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/* Double-precision AdvSIMD inverse tan
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 "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;
} 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),
};
#define SignMask v_u64 (0x8000000000000000)
#define TinyBound 0x3e10000000000000 /* asuint64(0x1p-30). */
#define BigBound 0x4340000000000000 /* asuint64(0x1p53). */
/* Fast implementation of vector atan.
Based on atan(x) ~ shift + z + z^3 * P(z^2) with reduction to [0,1] using
z=1/x and shift = pi/2. Maximum observed error is 2.27 ulps:
_ZGVnN2v_atan (0x1.0005af27c23e9p+0) got 0x1.9225645bdd7c1p-1
want 0x1.9225645bdd7c3p-1. */
float64x2_t VPCS_ATTR V_NAME_D1 (atan) (float64x2_t x)
{
const struct data *d = ptr_barrier (&data);
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);
/* Small cases, infs and nans are supported by our approximation technique,
but do not set fenv flags correctly. Only trigger special case if we need
fenv. */
uint64x2_t ix = vreinterpretq_u64_f64 (x);
uint64x2_t sign = vandq_u64 (ix, SignMask);
#if WANT_SIMD_EXCEPT
uint64x2_t ia12 = vandq_u64 (ix, v_u64 (0x7ff0000000000000));
uint64x2_t special = vcgtq_u64 (vsubq_u64 (ia12, v_u64 (TinyBound)),
v_u64 (BigBound - TinyBound));
/* If any lane is special, fall back to the scalar routine for all lanes. */
if (__glibc_unlikely (v_any_u64 (special)))
return v_call_f64 (atan, x, v_f64 (0), v_u64 (-1));
#endif
/* Argument reduction:
y := arctan(x) for x < 1
y := pi/2 + arctan(-1/x) for x > 1
Hence, use z=-1/a if x>=1, otherwise z=a. */
uint64x2_t red = vcagtq_f64 (x, v_f64 (1.0));
/* Avoid dependency in abs(x) in division (and comparison). */
float64x2_t z = vbslq_f64 (red, vdivq_f64 (v_f64 (1.0), x), x);
float64x2_t shift = vreinterpretq_f64_u64 (
vandq_u64 (red, vreinterpretq_u64_f64 (d->pi_over_2)));
/* Use absolute value only when needed (odd powers of z). */
float64x2_t az = vbslq_f64 (
SignMask, vreinterpretq_f64_u64 (vandq_u64 (SignMask, red)), z);
/* 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);
/* 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 y = vfmaq_f64 (p07, p819, x8);
/* Finalize. y = shift + z + z^3 * P(z^2). */
y = vfmaq_f64 (az, y, vmulq_f64 (z2, az));
y = vaddq_f64 (y, shift);
/* y = atan(x) if x>0, -atan(-x) otherwise. */
y = vreinterpretq_f64_u64 (veorq_u64 (vreinterpretq_u64_f64 (y), sign));
return y;
}
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