155682Smarkm 255682Smarkm/* @(#)k_cos.c 1.3 95/01/18 */ 355682Smarkm/* 455682Smarkm * ==================================================== 555682Smarkm * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. 655682Smarkm * 755682Smarkm * Developed at SunSoft, a Sun Microsystems, Inc. business. 855682Smarkm * Permission to use, copy, modify, and distribute this 955682Smarkm * software is freely granted, provided that this notice 1055682Smarkm * is preserved. 1155682Smarkm * ==================================================== 1255682Smarkm */ 1355682Smarkm 1455682Smarkm#include <sys/cdefs.h> 1555682Smarkm__FBSDID("$FreeBSD$"); 1655682Smarkm 1755682Smarkm/* 1855682Smarkm * __kernel_cos( x, y ) 1955682Smarkm * kernel cos function on [-pi/4, pi/4], pi/4 ~ 0.785398164 2055682Smarkm * Input x is assumed to be bounded by ~pi/4 in magnitude. 2155682Smarkm * Input y is the tail of x. 2255682Smarkm * 2355682Smarkm * Algorithm 2455682Smarkm * 1. Since cos(-x) = cos(x), we need only to consider positive x. 2555682Smarkm * 2. if x < 2^-27 (hx<0x3e400000 0), return 1 with inexact if x!=0. 2655682Smarkm * 3. cos(x) is approximated by a polynomial of degree 14 on 2755682Smarkm * [0,pi/4] 2855682Smarkm * 4 14 2955682Smarkm * cos(x) ~ 1 - x*x/2 + C1*x + ... + C6*x 3055682Smarkm * where the remez error is 3155682Smarkm * 3255682Smarkm * | 2 4 6 8 10 12 14 | -58 3355682Smarkm * |cos(x)-(1-.5*x +C1*x +C2*x +C3*x +C4*x +C5*x +C6*x )| <= 2 3455682Smarkm * | | 35178825Sdfr * 3655682Smarkm * 4 6 8 10 12 14 3755682Smarkm * 4. let r = C1*x +C2*x +C3*x +C4*x +C5*x +C6*x , then 3855682Smarkm * cos(x) ~ 1 - x*x/2 + r 3955682Smarkm * since cos(x+y) ~ cos(x) - sin(x)*y 4055682Smarkm * ~ cos(x) - x*y, 4155682Smarkm * a correction term is necessary in cos(x) and hence 4255682Smarkm * cos(x+y) = 1 - (x*x/2 - (r - x*y)) 4355682Smarkm * For better accuracy, rearrange to 4455682Smarkm * cos(x+y) ~ w + (tmp + (r-x*y)) 4555682Smarkm * where w = 1 - x*x/2 and tmp is a tiny correction term 4655682Smarkm * (1 - x*x/2 == w + tmp exactly in infinite precision). 47178825Sdfr * The exactness of w + tmp in infinite precision depends on w 4855682Smarkm * and tmp having the same precision as x. If they have extra 4955682Smarkm * precision due to compiler bugs, then the extra precision is 5055682Smarkm * only good provided it is retained in all terms of the final 5155682Smarkm * expression for cos(). Retention happens in all cases tested 5255682Smarkm * under FreeBSD, so don't pessimize things by forcibly clipping 5355682Smarkm * any extra precision in w. 5455682Smarkm */ 5555682Smarkm 5655682Smarkm#include "math.h" 5755682Smarkm#include "math_private.h" 5855682Smarkm 5955682Smarkmstatic const double 6055682Smarkmone = 1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */ 6155682SmarkmC1 = 4.16666666666666019037e-02, /* 0x3FA55555, 0x5555554C */ 6255682SmarkmC2 = -1.38888888888741095749e-03, /* 0xBF56C16C, 0x16C15177 */ 6355682SmarkmC3 = 2.48015872894767294178e-05, /* 0x3EFA01A0, 0x19CB1590 */ 6455682SmarkmC4 = -2.75573143513906633035e-07, /* 0xBE927E4F, 0x809C52AD */ 6555682SmarkmC5 = 2.08757232129817482790e-09, /* 0x3E21EE9E, 0xBDB4B1C4 */ 6655682SmarkmC6 = -1.13596475577881948265e-11; /* 0xBDA8FAE9, 0xBE8838D4 */ 6755682Smarkm 6855682Smarkmdouble 6955682Smarkm__kernel_cos(double x, double y) 7055682Smarkm{ 7155682Smarkm double hz,z,r,w; 7255682Smarkm 7355682Smarkm z = x*x; 7455682Smarkm w = z*z; 7555682Smarkm r = z*(C1+z*(C2+z*C3)) + w*w*(C4+z*(C5+z*C6)); 7655682Smarkm hz = 0.5*z; 7755682Smarkm w = one-hz; 7855682Smarkm return w + (((one-w)-hz) + (z*r-x*y)); 7955682Smarkm} 8055682Smarkm