Re: Optimize evaluation of symbolic expressions

*To*: mathgroup at smc.vnet.net*Subject*: [mg101999] Re: [mg101993] Optimize evaluation of symbolic expressions*From*: DrMajorBob <btreat1 at austin.rr.com>*Date*: Sun, 26 Jul 2009 03:54:18 -0400 (EDT)*References*: <200907250818.EAA29767@smc.vnet.net>*Reply-to*: drmajorbob at bigfoot.com

expr1 = {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)]}; expr2 = expr1 /. (1 + x^2 + y^2 -> s) {ArcTan[-1 + 2/s, 2 Sqrt[(x^2 + y^2)/s^2]], ArcTan[x/s, y/s]} f[x_, y_] = Block[{s = 1 + x^2 + y^2}, expr2]; f[a, b] {ArcTan[-1 + 2/(1 + a^2 + b^2), 2 Sqrt[(a^2 + b^2)/(1 + a^2 + b^2)^2]], ArcTan[a/(1 + a^2 + b^2), b/(1 + a^2 + b^2)]} Bobby On Sat, 25 Jul 2009 03:18:57 -0500, 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 > -- DrMajorBob at bigfoot.com

**References**:**Optimize evaluation of symbolic expressions***From:*Harrie Kraai <hakraai@xs4all.nl>

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