1*072a4ba8SAndrew Turner /*
2*072a4ba8SAndrew Turner * Double-precision scalar atan2(x) function.
3*072a4ba8SAndrew Turner *
4*072a4ba8SAndrew Turner * Copyright (c) 2021-2023, Arm Limited.
5*072a4ba8SAndrew Turner * SPDX-License-Identifier: MIT OR Apache-2.0 WITH LLVM-exception
6*072a4ba8SAndrew Turner */
7*072a4ba8SAndrew Turner
8*072a4ba8SAndrew Turner #include <stdbool.h>
9*072a4ba8SAndrew Turner
10*072a4ba8SAndrew Turner #include "atan_common.h"
11*072a4ba8SAndrew Turner #include "math_config.h"
12*072a4ba8SAndrew Turner #include "pl_sig.h"
13*072a4ba8SAndrew Turner #include "pl_test.h"
14*072a4ba8SAndrew Turner
15*072a4ba8SAndrew Turner #define Pi (0x1.921fb54442d18p+1)
16*072a4ba8SAndrew Turner #define PiOver2 (0x1.921fb54442d18p+0)
17*072a4ba8SAndrew Turner #define PiOver4 (0x1.921fb54442d18p-1)
18*072a4ba8SAndrew Turner #define SignMask (0x8000000000000000)
19*072a4ba8SAndrew Turner #define ExpMask (0x7ff0000000000000)
20*072a4ba8SAndrew Turner
21*072a4ba8SAndrew Turner /* We calculate atan2 by P(n/d), where n and d are similar to the input
22*072a4ba8SAndrew Turner arguments, and P is a polynomial. Evaluating P(x) requires calculating x^8,
23*072a4ba8SAndrew Turner which may underflow if n and d have very different magnitude.
24*072a4ba8SAndrew Turner POW8_EXP_UFLOW_BOUND is the lower bound of the difference in exponents of n
25*072a4ba8SAndrew Turner and d for which P underflows, and is used to special-case such inputs. */
26*072a4ba8SAndrew Turner #define POW8_EXP_UFLOW_BOUND 62
27*072a4ba8SAndrew Turner
28*072a4ba8SAndrew Turner static inline int64_t
biased_exponent(double f)29*072a4ba8SAndrew Turner biased_exponent (double f)
30*072a4ba8SAndrew Turner {
31*072a4ba8SAndrew Turner uint64_t fi = asuint64 (f);
32*072a4ba8SAndrew Turner return (fi & ExpMask) >> 52;
33*072a4ba8SAndrew Turner }
34*072a4ba8SAndrew Turner
35*072a4ba8SAndrew Turner /* Fast implementation of scalar atan2. Largest errors are when y and x are
36*072a4ba8SAndrew Turner close together. The greatest observed error is 2.28 ULP:
37*072a4ba8SAndrew Turner atan2(-0x1.5915b1498e82fp+732, 0x1.54d11ef838826p+732)
38*072a4ba8SAndrew Turner got -0x1.954f42f1fa841p-1 want -0x1.954f42f1fa843p-1. */
39*072a4ba8SAndrew Turner double
atan2(double y,double x)40*072a4ba8SAndrew Turner atan2 (double y, double x)
41*072a4ba8SAndrew Turner {
42*072a4ba8SAndrew Turner uint64_t ix = asuint64 (x);
43*072a4ba8SAndrew Turner uint64_t iy = asuint64 (y);
44*072a4ba8SAndrew Turner
45*072a4ba8SAndrew Turner uint64_t sign_x = ix & SignMask;
46*072a4ba8SAndrew Turner uint64_t sign_y = iy & SignMask;
47*072a4ba8SAndrew Turner
48*072a4ba8SAndrew Turner uint64_t iax = ix & ~SignMask;
49*072a4ba8SAndrew Turner uint64_t iay = iy & ~SignMask;
