xref: /illumos-gate/usr/src/lib/libm/common/R/atanf.c (revision 862246c1)

/*
 * CDDL HEADER START
 *
 * The contents of this file are subject to the terms of the
 * Common Development and Distribution License (the "License").
 * You may not use this file except in compliance with the License.
 *
 * You can obtain a copy of the license at usr/src/OPENSOLARIS.LICENSE
 * or http://www.opensolaris.org/os/licensing.
 * See the License for the specific language governing permissions
 * and limitations under the License.
 *
 * When distributing Covered Code, include this CDDL HEADER in each
 * file and include the License file at usr/src/OPENSOLARIS.LICENSE.
 * If applicable, add the following below this CDDL HEADER, with the
 * fields enclosed by brackets "[]" replaced with your own identifying
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/*
 * Copyright 2011 Nexenta Systems, Inc.  All rights reserved.
 */
/*
 * Copyright 2006 Sun Microsystems, Inc.  All rights reserved.
 * Use is subject to license terms.
 */

#pragma weak __atanf = atanf

/* INDENT OFF */
/*
 * float atanf(float x);
 * Table look-up algorithm
 * By K.C. Ng, March 9, 1989
 *
 * Algorithm.
 *
 * The algorithm is based on atan(x)=atan(y)+atan((x-y)/(1+x*y)).
 * We use poly1(x) to approximate atan(x) for x in [0,1/8] with
 * error (relative)
 * 	|(atan(x)-poly1(x))/x|<= 2^-115.94 	long double
 * 	|(atan(x)-poly1(x))/x|<= 2^-58.85	double
 * 	|(atan(x)-poly1(x))/x|<= 2^-25.53 	float
 * and use poly2(x) to approximate atan(x) for x in [0,1/65] with
 * error (absolute)
 *	|atan(x)-poly2(x)|<= 2^-122.15		long double
 *	|atan(x)-poly2(x)|<= 2^-64.79		double
 *	|atan(x)-poly2(x)|<= 2^-35.36		float
 * and use poly3(x) to approximate atan(x) for x in [1/8,7/16] with
 * error (relative, on for single precision)
 * 	|(atan(x)-poly1(x))/x|<= 2^-25.53 	float
 *
 * Here poly1-3 are odd polynomial with the following form:
 *		x + x^3*(a1+x^2*(a2+...))
 *
 * (0). Purge off Inf and NaN and 0
 * (1). Reduce x to positive by atan(x) = -atan(-x).
 * (2). For x <= 1/8, use
 *	(2.1) if x < 2^(-prec/2-2), atan(x) =  x  with inexact
 *	(2.2) Otherwise
 *		atan(x) = poly1(x)
 * (3). For x >= 8 then
 *	(3.1) if x >= 2^(prec+2),   atan(x) = atan(inf) - pio2lo
 *	(3.2) if x >= 2^(prec/3+2), atan(x) = atan(inf) - 1/x
 *	(3.3) if x >  65,           atan(x) = atan(inf) - poly2(1/x)
 *	(3.4) Otherwise,	    atan(x) = atan(inf) - poly1(1/x)
 *
 * (4). Now x is in (0.125, 8)
 *      Find y that match x to 4.5 bit after binary (easy).
 *	If iy is the high word of y, then
 *		single : j = (iy - 0x3e000000) >> 19
 *		(single is modified to (iy-0x3f000000)>>19)
 *		double : j = (iy - 0x3fc00000) >> 16
 *		quad   : j = (iy - 0x3ffc0000) >> 12
 *
 *	Let s = (x-y)/(1+x*y). Then
 *	atan(x) = atan(y) + poly1(s)
 *		= _TBL_r_atan_hi[j] + (_TBL_r_atan_lo[j] + poly2(s) )
 *
 *	Note. |s| <= 1.5384615385e-02 = 1/65. Maxium occurs at x = 1.03125
 *
 */

#include "libm.h"

extern const float _TBL_r_atan_hi[], _TBL_r_atan_lo[];
static const float
	big	=   1.0e37F,
	one	=   1.0F,
	p1	=  -3.333185951111688247225368498733544672172e-0001F,
	p2	=   1.969352894213455405211341983203180636021e-0001F,
	q1	=  -3.332921964095646819563419704110132937456e-0001F,
	a1	=  -3.333323465223893614063523351509338934592e-0001F,
	a2	=   1.999425625935277805494082274808174062403e-0001F,
	a3	=  -1.417547090509737780085769846290301788559e-0001F,
	a4	=   1.016250813871991983097273733227432685084e-0001F,
	a5	=  -5.137023693688358515753093811791755221805e-0002F,
	pio2hi	=   1.570796371e+0000F,
	pio2lo	=  -4.371139000e-0008F;
/* INDENT ON */

float
atanf(float xx) {
	float x, y, z, r, p, s;
	volatile double dummy __unused;
	int ix, iy, sign, j;

	x = xx;
	ix = *(int *) &x;
	sign = ix & 0x80000000;
	ix ^= sign;

	/* for |x| < 1/8 */
	if (ix < 0x3e000000) {
		if (ix < 0x38800000) {	/* if |x| < 2**(-prec/2-2) */
			dummy = big + x;	/* get inexact flag if x != 0 */
#ifdef lint
			dummy = dummy;
#endif
			return (x);
		}
		z = x * x;
		if (ix < 0x3c000000) {	/* if |x| < 2**(-prec/4-1) */
			x = x + (x * z) * p1;
			return (x);
		} else {
			x = x + (x * z) * (p1 + z * p2);
			return (x);
		}
	}

	/* for |x| >= 8.0 */
	if (ix >= 0x41000000) {
		*(int *) &x = ix;
		if (ix < 0x42820000) {	/* x <  65 */
			r = one / x;
			z = r * r;
			y = r * (one + z * (p1 + z * p2));	/* poly1 */
			y -= pio2lo;
		} else if (ix < 0x44800000) {	/* x <  2**(prec/3+2) */
			r = one / x;
			z = r * r;
			y = r * (one + z * q1);	/* poly2 */
			y -= pio2lo;
		} else if (ix < 0x4c800000) {	/* x <  2**(prec+2) */
			y = one / x - pio2lo;
		} else if (ix < 0x7f800000) {	/* x <  inf */
			y = -pio2lo;
		} else {		/* x is inf or NaN */
			if (ix > 0x7f800000) {
				return (x * x);	/* - -> * for Cheetah */
			}
			y = -pio2lo;
		}

		if (sign == 0)
			x = pio2hi - y;
		else
			x = y - pio2hi;
		return (x);
	}


	/* now x is between 1/8 and 8 */
	if (ix < 0x3f000000) {	/* between 1/8 and 1/2 */
		z = x * x;
		x = x + (x * z) * (a1 + z * (a2 + z * (a3 + z * (a4 +
			z * a5))));
		return (x);
	}
	*(int *) &x = ix;
	iy = (ix + 0x00040000) & 0x7ff80000;
	*(int *) &y = iy;
	j = (iy - 0x3f000000) >> 19;

	if (ix == iy)
		p = x - y;	/* p=0.0 */
	else {
		if (sign == 0)
			s = (x - y) / (one + x * y);
		else
			s = (y - x) / (one + x * y);
		z = s * s;
		p = s * (one + z * q1);
	}
	if (sign == 0) {
		r = p + _TBL_r_atan_lo[j];
		x = r + _TBL_r_atan_hi[j];
	} else {
		r = p - _TBL_r_atan_lo[j];
		x = r - _TBL_r_atan_hi[j];
	}
	return (x);
}