/*

 * ====================================================

 * Copyright 2004 Sun Microsystems, Inc.  All Rights Reserved.

 *

 * Permission to use, copy, modify, and distribute this

 * software is freely granted, provided that this notice

 * is preserved.

 * ====================================================

 */

/* __kernel_tan( x, y, k )

 * kernel tan function on ~[-pi/4, pi/4] (except on -0), 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. Callers must return tan(-0) = -0 without calling here since our

 *     odd polynomial is not evaluated in a way that preserves -0.

 *     Callers may do the optimization tan(x) ~ x for tiny x.

 *  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 "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

/* INDENT ON */

double

__kernel_tan(double x, double y, int iy) {

  double z, r, v, w, s;

  int32_t ix, hx;

  GET_HIGH_WORD(hx,x);

  ix = hx & 0x7fffffff;     /* high word of |x| */

  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);

  }

}