1/*
2 * Double-precision SVE acos(x) function.
3 *
4 * Copyright (c) 2023, Arm Limited.
5 * SPDX-License-Identifier: MIT OR Apache-2.0 WITH LLVM-exception
6 */
7
8#include "sv_math.h"
9#include "poly_sve_f64.h"
10#include "pl_sig.h"
11#include "pl_test.h"
12
13static const struct data
14{
15  float64_t poly[12];
16  float64_t pi, pi_over_2;
17} data = {
18  /* Polynomial approximation of  (asin(sqrt(x)) - sqrt(x)) / (x * sqrt(x))
19     on [ 0x1p-106, 0x1p-2 ], relative error: 0x1.c3d8e169p-57.  */
20  .poly = { 0x1.555555555554ep-3, 0x1.3333333337233p-4, 0x1.6db6db67f6d9fp-5,
21	    0x1.f1c71fbd29fbbp-6, 0x1.6e8b264d467d6p-6, 0x1.1c5997c357e9dp-6,
22	    0x1.c86a22cd9389dp-7, 0x1.856073c22ebbep-7, 0x1.fd1151acb6bedp-8,
23	    0x1.087182f799c1dp-6, -0x1.6602748120927p-7, 0x1.cfa0dd1f9478p-6, },
24  .pi = 0x1.921fb54442d18p+1,
25  .pi_over_2 = 0x1.921fb54442d18p+0,
26};
27
28/* Double-precision SVE implementation of vector acos(x).
29
30   For |x| in [0, 0.5], use an order 11 polynomial P such that the final
31   approximation of asin is an odd polynomial:
32
33     acos(x) ~ pi/2 - (x + x^3 P(x^2)).
34
35   The largest observed error in this region is 1.18 ulps,
36   _ZGVsMxv_acos (0x1.fbc5fe28ee9e3p-2) got 0x1.0d4d0f55667f6p+0
37				       want 0x1.0d4d0f55667f7p+0.
38
39   For |x| in [0.5, 1.0], use same approximation with a change of variable
40
41     acos(x) = y + y * z * P(z), with  z = (1-x)/2 and y = sqrt(z).
42
43   The largest observed error in this region is 1.52 ulps,
44   _ZGVsMxv_acos (0x1.24024271a500ap-1) got 0x1.ed82df4243f0dp-1
45				       want 0x1.ed82df4243f0bp-1.  */
46svfloat64_t SV_NAME_D1 (acos) (svfloat64_t x, const svbool_t pg)
47{
48  const struct data *d = ptr_barrier (&data);
49
50  svuint64_t sign = svand_x (pg, svreinterpret_u64 (x), 0x8000000000000000);
51  svfloat64_t ax = svabs_x (pg, x);
52
53  svbool_t a_gt_half = svacgt (pg, x, 0.5);
54
55  /* Evaluate polynomial Q(x) = z + z * z2 * P(z2) with
56     z2 = x ^ 2         and z = |x|     , if |x| < 0.5
57     z2 = (1 - |x|) / 2 and z = sqrt(z2), if |x| >= 0.5.  */
58  svfloat64_t z2 = svsel (a_gt_half, svmls_x (pg, sv_f64 (0.5), ax, 0.5),
59			  svmul_x (pg, x, x));
60  svfloat64_t z = svsqrt_m (ax, a_gt_half, z2);
61
62  /* Use a single polynomial approximation P for both intervals.  */
63  svfloat64_t z4 = svmul_x (pg, z2, z2);
64  svfloat64_t z8 = svmul_x (pg, z4, z4);
65  svfloat64_t z16 = svmul_x (pg, z8, z8);
66  svfloat64_t p = sv_estrin_11_f64_x (pg, z2, z4, z8, z16, d->poly);
67
68  /* Finalize polynomial: z + z * z2 * P(z2).  */
69  p = svmla_x (pg, z, svmul_x (pg, z, z2), p);
70
71  /* acos(|x|) = pi/2 - sign(x) * Q(|x|), for  |x| < 0.5
72	       = 2 Q(|x|)               , for  0.5 < x < 1.0
73	       = pi - 2 Q(|x|)          , for -1.0 < x < -0.5.  */
74  svfloat64_t y
75      = svreinterpret_f64 (svorr_x (pg, svreinterpret_u64 (p), sign));
76
77  svbool_t is_neg = svcmplt (pg, x, 0.0);
78  svfloat64_t off = svdup_f64_z (is_neg, d->pi);
79  svfloat64_t mul = svsel (a_gt_half, sv_f64 (2.0), sv_f64 (-1.0));
80  svfloat64_t add = svsel (a_gt_half, off, sv_f64 (d->pi_over_2));
81
82  return svmla_x (pg, add, mul, y);
83}
84
85PL_SIG (SV, D, 1, acos, -1.0, 1.0)
86PL_TEST_ULP (SV_NAME_D1 (acos), 1.02)
87PL_TEST_INTERVAL (SV_NAME_D1 (acos), 0, 0.5, 50000)
88PL_TEST_INTERVAL (SV_NAME_D1 (acos), 0.5, 1.0, 50000)
89PL_TEST_INTERVAL (SV_NAME_D1 (acos), 1.0, 0x1p11, 50000)
90PL_TEST_INTERVAL (SV_NAME_D1 (acos), 0x1p11, inf, 20000)
91PL_TEST_INTERVAL (SV_NAME_D1 (acos), -0, -inf, 20000)
92