xref: /aosp_15_r20/external/FP16/include/fp16/fp16.h (revision 5f32b7105932ea8520a0e8811c640f936367d707)

#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 = nonsign ? __builtin_clz(nonsign) : 32;
#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 = nonsign ? __builtin_clz(nonsign) : 32;
#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 uint32_t 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 */