#pragma once
#ifndef FP16_FP16_H
#define FP16_FP16_H
#if defined(__cplusplus) && (__cplusplus >= 201103L)
#include <cstdint>
#include <cmath>
#elif !defined(__OPENCL_VERSION__)
#include <stdint.h>
#include <math.h>
#endif
#ifdef _MSC_VER
#include <intrin.h>
#endif
#include "fp16/bitcasts.h"
/*
* Convert a 16-bit floating-point number in IEEE half-precision format, in bit representation, to
* a 32-bit floating-point number in IEEE single-precision format, in bit representation.
*
* @note The implementation doesn't use any floating-point operations.
*/
static inline uint32_t fp16_ieee_to_fp32_bits(uint16_t h) {
/*
* Extend the half-precision floating-point number to 32 bits and shift to the upper part of the 32-bit word:
* +---+-----+------------+-------------------+
* | S |EEEEE|MM MMMM MMMM|0000 0000 0000 0000|
* +---+-----+------------+-------------------+
* Bits 31 26-30 16-25 0-15
*
* S - sign bit, E - bits of the biased exponent, M - bits of the mantissa, 0 - zero bits.
*/
const uint32_t w = (uint32_t) h << 16;
/*
* Extract the sign of the input number into the high bit of the 32-bit word:
*
* +---+----------------------------------+
* | S |0000000 00000000 00000000 00000000|
* +---+----------------------------------+
* Bits 31 0-31
*/
const uint32_t sign = w & UINT32_C(0x80000000);
/*
* Extract mantissa and biased exponent of the input number into the bits 0-30 of the 32-bit word:
*
* +---+-----+------------+-------------------+
* | 0 |EEEEE|MM MMMM MMMM|0000 0000 0000 0000|
* +---+-----+------------+-------------------+
* Bits 30 27-31 17-26 0-16
*/
const uint32_t nonsign = w & UINT32_C(0x7FFFFFFF);
/*
* Renorm shift is the number of bits to shift mantissa left to make the half-precision number normalized.
* If the initial number is normalized, some of its high 6 bits (sign == 0 and 5-bit exponent) equals one.
* In this case renorm_shift == 0. If the number is denormalize, renorm_shift > 0. Note that if we shift
* denormalized nonsign by renorm_shift, the unit bit of mantissa will shift into exponent, turning the
* biased exponent into 1, and making mantissa normalized (i.e. without leading 1).
*/
#ifdef _MSC_VER
unsigned long nonsign_bsr;
_BitScanReverse(&nonsign_bsr, (unsigned long) nonsign);
uint32_t renorm_shift = (uint32_t) nonsign_bsr ^ 31;
#else
uint32_t renorm_shift = __builtin_clz(nonsign);
#endif
renorm_shift = renorm_shift > 5 ? renorm_shift - 5 : 0;
/*
* Iff half-precision number has exponent of 15, the addition overflows it into bit 31,
* and the subsequent shift turns the high 9 bits into 1. Thus
* inf_nan_mask ==
* 0x7F800000 if the half-precision number had exponent of 15 (i.e. was NaN or infinity)
* 0x00000000 otherwise
*/
const int32_t inf_nan_mask = ((int32_t) (nonsign + 0x04000000) >> 8) & INT32_C(0x7F800000);
/*
* Iff nonsign is 0, it overflows into 0xFFFFFFFF, turning bit 31 into 1. Otherwise, bit 31 remains 0.
* The signed shift right by 31 broadcasts bit 31 into all bits of the zero_mask. Thus
* zero_mask ==
* 0xFFFFFFFF if the half-precision number was zero (+0.0h or -0.0h)
* 0x00000000 otherwise
*/
const int32_t zero_mask = (int32_t) (nonsign - 1) >> 31;
/*
* 1. Shift nonsign left by renorm_shift to normalize it (if the input was denormal)
* 2. Shift nonsign right by 3 so the exponent (5 bits originally) becomes an 8-bit field and 10-bit mantissa
* shifts into the 10 high bits of the 23-bit mantissa of IEEE single-precision number.
* 3. Add 0x70 to the exponent (starting at bit 23) to compensate the different in exponent bias
* (0x7F for single-precision number less 0xF for half-precision number).
* 4. Subtract renorm_shift from the exponent (starting at bit 23) to account for renormalization. As renorm_shift
* is less than 0x70, this can be combined with step 3.
* 5. Binary OR with inf_nan_mask to turn the exponent into 0xFF if the input was NaN or infinity.
