/* * 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)); }