50*072a4ba8SAndrew Turner
51*072a4ba8SAndrew Turner bool xisnan = isnan (x);
52*072a4ba8SAndrew Turner if (unlikely (isnan (y) && !xisnan))
53*072a4ba8SAndrew Turner return __math_invalid (y);
54*072a4ba8SAndrew Turner if (unlikely (xisnan))
55*072a4ba8SAndrew Turner return __math_invalid (x);
56*072a4ba8SAndrew Turner
57*072a4ba8SAndrew Turner /* m = 2 * sign(x) + sign(y). */
58*072a4ba8SAndrew Turner uint32_t m = ((iy >> 63) & 1) | ((ix >> 62) & 2);
59*072a4ba8SAndrew Turner
60*072a4ba8SAndrew Turner int64_t exp_diff = biased_exponent (x) - biased_exponent (y);
61*072a4ba8SAndrew Turner
62*072a4ba8SAndrew Turner /* y = 0. */
63*072a4ba8SAndrew Turner if (iay == 0)
64*072a4ba8SAndrew Turner {
65*072a4ba8SAndrew Turner switch (m)
66*072a4ba8SAndrew Turner {
67*072a4ba8SAndrew Turner case 0:
68*072a4ba8SAndrew Turner case 1:
69*072a4ba8SAndrew Turner return y; /* atan(+-0,+anything)=+-0. */
70*072a4ba8SAndrew Turner case 2:
71*072a4ba8SAndrew Turner return Pi; /* atan(+0,-anything) = pi. */
72*072a4ba8SAndrew Turner case 3:
73*072a4ba8SAndrew Turner return -Pi; /* atan(-0,-anything) =-pi. */
74*072a4ba8SAndrew Turner }
75*072a4ba8SAndrew Turner }
76*072a4ba8SAndrew Turner /* Special case for (x, y) either on or very close to the y axis. Either x =
77*072a4ba8SAndrew Turner 0, or y is much larger than x (difference in exponents >=
78*072a4ba8SAndrew Turner POW8_EXP_UFLOW_BOUND). */
79*072a4ba8SAndrew Turner if (unlikely (iax == 0 || exp_diff <= -POW8_EXP_UFLOW_BOUND))
80*072a4ba8SAndrew Turner return sign_y ? -PiOver2 : PiOver2;
81*072a4ba8SAndrew Turner
82*072a4ba8SAndrew Turner /* Special case for either x is INF or (x, y) is very close to x axis and x is
83*072a4ba8SAndrew Turner negative. */
84*072a4ba8SAndrew Turner if (unlikely (iax == 0x7ff0000000000000
85*072a4ba8SAndrew Turner || (exp_diff >= POW8_EXP_UFLOW_BOUND && m >= 2)))
86*072a4ba8SAndrew Turner {
87*072a4ba8SAndrew Turner if (iay == 0x7ff0000000000000)
88*072a4ba8SAndrew Turner {
89*072a4ba8SAndrew Turner switch (m)
90*072a4ba8SAndrew Turner {
91*072a4ba8SAndrew Turner case 0:
92*072a4ba8SAndrew Turner return PiOver4; /* atan(+INF,+INF). */
93*072a4ba8SAndrew Turner case 1:
94*072a4ba8SAndrew Turner return -PiOver4; /* atan(-INF,+INF). */
95*072a4ba8SAndrew Turner case 2:
96*072a4ba8SAndrew Turner return 3.0 * PiOver4; /* atan(+INF,-INF). */
97*072a4ba8SAndrew Turner case 3:
98*072a4ba8SAndrew Turner return -3.0 * PiOver4; /* atan(-INF,-INF). */
99*072a4ba8SAndrew Turner }
100*072a4ba8SAndrew Turner }
101*072a4ba8SAndrew Turner else
102*072a4ba8SAndrew Turner {
103*072a4ba8SAndrew Turner switch (m)
104*072a4ba8SAndrew Turner {
105*072a4ba8SAndrew Turner case 0:
106*072a4ba8SAndrew Turner return 0.0; /* atan(+...,+INF). */
