        }
    } else if(nDoubleUnbiasedExponent == DOUBLE_EXPONENT_ZERO) {
        // 0 or a subnormal number  -- double biased exponent is 0
        uHalfBiasedExponent = 0;
        uHalfSignificand    = 0; // Any subnormal single will be too small to express as a half precision; TODO, is this really true?
    } else if(nDoubleUnbiasedExponent > HALF_EXPONENT_MAX) {
        // Exponent is too large to express in half-precision; round up to infinity; TODO, is this really true?
        uHalfBiasedExponent = HALF_EXPONENT_INF_OR_NAN + HALF_EXPONENT_BIAS;
        uHalfSignificand    = 0;
    } else if(nDoubleUnbiasedExponent < HALF_EXPONENT_MIN) {
        // Exponent is too small to express in half-precision; round down to zero
        uHalfBiasedExponent = HALF_EXPONENT_ZERO + HALF_EXPONENT_BIAS;
        // Difference between double normal exponent and the base exponent of a half subnormal
        const uint64_t uExpDiff = (uint64_t)-(nDoubleUnbiasedExponent - HALF_EXPONENT_MIN);
        // Also have to shift the significand by the difference in number of bits between a double and a half significand
        const uint64_t uSignificandBitsDiff = DOUBLE_NUM_SIGNIFICAND_BITS - HALF_NUM_SIGNIFICAND_BITS;
        // Add in the 1 that is implied in the significand of a normal number; it needs to be present in a subnormal
        const uint64_t uDoubleSignificandSubnormal = uDoubleSignificand + (0x01ULL << DOUBLE_NUM_SIGNIFICAND_BITS);
        uHalfSignificand = uDoubleSignificandSubnormal >> (uExpDiff + uSignificandBitsDiff);
    } else {
        // The normal case, exponent is in range for half-precision
        uHalfBiasedExponent = (uint32_t)(nDoubleUnbiasedExponent + HALF_EXPONENT_BIAS);
        uHalfSignificand    = uDoubleSignificand >> (DOUBLE_NUM_SIGNIFICAND_BITS - HALF_NUM_SIGNIFICAND_BITS);
    }
    uHalfSign = uDoubleSign;


    // Put the 3 values in the right place for a half precision
    const uint64_t uHalfPrecision =  uHalfSignificand |
                                    (uHalfBiasedExponent << HALF_EXPONENT_SHIFT) |
                                    (uHalfSign << HALF_SIGN_SHIFT);
    // Cast is safe because all the masks and shifts above work to make
    // a half precision value which is only 16 bits.
    return (uint16_t)uHalfPrecision;
}



// Public function; see ieee754.h
float IEEE754_HalfToFloat(uint16_t uHalfPrecision)
{
    // Pull out the three parts of the half-precision float
    // Do all the work in 32 bits because that is what the end result is
    // may give smaller code size and will keep static analyzers happier.
    const uint32_t uHalfSignificand      = uHalfPrecision & HALF_SIGNIFICAND_MASK;
    const int32_t  nHalfUnBiasedExponent = (int32_t)((uHalfPrecision & HALF_EXPONENT_MASK) >> HALF_EXPONENT_SHIFT) - HALF_EXPONENT_BIAS;
    const uint32_t uHalfSign             = (uHalfPrecision & HALF_SIGN_MASK) >> HALF_SIGN_SHIFT;


    // Make the three parts of the single-precision number
    uint32_t uSingleSignificand, uSingleSign, uSingleBiasedExponent;
    if(nHalfUnBiasedExponent == HALF_EXPONENT_ZERO) {
        // 0 or subnormal
        if(uHalfSignificand) {
            // Subnormal case
            uSingleBiasedExponent = -HALF_EXPONENT_BIAS + SINGLE_EXPONENT_BIAS +1;
            // A half-precision subnormal can always be converted to a normal single-precision float because the ranges line up
            uSingleSignificand = uHalfSignificand;
            // Shift bits from right of the decimal to left, reducing the exponent by 1 each time
            do {
                uSingleSignificand <<= 1;
                uSingleBiasedExponent--;
            } while ((uSingleSignificand & 0x400) == 0);
            uSingleSignificand &= HALF_SIGNIFICAND_MASK;
            uSingleSignificand <<= (SINGLE_NUM_SIGNIFICAND_BITS - HALF_NUM_SIGNIFICAND_BITS);
        } else {
            // Just zero
            uSingleBiasedExponent = SINGLE_EXPONENT_ZERO + SINGLE_EXPONENT_BIAS;
            uSingleSignificand = 0;
        }
    } else if(nHalfUnBiasedExponent == HALF_EXPONENT_INF_OR_NAN) {
        // NaN or Inifinity
        uSingleBiasedExponent = SINGLE_EXPONENT_INF_OR_NAN + SINGLE_EXPONENT_BIAS;
        if(uHalfSignificand) {
            // NaN
            // First preserve the NaN payload from half to single
            uSingleSignificand = uHalfSignificand & ~HALF_QUIET_NAN_BIT;
            if(uHalfSignificand & HALF_QUIET_NAN_BIT) {
                // Next, set qNaN if needed since half qNaN bit is not copied above
                uSingleSignificand |= SINGLE_QUIET_NAN_BIT;
            }
        } else {
            // Infinity
            uSingleSignificand = 0;
        }
    } else {
        // Normal number
        uSingleBiasedExponent = (uint32_t)(nHalfUnBiasedExponent + SINGLE_EXPONENT_BIAS);
        uSingleSignificand = uHalfSignificand << (SINGLE_NUM_SIGNIFICAND_BITS - HALF_NUM_SIGNIFICAND_BITS);
    }
    uSingleSign = uHalfSign;

