1*5a02ffc3SAndrew Turner /*
2*5a02ffc3SAndrew Turner * Double-precision vector asin(x) function.
3*5a02ffc3SAndrew Turner *
4*5a02ffc3SAndrew Turner * Copyright (c) 2023, Arm Limited.
5*5a02ffc3SAndrew Turner * SPDX-License-Identifier: MIT OR Apache-2.0 WITH LLVM-exception
6*5a02ffc3SAndrew Turner */
7*5a02ffc3SAndrew Turner
8*5a02ffc3SAndrew Turner #include "v_math.h"
9*5a02ffc3SAndrew Turner #include "poly_advsimd_f64.h"
10*5a02ffc3SAndrew Turner #include "pl_sig.h"
11*5a02ffc3SAndrew Turner #include "pl_test.h"
12*5a02ffc3SAndrew Turner
13*5a02ffc3SAndrew Turner static const struct data
14*5a02ffc3SAndrew Turner {
15*5a02ffc3SAndrew Turner float64x2_t poly[12];
16*5a02ffc3SAndrew Turner float64x2_t pi_over_2;
17*5a02ffc3SAndrew Turner uint64x2_t abs_mask;
18*5a02ffc3SAndrew Turner } data = {
19*5a02ffc3SAndrew Turner /* Polynomial approximation of (asin(sqrt(x)) - sqrt(x)) / (x * sqrt(x))
20*5a02ffc3SAndrew Turner on [ 0x1p-106, 0x1p-2 ], relative error: 0x1.c3d8e169p-57. */
21*5a02ffc3SAndrew Turner .poly = { V2 (0x1.555555555554ep-3), V2 (0x1.3333333337233p-4),
22*5a02ffc3SAndrew Turner V2 (0x1.6db6db67f6d9fp-5), V2 (0x1.f1c71fbd29fbbp-6),
23*5a02ffc3SAndrew Turner V2 (0x1.6e8b264d467d6p-6), V2 (0x1.1c5997c357e9dp-6),
24*5a02ffc3SAndrew Turner V2 (0x1.c86a22cd9389dp-7), V2 (0x1.856073c22ebbep-7),
25*5a02ffc3SAndrew Turner V2 (0x1.fd1151acb6bedp-8), V2 (0x1.087182f799c1dp-6),
26*5a02ffc3SAndrew Turner V2 (-0x1.6602748120927p-7), V2 (0x1.cfa0dd1f9478p-6), },
27*5a02ffc3SAndrew Turner .pi_over_2 = V2 (0x1.921fb54442d18p+0),
28*5a02ffc3SAndrew Turner .abs_mask = V2 (0x7fffffffffffffff),
29*5a02ffc3SAndrew Turner };
30*5a02ffc3SAndrew Turner
31*5a02ffc3SAndrew Turner #define AllMask v_u64 (0xffffffffffffffff)
32*5a02ffc3SAndrew Turner #define One (0x3ff0000000000000)
33*5a02ffc3SAndrew Turner #define Small (0x3e50000000000000) /* 2^-12. */
34*5a02ffc3SAndrew Turner
35*5a02ffc3SAndrew Turner #if WANT_SIMD_EXCEPT
36*5a02ffc3SAndrew Turner static float64x2_t VPCS_ATTR NOINLINE
special_case(float64x2_t x,float64x2_t y,uint64x2_t special)37*5a02ffc3SAndrew Turner special_case (float64x2_t x, float64x2_t y, uint64x2_t special)
38*5a02ffc3SAndrew Turner {
39*5a02ffc3SAndrew Turner return v_call_f64 (asin, x, y, special);
40*5a02ffc3SAndrew Turner }
41*5a02ffc3SAndrew Turner #endif
42*5a02ffc3SAndrew Turner
43*5a02ffc3SAndrew Turner /* Double-precision implementation of vector asin(x).
44*5a02ffc3SAndrew Turner
45*5a02ffc3SAndrew Turner For |x| < Small, approximate asin(x) by x. Small = 2^-12 for correct
46*5a02ffc3SAndrew Turner rounding. If WANT_SIMD_EXCEPT = 0, Small = 0 and we proceed with the
47*5a02ffc3SAndrew Turner following approximation.
48*5a02ffc3SAndrew Turner
49*5a02ffc3SAndrew Turner For |x| in [Small, 0.5], use an order 11 polynomial P such that the final
50*5a02ffc3SAndrew Turner approximation is an odd polynomial: asin(x) ~ x + x^3 P(x^2).
51*5a02ffc3SAndrew Turner
52*5a02ffc3SAndrew Turner The largest observed error in this region is 1.01 ulps,
53*5a02ffc3SAndrew Turner _ZGVnN2v_asin (0x1.da9735b5a9277p-2) got 0x1.ed78525a927efp-2
54*5a02ffc3SAndrew Turner want 0x1.ed78525a927eep-2.
55*5a02ffc3SAndrew Turner
56*5a02ffc3SAndrew Turner For |x| in [0.5, 1.0], use same approximation with a change of variable
57*5a02ffc3SAndrew Turner
58*5a02ffc3SAndrew Turner asin(x) = pi/2 - (y + y * z * P(z)), with z = (1-x)/2 and y = sqrt(z).
