Math

Bounds instead of estimates

Every other way of computing gives a number that is probably right. Interval arithmetic gives a pair that is certainly right, and the whole subject is what that costs. It costs sharpness, and the cost has a name: every occurrence of a variable is treated as a different variable, so x − x comes out as an interval two units wide. Which means the enclosure is a property of the FORMULA, not of the function - rewrite x(1 − x) as 0.25 − (x − 0.5)² and the width falls from 1 to the true 0.25; write x·x instead of x² in 1/(1 + 25x²) and the answer becomes the entire real line.

By Mykhailo Makarov

Actual application image

What happens

Every other module in this ring measures an error; this one refuses to. It computes with pairs of numbers guaranteed to contain the answer, rounding every endpoint outward, and the guarantee is the whole point: sharpness is traded for certainty. Feed x = [1, 2] into x − x and you get [−1, 1], not 0, because the arithmetic does not know the two x are the same - the dependency problem, and the reason intervals swell with every step. Divide by an interval containing zero and the honest answer is the whole line. Try the expressions, watch the widths grow, and see which rewritings keep them tight.

Equations used in the module

Equations
[a,b] + [c,d] = [↓(a+c), ↑(b+d)],  [a,b] − [c,d] = [↓(a−d), ↑(b−c)]
[a,b]·[c,d] = [↓min, ↑max] of the four products
[a,b]/[c,d] = the whole line when [c,d] contains 0
the dependency problem: x − x = [a−b, b−a], not 0

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