1*05a0b428SJohn Marino /* @(#)k_cos.c 5.1 93/09/24 */
2*05a0b428SJohn Marino /*
3*05a0b428SJohn Marino * ====================================================
4*05a0b428SJohn Marino * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
5*05a0b428SJohn Marino *
6*05a0b428SJohn Marino * Developed at SunPro, a Sun Microsystems, Inc. business.
7*05a0b428SJohn Marino * Permission to use, copy, modify, and distribute this
8*05a0b428SJohn Marino * software is freely granted, provided that this notice
9*05a0b428SJohn Marino * is preserved.
10*05a0b428SJohn Marino * ====================================================
11*05a0b428SJohn Marino */
12*05a0b428SJohn Marino
13*05a0b428SJohn Marino /*
14*05a0b428SJohn Marino * __kernel_cos( x, y )
15*05a0b428SJohn Marino * kernel cos function on [-pi/4, pi/4], pi/4 ~ 0.785398164
16*05a0b428SJohn Marino * Input x is assumed to be bounded by ~pi/4 in magnitude.
17*05a0b428SJohn Marino * Input y is the tail of x.
18*05a0b428SJohn Marino *
19*05a0b428SJohn Marino * Algorithm
20*05a0b428SJohn Marino * 1. Since cos(-x) = cos(x), we need only to consider positive x.
21*05a0b428SJohn Marino * 2. if x < 2^-27 (hx<0x3e400000 0), return 1 with inexact if x!=0.
22*05a0b428SJohn Marino * 3. cos(x) is approximated by a polynomial of degree 14 on
23*05a0b428SJohn Marino * [0,pi/4]
24*05a0b428SJohn Marino * 4 14
25*05a0b428SJohn Marino * cos(x) ~ 1 - x*x/2 + C1*x + ... + C6*x
26*05a0b428SJohn Marino * where the Remes error is
27*05a0b428SJohn Marino *
28*05a0b428SJohn Marino * | 2 4 6 8 10 12 14 | -58
29*05a0b428SJohn Marino * |cos(x)-(1-.5*x +C1*x +C2*x +C3*x +C4*x +C5*x +C6*x )| <= 2
30*05a0b428SJohn Marino * | |
31*05a0b428SJohn Marino *
32*05a0b428SJohn Marino * 4 6 8 10 12 14
33*05a0b428SJohn Marino * 4. let r = C1*x +C2*x +C3*x +C4*x +C5*x +C6*x , then
34*05a0b428SJohn Marino * cos(x) = 1 - x*x/2 + r
35*05a0b428SJohn Marino * since cos(x+y) ~ cos(x) - sin(x)*y
36*05a0b428SJohn Marino * ~ cos(x) - x*y,
37*05a0b428SJohn Marino * a correction term is necessary in cos(x) and hence
38*05a0b428SJohn Marino * cos(x+y) = 1 - (x*x/2 - (r - x*y))
39*05a0b428SJohn Marino * For better accuracy when x > 0.3, let qx = |x|/4 with
40*05a0b428SJohn Marino * the last 32 bits mask off, and if x > 0.78125, let qx = 0.28125.
41*05a0b428SJohn Marino * Then
42*05a0b428SJohn Marino * cos(x+y) = (1-qx) - ((x*x/2-qx) - (r-x*y)).
43*05a0b428SJohn Marino * Note that 1-qx and (x*x/2-qx) is EXACT here, and the
44*05a0b428SJohn Marino * magnitude of the latter is at least a quarter of x*x/2,
45*05a0b428SJohn Marino * thus, reducing the rounding error in the subtraction.
46*05a0b428SJohn Marino */
47*05a0b428SJohn Marino
48*05a0b428SJohn Marino #include "math.h"
49*05a0b428SJohn Marino #include "math_private.h"
50*05a0b428SJohn Marino
51*05a0b428SJohn Marino static const double
52*05a0b428SJohn Marino one = 1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */
53*05a0b428SJohn Marino C1 = 4.16666666666666019037e-02, /* 0x3FA55555, 0x5555554C */
54*05a0b428SJohn Marino C2 = -1.38888888888741095749e-03, /* 0xBF56C16C, 0x16C15177 */
55*05a0b428SJohn Marino C3 = 2.48015872894767294178e-05, /* 0x3EFA01A0, 0x19CB1590 */
56*05a0b428SJohn Marino C4 = -2.75573143513906633035e-07, /* 0xBE927E4F, 0x809C52AD */
57*05a0b428SJohn Marino C5 = 2.08757232129817482790e-09, /* 0x3E21EE9E, 0xBDB4B1C4 */
58*05a0b428SJohn Marino C6 = -1.13596475577881948265e-11; /* 0xBDA8FAE9, 0xBE8838D4 */
59*05a0b428SJohn Marino
60*05a0b428SJohn Marino double
__kernel_cos(double x,double y)61*05a0b428SJohn Marino __kernel_cos(double x, double y)
62*05a0b428SJohn Marino {
63*05a0b428SJohn Marino double a,hz,z,r,qx;
64*05a0b428SJohn Marino int32_t ix;
65*05a0b428SJohn Marino GET_HIGH_WORD(ix,x);
66*05a0b428SJohn Marino ix &= 0x7fffffff; /* ix = |x|'s high word*/
67*05a0b428SJohn Marino if(ix<0x3e400000) { /* if x < 2**27 */
68*05a0b428SJohn Marino if(((int)x)==0) return one; /* generate inexact */
69*05a0b428SJohn Marino }
70*05a0b428SJohn Marino z = x*x;
71*05a0b428SJohn Marino r = z*(C1+z*(C2+z*(C3+z*(C4+z*(C5+z*C6)))));
72*05a0b428SJohn Marino if(ix < 0x3FD33333) /* if |x| < 0.3 */
73*05a0b428SJohn Marino return one - (0.5*z - (z*r - x*y));
74*05a0b428SJohn Marino else {
75*05a0b428SJohn Marino if(ix > 0x3fe90000) { /* x > 0.78125 */
76*05a0b428SJohn Marino qx = 0.28125;
77*05a0b428SJohn Marino } else {
78*05a0b428SJohn Marino INSERT_WORDS(qx,ix-0x00200000,0); /* x/4 */
79*05a0b428SJohn Marino }
80*05a0b428SJohn Marino hz = 0.5*z-qx;
81*05a0b428SJohn Marino a = one-qx;
82*05a0b428SJohn Marino return a - (hz - (z*r-x*y));
83*05a0b428SJohn Marino }
84*05a0b428SJohn Marino }
85