1 /* Double-precision SVE inverse cos
3 Copyright (C) 2023-2024 Free Software Foundation, Inc.
4 This file is part of the GNU C Library.
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.
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.
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/>. */
21 #include "poly_sve_f64.h"
23 static const struct data
26 float64_t pi
, pi_over_2
;
28 /* Polynomial approximation of (asin(sqrt(x)) - sqrt(x)) / (x * sqrt(x))
29 on [ 0x1p-106, 0x1p-2 ], relative error: 0x1.c3d8e169p-57. */
30 .poly
= { 0x1.555555555554ep
-3, 0x1.3333333337233p
-4, 0x1.6db6db67f6d9fp
-5,
31 0x1.f1c71fbd29fbbp
-6, 0x1.6e8b264d467d6p
-6, 0x1.1c5997c357e9dp
-6,
32 0x1.c86a22cd9389dp
-7, 0x1.856073c22ebbep
-7, 0x1.fd1151acb6bedp
-8,
33 0x1.087182f799c1dp
-6, -0x1.6602748120927p
-7, 0x1.cfa0dd1f9478p
-6, },
34 .pi
= 0x1.921fb54442d18p
+1,
35 .pi_over_2
= 0x1.921fb54442d18p
+0,
38 /* Double-precision SVE implementation of vector acos(x).
40 For |x| in [0, 0.5], use an order 11 polynomial P such that the final
41 approximation of asin is an odd polynomial:
43 acos(x) ~ pi/2 - (x + x^3 P(x^2)).
45 The largest observed error in this region is 1.18 ulps,
46 _ZGVsMxv_acos (0x1.fbc5fe28ee9e3p-2) got 0x1.0d4d0f55667f6p+0
47 want 0x1.0d4d0f55667f7p+0.
49 For |x| in [0.5, 1.0], use same approximation with a change of variable
51 acos(x) = y + y * z * P(z), with z = (1-x)/2 and y = sqrt(z).
53 The largest observed error in this region is 1.52 ulps,
54 _ZGVsMxv_acos (0x1.24024271a500ap-1) got 0x1.ed82df4243f0dp-1
55 want 0x1.ed82df4243f0bp-1. */
56 svfloat64_t
SV_NAME_D1 (acos
) (svfloat64_t x
, const svbool_t pg
)
58 const struct data
*d
= ptr_barrier (&data
);
60 svuint64_t sign
= svand_x (pg
, svreinterpret_u64 (x
), 0x8000000000000000);
61 svfloat64_t ax
= svabs_x (pg
, x
);
63 svbool_t a_gt_half
= svacgt (pg
, x
, 0.5);
65 /* Evaluate polynomial Q(x) = z + z * z2 * P(z2) with
66 z2 = x ^ 2 and z = |x| , if |x| < 0.5
67 z2 = (1 - |x|) / 2 and z = sqrt(z2), if |x| >= 0.5. */
68 svfloat64_t z2
= svsel (a_gt_half
, svmls_x (pg
, sv_f64 (0.5), ax
, 0.5),
70 svfloat64_t z
= svsqrt_m (ax
, a_gt_half
, z2
);
72 /* Use a single polynomial approximation P for both intervals. */
73 svfloat64_t z4
= svmul_x (pg
, z2
, z2
);
74 svfloat64_t z8
= svmul_x (pg
, z4
, z4
);
75 svfloat64_t z16
= svmul_x (pg
, z8
, z8
);
76 svfloat64_t p
= sv_estrin_11_f64_x (pg
, z2
, z4
, z8
, z16
, d
->poly
);
78 /* Finalize polynomial: z + z * z2 * P(z2). */
79 p
= svmla_x (pg
, z
, svmul_x (pg
, z
, z2
), p
);
81 /* acos(|x|) = pi/2 - sign(x) * Q(|x|), for |x| < 0.5
82 = 2 Q(|x|) , for 0.5 < x < 1.0
83 = pi - 2 Q(|x|) , for -1.0 < x < -0.5. */
85 = svreinterpret_f64 (svorr_x (pg
, svreinterpret_u64 (p
), sign
));
87 svbool_t is_neg
= svcmplt (pg
, x
, 0.0);
88 svfloat64_t off
= svdup_f64_z (is_neg
, d
->pi
);
89 svfloat64_t mul
= svsel (a_gt_half
, sv_f64 (2.0), sv_f64 (-1.0));
90 svfloat64_t add
= svsel (a_gt_half
, off
, sv_f64 (d
->pi_over_2
));
92 return svmla_x (pg
, add
, mul
, y
);