1 /* @(#)k_cos.c 5.1 93/09/24 */ 2 /* 3 * ==================================================== 4 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. 5 * 6 * Developed at SunPro, a Sun Microsystems, Inc. business. 7 * Permission to use, copy, modify, and distribute this 8 * software is freely granted, provided that this notice 9 * is preserved. 10 * ==================================================== 11 */ 12 13 #ifndef lint 14 static char rcsid[] = "$FreeBSD$"; 15 #endif 16 17 /* 18 * __kernel_cos( x, y ) 19 * kernel cos function on [-pi/4, pi/4], pi/4 ~ 0.785398164 20 * Input x is assumed to be bounded by ~pi/4 in magnitude. 21 * Input y is the tail of x. 22 * 23 * Algorithm 24 * 1. Since cos(-x) = cos(x), we need only to consider positive x. 25 * 2. if x < 2^-27 (hx<0x3e400000 0), return 1 with inexact if x!=0. 26 * 3. cos(x) is approximated by a polynomial of degree 14 on 27 * [0,pi/4] 28 * 4 14 29 * cos(x) ~ 1 - x*x/2 + C1*x + ... + C6*x 30 * where the remez error is 31 * 32 * | 2 4 6 8 10 12 14 | -58 33 * |cos(x)-(1-.5*x +C1*x +C2*x +C3*x +C4*x +C5*x +C6*x )| <= 2 34 * | | 35 * 36 * 4 6 8 10 12 14 37 * 4. let r = C1*x +C2*x +C3*x +C4*x +C5*x +C6*x , then 38 * cos(x) = 1 - x*x/2 + r 39 * since cos(x+y) ~ cos(x) - sin(x)*y 40 * ~ cos(x) - x*y, 41 * a correction term is necessary in cos(x) and hence 42 * cos(x+y) = 1 - (x*x/2 - (r - x*y)) 43 * For better accuracy when x > 0.3, let qx = |x|/4 with 44 * the last 32 bits mask off, and if x > 0.78125, let qx = 0.28125. 45 * Then 46 * cos(x+y) = (1-qx) - ((x*x/2-qx) - (r-x*y)). 47 * Note that 1-qx and (x*x/2-qx) is EXACT here, and the 48 * magnitude of the latter is at least a quarter of x*x/2, 49 * thus, reducing the rounding error in the subtraction. 50 */ 51 52 #include "math.h" 53 #include "math_private.h" 54 55 static const double 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 double 65 __kernel_cos(double x, double y) 66 { 67 double a,hz,z,r,qx; 68 int32_t ix; 69 GET_HIGH_WORD(ix,x); 70 ix &= 0x7fffffff; /* ix = |x|'s high word*/ 71 if(ix<0x3e400000) { /* if x < 2**27 */ 72 if(((int)x)==0) return one; /* generate inexact */ 73 } 74 z = x*x; 75 r = z*(C1+z*(C2+z*(C3+z*(C4+z*(C5+z*C6))))); 76 if(ix < 0x3FD33333) /* if |x| < 0.3 */ 77 return one - (0.5*z - (z*r - x*y)); 78 else { 79 if(ix > 0x3fe90000) { /* x > 0.78125 */ 80 qx = 0.28125; 81 } else { 82 INSERT_WORDS(qx,ix-0x00200000,0); /* x/4 */ 83 } 84 hz = 0.5*z-qx; 85 a = one-qx; 86 return a - (hz - (z*r-x*y)); 87 } 88 } 89