The first thing that we're actually shown is Snape's modifications to the Draught of Living Death potion. Some of his modifications, like crushing the bean instead of cutting to release more juice, seem like reasonable things to try. Even adding juice from 1 extra bean seems like a reasonable thing to try.
However, the stirring directions (7 times clockwise, 1 time counterclockwise) seems absurdly specific. The directions say to stir twice clockwise. How do you get from "stir twice clockwise" to "stir 7 times clockwise and once counterclockwise"? How did he know to do 7 times rather than 6 or 8? Yeah, I know that 7 is a magically significant number, but 1 (the number of counterclockwise stirs) isn't, and neither is 2 (the number of stirs in the original directions), so really, I could ask how either of them came up with any of their numbers for stirs - it seems quite arbitrary.
Also, his Levicorups spell shows multiple attempts to get it right. How did he "figure it out" - guess at the incantation or something? What might his previous experiments have looked like, and what were the results likely to have been? Presumably, they had some effect given that he had written them down, but the fact that he crossed them out implies that successive attempts were better somehow. Better how, though?
TL;DR How did Snape figure out the stuff he had written in his book?