If your calculations are processor-dependent, that's a bad sign for your code. If your results really depend on things that can be altered by the specific floating-point implementation, you need to write code that's robust to changes in the way floating-point arithmetic is done, generally by tracking the uncertainty associated with each number in your calculation. (Obviously you don't need real-time performance since you're using cloud computing in the first place.) I'm not an expert on Mathematica, but it probably has such things built in if you go through the documentation, since Mathematica notebooks are supposed to exhibit reproduceable behavior on different machines. (Which is not to say that no matter what you write it's automatically going to be reproduceable.
Archiving hardware to get consistent results is mainly used when there are legal issues and some lawyer can jump in and say, "A-ha! This bit here is different, and therefore there's some kind of fraud going on!"