/*-------------------------------------------------------------------------
* drawElements Base Portability Library
* -------------------------------------
*
* Copyright 2014 The Android Open Source Project
*
* Licensed under the Apache License, Version 2.0 (the "License");
* you may not use this file except in compliance with the License.
* You may obtain a copy of the License at
*
* http://www.apache.org/licenses/LICENSE-2.0
*
* Unless required by applicable law or agreed to in writing, software
* distributed under the License is distributed on an "AS IS" BASIS,
* WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
* See the License for the specific language governing permissions and
* limitations under the License.
*
*//*!
* \file
* \brief 16-bit floating-point math.
*//*--------------------------------------------------------------------*/
#include "deFloat16.h"
DE_BEGIN_EXTERN_C
deFloat16 deFloat32To16 (float val32)
{
deUint32 sign;
int expotent;
deUint32 mantissa;
union
{
float f;
deUint32 u;
} x;
x.f = val32;
sign = (x.u >> 16u) & 0x00008000u;
expotent = (int)((x.u >> 23u) & 0x000000ffu) - (127 - 15);
mantissa = x.u & 0x007fffffu;
if (expotent <= 0)
{
if (expotent < -10)
{
/* Rounds to zero. */
return (deFloat16) sign;
}
/* Converted to denormalized half, add leading 1 to significand. */
mantissa = mantissa | 0x00800000u;
/* Round mantissa to nearest (10+e) */
{
deUint32 t = 14u - expotent;
deUint32 a = (1u << (t - 1u)) - 1u;
deUint32 b = (mantissa >> t) & 1u;
mantissa = (mantissa + a + b) >> t;
}
return (deFloat16) (sign | mantissa);
}
else if (expotent == 0xff - (127 - 15))
{
if (mantissa == 0u)
{
/* InF */
return (deFloat16) (sign | 0x7c00u);
}
else
{
/* NaN */
mantissa >>= 13u;
return (deFloat16) (sign | 0x7c00u | mantissa | (mantissa == 0u));
}
}
else
{
/* Normalized float. */
mantissa = mantissa + 0x00000fffu + ((mantissa >> 13u) & 1u);
if (mantissa & 0x00800000u)
{
/* Overflow in mantissa. */
mantissa = 0u;
expotent += 1;
}
if (expotent > 30)
{
/* \todo [pyry] Cause hw fp overflow */
return (deFloat16) (sign | 0x7c00u);
}
return (deFloat16) (sign | ((deUint32)expotent << 10u) | (mantissa >> 13u));
}
}
deFloat16 deFloat64To16 (double val64)
{
deUint64 sign;
long expotent;
deUint64 mantissa;
union
{
double f;
deUint64 u;
} x;
x.f = val64;
sign = (x.u >> 48u) & 0x00008000u;
expotent = (long int)((x.u >> 52u) & 0x000007ffu) - (1023 - 15);
mantissa = x.u & 0x00fffffffffffffu;
if (expotent <= 0)
{
if (expotent < -10)
{
/* Rounds to zero. */
return (deFloat16) sign;
}
/* Converted to denormalized half, add leading 1 to significand. */
mantissa = mantissa | 0x0010000000000000u;
/* Round mantissa to nearest (10+e) */
{
deUint64 t = 43u - expotent;
deUint64 a = (1u << (t - 1u)) - 1u;
deUint64 b = (mantissa >> t) & 1u;
mantissa = (mantissa + a + b) >> t;
}
return (deFloat16) (sign | mantissa);
}
else if (expotent == 0x7ff - (1023 - 15))
{
if (mantissa == 0u)
{
/* InF */
return (deFloat16) (sign | 0x7c00u);
}
else
{
/* NaN */
mantissa >>= 42u;
return (deFloat16) (sign | 0x7c00u | mantissa | (mantissa == 0u));
}
}
else
{
/* Normalized float. */
mantissa = mantissa + 0x000001ffffffffffu + ((mantissa >> 42u) & 1u);
if (mantissa & 0x010000000000000u)
{
/* Overflow in mantissa. */
mantissa = 0u;
expotent += 1;
}
if (expotent > 30)
{
return (deFloat16) (sign | 0x7c00u);
}
return (deFloat16) (sign | ((deUint32)expotent << 10u) | (mantissa >> 13u));
}
}
/*--------------------------------------------------------------------*//*!
* \brief Round the given number `val` to nearest even by discarding
* the last `numBitsToDiscard` bits.
