1 /*
2 * Double-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_f64.h"
12
13 static const struct data
14 {
15 float64x2_t pi_over_2;
16 float64x2_t poly[20];
17 } data = {
18 /* Coefficients of polynomial P such that atan(x)~x+x*P(x^2) on
19 the interval [2**-1022, 1.0]. */
20 .poly = { V2 (-0x1.5555555555555p-2), V2 (0x1.99999999996c1p-3),
21 V2 (-0x1.2492492478f88p-3), V2 (0x1.c71c71bc3951cp-4),
22 V2 (-0x1.745d160a7e368p-4), V2 (0x1.3b139b6a88ba1p-4),
23 V2 (-0x1.11100ee084227p-4), V2 (0x1.e1d0f9696f63bp-5),
24 V2 (-0x1.aebfe7b418581p-5), V2 (0x1.842dbe9b0d916p-5),
25 V2 (-0x1.5d30140ae5e99p-5), V2 (0x1.338e31eb2fbbcp-5),
26 V2 (-0x1.00e6eece7de8p-5), V2 (0x1.860897b29e5efp-6),
27 V2 (-0x1.0051381722a59p-6), V2 (0x1.14e9dc19a4a4ep-7),
28 V2 (-0x1.d0062b42fe3bfp-9), V2 (0x1.17739e210171ap-10),
29 V2 (-0x1.ab24da7be7402p-13), V2 (0x1.358851160a528p-16), },
30 .pi_over_2 = V2 (0x1.921fb54442d18p+0),
31 };
32
33 #define SignMask v_u64 (0x8000000000000000)
34
35 /* Special cases i.e. 0, infinity, NaN (fall back to scalar calls). */
36 static float64x2_t VPCS_ATTR NOINLINE
special_case(float64x2_t y,float64x2_t x,float64x2_t ret,uint64x2_t cmp)37 special_case (float64x2_t y, float64x2_t x, float64x2_t ret, uint64x2_t cmp)
38 {
39 return v_call2_f64 (atan2, y, x, ret, cmp);
40 }
41
42 /* Returns 1 if input is the bit representation of 0, infinity or nan. */
43 static inline uint64x2_t
zeroinfnan(uint64x2_t i)44 zeroinfnan (uint64x2_t i)
45 {
46 /* (2 * i - 1) >= (2 * asuint64 (INFINITY) - 1). */
47 return vcgeq_u64 (vsubq_u64 (vaddq_u64 (i, i), v_u64 (1)),
48 v_u64 (2 * asuint64 (INFINITY) - 1));
49 }
50
51 /* Fast implementation of vector atan2.
52 Maximum observed error is 2.8 ulps:
53 _ZGVnN2vv_atan2 (0x1.9651a429a859ap+5, 0x1.953075f4ee26p+5)
54 got 0x1.92d628ab678ccp-1
55 want 0x1.92d628ab678cfp-1. */
V_NAME_D2(atan2)56 float64x2_t VPCS_ATTR V_NAME_D2 (atan2) (float64x2_t y, float64x2_t x)
57 {
58 const struct data *data_ptr = ptr_barrier (&data);
59
60 uint64x2_t ix = vreinterpretq_u64_f64 (x);
61 uint64x2_t iy = vreinterpretq_u64_f64 (y);
62
63 uint64x2_t special_cases = vorrq_u64 (zeroinfnan (ix), zeroinfnan (iy));
64
65 uint64x2_t sign_x = vandq_u64 (ix, SignMask);
66 uint64x2_t sign_y = vandq_u64 (iy, SignMask);
67 uint64x2_t sign_xy = veorq_u64 (sign_x, sign_y);
68
69 float64x2_t ax = vabsq_f64 (x);
70 float64x2_t ay = vabsq_f64 (y);
71
72 uint64x2_t pred_xlt0 = vcltzq_f64 (x);
73 uint64x2_t pred_aygtax = vcgtq_f64 (ay, ax);
74
75 /* Set up z for call to atan. */
76 float64x2_t n = vbslq_f64 (pred_aygtax, vnegq_f64 (ax), ay);
77 float64x2_t d = vbslq_f64 (pred_aygtax, ay, ax);
78 float64x2_t z = vdivq_f64 (n, d);
79
80 /* Work out the correct shift. */
81 float64x2_t shift = vreinterpretq_f64_u64 (
82 vandq_u64 (pred_xlt0, vreinterpretq_u64_f64 (v_f64 (-2.0))));
83 shift = vbslq_f64 (pred_aygtax, vaddq_f64 (shift, v_f64 (1.0)), shift);
84 shift = vmulq_f64 (shift, data_ptr->pi_over_2);
85
86 /* Calculate the polynomial approximation.
87 Use split Estrin scheme for P(z^2) with deg(P)=19. Use split instead of
88 full scheme to avoid underflow in x^16.
89 The order 19 polynomial P approximates
90 (atan(sqrt(x))-sqrt(x))/x^(3/2). */
91 float64x2_t z2 = vmulq_f64 (z, z);
92 float64x2_t x2 = vmulq_f64 (z2, z2);
93 float64x2_t x4 = vmulq_f64 (x2, x2);
94 float64x2_t x8 = vmulq_f64 (x4, x4);
95 float64x2_t ret
96 = vfmaq_f64 (v_estrin_7_f64 (z2, x2, x4, data_ptr->poly),
97 v_estrin_11_f64 (z2, x2, x4, x8, data_ptr->poly + 8), x8);
98
99 /* Finalize. y = shift + z + z^3 * P(z^2). */
100 ret = vfmaq_f64 (z, ret, vmulq_f64 (z2, z));
101 ret = vaddq_f64 (ret, shift);
102
103 /* Account for the sign of x and y. */
104 ret = vreinterpretq_f64_u64 (
105 veorq_u64 (vreinterpretq_u64_f64 (ret), sign_xy));
106
107 if (unlikely (v_any_u64 (special_cases)))
108 return special_case (y, x, ret, special_cases);
109
110 return ret;
111 }
112
113 /* Arity of 2 means no mathbench entry emitted. See test/mathbench_funcs.h. */
114 PL_SIG (V, D, 2, atan2)
115 // TODO tighten this once __v_atan2 is fixed
116 PL_TEST_ULP (V_NAME_D2 (atan2), 2.9)
117 PL_TEST_INTERVAL (V_NAME_D2 (atan2), -10.0, 10.0, 50000)
118 PL_TEST_INTERVAL (V_NAME_D2 (atan2), -1.0, 1.0, 40000)
119 PL_TEST_INTERVAL (V_NAME_D2 (atan2), 0.0, 1.0, 40000)
120 PL_TEST_INTERVAL (V_NAME_D2 (atan2), 1.0, 100.0, 40000)
121 PL_TEST_INTERVAL (V_NAME_D2 (atan2), 1e6, 1e32, 40000)
122