Using a computer algebra system to optimize mathematical expressions

This is something I've been wondering for years. Software like Mathematica is great at manipulating expressions into simplified, factorized, and other forms. I'm wondering if there's a way, theoretically and/or practically, to find the form that has the fewest operations. The next step would be to prefer operations that are faster (ie. multiply instead of divide). Lastly, to find a form that maximizes extraction of repetitive subexpressions, so that the subexpressions can be evaluated once and substituted for potentially significant performance gains. Has any research been done in this area? Thanks.

This doesn't completely answer your question, but you ask if research has been done on this.

This is an area called Symbolic Algebra. I'm sure there are many such research groups, but here is one:

https://www.scg.uwaterloo.ca/

They're the group at Waterloo which developed the Maple software (similar to Mathematica).

StackExchange Computer Science Q#12092, answer score: 2