1 2 /* @(#)k_cos.c 1.3 95/01/18 */ 3 /* 4 * ==================================================== 5 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. 6 * 7 * Developed at SunSoft, a Sun Microsystems, Inc. business. 8 * Permission to use, copy, modify, and distribute this 9 * software is freely granted, provided that this notice 10 * is preserved. 11 * ==================================================== 12 */ 13 14 #include <sys/cdefs.h> 15 __FBSDID("$FreeBSD$"); 16 17 /* 18 * __kernel_cos( x, y ) 19 * kernel cos function on [-pi/4, pi/4], pi/4 ~ 0.785398164 20 * Input x is assumed to be bounded by ~pi/4 in magnitude. 21 * Input y is the tail of x. 22 * 23 * Algorithm 24 * 1. Since cos(-x) = cos(x), we need only to consider positive x. 25 * 2. if x < 2^-27 (hx<0x3e400000 0), return 1 with inexact if x!=0. 26 * 3. cos(x) is approximated by a polynomial of degree 14 on 27 * [0,pi/4] 28 * 4 14 29 * cos(x) ~ 1 - x*x/2 + C1*x + ... + C6*x 30 * where the remez error is 31 * 32 * | 2 4 6 8 10 12 14 | -58 33 * |cos(x)-(1-.5*x +C1*x +C2*x +C3*x +C4*x +C5*x +C6*x )| <= 2 34 * | | 35 * 36 * 4 6 8 10 12 14 37 * 4. let r = C1*x +C2*x +C3*x +C4*x +C5*x +C6*x , then 38 * cos(x) ~ 1 - x*x/2 + r 39 * since cos(x+y) ~ cos(x) - sin(x)*y 40 * ~ cos(x) - x*y, 41 * a correction term is necessary in cos(x) and hence 42 * cos(x+y) = 1 - (x*x/2 - (r - x*y)) 43 * For better accuracy, rearrange to 44 * cos(x+y) ~ w + (tmp + (r-x*y)) 45 * where w = 1 - x*x/2 and tmp is a tiny correction term 46 * (1 - x*x/2 == w + tmp exactly in infinite precision). 47 * The exactness of w + tmp in infinite precision depends on w 48 * and tmp having the same precision as x. If they have extra 49 * precision due to compiler bugs, then the extra precision is 50 * only good provided it is retained in all terms of the final 51 * expression for cos(). Retention happens in all cases tested 52 * under FreeBSD, so don't pessimize things by forcibly clipping 53 * any extra precision in w. 54 */ 55 56 #include "math.h" 57 #include "math_private.h" 58 59 static const double 60 one = 1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */ 61 C1 = 4.16666666666666019037e-02, /* 0x3FA55555, 0x5555554C */ 62 C2 = -1.38888888888741095749e-03, /* 0xBF56C16C, 0x16C15177 */ 63 C3 = 2.48015872894767294178e-05, /* 0x3EFA01A0, 0x19CB1590 */ 64 C4 = -2.75573143513906633035e-07, /* 0xBE927E4F, 0x809C52AD */ 65 C5 = 2.08757232129817482790e-09, /* 0x3E21EE9E, 0xBDB4B1C4 */ 66 C6 = -1.13596475577881948265e-11; /* 0xBDA8FAE9, 0xBE8838D4 */ 67 68 double 69 __kernel_cos(double x, double y) 70 { 71 double hz,z,r,w; 72 73 z = x*x; 74 w = z*z; 75 r = z*(C1+z*(C2+z*C3)) + w*w*(C4+z*(C5+z*C6)); 76 hz = 0.5*z; 77 w = one-hz; 78 return w + (((one-w)-hz) + (z*r-x*y)); 79 } 80