1*f3087befSAndrew Turner /*
2*f3087befSAndrew Turner * Single-precision vector acos(x) function.
3*f3087befSAndrew Turner *
4*f3087befSAndrew Turner * Copyright (c) 2023-2024, Arm Limited.
5*f3087befSAndrew Turner * SPDX-License-Identifier: MIT OR Apache-2.0 WITH LLVM-exception
6*f3087befSAndrew Turner */
7*f3087befSAndrew Turner
8*f3087befSAndrew Turner #include "v_math.h"
9*f3087befSAndrew Turner #include "v_poly_f32.h"
10*f3087befSAndrew Turner #include "test_sig.h"
11*f3087befSAndrew Turner #include "test_defs.h"
12*f3087befSAndrew Turner
13*f3087befSAndrew Turner static const struct data
14*f3087befSAndrew Turner {
15*f3087befSAndrew Turner float32x4_t poly[5];
16*f3087befSAndrew Turner float32x4_t pi_over_2f, pif;
17*f3087befSAndrew Turner } data = {
18*f3087befSAndrew Turner /* Polynomial approximation of (asin(sqrt(x)) - sqrt(x)) / (x * sqrt(x)) on
19*f3087befSAndrew Turner [ 0x1p-24 0x1p-2 ] order = 4 rel error: 0x1.00a23bbp-29 . */
20*f3087befSAndrew Turner .poly = { V4 (0x1.55555ep-3), V4 (0x1.33261ap-4), V4 (0x1.70d7dcp-5),
21*f3087befSAndrew Turner V4 (0x1.b059dp-6), V4 (0x1.3af7d8p-5) },
22*f3087befSAndrew Turner .pi_over_2f = V4 (0x1.921fb6p+0f),
23*f3087befSAndrew Turner .pif = V4 (0x1.921fb6p+1f),
24*f3087befSAndrew Turner };
25*f3087befSAndrew Turner
26*f3087befSAndrew Turner #define AbsMask 0x7fffffff
27*f3087befSAndrew Turner #define Half 0x3f000000
28*f3087befSAndrew Turner #define One 0x3f800000
29*f3087befSAndrew Turner #define Small 0x32800000 /* 2^-26. */
30*f3087befSAndrew Turner
31*f3087befSAndrew Turner #if WANT_SIMD_EXCEPT
32*f3087befSAndrew Turner static float32x4_t VPCS_ATTR NOINLINE
special_case(float32x4_t x,float32x4_t y,uint32x4_t special)33*f3087befSAndrew Turner special_case (float32x4_t x, float32x4_t y, uint32x4_t special)
34*f3087befSAndrew Turner {
35*f3087befSAndrew Turner return v_call_f32 (acosf, x, y, special);
36*f3087befSAndrew Turner }
37*f3087befSAndrew Turner #endif
38*f3087befSAndrew Turner
39*f3087befSAndrew Turner /* Single-precision implementation of vector acos(x).
40*f3087befSAndrew Turner
41*f3087befSAndrew Turner For |x| < Small, approximate acos(x) by pi/2 - x. Small = 2^-26 for correct
42*f3087befSAndrew Turner rounding.
43*f3087befSAndrew Turner If WANT_SIMD_EXCEPT = 0, Small = 0 and we proceed with the following
44*f3087befSAndrew Turner approximation.
45*f3087befSAndrew Turner
46*f3087befSAndrew Turner For |x| in [Small, 0.5], use order 4 polynomial P such that the final
47*f3087befSAndrew Turner approximation of asin is an odd polynomial:
48*f3087befSAndrew Turner
49*f3087befSAndrew Turner acos(x) ~ pi/2 - (x + x^3 P(x^2)).
50*f3087befSAndrew Turner
51*f3087befSAndrew Turner The largest observed error in this region is 1.26 ulps,
52*f3087befSAndrew Turner _ZGVnN4v_acosf (0x1.843bfcp-2) got 0x1.2e934cp+0 want 0x1.2e934ap+0.
53*f3087befSAndrew Turner
54*f3087befSAndrew Turner For |x| in [0.5, 1.0], use same approximation with a change of variable
55*f3087befSAndrew Turner
56*f3087befSAndrew Turner acos(x) = y + y * z * P(z), with z = (1-x)/2 and y = sqrt(z).
