4.3BSD/usr/src/usr.lib/libm/IEEE/atan2.c
/*
* Copyright (c) 1985 Regents of the University of California.
*
* Use and reproduction of this software are granted in accordance with
* the terms and conditions specified in the Berkeley Software License
* Agreement (in particular, this entails acknowledgement of the programs'
* source, and inclusion of this notice) with the additional understanding
* that all recipients should regard themselves as participants in an
* ongoing research project and hence should feel obligated to report
* their experiences (good or bad) with these elementary function codes,
* using "sendbug 4bsd-bugs@BERKELEY", to the authors.
*/
#ifndef lint
static char sccsid[] = "@(#)atan2.c 1.3 (Berkeley) 8/21/85";
#endif not lint
/* 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.
*/
static double
#ifdef VAX /* VAX D format */
athfhi = 4.6364760900080611433E-1 , /*Hex 2^ -1 * .ED63382B0DDA7B */
athflo = 1.9338828231967579916E-19 , /*Hex 2^-62 * .E450059CFE92C0 */
PIo4 = 7.8539816339744830676E-1 , /*Hex 2^ 0 * .C90FDAA22168C2 */
at1fhi = 9.8279372324732906796E-1 , /*Hex 2^ 0 * .FB985E940FB4D9 */
at1flo = -3.5540295636764633916E-18 , /*Hex 2^-57 * -.831EDC34D6EAEA */
PIo2 = 1.5707963267948966135E0 , /*Hex 2^ 1 * .C90FDAA22168C2 */
PI = 3.1415926535897932270E0 , /*Hex 2^ 2 * .C90FDAA22168C2 */
a1 = 3.3333333333333473730E-1 , /*Hex 2^ -1 * .AAAAAAAAAAAB75 */
a2 = -2.0000000000017730678E-1 , /*Hex 2^ -2 * -.CCCCCCCCCD946E */
a3 = 1.4285714286694640301E-1 , /*Hex 2^ -2 * .92492492744262 */
a4 = -1.1111111135032672795E-1 , /*Hex 2^ -3 * -.E38E38EBC66292 */
a5 = 9.0909091380563043783E-2 , /*Hex 2^ -3 * .BA2E8BB31BD70C */
a6 = -7.6922954286089459397E-2 , /*Hex 2^ -3 * -.9D89C827C37F18 */
a7 = 6.6663180891693915586E-2 , /*Hex 2^ -3 * .8886B4AE379E58 */
a8 = -5.8772703698290408927E-2 , /*Hex 2^ -4 * -.F0BBA58481A942 */
a9 = 5.2170707402812969804E-2 , /*Hex 2^ -4 * .D5B0F3A1AB13AB */
a10 = -4.4895863157820361210E-2 , /*Hex 2^ -4 * -.B7E4B97FD1048F */
a11 = 3.3006147437343875094E-2 , /*Hex 2^ -4 * .8731743CF72D87 */
a12 = -1.4614844866464185439E-2 ; /*Hex 2^ -6 * -.EF731A2F3476D9 */
#else /* IEEE double */
athfhi = 4.6364760900080609352E-1 , /*Hex 2^ -2 * 1.DAC670561BB4F */
athflo = 4.6249969567426939759E-18 , /*Hex 2^-58 * 1.5543B8F253271 */
PIo4 = 7.8539816339744827900E-1 , /*Hex 2^ -1 * 1.921FB54442D18 */
at1fhi = 9.8279372324732905408E-1 , /*Hex 2^ -1 * 1.F730BD281F69B */
at1flo = -2.4407677060164810007E-17 , /*Hex 2^-56 * -1.C23DFEFEAE6B5 */
PIo2 = 1.5707963267948965580E0 , /*Hex 2^ 0 * 1.921FB54442D18 */
PI = 3.1415926535897931160E0 , /*Hex 2^ 1 * 1.921FB54442D18 */
a1 = 3.3333333333333942106E-1 , /*Hex 2^ -2 * 1.55555555555C3 */
a2 = -1.9999999999979536924E-1 , /*Hex 2^ -3 * -1.9999999997CCD */
a3 = 1.4285714278004377209E-1 , /*Hex 2^ -3 * 1.24924921EC1D7 */
a4 = -1.1111110579344973814E-1 , /*Hex 2^ -4 * -1.C71C7059AF280 */
a5 = 9.0908906105474668324E-2 , /*Hex 2^ -4 * 1.745CE5AA35DB2 */
a6 = -7.6919217767468239799E-2 , /*Hex 2^ -4 * -1.3B0FA54BEC400 */
a7 = 6.6614695906082474486E-2 , /*Hex 2^ -4 * 1.10DA924597FFF */
a8 = -5.8358371008508623523E-2 , /*Hex 2^ -5 * -1.DE125FDDBD793 */
a9 = 4.9850617156082015213E-2 , /*Hex 2^ -5 * 1.9860524BDD807 */
a10 = -3.6700606902093604877E-2 , /*Hex 2^ -5 * -1.2CA6C04C6937A */
a11 = 1.6438029044759730479E-2 ; /*Hex 2^ -6 * 1.0D52174A1BB54 */
#endif
double atan2(y,x)
double y,x;
{
static double zero=0, one=1, small=1.0E-9, big=1.0E18;
double copysign(),logb(),scalb(),t,z,signy,signx,hi,lo;
int finite(), k,m;
/* if x or y is NAN */
if(x!=x) return(x); if(y!=y) return(y);
/* copy down the sign of y and x */
signy = copysign(one,y) ;
signx = copysign(one,x) ;
/* if x is 1.0, goto begin */
if(x==1) { y=copysign(y,one); t=y; if(finite(t)) goto begin;}
/* when y = 0 */
if(y==zero) return((signx==one)?y:copysign(PI,signy));
/* when x = 0 */
if(x==zero) return(copysign(PIo2,signy));
/* when x is INF */
if(!finite(x))
if(!finite(y))
return(copysign((signx==one)?PIo4:3*PIo4,signy));
else
return(copysign((signx==one)?zero:PI,signy));
/* when y is INF */
if(!finite(y)) return(copysign(PIo2,signy));
/* compute y/x */
x=copysign(x,one);
y=copysign(y,one);
if((m=(k=logb(y))-logb(x)) > 60) t=big+big;
else if(m < -80 ) t=y/x;
else { t = y/x ; y = scalb(y,-k); x=scalb(x,-k); }
/* 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;
#ifdef VAX
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 /* IEEE double */
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
z = lo - z; z += t; z += hi;
return(copysign((signx>zero)?z:PI-z,signy));
}