* 6. Binary ANDNOT with zero_mask to turn the mantissa and exponent into zero if the input was zero.
* 7. Combine with the sign of the input number.
*/
return sign | ((((nonsign << renorm_shift >> 3) + ((0x70 - renorm_shift) << 23)) | inf_nan_mask) & ~zero_mask);
}
/*
* Convert a 16-bit floating-point number in IEEE half-precision format, in bit representation, to
* a 32-bit floating-point number in IEEE single-precision format.
*
* @note The implementation relies on IEEE-like (no assumption about rounding mode and no operations on denormals)
* floating-point operations and bitcasts between integer and floating-point variables.
*/
static inline float fp16_ieee_to_fp32_value(uint16_t h) {
/*
* Extend the half-precision floating-point number to 32 bits and shift to the upper part of the 32-bit word:
* +---+-----+------------+-------------------+
* | S |EEEEE|MM MMMM MMMM|0000 0000 0000 0000|
* +---+-----+------------+-------------------+
* Bits 31 26-30 16-25 0-15
*
* S - sign bit, E - bits of the biased exponent, M - bits of the mantissa, 0 - zero bits.
*/
const uint32_t w = (uint32_t) h << 16;
/*
* Extract the sign of the input number into the high bit of the 32-bit word:
*
* +---+----------------------------------+
* | S |0000000 00000000 00000000 00000000|
* +---+----------------------------------+
* Bits 31 0-31
*/
const uint32_t sign = w & UINT32_C(0x80000000);
/*
* Extract mantissa and biased exponent of the input number into the high bits of the 32-bit word:
*
* +-----+------------+---------------------+
* |EEEEE|MM MMMM MMMM|0 0000 0000 0000 0000|
* +-----+------------+---------------------+
* Bits 27-31 17-26 0-16
*/
const uint32_t two_w = w + w;
/*
* Shift mantissa and exponent into bits 23-28 and bits 13-22 so they become mantissa and exponent
* of a single-precision floating-point number:
*
* S|Exponent | Mantissa
* +-+---+-----+------------+----------------+
* |0|000|EEEEE|MM MMMM MMMM|0 0000 0000 0000|
* +-+---+-----+------------+----------------+
* Bits | 23-31 | 0-22
*
* Next, there are some adjustments to the exponent:
* - The exponent needs to be corrected by the difference in exponent bias between single-precision and half-precision
* formats (0x7F - 0xF = 0x70)
* - Inf and NaN values in the inputs should become Inf and NaN values after conversion to the single-precision number.
* Therefore, if the biased exponent of the half-precision input was 0x1F (max possible value), the biased exponent
* of the single-precision output must be 0xFF (max possible value). We do this correction in two steps:
* - First, we adjust the exponent by (0xFF - 0x1F) = 0xE0 (see exp_offset below) rather than by 0x70 suggested
* by the difference in the exponent bias (see above).
* - Then we multiply the single-precision result of exponent adjustment by 2**(-112) to reverse the effect of
* exponent adjustment by 0xE0 less the necessary exponent adjustment by 0x70 due to difference in exponent bias.
* The floating-point multiplication hardware would ensure than Inf and NaN would retain their value on at least
* partially IEEE754-compliant implementations.
*
* Note that the above operations do not handle denormal inputs (where biased exponent == 0). However, they also do not
* operate on denormal inputs, and do not produce denormal results.
*/
const uint32_t exp_offset = UINT32_C(0xE0) << 23;
#if defined(__STDC_VERSION__) && (__STDC_VERSION__ >= 199901L) || defined(__GNUC__) && !defined(__STRICT_ANSI__)
const float exp_scale = 0x1.0p-112f;
#else
const float exp_scale = fp32_from_bits(UINT32_C(0x7800000));
#endif
const float normalized_value = fp32_from_bits((two_w >> 4) + exp_offset) * exp_scale;
/*
* Convert denormalized half-precision inputs into single-precision results (always normalized).
* Zero inputs are also handled here.
*
* In a denormalized number the biased exponent is zero, and mantissa has on-zero bits.
* First, we shift mantissa into bits 0-9 of the 32-bit word.
*
* zeros | mantissa
* +---------------------------+------------+
* |0000 0000 0000 0000 0000 00|MM MMMM MMMM|
* +---------------------------+------------+
* Bits 10-31 0-9
*
* Now, remember that denormalized half-precision numbers are represented as:
* FP16 = mantissa * 2**(-24).
