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volk_sse_intrinsics.h
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/* -*- c++ -*- */
/*
* Copyright 2015 Free Software Foundation, Inc.
* Copyright 2023-2026 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 SSE intrinsics of intrinsics.
* They should be used in VOLK kernels to avoid copy-pasta.
*/
#ifndef INCLUDE_VOLK_VOLK_SSE_INTRINSICS_H_
#define INCLUDE_VOLK_VOLK_SSE_INTRINSICS_H_
#include <emmintrin.h>
#include <xmmintrin.h>
/*
* Newton-Raphson refined reciprocal square root: 1/sqrt(a)
* One iteration doubles precision from ~12-bit to ~24-bit
* x1 = x0 * (1.5 - 0.5 * a * x0^2)
* Handles edge cases: +0 → +Inf, +Inf → 0
*/
static inline __m128 _mm_rsqrt_nr_ps(const __m128 a)
{
const __m128 HALF = _mm_set1_ps(0.5f);
const __m128 THREE_HALFS = _mm_set1_ps(1.5f);
const __m128 x0 = _mm_rsqrt_ps(a); // +Inf for +0, 0 for +Inf
// Newton-Raphson: x1 = x0 * (1.5 - 0.5 * a * x0^2)
__m128 x1 = _mm_mul_ps(
x0, _mm_sub_ps(THREE_HALFS, _mm_mul_ps(HALF, _mm_mul_ps(_mm_mul_ps(x0, x0), a))));
// For +0 and +Inf inputs, x0 is correct but NR produces NaN due to Inf*0
// Blend: use x0 where a == +0 or a == +Inf, else use x1
__m128i a_si = _mm_castps_si128(a);
__m128i zero_mask = _mm_cmpeq_epi32(a_si, _mm_setzero_si128());
__m128i inf_mask = _mm_cmpeq_epi32(a_si, _mm_set1_epi32(0x7F800000));
__m128 special_mask = _mm_castsi128_ps(_mm_or_si128(zero_mask, inf_mask));
// SSE2-compatible blend: (x0 & mask) | (x1 & ~mask)
return _mm_or_ps(_mm_and_ps(special_mask, x0), _mm_andnot_ps(special_mask, x1));
}
/*
* 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 __m128 _mm_arctan_poly_sse(const __m128 x)
{
const __m128 a1 = _mm_set1_ps(+0x1.ffffeap-1f);
const __m128 a3 = _mm_set1_ps(-0x1.55437p-2f);
const __m128 a5 = _mm_set1_ps(+0x1.972be6p-3f);
const __m128 a7 = _mm_set1_ps(-0x1.1436ap-3f);
const __m128 a9 = _mm_set1_ps(+0x1.5785aap-4f);
const __m128 a11 = _mm_set1_ps(-0x1.2f3004p-5f);
const __m128 a13 = _mm_set1_ps(+0x1.01a37cp-7f);
const __m128 x_times_x = _mm_mul_ps(x, x);
__m128 arctan;
arctan = a13;
arctan = _mm_mul_ps(x_times_x, arctan);
arctan = _mm_add_ps(arctan, a11);
arctan = _mm_mul_ps(x_times_x, arctan);
arctan = _mm_add_ps(arctan, a9);
arctan = _mm_mul_ps(x_times_x, arctan);
arctan = _mm_add_ps(arctan, a7);
arctan = _mm_mul_ps(x_times_x, arctan);
arctan = _mm_add_ps(arctan, a5);
arctan = _mm_mul_ps(x_times_x, arctan);
arctan = _mm_add_ps(arctan, a3);
arctan = _mm_mul_ps(x_times_x, arctan);
arctan = _mm_add_ps(arctan, a1);
arctan = _mm_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 __m128 _mm_arcsin_poly_sse(const __m128 x)
{
const __m128 c0 = _mm_set1_ps(0x1.ffffcep-1f);
const __m128 c1 = _mm_set1_ps(0x1.55b648p-3f);
const __m128 c2 = _mm_set1_ps(0x1.24d192p-4f);
const __m128 c3 = _mm_set1_ps(0x1.0a788p-4f);
const __m128 u = _mm_mul_ps(x, x);
__m128 p = c3;
p = _mm_mul_ps(u, p);
p = _mm_add_ps(p, c2);
p = _mm_mul_ps(u, p);
p = _mm_add_ps(p, c1);
p = _mm_mul_ps(u, p);
p = _mm_add_ps(p, c0);
return _mm_mul_ps(x, p);
