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/* Double-precision AdvSIMD atan2
Copyright (C) 2023-2025 Free Software Foundation, Inc.
This file is part of the GNU C Library.
The GNU C Library is free software; you can redistribute it and/or
modify it under the terms of the GNU Lesser General Public
License as published by the Free Software Foundation; either
version 2.1 of the License, or (at your option) any later version.
The GNU C Library is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
Lesser General Public License for more details.
You should have received a copy of the GNU Lesser General Public
License along with the GNU C Library; if not, see
<https://www.gnu.org/licenses/>. */
#include "math_config.h"
#include "v_math.h"
#include "poly_advsimd_f64.h"
static const struct data
{
float64x2_t c0, c2, c4, c6, c8, c10, c12, c14, c16, c18;
float64x2_t pi_over_2;
double c1, c3, c5, c7, c9, c11, c13, c15, c17, c19;
uint64x2_t zeroinfnan, minustwo;
} data = {
/* Coefficients of polynomial P such that atan(x)~x+x*P(x^2) on
[2**-1022, 1.0]. */
.c0 = V2 (-0x1.5555555555555p-2),
.c1 = 0x1.99999999996c1p-3,
.c2 = V2 (-0x1.2492492478f88p-3),
.c3 = 0x1.c71c71bc3951cp-4,
.c4 = V2 (-0x1.745d160a7e368p-4),
.c5 = 0x1.3b139b6a88ba1p-4,
.c6 = V2 (-0x1.11100ee084227p-4),
.c7 = 0x1.e1d0f9696f63bp-5,
.c8 = V2 (-0x1.aebfe7b418581p-5),
.c9 = 0x1.842dbe9b0d916p-5,
.c10 = V2 (-0x1.5d30140ae5e99p-5),
.c11 = 0x1.338e31eb2fbbcp-5,
.c12 = V2 (-0x1.00e6eece7de8p-5),
.c13 = 0x1.860897b29e5efp-6,
.c14 = V2 (-0x1.0051381722a59p-6),
.c15 = 0x1.14e9dc19a4a4ep-7,
.c16 = V2 (-0x1.d0062b42fe3bfp-9),
.c17 = 0x1.17739e210171ap-10,
.c18 = V2 (-0x1.ab24da7be7402p-13),
.c19 = 0x1.358851160a528p-16,
.pi_over_2 = V2 (0x1.921fb54442d18p+0),
.zeroinfnan = V2 (2 * 0x7ff0000000000000ul - 1),
.minustwo = V2 (0xc000000000000000),
};
#define SignMask v_u64 (0x8000000000000000)
/* Special cases i.e. 0, infinity, NaN (fall back to scalar calls). */
static float64x2_t VPCS_ATTR NOINLINE
special_case (float64x2_t y, float64x2_t x, float64x2_t ret,
uint64x2_t sign_xy, uint64x2_t cmp)
{
/* Account for the sign of x and y. */
ret = vreinterpretq_f64_u64 (
veorq_u64 (vreinterpretq_u64_f64 (ret), sign_xy));
return v_call2_f64 (atan2, y, x, ret, cmp);
}
/* Returns 1 if input is the bit representation of 0, infinity or nan. */
static inline uint64x2_t
zeroinfnan (uint64x2_t i, const struct data *d)
{
/* (2 * i - 1) >= (2 * asuint64 (INFINITY) - 1). */
return vcgeq_u64 (vsubq_u64 (vaddq_u64 (i, i), v_u64 (1)), d->zeroinfnan);
}
/* Fast implementation of vector atan2.
