It's not recognizing a loop, but I've always liked the "turn integer division by a constant into a multiplication by a magic number" optimization: https://godbolt.org/z/Z0FgPW

This and many many other bit-twiddling hacks are described in Hacker's Delight: https://hackersdelight.org/. Compiler developers tend to go through this book and implement all the ones that seem interesting to them.