/* $NetBSD: n_atan2.c,v 1.7 2014/10/10 20:58:09 martin Exp $ */
/*
* Copyright (c) 1985, 1993
* The Regents of the University of California. All rights reserved.
*
* Redistribution and use in source and binary forms, with or without
* modification, are permitted provided that the following conditions
* are met:
* 1. Redistributions of source code must retain the above copyright
* notice, this list of conditions and the following disclaimer.
* 2. Redistributions in binary form must reproduce the above copyright
* notice, this list of conditions and the following disclaimer in the
* documentation and/or other materials provided with the distribution.
* 3. Neither the name of the University nor the names of its contributors
* may be used to endorse or promote products derived from this software
* without specific prior written permission.
*
* THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND
* ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
* IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
* ARE DISCLAIMED. IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE
* FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
* DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
* OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
* HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
* LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
* OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
* SUCH DAMAGE.
*/
/* ATAN2(Y,X)
* RETURN ARG (X+iY)
* DOUBLE PRECISION (VAX D format 56 bits, IEEE DOUBLE 53 BITS)
* CODED IN C BY K.C. NG, 1/8/85;
* REVISED BY K.C. NG on 2/7/85, 2/13/85, 3/7/85, 3/30/85, 6/29/85.
*
* Required system supported functions :
* copysign(x,y)
* scalb(x,y)
* logb(x)
*
* Method :
* 1. Reduce y to positive by atan2(y,x)=-atan2(-y,x).
* 2. Reduce x to positive by (if x and y are unexceptional):
* ARG (x+iy) = arctan(y/x) ... if x > 0,
* ARG (x+iy) = pi - arctan[y/(-x)] ... if x < 0,
* 3. According to the integer k=4t+0.25 truncated , t=y/x, the argument
* is further reduced to one of the following intervals and the
* arctangent of y/x is evaluated by the corresponding formula:
*
* [0,7/16] atan(y/x) = t - t^3*(a1+t^2*(a2+...(a10+t^2*a11)...)
* [7/16,11/16] atan(y/x) = atan(1/2) + atan( (y-x/2)/(x+y/2) )
* [11/16.19/16] atan(y/x) = atan( 1 ) + atan( (y-x)/(x+y) )
* [19/16,39/16] atan(y/x) = atan(3/2) + atan( (y-1.5x)/(x+1.5y) )
* [39/16,INF] atan(y/x) = atan(INF) + atan( -x/y )
*
* Special cases:
* Notations: atan2(y,x) == ARG (x+iy) == ARG(x,y).
*
* ARG( NAN , (anything) ) is NaN;
* ARG( (anything), NaN ) is NaN;
* ARG(+(anything but NaN), +-0) is +-0 ;
* ARG(-(anything but NaN), +-0) is +-PI ;
* ARG( 0, +-(anything but 0 and NaN) ) is +-PI/2;
* ARG( +INF,+-(anything but INF and NaN) ) is +-0 ;
* ARG( -INF,+-(anything but INF and NaN) ) is +-PI;
* ARG( +INF,+-INF ) is +-PI/4 ;
* ARG( -INF,+-INF ) is +-3PI/4;
* ARG( (anything but,0,NaN, and INF),+-INF ) is +-PI/2;
*
* Accuracy:
* atan2(y,x) returns (PI/pi) * the exact ARG (x+iy) nearly rounded,
* where
*
* in decimal:
* pi = 3.141592653589793 23846264338327 .....
* 53 bits PI = 3.141592653589793 115997963 ..... ,
* 56 bits PI = 3.141592653589793 227020265 ..... ,
*
* in hexadecimal:
* pi = 3.243F6A8885A308D313198A2E....
* 53 bits PI = 3.243F6A8885A30 = 2 * 1.921FB54442D18 error=.276ulps
* 56 bits PI = 3.243F6A8885A308 = 4 * .C90FDAA22168C2 error=.206ulps
*
* In a test run with 356,000 random argument on [-1,1] * [-1,1] on a
* VAX, the maximum observed error was 1.41 ulps (units of the last place)
* compared with (PI/pi)*(the exact ARG(x+iy)).
*
* Note:
* We use machine PI (the true pi rounded) in place of the actual
* value of pi for all the trig and inverse trig functions. In general,
* if trig is one of sin, cos, tan, then computed trig(y) returns the
* exact trig(y*pi/PI) nearly rounded; correspondingly, computed arctrig
* returns the exact arctrig(y)*PI/pi nearly rounded. These guarantee the
* trig functions have period PI, and trig(arctrig(x)) returns x for
* all critical values x.
*
* Constants:
* The hexadecimal values are the intended ones for the following constants.
* The decimal values may be used, provided that the compiler will convert
* from decimal to binary accurately enough to produce the hexadecimal values
* shown.
*/
/* begin argument reduction */
begin:
if (t < 2.4375) {
/* truncate 4(t+1/16) to integer for branching */
k = 4 * (t+0.0625);
switch (k) {
/* t is in [0,7/16] */
case 0:
case 1:
if (t < small)
{ big + small ; /* raise inexact flag */
return (copysign((signx>zero)?t:PI-t,signy)); }
hi = zero; lo = zero; break;
/* t is in [7/16,11/16] */
case 2:
hi = athfhi; lo = athflo;
z = x+x;
t = ( (y+y) - x ) / ( z + y ); break;
/* t is in [11/16,19/16] */
case 3:
case 4:
hi = PIo4; lo = zero;
t = ( y - x ) / ( x + y ); break;
/* t is in [19/16,39/16] */
default:
hi = at1fhi; lo = at1flo;
z = y-x; y=y+y+y; t = x+x;
t = ( (z+z)-x ) / ( t + y ); break;
}
}
/* end of if (t < 2.4375) */
else
{
hi = PIo2; lo = zero;
/* t is in [2.4375, big] */
if (t <= big) t = - x / y;
/* t is in [big, INF] */
else
{ big+small; /* raise inexact flag */
t = zero; }
}
/* end of argument reduction */
/* compute atan(t) for t in [-.4375, .4375] */
z = t*t;
#if defined(__vax__)||defined(tahoe)
z = t*(z*(a1+z*(a2+z*(a3+z*(a4+z*(a5+z*(a6+z*(a7+z*(a8+
z*(a9+z*(a10+z*(a11+z*a12))))))))))));
#else /* defined(__vax__)||defined(tahoe) */
z = t*(z*(a1+z*(a2+z*(a3+z*(a4+z*(a5+z*(a6+z*(a7+z*(a8+
z*(a9+z*(a10+z*a11)))))))))));
#endif /* defined(__vax__)||defined(tahoe) */
z = lo - z; z += t; z += hi;
