1 2 /* @(#)k_cos.c 1.3 95/01/18 */ 3 /* 4 * ==================================================== 5 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. 6 * 7 * Developed at SunSoft, a Sun Microsystems, Inc. business. 8 * Permission to use, copy, modify, and distribute this 9 * software is freely granted, provided that this notice 10 * is preserved. 11 * ==================================================== 12 */ 13 14 /* 15 * __kernel_cos( x, y ) 16 * kernel cos function on [-pi/4, pi/4], pi/4 ~ 0.785398164 17 * Input x is assumed to be bounded by ~pi/4 in magnitude. 18 * Input y is the tail of x. 19 * 20 * Algorithm 21 * 1. Since ieee_cos(-x) = ieee_cos(x), we need only to consider positive x. 22 * 2. if x < 2^-27 (hx<0x3e400000 0), return 1 with inexact if x!=0. 23 * 3. ieee_cos(x) is approximated by a polynomial of degree 14 on 24 * [0,pi/4] 25 * 4 14 26 * cos(x) ~ 1 - x*x/2 + C1*x + ... + C6*x 27 * where the remez error is 28 * 29 * | 2 4 6 8 10 12 14 | -58 30 * |ieee_cos(x)-(1-.5*x +C1*x +C2*x +C3*x +C4*x +C5*x +C6*x )| <= 2 31 * | | 32 * 33 * 4 6 8 10 12 14 34 * 4. let r = C1*x +C2*x +C3*x +C4*x +C5*x +C6*x , then 35 * ieee_cos(x) = 1 - x*x/2 + r 36 * since ieee_cos(x+y) ~ ieee_cos(x) - ieee_sin(x)*y 37 * ~ ieee_cos(x) - x*y, 38 * a correction term is necessary in ieee_cos(x) and hence 39 * cos(x+y) = 1 - (x*x/2 - (r - x*y)) 40 * For better accuracy when x > 0.3, let qx = |x|/4 with 41 * the last 32 bits mask off, and if x > 0.78125, let qx = 0.28125. 42 * Then 43 * cos(x+y) = (1-qx) - ((x*x/2-qx) - (r-x*y)). 44 * Note that 1-qx and (x*x/2-qx) is EXACT here, and the 45 * magnitude of the latter is at least a quarter of x*x/2, 46 * thus, reducing the rounding error in the subtraction. 47 */ 48 49 #include "fdlibm.h" 50 51 #ifdef __STDC__ 52 static const double 53 #else 54 static double 55 #endif 56 one = 1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */ 57 C1 = 4.16666666666666019037e-02, /* 0x3FA55555, 0x5555554C */ 58 C2 = -1.38888888888741095749e-03, /* 0xBF56C16C, 0x16C15177 */ 59 C3 = 2.48015872894767294178e-05, /* 0x3EFA01A0, 0x19CB1590 */ 60 C4 = -2.75573143513906633035e-07, /* 0xBE927E4F, 0x809C52AD */ 61 C5 = 2.08757232129817482790e-09, /* 0x3E21EE9E, 0xBDB4B1C4 */ 62 C6 = -1.13596475577881948265e-11; /* 0xBDA8FAE9, 0xBE8838D4 */ 63 64 #ifdef __STDC__ __kernel_cos(double x,double y)65 double __kernel_cos(double x, double y) 66 #else 67 double __kernel_cos(x, y) 68 double x,y; 69 #endif 70 { 71 double a,hz,z,r,qx; 72 int ix; 73 ix = __HI(x)&0x7fffffff; /* ix = |x|'s high word*/ 74 if(ix<0x3e400000) { /* if x < 2**27 */ 75 if(((int)x)==0) return one; /* generate inexact */ 76 } 77 z = x*x; 78 r = z*(C1+z*(C2+z*(C3+z*(C4+z*(C5+z*C6))))); 79 if(ix < 0x3FD33333) /* if |x| < 0.3 */ 80 return one - (0.5*z - (z*r - x*y)); 81 else { 82 if(ix > 0x3fe90000) { /* x > 0.78125 */ 83 qx = 0.28125; 84 } else { 85 __HI(qx) = ix-0x00200000; /* x/4 */ 86 __LO(qx) = 0; 87 } 88 hz = 0.5*z-qx; 89 a = one-qx; 90 return a - (hz - (z*r-x*y)); 91 } 92 } 93