Maximum observed error is 2.8 ulps:
_ZGVnN2vv_atan2 (0x1.9651a429a859ap+5, 0x1.953075f4ee26p+5)
got 0x1.92d628ab678ccp-1
want 0x1.92d628ab678cfp-1. */
float64x2_t VPCS_ATTR V_NAME_D2 (atan2) (float64x2_t y, float64x2_t x)
{
const struct data *d = ptr_barrier (&data);
uint64x2_t ix = vreinterpretq_u64_f64 (x);
uint64x2_t iy = vreinterpretq_u64_f64 (y);
uint64x2_t special_cases
= vorrq_u64 (zeroinfnan (ix, d), zeroinfnan (iy, d));
uint64x2_t sign_x = vandq_u64 (ix, SignMask);
uint64x2_t sign_y = vandq_u64 (iy, SignMask);
uint64x2_t sign_xy = veorq_u64 (sign_x, sign_y);
float64x2_t ax = vabsq_f64 (x);
float64x2_t ay = vabsq_f64 (y);
uint64x2_t pred_xlt0 = vcltzq_f64 (x);
uint64x2_t pred_aygtax = vcagtq_f64 (y, x);
/* Set up z for call to atan. */
float64x2_t n = vbslq_f64 (pred_aygtax, vnegq_f64 (ax), ay);
float64x2_t q = vbslq_f64 (pred_aygtax, ay, ax);
float64x2_t z = vdivq_f64 (n, q);
/* Work out the correct shift. */
float64x2_t shift
= vreinterpretq_f64_u64 (vandq_u64 (pred_xlt0, d->minustwo));
shift = vbslq_f64 (pred_aygtax, vaddq_f64 (shift, v_f64 (1.0)), shift);
shift = vmulq_f64 (shift, d->pi_over_2);
/* Calculate the polynomial approximation.
Use split Estrin scheme for P(z^2) with deg(P)=19. Use split instead of
full scheme to avoid underflow in x^16.
The order 19 polynomial P approximates
(atan(sqrt(x))-sqrt(x))/x^(3/2). */
float64x2_t z2 = vmulq_f64 (z, z);
float64x2_t x2 = vmulq_f64 (z2, z2);
float64x2_t x4 = vmulq_f64 (x2, x2);
float64x2_t x8 = vmulq_f64 (x4, x4);
float64x2_t c13 = vld1q_f64 (&d->c1);
float64x2_t c57 = vld1q_f64 (&d->c5);
float64x2_t c911 = vld1q_f64 (&d->c9);
float64x2_t c1315 = vld1q_f64 (&d->c13);
float64x2_t c1719 = vld1q_f64 (&d->c17);
/* estrin_7. */
float64x2_t p01 = vfmaq_laneq_f64 (d->c0, z2, c13, 0);
float64x2_t p23 = vfmaq_laneq_f64 (d->c2, z2, c13, 1);
float64x2_t p03 = vfmaq_f64 (p01, x2, p23);
float64x2_t p45 = vfmaq_laneq_f64 (d->c4, z2, c57, 0);
float64x2_t p67 = vfmaq_laneq_f64 (d->c6, z2, c57, 1);
float64x2_t p47 = vfmaq_f64 (p45, x2, p67);
float64x2_t p07 = vfmaq_f64 (p03, x4, p47);
/* estrin_11. */
float64x2_t p89 = vfmaq_laneq_f64 (d->c8, z2, c911, 0);
float64x2_t p1011 = vfmaq_laneq_f64 (d->c10, z2, c911, 1);
float64x2_t p811 = vfmaq_f64 (p89, x2, p1011);
float64x2_t p1213 = vfmaq_laneq_f64 (d->c12, z2, c1315, 0);
float64x2_t p1415 = vfmaq_laneq_f64 (d->c14, z2, c1315, 1);
float64x2_t p1215 = vfmaq_f64 (p1213, x2, p1415);
float64x2_t p1617 = vfmaq_laneq_f64 (d->c16, z2, c1719, 0);
float64x2_t p1819 = vfmaq_laneq_f64 (d->c18, z2, c1719, 1);
float64x2_t p1619 = vfmaq_f64 (p1617, x2, p1819);
float64x2_t p815 = vfmaq_f64 (p811, x4, p1215);
float64x2_t p819 = vfmaq_f64 (p815, x8, p1619);
float64x2_t ret = vfmaq_f64 (p07, p819, x8);
/* Finalize. y = shift + z + z^3 * P(z^2). */
ret = vfmaq_f64 (z, ret, vmulq_f64 (z2, z));
ret = vaddq_f64 (ret, shift);
if (__glibc_unlikely (v_any_u64 (special_cases)))
return special_case (y, x, ret, sign_xy, special_cases);
/* Account for the sign of x and y. */
ret = vreinterpretq_f64_u64 (
veorq_u64 (vreinterpretq_u64_f64 (ret), sign_xy));
return ret;
}
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