k_tan.c

/*
 *                               POK header
 * 
 * The following file is a part of the POK project. Any modification should
 * made according to the POK licence. You CANNOT use this file or a part of
 * this file is this part of a file for your own project
 *
 * For more information on the POK licence, please see our LICENCE FILE
 *
 * Please follow the coding guidelines described in doc/CODING_GUIDELINES
 *
 *                                      Copyright (c) 2007-2009 POK team 
 *
 * Created by julien on Sat Jan 31 20:12:07 2009 
 */

/* @(#)k_tan.c 5.1 93/09/24 */
/*
 * ====================================================
 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
 *
 * Developed at SunPro, a Sun Microsystems, Inc. business.
 * Permission to use, copy, modify, and distribute this
 * software is freely granted, provided that this notice
 * is preserved.
 * ====================================================
 */

#ifdef POK_NEEDS_LIBMATH

/* __kernel_tan( x, y, k )
 * kernel tan function on [-pi/4, pi/4], pi/4 ~ 0.7854
 * Input x is assumed to be bounded by ~pi/4 in magnitude.
 * Input y is the tail of x.
 * Input k indicates whether tan (if k=1) or
 * -1/tan (if k= -1) is returned.
 *
 * Algorithm
 *      1. Since tan(-x) = -tan(x), we need only to consider positive x.
 *      2. if x < 2^-28 (hx<0x3e300000 0), return x with inexact if x!=0.
 *      3. tan(x) is approximated by a odd polynomial of degree 27 on
 *         [0,0.67434]
 *                               3             27
 *              tan(x) ~ x + T1*x + ... + T13*x
 *         where
 *
 *              |tan(x)         2     4            26   |     -59.2
 *              |----- - (1+T1*x +T2*x +.... +T13*x    )| <= 2
 *              |  x                                    |
 *
 *         Note: tan(x+y) = tan(x) + tan'(x)*y
 *                        ~ tan(x) + (1+x*x)*y
 *         Therefore, for better accuracy in computing tan(x+y), let
 *                   3      2      2       2       2
 *              r = x *(T2+x *(T3+x *(...+x *(T12+x *T13))))
 *         then
 *                                  3    2
 *              tan(x+y) = x + (T1*x + (x *(r+y)+y))
 *
 *      4. For x in [0.67434,pi/4],  let y = pi/4 - x, then
 *              tan(x) = tan(pi/4-y) = (1-tan(y))/(1+tan(y))
 *                     = 1 - 2*(tan(y) - (tan(y)^2)/(1+tan(y)))
 */

#include <libm.h>
#include "math_private.h"

static const double xxx[] = {
                 3.33333333333334091986e-01,    /* 3FD55555, 55555563 */
                 1.33333333333201242699e-01,    /* 3FC11111, 1110FE7A */
                 5.39682539762260521377e-02,    /* 3FABA1BA, 1BB341FE */
                 2.18694882948595424599e-02,    /* 3F9664F4, 8406D637 */
                 8.86323982359930005737e-03,    /* 3F8226E3, E96E8493 */
                 3.59207910759131235356e-03,    /* 3F6D6D22, C9560328 */
                 1.45620945432529025516e-03,    /* 3F57DBC8, FEE08315 */
                 5.88041240820264096874e-04,    /* 3F4344D8, F2F26501 */
                 2.46463134818469906812e-04,    /* 3F3026F7, 1A8D1068 */
                 7.81794442939557092300e-05,    /* 3F147E88, A03792A6 */
                 7.14072491382608190305e-05,    /* 3F12B80F, 32F0A7E9 */
                -1.85586374855275456654e-05,    /* BEF375CB, DB605373 */
                 2.59073051863633712884e-05,    /* 3EFB2A70, 74BF7AD4 */
/* one */        1.00000000000000000000e+00,    /* 3FF00000, 00000000 */
/* pio4 */       7.85398163397448278999e-01,    /* 3FE921FB, 54442D18 */
/* pio4lo */     3.06161699786838301793e-17     /* 3C81A626, 33145C07 */
};
#define one     xxx[13]
#define pio4    xxx[14]
#define pio4lo  xxx[15]
#define T       xxx

double
__kernel_tan(double x, double y, int iy)
{
        double z, r, v, w, s;
        int32_t ix, hx;

        GET_HIGH_WORD(hx, x);   /* high word of x */
        ix = hx & 0x7fffffff;                   /* high word of |x| */
        if (ix < 0x3e300000) {                  /* x < 2**-28 */
                if ((int) x == 0) {             /* generate inexact */
                        uint32_t low;
                        GET_LOW_WORD(low, x);
                        if(((ix | low) | (iy + 1)) == 0)
                                return one / fabs(x);
                        else {
                                if (iy == 1)
                                        return x;
                                else {  /* compute -1 / (x+y) carefully */
                                        double a, t;

                                        z = w = x + y;
                                        SET_LOW_WORD(z, 0);
                                        v = y - (z - x);
                                        t = a = -one / w;
                                        SET_LOW_WORD(t, 0);
                                        s = one + t * z;
                                        return t + a * (s + t * v);
                                }
                        }
                }
        }
        if (ix >= 0x3FE59428) { /* |x| >= 0.6744 */
                if (hx < 0) {
                        x = -x;
                        y = -y;
                }
                z = pio4 - x;
                w = pio4lo - y;
                x = z + w;
                y = 0.0;
        }
        z = x * x;
        w = z * z;
        /*
         * Break x^5*(T[1]+x^2*T[2]+...) into
         * x^5(T[1]+x^4*T[3]+...+x^20*T[11]) +
         * x^5(x^2*(T[2]+x^4*T[4]+...+x^22*[T12]))
         */
        r = T[1] + w * (T[3] + w * (T[5] + w * (T[7] + w * (T[9] +
                w * T[11]))));
        v = z * (T[2] + w * (T[4] + w * (T[6] + w * (T[8] + w * (T[10] +
                w * T[12])))));
        s = z * x;
        r = y + z * (s * (r + v) + y);
        r += T[0] * s;
        w = x + r;
        if (ix >= 0x3FE59428) {
                v = (double) iy;
                return (double) (1 - ((hx >> 30) & 2)) *
                        (v - 2.0 * (x - (w * w / (w + v) - r)));
        }
        if (iy == 1)
                return w;
        else {
                /*
                 * if allow error up to 2 ulp, simply return
                 * -1.0 / (x+r) here
                 */
                /* compute -1.0 / (x+r) accurately */
                double a, t;
                z = w;
                SET_LOW_WORD(z, 0);
                v = r - (z - x);        /* z+v = r+x */
                t = a = -1.0 / w;       /* a = -1.0/w */
                SET_LOW_WORD(t, 0);
                s = 1.0 + t * z;
                return t + a * (s + t * v);
        }
}

#endif