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