* The trick is to construct a normalized single-precision number with the same mantissa and thehalf-precision input
* and with an exponent which would scale the corresponding mantissa bits to 2**(-24).
* A normalized single-precision floating-point number is represented as:
* FP32 = (1 + mantissa * 2**(-23)) * 2**(exponent - 127)
* Therefore, when the biased exponent is 126, a unit change in the mantissa of the input denormalized half-precision
* number causes a change of the constructud single-precision number by 2**(-24), i.e. the same ammount.
*
* The last step is to adjust the bias of the constructed single-precision number. When the input half-precision number
* is zero, the constructed single-precision number has the value of
* FP32 = 1 * 2**(126 - 127) = 2**(-1) = 0.5
* Therefore, we need to subtract 0.5 from the constructed single-precision number to get the numerical equivalent of
* the input half-precision number.
*/
const uint32_t magic_mask = UINT32_C(126) << 23;
const float magic_bias = 0.5f;
const float denormalized_value = fp32_from_bits((two_w >> 17) | magic_mask) - magic_bias;
/*
* - Choose either results of conversion of input as a normalized number, or as a denormalized number, depending on the
* input exponent. The variable two_w contains input exponent in bits 27-31, therefore if its smaller than 2**27, the
* input is either a denormal number, or zero.
* - Combine the result of conversion of exponent and mantissa with the sign of the input number.
*/
const uint32_t denormalized_cutoff = UINT32_C(1) << 27;
const uint32_t result = sign |
(two_w < denormalized_cutoff ? fp32_to_bits(denormalized_value) : fp32_to_bits(normalized_value));
return fp32_from_bits(result);
}
/*
* Convert a 32-bit floating-point number in IEEE single-precision format to a 16-bit floating-point number in
* IEEE half-precision format, in bit representation.
*
* @note The implementation relies on IEEE-like (no assumption about rounding mode and no operations on denormals)
* floating-point operations and bitcasts between integer and floating-point variables.
*/
static inline uint16_t fp16_ieee_from_fp32_value(float f) {
#if defined(__STDC_VERSION__) && (__STDC_VERSION__ >= 199901L) || defined(__GNUC__) && !defined(__STRICT_ANSI__)
const float scale_to_inf = 0x1.0p+112f;
const float scale_to_zero = 0x1.0p-110f;
#else
const float scale_to_inf = fp32_from_bits(UINT32_C(0x77800000));
const float scale_to_zero = fp32_from_bits(UINT32_C(0x08800000));
#endif
float base = (fabsf(f) * scale_to_inf) * scale_to_zero;
const uint32_t w = fp32_to_bits(f);
const uint32_t shl1_w = w + w;
const uint32_t sign = w & UINT32_C(0x80000000);
uint32_t bias = shl1_w & UINT32_C(0xFF000000);
if (bias < UINT32_C(0x71000000)) {
bias = UINT32_C(0x71000000);
}
base = fp32_from_bits((bias >> 1) + UINT32_C(0x07800000)) + base;
const uint32_t bits = fp32_to_bits(base);
const uint32_t exp_bits = (bits >> 13) & UINT32_C(0x00007C00);
const uint32_t mantissa_bits = bits & UINT32_C(0x00000FFF);
const uint32_t nonsign = exp_bits + mantissa_bits;
return (sign >> 16) | (shl1_w > UINT32_C(0xFF000000) ? UINT16_C(0x7E00) : nonsign);
}
/*
* Convert a 16-bit floating-point number in ARM alternative half-precision format, in bit representation, to
* a 32-bit floating-point number in IEEE single-precision format, in bit representation.
*
* @note The implementation doesn't use any floating-point operations.
*/
static inline uint32_t fp16_alt_to_fp32_bits(uint16_t h) {
/*
* Extend the half-precision floating-point number to 32 bits and shift to the upper part of the 32-bit word:
* +---+-----+------------+-------------------+
* | S |EEEEE|MM MMMM MMMM|0000 0000 0000 0000|
* +---+-----+------------+-------------------+
* Bits 31 26-30 16-25 0-15
*
* S - sign bit, E - bits of the biased exponent, M - bits of the mantissa, 0 - zero bits.