* \param val value to round
* \param numBitsToDiscard number of (least significant) bits to discard
* \return The rounded value with the last `numBitsToDiscard` removed
*//*--------------------------------------------------------------------*/
static deUint32 roundToNearestEven (deUint32 val, const deUint32 numBitsToDiscard)
{
const deUint32 lastBits = val & ((1 << numBitsToDiscard) - 1);
const deUint32 headBit = val & (1 << (numBitsToDiscard - 1));
DE_ASSERT(numBitsToDiscard > 0 && numBitsToDiscard < 32); /* Make sure no overflow. */
val >>= numBitsToDiscard;
if (headBit == 0)
{
return val;
}
else if (headBit == lastBits)
{
if ((val & 0x1) == 0x1)
{
return val + 1;
}
else
{
return val;
}
}
else
{
return val + 1;
}
}
deFloat16 deFloat32To16Round (float val32, deRoundingMode mode)
{
union
{
float f; /* Interpret as 32-bit float */
deUint32 u; /* Interpret as 32-bit unsigned integer */
} x;
deUint32 sign; /* sign : 0000 0000 0000 0000 X000 0000 0000 0000 */
deUint32 exp32; /* exp32: biased exponent for 32-bit floats */
int exp16; /* exp16: biased exponent for 16-bit floats */
deUint32 mantissa;
/* We only support these two rounding modes for now */
DE_ASSERT(mode == DE_ROUNDINGMODE_TO_ZERO || mode == DE_ROUNDINGMODE_TO_NEAREST_EVEN);
x.f = val32;
sign = (x.u >> 16u) & 0x00008000u;
exp32 = (x.u >> 23u) & 0x000000ffu;
exp16 = (int) (exp32) - 127 + 15; /* 15/127: exponent bias for 16-bit/32-bit floats */
mantissa = x.u & 0x007fffffu;
/* Case: zero and denormalized floats */
if (exp32 == 0)
{
/* Denormalized floats are < 2^(1-127), not representable in 16-bit floats, rounding to zero. */
return (deFloat16) sign;
}
/* Case: Inf and NaN */
else if (exp32 == 0x000000ffu)
{
if (mantissa == 0u)
{
/* Inf */
return (deFloat16) (sign | 0x7c00u);
}
else
{
/* NaN */
mantissa >>= 13u; /* 16-bit floats has 10-bit for mantissa, 13-bit less than 32-bit floats. */
/* Make sure we don't turn NaN into zero by | (mantissa == 0). */
return (deFloat16) (sign | 0x7c00u | mantissa | (mantissa == 0u));
}
}
/* The following are cases for normalized floats.
*
* * If exp16 is less than 0, we are experiencing underflow for the exponent. To encode this underflowed exponent,
* we can only shift the mantissa further right.
* The real exponent is exp16 - 15. A denormalized 16-bit float can represent -14 via its exponent.
* Note that the most significant bit in the mantissa of a denormalized float is already -1 as for exponent.
* So, we just need to right shift the mantissa -exp16 bits.
* * If exp16 is 0, mantissa shifting requirement is similar to the above.
* * If exp16 is greater than 30 (0b11110), we are experiencing overflow for the exponent of 16-bit normalized floats.
*/
/* Case: normalized floats -> zero */
else if (exp16 < -10)
{
/* 16-bit floats have only 10 bits for mantissa. Minimal 16-bit denormalized float is (2^-10) * (2^-14). */
/* Expecting a number < (2^-10) * (2^-14) here, not representable, round to zero. */
return (deFloat16) sign;
}
/* Case: normalized floats -> zero and denormalized halfs */
else if (exp16 <= 0)
{
/* Add the implicit leading 1 in mormalized float to mantissa. */
mantissa |= 0x00800000u;
/* We have a (23 + 1)-bit mantissa, but 16-bit floats only expect 10-bit mantissa.
* Need to discard the last 14-bits considering rounding mode.
* We also need to shift right -exp16 bits to encode the underflowed exponent.
*/
if (mode == DE_ROUNDINGMODE_TO_ZERO)
{
mantissa >>= (14 - exp16);
}
else
{
/* mantissa in the above may exceed 10-bits, in which case overflow happens.
* The overflowed bit is automatically carried to exponent then.
*/
mantissa = roundToNearestEven(mantissa, 14 - exp16);
}
return (deFloat16) (sign | mantissa);
}
/* Case: normalized floats -> normalized floats */
else if (exp16 <= 30)
{
if (mode == DE_ROUNDINGMODE_TO_ZERO)
{
return (deFloat16) (sign | ((deUint32)exp16 << 10u) | (mantissa >> 13u));
}
else
{
mantissa = roundToNearestEven(mantissa, 13);
/* Handle overflow. exp16 may overflow (and become Inf) itself, but that's correct. */
exp16 = (exp16 << 10u) + (mantissa & (1 << 10));
mantissa &= (1u << 10) - 1;
return (deFloat16) (sign | ((deUint32) exp16) | mantissa);
}
}
/* Case: normalized floats (too large to be representable as 16-bit floats) */
else
{
/* According to IEEE Std 754-2008 Section 7.4,
* * roundTiesToEven and roundTiesToAway carry all overflows to Inf with the sign
* of the intermediate result.
