1/* Double-precision AdvSIMD inverse tan
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 [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#define TinyBound 0x3e10000000000000 /* asuint64(0x1p-30). */
45#define BigBound 0x4340000000000000 /* asuint64(0x1p53). */
46
47/* Fast implementation of vector atan.
48 Based on atan(x) ~ shift + z + z^3 * P(z^2) with reduction to [0,1] using
49 z=1/x and shift = pi/2. Maximum observed error is 2.27 ulps:
50 _ZGVnN2v_atan (0x1.0005af27c23e9p+0) got 0x1.9225645bdd7c1p-1
51 want 0x1.9225645bdd7c3p-1. */
52float64x2_t VPCS_ATTR V_NAME_D1 (atan) (float64x2_t x)
53{
54 const struct data *d = ptr_barrier (&data);
55
56 /* Small cases, infs and nans are supported by our approximation technique,
57 but do not set fenv flags correctly. Only trigger special case if we need
58 fenv. */
59 uint64x2_t ix = vreinterpretq_u64_f64 (x);
60 uint64x2_t sign = vandq_u64 (ix, SignMask);
61
62#if WANT_SIMD_EXCEPT
63 uint64x2_t ia12 = vandq_u64 (ix, v_u64 (0x7ff0000000000000));
64 uint64x2_t special = vcgtq_u64 (vsubq_u64 (ia12, v_u64 (TinyBound)),
65 v_u64 (BigBound - TinyBound));
66 /* If any lane is special, fall back to the scalar routine for all lanes. */
67 if (__glibc_unlikely (v_any_u64 (special)))
68 return v_call_f64 (atan, x, v_f64 (0), v_u64 (-1));
69#endif
70
71 /* Argument reduction:
72 y := arctan(x) for x < 1
73 y := pi/2 + arctan(-1/x) for x > 1
74 Hence, use z=-1/a if x>=1, otherwise z=a. */
75 uint64x2_t red = vcagtq_f64 (x, v_f64 (1.0));
76 /* Avoid dependency in abs(x) in division (and comparison). */
77 float64x2_t z = vbslq_f64 (red, vdivq_f64 (v_f64 (1.0), x), x);
78 float64x2_t shift = vreinterpretq_f64_u64 (
79 vandq_u64 (red, vreinterpretq_u64_f64 (d->pi_over_2)));
80 /* Use absolute value only when needed (odd powers of z). */
81 float64x2_t az = vbslq_f64 (
82 SignMask, vreinterpretq_f64_u64 (vandq_u64 (SignMask, red)), z);
83
84 /* Calculate the polynomial approximation.
85 Use split Estrin scheme for P(z^2) with deg(P)=19. Use split instead of
86 full scheme to avoid underflow in x^16.
87 The order 19 polynomial P approximates
88 (atan(sqrt(x))-sqrt(x))/x^(3/2). */
89 float64x2_t z2 = vmulq_f64 (z, z);
90 float64x2_t x2 = vmulq_f64 (z2, z2);
91 float64x2_t x4 = vmulq_f64 (x2, x2);
92 float64x2_t x8 = vmulq_f64 (x4, x4);
93 float64x2_t y
94 = vfmaq_f64 (v_estrin_7_f64 (z2, x2, x4, d->poly),
95 v_estrin_11_f64 (z2, x2, x4, x8, d->poly + 8), x8);
96
97 /* Finalize. y = shift + z + z^3 * P(z^2). */
98 y = vfmaq_f64 (az, y, vmulq_f64 (z2, az));
99 y = vaddq_f64 (y, shift);
100
101 /* y = atan(x) if x>0, -atan(-x) otherwise. */
102 y = vreinterpretq_f64_u64 (veorq_u64 (vreinterpretq_u64_f64 (y), sign));
103 return y;
104}
105

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