1/* Single-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_f32.h"
22
23static const struct data
24{
25 float32x4_t poly[8];
26 float32x4_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 Generated using fpminimax between FLT_MIN and 1. */
31 .poly = { V4 (-0x1.55555p-2f), V4 (0x1.99935ep-3f), V4 (-0x1.24051ep-3f),
32 V4 (0x1.bd7368p-4f), V4 (-0x1.491f0ep-4f), V4 (0x1.93a2c0p-5f),
33 V4 (-0x1.4c3c60p-6f), V4 (0x1.01fd88p-8f) },
34 .pi_over_2 = V4 (0x1.921fb6p+0f),
35};
36
37#define SignMask v_u32 (0x80000000)
38
39#define P(i) d->poly[i]
40
41#define TinyBound 0x30800000 /* asuint(0x1p-30). */
42#define BigBound 0x4e800000 /* asuint(0x1p30). */
43
44#if WANT_SIMD_EXCEPT
45static float32x4_t VPCS_ATTR NOINLINE
46special_case (float32x4_t x, float32x4_t y, uint32x4_t special)
47{
48 return v_call_f32 (atanf, x, y, special);
49}
50#endif
51
52/* Fast implementation of vector atanf based on
53 atan(x) ~ shift + z + z^3 * P(z^2) with reduction to [0,1]
54 using z=-1/x and shift = pi/2. Maximum observed error is 2.9ulps:
55 _ZGVnN4v_atanf (0x1.0468f6p+0) got 0x1.967f06p-1 want 0x1.967fp-1. */
56float32x4_t VPCS_ATTR NOINLINE V_NAME_F1 (atan) (float32x4_t x)
57{
58 const struct data *d = ptr_barrier (&data);
59
60 /* Small cases, infs and nans are supported by our approximation technique,
61 but do not set fenv flags correctly. Only trigger special case if we need
62 fenv. */
63 uint32x4_t ix = vreinterpretq_u32_f32 (x);
64 uint32x4_t sign = vandq_u32 (ix, SignMask);
65
66#if WANT_SIMD_EXCEPT
67 uint32x4_t ia = vandq_u32 (ix, v_u32 (0x7ff00000));
68 uint32x4_t special = vcgtq_u32 (vsubq_u32 (ia, v_u32 (TinyBound)),
69 v_u32 (BigBound - TinyBound));
70 /* If any lane is special, fall back to the scalar routine for all lanes. */
71 if (__glibc_unlikely (v_any_u32 (special)))
72 return special_case (x, x, v_u32 (-1));
73#endif
74
75 /* Argument reduction:
76 y := arctan(x) for x < 1
77 y := pi/2 + arctan(-1/x) for x > 1
78 Hence, use z=-1/a if x>=1, otherwise z=a. */
79 uint32x4_t red = vcagtq_f32 (x, v_f32 (1.0));
80 /* Avoid dependency in abs(x) in division (and comparison). */
81 float32x4_t z = vbslq_f32 (red, vdivq_f32 (v_f32 (1.0f), x), x);
82 float32x4_t shift = vreinterpretq_f32_u32 (
83 vandq_u32 (red, vreinterpretq_u32_f32 (d->pi_over_2)));
84 /* Use absolute value only when needed (odd powers of z). */
85 float32x4_t az = vbslq_f32 (
86 SignMask, vreinterpretq_f32_u32 (vandq_u32 (SignMask, red)), z);
87
88 /* Calculate the polynomial approximation.
89 Use 2-level Estrin scheme for P(z^2) with deg(P)=7. However,
90 a standard implementation using z8 creates spurious underflow
91 in the very last fma (when z^8 is small enough).
92 Therefore, we split the last fma into a mul and an fma.
93 Horner and single-level Estrin have higher errors that exceed
94 threshold. */
95 float32x4_t z2 = vmulq_f32 (z, z);
96 float32x4_t z4 = vmulq_f32 (z2, z2);
97
98 float32x4_t y = vfmaq_f32 (
99 v_pairwise_poly_3_f32 (z2, z4, d->poly), z4,
100 vmulq_f32 (z4, v_pairwise_poly_3_f32 (z2, z4, d->poly + 4)));
101
102 /* y = shift + z * P(z^2). */
103 y = vaddq_f32 (vfmaq_f32 (az, y, vmulq_f32 (z2, az)), shift);
104
105 /* y = atan(x) if x>0, -atan(-x) otherwise. */
106 y = vreinterpretq_f32_u32 (veorq_u32 (vreinterpretq_u32_f32 (y), sign));
107
108 return y;
109}
110libmvec_hidden_def (V_NAME_F1 (atan))
111HALF_WIDTH_ALIAS_F1 (atan)
112

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