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Optimize evaluation of symbolic expressions

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  • Subject: [mg101993] Optimize evaluation of symbolic expressions
  • From: Harrie Kraai <hakraai at>
  • Date: Sat, 25 Jul 2009 04:18:57 -0400 (EDT)

Hello all,

With a specific example I countered today I want to introduce a question 
that I have had for a longer time.

 From some composition of functions (and a subsequent FullSimplify) I 
get the following expression:

{ArcTan[-1 + 2/(1 + x^2 + y^2), 2 Sqrt[(x^2 + y^2)/(1 + x^2 + y^2)^2]],
  ArcTan[x/(1 + x^2 + y^2), y/(1 + x^2 + y^2)]}

which I need to evaluate many times inside a Manipulate.

Now, how does Mathematica evaluate this?

If I would program this myself, I would, of course, evaluate the 
subexpression (1 + x^2 + y^2) first and use this result 4 times in the 
subsequent calculations.

It looks to me like Mathematica is not able to or at least does not make 
this optimization. (Find out using 'Trace').  Perhaps this is difficult 
to do in a general sense because any subexpression may have 
side-effects. However, in this case it should be possible to instruct 
Mathematica to collect common subexpressions first. It would surely make 
calculations much faster.
The symbolic calculations have led to an expression that is 
"Simplify"-ed  in terms of reading (perhaps) but not in terms of 
evaluation. Is there, or should there be a function that translates 
expressions to a form that is optimized for evaluation?

This appears to me as a frequently encountered problem. How do we go 
about it?

Thanks for your advice and/or thoughts.


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