1/* Single-precision SVE 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 "sv_math.h"
21#include "poly_sve_f32.h"
22
23static const struct data
24{
25 float32_t poly[8];
26 float32_t pi_over_2;
27} data = {
28 /* Coefficients of polynomial P such that atan(x)~x+x*P(x^2) on
29 [2**-128, 1.0]. */
30 .poly = { -0x1.55555p-2f, 0x1.99935ep-3f, -0x1.24051ep-3f, 0x1.bd7368p-4f,
31 -0x1.491f0ep-4f, 0x1.93a2c0p-5f, -0x1.4c3c60p-6f, 0x1.01fd88p-8f },
32 .pi_over_2 = 0x1.921fb6p+0f,
33};
34
35#define SignMask (0x80000000)
36
37/* Fast implementation of SVE atanf based on
38 atan(x) ~ shift + z + z^3 * P(z^2) with reduction to [0,1] using
39 z=-1/x and shift = pi/2.
40 Largest observed error is 2.9 ULP, close to +/-1.0:
41 _ZGVsMxv_atanf (0x1.0468f6p+0) got -0x1.967f06p-1
42 want -0x1.967fp-1. */
43svfloat32_t SV_NAME_F1 (atan) (svfloat32_t x, const svbool_t pg)
44{
45 const struct data *d = ptr_barrier (&data);
46
47 /* No need to trigger special case. Small cases, infs and nans
48 are supported by our approximation technique. */
49 svuint32_t ix = svreinterpret_u32 (x);
50 svuint32_t sign = svand_x (pg, ix, SignMask);
51
52 /* Argument reduction:
53 y := arctan(x) for x < 1
54 y := pi/2 + arctan(-1/x) for x > 1
55 Hence, use z=-1/a if x>=1, otherwise z=a. */
56 svbool_t red = svacgt (pg, x, 1.0f);
57 /* Avoid dependency in abs(x) in division (and comparison). */
58 svfloat32_t z = svsel (red, svdiv_x (pg, sv_f32 (x: 1.0f), x), x);
59 /* Use absolute value only when needed (odd powers of z). */
60 svfloat32_t az = svabs_x (pg, z);
61 az = svneg_m (az, red, az);
62
63 /* Use split Estrin scheme for P(z^2) with deg(P)=7. */
64 svfloat32_t z2 = svmul_x (pg, z, z);
65 svfloat32_t z4 = svmul_x (pg, z2, z2);
66 svfloat32_t z8 = svmul_x (pg, z4, z4);
67
68 svfloat32_t y = sv_estrin_7_f32_x (pg, x: z2, x2: z4, x4: z8, poly: d->poly);
69
70 /* y = shift + z + z^3 * P(z^2). */
71 svfloat32_t z3 = svmul_x (pg, z2, az);
72 y = svmla_x (pg, az, z3, y);
73
74 /* Apply shift as indicated by 'red' predicate. */
75 y = svadd_m (red, y, sv_f32 (x: d->pi_over_2));
76
77 /* y = atan(x) if x>0, -atan(-x) otherwise. */
78 return svreinterpret_f32 (sveor_x (pg, svreinterpret_u32 (y), sign));
79}
80

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