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1
2/* @(#)k_cos.c 1.3 95/01/18 */
3/* \$FreeBSD: head/lib/msun/src/k_cos.c 176408 2008-02-19 12:54:14Z bde \$ */
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4/*
5 * ====================================================
7 *
6ff43c94 8 * Developed at SunSoft, a Sun Microsystems, Inc. business.
b34b60bc 9 * Permission to use, copy, modify, and distribute this
6ff43c94 10 * software is freely granted, provided that this notice
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11 * is preserved.
12 * ====================================================
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13 */
14
15/*
16 * __kernel_cos( x, y )
17 * kernel cos function on [-pi/4, pi/4], pi/4 ~ 0.785398164
18 * Input x is assumed to be bounded by ~pi/4 in magnitude.
6ff43c94 19 * Input y is the tail of x.
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20 *
21 * Algorithm
22 * 1. Since cos(-x) = cos(x), we need only to consider positive x.
23 * 2. if x < 2^-27 (hx<0x3e400000 0), return 1 with inexact if x!=0.
24 * 3. cos(x) is approximated by a polynomial of degree 14 on
25 * [0,pi/4]
26 * 4 14
27 * cos(x) ~ 1 - x*x/2 + C1*x + ... + C6*x
28 * where the remez error is
6ff43c94 29 *
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30 * | 2 4 6 8 10 12 14 | -58
31 * |cos(x)-(1-.5*x +C1*x +C2*x +C3*x +C4*x +C5*x +C6*x )| <= 2
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32 * | |
33 *
34 * 4 6 8 10 12 14
b34b60bc 35 * 4. let r = C1*x +C2*x +C3*x +C4*x +C5*x +C6*x , then
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36 * cos(x) ~ 1 - x*x/2 + r
37 * since cos(x+y) ~ cos(x) - sin(x)*y
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38 * ~ cos(x) - x*y,
39 * a correction term is necessary in cos(x) and hence
40 * cos(x+y) = 1 - (x*x/2 - (r - x*y))
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41 * For better accuracy, rearrange to
42 * cos(x+y) ~ w + (tmp + (r-x*y))
43 * where w = 1 - x*x/2 and tmp is a tiny correction term
44 * (1 - x*x/2 == w + tmp exactly in infinite precision).
45 * The exactness of w + tmp in infinite precision depends on w
46 * and tmp having the same precision as x. If they have extra
47 * precision due to compiler bugs, then the extra precision is
48 * only good provided it is retained in all terms of the final
49 * expression for cos(). Retention happens in all cases tested
50 * under FreeBSD, so don't pessimize things by forcibly clipping
51 * any extra precision in w.
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52 */
53
6ff43c94 54#include "math.h"
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55#include "math_private.h"
56
57static const double
58one = 1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */
59C1 = 4.16666666666666019037e-02, /* 0x3FA55555, 0x5555554C */
60C2 = -1.38888888888741095749e-03, /* 0xBF56C16C, 0x16C15177 */
61C3 = 2.48015872894767294178e-05, /* 0x3EFA01A0, 0x19CB1590 */
62C4 = -2.75573143513906633035e-07, /* 0xBE927E4F, 0x809C52AD */
63C5 = 2.08757232129817482790e-09, /* 0x3E21EE9E, 0xBDB4B1C4 */
64C6 = -1.13596475577881948265e-11; /* 0xBDA8FAE9, 0xBE8838D4 */
65
66double
67__kernel_cos(double x, double y)
68{
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69 double hz,z,r,w;
70
b34b60bc 71 z = x*x;
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72 w = z*z;
73 r = z*(C1+z*(C2+z*C3)) + w*w*(C4+z*(C5+z*C6));
74 hz = 0.5*z;
75 w = one-hz;
76 return w + (((one-w)-hz) + (z*r-x*y));
b34b60bc 77}