1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
|
/*
* Copyright (c) 2014 Advanced Micro Devices, Inc.
*
* Permission is hereby granted, free of charge, to any person obtaining a copy
* of this software and associated documentation files (the "Software"), to deal
* in the Software without restriction, including without limitation the rights
* to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
* copies of the Software, and to permit persons to whom the Software is
* furnished to do so, subject to the following conditions:
*
* The above copyright notice and this permission notice shall be included in
* all copies or substantial portions of the Software.
*
* THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
* IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
* FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
* AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
* LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
* OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN
* THE SOFTWARE.
*/
_CL_OVERLOADABLE vtype atan2(vtype y, vtype x)
{
const vtype pi = (vtype)3.1415926535897932e+00; /* 0x400921fb54442d18 */
const vtype piby2 = (vtype)1.5707963267948966e+00; /* 0x3ff921fb54442d18 */
const vtype piby4 = (vtype)7.8539816339744831e-01; /* 0x3fe921fb54442d18 */
const vtype three_piby4 = (vtype)2.3561944901923449e+00; /* 0x4002d97c7f3321d2 */
const vtype pi_head = (vtype)3.1415926218032836e+00; /* 0x400921fb50000000 */
const vtype pi_tail = (vtype)3.1786509547056392e-08; /* 0x3e6110b4611a6263 */
const vtype piby2_head = (vtype)1.5707963267948965e+00; /* 0x3ff921fb54442d18 */
const vtype piby2_tail = (vtype)6.1232339957367660e-17; /* 0x3c91a62633145c07 */
vtype x2 = x;
itype xneg = as_itype(x) < 0;
itype xexp = (as_itype(x) >> 252) & (itype)0x7ff;
vtype y2 = y;
itype yneg = as_itype(y) < 0;
itype yexp = (as_itype(y) >> 252) & (itype)0x7ff;
itype cond2 = (xexp < (itype)1021) & (yexp < (itype)1021);
itype diffexp = yexp - xexp;
// Scale up both x and y if they are both below 1/4
vtype x1 = ldexp(x, (inttype)1024);
itype xexp1 = (as_itype(x1) >> 52) & (itype)0x7ff;
vtype y1 = ldexp(y, (inttype)1024);
itype yexp1 = (as_itype(y1) >> 52) & (itype)0x7ff;
itype diffexp1 = yexp1 - xexp1;
diffexp = cond2 ? diffexp1 : diffexp;
x = cond2 ? x1 : x;
y = cond2 ? y1 : y;
// General case: take absolute values of arguments
vtype u = fabs(x);
vtype v = fabs(y);
// Swap u and v if necessary to obtain 0 < v < u. Compute v/u.
itype swap_vu = u < v;
vtype uu = u;
u = swap_vu ? v : u;
v = swap_vu ? uu : v;
vtype vbyu = v / u;
vtype q1, q2;
// General values of v/u. Use a look-up table and series expansion.
{
vtype val = (vbyu > (vtype)0.0625) ? vbyu : (vtype)0.063;
inttype index = convert_inttype(fma((vtype)256.0, val, (vtype)0.5));
vtype2 tv = USE_TABLE(atan_jby256_tbl, index - 16);
vtype s0 = tv.lo;
vtype s1 = tv.hi;
q1 = s0;
q2 = s1;
vtype c = convert_vtype(index) * (vtype)0x1.0p-8;
// We're going to scale u and v by 2^(-u_exponent) to bring them close to 1
// u_exponent could be EMAX so we have to do it in 2 steps
inttype m = -(convert_inttype(as_utype(u) >> EXPSHIFTBITS_DP64) - (inttype)EXPBIAS_DP64);
vtype um = ldexp(u, m);
vtype vm = ldexp(v, m);
// 26 leading bits of u
vtype u1 = as_vtype(as_utype(um) & (utype)0xfffffffff8000000UL);
vtype u2 = um - u1;
vtype r = MATH_DIVIDE(fma(-c, u2, fma(-c, u1, vm)), fma(c, vm, um));
// Polynomial approximation to atan(r)
vtype s = r * r;
q2 = q2 + fma((s * fma(-s, (vtype)0.19999918038989143496, (vtype)0.33333333333224095522)), -r, r);
}
vtype q3, q4;
{
q3 = (vtype)0.0;
q4 = vbyu;
}
vtype q5, q6;
{
vtype u1 = as_vtype(as_utype(u) & (utype)0xffffffff00000000UL);
vtype u2 = u - u1;
vtype vu1 = as_vtype(as_utype(vbyu) & (utype)0xffffffff00000000UL);
vtype vu2 = vbyu - vu1;
q5 = 0.0;
vtype s = vbyu * vbyu;
q6 = vbyu + fma(-vbyu * s,
fma(-s,
fma(-s,
fma(-s,
fma(-s, (vtype)0.90029810285449784439E-01,
(vtype)0.11110736283514525407),
(vtype)0.14285713561807169030),
(vtype)0.19999999999393223405),
(vtype)0.33333333333333170500),
MATH_DIVIDE(fma(-u, vu2, fma(-u2, vu1, fma(-u1, vu1, v))), u));
}
q3 = (vbyu < (vtype)0x1.d12ed0af1a27fp-27) ? q3 : q5;
q4 = (vbyu < (vtype)0x1.d12ed0af1a27fp-27) ? q4 : q6;
q1 = (vbyu > (vtype)0.0625) ? q1 : q3;
q2 = (vbyu > (vtype)0.0625) ? q2 : q4;
// Tidy-up according to which quadrant the arguments lie in
vtype res1, res2, res3, res4;
q1 = swap_vu ? piby2_head - q1 : q1;
q2 = swap_vu ? piby2_tail - q2 : q2;
q1 = xneg ? pi_head - q1 : q1;
q2 = xneg ? pi_tail - q2 : q2;
q1 = q1 + q2;
res4 = yneg ? -q1 : q1;
res1 = yneg ? -three_piby4 : three_piby4;
res2 = yneg ? -piby4 : piby4;
res3 = xneg ? res1 : res2;
res3 = (isinf(x2) & isinf(y2)) ? res3 : res4;
res1 = yneg ? -pi : pi;
// abs(x)/abs(y) > 2^56 and x < 0
res3 = ((diffexp < (itype)-56) & xneg) ? res1 : res3;
res4 = MATH_DIVIDE(y, x);
// x positive and dominant over y by a factor of 2^28
res3 = ((diffexp < (itype)-28) & (xneg == (itype)0)) ? res4 : res3;
// abs(y)/abs(x) > 2^56
res4 = yneg ? -piby2 : piby2; // atan(y/x) is insignificant compared to piby2
res3 = (diffexp > (itype)56) ? res4 : res3;
res3 = (x2 == (vtype)0.0) ? res4 : res3; // Zero x gives +- pi/2 depending on sign of y
res4 = xneg ? res1 : y2;
res3 = (y2 == (vtype)0.0) ? res4 : res3; // Zero y gives +-0 for positive x and +-pi for negative x
res3 = isnan(y2) ? y2 : res3;
res3 = isnan(x2) ? x2 : res3;
return res3;
}
|
