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 /* @(#)k_cos.c 1.3 95/01/18 */ /* * ==================================================== * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. * * Developed at SunSoft, a Sun Microsystems, Inc. business. * Permission to use, copy, modify, and distribute this * software is freely granted, provided that this notice * is preserved. * ==================================================== */ #ifndef lint static char rcsid[] = "\$FreeBSD: src/lib/msun/src/k_cos.c,v 1.10 2005/10/26 12:36:18 bde Exp \$"; #endif /* * __kernel_cos( x, y ) * kernel cos function on [-pi/4, pi/4], pi/4 ~ 0.785398164 * Input x is assumed to be bounded by ~pi/4 in magnitude. * Input y is the tail of x. * * Algorithm * 1. Since cos(-x) = cos(x), we need only to consider positive x. * 2. if x < 2^-27 (hx<0x3e400000 0), return 1 with inexact if x!=0. * 3. cos(x) is approximated by a polynomial of degree 14 on * [0,pi/4] * 4 14 * cos(x) ~ 1 - x*x/2 + C1*x + ... + C6*x * where the remez error is * * | 2 4 6 8 10 12 14 | -58 * |cos(x)-(1-.5*x +C1*x +C2*x +C3*x +C4*x +C5*x +C6*x )| <= 2 * | | * * 4 6 8 10 12 14 * 4. let r = C1*x +C2*x +C3*x +C4*x +C5*x +C6*x , then * cos(x) ~ 1 - x*x/2 + r * since cos(x+y) ~ cos(x) - sin(x)*y * ~ cos(x) - x*y, * a correction term is necessary in cos(x) and hence * cos(x+y) = 1 - (x*x/2 - (r - x*y)) * For better accuracy, rearrange to * cos(x+y) ~ w + (tmp + (r-x*y)) * where w = 1 - x*x/2 and tmp is a tiny correction term * (1 - x*x/2 == w + tmp exactly in infinite precision). * The exactness of w + tmp in infinite precision depends on w * and tmp having the same precision as x. If they have extra * precision due to compiler bugs, then the extra precision is * only good provided it is retained in all terms of the final * expression for cos(). Retention happens in all cases tested * under FreeBSD, so don't pessimize things by forcibly clipping * any extra precision in w. */ #include "math.h" #include "math_private.h" static const double one = 1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */ C1 = 4.16666666666666019037e-02, /* 0x3FA55555, 0x5555554C */ C2 = -1.38888888888741095749e-03, /* 0xBF56C16C, 0x16C15177 */ C3 = 2.48015872894767294178e-05, /* 0x3EFA01A0, 0x19CB1590 */ C4 = -2.75573143513906633035e-07, /* 0xBE927E4F, 0x809C52AD */ C5 = 2.08757232129817482790e-09, /* 0x3E21EE9E, 0xBDB4B1C4 */ C6 = -1.13596475577881948265e-11; /* 0xBDA8FAE9, 0xBE8838D4 */ double __kernel_cos(double x, double y) { double hz,z,r,w; z = x*x; r = z*(C1+z*(C2+z*(C3+z*(C4+z*(C5+z*C6))))); hz = (float)0.5*z; w = one-hz; return w + (((one-w)-hz) + (z*r-x*y)); }