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