xref: /csrg-svn/old/libm/libm/IEEE/atan2.c (revision 31705)
121053Smiriam /*
221053Smiriam  * Copyright (c) 1985 Regents of the University of California.
321053Smiriam  *
421053Smiriam  * Use and reproduction of this software are granted  in  accordance  with
521053Smiriam  * the terms and conditions specified in  the  Berkeley  Software  License
621053Smiriam  * Agreement (in particular, this entails acknowledgement of the programs'
721053Smiriam  * source, and inclusion of this notice) with the additional understanding
821053Smiriam  * that  all  recipients  should regard themselves as participants  in  an
921053Smiriam  * ongoing  research  project and hence should  feel  obligated  to report
1021053Smiriam  * their  experiences (good or bad) with these elementary function  codes,
1121053Smiriam  * using "sendbug 4bsd-bugs@BERKELEY", to the authors.
1221053Smiriam  */
1321053Smiriam 
1421053Smiriam #ifndef lint
15*31705Ssam static char sccsid[] = "@(#)atan2.c	1.4 (Berkeley) 06/29/87";
1621053Smiriam #endif not lint
1721053Smiriam 
1821053Smiriam /* ATAN2(Y,X)
1921053Smiriam  * RETURN ARG (X+iY)
2021053Smiriam  * DOUBLE PRECISION (VAX D format 56 bits, IEEE DOUBLE 53 BITS)
2121053Smiriam  * CODED IN C BY K.C. NG, 1/8/85;
2224366Smiriam  * REVISED BY K.C. NG on 2/7/85, 2/13/85, 3/7/85, 3/30/85, 6/29/85.
2321053Smiriam  *
2421053Smiriam  * Required system supported functions :
2521053Smiriam  *	copysign(x,y)
2621053Smiriam  *	scalb(x,y)
2721053Smiriam  *	logb(x)
2821053Smiriam  *
2921053Smiriam  * Method :
3021053Smiriam  *	1. Reduce y to positive by atan2(y,x)=-atan2(-y,x).
3121053Smiriam  *	2. Reduce x to positive by (if x and y are unexceptional):
3221053Smiriam  *		ARG (x+iy) = arctan(y/x)   	   ... if x > 0,
3321053Smiriam  *		ARG (x+iy) = pi - arctan[y/(-x)]   ... if x < 0,
3421053Smiriam  *	3. According to the integer k=4t+0.25 truncated , t=y/x, the argument
3521053Smiriam  *	   is further reduced to one of the following intervals and the
3621053Smiriam  *	   arctangent of y/x is evaluated by the corresponding formula:
3721053Smiriam  *
3821053Smiriam  *         [0,7/16]	   atan(y/x) = t - t^3*(a1+t^2*(a2+...(a10+t^2*a11)...)
3921053Smiriam  *	   [7/16,11/16]    atan(y/x) = atan(1/2) + atan( (y-x/2)/(x+y/2) )
4021053Smiriam  *	   [11/16.19/16]   atan(y/x) = atan( 1 ) + atan( (y-x)/(x+y) )
4121053Smiriam  *	   [19/16,39/16]   atan(y/x) = atan(3/2) + atan( (y-1.5x)/(x+1.5y) )
4221053Smiriam  *	   [39/16,INF]     atan(y/x) = atan(INF) + atan( -x/y )
4321053Smiriam  *
4421053Smiriam  * Special cases:
4521053Smiriam  * Notations: atan2(y,x) == ARG (x+iy) == ARG(x,y).
4621053Smiriam  *
4721053Smiriam  *	ARG( NAN , (anything) ) is NaN;
4821053Smiriam  *	ARG( (anything), NaN ) is NaN;
4921053Smiriam  *	ARG(+(anything but NaN), +-0) is +-0  ;
5021053Smiriam  *	ARG(-(anything but NaN), +-0) is +-PI ;
5121053Smiriam  *	ARG( 0, +-(anything but 0 and NaN) ) is +-PI/2;
5221053Smiriam  *	ARG( +INF,+-(anything but INF and NaN) ) is +-0 ;
5321053Smiriam  *	ARG( -INF,+-(anything but INF and NaN) ) is +-PI;
5421053Smiriam  *	ARG( +INF,+-INF ) is +-PI/4 ;
5521053Smiriam  *	ARG( -INF,+-INF ) is +-3PI/4;
5621053Smiriam  *	ARG( (anything but,0,NaN, and INF),+-INF ) is +-PI/2;
5721053Smiriam  *
5821053Smiriam  * Accuracy:
5921053Smiriam  *	atan2(y,x) returns (PI/pi) * the exact ARG (x+iy) nearly rounded,
6021053Smiriam  *	where
6121053Smiriam  *
6221053Smiriam  *	in decimal:
6321053Smiriam  *		pi = 3.141592653589793 23846264338327 .....
