libm/src/k_tan.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 */
#ifndef lint
static char rcsid[] = "$FreeBSD: src/lib/msun/src/k_tan.c,v 1.12 2005/11/02 14:01:45 bde Exp $";
#endif

/* __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.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 */
		 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);
	}
}