1*412f47f9SXin Li /*
2*412f47f9SXin Li * Double-precision atan(x) function.
3*412f47f9SXin Li *
4*412f47f9SXin Li * Copyright (c) 2022-2023, Arm Limited.
5*412f47f9SXin Li * SPDX-License-Identifier: MIT OR Apache-2.0 WITH LLVM-exception
6*412f47f9SXin Li */
7*412f47f9SXin Li
8*412f47f9SXin Li #include "pl_sig.h"
9*412f47f9SXin Li #include "pl_test.h"
10*412f47f9SXin Li #include "atan_common.h"
11*412f47f9SXin Li
12*412f47f9SXin Li #define AbsMask 0x7fffffffffffffff
13*412f47f9SXin Li #define PiOver2 0x1.921fb54442d18p+0
14*412f47f9SXin Li #define TinyBound 0x3e1 /* top12(asuint64(0x1p-30)). */
15*412f47f9SXin Li #define BigBound 0x434 /* top12(asuint64(0x1p53)). */
16*412f47f9SXin Li #define OneTop 0x3ff
17*412f47f9SXin Li
18*412f47f9SXin Li /* Fast implementation of double-precision atan.
19*412f47f9SXin Li Based on atan(x) ~ shift + z + z^3 * P(z^2) with reduction to [0,1] using
20*412f47f9SXin Li z=1/x and shift = pi/2. Maximum observed error is 2.27 ulps:
21*412f47f9SXin Li atan(0x1.0005af27c23e9p+0) got 0x1.9225645bdd7c1p-1
22*412f47f9SXin Li want 0x1.9225645bdd7c3p-1. */
23*412f47f9SXin Li double
atan(double x)24*412f47f9SXin Li atan (double x)
25*412f47f9SXin Li {
26*412f47f9SXin Li uint64_t ix = asuint64 (x);
27*412f47f9SXin Li uint64_t sign = ix & ~AbsMask;
28*412f47f9SXin Li uint64_t ia = ix & AbsMask;
29*412f47f9SXin Li uint32_t ia12 = ia >> 52;
30*412f47f9SXin Li
31*412f47f9SXin Li if (unlikely (ia12 >= BigBound || ia12 < TinyBound))
32*412f47f9SXin Li {
33*412f47f9SXin Li if (ia12 < TinyBound)
34*412f47f9SXin Li /* Avoid underflow by returning x. */
35*412f47f9SXin Li return x;
36*412f47f9SXin Li if (ia > 0x7ff0000000000000)
37*412f47f9SXin Li /* Propagate NaN. */
38*412f47f9SXin Li return __math_invalid (x);
39*412f47f9SXin Li /* atan(x) rounds to PiOver2 for large x. */
40*412f47f9SXin Li return asdouble (asuint64 (PiOver2) ^ sign);
41*412f47f9SXin Li }
42*412f47f9SXin Li
43*412f47f9SXin Li double z, az, shift;
44*412f47f9SXin Li if (ia12 >= OneTop)
45*412f47f9SXin Li {
46*412f47f9SXin Li /* For x > 1, use atan(x) = pi / 2 + atan(-1 / x). */
47*412f47f9SXin Li z = -1.0 / x;
48*412f47f9SXin Li shift = PiOver2;
49*412f47f9SXin Li /* Use absolute value only when needed (odd powers of z). */
50*412f47f9SXin Li az = -fabs (z);
51*412f47f9SXin Li }
52*412f47f9SXin Li else
53*412f47f9SXin Li {
54*412f47f9SXin Li /* For x < 1, approximate atan(x) directly. */
55*412f47f9SXin Li z = x;
56*412f47f9SXin Li shift = 0;
57*412f47f9SXin Li az = asdouble (ia);
58*412f47f9SXin Li }
59*412f47f9SXin Li
60*412f47f9SXin Li /* Calculate polynomial, shift + z + z^3 * P(z^2). */
61*412f47f9SXin Li double y = eval_poly (z, az, shift);
62*412f47f9SXin Li /* Copy sign. */
63*412f47f9SXin Li return asdouble (asuint64 (y) ^ sign);
64*412f47f9SXin Li }
65*412f47f9SXin Li
66*412f47f9SXin Li PL_SIG (S, D, 1, atan, -10.0, 10.0)
67*412f47f9SXin Li PL_TEST_ULP (atan, 1.78)
68*412f47f9SXin Li PL_TEST_INTERVAL (atan, 0, 0x1p-30, 10000)
69*412f47f9SXin Li PL_TEST_INTERVAL (atan, -0, -0x1p-30, 1000)
70*412f47f9SXin Li PL_TEST_INTERVAL (atan, 0x1p-30, 0x1p53, 900000)
71*412f47f9SXin Li PL_TEST_INTERVAL (atan, -0x1p-30, -0x1p53, 90000)
72*412f47f9SXin Li PL_TEST_INTERVAL (atan, 0x1p53, inf, 10000)
73*412f47f9SXin Li PL_TEST_INTERVAL (atan, -0x1p53, -inf, 1000)
74