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Equation 1 · Part 3 · How Mathematics, Proof, and Scientific Computation Actually Work

Symbol delta

fl(xopy)=(xopy)(1+δ),∣δ∣≤u,\mathrm{fl}(x \mathbin{\text{op}} y) = (x \mathbin{\text{op}} y)(1 + \delta), \qquad |\delta| \le u,
δ\delta

What this part means

delta is one of the signed contributions combined to compute the quantity on the left.

Its job in the formula

delta is one of the signed contributions combined to compute the quantity on the left.

The passage around this formula

Computers represent real numbers with a finite number of bits, following a specification — IEEE 754 — that fixes the format and the rounding behavior of arithmetic exactly enough that different compliant hardware and software give reproducible results for the same operations [ 5 ] . The standard’s core guarantee is deceptively narrow: each individual elementary operation is correctly rounded , meaning the result is the closest representable floating-point number to the true mathematical result of that one operation. For a single addition or multiplication, this means fl(xopy)=(xopy)(1+δ),∣δ∣≤u\mathrm{fl}(x \mathbin{\text{op}} y) = (x \mathbin{\text{op}} y)(1 + \delta), \qquad |\delta| \le u. where u is the unit roundoff — roughly 10^{-16} for standard double precision — and op\mathrm{op} is…

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Learn the underlying idea

A variable is a named place for a value. Its letter is a local label: x can mean position in one formula and a data point in another.

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Sources cited in the surrounding passage

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