6421053Smiriam  *    53 bits   PI = 3.141592653589793 115997963 ..... ,
6521053Smiriam  *    56 bits   PI = 3.141592653589793 227020265 ..... ,
6621053Smiriam  *
6721053Smiriam  *	in hexadecimal:
6821053Smiriam  *		pi = 3.243F6A8885A308D313198A2E....
6921053Smiriam  *    53 bits   PI = 3.243F6A8885A30  =  2 * 1.921FB54442D18	error=.276ulps
7021053Smiriam  *    56 bits   PI = 3.243F6A8885A308 =  4 * .C90FDAA22168C2    error=.206ulps
7121053Smiriam  *
7221053Smiriam  *	In a test run with 356,000 random argument on [-1,1] * [-1,1] on a
7321053Smiriam  *	VAX, the maximum observed error was 1.41 ulps (units of the last place)
7421053Smiriam  *	compared with (PI/pi)*(the exact ARG(x+iy)).
7521053Smiriam  *
7621053Smiriam  * Note:
7721053Smiriam  *	We use machine PI (the true pi rounded) in place of the actual
7821053Smiriam  *	value of pi for all the trig and inverse trig functions. In general,
7921053Smiriam  *	if trig is one of sin, cos, tan, then computed trig(y) returns the
8021053Smiriam  *	exact trig(y*pi/PI) nearly rounded; correspondingly, computed arctrig
8121053Smiriam  *	returns the exact arctrig(y)*PI/pi nearly rounded. These guarantee the
8221053Smiriam  *	trig functions have period PI, and trig(arctrig(x)) returns x for
8321053Smiriam  *	all critical values x.
8421053Smiriam  *
8521053Smiriam  * Constants:
8621053Smiriam  * The hexadecimal values are the intended ones for the following constants.
8721053Smiriam  * The decimal values may be used, provided that the compiler will convert
8821053Smiriam  * from decimal to binary accurately enough to produce the hexadecimal values
8921053Smiriam  * shown.
9021053Smiriam  */
9121053Smiriam 
9221053Smiriam static double
93*31705Ssam #if defined(VAX) || defined(TAHOE) 	/* VAX D format */
9421053Smiriam athfhi =  4.6364760900080611433E-1    , /*Hex  2^ -1   *  .ED63382B0DDA7B */
9521053Smiriam athflo =  1.9338828231967579916E-19   , /*Hex  2^-62   *  .E450059CFE92C0 */
9621053Smiriam PIo4   =  7.8539816339744830676E-1    , /*Hex  2^  0   *  .C90FDAA22168C2 */
9721053Smiriam at1fhi =  9.8279372324732906796E-1    , /*Hex  2^  0   *  .FB985E940FB4D9 */
9821053Smiriam at1flo = -3.5540295636764633916E-18   , /*Hex  2^-57   * -.831EDC34D6EAEA */
9921053Smiriam PIo2   =  1.5707963267948966135E0     , /*Hex  2^  1   *  .C90FDAA22168C2 */
10021053Smiriam PI     =  3.1415926535897932270E0     , /*Hex  2^  2   *  .C90FDAA22168C2 */
10121053Smiriam a1     =  3.3333333333333473730E-1    , /*Hex  2^ -1   *  .AAAAAAAAAAAB75 */
10221053Smiriam a2     = -2.0000000000017730678E-1    , /*Hex  2^ -2   * -.CCCCCCCCCD946E */
10321053Smiriam a3     =  1.4285714286694640301E-1    , /*Hex  2^ -2   *  .92492492744262 */
10421053Smiriam a4     = -1.1111111135032672795E-1    , /*Hex  2^ -3   * -.E38E38EBC66292 */
10521053Smiriam a5     =  9.0909091380563043783E-2    , /*Hex  2^ -3   *  .BA2E8BB31BD70C */
10621053Smiriam a6     = -7.6922954286089459397E-2    , /*Hex  2^ -3   * -.9D89C827C37F18 */
10721053Smiriam a7     =  6.6663180891693915586E-2    , /*Hex  2^ -3   *  .8886B4AE379E58 */
10821053Smiriam a8     = -5.8772703698290408927E-2    , /*Hex  2^ -4   * -.F0BBA58481A942 */
