The multiplicative complexity of interval checking

TL;DR

This paper precisely calculates the minimal AND-gate complexity for interval checking, revealing that it can be as low as or lower than a single comparison, challenging assumptions about its computational cost.

Contribution

It provides the exact AND-gate cost for interval checking with constant bounds, showing it can be less than or equal to a single comparison.

Findings

Interval checking cost never exceeds one comparison.

In some cases, interval checking requires fewer gates than a comparison.

The results challenge traditional views on the complexity of interval checks.

Abstract

We determine the exact AND-gate cost of checking if $a\leq x < b$, where $a$ and $b$ are constant integers. Perhaps surprisingly, we find that the cost of interval checking never exceeds that of a single comparison and, in some cases, it is even lower.