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
|
/* Used by sinf, cosf and sincosf functions.
Copyright (C) 2017-2018 Free Software Foundation, Inc.
This file is part of the GNU C Library.
The GNU C Library is free software; you can redistribute it and/or
modify it under the terms of the GNU Lesser General Public
License as published by the Free Software Foundation; either
version 2.1 of the License, or (at your option) any later version.
The GNU C Library is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
Lesser General Public License for more details.
You should have received a copy of the GNU Lesser General Public
License along with the GNU C Library; if not, see
<http://www.gnu.org/licenses/>. */
/* Chebyshev constants for cos, range -PI/4 - PI/4. */
static const double C0 = -0x1.ffffffffe98aep-2;
static const double C1 = 0x1.55555545c50c7p-5;
static const double C2 = -0x1.6c16b348b6874p-10;
static const double C3 = 0x1.a00eb9ac43ccp-16;
static const double C4 = -0x1.23c97dd8844d7p-22;
/* Chebyshev constants for sin, range -PI/4 - PI/4. */
static const double S0 = -0x1.5555555551cd9p-3;
static const double S1 = 0x1.1111110c2688bp-7;
static const double S2 = -0x1.a019f8b4bd1f9p-13;
static const double S3 = 0x1.71d7264e6b5b4p-19;
static const double S4 = -0x1.a947e1674b58ap-26;
/* Chebyshev constants for sin, range 2^-27 - 2^-5. */
static const double SS0 = -0x1.555555543d49dp-3;
static const double SS1 = 0x1.110f475cec8c5p-7;
/* Chebyshev constants for cos, range 2^-27 - 2^-5. */
static const double CC0 = -0x1.fffffff5cc6fdp-2;
static const double CC1 = 0x1.55514b178dac5p-5;
/* PI/2 with 98 bits of accuracy. */
static const double PI_2_hi = 0x1.921fb544p+0;
static const double PI_2_lo = 0x1.0b4611a626332p-34;
static const double SMALL = 0x1p-50; /* 2^-50. */
static const double inv_PI_4 = 0x1.45f306dc9c883p+0; /* 4/PI. */
#define FLOAT_EXPONENT_SHIFT 23
#define FLOAT_EXPONENT_BIAS 127
static const double pio2_table[] = {
0 * M_PI_2,
1 * M_PI_2,
2 * M_PI_2,
3 * M_PI_2,
4 * M_PI_2,
5 * M_PI_2
};
static const double invpio4_table[] = {
0x0p+0,
0x1.45f306cp+0,
0x1.c9c882ap-28,
0x1.4fe13a8p-58,
0x1.f47d4dp-85,
0x1.bb81b6cp-112,
0x1.4acc9ep-142,
0x1.0e4107cp-169
};
static const double ones[] = { 1.0, -1.0 };
/* Compute the sine value using Chebyshev polynomials where
THETA is the range reduced absolute value of the input
and it is less than Pi/4,
N is calculated as trunc(|x|/(Pi/4)) + 1 and it is used to decide
whether a sine or cosine approximation is more accurate and
SIGNBIT is used to add the correct sign after the Chebyshev
polynomial is computed. */
static inline float
reduced_sin (const double theta, const unsigned int n,
const unsigned int signbit)
{
double sx;
const double theta2 = theta * theta;
/* We are operating on |x|, so we need to add back the original
signbit for sinf. */
double sign;
/* Determine positive or negative primary interval. */
sign = ones[((n >> 2) & 1) ^ signbit];
/* Are we in the primary interval of sin or cos? */
if ((n & 2) == 0)
{
/* Here sinf() is calculated using sin Chebyshev polynomial:
x+x^3*(S0+x^2*(S1+x^2*(S2+x^2*(S3+x^2*S4)))). */
sx = S3 + theta2 * S4; /* S3+x^2*S4. */
sx = S2 + theta2 * sx; /* S2+x^2*(S3+x^2*S4). */
sx = S1 + theta2 * sx; /* S1+x^2*(S2+x^2*(S3+x^2*S4)). */
sx = S0 + theta2 * sx; /* S0+x^2*(S1+x^2*(S2+x^2*(S3+x^2*S4))). */
sx = theta + theta * theta2 * sx;
}
else
{
/* Here sinf() is calculated using cos Chebyshev polynomial:
1.0+x^2*(C0+x^2*(C1+x^2*(C2+x^2*(C3+x^2*C4)))). */
sx = C3 + theta2 * C4; /* C3+x^2*C4. */
sx = C2 + theta2 * sx; /* C2+x^2*(C3+x^2*C4). */
sx = C1 + theta2 * sx; /* C1+x^2*(C2+x^2*(C3+x^2*C4)). */
sx = C0 + theta2 * sx; /* C0+x^2*(C1+x^2*(C2+x^2*(C3+x^2*C4))). */
sx = 1.0 + theta2 * sx;
}
/* Add in the signbit and assign the result. */
return sign * sx;
}
/* Compute the cosine value using Chebyshev polynomials where
THETA is the range reduced absolute value of the input
and it is less than Pi/4,
N is calculated as trunc(|x|/(Pi/4)) + 1 and it is used to decide
whether a sine or cosine approximation is more accurate and
the sign of the result. */
static inline float
reduced_cos (double theta, unsigned int n)
{
double sign, cx;
const double theta2 = theta * theta;
/* Determine positive or negative primary interval. */
n += 2;
sign = ones[(n >> 2) & 1];
/* Are we in the primary interval of sin or cos? */
if ((n & 2) == 0)
{
/* Here cosf() is calculated using sin Chebyshev polynomial:
x+x^3*(S0+x^2*(S1+x^2*(S2+x^2*(S3+x^2*S4)))). */
cx = S3 + theta2 * S4;
cx = S2 + theta2 * cx;
cx = S1 + theta2 * cx;
cx = S0 + theta2 * cx;
cx = theta + theta * theta2 * cx;
}
else
{
/* Here cosf() is calculated using cos Chebyshev polynomial:
1.0+x^2*(C0+x^2*(C1+x^2*(C2+x^2*(C3+x^2*C4)))). */
cx = C3 + theta2 * C4;
cx = C2 + theta2 * cx;
cx = C1 + theta2 * cx;
cx = C0 + theta2 * cx;
cx = 1. + theta2 * cx;
}
return sign * cx;
}
|