10921053Smiriam a9     =  5.2170707402812969804E-2    , /*Hex  2^ -4   *  .D5B0F3A1AB13AB */
11021053Smiriam a10    = -4.4895863157820361210E-2    , /*Hex  2^ -4   * -.B7E4B97FD1048F */
11121053Smiriam a11    =  3.3006147437343875094E-2    , /*Hex  2^ -4   *  .8731743CF72D87 */
11221053Smiriam a12    = -1.4614844866464185439E-2    ; /*Hex  2^ -6   * -.EF731A2F3476D9 */
11321053Smiriam #else 	/* IEEE double */
11421053Smiriam athfhi =  4.6364760900080609352E-1    , /*Hex  2^ -2   *  1.DAC670561BB4F */
11521053Smiriam athflo =  4.6249969567426939759E-18   , /*Hex  2^-58   *  1.5543B8F253271 */
11621053Smiriam PIo4   =  7.8539816339744827900E-1    , /*Hex  2^ -1   *  1.921FB54442D18 */
11721053Smiriam at1fhi =  9.8279372324732905408E-1    , /*Hex  2^ -1   *  1.F730BD281F69B */
11821053Smiriam at1flo = -2.4407677060164810007E-17   , /*Hex  2^-56   * -1.C23DFEFEAE6B5 */
11921053Smiriam PIo2   =  1.5707963267948965580E0     , /*Hex  2^  0   *  1.921FB54442D18 */
12021053Smiriam PI     =  3.1415926535897931160E0     , /*Hex  2^  1   *  1.921FB54442D18 */
12121053Smiriam a1     =  3.3333333333333942106E-1    , /*Hex  2^ -2   *  1.55555555555C3 */
12221053Smiriam a2     = -1.9999999999979536924E-1    , /*Hex  2^ -3   * -1.9999999997CCD */
12321053Smiriam a3     =  1.4285714278004377209E-1    , /*Hex  2^ -3   *  1.24924921EC1D7 */
12421053Smiriam a4     = -1.1111110579344973814E-1    , /*Hex  2^ -4   * -1.C71C7059AF280 */
12521053Smiriam a5     =  9.0908906105474668324E-2    , /*Hex  2^ -4   *  1.745CE5AA35DB2 */
12621053Smiriam a6     = -7.6919217767468239799E-2    , /*Hex  2^ -4   * -1.3B0FA54BEC400 */
12721053Smiriam a7     =  6.6614695906082474486E-2    , /*Hex  2^ -4   *  1.10DA924597FFF */
12821053Smiriam a8     = -5.8358371008508623523E-2    , /*Hex  2^ -5   * -1.DE125FDDBD793 */
12921053Smiriam a9     =  4.9850617156082015213E-2    , /*Hex  2^ -5   *  1.9860524BDD807 */
13021053Smiriam a10    = -3.6700606902093604877E-2    , /*Hex  2^ -5   * -1.2CA6C04C6937A */
13121053Smiriam a11    =  1.6438029044759730479E-2    ; /*Hex  2^ -6   *  1.0D52174A1BB54 */
13221053Smiriam #endif
13321053Smiriam 
atan2(y,x)13421053Smiriam double atan2(y,x)
13521053Smiriam double  y,x;
13621053Smiriam {
13721053Smiriam 	static double zero=0, one=1, small=1.0E-9, big=1.0E18;
13821053Smiriam 	double copysign(),logb(),scalb(),t,z,signy,signx,hi,lo;
13924366Smiriam 	int finite(), k,m;
14021053Smiriam 
14121053Smiriam     /* if x or y is NAN */
14221053Smiriam 	if(x!=x) return(x); if(y!=y) return(y);
14321053Smiriam 
14421053Smiriam     /* copy down the sign of y and x */
14521053Smiriam 	signy = copysign(one,y) ;
14621053Smiriam 	signx = copysign(one,x) ;
14721053Smiriam 
14821053Smiriam     /* if x is 1.0, goto begin */
14921053Smiriam 	if(x==1) { y=copysign(y,one); t=y; if(finite(t)) goto begin;}
15021053Smiriam 
15121053Smiriam     /* when y = 0 */
15221053Smiriam 	if(y==zero) return((signx==one)?y:copysign(PI,signy));
15321053Smiriam 
15421053Smiriam     /* when x = 0 */
15521053Smiriam 	if(x==zero) return(copysign(PIo2,signy));
15621053Smiriam 
15721053Smiriam     /* when x is INF */
