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.  */
23double
24atan (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
66PL_SIG (S, D, 1, atan, -10.0, 10.0)
67PL_TEST_ULP (atan, 1.78)
68PL_TEST_INTERVAL (atan, 0, 0x1p-30, 10000)
69PL_TEST_INTERVAL (atan, -0, -0x1p-30, 1000)
70PL_TEST_INTERVAL (atan, 0x1p-30, 0x1p53, 900000)
71PL_TEST_INTERVAL (atan, -0x1p-30, -0x1p53, 90000)
72PL_TEST_INTERVAL (atan, 0x1p53, inf, 10000)
73PL_TEST_INTERVAL (atan, -0x1p53, -inf, 1000)
74