1*412f47f9SXin Li /*
2*412f47f9SXin Li * Double-precision SVE asin(x) function.
3*412f47f9SXin Li *
4*412f47f9SXin Li * Copyright (c) 2023-2024, 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_f64.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 float64_t poly[12];
16*412f47f9SXin Li float64_t pi_over_2f;
17*412f47f9SXin Li } data = {
18*412f47f9SXin Li /* Polynomial approximation of (asin(sqrt(x)) - sqrt(x)) / (x * sqrt(x))
19*412f47f9SXin Li on [ 0x1p-106, 0x1p-2 ], relative error: 0x1.c3d8e169p-57. */
20*412f47f9SXin Li .poly = { 0x1.555555555554ep-3, 0x1.3333333337233p-4,
21*412f47f9SXin Li 0x1.6db6db67f6d9fp-5, 0x1.f1c71fbd29fbbp-6,
22*412f47f9SXin Li 0x1.6e8b264d467d6p-6, 0x1.1c5997c357e9dp-6,
23*412f47f9SXin Li 0x1.c86a22cd9389dp-7, 0x1.856073c22ebbep-7,
24*412f47f9SXin Li 0x1.fd1151acb6bedp-8, 0x1.087182f799c1dp-6,
25*412f47f9SXin Li -0x1.6602748120927p-7, 0x1.cfa0dd1f9478p-6, },
26*412f47f9SXin Li .pi_over_2f = 0x1.921fb54442d18p+0,
27*412f47f9SXin Li };
28*412f47f9SXin Li
29*412f47f9SXin Li #define P(i) sv_f64 (d->poly[i])
30*412f47f9SXin Li
31*412f47f9SXin Li /* Double-precision SVE implementation of vector asin(x).
32*412f47f9SXin Li
33*412f47f9SXin Li For |x| in [0, 0.5], use an order 11 polynomial P such that the final
34*412f47f9SXin Li approximation is an odd polynomial: asin(x) ~ x + x^3 P(x^2).
35*412f47f9SXin Li
36*412f47f9SXin Li The largest observed error in this region is 0.52 ulps,
37*412f47f9SXin Li _ZGVsMxv_asin(0x1.d95ae04998b6cp-2) got 0x1.ec13757305f27p-2
38*412f47f9SXin Li want 0x1.ec13757305f26p-2.
39*412f47f9SXin Li
40*412f47f9SXin Li For |x| in [0.5, 1.0], use same approximation with a change of variable
41*412f47f9SXin Li
42*412f47f9SXin Li asin(x) = pi/2 - (y + y * z * P(z)), with z = (1-x)/2 and y = sqrt(z).
43*412f47f9SXin Li
44*412f47f9SXin Li The largest observed error in this region is 2.69 ulps,
45*412f47f9SXin Li _ZGVsMxv_asin (0x1.044e8cefee301p-1) got 0x1.1111dd54ddf96p-1
46*412f47f9SXin Li want 0x1.1111dd54ddf99p-1. */
SV_NAME_D1(asin)47*412f47f9SXin Li svfloat64_t SV_NAME_D1 (asin) (svfloat64_t x, const svbool_t pg)
48*412f47f9SXin Li {
49*412f47f9SXin Li const struct data *d = ptr_barrier (&data);
50*412f47f9SXin Li
51*412f47f9SXin Li svuint64_t sign = svand_x (pg, svreinterpret_u64 (x), 0x8000000000000000);
52*412f47f9SXin Li svfloat64_t ax = svabs_x (pg, x);
53*412f47f9SXin Li svbool_t a_ge_half = svacge (pg, x, 0.5);
54*412f47f9SXin Li
55*412f47f9SXin Li /* Evaluate polynomial Q(x) = y + y * z * P(z) with
56*412f47f9SXin Li z = x ^ 2 and y = |x| , if |x| < 0.5
57*412f47f9SXin Li z = (1 - |x|) / 2 and y = sqrt(z), if |x| >= 0.5. */
58*412f47f9SXin Li svfloat64_t z2 = svsel (a_ge_half, svmls_x (pg, sv_f64 (0.5), ax, 0.5),
59*412f47f9SXin Li svmul_x (pg, x, x));
60*412f47f9SXin Li svfloat64_t z = svsqrt_m (ax, a_ge_half, z2);
61*412f47f9SXin Li
62*412f47f9SXin Li /* Use a single polynomial approximation P for both intervals. */
63*412f47f9SXin Li svfloat64_t z4 = svmul_x (pg, z2, z2);
64*412f47f9SXin Li svfloat64_t z8 = svmul_x (pg, z4, z4);
65*412f47f9SXin Li svfloat64_t z16 = svmul_x (pg, z8, z8);
66*412f47f9SXin Li svfloat64_t p = sv_estrin_11_f64_x (pg, z2, z4, z8, z16, d->poly);
67*412f47f9SXin Li /* Finalize polynomial: z + z * z2 * P(z2). */
68*412f47f9SXin Li p = svmla_x (pg, z, svmul_x (pg, z, z2), p);
69*412f47f9SXin Li
70*412f47f9SXin Li /* asin(|x|) = Q(|x|) , for |x| < 0.5
71*412f47f9SXin Li = pi/2 - 2 Q(|x|), for |x| >= 0.5. */
72*412f47f9SXin Li svfloat64_t y = svmad_m (a_ge_half, p, sv_f64 (-2.0), d->pi_over_2f);
73*412f47f9SXin Li
74*412f47f9SXin Li /* Copy sign. */
75*412f47f9SXin Li return svreinterpret_f64 (svorr_x (pg, svreinterpret_u64 (y), sign));
76*412f47f9SXin Li }
77*412f47f9SXin Li
78*412f47f9SXin Li PL_SIG (SV, D, 1, asin, -1.0, 1.0)
79*412f47f9SXin Li PL_TEST_ULP (SV_NAME_D1 (asin), 2.20)
80*412f47f9SXin Li PL_TEST_INTERVAL (SV_NAME_D1 (asin), 0, 0.5, 50000)
81*412f47f9SXin Li PL_TEST_INTERVAL (SV_NAME_D1 (asin), 0.5, 1.0, 50000)
82*412f47f9SXin Li PL_TEST_INTERVAL (SV_NAME_D1 (asin), 1.0, 0x1p11, 50000)
83*412f47f9SXin Li PL_TEST_INTERVAL (SV_NAME_D1 (asin), 0x1p11, inf, 20000)
84*412f47f9SXin Li PL_TEST_INTERVAL (SV_NAME_D1 (asin), -0, -inf, 20000)
85