15821915Smiriam 	if(!finite(x))
15921915Smiriam 	    if(!finite(y))
16021053Smiriam 		return(copysign((signx==one)?PIo4:3*PIo4,signy));
16121053Smiriam 	    else
16221053Smiriam 		return(copysign((signx==one)?zero:PI,signy));
16321053Smiriam 
16421053Smiriam     /* when y is INF */
16521915Smiriam 	if(!finite(y)) return(copysign(PIo2,signy));
16621053Smiriam 
16721053Smiriam 
16821053Smiriam     /* compute y/x */
16921053Smiriam 	x=copysign(x,one);
17021053Smiriam 	y=copysign(y,one);
17124366Smiriam 	if((m=(k=logb(y))-logb(x)) > 60) t=big+big;
17224366Smiriam 	    else if(m < -80 ) t=y/x;
17321053Smiriam 	    else { t = y/x ; y = scalb(y,-k); x=scalb(x,-k); }
17421053Smiriam 
17521053Smiriam     /* begin argument reduction */
17621053Smiriam begin:
17721053Smiriam 	if (t < 2.4375) {
17821053Smiriam 
17921053Smiriam 	/* truncate 4(t+1/16) to integer for branching */
18021053Smiriam 	    k = 4 * (t+0.0625);
18121053Smiriam 	    switch (k) {
18221053Smiriam 
18321053Smiriam 	    /* t is in [0,7/16] */
18421053Smiriam 	    case 0:
18521053Smiriam 	    case 1:
18621053Smiriam 		if (t < small)
18721053Smiriam 		    { big + small ;  /* raise inexact flag */
18821053Smiriam 		      return (copysign((signx>zero)?t:PI-t,signy)); }
18921053Smiriam 
19021053Smiriam 		hi = zero;  lo = zero;  break;
19121053Smiriam 
19221053Smiriam 	    /* t is in [7/16,11/16] */
19321053Smiriam 	    case 2:
19421053Smiriam 		hi = athfhi; lo = athflo;
19521053Smiriam 		z = x+x;
19621053Smiriam 		t = ( (y+y) - x ) / ( z +  y ); break;
19721053Smiriam 
19821053Smiriam 	    /* t is in [11/16,19/16] */
19921053Smiriam 	    case 3:
20021053Smiriam 	    case 4:
20121053Smiriam 		hi = PIo4; lo = zero;
20221053Smiriam 		t = ( y - x ) / ( x + y ); break;
20321053Smiriam 
20421053Smiriam 	    /* t is in [19/16,39/16] */
20521053Smiriam 	    default:
20621053Smiriam 		hi = at1fhi; lo = at1flo;
20721053Smiriam 		z = y-x; y=y+y+y; t = x+x;
20821053Smiriam 		t = ( (z+z)-x ) / ( t + y ); break;
20921053Smiriam 	    }
21021053Smiriam 	}
21121053Smiriam 	/* end of if (t < 2.4375) */
21221053Smiriam 
21321053Smiriam 	else
21421053Smiriam 	{
21521053Smiriam 	    hi = PIo2; lo = zero;
21621053Smiriam 
21721053Smiriam 	    /* t is in [2.4375, big] */
21821053Smiriam 	    if (t <= big)  t = - x / y;
21921053Smiriam 
22021053Smiriam 	    /* t is in [big, INF] */
22121053Smiriam 	    else
22221053Smiriam 	      { big+small;	/* raise inexact flag */
22321053Smiriam 		t = zero; }
22421053Smiriam 	}
22521053Smiriam     /* end of argument reduction */
22621053Smiriam 
22721053Smiriam     /* compute atan(t) for t in [-.4375, .4375] */
22821053Smiriam 	z = t*t;
229*31705Ssam #if defined(VAX) || defined(TAHOE)
23021053Smiriam 	z = t*(z*(a1+z*(a2+z*(a3+z*(a4+z*(a5+z*(a6+z*(a7+z*(a8+
23121053Smiriam 			z*(a9+z*(a10+z*(a11+z*a12))))))))))));
23221053Smiriam #else	/* IEEE double */
23321053Smiriam 	z = t*(z*(a1+z*(a2+z*(a3+z*(a4+z*(a5+z*(a6+z*(a7+z*(a8+
23421053Smiriam 			z*(a9+z*(a10+z*a11)))))))))));
23521053Smiriam #endif
23621053Smiriam 	z = lo - z; z += t; z += hi;
23721053Smiriam 
23821053Smiriam 	return(copysign((signx>zero)?z:PI-z,signy));
23921053Smiriam }
240