# code: 9ferno

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```/* derived from /netlib/fdlibm */

/* @(#)k_tan.c 1.3 95/01/18 */
/*
* ====================================================
*
* Developed at SunSoft, 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"
static const double
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 */
};

double __kernel_tan(double x, double y, int iy)
{
double z,r,v,w,s;
int ix,hx;
hx = __HI(x);	/* high word of x */
ix = hx&0x7fffffff;	/* high word of |x| */
if(ix<0x3e300000)			/* x < 2**-28 */
{if((int)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);
}
}
```