1 /*
2  * Single-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 "atanf_common.h"
9 #include "pl_sig.h"
10 #include "pl_test.h"
11 
12 #define PiOver2 0x1.921fb6p+0f
13 #define AbsMask 0x7fffffff
14 #define TinyBound 0x30800000 /* asuint(0x1p-30).  */
15 #define BigBound 0x4e800000  /* asuint(0x1p30).  */
16 #define One 0x3f800000
17 
18 /* Approximation of single-precision atan(x) based on
19    atan(x) ~ shift + z + z^3 * P(z^2) with reduction to [0,1]
20    using z=-1/x and shift = pi/2.
21    Maximum error is 2.88 ulps:
22    atanf(0x1.0565ccp+0) got 0x1.97771p-1
23 		       want 0x1.97770ap-1.  */
24 float
25 atanf (float x)
26 {
27   uint32_t ix = asuint (x);
28   uint32_t sign = ix & ~AbsMask;
29   uint32_t ia = ix & AbsMask;
30 
31   if (unlikely (ia < TinyBound))
32     /* Avoid underflow by returning x.  */
33     return x;
34 
35   if (unlikely (ia > BigBound))
36     {
37       if (ia > 0x7f800000)
38 	/* Propagate NaN.  */
39 	return __math_invalidf (x);
40       /* atan(x) rounds to PiOver2 for large x.  */
41       return asfloat (asuint (PiOver2) ^ sign);
42     }
43 
44   float z, az, shift;
45   if (ia > One)
46     {
47       /* For x > 1, use atan(x) = pi / 2 + atan(-1 / x).  */
48       z = -1.0f / x;
49       shift = PiOver2;
50       /* Use absolute value only when needed (odd powers of z).  */
51       az = -fabsf (z);
52     }
53   else
54     {
55       /* For x < 1, approximate atan(x) directly.  */
56       z = x;
57       az = asfloat (ia);
58       shift = 0;
59     }
60 
61   /* Calculate polynomial, shift + z + z^3 * P(z^2).  */
62   float y = eval_poly (z, az, shift);
63   /* Copy sign.  */
64   return asfloat (asuint (y) ^ sign);
65 }
66 
67 PL_SIG (S, F, 1, atan, -10.0, 10.0)
68 PL_TEST_ULP (atanf, 2.38)
69 PL_TEST_INTERVAL (atanf, 0, 0x1p-30, 5000)
70 PL_TEST_INTERVAL (atanf, -0, -0x1p-30, 5000)
71 PL_TEST_INTERVAL (atanf, 0x1p-30, 1, 40000)
72 PL_TEST_INTERVAL (atanf, -0x1p-30, -1, 40000)
73 PL_TEST_INTERVAL (atanf, 1, 0x1p30, 40000)
74 PL_TEST_INTERVAL (atanf, -1, -0x1p30, 40000)
75 PL_TEST_INTERVAL (atanf, 0x1p30, inf, 1000)
76 PL_TEST_INTERVAL (atanf, -0x1p30, -inf, 1000)
77