178 lines
7.4 KiB
C++
178 lines
7.4 KiB
C++
// SPDX - FileCopyrightText : 2017 - 2024 Marat Dukhan
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// SPDX - FileCopyrightText : 2025 Advanced Micro Devices, Inc.
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//
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// SPDX - License - Identifier : MIT
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#pragma once
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#ifndef FLASHINFER_FP16_H
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#define FLASHINFER_FP16_H
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#include <bit>
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#include <boost/math/ccmath/fabs.hpp>
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#include <cstdint>
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#include <limits>
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/*
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* Convert a 32-bit floating-point number in IEEE single-precision format to a
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* 16-bit floating-point number in IEEE half-precision format, in bit
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* representation.
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*
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* @note The implementation relies on IEEE-like (no assumption about rounding
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* mode and no operations on denormals) floating-point operations and bitcasts
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* between integer and floating-point variables.
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*/
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static constexpr uint16_t fp16_ieee_from_fp32_value(float f) {
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const float scale_to_inf = std::bit_cast<float>(UINT32_C(0x77800000));
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const float scale_to_zero = std::bit_cast<float>(UINT32_C(0x08800000));
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const float saturated_f = boost::math::ccmath::fabs<float>(f) * scale_to_inf;
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float base = saturated_f * scale_to_zero;
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const uint32_t w = std::bit_cast<uint32_t>(f);
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const uint32_t shl1_w = w + w;
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const uint32_t sign = w & UINT32_C(0x80000000);
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uint32_t bias = shl1_w & UINT32_C(0xFF000000);
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if (bias < UINT32_C(0x71000000)) {
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bias = UINT32_C(0x71000000);
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}
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base = std::bit_cast<float>((bias >> 1) + UINT32_C(0x07800000)) + base;
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const uint32_t bits = std::bit_cast<uint32_t>(base);
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const uint32_t exp_bits = (bits >> 13) & UINT32_C(0x00007C00);
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const uint32_t mantissa_bits = bits & UINT32_C(0x00000FFF);
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const uint32_t nonsign = exp_bits + mantissa_bits;
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return (sign >> 16) | (shl1_w > UINT32_C(0xFF000000) ? UINT16_C(0x7E00) : nonsign);
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}
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static constexpr float fp16_ieee_to_fp32_value(uint16_t h) {
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/*
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* Extend the half-precision floating-point number to 32 bits and shift to
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* the upper part of the 32-bit word:
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* +---+-----+------------+-------------------+
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* | S |EEEEE|MM MMMM MMMM|0000 0000 0000 0000|
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* +---+-----+------------+-------------------+
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* Bits 31 26-30 16-25 0-15
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*
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* S - sign bit, E - bits of the biased exponent, M - bits of the mantissa,
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* 0 - zero bits.
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*/
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const uint32_t w = (uint32_t)h << 16;
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/*
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* Extract the sign of the input number into the high bit of the 32-bit
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* word:
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*
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* +---+----------------------------------+
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* | S |0000000 00000000 00000000 00000000|
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* +---+----------------------------------+
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* Bits 31 0-31
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*/
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const uint32_t sign = w & UINT32_C(0x80000000);
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/*
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* Extract mantissa and biased exponent of the input number into the high
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* bits of the 32-bit word:
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*
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* +-----+------------+---------------------+
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* |EEEEE|MM MMMM MMMM|0 0000 0000 0000 0000|
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* +-----+------------+---------------------+
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* Bits 27-31 17-26 0-16
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*/
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const uint32_t two_w = w + w;
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/*
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* Shift mantissa and exponent into bits 23-28 and bits 13-22 so they become
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* mantissa and exponent of a single-precision floating-point number:
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*
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* S|Exponent | Mantissa
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* +-+---+-----+------------+----------------+
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* |0|000|EEEEE|MM MMMM MMMM|0 0000 0000 0000|
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* +-+---+-----+------------+----------------+
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* Bits | 23-31 | 0-22
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*
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* Next, there are some adjustments to the exponent:
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* - The exponent needs to be corrected by the difference in exponent bias
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* between single-precision and half-precision
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* formats (0x7F - 0xF = 0x70)
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* - Inf and NaN values in the inputs should become Inf and NaN values after
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* conversion to the single-precision number.
