Three instructions to turn a day count into a weekday

Benjamin Joffe has published a collection of day-of-week algorithms that he benchmarks at roughly 0.3–0.5× the time of Cassio Neri's 2024 method, itself the current reference implementation and already faster than the naive double-modulo or Rust's rem_euclid. Several of the new functions have a latency of one multiplication plus two cycles.

The core trick exploits that 7 is a Mersenne number (2³ − 1), which permits the identity N % 7 = floor(N * 8 / 7) % 8. Multiplying by an approximation of 8/7 and taking the low bits, with a right shift of exactly three less than the register width, leaves the modulo for free in the remaining three bits. An added constant rotates the output to align with the Unix epoch falling on a Thursday. Because that constant is only a rotation, producing ISO weekdays (1–7) instead of Unix (0–6) costs nothing — same instructions, different constant.

The simplest version is three operations but valid only over about ±242,000 years; full 32-bit range needs 64-bit widening or one of three other variants. The article notes the technique is already in use: Rust's Jiff date library, which spans ±10,000 years, saw a 40% improvement in nth_weekday_of_month.

Relevance is narrow — date libraries, database engines, compilers. Joffe himself recommends plain (rd + 4) % 7 where maintainability matters. Benchmarks come from the author on AMD Ryzen 9 and Apple M4 Pro; no independent verification is cited.