StdLib/LibC/Math/k_tan.c

/* @(#)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.
 * ====================================================
 */
#include  <LibConfig.h>
#include  <sys/EfiCdefs.h>
#if defined(LIBM_SCCS) && !defined(lint)
__RCSID("$NetBSD: k_tan.c,v 1.12 2004/07/22 18:24:09 drochner Exp $");
#endif

/* __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 "math.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 */
      u_int32_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);
  }
}