prime/src/Runtime/k_tan.c

/* @(#)k_tan.c 1.2 95/01/04 */
/*
 * ====================================================
 * 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.
 * ====================================================
 */

/* __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 "fdlibm.h"

#ifdef __STDC__
static const double
#else
static double
#endif
    one = 1.00000000000000000000e+00,    /* 0x3FF00000, 0x00000000 */
    pio4 = 7.85398163397448278999e-01,   /* 0x3FE921FB, 0x54442D18 */
    pio4lo = 3.06161699786838301793e-17, /* 0x3C81A626, 0x33145C07 */
    T[] = {
        3.33333333333334091986e-01,  /* 0x3FD55555, 0x55555563 */
        1.33333333333201242699e-01,  /* 0x3FC11111, 0x1110FE7A */
        5.39682539762260521377e-02,  /* 0x3FABA1BA, 0x1BB341FE */
        2.18694882948595424599e-02,  /* 0x3F9664F4, 0x8406D637 */
        8.86323982359930005737e-03,  /* 0x3F8226E3, 0xE96E8493 */
        3.59207910759131235356e-03,  /* 0x3F6D6D22, 0xC9560328 */
        1.45620945432529025516e-03,  /* 0x3F57DBC8, 0xFEE08315 */
        5.88041240820264096874e-04,  /* 0x3F4344D8, 0xF2F26501 */
        2.46463134818469906812e-04,  /* 0x3F3026F7, 0x1A8D1068 */
        7.81794442939557092300e-05,  /* 0x3F147E88, 0xA03792A6 */
        7.14072491382608190305e-05,  /* 0x3F12B80F, 0x32F0A7E9 */
        -1.85586374855275456654e-05, /* 0xBEF375CB, 0xDB605373 */
        2.59073051863633712884e-05,  /* 0x3EFB2A70, 0x74BF7AD4 */
};

#ifdef __STDC__
double __kernel_tan(double x, double y, _INT32 iy)
#else
double __kernel_tan(x, y, iy)
double x, y;
_INT32 iy;
#endif
{
  double z, r, v, w, s;
  _INT32 ix, hx;
  hx = __HI(x);         /* high word of x */
  ix = hx & 0x7fffffff; /* high word of |x| */
  if (ix < 0x3e300000)  /* x < 2**-28 */
  {
    if ((_INT32)x == 0) { /* generate inexact */
      if (((ix | __LO(x)) | (iy + 1)) == 0)
        return one / fabs(x);
      else
        return (iy == 1) ? x : -one / 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;
    __LO(z) = 0;
    v = r - (z - x);  /* z+v = r+x */
    t = a = -1.0 / w; /* a = -1.0/w */
    __LO(t) = 0;
    s = 1.0 + t * z;
    return t + a * (s + t * v);
  }
}