107*072a4ba8SAndrew Turner case 1:
108*072a4ba8SAndrew Turner return -0.0; /* atan(-...,+INF). */
109*072a4ba8SAndrew Turner case 2:
110*072a4ba8SAndrew Turner return Pi; /* atan(+...,-INF). */
111*072a4ba8SAndrew Turner case 3:
112*072a4ba8SAndrew Turner return -Pi; /* atan(-...,-INF). */
113*072a4ba8SAndrew Turner }
114*072a4ba8SAndrew Turner }
115*072a4ba8SAndrew Turner }
116*072a4ba8SAndrew Turner /* y is INF. */
117*072a4ba8SAndrew Turner if (iay == 0x7ff0000000000000)
118*072a4ba8SAndrew Turner return sign_y ? -PiOver2 : PiOver2;
119*072a4ba8SAndrew Turner
120*072a4ba8SAndrew Turner uint64_t sign_xy = sign_x ^ sign_y;
121*072a4ba8SAndrew Turner
122*072a4ba8SAndrew Turner double ax = asdouble (iax);
123*072a4ba8SAndrew Turner double ay = asdouble (iay);
124*072a4ba8SAndrew Turner uint64_t pred_aygtax = (ay > ax);
125*072a4ba8SAndrew Turner
126*072a4ba8SAndrew Turner /* Set up z for call to atan. */
127*072a4ba8SAndrew Turner double n = pred_aygtax ? -ax : ay;
128*072a4ba8SAndrew Turner double d = pred_aygtax ? ay : ax;
129*072a4ba8SAndrew Turner double z = n / d;
130*072a4ba8SAndrew Turner
131*072a4ba8SAndrew Turner double ret;
132*072a4ba8SAndrew Turner if (unlikely (m < 2 && exp_diff >= POW8_EXP_UFLOW_BOUND))
133*072a4ba8SAndrew Turner {
134*072a4ba8SAndrew Turner /* If (x, y) is very close to x axis and x is positive, the polynomial
135*072a4ba8SAndrew Turner will underflow and evaluate to z. */
136*072a4ba8SAndrew Turner ret = z;
137*072a4ba8SAndrew Turner }
138*072a4ba8SAndrew Turner else
139*072a4ba8SAndrew Turner {
140*072a4ba8SAndrew Turner /* Work out the correct shift. */
141*072a4ba8SAndrew Turner double shift = sign_x ? -2.0 : 0.0;
142*072a4ba8SAndrew Turner shift = pred_aygtax ? shift + 1.0 : shift;
143*072a4ba8SAndrew Turner shift *= PiOver2;
144*072a4ba8SAndrew Turner
145*072a4ba8SAndrew Turner ret = eval_poly (z, z, shift);
146*072a4ba8SAndrew Turner }
147*072a4ba8SAndrew Turner
148*072a4ba8SAndrew Turner /* Account for the sign of x and y. */
149*072a4ba8SAndrew Turner return asdouble (asuint64 (ret) ^ sign_xy);
150*072a4ba8SAndrew Turner }
151*072a4ba8SAndrew Turner
152*072a4ba8SAndrew Turner /* Arity of 2 means no mathbench entry emitted. See test/mathbench_funcs.h. */
153*072a4ba8SAndrew Turner PL_SIG (S, D, 2, atan2)
154*072a4ba8SAndrew Turner PL_TEST_ULP (atan2, 1.78)
155*072a4ba8SAndrew Turner PL_TEST_INTERVAL (atan2, -10.0, 10.0, 50000)
156*072a4ba8SAndrew Turner PL_TEST_INTERVAL (atan2, -1.0, 1.0, 40000)
157*072a4ba8SAndrew Turner PL_TEST_INTERVAL (atan2, 0.0, 1.0, 40000)
158*072a4ba8SAndrew Turner PL_TEST_INTERVAL (atan2, 1.0, 100.0, 40000)
159*072a4ba8SAndrew Turner PL_TEST_INTERVAL (atan2, 1e6, 1e32, 40000)
160