*/
const uint32_t w = (uint32_t) h << 16;
/*
* Extract the sign of the input number into the high bit of the 32-bit word:
*
* +---+----------------------------------+
* | S |0000000 00000000 00000000 00000000|
* +---+----------------------------------+
* Bits 31 0-31
*/
const uint32_t sign = w & UINT32_C(0x80000000);
/*
* Extract mantissa and biased exponent of the input number into the bits 0-30 of the 32-bit word:
*
* +---+-----+------------+-------------------+
* | 0 |EEEEE|MM MMMM MMMM|0000 0000 0000 0000|
* +---+-----+------------+-------------------+
* Bits 30 27-31 17-26 0-16
*/
const uint32_t nonsign = w & UINT32_C(0x7FFFFFFF);
/*
* Renorm shift is the number of bits to shift mantissa left to make the half-precision number normalized.
* If the initial number is normalized, some of its high 6 bits (sign == 0 and 5-bit exponent) equals one.
* In this case renorm_shift == 0. If the number is denormalize, renorm_shift > 0. Note that if we shift
* denormalized nonsign by renorm_shift, the unit bit of mantissa will shift into exponent, turning the
* biased exponent into 1, and making mantissa normalized (i.e. without leading 1).
*/
#ifdef _MSC_VER
unsigned long nonsign_bsr;
_BitScanReverse(&nonsign_bsr, (unsigned long) nonsign);
uint32_t renorm_shift = (uint32_t) nonsign_bsr ^ 31;
#else
uint32_t renorm_shift = __builtin_clz(nonsign);
#endif
renorm_shift = renorm_shift > 5 ? renorm_shift - 5 : 0;
/*
* Iff nonsign is 0, it overflows into 0xFFFFFFFF, turning bit 31 into 1. Otherwise, bit 31 remains 0.
* The signed shift right by 31 broadcasts bit 31 into all bits of the zero_mask. Thus
* zero_mask ==
* 0xFFFFFFFF if the half-precision number was zero (+0.0h or -0.0h)
* 0x00000000 otherwise
*/
const int32_t zero_mask = (int32_t) (nonsign - 1) >> 31;
/*
* 1. Shift nonsign left by renorm_shift to normalize it (if the input was denormal)
* 2. Shift nonsign right by 3 so the exponent (5 bits originally) becomes an 8-bit field and 10-bit mantissa
* shifts into the 10 high bits of the 23-bit mantissa of IEEE single-precision number.
* 3. Add 0x70 to the exponent (starting at bit 23) to compensate the different in exponent bias
* (0x7F for single-precision number less 0xF for half-precision number).
* 4. Subtract renorm_shift from the exponent (starting at bit 23) to account for renormalization. As renorm_shift
* is less than 0x70, this can be combined with step 3.
* 5. Binary ANDNOT with zero_mask to turn the mantissa and exponent into zero if the input was zero.
* 6. Combine with the sign of the input number.
*/
return sign | (((nonsign << renorm_shift >> 3) + ((0x70 - renorm_shift) << 23)) & ~zero_mask);
}
/*
* Convert a 16-bit floating-point number in ARM alternative half-precision format, in bit representation, to
* a 32-bit floating-point number in IEEE single-precision format.
*
* @note The implementation relies on IEEE-like (no assumption about rounding mode and no operations on denormals)
* floating-point operations and bitcasts between integer and floating-point variables.
*/
static inline float fp16_alt_to_fp32_value(uint16_t h) {
/*
* Extend the half-precision floating-point number to 32 bits and shift to the upper part of the 32-bit word:
* +---+-----+------------+-------------------+
* | S |EEEEE|MM MMMM MMMM|0000 0000 0000 0000|
* +---+-----+------------+-------------------+
* Bits 31 26-30 16-25 0-15
*
* S - sign bit, E - bits of the biased exponent, M - bits of the mantissa, 0 - zero bits.
*/
const uint32_t w = (uint32_t) h << 16;
/*
* Extract the sign of the input number into the high bit of the 32-bit word:
*
* +---+----------------------------------+
* | S |0000000 00000000 00000000 00000000|
* +---+----------------------------------+
* Bits 31 0-31
*/
const uint32_t sign = w & UINT32_C(0x80000000);
/*
* Extract mantissa and biased exponent of the input number into the high bits of the 32-bit word:
*
* +-----+------------+---------------------+
* |EEEEE|MM MMMM MMMM|0 0000 0000 0000 0000|
* +-----+------------+---------------------+
* Bits 27-31 17-26 0-16
*/
const uint32_t two_w = w + w;
/*
* Shift mantissa and exponent into bits 23-28 and bits 13-22 so they become mantissa and exponent
* of a single-precision floating-point number:
*
* S|Exponent | Mantissa
* +-+---+-----+------------+----------------+
* |0|000|EEEEE|MM MMMM MMMM|0 0000 0000 0000|
* +-+---+-----+------------+----------------+
* Bits | 23-31 | 0-22
*
* Next, the exponent is adjusted for the difference in exponent bias between single-precision and half-precision
* formats (0x7F - 0xF = 0x70). This operation never overflows or generates non-finite values, as the largest
* half-precision exponent is 0x1F and after the adjustment is can not exceed 0x8F < 0xFE (largest single-precision
* exponent for non-finite values).
