1 /*
2 * Single-precision vector asin(x) function.
3 *
4 * Copyright (c) 2023-2024, Arm Limited.
5 * SPDX-License-Identifier: MIT OR Apache-2.0 WITH LLVM-exception
6 */
7
8 #include "v_math.h"
9 #include "v_poly_f32.h"
10 #include "test_sig.h"
11 #include "test_defs.h"
12
13 static const struct data
14 {
15 float32x4_t poly[5];
16 float32x4_t pi_over_2f;
17 } data = {
18 /* Polynomial approximation of (asin(sqrt(x)) - sqrt(x)) / (x * sqrt(x)) on
19 [ 0x1p-24 0x1p-2 ] order = 4 rel error: 0x1.00a23bbp-29 . */
20 .poly = { V4 (0x1.55555ep-3), V4 (0x1.33261ap-4), V4 (0x1.70d7dcp-5),
21 V4 (0x1.b059dp-6), V4 (0x1.3af7d8p-5) },
22 .pi_over_2f = V4 (0x1.921fb6p+0f),
23 };
24
25 #define AbsMask 0x7fffffff
26 #define Half 0x3f000000
27 #define One 0x3f800000
28 #define Small 0x39800000 /* 2^-12. */
29
30 #if WANT_SIMD_EXCEPT
31 static float32x4_t VPCS_ATTR NOINLINE
special_case(float32x4_t x,float32x4_t y,uint32x4_t special)32 special_case (float32x4_t x, float32x4_t y, uint32x4_t special)
33 {
34 return v_call_f32 (asinf, x, y, special);
35 }
36 #endif
37
38 /* Single-precision implementation of vector asin(x).
39
40 For |x| < Small, approximate asin(x) by x. Small = 2^-12 for correct
41 rounding. If WANT_SIMD_EXCEPT = 0, Small = 0 and we proceed with the
42 following approximation.
43
44 For |x| in [Small, 0.5], use order 4 polynomial P such that the final
45 approximation is an odd polynomial: asin(x) ~ x + x^3 P(x^2).
46
47 The largest observed error in this region is 0.83 ulps,
48 _ZGVnN4v_asinf (0x1.ea00f4p-2) got 0x1.fef15ep-2 want 0x1.fef15cp-2.
49
50 For |x| in [0.5, 1.0], use same approximation with a change of variable
51
52 asin(x) = pi/2 - (y + y * z * P(z)), with z = (1-x)/2 and y = sqrt(z).
53
54 The largest observed error in this region is 2.41 ulps,
55 _ZGVnN4v_asinf (0x1.00203ep-1) got 0x1.0c3a64p-1 want 0x1.0c3a6p-1. */
V_NAME_F1(asin)56 float32x4_t VPCS_ATTR NOINLINE V_NAME_F1 (asin) (float32x4_t x)
57 {
58 const struct data *d = ptr_barrier (&data);
59
60 uint32x4_t ix = vreinterpretq_u32_f32 (x);
61 uint32x4_t ia = vandq_u32 (ix, v_u32 (AbsMask));
62
63 #if WANT_SIMD_EXCEPT
64 /* Special values need to be computed with scalar fallbacks so
65 that appropriate fp exceptions are raised. */
66 uint32x4_t special
67 = vcgtq_u32 (vsubq_u32 (ia, v_u32 (Small)), v_u32 (One - Small));
68 if (unlikely (v_any_u32 (special)))
69 return special_case (x, x, v_u32 (0xffffffff));
70 #endif
71
72 float32x4_t ax = vreinterpretq_f32_u32 (ia);
73 uint32x4_t a_lt_half = vcltq_u32 (ia, v_u32 (Half));
74
75 /* Evaluate polynomial Q(x) = y + y * z * P(z) with
76 z = x ^ 2 and y = |x| , if |x| < 0.5
77 z = (1 - |x|) / 2 and y = sqrt(z), if |x| >= 0.5. */
78 float32x4_t z2 = vbslq_f32 (a_lt_half, vmulq_f32 (x, x),
79 vfmsq_n_f32 (v_f32 (0.5), ax, 0.5));
80 float32x4_t z = vbslq_f32 (a_lt_half, ax, vsqrtq_f32 (z2));
81
82 /* Use a single polynomial approximation P for both intervals. */
83 float32x4_t p = v_horner_4_f32 (z2, d->poly);
84 /* Finalize polynomial: z + z * z2 * P(z2). */
85 p = vfmaq_f32 (z, vmulq_f32 (z, z2), p);
86
87 /* asin(|x|) = Q(|x|) , for |x| < 0.5
88 = pi/2 - 2 Q(|x|), for |x| >= 0.5. */
89 float32x4_t y
90 = vbslq_f32 (a_lt_half, p, vfmsq_n_f32 (d->pi_over_2f, p, 2.0));
91
92 /* Copy sign. */
93 return vbslq_f32 (v_u32 (AbsMask), y, x);
94 }
95
96 HALF_WIDTH_ALIAS_F1 (asin)
97
98 TEST_SIG (V, F, 1, asin, -1.0, 1.0)
99 TEST_ULP (V_NAME_F1 (asin), 1.91)
100 TEST_DISABLE_FENV_IF_NOT (V_NAME_F1 (asin), WANT_SIMD_EXCEPT)
101 TEST_INTERVAL (V_NAME_F1 (asin), 0, 0x1p-12, 5000)
102 TEST_INTERVAL (V_NAME_F1 (asin), 0x1p-12, 0.5, 50000)
103 TEST_INTERVAL (V_NAME_F1 (asin), 0.5, 1.0, 50000)
104 TEST_INTERVAL (V_NAME_F1 (asin), 1.0, 0x1p11, 50000)
105 TEST_INTERVAL (V_NAME_F1 (asin), 0x1p11, inf, 20000)
106 TEST_INTERVAL (V_NAME_F1 (asin), -0, -inf, 20000)
107