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I have a longer analytic expression in several variables containing special functions and others. Does Mathematica bring tools - or are there any packages - to examine the stability when I evaluate the expression numerically. I want to understand for which variables the accuracy of the numerical result is lowest in order to see how I can rewrite the expression to improve the result at finite precision. Note that the goal is not to obtain a high accuracy result by telling Mathematica to use more digits. I rather want to have an expression which evaluates accurately using a relatively small number of digits. Since it is not obvious how to achieve this because of the length of the expression I would like to have some help from Mathematica telling me how to optimize the expression.

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Thanks, but that is not really what I am looking after. I also don't really want to examine specific examples. I rather would like to know if there are tools which help me to tell for which variables x,y,z,... f(x,y,z,...) has which largest uncertainty if evaluated at finite precision. Here f is a generic, possibly scalar, function. Perhaps later I would like to find out why. I am not interested in AccuracyGoal, PrecisionGoal, WorkingPrecision etc. –  highsciguy Jan 4 at 16:02
    
It is a good question, but I think the answer is probably "no". If you have the time and inclination, it might be an interesting project to implement this. But, it might not be easy. One of the major arguments for multiprecision arithmetic (in Mathematica particularly) is that it is often cheaper/more practical just to use a higher precision than to perform this careful stability analysis. –  Oleksandr R. Jan 4 at 19:58

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