}
static inline __m128 _mm_magnitudesquared_ps(__m128 cplxValue1, __m128 cplxValue2)
{
__m128 iValue, qValue;
// Arrange in i1i2i3i4 format
iValue = _mm_shuffle_ps(cplxValue1, cplxValue2, _MM_SHUFFLE(2, 0, 2, 0));
// Arrange in q1q2q3q4 format
qValue = _mm_shuffle_ps(cplxValue1, cplxValue2, _MM_SHUFFLE(3, 1, 3, 1));
iValue = _mm_mul_ps(iValue, iValue); // Square the I values
qValue = _mm_mul_ps(qValue, qValue); // Square the Q Values
return _mm_add_ps(iValue, qValue); // Add the I2 and Q2 values
}
static inline __m128 _mm_magnitude_ps(__m128 cplxValue1, __m128 cplxValue2)
{
return _mm_sqrt_ps(_mm_magnitudesquared_ps(cplxValue1, cplxValue2));
}
static inline __m128 _mm_scaled_norm_dist_ps_sse(const __m128 symbols0,
const __m128 symbols1,
const __m128 points0,
const __m128 points1,
const __m128 scalar)
{
// calculate scalar * |x - y|^2
const __m128 diff0 = _mm_sub_ps(symbols0, points0);
const __m128 diff1 = _mm_sub_ps(symbols1, points1);
const __m128 norms = _mm_magnitudesquared_ps(diff0, diff1);
return _mm_mul_ps(norms, scalar);
}
static inline __m128 _mm_accumulate_square_sum_ps(
__m128 sq_acc, __m128 acc, __m128 val, __m128 rec, __m128 aux)
{
aux = _mm_mul_ps(aux, val);
aux = _mm_sub_ps(aux, acc);
aux = _mm_mul_ps(aux, aux);
aux = _mm_mul_ps(aux, rec);
return _mm_add_ps(sq_acc, aux);
}
/*
* 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 __m128 _mm_sin_poly_sse(const __m128 x)
{
const __m128 s1 = _mm_set1_ps(-0x1.555552p-3f);
const __m128 s2 = _mm_set1_ps(+0x1.110be2p-7f);
const __m128 s3 = _mm_set1_ps(-0x1.9ab22ap-13f);
const __m128 x2 = _mm_mul_ps(x, x);
const __m128 x3 = _mm_mul_ps(x2, x);
__m128 poly = _mm_add_ps(_mm_mul_ps(x2, s3), s2);
poly = _mm_add_ps(_mm_mul_ps(x2, poly), s1);
return _mm_add_ps(_mm_mul_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 __m128 _mm_cos_poly_sse(const __m128 x)
{
const __m128 c1 = _mm_set1_ps(-0x1.fffff4p-2f);
const __m128 c2 = _mm_set1_ps(+0x1.554a46p-5f);
const __m128 c3 = _mm_set1_ps(-0x1.661be2p-10f);
const __m128 one = _mm_set1_ps(1.0f);
const __m128 x2 = _mm_mul_ps(x, x);
__m128 poly = _mm_add_ps(_mm_mul_ps(x2, c3), c2);
poly = _mm_add_ps(_mm_mul_ps(x2, poly), c1);
return _mm_add_ps(_mm_mul_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
*/
static inline __m128 _mm_log2_poly_sse(const __m128 x)
{
const __m128 c0 = _mm_set1_ps(+0x1.a8a726p+1f);
const __m128 c1 = _mm_set1_ps(-0x1.0b7f7ep+2f);
const __m128 c2 = _mm_set1_ps(+0x1.05d9ccp+2f);
const __m128 c3 = _mm_set1_ps(-0x1.4d476cp+1f);
const __m128 c4 = _mm_set1_ps(+0x1.04fc3ap+0f);
const __m128 c5 = _mm_set1_ps(-0x1.c97982p-3f);
const __m128 c6 = _mm_set1_ps(+0x1.57aa42p-6f);
// Horner's method: c0 + x*(c1 + x*(c2 + ...))
__m128 poly = c6;
poly = _mm_add_ps(_mm_mul_ps(poly, x), c5);
poly = _mm_add_ps(_mm_mul_ps(poly, x), c4);
poly = _mm_add_ps(_mm_mul_ps(poly, x), c3);
poly = _mm_add_ps(_mm_mul_ps(poly, x), c2);
poly = _mm_add_ps(_mm_mul_ps(poly, x), c1);
poly = _mm_add_ps(_mm_mul_ps(poly, x), c0);
return poly;
}
#endif /* INCLUDE_VOLK_VOLK_SSE_INTRINSICS_H_ */