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glibc  2.9
k_tan.c
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00001 /* @(#)k_tan.c 5.1 93/09/24 */
00002 /*
00003  * ====================================================
00004  * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
00005  *
00006  * Developed at SunPro, a Sun Microsystems, Inc. business.
00007  * Permission to use, copy, modify, and distribute this
00008  * software is freely granted, provided that this notice
00009  * is preserved.
00010  * ====================================================
00011  */
00012 /* Modified by Naohiko Shimizu/Tokai University, Japan 1997/08/25,
00013    for performance improvement on pipelined processors.
00014 */
00015 
00016 #if defined(LIBM_SCCS) && !defined(lint)
00017 static char rcsid[] = "$NetBSD: k_tan.c,v 1.8 1995/05/10 20:46:37 jtc Exp $";
00018 #endif
00019 
00020 /* __kernel_tan( x, y, k )
00021  * kernel tan function on [-pi/4, pi/4], pi/4 ~ 0.7854
00022  * Input x is assumed to be bounded by ~pi/4 in magnitude.
00023  * Input y is the tail of x.
00024  * Input k indicates whether tan (if k=1) or
00025  * -1/tan (if k= -1) is returned.
00026  *
00027  * Algorithm
00028  *     1. Since tan(-x) = -tan(x), we need only to consider positive x.
00029  *     2. if x < 2^-28 (hx<0x3e300000 0), return x with inexact if x!=0.
00030  *     3. tan(x) is approximated by a odd polynomial of degree 27 on
00031  *        [0,0.67434]
00032  *                            3             27
00033  *            tan(x) ~ x + T1*x + ... + T13*x
00034  *        where
00035  *
00036  *             |tan(x)         2     4            26   |     -59.2
00037  *             |----- - (1+T1*x +T2*x +.... +T13*x    )| <= 2
00038  *             |  x                              |
00039  *
00040  *        Note: tan(x+y) = tan(x) + tan'(x)*y
00041  *                      ~ tan(x) + (1+x*x)*y
00042  *        Therefore, for better accuracy in computing tan(x+y), let
00043  *                 3      2      2       2       2
00044  *            r = x *(T2+x *(T3+x *(...+x *(T12+x *T13))))
00045  *        then
00046  *                              3    2
00047  *            tan(x+y) = x + (T1*x + (x *(r+y)+y))
00048  *
00049  *      4. For x in [0.67434,pi/4],  let y = pi/4 - x, then
00050  *            tan(x) = tan(pi/4-y) = (1-tan(y))/(1+tan(y))
00051  *                   = 1 - 2*(tan(y) - (tan(y)^2)/(1+tan(y)))
00052  */
00053 
00054 #include "math.h"
00055 #include "math_private.h"
00056 #ifdef __STDC__
00057 static const double
00058 #else
00059 static double
00060 #endif
00061 one   =  1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */
00062 pio4  =  7.85398163397448278999e-01, /* 0x3FE921FB, 0x54442D18 */
00063 pio4lo=  3.06161699786838301793e-17, /* 0x3C81A626, 0x33145C07 */
00064 T[] =  {
00065   3.33333333333334091986e-01, /* 0x3FD55555, 0x55555563 */
00066   1.33333333333201242699e-01, /* 0x3FC11111, 0x1110FE7A */
00067   5.39682539762260521377e-02, /* 0x3FABA1BA, 0x1BB341FE */
00068   2.18694882948595424599e-02, /* 0x3F9664F4, 0x8406D637 */
00069   8.86323982359930005737e-03, /* 0x3F8226E3, 0xE96E8493 */
00070   3.59207910759131235356e-03, /* 0x3F6D6D22, 0xC9560328 */
00071   1.45620945432529025516e-03, /* 0x3F57DBC8, 0xFEE08315 */
00072   5.88041240820264096874e-04, /* 0x3F4344D8, 0xF2F26501 */
00073   2.46463134818469906812e-04, /* 0x3F3026F7, 0x1A8D1068 */
00074   7.81794442939557092300e-05, /* 0x3F147E88, 0xA03792A6 */
00075   7.14072491382608190305e-05, /* 0x3F12B80F, 0x32F0A7E9 */
00076  -1.85586374855275456654e-05, /* 0xBEF375CB, 0xDB605373 */
00077   2.59073051863633712884e-05, /* 0x3EFB2A70, 0x74BF7AD4 */
00078 };
00079 
00080 #ifdef __STDC__
00081        double __kernel_tan(double x, double y, int iy)
00082 #else
00083        double __kernel_tan(x, y, iy)
00084        double x,y; int iy;
00085 #endif
00086 {
00087        double z,r,v,w,s,r1,r2,r3,v1,v2,v3,w2,w4;
00088        int32_t ix,hx;
00089        GET_HIGH_WORD(hx,x);
00090        ix = hx&0x7fffffff;  /* high word of |x| */
00091        if(ix<0x3e300000)                  /* x < 2**-28 */
00092            {if((int)x==0) {               /* generate inexact */
00093                u_int32_t low;
00094               GET_LOW_WORD(low,x);
00095               if(((ix|low)|(iy+1))==0) return one/fabs(x);
00096               else return (iy==1)? x: -one/x;
00097            }
00098            }
00099        if(ix>=0x3FE59428) {                      /* |x|>=0.6744 */
00100            if(hx<0) {x = -x; y = -y;}
00101            z = pio4-x;
00102            w = pio4lo-y;
00103            x = z+w; y = 0.0;
00104        }
00105        z      =  x*x;
00106        w      =  z*z;
00107     /* Break x^5*(T[1]+x^2*T[2]+...) into
00108      *   x^5(T[1]+x^4*T[3]+...+x^20*T[11]) +
00109      *   x^5(x^2*(T[2]+x^4*T[4]+...+x^22*[T12]))
00110      */
00111 #ifdef DO_NOT_USE_THIS
00112        r = T[1]+w*(T[3]+w*(T[5]+w*(T[7]+w*(T[9]+w*T[11]))));
00113        v = z*(T[2]+w*(T[4]+w*(T[6]+w*(T[8]+w*(T[10]+w*T[12])))));
00114 #else
00115        v1 = T[10]+w*T[12]; w2=w*w;
00116        v2 = T[6]+w*T[8]; w4=w2*w2;
00117        v3 = T[2]+w*T[4]; v1=z*v1;
00118        r1 = T[9]+w*T[11]; v2=z*v2;
00119        r2 = T[5]+w*T[7]; v3=z*v3;
00120        r3 = T[1]+w*T[3];
00121        v  = v3 + w2*v2 + w4*v1;
00122        r = r3 + w2*r2 + w4*r1;
00123 #endif
00124        s = z*x;
00125        r = y + z*(s*(r+v)+y);
00126        r += T[0]*s;
00127        w = x+r;
00128        if(ix>=0x3FE59428) {
00129            v = (double)iy;
00130            return (double)(1-((hx>>30)&2))*(v-2.0*(x-(w*w/(w+v)-r)));
00131        }
00132        if(iy==1) return w;
00133        else {        /* if allow error up to 2 ulp,
00134                         simply return -1.0/(x+r) here */
00135      /*  compute -1.0/(x+r) accurately */
00136            double a,t;
00137            z  = w;
00138            SET_LOW_WORD(z,0);
00139            v  = r-(z - x);  /* z+v = r+x */
00140            t = a  = -1.0/w; /* a = -1.0/w */
00141            SET_LOW_WORD(t,0);
00142            s  = 1.0+t*z;
00143            return t+a*(s+t*v);
00144        }
00145 }