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