What is the cheapest way to register a software patent?

You can file a provisional patent to lock in the priority date: have one year to file the utility after the provisional, so it gives you one year time. Provisionals are not as formal as a utility, and are meant to lock in a priority date to give you more time to make decisions (for your exact purpose). Depending on the entity you are (large, small, micro) the filing fees vary. With micro the provisional filing fee is around $65. Compared to $450 to file a utility for a micro entity. That is only for the filing fees.
The main cost is in a professional drafting the application. A provisional only protects you for what it discloses (ideally needs to be as complete as possible). Some find that the cost to have a good provisional is almost as much as filing a utility. Varies a lot depending on who drafts it, how complex it is, etc. But you are looking at thousands and not hundreds.

Consider talking to a professional who can walk you through options knowing the details and your goals better.
Consider filing a provisional &/or consider doing a prior art search. That way if an art search turns up art you weren't aware of, that is very material to you being able to get the claims you want, you may decide to not file a utility at all etc.

This is for the US of course!
Hope this helps!
Mary