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/*
* C-implementation of Float16 - https://github.com/Maratyszcza/FP16
*
* Version: 0a92994d729ff76a58f692d3028ca1b64b145d91
*
* The MIT License (MIT)
*
* Copyright (c) 2017 Facebook Inc.
* Copyright (c) 2017 Georgia Institute of Technology
* Copyright 2019 Google LLC
*
* Permission is hereby granted, free of charge, to any person obtaining a copy of this
* software and associated documentation files (the "Software"), to deal in the Software
* without restriction, including without limitation the rights to use, copy, modify,
* merge, publish, distribute, sublicense, and/or sell copies of the Software, and to
* permit persons to whom the Software is furnished to do so, subject to the following
* conditions:
*
* The above copyright notice and this permission notice shall be included in all copies
* or substantial portions of the Software.
*
* THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED,
* INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PARTICULAR
* PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE
* FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR
* OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER
* DEALINGS IN THE SOFTWARE.
*/
/*
* The following changes were done to adapt this to Erlang/OTP conventions:
*
* 1. uint16_t and uint32_t have been rewritten to Uint16 and Uint32
* 2. Mixed declarations were moved to the top to avoid warnings
* 3. inline was rewritten as ERTS_INLINE
* 4. UINT16_C(x) and UINT32_C(x) were rewritten to xU as we don't support 16bits platform
*/
static ERTS_INLINE float fp32_from_bits(Uint32 w) {
union {
Uint32 as_bits;
float as_value;
} fp32 = { w };
return fp32.as_value;
}
static ERTS_INLINE Uint32 fp32_to_bits(float f) {
union {
float as_value;
Uint32 as_bits;
} fp32 = { f };
return fp32.as_bits;
}
/*
* 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 ERTS_INLINE Uint16 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(0x77800000U);
const float scale_to_zero = fp32_from_bits(0x08800000U);
#endif
float base = (fabsf(f) * scale_to_inf) * scale_to_zero;
const Uint32 w = fp32_to_bits(f);
const Uint32 shl1_w = w + w;
const Uint32 sign = w & 0x80000000U;
Uint32 bits, exp_bits, mantissa_bits, nonsign, bias;
bias = shl1_w & 0xFF000000U;
if (bias < 0x71000000U) {
bias = 0x71000000U;
}
base = fp32_from_bits((bias >> 1) + 0x07800000U) + base;
bits = fp32_to_bits(base);
exp_bits = (bits >> 13) & 0x00007C00U;
mantissa_bits = bits & 0x00000FFFU;
nonsign = exp_bits + mantissa_bits;
return (sign >> 16) | (shl1_w > 0xFF000000U ? 0x7E00U : nonsign);
}
/*
* 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 ERTS_INLINE float fp16_ieee_to_fp32_value(Uint16 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 w = (Uint32) 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 sign = w & 0x80000000U;
/*
* 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 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 exp_offset = 0xE0U << 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(0x7800000U);
#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 amount.
*
* 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 magic_mask = 126U << 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 denormalized_cutoff = 1U << 27;
const Uint32 result = sign |
(two_w < denormalized_cutoff ? fp32_to_bits(denormalized_value) : fp32_to_bits(normalized_value));
return fp32_from_bits(result);
}
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