| /* 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)); |
| } |