1 /* derived from /netlib/fdlibm */ 2 3 /* @(#)k_cos.c 1.3 95/01/18 */ 4 /* 5 * ==================================================== 6 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. 7 * 8 * Developed at SunSoft, a Sun Microsystems, Inc. business. 9 * Permission to use, copy, modify, and distribute this 10 * software is freely granted, provided that this notice 11 * is preserved. 12 * ==================================================== 13 */ 14 15 /* 16 * __kernel_cos( x, y ) 17 * kernel cos function on [-pi/4, pi/4], pi/4 ~ 0.785398164 18 * Input x is assumed to be bounded by ~pi/4 in magnitude. 19 * Input y is the tail of x. 20 * 21 * Algorithm 22 * 1. Since cos(-x) = cos(x), we need only to consider positive x. 23 * 2. if x < 2^-27 (hx<0x3e400000 0), return 1 with inexact if x!=0. 24 * 3. cos(x) is approximated by a polynomial of degree 14 on 25 * [0,pi/4] 26 * 4 14 27 * cos(x) ~ 1 - x*x/2 + C1*x + ... + C6*x 28 * where the remez error is 29 * 30 * | 2 4 6 8 10 12 14 | -58 31 * |cos(x)-(1-.5*x +C1*x +C2*x +C3*x +C4*x +C5*x +C6*x )| <= 2 32 * | | 33 * 34 * 4 6 8 10 12 14 35 * 4. let r = C1*x +C2*x +C3*x +C4*x +C5*x +C6*x , then 36 * cos(x) = 1 - x*x/2 + r 37 * since cos(x+y) ~ cos(x) - sin(x)*y 38 * ~ cos(x) - x*y, 39 * a correction term is necessary in cos(x) and hence 40 * cos(x+y) = 1 - (x*x/2 - (r - x*y)) 41 * For better accuracy when x > 0.3, let qx = |x|/4 with 42 * the last 32 bits mask off, and if x > 0.78125, let qx = 0.28125. 43 * Then 44 * cos(x+y) = (1-qx) - ((x*x/2-qx) - (r-x*y)). 45 * Note that 1-qx and (x*x/2-qx) is EXACT here, and the 46 * magnitude of the latter is at least a quarter of x*x/2, 47 * thus, reducing the rounding error in the subtraction. 48 */ 49 50 #include "fdlibm.h" 51 52 static const double 53 one = 1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */ 54 C1 = 4.16666666666666019037e-02, /* 0x3FA55555, 0x5555554C */ 55 C2 = -1.38888888888741095749e-03, /* 0xBF56C16C, 0x16C15177 */ 56 C3 = 2.48015872894767294178e-05, /* 0x3EFA01A0, 0x19CB1590 */ 57 C4 = -2.75573143513906633035e-07, /* 0xBE927E4F, 0x809C52AD */ 58 C5 = 2.08757232129817482790e-09, /* 0x3E21EE9E, 0xBDB4B1C4 */ 59 C6 = -1.13596475577881948265e-11; /* 0xBDA8FAE9, 0xBE8838D4 */ 60 61 double __kernel_cos(double x, double y) 62 { 63 double a,hz,z,r,qx; 64 int ix; 65 ix = __HI(x)&0x7fffffff; /* ix = |x|'s high word*/ 66 if(ix<0x3e400000) { /* if x < 2**27 */ 67 if(((int)x)==0) return one; /* generate inexact */ 68 } 69 z = x*x; 70 r = z*(C1+z*(C2+z*(C3+z*(C4+z*(C5+z*C6))))); 71 if(ix < 0x3FD33333) /* if |x| < 0.3 */ 72 return one - (0.5*z - (z*r - x*y)); 73 else { 74 if(ix > 0x3fe90000) { /* x > 0.78125 */ 75 qx = 0.28125; 76 } else { 77 __HI(qx) = ix-0x00200000; /* x/4 */ 78 __LO(qx) = 0; 79 } 80 hz = 0.5*z-qx; 81 a = one-qx; 82 return a - (hz - (z*r-x*y)); 83 } 84 } 85