k_cos.c (2900B)
1 /* 2 * ==================================================== 3 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. 4 * 5 * Developed at SunPro, a Sun Microsystems, Inc. business. 6 * Permission to use, copy, modify, and distribute this 7 * software is freely granted, provided that this notice 8 * is preserved. 9 * ==================================================== 10 */ 11 12 /* 13 * __kernel_cos( x, y ) 14 * kernel cos function on [-pi/4, pi/4], pi/4 ~ 0.785398164 15 * Input x is assumed to be bounded by ~pi/4 in magnitude. 16 * Input y is the tail of x. 17 * 18 * Algorithm 19 * 1. Since cos(-x) = cos(x), we need only to consider positive x. 20 * 2. if x < 2^-27 (hx<0x3e400000 0), return 1 with inexact if x!=0. 21 * 3. cos(x) is approximated by a polynomial of degree 14 on 22 * [0,pi/4] 23 * 4 14 24 * cos(x) ~ 1 - x*x/2 + C1*x + ... + C6*x 25 * where the remez error is 26 * 27 * | 2 4 6 8 10 12 14 | -58 28 * |cos(x)-(1-.5*x +C1*x +C2*x +C3*x +C4*x +C5*x +C6*x )| <= 2 29 * | | 30 * 31 * 4 6 8 10 12 14 32 * 4. let r = C1*x +C2*x +C3*x +C4*x +C5*x +C6*x , then 33 * cos(x) = 1 - x*x/2 + r 34 * since cos(x+y) ~ cos(x) - sin(x)*y 35 * ~ cos(x) - x*y, 36 * a correction term is necessary in cos(x) and hence 37 * cos(x+y) = 1 - (x*x/2 - (r - x*y)) 38 * For better accuracy when x > 0.3, let qx = |x|/4 with 39 * the last 32 bits mask off, and if x > 0.78125, let qx = 0.28125. 40 * Then 41 * cos(x+y) = (1-qx) - ((x*x/2-qx) - (r-x*y)). 42 * Note that 1-qx and (x*x/2-qx) is EXACT here, and the 43 * magnitude of the latter is at least a quarter of x*x/2, 44 * thus, reducing the rounding error in the subtraction. 45 */ 46 47 #include "math_libm.h" 48 #include "math_private.h" 49 50 static const double 51 one = 1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */ 52 C1 = 4.16666666666666019037e-02, /* 0x3FA55555, 0x5555554C */ 53 C2 = -1.38888888888741095749e-03, /* 0xBF56C16C, 0x16C15177 */ 54 C3 = 2.48015872894767294178e-05, /* 0x3EFA01A0, 0x19CB1590 */ 55 C4 = -2.75573143513906633035e-07, /* 0xBE927E4F, 0x809C52AD */ 56 C5 = 2.08757232129817482790e-09, /* 0x3E21EE9E, 0xBDB4B1C4 */ 57 C6 = -1.13596475577881948265e-11; /* 0xBDA8FAE9, 0xBE8838D4 */ 58 59 double attribute_hidden __kernel_cos(double x, double y) 60 { 61 double a,hz,z,r,qx; 62 int32_t ix; 63 GET_HIGH_WORD(ix,x); 64 ix &= 0x7fffffff; /* ix = |x|'s high word*/ 65 if(ix<0x3e400000) { /* if x < 2**27 */ 66 if(((int)x)==0) return one; /* generate inexact */ 67 } 68 z = x*x; 69 r = z*(C1+z*(C2+z*(C3+z*(C4+z*(C5+z*C6))))); 70 if(ix < 0x3FD33333) /* if |x| < 0.3 */ 71 return one - (0.5*z - (z*r - x*y)); 72 else { 73 if(ix > 0x3fe90000) { /* x > 0.78125 */ 74 qx = 0.28125; 75 } else { 76 INSERT_WORDS(qx,ix-0x00200000,0); /* x/4 */ 77 } 78 hz = 0.5*z-qx; 79 a = one-qx; 80 return a - (hz - (z*r-x*y)); 81 } 82 }