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volk_avx2_fma_intrinsics.h
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/* -*- c++ -*- */
/*
* Copyright 2023 - 2025 Magnus Lundmark <magnuslundmark@gmail.com>
*
* This file is part of VOLK
*
* SPDX-License-Identifier: LGPL-3.0-or-later
*/
/*
* This file is intended to hold AVX2 FMA intrinsics.
* They should be used in VOLK kernels to avoid copy-paste.
*/
#ifndef INCLUDE_VOLK_VOLK_AVX2_FMA_INTRINSICS_H_
#define INCLUDE_VOLK_VOLK_AVX2_FMA_INTRINSICS_H_
#include <immintrin.h>
/*
* Approximate arctan(x) via polynomial expansion
* on the interval [-1, 1]
*
* Maximum relative error ~6.5e-7
* Polynomial evaluated via Horner's method
*/
static inline __m256 _mm256_arctan_poly_avx2_fma(const __m256 x)
{
const __m256 a1 = _mm256_set1_ps(+0x1.ffffeap-1f);
const __m256 a3 = _mm256_set1_ps(-0x1.55437p-2f);
const __m256 a5 = _mm256_set1_ps(+0x1.972be6p-3f);
const __m256 a7 = _mm256_set1_ps(-0x1.1436ap-3f);
const __m256 a9 = _mm256_set1_ps(+0x1.5785aap-4f);
const __m256 a11 = _mm256_set1_ps(-0x1.2f3004p-5f);
const __m256 a13 = _mm256_set1_ps(+0x1.01a37cp-7f);
const __m256 x_times_x = _mm256_mul_ps(x, x);
__m256 arctan;
arctan = a13;
arctan = _mm256_fmadd_ps(x_times_x, arctan, a11);
arctan = _mm256_fmadd_ps(x_times_x, arctan, a9);
arctan = _mm256_fmadd_ps(x_times_x, arctan, a7);
arctan = _mm256_fmadd_ps(x_times_x, arctan, a5);
arctan = _mm256_fmadd_ps(x_times_x, arctan, a3);
arctan = _mm256_fmadd_ps(x_times_x, arctan, a1);
arctan = _mm256_mul_ps(x, arctan);
return arctan;
}
/*
* Approximate arcsin(x) via polynomial expansion
* P(u) such that asin(x) = x * P(x^2) on |x| <= 0.5
*
* Maximum relative error ~1.5e-6
* Polynomial evaluated via Horner's method
*/
static inline __m256 _mm256_arcsin_poly_avx2_fma(const __m256 x)
{
const __m256 c0 = _mm256_set1_ps(0x1.ffffcep-1f);
const __m256 c1 = _mm256_set1_ps(0x1.55b648p-3f);
const __m256 c2 = _mm256_set1_ps(0x1.24d192p-4f);
const __m256 c3 = _mm256_set1_ps(0x1.0a788p-4f);
const __m256 u = _mm256_mul_ps(x, x);
__m256 p = c3;
p = _mm256_fmadd_ps(u, p, c2);
p = _mm256_fmadd_ps(u, p, c1);
p = _mm256_fmadd_ps(u, p, c0);
return _mm256_mul_ps(x, p);
}
/*
* Minimax polynomial for sin(x) on [-pi/4, pi/4]
* Coefficients via Remez algorithm (Sollya)
* Max |error| < 7.3e-9
* sin(x) = x + x^3 * (s1 + x^2 * (s2 + x^2 * s3))
*/
static inline __m256 _mm256_sin_poly_avx2_fma(const __m256 x)
{
const __m256 s1 = _mm256_set1_ps(-0x1.555552p-3f);
const __m256 s2 = _mm256_set1_ps(+0x1.110be2p-7f);
const __m256 s3 = _mm256_set1_ps(-0x1.9ab22ap-13f);
const __m256 x2 = _mm256_mul_ps(x, x);
const __m256 x3 = _mm256_mul_ps(x2, x);
__m256 poly = _mm256_fmadd_ps(x2, s3, s2);
poly = _mm256_fmadd_ps(x2, poly, s1);
return _mm256_fmadd_ps(x3, poly, x);
}
/*
* Minimax polynomial for cos(x) on [-pi/4, pi/4]
* Coefficients via Remez algorithm (Sollya)
* Max |error| < 1.1e-7
* cos(x) = 1 + x^2 * (c1 + x^2 * (c2 + x^2 * c3))
*/
static inline __m256 _mm256_cos_poly_avx2_fma(const __m256 x)
{
const __m256 c1 = _mm256_set1_ps(-0x1.fffff4p-2f);
const __m256 c2 = _mm256_set1_ps(+0x1.554a46p-5f);
const __m256 c3 = _mm256_set1_ps(-0x1.661be2p-10f);
const __m256 one = _mm256_set1_ps(1.0f);
const __m256 x2 = _mm256_mul_ps(x, x);
__m256 poly = _mm256_fmadd_ps(x2, c3, c2);
poly = _mm256_fmadd_ps(x2, poly, c1);
return _mm256_fmadd_ps(x2, poly, one);
}
/*
* Polynomial coefficients for log2(x)/(x-1) on [1, 2]
* Generated with Sollya: remez(log2(x)/(x-1), 6, [1+1b-20, 2])
* Max error: ~1.55e-6
*
* Usage: log2(x) ≈ poly(x) * (x - 1) for x ∈ [1, 2]
* Polynomial evaluated via Horner's method with FMA
*/
static inline __m256 _mm256_log2_poly_avx2_fma(const __m256 x)
{
const __m256 c0 = _mm256_set1_ps(+0x1.a8a726p+1f);
const __m256 c1 = _mm256_set1_ps(-0x1.0b7f7ep+2f);
const __m256 c2 = _mm256_set1_ps(+0x1.05d9ccp+2f);
const __m256 c3 = _mm256_set1_ps(-0x1.4d476cp+1f);
const __m256 c4 = _mm256_set1_ps(+0x1.04fc3ap+0f);
const __m256 c5 = _mm256_set1_ps(-0x1.c97982p-3f);
const __m256 c6 = _mm256_set1_ps(+0x1.57aa42p-6f);
// Horner's method with FMA: c0 + x*(c1 + x*(c2 + ...))
__m256 poly = c6;
poly = _mm256_fmadd_ps(poly, x, c5);
poly = _mm256_fmadd_ps(poly, x, c4);
poly = _mm256_fmadd_ps(poly, x, c3);
poly = _mm256_fmadd_ps(poly, x, c2);
poly = _mm256_fmadd_ps(poly, x, c1);
poly = _mm256_fmadd_ps(poly, x, c0);
return poly;
}
#endif /* INCLUDE_VOLK_VOLK_AVX2_FMA_INTRINSICS_H_ */