1*412f47f9SXin Li /*
2*412f47f9SXin Li * Single-precision SVE acos(x) function.
3*412f47f9SXin Li *
4*412f47f9SXin Li * Copyright (c) 2023, Arm Limited.
5*412f47f9SXin Li * SPDX-License-Identifier: MIT OR Apache-2.0 WITH LLVM-exception
6*412f47f9SXin Li */
7*412f47f9SXin Li
8*412f47f9SXin Li #include "sv_math.h"
9*412f47f9SXin Li #include "poly_sve_f32.h"
10*412f47f9SXin Li #include "pl_sig.h"
11*412f47f9SXin Li #include "pl_test.h"
12*412f47f9SXin Li
13*412f47f9SXin Li static const struct data
14*412f47f9SXin Li {
15*412f47f9SXin Li float32_t poly[5];
16*412f47f9SXin Li float32_t pi, pi_over_2;
17*412f47f9SXin Li } data = {
18*412f47f9SXin Li /* Polynomial approximation of (asin(sqrt(x)) - sqrt(x)) / (x * sqrt(x)) on
19*412f47f9SXin Li [ 0x1p-24 0x1p-2 ] order = 4 rel error: 0x1.00a23bbp-29 . */
20*412f47f9SXin Li .poly = { 0x1.55555ep-3, 0x1.33261ap-4, 0x1.70d7dcp-5, 0x1.b059dp-6,
21*412f47f9SXin Li 0x1.3af7d8p-5, },
22*412f47f9SXin Li .pi = 0x1.921fb6p+1f,
23*412f47f9SXin Li .pi_over_2 = 0x1.921fb6p+0f,
24*412f47f9SXin Li };
25*412f47f9SXin Li
26*412f47f9SXin Li /* Single-precision SVE implementation of vector acos(x).
27*412f47f9SXin Li
28*412f47f9SXin Li For |x| in [0, 0.5], use order 4 polynomial P such that the final
29*412f47f9SXin Li approximation of asin is an odd polynomial:
30*412f47f9SXin Li
31*412f47f9SXin Li acos(x) ~ pi/2 - (x + x^3 P(x^2)).
32*412f47f9SXin Li
33*412f47f9SXin Li The largest observed error in this region is 1.16 ulps,
34*412f47f9SXin Li _ZGVsMxv_acosf(0x1.ffbeccp-2) got 0x1.0c27f8p+0
35*412f47f9SXin Li want 0x1.0c27f6p+0.
36*412f47f9SXin Li
37*412f47f9SXin Li For |x| in [0.5, 1.0], use same approximation with a change of variable
38*412f47f9SXin Li
39*412f47f9SXin Li acos(x) = y + y * z * P(z), with z = (1-x)/2 and y = sqrt(z).
40*412f47f9SXin Li
41*412f47f9SXin Li The largest observed error in this region is 1.32 ulps,
42*412f47f9SXin Li _ZGVsMxv_acosf (0x1.15ba56p-1) got 0x1.feb33p-1
43*412f47f9SXin Li want 0x1.feb32ep-1. */
SV_NAME_F1(acos)44*412f47f9SXin Li svfloat32_t SV_NAME_F1 (acos) (svfloat32_t x, const svbool_t pg)
45*412f47f9SXin Li {
46*412f47f9SXin Li const struct data *d = ptr_barrier (&data);
47*412f47f9SXin Li
48*412f47f9SXin Li svuint32_t sign = svand_x (pg, svreinterpret_u32 (x), 0x80000000);
49*412f47f9SXin Li svfloat32_t ax = svabs_x (pg, x);
50*412f47f9SXin Li svbool_t a_gt_half = svacgt (pg, x, 0.5);
51*412f47f9SXin Li
52*412f47f9SXin Li /* Evaluate polynomial Q(x) = z + z * z2 * P(z2) with
53*412f47f9SXin Li z2 = x ^ 2 and z = |x| , if |x| < 0.5
54*412f47f9SXin Li z2 = (1 - |x|) / 2 and z = sqrt(z2), if |x| >= 0.5. */
55*412f47f9SXin Li svfloat32_t z2 = svsel (a_gt_half, svmls_x (pg, sv_f32 (0.5), ax, 0.5),
56*412f47f9SXin Li svmul_x (pg, x, x));
57*412f47f9SXin Li svfloat32_t z = svsqrt_m (ax, a_gt_half, z2);
58*412f47f9SXin Li
59*412f47f9SXin Li /* Use a single polynomial approximation P for both intervals. */
60*412f47f9SXin Li svfloat32_t p = sv_horner_4_f32_x (pg, z2, d->poly);
61*412f47f9SXin Li /* Finalize polynomial: z + z * z2 * P(z2). */
62*412f47f9SXin Li p = svmla_x (pg, z, svmul_x (pg, z, z2), p);
63*412f47f9SXin Li
64*412f47f9SXin Li /* acos(|x|) = pi/2 - sign(x) * Q(|x|), for |x| < 0.5
65*412f47f9SXin Li = 2 Q(|x|) , for 0.5 < x < 1.0
66*412f47f9SXin Li = pi - 2 Q(|x|) , for -1.0 < x < -0.5. */
67*412f47f9SXin Li svfloat32_t y
68*412f47f9SXin Li = svreinterpret_f32 (svorr_x (pg, svreinterpret_u32 (p), sign));
69*412f47f9SXin Li
70*412f47f9SXin Li svbool_t is_neg = svcmplt (pg, x, 0.0);
71*412f47f9SXin Li svfloat32_t off = svdup_f32_z (is_neg, d->pi);
72*412f47f9SXin Li svfloat32_t mul = svsel (a_gt_half, sv_f32 (2.0), sv_f32 (-1.0));
73*412f47f9SXin Li svfloat32_t add = svsel (a_gt_half, off, sv_f32 (d->pi_over_2));
74*412f47f9SXin Li
75*412f47f9SXin Li return svmla_x (pg, add, mul, y);
76*412f47f9SXin Li }
77*412f47f9SXin Li
78*412f47f9SXin Li PL_SIG (SV, F, 1, acos, -1.0, 1.0)
79*412f47f9SXin Li PL_TEST_ULP (SV_NAME_F1 (acos), 0.82)
80*412f47f9SXin Li PL_TEST_INTERVAL (SV_NAME_F1 (acos), 0, 0.5, 50000)
81*412f47f9SXin Li PL_TEST_INTERVAL (SV_NAME_F1 (acos), 0.5, 1.0, 50000)
82*412f47f9SXin Li PL_TEST_INTERVAL (SV_NAME_F1 (acos), 1.0, 0x1p11, 50000)
83*412f47f9SXin Li PL_TEST_INTERVAL (SV_NAME_F1 (acos), 0x1p11, inf, 20000)
84*412f47f9SXin Li PL_TEST_INTERVAL (SV_NAME_F1 (acos), -0, -inf, 20000)
85