    // Shift the three parts of the single-precision into place
    const uint32_t uSinglePrecision = uSingleSignificand |
                                     (uSingleBiasedExponent << SINGLE_EXPONENT_SHIFT) |
                                     (uSingleSign << SINGLE_SIGN_SHIFT);

    return CopyUint32ToFloat(uSinglePrecision);
}


// Public function; see ieee754.h
double IEEE754_HalfToDouble(uint16_t uHalfPrecision)
{
    // Pull out the three parts of the half-precision float
    // Do all the work in 64 bits because that is what the end result is
    // may give smaller code size and will keep static analyzers happier.
    const uint64_t uHalfSignificand      = uHalfPrecision & HALF_SIGNIFICAND_MASK;
    const int64_t  nHalfUnBiasedExponent = (int64_t)((uHalfPrecision & HALF_EXPONENT_MASK) >> HALF_EXPONENT_SHIFT) - HALF_EXPONENT_BIAS;
    const uint64_t uHalfSign             = (uHalfPrecision & HALF_SIGN_MASK) >> HALF_SIGN_SHIFT;


    // Make the three parts of hte single-precision number
    uint64_t uDoubleSignificand, uDoubleSign, uDoubleBiasedExponent;
    if(nHalfUnBiasedExponent == HALF_EXPONENT_ZERO) {
        // 0 or subnormal
        uDoubleBiasedExponent = DOUBLE_EXPONENT_ZERO + DOUBLE_EXPONENT_BIAS;
        if(uHalfSignificand) {
            // Subnormal case
            uDoubleBiasedExponent = -HALF_EXPONENT_BIAS + DOUBLE_EXPONENT_BIAS +1;
            // A half-precision subnormal can always be converted to a normal double-precision float because the ranges line up
            uDoubleSignificand = uHalfSignificand;
            // Shift bits from right of the decimal to left, reducing the exponent by 1 each time
            do {
                uDoubleSignificand <<= 1;
                uDoubleBiasedExponent--;
            } while ((uDoubleSignificand & 0x400) == 0);
            uDoubleSignificand &= HALF_SIGNIFICAND_MASK;
            uDoubleSignificand <<= (DOUBLE_NUM_SIGNIFICAND_BITS - HALF_NUM_SIGNIFICAND_BITS);
        } else {
            // Just zero
            uDoubleSignificand = 0;
        }
    } else if(nHalfUnBiasedExponent == HALF_EXPONENT_INF_OR_NAN) {
        // NaN or Inifinity
        uDoubleBiasedExponent = DOUBLE_EXPONENT_INF_OR_NAN + DOUBLE_EXPONENT_BIAS;
        if(uHalfSignificand) {
            // NaN
            // First preserve the NaN payload from half to single
            uDoubleSignificand = uHalfSignificand & ~HALF_QUIET_NAN_BIT;
            if(uHalfSignificand & HALF_QUIET_NAN_BIT) {
                // Next, set qNaN if needed since half qNaN bit is not copied above
                uDoubleSignificand |= DOUBLE_QUIET_NAN_BIT;
            }
        } else {
            // Infinity
            uDoubleSignificand = 0;
        }
    } else {
        // Normal number
        uDoubleBiasedExponent = (uint64_t)(nHalfUnBiasedExponent + DOUBLE_EXPONENT_BIAS);
        uDoubleSignificand    = uHalfSignificand << (DOUBLE_NUM_SIGNIFICAND_BITS - HALF_NUM_SIGNIFICAND_BITS);
    }
    uDoubleSign = uHalfSign;


