interval arithmetic

Ordinary floating-point computing gives you a single number and a vague hope that it is close to the truth. Interval arithmetic gives you a guarantee instead: rather than carrying one approximate value, it carries an entire interval [low, high] that is mathematically certain to contain the true value. Think of tracking not a dot but a guaranteed-enclosing box that shrinks or grows as the computation proceeds, so the final box rigorously brackets the exact answer.

The mechanics replace each number by an interval and redefine the arithmetic so the result interval is guaranteed to enclose every possible exact result. For example [a, b] + [c, d] = [a + c, b + d], and [a, b] times [c, d] is the smallest interval containing all products of one endpoint from each. To stay rigorous on a real computer, each lower bound is rounded DOWN and each upper bound rounded UP (outward rounding), so floating-point rounding can never let the true value escape the interval. Run a whole algorithm this way and the output interval is a certified, machine-proven bound on the answer — a form of a posteriori error control.

Be candid about the trade-off. Interval arithmetic gives rigorous, guaranteed bounds, which is invaluable for computer-assisted proofs and safety-critical results — but the naive version suffers the 'dependency problem': because it forgets that the same variable appears more than once, the intervals can grow pessimistically wide and overstate the true uncertainty (e.g. x - x is computed as a wide interval, not exactly 0). It is also several times more expensive than plain floating-point. Skilled use (centred forms, splitting domains) tightens the bounds; it is a specialist tool for when a certified enclosure, not just a good estimate, is required.