/* $OpenBSD: n_atan2.c,v 1.13 2009/10/27 23:59:29 deraadt Exp $ */ /* $NetBSD: n_atan2.c,v 1.1 1995/10/10 23:36:37 ragge 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) * scalbn(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. */ #include #include #include "mathimpl.h" static const double athfhi = 4.6364760900080611433E-1; static const double athflo = 1.9338828231967579916E-19; static const double PIo4 = 7.8539816339744830676E-1; static const double at1fhi = 9.8279372324732906796E-1; static const double at1flo = -3.5540295636764633916E-18; static const double PIo2 = 1.5707963267948966135E0; static const double PI = 3.1415926535897932270E0; static const double a1 = 3.3333333333333473730E-1; static const double a2 = -2.0000000000017730678E-1; static const double a3 = 1.4285714286694640301E-1; static const double a4 = -1.1111111135032672795E-1; static const double a5 = 9.0909091380563043783E-2; static const double a6 = -7.6922954286089459397E-2; static const double a7 = 6.6663180891693915586E-2; static const double a8 = -5.8772703698290408927E-2; static const double a9 = 5.2170707402812969804E-2; static const double a10 = -4.4895863157820361210E-2; static const double a11 = 3.3006147437343875094E-2; static const double a12 = -1.4614844866464185439E-2; double atan2(double y, double x) { static const double zero=0, one=1, small=1.0E-9, big=1.0E18; double t,z,signy,signx,hi,lo; int k,m; /* if x or y is NAN */ if (isnan(x)) return (x); if (isnan(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 = scalbn(y,-k); x=scalbn(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) { if (big + small > 0.0) /* 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 { if (big + small > 0.0) /* raise inexact flag */ t = zero; } } /* end of argument reduction */ /* compute atan(t) for t in [-.4375, .4375] */ z = t*t; #if defined(__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 /* defined(__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))))))))))); #endif /* defined(__vax__) */ z = lo - z; z += t; z += hi; return(copysign((signx>zero)?z:PI-z,signy)); } #ifdef __weak_alias __weak_alias(atan2l, atan2); #endif /* __weak_alias */