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// SPDX - FileCopyrightText : 2025 Advanced Micro Devices, Inc.
//
// SPDX - License - Identifier : MIT
#pragma once
#ifndef FLASHINFER_FP16_H
#define FLASHINFER_FP16_H
#include <bit>
#include <boost/math/ccmath/fabs.hpp>
#include <cstdint>
#include <limits>
/*
* 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 constexpr uint16_t fp16_ieee_from_fp32_value(float f) {
const float scale_to_inf = std::bit_cast<float>(UINT32_C(0x77800000));
const float scale_to_zero = std::bit_cast<float>(UINT32_C(0x08800000));
const float saturated_f = boost::math::ccmath::fabs<float>(f) * scale_to_inf;
float base = saturated_f * scale_to_zero;
const uint32_t w = std::bit_cast<uint32_t>(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 = std::bit_cast<float>((bias >> 1) + UINT32_C(0x07800000)) + base;
const uint32_t bits = std::bit_cast<uint32_t>(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);
}
static constexpr 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;
const float exp_scale = std::bit_cast<float>(UINT32_C(0x7800000));
const float normalized_value = std::bit_cast<float>((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 = std::bit_cast<float>((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 ? std::bit_cast<uint32_t>(denormalized_value)
: std::bit_cast<uint32_t>(normalized_value));
return std::bit_cast<float>(result);
#endif
}
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