* * roundTowardZero carries all overflows to the format’s largest finite number
* with the sign of the intermediate result.
*/
if (mode == DE_ROUNDINGMODE_TO_ZERO)
{
return (deFloat16) (sign | 0x7bffu); /* 111 1011 1111 1111 */
}
else
{
return (deFloat16) (sign | (0x1f << 10));
}
}
/* Make compiler happy */
return (deFloat16) 0;
}
/*--------------------------------------------------------------------*//*!
* \brief Round the given number `val` to nearest even by discarding
* the last `numBitsToDiscard` bits.
* \param val value to round
* \param numBitsToDiscard number of (least significant) bits to discard
* \return The rounded value with the last `numBitsToDiscard` removed
*//*--------------------------------------------------------------------*/
static deUint64 roundToNearestEven64 (deUint64 val, const deUint64 numBitsToDiscard)
{
const deUint64 lastBits = val & (((deUint64)1 << numBitsToDiscard) - 1);
const deUint64 headBit = val & ((deUint64)1 << (numBitsToDiscard - 1));
DE_ASSERT(numBitsToDiscard > 0 && numBitsToDiscard < 64); /* Make sure no overflow. */
val >>= numBitsToDiscard;
if (headBit == 0)
{
return val;
}
else if (headBit == lastBits)
{
if ((val & 0x1) == 0x1)
{
return val + 1;
}
else
{
return val;
}
}
else
{
return val + 1;
}
}
deFloat16 deFloat64To16Round (double val64, deRoundingMode mode)
{
union
{
double f; /* Interpret as 64-bit float */
deUint64 u; /* Interpret as 64-bit unsigned integer */
} x;
deUint64 sign; /* sign : 0000 0000 0000 0000 X000 0000 0000 0000 */
deUint64 exp64; /* exp32: biased exponent for 64-bit floats */
int exp16; /* exp16: biased exponent for 16-bit floats */
deUint64 mantissa;
/* We only support these two rounding modes for now */
DE_ASSERT(mode == DE_ROUNDINGMODE_TO_ZERO || mode == DE_ROUNDINGMODE_TO_NEAREST_EVEN);
x.f = val64;
sign = (x.u >> 48u) & 0x00008000u;
exp64 = (x.u >> 52u) & 0x000007ffu;
exp16 = (int) (exp64) - 1023 + 15; /* 15/127: exponent bias for 16-bit/32-bit floats */
mantissa = x.u & 0x00fffffffffffffu;
/* Case: zero and denormalized floats */
if (exp64 == 0)
{
/* Denormalized floats are < 2^(1-1023), not representable in 16-bit floats, rounding to zero. */
return (deFloat16) sign;
}
/* Case: Inf and NaN */
else if (exp64 == 0x000007ffu)
{
if (mantissa == 0u)
{
/* Inf */
return (deFloat16) (sign | 0x7c00u);
}
else
{
/* NaN */
mantissa >>= 42u; /* 16-bit floats has 10-bit for mantissa, 42-bit less than 64-bit floats. */
/* Make sure we don't turn NaN into zero by | (mantissa == 0). */
return (deFloat16) (sign | 0x7c00u | mantissa | (mantissa == 0u));
}
}
/* The following are cases for normalized floats.
*
* * If exp16 is less than 0, we are experiencing underflow for the exponent. To encode this underflowed exponent,
* we can only shift the mantissa further right.
* The real exponent is exp16 - 15. A denormalized 16-bit float can represent -14 via its exponent.
* Note that the most significant bit in the mantissa of a denormalized float is already -1 as for exponent.
* So, we just need to right shift the mantissa -exp16 bits.
* * If exp16 is 0, mantissa shifting requirement is similar to the above.
* * If exp16 is greater than 30 (0b11110), we are experiencing overflow for the exponent of 16-bit normalized floats.
*/
/* Case: normalized floats -> zero */
else if (exp16 < -10)
{
/* 16-bit floats have only 10 bits for mantissa. Minimal 16-bit denormalized float is (2^-10) * (2^-14). */
/* Expecting a number < (2^-10) * (2^-14) here, not representable, round to zero. */
return (deFloat16) sign;
}
/* Case: normalized floats -> zero and denormalized halfs */
else if (exp16 <= 0)
{
/* Add the implicit leading 1 in mormalized float to mantissa. */
mantissa |= 0x0010000000000000u;
/* We have a (23 + 1)-bit mantissa, but 16-bit floats only expect 10-bit mantissa.
