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 ```/* origin: FreeBSD /usr/src/lib/msun/src/k_cos.c */ ``` ```/* ``` ``` * ==================================================== ``` ``` * 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. ``` ``` * ==================================================== ``` ``` */ ``` ```/* ``` ``` * __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 "libm.h" ``` ``` ``` ```static const double ``` ```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 __cos(double x, double y) ``` ```{ ``` ``` double_t hz,z,r,w; ``` ``` ``` ``` z = x*x; ``` ``` w = z*z; ``` ``` r = z*(C1+z*(C2+z*C3)) + w*w*(C4+z*(C5+z*C6)); ``` ``` hz = 0.5*z; ``` ``` w = 1.0-hz; ``` ``` return w + (((1.0-w)-hz) + (z*r-x*y)); ``` ```} ``` ``` ```