You can do formal verification with tools like Coq using a laptop. The real issue is that writing a formal specification is hard even for small problems; writing a formal spec for something like TLS would be harder than actually implementing TLS.

Cool! I do hardware, I guess software is probably easier to compute.

Ah, you might have been thinking of techniques basic on SAT and ATPG solving. What makes software different is this:

https://en.wikipedia.org/wiki/Curry-Howard_Correspondence

Basically, checking a proof of correctness for software is equivalent to type checking (for a very fancy type system).