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k_cos.c
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1 /* @(#)k_cos.c 5.1 93/09/24 */
2 /*
3 * ====================================================
4 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
5 *
6 * Developed at SunPro, a Sun Microsystems, Inc. business.
7 * Permission to use, copy, modify, and distribute this
8 * software is freely granted, provided that this notice
9 * is preserved.
10 * ====================================================
11 */
12
13 #if defined(LIBM_SCCS) && !defined(lint)
14 static const char rcsid[] =
15 "$NetBSD: k_cos.c,v 1.8 1995/05/10 20:46:22 jtc Exp $";
16 #endif
17
18 /*
19 * __kernel_cos( x, y )
20 * kernel cos function on [-pi/4, pi/4], pi/4 ~ 0.785398164
21 * Input x is assumed to be bounded by ~pi/4 in magnitude.
22 * Input y is the tail of x.
23 *
24 * Algorithm
25 * 1. Since cos(-x) = cos(x), we need only to consider positive x.
26 * 2. if x < 2^-27 (hx<0x3e400000 0), return 1 with inexact if x!=0.
27 * 3. cos(x) is approximated by a polynomial of degree 14 on
28 * [0,pi/4]
29 * 4 14
30 * cos(x) ~ 1 - x*x/2 + C1*x + ... + C6*x
31 * where the remez error is
32 *
33 * | 2 4 6 8 10 12 14 | -58
34 * |cos(x)-(1-.5*x +C1*x +C2*x +C3*x +C4*x +C5*x +C6*x )| <= 2
35 * | |
36 *
37 * 4 6 8 10 12 14
38 * 4. let r = C1*x +C2*x +C3*x +C4*x +C5*x +C6*x , then
39 * cos(x) = 1 - x*x/2 + r
40 * since cos(x+y) ~ cos(x) - sin(x)*y
41 * ~ cos(x) - x*y,
42 * a correction term is necessary in cos(x) and hence
43 * cos(x+y) = 1 - (x*x/2 - (r - x*y))
44 * For better accuracy when x > 0.3, let qx = |x|/4 with
45 * the last 32 bits mask off, and if x > 0.78125, let qx = 0.28125.
46 * Then
47 * cos(x+y) = (1-qx) - ((x*x/2-qx) - (r-x*y)).
48 * Note that 1-qx and (x*x/2-qx) is EXACT here, and the
49 * magnitude of the latter is at least a quarter of x*x/2,
50 * thus, reducing the rounding error in the subtraction.
51 */
52
53 #include "math_libm.h"
54 #include "math_private.h"
55
56 #ifdef __STDC__
57 static const double
58 #else
59 static double
60 #endif
61 one = 1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */
62 C1 = 4.16666666666666019037e-02, /* 0x3FA55555, 0x5555554C */
63 C2 = -1.38888888888741095749e-03, /* 0xBF56C16C, 0x16C15177 */
64 C3 = 2.48015872894767294178e-05, /* 0x3EFA01A0, 0x19CB1590 */
65 C4 = -2.75573143513906633035e-07, /* 0xBE927E4F, 0x809C52AD */
66 C5 = 2.08757232129817482790e-09, /* 0x3E21EE9E, 0xBDB4B1C4 */
67 C6 = -1.13596475577881948265e-11; /* 0xBDA8FAE9, 0xBE8838D4 */
68
69 #ifdef __STDC__
70 double attribute_hidden
71 __kernel_cos(double x, double y)
72 #else
73 double attribute_hidden
74 __kernel_cos(x, y)
75 double x, y;
76 #endif
77 {
78 double a, hz, z, r, qx;
79 int32_t ix;
80 GET_HIGH_WORD(ix, x);
81 ix &= 0x7fffffff; /* ix = |x|'s high word */
82 if (ix < 0x3e400000) { /* if x < 2**27 */
83 if (((int) x) == 0)
84 return one; /* generate inexact */
85 }
86 z = x * x;
87 r = z * (C1 + z * (C2 + z * (C3 + z * (C4 + z * (C5 + z * C6)))));
88 if (ix < 0x3FD33333) /* if |x| < 0.3 */
89 return one - (0.5 * z - (z * r - x * y));
90 else {
91 if (ix > 0x3fe90000) { /* x > 0.78125 */
92 qx = 0.28125;
93 } else {
94 INSERT_WORDS(qx, ix - 0x00200000, 0); /* x/4 */
95 }
96 hz = 0.5 * z - qx;
97 a = one - qx;
98 return a - (hz - (z * r - x * y));
99 }
100 }
GET_HIGH_WORD
#define GET_HIGH_WORD(i, d)
Definition: math_private.h:102
x
EGLSurface EGLint x
Definition: eglext.h:293
one
static double one
Definition: k_cos.c:61
a
GLboolean GLboolean GLboolean GLboolean a
Definition: glew.h:8736
int32_t
long int32_t
Definition: types.h:9
C2
static double C2
Definition: k_cos.c:63
C5
static double C5
Definition: k_cos.c:66
__kernel_cos
double __kernel_cos(double, double) attribute_hidden
C4
static double C4
Definition: k_cos.c:65
math_libm.h
C3
static double C3
Definition: k_cos.c:64
y
EGLSurface EGLint EGLint y
Definition: eglext.h:293
C6
static double C6
Definition: k_cos.c:67
attribute_hidden
double attribute_hidden
ix
int32_t ix
Definition: e_rem_pio2.c:100
C1
static double C1
Definition: k_cos.c:62
r
GLdouble GLdouble GLdouble r
Definition: glew.h:1392
z
GLint GLint GLint GLint z
Definition: gl2ext.h:1214
INSERT_WORDS
INSERT_WORDS(z, ix0, ix1)
math_private.h