1 /*
2 * Double-precision scalar atan2(x) function.
3 *
4 * Copyright (c) 2021-2024, Arm Limited.
5 * SPDX-License-Identifier: MIT OR Apache-2.0 WITH LLVM-exception
6 */
7
8 #include <stdbool.h>
9
10 #include "atan_common.h"
11 #include "math_config.h"
12 #include "test_sig.h"
13 #include "test_defs.h"
14
15 #define Pi (0x1.921fb54442d18p+1)
16 #define PiOver2 (0x1.921fb54442d18p+0)
17 #define PiOver4 (0x1.921fb54442d18p-1)
18 #define SignMask (0x8000000000000000)
19 #define ExpMask (0x7ff0000000000000)
20
21 /* We calculate atan2 by P(n/d), where n and d are similar to the input
22 arguments, and P is a polynomial. Evaluating P(x) requires calculating x^8,
23 which may underflow if n and d have very different magnitude.
24 POW8_EXP_UFLOW_BOUND is the lower bound of the difference in exponents of n
25 and d for which P underflows, and is used to special-case such inputs. */
26 #define POW8_EXP_UFLOW_BOUND 62
27
28 static inline int64_t
biased_exponent(double f)29 biased_exponent (double f)
30 {
31 uint64_t fi = asuint64 (f);
32 return (fi & ExpMask) >> 52;
33 }
34
35 /* Fast implementation of scalar atan2. Largest errors are when y and x are
36 close together. The greatest observed error is 2.28 ULP:
37 atan2(-0x1.5915b1498e82fp+732, 0x1.54d11ef838826p+732)
38 got -0x1.954f42f1fa841p-1 want -0x1.954f42f1fa843p-1. */
39 double
atan2(double y,double x)40 atan2 (double y, double x)
41 {
42 uint64_t ix = asuint64 (x);
43 uint64_t iy = asuint64 (y);
44
45 uint64_t sign_x = ix & SignMask;
46 uint64_t sign_y = iy & SignMask;
47
48 uint64_t iax = ix & ~SignMask;
49 uint64_t iay = iy & ~SignMask;
50
51 bool xisnan = isnan (x);
52 if (unlikely (isnan (y) && !xisnan))
53 return __math_invalid (y);
54 if (unlikely (xisnan))
55 return __math_invalid (x);
56
57 /* m = 2 * sign(x) + sign(y). */
58 uint32_t m = ((iy >> 63) & 1) | ((ix >> 62) & 2);
59
60 int64_t exp_diff = biased_exponent (x) - biased_exponent (y);
61
62 /* y = 0. */
63 if (iay == 0)
64 {
65 switch (m)
66 {
67 case 0:
68 case 1:
69 return y; /* atan(+-0,+anything)=+-0. */
70 case 2:
71 return Pi; /* atan(+0,-anything) = pi. */
72 case 3:
73 return -Pi; /* atan(-0,-anything) =-pi. */
74 }
75 }
76 /* Special case for (x, y) either on or very close to the y axis. Either x =
77 0, or y is much larger than x (difference in exponents >=
78 POW8_EXP_UFLOW_BOUND). */
79 if (unlikely (iax == 0 || exp_diff <= -POW8_EXP_UFLOW_BOUND))
80 return sign_y ? -PiOver2 : PiOver2;
81
82 /* Special case for either x is INF or (x, y) is very close to x axis and x
83 is negative. */
84 if (unlikely (iax == 0x7ff0000000000000
85 || (exp_diff >= POW8_EXP_UFLOW_BOUND && m >= 2)))
86 {
87 if (iay == 0x7ff0000000000000)
88 {
89 switch (m)
90 {
91 case 0:
92 return PiOver4; /* atan(+INF,+INF). */
93 case 1:
94 return -PiOver4; /* atan(-INF,+INF). */
95 case 2:
96 return 3.0 * PiOver4; /* atan(+INF,-INF). */
97 case 3:
98 return -3.0 * PiOver4; /* atan(-INF,-INF). */
99 }
100 }
101 else
102 {
103 switch (m)
104 {
105 case 0:
106 return 0.0; /* atan(+...,+INF). */
107 case 1:
108 return -0.0; /* atan(-...,+INF). */
109 case 2:
110 return Pi; /* atan(+...,-INF). */
111 case 3:
112 return -Pi; /* atan(-...,-INF). */
113 }
114 }
115 }
116 /* y is INF. */
117 if (iay == 0x7ff0000000000000)
118 return sign_y ? -PiOver2 : PiOver2;
119
120 uint64_t sign_xy = sign_x ^ sign_y;
121
122 double ax = asdouble (iax);
123 double ay = asdouble (iay);
124 uint64_t pred_aygtax = (ay > ax);
125
126 /* Set up z for call to atan. */
127 double n = pred_aygtax ? -ax : ay;
128 double d = pred_aygtax ? ay : ax;
129 double z = n / d;
130
131 double ret;
132 if (unlikely (m < 2 && exp_diff >= POW8_EXP_UFLOW_BOUND))
133 {
134 /* If (x, y) is very close to x axis and x is positive, the polynomial
135 will underflow and evaluate to z. */
136 ret = z;
137 }
138 else
139 {
140 /* Work out the correct shift. */
141 double shift = sign_x ? -2.0 : 0.0;
142 shift = pred_aygtax ? shift + 1.0 : shift;
143 shift *= PiOver2;
144
145 ret = eval_poly (z, z, shift);
146 }
147
148 /* Account for the sign of x and y. */
149 return asdouble (asuint64 (ret) ^ sign_xy);
150 }
151
152 /* Arity of 2 means no mathbench entry emitted. See test/mathbench_funcs.h. */
153 TEST_SIG (S, D, 2, atan2)
154 TEST_ULP (atan2, 1.78)
155 TEST_INTERVAL (atan2, -10.0, 10.0, 50000)
156 TEST_INTERVAL (atan2, -1.0, 1.0, 40000)
157 TEST_INTERVAL (atan2, 0.0, 1.0, 40000)
158 TEST_INTERVAL (atan2, 1.0, 100.0, 40000)
159 TEST_INTERVAL (atan2, 1e6, 1e32, 40000)
160