1/*
2 * Single-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_f32.h"
10#include "pl_sig.h"
11#include "pl_test.h"
12
13static 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
31static float32x4_t VPCS_ATTR NOINLINE
32special_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.  */
56float32x4_t VPCS_ATTR 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
96PL_SIG (V, F, 1, asin, -1.0, 1.0)
97PL_TEST_ULP (V_NAME_F1 (asin), 1.91)
98PL_TEST_EXPECT_FENV (V_NAME_F1 (asin), WANT_SIMD_EXCEPT)
99PL_TEST_INTERVAL (V_NAME_F1 (asin), 0, 0x1p-12, 5000)
100PL_TEST_INTERVAL (V_NAME_F1 (asin), 0x1p-12, 0.5, 50000)
101PL_TEST_INTERVAL (V_NAME_F1 (asin), 0.5, 1.0, 50000)
102PL_TEST_INTERVAL (V_NAME_F1 (asin), 1.0, 0x1p11, 50000)
103PL_TEST_INTERVAL (V_NAME_F1 (asin), 0x1p11, inf, 20000)
104PL_TEST_INTERVAL (V_NAME_F1 (asin), -0, -inf, 20000)
105