Module fixed

Fixed-point intervals at a chosen precision, with certified sin and cos: the tier for values no finite arithmetic holds exactly.

A FixedInterval is [lo, hi] * 2^-bits with big-integer bounds. Every operation rounds its bounds outward, so the true value stays inside. Unlike Dyadic the result is not exact, but its width shrinks as bits grows: a sign the interval cannot decide at one precision is asked again at a higher one, and any nonzero value is decided at some precision. Deciding that a value is exactly zero is the caller’s business (by an identity, as for harmonics of dyadic angles).

sin and cos of a dyadic angle are summed from their Taylor series after halving the angle below 1/16, with the Lagrange remainder added to the bounds, and doubled back up in interval arithmetic.

Structs

FixedInterval

A real number between lo * 2^-bits and hi * 2^-bits.