1 /* 2 * Copyright (c) 1998, 2001, Oracle and/or its affiliates. All rights reserved. 3 * DO NOT ALTER OR REMOVE COPYRIGHT NOTICES OR THIS FILE HEADER. 4 * 5 * This code is free software; you can redistribute it and/or modify it 6 * under the terms of the GNU General Public License version 2 only, as 7 * published by the Free Software Foundation. Oracle designates this 8 * particular file as subject to the "Classpath" exception as provided 9 * by Oracle in the LICENSE file that accompanied this code. 10 * 11 * This code is distributed in the hope that it will be useful, but WITHOUT 12 * ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or 13 * FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License 14 * version 2 for more details (a copy is included in the LICENSE file that 15 * accompanied this code). 16 * 17 * You should have received a copy of the GNU General Public License version 18 * 2 along with this work; if not, write to the Free Software Foundation, 19 * Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301 USA. 20 * 21 * Please contact Oracle, 500 Oracle Parkway, Redwood Shores, CA 94065 USA 22 * or visit www.oracle.com if you need additional information or have any 23 * questions. 24 */ 25 26 /* 27 * __kernel_cos( x, y ) 28 * kernel cos function on [-pi/4, pi/4], pi/4 ~ 0.785398164 29 * Input x is assumed to be bounded by ~pi/4 in magnitude. 30 * Input y is the tail of x. 31 * 32 * Algorithm 33 * 1. Since cos(-x) = cos(x), we need only to consider positive x. 34 * 2. if x < 2^-27 (hx<0x3e400000 0), return 1 with inexact if x!=0. 35 * 3. cos(x) is approximated by a polynomial of degree 14 on 36 * [0,pi/4] 37 * 4 14 38 * cos(x) ~ 1 - x*x/2 + C1*x + ... + C6*x 39 * where the remez error is 40 * 41 * | 2 4 6 8 10 12 14 | -58 42 * |cos(x)-(1-.5*x +C1*x +C2*x +C3*x +C4*x +C5*x +C6*x )| <= 2 43 * | | 44 * 45 * 4 6 8 10 12 14 46 * 4. let r = C1*x +C2*x +C3*x +C4*x +C5*x +C6*x , then 47 * cos(x) = 1 - x*x/2 + r 48 * since cos(x+y) ~ cos(x) - sin(x)*y 49 * ~ cos(x) - x*y, 50 * a correction term is necessary in cos(x) and hence 51 * cos(x+y) = 1 - (x*x/2 - (r - x*y)) 52 * For better accuracy when x > 0.3, let qx = |x|/4 with 53 * the last 32 bits mask off, and if x > 0.78125, let qx = 0.28125. 54 * Then 55 * cos(x+y) = (1-qx) - ((x*x/2-qx) - (r-x*y)). 56 * Note that 1-qx and (x*x/2-qx) is EXACT here, and the 57 * magnitude of the latter is at least a quarter of x*x/2, 58 * thus, reducing the rounding error in the subtraction. 59 */ 60 61 #include "fdlibm.h" 62 63 #ifdef __STDC__ 64 static const double 65 #else 66 static double 67 #endif 68 one = 1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */ 69 C1 = 4.16666666666666019037e-02, /* 0x3FA55555, 0x5555554C */ 70 C2 = -1.38888888888741095749e-03, /* 0xBF56C16C, 0x16C15177 */ 71 C3 = 2.48015872894767294178e-05, /* 0x3EFA01A0, 0x19CB1590 */ 72 C4 = -2.75573143513906633035e-07, /* 0xBE927E4F, 0x809C52AD */ 73 C5 = 2.08757232129817482790e-09, /* 0x3E21EE9E, 0xBDB4B1C4 */ 74 C6 = -1.13596475577881948265e-11; /* 0xBDA8FAE9, 0xBE8838D4 */ 75 76 #ifdef __STDC__ __kernel_cos(double x,double y)77 double __kernel_cos(double x, double y) 78 #else 79 double __kernel_cos(x, y) 80 double x,y; 81 #endif 82 { 83 double a,hz,z,r,qx; 84 int ix; 85 ix = __HI(x)&0x7fffffff; /* ix = |x|'s high word*/ 86 if(ix<0x3e400000) { /* if x < 2**27 */ 87 if(((int)x)==0) return one; /* generate inexact */ 88 } 89 z = x*x; 90 r = z*(C1+z*(C2+z*(C3+z*(C4+z*(C5+z*C6))))); 91 if(ix < 0x3FD33333) /* if |x| < 0.3 */ 92 return one - (0.5*z - (z*r - x*y)); 93 else { 94 if(ix > 0x3fe90000) { /* x > 0.78125 */ 95 qx = 0.28125; 96 } else { 97 __HI(qx) = ix-0x00200000; /* x/4 */ 98 __LO(qx) = 0; 99 } 100 hz = 0.5*z-qx; 101 a = one-qx; 102 return a - (hz - (z*r-x*y)); 103 } 104 } 105