Re: Optimize evaluation of symbolic expressions

*To*: mathgroup at smc.vnet.net*Subject*: [mg101995] Re: [mg101993] Optimize evaluation of symbolic expressions*From*: Bob Hanlon <hanlonr at cox.net>*Date*: Sun, 26 Jul 2009 03:53:35 -0400 (EDT)*Reply-to*: hanlonr at cox.net

You need to help it expr = {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)]}; Note that if x and y are Real, this can be further simplified Simplify[expr, Element[{x, y}, Reals]] {ArcTan[-x^2 - y^2 + 1, 2*Sqrt[x^2 + y^2]], ArcTan[x, y]} In the general case, Simplify[expr, t == x^2 + y^2 + 1] {ArcTan[2/t - 1, 2*Sqrt[(t - 1)/t^2]], ArcTan[x/t, y/t]} Manipulate[Module[{t = x^2 + y^2 + 1}, {ArcTan[2/t - 1, 2*Sqrt[(t - 1)/t^2]], ArcTan[x/t, y/t]}], {x, -1, 1, .01, Appearance -> "Labeled"}, {y, -1, 1, .01, Appearance -> "Labeled"}] Bob Hanlon ---- Harrie Kraai <hakraai at xs4all.nl> wrote: ============= 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. Harrie

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**Re: Optimize evaluation of symbolic expressions**

**Re: Optimize evaluation of symbolic expressions**