* 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.
static char sccsid
[] = "@(#)atan2.c 1.3 (Berkeley) 8/21/85";
* 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 :
* 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 )
* 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;
* atan2(y,x) returns (PI/pi) * the exact ARG (x+iy) nearly rounded,
* pi = 3.141592653589793 23846264338327 .....
* 53 bits PI = 3.141592653589793 115997963 ..... ,
* 56 bits PI = 3.141592653589793 227020265 ..... ,
* 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)).
* 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
* 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
#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 */
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 */
static double zero
=0, one
=1, small
=1.0E-9, big
=1.0E18
;
double copysign(),logb(),scalb(),t
,z
,signy
,signx
,hi
,lo
;
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
;}
if(y
==zero
) return((signx
==one
)?y
:copysign(PI
,signy
));
if(x
==zero
) return(copysign(PIo2
,signy
));
return(copysign((signx
==one
)?PIo4
:3*PIo4
,signy
));
return(copysign((signx
==one
)?zero
:PI
,signy
));
if(!finite(y
)) return(copysign(PIo2
,signy
));
if((m
=(k
=logb(y
))-logb(x
)) > 60) t
=big
+big
;
else { t
= y
/x
; y
= scalb(y
,-k
); x
=scalb(x
,-k
); }
/* begin argument reduction */
/* truncate 4(t+1/16) to integer for branching */
{ 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] */
hi
= athfhi
; lo
= athflo
;
t
= ( (y
+y
) - x
) / ( z
+ y
); break;
/* t is in [11/16,19/16] */
t
= ( y
- x
) / ( x
+ y
); break;
/* t is in [19/16,39/16] */
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) */
/* t is in [2.4375, big] */
if (t
<= big
) t
= - x
/ y
;
{ big
+small
; /* raise inexact flag */
/* end of argument reduction */
/* compute atan(t) for t in [-.4375, .4375] */
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
))))))))))));
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
= lo
- z
; z
+= t
; z
+= hi
;
return(copysign((signx
>zero
)?z
:PI
-z
,signy
));