*
* Note that this operation does not handle denormal inputs (where biased exponent == 0). However, they also do not
* operate on denormal inputs, and do not produce denormal results.
*/
const float exp_offset = UINT32_C(0x70) << 23;
const float normalized_value = fp32_from_bits((two_w >> 4) + exp_offset);
/*
* Convert denormalized half-precision inputs into single-precision results (always normalized).
* Zero inputs are also handled here.
*
* In a denormalized number the biased exponent is zero, and mantissa has on-zero bits.
* First, we shift mantissa into bits 0-9 of the 32-bit word.
*
* zeros | mantissa
* +---------------------------+------------+
* |0000 0000 0000 0000 0000 00|MM MMMM MMMM|
* +---------------------------+------------+
* Bits 10-31 0-9
*
* Now, remember that denormalized half-precision numbers are represented as:
* FP16 = mantissa * 2**(-24).
* The trick is to construct a normalized single-precision number with the same mantissa and thehalf-precision input
* and with an exponent which would scale the corresponding mantissa bits to 2**(-24).
* A normalized single-precision floating-point number is represented as:
* FP32 = (1 + mantissa * 2**(-23)) * 2**(exponent - 127)
* Therefore, when the biased exponent is 126, a unit change in the mantissa of the input denormalized half-precision
* number causes a change of the constructud single-precision number by 2**(-24), i.e. the same ammount.
*
* The last step is to adjust the bias of the constructed single-precision number. When the input half-precision number
* is zero, the constructed single-precision number has the value of
* FP32 = 1 * 2**(126 - 127) = 2**(-1) = 0.5
* Therefore, we need to subtract 0.5 from the constructed single-precision number to get the numerical equivalent of
* the input half-precision number.
*/
const uint32_t magic_mask = UINT32_C(126) << 23;
const float magic_bias = 0.5f;
const float denormalized_value = fp32_from_bits((two_w >> 17) | magic_mask) - magic_bias;
/*
* - Choose either results of conversion of input as a normalized number, or as a denormalized number, depending on the
* input exponent. The variable two_w contains input exponent in bits 27-31, therefore if its smaller than 2**27, the
* input is either a denormal number, or zero.
* - Combine the result of conversion of exponent and mantissa with the sign of the input number.
*/
const uint32_t denormalized_cutoff = UINT32_C(1) << 27;
const uint32_t result = sign |
(two_w < denormalized_cutoff ? fp32_to_bits(denormalized_value) : fp32_to_bits(normalized_value));
return fp32_from_bits(result);
}
/*
* Convert a 32-bit floating-point number in IEEE single-precision format to a 16-bit floating-point number in
* ARM alternative half-precision format, in bit representation.
*
* @note The implementation relies on IEEE-like (no assumption about rounding mode and no operations on denormals)
* floating-point operations and bitcasts between integer and floating-point variables.
*/
static inline uint16_t fp16_alt_from_fp32_value(float f) {
const uint32_t w = fp32_to_bits(f);
const uint32_t sign = w & UINT32_C(0x80000000);
const uint32_t shl1_w = w + w;
const uint32_t shl1_max_fp16_fp32 = UINT32_C(0x8FFFC000);
const uint32_t shl1_base = shl1_w > shl1_max_fp16_fp32 ? shl1_max_fp16_fp32 : shl1_w;
uint32_t shl1_bias = shl1_base & UINT32_C(0xFF000000);
const uint32_t exp_difference = 23 - 10;
const uint32_t shl1_bias_min = (127 - 1 - exp_difference) << 24;
if (shl1_bias < shl1_bias_min) {
shl1_bias = shl1_bias_min;
}
const float bias = fp32_from_bits((shl1_bias >> 1) + ((exp_difference + 2) << 23));
const float base = fp32_from_bits((shl1_base >> 1) + (2 << 23)) + bias;
const uint32_t exp_f = fp32_to_bits(base) >> 13;
return (sign >> 16) | ((exp_f & UINT32_C(0x00007C00)) + (fp32_to_bits(base) & UINT32_C(0x00000FFF)));
}
#endif /* FP16_FP16_H */