1 /*
2 * Double-precision vector atan(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
12 #if V_SUPPORTED
13
14 #include "atan_common.h"
15
16 #define PiOver2 v_f64 (0x1.921fb54442d18p+0)
17 #define AbsMask v_u64 (0x7fffffffffffffff)
18 #define TinyBound 0x3e1 /* top12(asuint64(0x1p-30)). */
19 #define BigBound 0x434 /* top12(asuint64(0x1p53)). */
20
21 /* Fast implementation of vector atan.
22 Based on atan(x) ~ shift + z + z^3 * P(z^2) with reduction to [0,1] using
23 z=1/x and shift = pi/2. Maximum observed error is 2.27 ulps:
24 __v_atan(0x1.0005af27c23e9p+0) got 0x1.9225645bdd7c1p-1
25 want 0x1.9225645bdd7c3p-1. */
26 VPCS_ATTR
V_NAME(atan)27 v_f64_t V_NAME (atan) (v_f64_t x)
28 {
29 /* Small cases, infs and nans are supported by our approximation technique,
30 but do not set fenv flags correctly. Only trigger special case if we need
31 fenv. */
32 v_u64_t ix = v_as_u64_f64 (x);
33 v_u64_t sign = ix & ~AbsMask;
34
35 #if WANT_SIMD_EXCEPT
36 v_u64_t ia12 = (ix >> 52) & 0x7ff;
37 v_u64_t special = v_cond_u64 (ia12 - TinyBound > BigBound - TinyBound);
38 /* If any lane is special, fall back to the scalar routine for all lanes. */
39 if (unlikely (v_any_u64 (special)))
40 return v_call_f64 (atan, x, v_f64 (0), v_u64 (-1));
41 #endif
42
43 /* Argument reduction:
44 y := arctan(x) for x < 1
45 y := pi/2 + arctan(-1/x) for x > 1
46 Hence, use z=-1/a if x>=1, otherwise z=a. */
47 v_u64_t red = v_cagt_f64 (x, v_f64 (1.0));
48 /* Avoid dependency in abs(x) in division (and comparison). */
49 v_f64_t z = v_sel_f64 (red, v_div_f64 (v_f64 (-1.0), x), x);
50 v_f64_t shift = v_sel_f64 (red, PiOver2, v_f64 (0.0));
51 /* Use absolute value only when needed (odd powers of z). */
52 v_f64_t az = v_abs_f64 (z);
53 az = v_sel_f64 (red, -az, az);
54
55 /* Calculate the polynomial approximation. */
56 v_f64_t y = eval_poly (z, az, shift);
57
58 /* y = atan(x) if x>0, -atan(-x) otherwise. */
59 y = v_as_f64_u64 (v_as_u64_f64 (y) ^ sign);
60 return y;
61 }
62 VPCS_ALIAS
63
64 PL_SIG (V, D, 1, atan, -10.0, 10.0)
65 PL_TEST_ULP (V_NAME (atan), 1.78)
66 PL_TEST_EXPECT_FENV (V_NAME (atan), WANT_SIMD_EXCEPT)
67 PL_TEST_INTERVAL (V_NAME (atan), 0, 0x1p-30, 10000)
68 PL_TEST_INTERVAL (V_NAME (atan), -0, -0x1p-30, 1000)
69 PL_TEST_INTERVAL (V_NAME (atan), 0x1p-30, 0x1p53, 900000)
70 PL_TEST_INTERVAL (V_NAME (atan), -0x1p-30, -0x1p53, 90000)
71 PL_TEST_INTERVAL (V_NAME (atan), 0x1p53, inf, 10000)
72 PL_TEST_INTERVAL (V_NAME (atan), -0x1p53, -inf, 1000)
73
74 #endif
75