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k_tan.c
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00001 /* -*- Mode: C; tab-width: 8; indent-tabs-mode: nil; c-basic-offset: 4 -*-
00002  *
00003  * ***** BEGIN LICENSE BLOCK *****
00004  * Version: MPL 1.1/GPL 2.0/LGPL 2.1
00005  *
00006  * The contents of this file are subject to the Mozilla Public License Version
00007  * 1.1 (the "License"); you may not use this file except in compliance with
00008  * the License. You may obtain a copy of the License at
00009  * http://www.mozilla.org/MPL/
00010  *
00011  * Software distributed under the License is distributed on an "AS IS" basis,
00012  * WITHOUT WARRANTY OF ANY KIND, either express or implied. See the License
00013  * for the specific language governing rights and limitations under the
00014  * License.
00015  *
00016  * The Original Code is Mozilla Communicator client code, released
00017  * March 31, 1998.
00018  *
00019  * The Initial Developer of the Original Code is
00020  * Sun Microsystems, Inc.
00021  * Portions created by the Initial Developer are Copyright (C) 1998
00022  * the Initial Developer. All Rights Reserved.
00023  *
00024  * Contributor(s):
00025  *
00026  * Alternatively, the contents of this file may be used under the terms of
00027  * either of the GNU General Public License Version 2 or later (the "GPL"),
00028  * or the GNU Lesser General Public License Version 2.1 or later (the "LGPL"),
00029  * in which case the provisions of the GPL or the LGPL are applicable instead
00030  * of those above. If you wish to allow use of your version of this file only
00031  * under the terms of either the GPL or the LGPL, and not to allow others to
00032  * use your version of this file under the terms of the MPL, indicate your
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00034  * and other provisions required by the GPL or the LGPL. If you do not delete
00035  * the provisions above, a recipient may use your version of this file under
00036  * the terms of any one of the MPL, the GPL or the LGPL.
00037  *
00038  * ***** END LICENSE BLOCK ***** */
00039 
00040 /* @(#)k_tan.c 1.3 95/01/18 */
00041 /*
00042  * ====================================================
00043  * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
00044  *
00045  * Developed at SunSoft, a Sun Microsystems, Inc. business.
00046  * Permission to use, copy, modify, and distribute this
00047  * software is freely granted, provided that this notice 
00048  * is preserved.
00049  * ====================================================
00050  */
00051 
00052 /* __kernel_tan( x, y, k )
00053  * kernel tan function on [-pi/4, pi/4], pi/4 ~ 0.7854
00054  * Input x is assumed to be bounded by ~pi/4 in magnitude.
00055  * Input y is the tail of x.
00056  * Input k indicates whether tan (if k=1) or 
00057  * -1/tan (if k= -1) is returned.
00058  *
00059  * Algorithm
00060  *     1. Since tan(-x) = -tan(x), we need only to consider positive x. 
00061  *     2. if x < 2^-28 (hx<0x3e300000 0), return x with inexact if x!=0.
00062  *     3. tan(x) is approximated by a odd polynomial of degree 27 on
00063  *        [0,0.67434]
00064  *                            3             27
00065  *            tan(x) ~ x + T1*x + ... + T13*x
00066  *        where
00067  *     
00068  *             |tan(x)         2     4            26   |     -59.2
00069  *             |----- - (1+T1*x +T2*x +.... +T13*x    )| <= 2
00070  *             |  x                              | 
00071  * 
00072  *        Note: tan(x+y) = tan(x) + tan'(x)*y
00073  *                      ~ tan(x) + (1+x*x)*y
00074  *        Therefore, for better accuracy in computing tan(x+y), let 
00075  *                 3      2      2       2       2
00076  *            r = x *(T2+x *(T3+x *(...+x *(T12+x *T13))))
