Leigh worried:
>  My only question is how accurately D. performs 
>  numerical differentiation. 

I know nothing of numerical analysis, and remember little of calculus, but I 
also recall some complaints on the Forum along the lines that  D.  loses 
precision quickly.

To that end, you may want to (A) get some test data (perhaps generated by 
Excel) and (B) try various formulations of the gammas to find the set whose 
output most closely matches that test data.

Here's a script that automatically generates and compares a bunch of 
theoretically-equivalent gammas functions:

   http://www.jsoftware.com/svn/DanBron/trunk/environment/generate_j.ijs

It is merely an example.  It has some shortcomings:

   (A)  Its test data is not independent.  In real life,
        you'd want data which you know to be normative.

        This would usually come from the spec or the 
        application you're trying to emulate; in this case, 
        Excel.
         
        It appears my Excel does not have lngamma & derivatives.

   (B)  The formulations are all naive: they all use D.  .
        If  D.  turns out to be workable, then that's not a
        problem.  But if my memory is correct and your fears
        are founded, then indeed you need a different approach.

        As I said, my calculus is rusty; I cannot help you on
        that front.

Jose Quintana:  if you're reading this, the script above is an example of the 
"auto-generation" technique, which you requested a while back:

    http://www.jsoftware.com/pipermail/programming/2007-March/005444.html

-Dan
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