Quelle k_tan.cpp Sprache: C

/* @(#)k_tan.c 1.5 04/04/22 SMI */

/*
* ====================================================
* 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.
* ====================================================
*/

/* INDENT OFF */
//#include <sys/cdefs.h>
//__FBSDID("$FreeBSD$");

/* __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);
}
}

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