I have to manipulate huge expressions that are rational functions of many (∼30) variables with integer coefficients. Storing them just as a ratio of two polynomials would be impractical. But they can be represented as multi-level (∼140 levels) nested sums of fractions. Storing them in this form still would be impractical (billions of nodes), but, luckily, they have lots of repeated sub-expressions, so I store them as expressions with Experimental`OptimizedExpression head, where all repeated parts are recursively extracted and denoted by auxiliary intermediate variables. In this form my expressions only have a few thousands of nodes.
To convert expressions to this optimized form I use Experimental`OptimizeExpression function, with some additional custom post-processing. Often, I have a list of expressions, but as they usually have many shared repeated parts, the list can be efficiently represented as a single optimized expression where all shared parts are extracted.
Now, I need to do some symbolic computations with optimized expressions: addition, multiplication, division and a substitution of one optimized expression for a variable in another. In all these cases, I want to get a single optimized expression as a result, where each shared repeated part among operands is extracted and denoted by a single variable.
For smaller expressions, this can be done by a brute-force approach: expand the operands first, perform the operation as usual and, finally, apply Experimental`OptimizeExpression to the result to get an optimized form. For larger expressions, it becomes too slow. I was about to write my custom algorithm for extracting shared parts from several optimized expressions without fully expanding them. But I thought that other people, likely, already have faced a similar problem. So, I would appreciate if anybody has an existing solution, and is willing to share it. Or, maybe, there is already a function in Mathematica that can do it?
