'''Purpose:''' Explain the math function '''atan'''.

----
The call,

    [expr { atan( $x ) }]

returns the arcus tangent (inverse tangent) of the number, $x.  
The resulting angle is measured in radians.

----
The most common reason for computing an arctangent is to determine 
the angle from the positive x-axis to a vector in the plane, but
for that purpose it is much better to use [atan2]. 
Consider the point (x,y) = (-1,1). The angle to this point is 3/4*pi 
and the tangent for this vector is -1, but
 % expr atan(-1)
 -0.785398
i.e., the angle -1/4*pi, on account on the fact that this angle
also has tangent -1. [atan2] can distinguish the two:
 % expr atan2(1,-1)
 2.35619
 % expr atan2(-1,1)
 -0.785398

----
atan provides a handy way to ask Tcl for the value of pi: See [pi]

 % expr {atan(1) * 4}
 3.1415926535897931

Actually, using acos() is (slightly) more efficient:
 % set tcl_precision 17
 17
 % expr {acos(-1)}
 3.1415926535897931

''Does anyone have any data on which method is preferable from a numerical point of view?''

[IDG] Both contain the assumption that the transcendental functions are accurate to the last ulp. In many math libraries this is not so. I think you are safer with a string representation

set pi 3.1415926535897931
----
atan is also available in [Tclx].

----
[Math function help] - [Arts and Crafts of Tcl-Tk Programming]
- [Category Mathematics]