Question: Efficient Extreme Value Analysis

I have a question that I hope someone can help with. We perform extreme value analysis on expressions. My this I mean that we evaluate expressions for each combination of min or max values of each variable in the expression. For example, the expression a*b+c*d has 4 variables and we would evaluate the expression for all of the 16 combinations of a,b,c, and d to determine the worst-case result. I understand that the worst case results may not occur at the extreme values of the variables, but that is a different question.

 What I would like to know is if there is a way to reduce the number of calculations by splitting the expression into independent parts. In the example above, the expression consists of 2 parts a*b and c*d. These parts could be evaluated separately. Each has 2 variables, and each would therefore requires  4 combinations of variables to determine the extreme values of the results. So in this case, the number of calculations is 4+4=8, which is half of the initial 16  runs as would be done with a full-factorial analysis. Also, because each calculation is done on only part of the original expression, each calculation takes less time.

Is there a way to find the independent parts of an arbitrary expression so that each independent part could be evaluated separately to speed processing?


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