It's silly to use Big-O to describe the number of comparisons when the goal is to analyze performance. Comparisons cost 1 cycle, and you can do many of them per cycle with instruction level parallelism and SIMD. The main bottleneck, the actual source of slowness, is memory. It costs thousands of cycles to go to memory, sometimes tens of thousands or hundreds of thousands (if you need a TLB walk or OS interrupt). If you want to use Big-O, use it to estimate the number of cache misses.

I'd probably go with a custom perfect hash function. And Phil Bagwell's popcount trick. That would be faster than everyone else's solution that involves multiple lookups in memory.

The CPU is fast, memory is slow.