|
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
|
#ifndef DataFormats_Math_FastMath_h
#define DataFormats_Math_FastMath_h
// faster function will a limited precision
#include <cmath>
#include <utility>
#ifdef __SSE2__
#include <emmintrin.h>
#endif
namespace fastmath {
inline float invSqrt(float in) {
#ifndef __SSE2__
return 1.f / std::sqrt(in);
#else
float out;
_mm_store_ss(&out, _mm_rsqrt_ss(_mm_load_ss(&in))); // compiles to movss, rsqrtss, movss
// return out; // already good enough!
return out * (1.5f - 0.5f * in * out * out); // One (more?) round of Newton's method
#endif
}
inline double invSqrt(double in) { return 1. / std::sqrt(in); }
} // namespace fastmath
namespace fastmath_details {
const double _2pi = (2.0 * 3.1415926535897932384626434);
const float _2pif = float(_2pi);
extern float atanbuf_[257 * 2];
extern double datanbuf_[513 * 2];
} // namespace fastmath_details
namespace fastmath {
// =====================================================================
// arctan, single-precision; returns phi and r (or 1/r if overR=true)
// =====================================================================
inline std::pair<float, float> atan2r(float y_, float x_, bool overR = false) {
using namespace fastmath_details;
float mag2 = x_ * x_ + y_ * y_;
if (!(mag2 > 0)) {
return std::pair<float, float>(0.f, 0.f);
} // degenerate case
// float r_ = std::sqrt(mag2);
float rinv = invSqrt(mag2);
unsigned int flags = 0;
float x, y;
union {
float f;
int i;
} yp;
yp.f = 32768.f;
if (y_ < 0) {
flags |= 4;
y_ = -y_;
}
if (x_ < 0) {
flags |= 2;
x_ = -x_;
}
if (y_ > x_) {
flags |= 1;
x = rinv * y_;
y = rinv * x_;
yp.f += y;
} else {
x = rinv * x_;
y = rinv * y_;
yp.f += y;
}
int ind = (yp.i & 0x01FF) * 2;
float* asbuf = (float*)(atanbuf_ + ind);
float sv = yp.f - 32768.f;
float cv = asbuf[0];
float asv = asbuf[1];
sv = y * cv - x * sv; // delta sin value
// ____ compute arcsin directly
float asvd = 6.f + sv * sv;
sv *= float(1.0f / 6.0f);
float th = asv + asvd * sv;
if (flags & 1) {
th = (_2pif / 4.f) - th;
}
if (flags & 2) {
th = (_2pif / 2.f) - th;
}
if (flags & 4) {
th = -th;
}
return std::pair<float, float>(th, overR ? rinv : rinv * mag2);
}
// =====================================================================
// arctan, double-precision; returns phi and r (or 1/r if overR=true)
// =====================================================================
inline std::pair<double, double> atan2r(double y_, double x_, bool overR = false) {
using namespace fastmath_details;
// assert(ataninited);
double mag2 = x_ * x_ + y_ * y_;
if (!(mag2 > 0)) {
return std::pair<double, double>(0., 0.);
} // degenerate case
double r_ = std::sqrt(mag2);
double rinv = 1. / r_;
unsigned int flags = 0;
double x, y;
const double _2p43 = 65536.0 * 65536.0 * 2048.0;
union {
double d;
int i[2];
} yp;
yp.d = _2p43;
if (y_ < 0) {
flags |= 4;
y_ = -y_;
}
if (x_ < 0) {
flags |= 2;
x_ = -x_;
}
if (y_ > x_) {
flags |= 1;
x = rinv * y_;
y = rinv * x_;
yp.d += y;
} else {
x = rinv * x_;
y = rinv * y_;
yp.d += y;
}
int ind = (yp.i[0] & 0x03FF) * 2; // 0 for little indian
double* dasbuf = (double*)(datanbuf_ + ind);
double sv = yp.d - _2p43; // index fraction
double cv = dasbuf[0];
double asv = dasbuf[1];
sv = y * cv - x * sv; // delta sin value
// double sv = y *(cv-x);
// ____ compute arcsin directly
double asvd = 6 + sv * sv;
sv *= double(1.0 / 6.0);
double th = asv + asvd * sv;
if (flags & 1) {
th = (_2pi / 4) - th;
}
if (flags & 2) {
th = (_2pi / 2) - th;
}
if (flags & 4) {
th = -th;
}
return std::pair<double, double>(th, overR ? rinv : r_);
}
// return eta phi saving some computation
template <typename T>
inline std::pair<T, T> etaphi(T x, T y, T z) {
std::pair<T, T> por = atan2r(y, x, true);
x = z * por.second;
return std::pair<float, float>(std::log(x + std::sqrt(x * x + T(1))), por.first);
}
} // namespace fastmath
#endif
|