57*f3087befSAndrew Turner
58*f3087befSAndrew Turner The largest observed error in this region is 1.32 ulps,
59*f3087befSAndrew Turner _ZGVnN4v_acosf (0x1.15ba56p-1) got 0x1.feb33p-1
60*f3087befSAndrew Turner want 0x1.feb32ep-1. */
V_NAME_F1(acos)61*f3087befSAndrew Turner float32x4_t VPCS_ATTR NOINLINE V_NAME_F1 (acos) (float32x4_t x)
62*f3087befSAndrew Turner {
63*f3087befSAndrew Turner const struct data *d = ptr_barrier (&data);
64*f3087befSAndrew Turner
65*f3087befSAndrew Turner uint32x4_t ix = vreinterpretq_u32_f32 (x);
66*f3087befSAndrew Turner uint32x4_t ia = vandq_u32 (ix, v_u32 (AbsMask));
67*f3087befSAndrew Turner
68*f3087befSAndrew Turner #if WANT_SIMD_EXCEPT
69*f3087befSAndrew Turner /* A single comparison for One, Small and QNaN. */
70*f3087befSAndrew Turner uint32x4_t special
71*f3087befSAndrew Turner = vcgtq_u32 (vsubq_u32 (ia, v_u32 (Small)), v_u32 (One - Small));
72*f3087befSAndrew Turner if (unlikely (v_any_u32 (special)))
73*f3087befSAndrew Turner return special_case (x, x, v_u32 (0xffffffff));
74*f3087befSAndrew Turner #endif
75*f3087befSAndrew Turner
76*f3087befSAndrew Turner float32x4_t ax = vreinterpretq_f32_u32 (ia);
77*f3087befSAndrew Turner uint32x4_t a_le_half = vcleq_u32 (ia, v_u32 (Half));
78*f3087befSAndrew Turner
79*f3087befSAndrew Turner /* Evaluate polynomial Q(x) = z + z * z2 * P(z2) with
80*f3087befSAndrew Turner z2 = x ^ 2 and z = |x| , if |x| < 0.5
81*f3087befSAndrew Turner z2 = (1 - |x|) / 2 and z = sqrt(z2), if |x| >= 0.5. */
82*f3087befSAndrew Turner float32x4_t z2 = vbslq_f32 (a_le_half, vmulq_f32 (x, x),
83*f3087befSAndrew Turner vfmsq_n_f32 (v_f32 (0.5), ax, 0.5));
84*f3087befSAndrew Turner float32x4_t z = vbslq_f32 (a_le_half, ax, vsqrtq_f32 (z2));
85*f3087befSAndrew Turner
86*f3087befSAndrew Turner /* Use a single polynomial approximation P for both intervals. */
87*f3087befSAndrew Turner float32x4_t p = v_horner_4_f32 (z2, d->poly);
88*f3087befSAndrew Turner /* Finalize polynomial: z + z * z2 * P(z2). */
89*f3087befSAndrew Turner p = vfmaq_f32 (z, vmulq_f32 (z, z2), p);
90*f3087befSAndrew Turner
91*f3087befSAndrew Turner /* acos(|x|) = pi/2 - sign(x) * Q(|x|), for |x| < 0.5
92*f3087befSAndrew Turner = 2 Q(|x|) , for 0.5 < x < 1.0
93*f3087befSAndrew Turner = pi - 2 Q(|x|) , for -1.0 < x < -0.5. */
94*f3087befSAndrew Turner float32x4_t y = vbslq_f32 (v_u32 (AbsMask), p, x);
95*f3087befSAndrew Turner
96*f3087befSAndrew Turner uint32x4_t is_neg = vcltzq_f32 (x);
97*f3087befSAndrew Turner float32x4_t off = vreinterpretq_f32_u32 (
98*f3087befSAndrew Turner vandq_u32 (vreinterpretq_u32_f32 (d->pif), is_neg));
99*f3087befSAndrew Turner float32x4_t mul = vbslq_f32 (a_le_half, v_f32 (-1.0), v_f32 (2.0));
100*f3087befSAndrew Turner float32x4_t add = vbslq_f32 (a_le_half, d->pi_over_2f, off);
101*f3087befSAndrew Turner
102*f3087befSAndrew Turner return vfmaq_f32 (add, mul, y);
103*f3087befSAndrew Turner }
104*f3087befSAndrew Turner
105*f3087befSAndrew Turner HALF_WIDTH_ALIAS_F1 (acos)
106*f3087befSAndrew Turner
107*f3087befSAndrew Turner TEST_SIG (V, F, 1, acos, -1.0, 1.0)
108*f3087befSAndrew Turner TEST_ULP (V_NAME_F1 (acos), 0.82)
109*f3087befSAndrew Turner TEST_DISABLE_FENV_IF_NOT (V_NAME_F1 (acos), WANT_SIMD_EXCEPT)
110*f3087befSAndrew Turner TEST_INTERVAL (V_NAME_F1 (acos), 0, 0x1p-26, 5000)
111*f3087befSAndrew Turner TEST_INTERVAL (V_NAME_F1 (acos), 0x1p-26, 0.5, 50000)
112*f3087befSAndrew Turner TEST_INTERVAL (V_NAME_F1 (acos), 0.5, 1.0, 50000)
113*f3087befSAndrew Turner TEST_INTERVAL (V_NAME_F1 (acos), 1.0, 0x1p11, 50000)
114*f3087befSAndrew Turner TEST_INTERVAL (V_NAME_F1 (acos), 0x1p11, inf, 20000)
115*f3087befSAndrew Turner TEST_INTERVAL (V_NAME_F1 (acos), -0, -inf, 20000)
116