00077  *        then
00078  *                              3    2
00079  *            tan(x+y) = x + (T1*x + (x *(r+y)+y))
00080  *
00081  *      4. For x in [0.67434,pi/4],  let y = pi/4 - x, then
00082  *            tan(x) = tan(pi/4-y) = (1-tan(y))/(1+tan(y))
00083  *                   = 1 - 2*(tan(y) - (tan(y)^2)/(1+tan(y)))
00084  */
00085 
00086 #include "fdlibm.h"
00087 #ifdef __STDC__
00088 static const double 
00089 #else
00090 static double 
00091 #endif
00092 one   =  1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */
00093 pio4  =  7.85398163397448278999e-01, /* 0x3FE921FB, 0x54442D18 */
00094 pio4lo=  3.06161699786838301793e-17, /* 0x3C81A626, 0x33145C07 */
00095 T[] =  {
00096   3.33333333333334091986e-01, /* 0x3FD55555, 0x55555563 */
00097   1.33333333333201242699e-01, /* 0x3FC11111, 0x1110FE7A */
00098   5.39682539762260521377e-02, /* 0x3FABA1BA, 0x1BB341FE */
00099   2.18694882948595424599e-02, /* 0x3F9664F4, 0x8406D637 */
00100   8.86323982359930005737e-03, /* 0x3F8226E3, 0xE96E8493 */
00101   3.59207910759131235356e-03, /* 0x3F6D6D22, 0xC9560328 */
00102   1.45620945432529025516e-03, /* 0x3F57DBC8, 0xFEE08315 */
00103   5.88041240820264096874e-04, /* 0x3F4344D8, 0xF2F26501 */
00104   2.46463134818469906812e-04, /* 0x3F3026F7, 0x1A8D1068 */
00105   7.81794442939557092300e-05, /* 0x3F147E88, 0xA03792A6 */
00106   7.14072491382608190305e-05, /* 0x3F12B80F, 0x32F0A7E9 */
00107  -1.85586374855275456654e-05, /* 0xBEF375CB, 0xDB605373 */
00108   2.59073051863633712884e-05, /* 0x3EFB2A70, 0x74BF7AD4 */
00109 };
00110 
00111 #ifdef __STDC__
00112        double __kernel_tan(double x, double y, int iy)
00113 #else
00114        double __kernel_tan(x, y, iy)
00115        double x,y; int iy;
00116 #endif
00117 {
00118         fd_twoints u;
00119        double z,r,v,w,s;
00120        int ix,hx;
00121         u.d = x;
00122        hx = __HI(u); /* high word of x */
00123        ix = hx&0x7fffffff;  /* high word of |x| */
00124        if(ix<0x3e300000)                  /* x < 2**-28 */
00125            {if((int)x==0) {               /* generate inexact */
00126                 u.d =x;
00127               if(((ix|__LO(u))|(iy+1))==0) return one/fd_fabs(x);
00128               else return (iy==1)? x: -one/x;
00129            }
00130            }
00131        if(ix>=0x3FE59428) {                      /* |x|>=0.6744 */
00132            if(hx<0) {x = -x; y = -y;}
00133            z = pio4-x;
00134            w = pio4lo-y;
00135            x = z+w; y = 0.0;
00136        }
00137        z      =  x*x;
00138        w      =  z*z;
00139     /* Break x^5*(T[1]+x^2*T[2]+...) into
00140      *   x^5(T[1]+x^4*T[3]+...+x^20*T[11]) +
00141      *   x^5(x^2*(T[2]+x^4*T[4]+...+x^22*[T12]))
00142      */
00143        r = T[1]+w*(T[3]+w*(T[5]+w*(T[7]+w*(T[9]+w*T[11]))));
00144        v = z*(T[2]+w*(T[4]+w*(T[6]+w*(T[8]+w*(T[10]+w*T[12])))));
00145        s = z*x;
00146        r = y + z*(s*(r+v)+y);
00147        r += T[0]*s;
00148        w = x+r;
00149        if(ix>=0x3FE59428) {
00150            v = (double)iy;
00151            return (double)(1-((hx>>30)&2))*(v-2.0*(x-(w*w/(w+v)-r)));
00152        }
00153        if(iy==1) return w;
00154        else {        /* if allow error up to 2 ulp, 
00155                         simply return -1.0/(x+r) here */
00156      /*  compute -1.0/(x+r) accurately */
00157            double a,t;
00158            z  = w;
00159             u.d = z;
00160            __LO(u) = 0;
00161             z = u.d;
00162            v  = r-(z - x);  /* z+v = r+x */
00163            t = a  = -1.0/w; /* a = -1.0/w */
00164             u.d = t;
00165            __LO(u) = 0;
00166             t = u.d;
00167            s  = 1.0+t*z;
00168            return t+a*(s+t*v);
00169        }
00170 }