Whether saving information makes something faster depends on how much time it takes to retrieve from storage and how much time it takes to recalculate the variables on the fly.
As a general rule, it takes something on the order of fractions of nanoseconds to get something stored in cache or to perform a CPU instruction. Going to RAM can take anywhere from ~1 nanosecond to ~20 nanoseconds, depending on the speed of the memory and if it's being accessed in a contiguous block.
Getting something from disk can take as much as a millisecond. This time is greatly reduced if you're reading a large amount of contiguous stuff in at once.
Whether you can make a program faster by storing intermediate values and precalculating some parts depends on how much time it takes to calculate those values in the first place. There's no simple, general answer to "how much can I gain by storing intermediate values?".