1/* Double-precision AdvSIMD atan2
2
3 Copyright (C) 2023-2024 Free Software Foundation, Inc.
4 This file is part of the GNU C Library.
5
6 The GNU C Library is free software; you can redistribute it and/or
7 modify it under the terms of the GNU Lesser General Public
8 License as published by the Free Software Foundation; either
9 version 2.1 of the License, or (at your option) any later version.
10
11 The GNU C Library is distributed in the hope that it will be useful,
12 but WITHOUT ANY WARRANTY; without even the implied warranty of
13 MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
14 Lesser General Public License for more details.
15
16 You should have received a copy of the GNU Lesser General Public
17 License along with the GNU C Library; if not, see
18 <https://www.gnu.org/licenses/>. */
19
20#include "v_math.h"
21#include "poly_advsimd_f64.h"
22
23static const struct data
24{
25 float64x2_t pi_over_2;
26 float64x2_t poly[20];
27} data = {
28 /* Coefficients of polynomial P such that atan(x)~x+x*P(x^2) on
29 the interval [2**-1022, 1.0]. */
30 .poly = { V2 (-0x1.5555555555555p-2), V2 (0x1.99999999996c1p-3),
31 V2 (-0x1.2492492478f88p-3), V2 (0x1.c71c71bc3951cp-4),
32 V2 (-0x1.745d160a7e368p-4), V2 (0x1.3b139b6a88ba1p-4),
33 V2 (-0x1.11100ee084227p-4), V2 (0x1.e1d0f9696f63bp-5),
34 V2 (-0x1.aebfe7b418581p-5), V2 (0x1.842dbe9b0d916p-5),
35 V2 (-0x1.5d30140ae5e99p-5), V2 (0x1.338e31eb2fbbcp-5),
36 V2 (-0x1.00e6eece7de8p-5), V2 (0x1.860897b29e5efp-6),
37 V2 (-0x1.0051381722a59p-6), V2 (0x1.14e9dc19a4a4ep-7),
38 V2 (-0x1.d0062b42fe3bfp-9), V2 (0x1.17739e210171ap-10),
39 V2 (-0x1.ab24da7be7402p-13), V2 (0x1.358851160a528p-16), },
40 .pi_over_2 = V2 (0x1.921fb54442d18p+0),
41};
42
43#define SignMask v_u64 (0x8000000000000000)
44
45/* Special cases i.e. 0, infinity, NaN (fall back to scalar calls). */
46static float64x2_t VPCS_ATTR NOINLINE
47special_case (float64x2_t y, float64x2_t x, float64x2_t ret, uint64x2_t cmp)
48{
49 return v_call2_f64 (atan2, y, x, ret, cmp);
50}
51
52/* Returns 1 if input is the bit representation of 0, infinity or nan. */
53static inline uint64x2_t
54zeroinfnan (uint64x2_t i)
55{
56 /* (2 * i - 1) >= (2 * asuint64 (INFINITY) - 1). */
57 return vcgeq_u64 (vsubq_u64 (vaddq_u64 (i, i), v_u64 (1)),
58 v_u64 (2 * asuint64 (INFINITY) - 1));
59}
60
61/* Fast implementation of vector atan2.
62 Maximum observed error is 2.8 ulps:
63 _ZGVnN2vv_atan2 (0x1.9651a429a859ap+5, 0x1.953075f4ee26p+5)
64 got 0x1.92d628ab678ccp-1
65 want 0x1.92d628ab678cfp-1. */
66float64x2_t VPCS_ATTR V_NAME_D2 (atan2) (float64x2_t y, float64x2_t x)
67{
68 const struct data *data_ptr = ptr_barrier (&data);
69
70 uint64x2_t ix = vreinterpretq_u64_f64 (x);
71 uint64x2_t iy = vreinterpretq_u64_f64 (y);
72
73 uint64x2_t special_cases = vorrq_u64 (zeroinfnan (ix), zeroinfnan (iy));
74
75 uint64x2_t sign_x = vandq_u64 (ix, SignMask);
76 uint64x2_t sign_y = vandq_u64 (iy, SignMask);
77 uint64x2_t sign_xy = veorq_u64 (sign_x, sign_y);
78
79 float64x2_t ax = vabsq_f64 (x);
80 float64x2_t ay = vabsq_f64 (y);
81
82 uint64x2_t pred_xlt0 = vcltzq_f64 (x);
83 uint64x2_t pred_aygtax = vcgtq_f64 (ay, ax);
84
85 /* Set up z for call to atan. */
86 float64x2_t n = vbslq_f64 (pred_aygtax, vnegq_f64 (ax), ay);
87 float64x2_t d = vbslq_f64 (pred_aygtax, ay, ax);
88 float64x2_t z = vdivq_f64 (n, d);
89
90 /* Work out the correct shift. */
91 float64x2_t shift = vreinterpretq_f64_u64 (
92 vandq_u64 (pred_xlt0, vreinterpretq_u64_f64 (v_f64 (-2.0))));
93 shift = vbslq_f64 (pred_aygtax, vaddq_f64 (shift, v_f64 (1.0)), shift);
94 shift = vmulq_f64 (shift, data_ptr->pi_over_2);
95
96 /* Calculate the polynomial approximation.
97 Use split Estrin scheme for P(z^2) with deg(P)=19. Use split instead of
98 full scheme to avoid underflow in x^16.
99 The order 19 polynomial P approximates
100 (atan(sqrt(x))-sqrt(x))/x^(3/2). */
101 float64x2_t z2 = vmulq_f64 (z, z);
102 float64x2_t x2 = vmulq_f64 (z2, z2);
103 float64x2_t x4 = vmulq_f64 (x2, x2);
104 float64x2_t x8 = vmulq_f64 (x4, x4);
105 float64x2_t ret
106 = vfmaq_f64 (v_estrin_7_f64 (z2, x2, x4, data_ptr->poly),
107 v_estrin_11_f64 (z2, x2, x4, x8, data_ptr->poly + 8), x8);
108
109 /* Finalize. y = shift + z + z^3 * P(z^2). */
110 ret = vfmaq_f64 (z, ret, vmulq_f64 (z2, z));
111 ret = vaddq_f64 (ret, shift);
112
113 /* Account for the sign of x and y. */
114 ret = vreinterpretq_f64_u64 (
115 veorq_u64 (vreinterpretq_u64_f64 (ret), sign_xy));
116
117 if (__glibc_unlikely (v_any_u64 (special_cases)))
118 return special_case (y, x, ret, special_cases);
119
120 return ret;
121}
122

source code of glibc/sysdeps/aarch64/fpu/atan2_advsimd.c