1/*
2 * Single-precision vector acos(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, pif;
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  .pif = V4 (0x1.921fb6p+1f),
24};
25
26#define AbsMask 0x7fffffff
27#define Half 0x3f000000
28#define One 0x3f800000
29#define Small 0x32800000 /* 2^-26.  */
30
31#if WANT_SIMD_EXCEPT
32static float32x4_t VPCS_ATTR NOINLINE
33special_case (float32x4_t x, float32x4_t y, uint32x4_t special)
34{
35  return v_call_f32 (acosf, x, y, special);
36}
37#endif
38
39/* Single-precision implementation of vector acos(x).
40
41   For |x| < Small, approximate acos(x) by pi/2 - x. Small = 2^-26 for correct
42   rounding.
43   If WANT_SIMD_EXCEPT = 0, Small = 0 and we proceed with the following
44   approximation.
45
46   For |x| in [Small, 0.5], use order 4 polynomial P such that the final
47   approximation of asin is an odd polynomial:
48
49     acos(x) ~ pi/2 - (x + x^3 P(x^2)).
50
51    The largest observed error in this region is 1.26 ulps,
52      _ZGVnN4v_acosf (0x1.843bfcp-2) got 0x1.2e934cp+0 want 0x1.2e934ap+0.
53
54    For |x| in [0.5, 1.0], use same approximation with a change of variable
55
56      acos(x) = y + y * z * P(z), with  z = (1-x)/2 and y = sqrt(z).
57
58   The largest observed error in this region is 1.32 ulps,
59   _ZGVnN4v_acosf (0x1.15ba56p-1) got 0x1.feb33p-1
60			   want 0x1.feb32ep-1.  */
61float32x4_t VPCS_ATTR V_NAME_F1 (acos) (float32x4_t x)
62{
63  const struct data *d = ptr_barrier (&data);
64
65  uint32x4_t ix = vreinterpretq_u32_f32 (x);
66  uint32x4_t ia = vandq_u32 (ix, v_u32 (AbsMask));
67
68#if WANT_SIMD_EXCEPT
69  /* A single comparison for One, Small and QNaN.  */
70  uint32x4_t special
71      = vcgtq_u32 (vsubq_u32 (ia, v_u32 (Small)), v_u32 (One - Small));
72  if (unlikely (v_any_u32 (special)))
73    return special_case (x, x, v_u32 (0xffffffff));
74#endif
75
76  float32x4_t ax = vreinterpretq_f32_u32 (ia);
77  uint32x4_t a_le_half = vcleq_u32 (ia, v_u32 (Half));
78
79  /* Evaluate polynomial Q(x) = z + z * z2 * P(z2) with
80     z2 = x ^ 2         and z = |x|     , if |x| < 0.5
81     z2 = (1 - |x|) / 2 and z = sqrt(z2), if |x| >= 0.5.  */
82  float32x4_t z2 = vbslq_f32 (a_le_half, vmulq_f32 (x, x),
83			      vfmsq_n_f32 (v_f32 (0.5), ax, 0.5));
84  float32x4_t z = vbslq_f32 (a_le_half, ax, vsqrtq_f32 (z2));
85
86  /* Use a single polynomial approximation P for both intervals.  */
87  float32x4_t p = v_horner_4_f32 (z2, d->poly);
88  /* Finalize polynomial: z + z * z2 * P(z2).  */
89  p = vfmaq_f32 (z, vmulq_f32 (z, z2), p);
90
91  /* acos(|x|) = pi/2 - sign(x) * Q(|x|), for  |x| < 0.5
92	       = 2 Q(|x|)               , for  0.5 < x < 1.0
93	       = pi - 2 Q(|x|)          , for -1.0 < x < -0.5.  */
94  float32x4_t y = vbslq_f32 (v_u32 (AbsMask), p, x);
95
96  uint32x4_t is_neg = vcltzq_f32 (x);
97  float32x4_t off = vreinterpretq_f32_u32 (
98      vandq_u32 (vreinterpretq_u32_f32 (d->pif), is_neg));
99  float32x4_t mul = vbslq_f32 (a_le_half, v_f32 (-1.0), v_f32 (2.0));
100  float32x4_t add = vbslq_f32 (a_le_half, d->pi_over_2f, off);
101
102  return vfmaq_f32 (add, mul, y);
103}
104
105PL_SIG (V, F, 1, acos, -1.0, 1.0)
106PL_TEST_ULP (V_NAME_F1 (acos), 0.82)
107PL_TEST_EXPECT_FENV (V_NAME_F1 (acos), WANT_SIMD_EXCEPT)
108PL_TEST_INTERVAL (V_NAME_F1 (acos), 0, 0x1p-26, 5000)
109PL_TEST_INTERVAL (V_NAME_F1 (acos), 0x1p-26, 0.5, 50000)
110PL_TEST_INTERVAL (V_NAME_F1 (acos), 0.5, 1.0, 50000)
111PL_TEST_INTERVAL (V_NAME_F1 (acos), 1.0, 0x1p11, 50000)
112PL_TEST_INTERVAL (V_NAME_F1 (acos), 0x1p11, inf, 20000)
113PL_TEST_INTERVAL (V_NAME_F1 (acos), -0, -inf, 20000)
114