* Need to discard the last 14-bits considering rounding mode.
* We also need to shift right -exp16 bits to encode the underflowed exponent.
*/
if (mode == DE_ROUNDINGMODE_TO_ZERO)
{
mantissa >>= (43 - exp16);
}
else
{
/* mantissa in the above may exceed 10-bits, in which case overflow happens.
* The overflowed bit is automatically carried to exponent then.
*/
mantissa = roundToNearestEven64(mantissa, 43 - exp16);
}
return (deFloat16) (sign | mantissa);
}
/* Case: normalized floats -> normalized floats */
else if (exp16 <= 30)
{
if (mode == DE_ROUNDINGMODE_TO_ZERO)
{
return (deFloat16) (sign | ((deUint32)exp16 << 10u) | (mantissa >> 42u));
}
else
{
mantissa = roundToNearestEven64(mantissa, 42);
/* Handle overflow. exp16 may overflow (and become Inf) itself, but that's correct. */
exp16 = (exp16 << 10u) + (deFloat16)(mantissa & (1 << 10));
mantissa &= (1u << 10) - 1;
return (deFloat16) (sign | ((deUint32) exp16) | mantissa);
}
}
/* Case: normalized floats (too large to be representable as 16-bit floats) */
else
{
/* According to IEEE Std 754-2008 Section 7.4,
* * roundTiesToEven and roundTiesToAway carry all overflows to Inf with the sign
* of the intermediate result.
* * roundTowardZero carries all overflows to the format’s largest finite number
* with the sign of the intermediate result.
*/
if (mode == DE_ROUNDINGMODE_TO_ZERO)
{
return (deFloat16) (sign | 0x7bffu); /* 111 1011 1111 1111 */
}
else
{
return (deFloat16) (sign | (0x1f << 10));
}
}
/* Make compiler happy */
return (deFloat16) 0;
}
float deFloat16To32 (deFloat16 val16)
{
deUint32 sign;
deUint32 expotent;
deUint32 mantissa;
union
{
float f;
deUint32 u;
} x;
x.u = 0u;
sign = ((deUint32)val16 >> 15u) & 0x00000001u;
expotent = ((deUint32)val16 >> 10u) & 0x0000001fu;
mantissa = (deUint32)val16 & 0x000003ffu;
if (expotent == 0u)
{
if (mantissa == 0u)
{
/* +/- 0 */
x.u = sign << 31u;
return x.f;
}
else
{
/* Denormalized, normalize it. */
while (!(mantissa & 0x00000400u))
{
mantissa <<= 1u;
expotent -= 1u;
}
expotent += 1u;
mantissa &= ~0x00000400u;
}
}
else if (expotent == 31u)
{
if (mantissa == 0u)
{
/* +/- InF */
x.u = (sign << 31u) | 0x7f800000u;
return x.f;
}
else
{
/* +/- NaN */
x.u = (sign << 31u) | 0x7f800000u | (mantissa << 13u);
return x.f;
}
}
expotent = expotent + (127u - 15u);
mantissa = mantissa << 13u;
x.u = (sign << 31u) | (expotent << 23u) | mantissa;
return x.f;
}
double deFloat16To64 (deFloat16 val16)
{
deUint64 sign;
deUint64 expotent;
deUint64 mantissa;
union
{
double f;
deUint64 u;
} x;
x.u = 0u;
sign = ((deUint32)val16 >> 15u) & 0x00000001u;
expotent = ((deUint32)val16 >> 10u) & 0x0000001fu;
mantissa = (deUint32)val16 & 0x000003ffu;
if (expotent == 0u)
{
if (mantissa == 0u)
{
/* +/- 0 */
x.u = sign << 63u;
return x.f;
}
else
{
/* Denormalized, normalize it. */
while (!(mantissa & 0x00000400u))
{
mantissa <<= 1u;
expotent -= 1u;
}
expotent += 1u;
mantissa &= ~0x00000400u;
}
}
else if (expotent == 31u)
{
if (mantissa == 0u)
{
/* +/- InF */
x.u = (sign << 63u) | 0x7ff0000000000000u;
return x.f;
}
else
{
/* +/- NaN */
x.u = (sign << 63u) | 0x7ff0000000000000u | (mantissa << 42u);
return x.f;
}
}
expotent = expotent + (1023u - 15u);
mantissa = mantissa << 42u;
x.u = (sign << 63u) | (expotent << 52u) | mantissa;
return x.f;
}
DE_END_EXTERN_C