1*412f47f9SXin Li /*
2*412f47f9SXin Li * Single-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 "atanf_common.h"
9*412f47f9SXin Li #include "pl_sig.h"
10*412f47f9SXin Li #include "pl_test.h"
11*412f47f9SXin Li
12*412f47f9SXin Li #define PiOver2 0x1.921fb6p+0f
13*412f47f9SXin Li #define AbsMask 0x7fffffff
14*412f47f9SXin Li #define TinyBound 0x30800000 /* asuint(0x1p-30). */
15*412f47f9SXin Li #define BigBound 0x4e800000 /* asuint(0x1p30). */
16*412f47f9SXin Li #define One 0x3f800000
17*412f47f9SXin Li
18*412f47f9SXin Li /* Approximation of single-precision atan(x) based on
19*412f47f9SXin Li atan(x) ~ shift + z + z^3 * P(z^2) with reduction to [0,1]
20*412f47f9SXin Li using z=-1/x and shift = pi/2.
21*412f47f9SXin Li Maximum error is 2.88 ulps:
22*412f47f9SXin Li atanf(0x1.0565ccp+0) got 0x1.97771p-1
23*412f47f9SXin Li want 0x1.97770ap-1. */
24*412f47f9SXin Li float
atanf(float x)25*412f47f9SXin Li atanf (float x)
26*412f47f9SXin Li {
27*412f47f9SXin Li uint32_t ix = asuint (x);
28*412f47f9SXin Li uint32_t sign = ix & ~AbsMask;
29*412f47f9SXin Li uint32_t ia = ix & AbsMask;
30*412f47f9SXin Li
31*412f47f9SXin Li if (unlikely (ia < TinyBound))
32*412f47f9SXin Li /* Avoid underflow by returning x. */
33*412f47f9SXin Li return x;
34*412f47f9SXin Li
35*412f47f9SXin Li if (unlikely (ia > BigBound))
36*412f47f9SXin Li {
37*412f47f9SXin Li if (ia > 0x7f800000)
38*412f47f9SXin Li /* Propagate NaN. */
39*412f47f9SXin Li return __math_invalidf (x);
40*412f47f9SXin Li /* atan(x) rounds to PiOver2 for large x. */
41*412f47f9SXin Li return asfloat (asuint (PiOver2) ^ sign);
42*412f47f9SXin Li }
43*412f47f9SXin Li
44*412f47f9SXin Li float z, az, shift;
45*412f47f9SXin Li if (ia > One)
46*412f47f9SXin Li {
47*412f47f9SXin Li /* For x > 1, use atan(x) = pi / 2 + atan(-1 / x). */
48*412f47f9SXin Li z = -1.0f / x;
49*412f47f9SXin Li shift = PiOver2;
50*412f47f9SXin Li /* Use absolute value only when needed (odd powers of z). */
51*412f47f9SXin Li az = -fabsf (z);
52*412f47f9SXin Li }
53*412f47f9SXin Li else
54*412f47f9SXin Li {
55*412f47f9SXin Li /* For x < 1, approximate atan(x) directly. */
56*412f47f9SXin Li z = x;
57*412f47f9SXin Li az = asfloat (ia);
58*412f47f9SXin Li shift = 0;
59*412f47f9SXin Li }
60*412f47f9SXin Li
61*412f47f9SXin Li /* Calculate polynomial, shift + z + z^3 * P(z^2). */
62*412f47f9SXin Li float y = eval_poly (z, az, shift);
63*412f47f9SXin Li /* Copy sign. */
64*412f47f9SXin Li return asfloat (asuint (y) ^ sign);
65*412f47f9SXin Li }
66*412f47f9SXin Li
67*412f47f9SXin Li PL_SIG (S, F, 1, atan, -10.0, 10.0)
68*412f47f9SXin Li PL_TEST_ULP (atanf, 2.38)
69*412f47f9SXin Li PL_TEST_SYM_INTERVAL (atanf, 0, 0x1p-30, 5000)
70*412f47f9SXin Li PL_TEST_SYM_INTERVAL (atanf, 0x1p-30, 1, 40000)
71*412f47f9SXin Li PL_TEST_SYM_INTERVAL (atanf, 1, 0x1p30, 40000)
72*412f47f9SXin Li PL_TEST_SYM_INTERVAL (atanf, 0x1p30, inf, 1000)
73