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* Therefore, if the biased exponent of the half-precision input was 0x1F
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* (max possible value), the biased exponent
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* of the single-precision output must be 0xFF (max possible value). We do
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* this correction in two steps:
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* - First, we adjust the exponent by (0xFF - 0x1F) = 0xE0 (see exp_offset
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* below) rather than by 0x70 suggested
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* by the difference in the exponent bias (see above).
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* - Then we multiply the single-precision result of exponent adjustment
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* by 2**(-112) to reverse the effect of
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* exponent adjustment by 0xE0 less the necessary exponent adjustment by
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* 0x70 due to difference in exponent bias.
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* The floating-point multiplication hardware would ensure than Inf and
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* NaN would retain their value on at least
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* partially IEEE754-compliant implementations.
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*
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* Note that the above operations do not handle denormal inputs (where
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* biased exponent == 0). However, they also do not operate on denormal
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* inputs, and do not produce denormal results.
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*/
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const uint32_t exp_offset = UINT32_C(0xE0) << 23;
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const float exp_scale = std::bit_cast<float>(UINT32_C(0x7800000));
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const float normalized_value = std::bit_cast<float>((two_w >> 4) + exp_offset) * exp_scale;
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/*
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* Convert denormalized half-precision inputs into single-precision results
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* (always normalized).
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* Zero inputs are also handled here.
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*
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* In a denormalized number the biased exponent is zero, and mantissa has
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* on-zero bits.
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* First, we shift mantissa into bits 0-9 of the 32-bit word.
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*
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* zeros | mantissa
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* +---------------------------+------------+
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* |0000 0000 0000 0000 0000 00|MM MMMM MMMM|
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* +---------------------------+------------+
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* Bits 10-31 0-9
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*
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* Now, remember that denormalized half-precision numbers are represented
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* as:
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* FP16 = mantissa * 2**(-24).
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* The trick is to construct a normalized single-precision number with the
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* same mantissa and thehalf-precision input
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* and with an exponent which would scale the corresponding mantissa bits
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* to 2**(-24).
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* A normalized single-precision floating-point number is represented as:
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* FP32 = (1 + mantissa * 2**(-23)) * 2**(exponent - 127)
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* Therefore, when the biased exponent is 126, a unit change in the mantissa
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* of the input denormalized half-precision
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* number causes a change of the constructud single-precision number by
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* 2**(-24), i.e. the same ammount.
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*
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* The last step is to adjust the bias of the constructed single-precision
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* number. When the input half-precision number
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* is zero, the constructed single-precision number has the value of
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* FP32 = 1 * 2**(126 - 127) = 2**(-1) = 0.5
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* Therefore, we need to subtract 0.5 from the constructed single-precision
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* number to get the numerical equivalent of
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* the input half-precision number.
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*/
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const uint32_t magic_mask = UINT32_C(126) << 23;
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const float magic_bias = 0.5f;
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const float denormalized_value = std::bit_cast<float>((two_w >> 17) | magic_mask) - magic_bias;
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/*
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* - Choose either results of conversion of input as a normalized number, or
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* as a denormalized number, depending on the
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* input exponent. The variable two_w contains input exponent in bits
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* 27-31, therefore if its smaller than 2**27, the
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* input is either a denormal number, or zero.
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* - Combine the result of conversion of exponent and mantissa with the sign
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* of the input number.
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*/
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const uint32_t denormalized_cutoff = UINT32_C(1) << 27;
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const uint32_t result =
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sign | (two_w < denormalized_cutoff ? std::bit_cast<uint32_t>(denormalized_value)
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: std::bit_cast<uint32_t>(normalized_value));
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return std::bit_cast<float>(result);
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#endif
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}
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