#pragma ident "@(#)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 */

/* __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 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;

	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 {

				if (iy == 1)

					return x;

				else {	/* compute -1 / (x+y) carefully */

					double a, t;

					z = w = x + y;

					__LO(z) = 0;

					v = y - (z - x);

					t = a = -one / w;

					__LO(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;

		__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);

	}

}