59*5a02ffc3SAndrew Turner
60*5a02ffc3SAndrew Turner The largest observed error in this region is 2.69 ulps,
61*5a02ffc3SAndrew Turner _ZGVnN2v_asin (0x1.044ac9819f573p-1) got 0x1.110d7e85fdd5p-1
62*5a02ffc3SAndrew Turner want 0x1.110d7e85fdd53p-1. */
V_NAME_D1(asin)63*5a02ffc3SAndrew Turner float64x2_t VPCS_ATTR V_NAME_D1 (asin) (float64x2_t x)
64*5a02ffc3SAndrew Turner {
65*5a02ffc3SAndrew Turner const struct data *d = ptr_barrier (&data);
66*5a02ffc3SAndrew Turner
67*5a02ffc3SAndrew Turner float64x2_t ax = vabsq_f64 (x);
68*5a02ffc3SAndrew Turner
69*5a02ffc3SAndrew Turner #if WANT_SIMD_EXCEPT
70*5a02ffc3SAndrew Turner /* Special values need to be computed with scalar fallbacks so
71*5a02ffc3SAndrew Turner that appropriate exceptions are raised. */
72*5a02ffc3SAndrew Turner uint64x2_t special
73*5a02ffc3SAndrew Turner = vcgtq_u64 (vsubq_u64 (vreinterpretq_u64_f64 (ax), v_u64 (Small)),
74*5a02ffc3SAndrew Turner v_u64 (One - Small));
75*5a02ffc3SAndrew Turner if (unlikely (v_any_u64 (special)))
76*5a02ffc3SAndrew Turner return special_case (x, x, AllMask);
77*5a02ffc3SAndrew Turner #endif
78*5a02ffc3SAndrew Turner
79*5a02ffc3SAndrew Turner uint64x2_t a_lt_half = vcltq_f64 (ax, v_f64 (0.5));
80*5a02ffc3SAndrew Turner
81*5a02ffc3SAndrew Turner /* Evaluate polynomial Q(x) = y + y * z * P(z) with
82*5a02ffc3SAndrew Turner z = x ^ 2 and y = |x| , if |x| < 0.5
83*5a02ffc3SAndrew Turner z = (1 - |x|) / 2 and y = sqrt(z), if |x| >= 0.5. */
84*5a02ffc3SAndrew Turner float64x2_t z2 = vbslq_f64 (a_lt_half, vmulq_f64 (x, x),
85*5a02ffc3SAndrew Turner vfmsq_n_f64 (v_f64 (0.5), ax, 0.5));
86*5a02ffc3SAndrew Turner float64x2_t z = vbslq_f64 (a_lt_half, ax, vsqrtq_f64 (z2));
87*5a02ffc3SAndrew Turner
88*5a02ffc3SAndrew Turner /* Use a single polynomial approximation P for both intervals. */
89*5a02ffc3SAndrew Turner float64x2_t z4 = vmulq_f64 (z2, z2);
90*5a02ffc3SAndrew Turner float64x2_t z8 = vmulq_f64 (z4, z4);
91*5a02ffc3SAndrew Turner float64x2_t z16 = vmulq_f64 (z8, z8);
92*5a02ffc3SAndrew Turner float64x2_t p = v_estrin_11_f64 (z2, z4, z8, z16, d->poly);
93*5a02ffc3SAndrew Turner
94*5a02ffc3SAndrew Turner /* Finalize polynomial: z + z * z2 * P(z2). */
95*5a02ffc3SAndrew Turner p = vfmaq_f64 (z, vmulq_f64 (z, z2), p);
96*5a02ffc3SAndrew Turner
97*5a02ffc3SAndrew Turner /* asin(|x|) = Q(|x|) , for |x| < 0.5
98*5a02ffc3SAndrew Turner = pi/2 - 2 Q(|x|), for |x| >= 0.5. */
99*5a02ffc3SAndrew Turner float64x2_t y = vbslq_f64 (a_lt_half, p, vfmsq_n_f64 (d->pi_over_2, p, 2.0));
100*5a02ffc3SAndrew Turner
101*5a02ffc3SAndrew Turner /* Copy sign. */
102*5a02ffc3SAndrew Turner return vbslq_f64 (d->abs_mask, y, x);
103*5a02ffc3SAndrew Turner }
104*5a02ffc3SAndrew Turner
105*5a02ffc3SAndrew Turner PL_SIG (V, D, 1, asin, -1.0, 1.0)
106*5a02ffc3SAndrew Turner PL_TEST_ULP (V_NAME_D1 (asin), 2.19)
107*5a02ffc3SAndrew Turner PL_TEST_EXPECT_FENV (V_NAME_D1 (asin), WANT_SIMD_EXCEPT)
108*5a02ffc3SAndrew Turner PL_TEST_INTERVAL (V_NAME_D1 (asin), 0, Small, 5000)
109*5a02ffc3SAndrew Turner PL_TEST_INTERVAL (V_NAME_D1 (asin), Small, 0.5, 50000)
110*5a02ffc3SAndrew Turner PL_TEST_INTERVAL (V_NAME_D1 (asin), 0.5, 1.0, 50000)
111*5a02ffc3SAndrew Turner PL_TEST_INTERVAL (V_NAME_D1 (asin), 1.0, 0x1p11, 50000)
112*5a02ffc3SAndrew Turner PL_TEST_INTERVAL (V_NAME_D1 (asin), 0x1p11, inf, 20000)
113*5a02ffc3SAndrew Turner PL_TEST_INTERVAL (V_NAME_D1 (asin), -0, -inf, 20000)
114