# k_tan_freeBSD.c   [plain text]

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

/*
* ====================================================
*
* Permission to use, copy, modify, and distribute this
* software is freely granted, provided that this notice
* is preserved.
* ====================================================
*/

/* INDENT OFF */
#ifndef lint
static char rcsid[] = "\$FreeBSD: src/lib/msun/src/k_tan.c,v 1.10 2005/02/04 18:26:06 das 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 */
/* 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 < 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);
}
}
```