Even something very rudimentary is fine. Like entering and evaluating something like x+1, where x is a variable I get to specify.
I was thinking of writing my own using Strings for each formula, regex for parsing them, etc., but let's assume my goal is to be as lazy as I can possibly get away with, and that if there's another option (especially a formal one), I'd rather try to use that instead, first.
Is there anything out there that does something like this?
You can think about using scripting from Java 6. The reference JDK comes with a JavaScript implementation, and you can plug in other languages too.
I've used JEP with success a couple of years ago and it still seems to be actively developed. I believe they had a more permissive license back in those days, but it's probably cheaper than coding it yourself (depending on your exact needs and skills).
I've built a java library to do exactly this as a learning example for Lex/YACC, for myself. It was incorporated into a larger project, where it seems to work well. Were I to start the larger project again, I'd follow the advice above and use the Java scripting engine.
I know the following libraries:
MathEclipse Parser (Apache License) supports Mathematica like syntax
JParser (BSD License)
Jep - Java Math Expression Parser (commercial - older versions under GNU Public License?)
JFormula (commercial)
JEval (Apache License)
You could try LibFormula, a formula eavluation library that's based on OpenDocument's OpenFormula standard.
OpenFormula was written for spreadsheet applications and uses a syntax close to Excel-formulas. More info can be found in the Pentaho-Wiki
Sample:
=[x]+1