1 /*
2 * Single-precision vector atan2(x) function.
3 *
4 * Copyright (c) 2021-2023, Arm Limited.
5 * SPDX-License-Identifier: MIT OR Apache-2.0 WITH LLVM-exception
6 */
7
8 #include "v_math.h"
9 #include "pl_sig.h"
10 #include "pl_test.h"
11 #include "poly_advsimd_f32.h"
12
13 static const struct data
14 {
15 float32x4_t poly[8];
16 float32x4_t pi_over_2;
17 } data = {
18 /* Coefficients of polynomial P such that atan(x)~x+x*P(x^2) on
19 [2**-128, 1.0].
20 Generated using fpminimax between FLT_MIN and 1. */
21 .poly = { V4 (-0x1.55555p-2f), V4 (0x1.99935ep-3f), V4 (-0x1.24051ep-3f),
22 V4 (0x1.bd7368p-4f), V4 (-0x1.491f0ep-4f), V4 (0x1.93a2c0p-5f),
23 V4 (-0x1.4c3c60p-6f), V4 (0x1.01fd88p-8f) },
24 .pi_over_2 = V4 (0x1.921fb6p+0f),
25 };
26
27 #define SignMask v_u32 (0x80000000)
28
29 /* Special cases i.e. 0, infinity and nan (fall back to scalar calls). */
30 static float32x4_t VPCS_ATTR NOINLINE
special_case(float32x4_t y,float32x4_t x,float32x4_t ret,uint32x4_t cmp)31 special_case (float32x4_t y, float32x4_t x, float32x4_t ret, uint32x4_t cmp)
32 {
33 return v_call2_f32 (atan2f, y, x, ret, cmp);
34 }
35
36 /* Returns 1 if input is the bit representation of 0, infinity or nan. */
37 static inline uint32x4_t
zeroinfnan(uint32x4_t i)38 zeroinfnan (uint32x4_t i)
39 {
40 /* 2 * i - 1 >= 2 * 0x7f800000lu - 1. */
41 return vcgeq_u32 (vsubq_u32 (vmulq_n_u32 (i, 2), v_u32 (1)),
42 v_u32 (2 * 0x7f800000lu - 1));
43 }
44
45 /* Fast implementation of vector atan2f. Maximum observed error is
46 2.95 ULP in [0x1.9300d6p+6 0x1.93c0c6p+6] x [0x1.8c2dbp+6 0x1.8cea6p+6]:
47 _ZGVnN4vv_atan2f (0x1.93836cp+6, 0x1.8cae1p+6) got 0x1.967f06p-1
48 want 0x1.967f00p-1. */
V_NAME_F2(atan2)49 float32x4_t VPCS_ATTR V_NAME_F2 (atan2) (float32x4_t y, float32x4_t x)
50 {
51 const struct data *data_ptr = ptr_barrier (&data);
52
53 uint32x4_t ix = vreinterpretq_u32_f32 (x);
54 uint32x4_t iy = vreinterpretq_u32_f32 (y);
55
56 uint32x4_t special_cases = vorrq_u32 (zeroinfnan (ix), zeroinfnan (iy));
57
58 uint32x4_t sign_x = vandq_u32 (ix, SignMask);
59 uint32x4_t sign_y = vandq_u32 (iy, SignMask);
60 uint32x4_t sign_xy = veorq_u32 (sign_x, sign_y);
61
62 float32x4_t ax = vabsq_f32 (x);
63 float32x4_t ay = vabsq_f32 (y);
64
65 uint32x4_t pred_xlt0 = vcltzq_f32 (x);
66 uint32x4_t pred_aygtax = vcgtq_f32 (ay, ax);
67
68 /* Set up z for call to atanf. */
69 float32x4_t n = vbslq_f32 (pred_aygtax, vnegq_f32 (ax), ay);
70 float32x4_t d = vbslq_f32 (pred_aygtax, ay, ax);
71 float32x4_t z = vdivq_f32 (n, d);
72
73 /* Work out the correct shift. */
74 float32x4_t shift = vreinterpretq_f32_u32 (
75 vandq_u32 (pred_xlt0, vreinterpretq_u32_f32 (v_f32 (-2.0f))));
76 shift = vbslq_f32 (pred_aygtax, vaddq_f32 (shift, v_f32 (1.0f)), shift);
77 shift = vmulq_f32 (shift, data_ptr->pi_over_2);
78
79 /* Calculate the polynomial approximation.
80 Use 2-level Estrin scheme for P(z^2) with deg(P)=7. However,
81 a standard implementation using z8 creates spurious underflow
82 in the very last fma (when z^8 is small enough).
83 Therefore, we split the last fma into a mul and an fma.
84 Horner and single-level Estrin have higher errors that exceed
85 threshold. */
86 float32x4_t z2 = vmulq_f32 (z, z);
87 float32x4_t z4 = vmulq_f32 (z2, z2);
88
89 float32x4_t ret = vfmaq_f32 (
90 v_pairwise_poly_3_f32 (z2, z4, data_ptr->poly), z4,
91 vmulq_f32 (z4, v_pairwise_poly_3_f32 (z2, z4, data_ptr->poly + 4)));
92
93 /* y = shift + z * P(z^2). */
94 ret = vaddq_f32 (vfmaq_f32 (z, ret, vmulq_f32 (z2, z)), shift);
95
96 /* Account for the sign of y. */
97 ret = vreinterpretq_f32_u32 (
98 veorq_u32 (vreinterpretq_u32_f32 (ret), sign_xy));
99
100 if (unlikely (v_any_u32 (special_cases)))
101 {
102 return special_case (y, x, ret, special_cases);
103 }
104
105 return ret;
106 }
107
108 /* Arity of 2 means no mathbench entry emitted. See test/mathbench_funcs.h. */
109 PL_SIG (V, F, 2, atan2)
110 PL_TEST_ULP (V_NAME_F2 (atan2), 2.46)
111 PL_TEST_INTERVAL (V_NAME_F2 (atan2), -10.0, 10.0, 50000)
112 PL_TEST_INTERVAL (V_NAME_F2 (atan2), -1.0, 1.0, 40000)
113 PL_TEST_INTERVAL (V_NAME_F2 (atan2), 0.0, 1.0, 40000)
114 PL_TEST_INTERVAL (V_NAME_F2 (atan2), 1.0, 100.0, 40000)
115 PL_TEST_INTERVAL (V_NAME_F2 (atan2), 1e6, 1e32, 40000)
116