    // Shift the 3 parts into place as a double-precision
    const uint64_t uDouble = uDoubleSignificand |
                            (uDoubleBiasedExponent << DOUBLE_EXPONENT_SHIFT) |
                            (uDoubleSign << DOUBLE_SIGN_SHIFT);
    return CopyUint64ToDouble(uDouble);
}


// Public function; see ieee754.h
IEEE754_union IEEE754_FloatToSmallest(float f)
{
    IEEE754_union result;

    // Pull the neeed two parts out of the single-precision float
    const uint32_t uSingle = CopyFloatToUint32(f);
    const int32_t  nSingleExponent    = (int32_t)((uSingle & SINGLE_EXPONENT_MASK) >> SINGLE_EXPONENT_SHIFT) - SINGLE_EXPONENT_BIAS;
    const uint32_t uSingleSignificand =   uSingle & SINGLE_SIGNIFICAND_MASK;

    // Bit mask that is the significand bits that would be lost when converting
    // from single-precision to half-precision
    const uint64_t uDroppedSingleBits = SINGLE_SIGNIFICAND_MASK >> HALF_NUM_SIGNIFICAND_BITS;

    // Optimizer will re organize so there is only one call to IEEE754_FloatToHalf()
    if(uSingle == 0) {
        // Value is 0.0000, not a a subnormal
        result.uSize = IEEE754_UNION_IS_HALF;
        result.uValue  = IEEE754_FloatToHalf(f);
    } else if(nSingleExponent == SINGLE_EXPONENT_INF_OR_NAN) {
        // NaN, +/- infinity
        result.uSize = IEEE754_UNION_IS_HALF;
        result.uValue  = IEEE754_FloatToHalf(f);
    } else if((nSingleExponent >= HALF_EXPONENT_MIN) && nSingleExponent <= HALF_EXPONENT_MAX && (!(uSingleSignificand & uDroppedSingleBits))) {
        // Normal number in exponent range and precision won't be lost
        result.uSize = IEEE754_UNION_IS_HALF;
        result.uValue  = IEEE754_FloatToHalf(f);
    } else {
        // Subnormal, exponent out of range, or precision will be lost
        result.uSize = IEEE754_UNION_IS_SINGLE;
        result.uValue  = uSingle;
    }

    return result;
}

// Public function; see ieee754.h
IEEE754_union IEEE754_DoubleToSmallestInternal(double d, int bAllowHalfPrecision)
{
    IEEE754_union result;

    // Pull the needed two parts out of the double-precision float
    const uint64_t uDouble = CopyDoubleToUint64(d);
    const int64_t  nDoubleExponent     = (int64_t)((uDouble & DOUBLE_EXPONENT_MASK) >> DOUBLE_EXPONENT_SHIFT) - DOUBLE_EXPONENT_BIAS;
    const uint64_t uDoubleSignificand  =   uDouble & DOUBLE_SIGNIFICAND_MASK;

    // Masks to check whether dropped significand bits are zero or not
    const uint64_t uDroppedDoubleBits = DOUBLE_SIGNIFICAND_MASK >> HALF_NUM_SIGNIFICAND_BITS;
    const uint64_t uDroppedSingleBits = DOUBLE_SIGNIFICAND_MASK >> SINGLE_NUM_SIGNIFICAND_BITS;

    // The various cases
    if(d == 0.0) { // Take care of positive and negative zero
        // Value is 0.0000, not a a subnormal
        result.uSize  = IEEE754_UNION_IS_HALF;
        result.uValue = IEEE754_DoubleToHalf(d);
    } else if(nDoubleExponent == DOUBLE_EXPONENT_INF_OR_NAN) {
        // NaN, +/- infinity
        result.uSize  = IEEE754_UNION_IS_HALF;
        result.uValue = IEEE754_DoubleToHalf(d);
    } else if(bAllowHalfPrecision && (nDoubleExponent >= HALF_EXPONENT_MIN) && nDoubleExponent <= HALF_EXPONENT_MAX && (!(uDoubleSignificand & uDroppedDoubleBits))) {
        // Can convert to half without precision loss
        result.uSize  = IEEE754_UNION_IS_HALF;
        result.uValue = IEEE754_DoubleToHalf(d);
    } else if((nDoubleExponent >= SINGLE_EXPONENT_MIN) && nDoubleExponent <= SINGLE_EXPONENT_MAX && (!(uDoubleSignificand & uDroppedSingleBits))) {
        // Can convert to single without precision loss
        result.uSize  = IEEE754_UNION_IS_SINGLE;
        result.uValue = CopyFloatToUint32((float)d);
    } else {
        // Can't convert without precision loss
        result.uSize  = IEEE754_UNION_IS_DOUBLE;
        result.uValue = uDouble;
    }

    return result;
}