= Round-off error
= Roundoff error
{synonym}
<Round-off error> arises when numbers or arithmetic results are rounded to a finite set of representable values. Its scale is associated with <machine epsilon>, and its effects depend on subsequent operations. Even a formula with zero <truncation error> can lose information. In decimal arithmetic rounded to three significant digits, $1.234$ becomes $1.23$ and $1.233$ becomes $1.23$: subtracting the stored numbers gives zero although the exact difference is $0.001$. The subtraction exposes errors already present in its nearly equal operands.
This example also has a large <condition number>: independently perturbing the two inputs of $a-b$ can produce a relative amplification $(|a|+|b|)/|a-b|=2467$. High sensitivity belongs to the input-output problem; <numerical stability> concerns the arithmetic used to solve it. Conversely, on a common binary system with 53 significant bits and rounding to nearest, $(10^{16}+1)-10^{16}$ evaluates to zero while $(10^{16}-10^{16})+1$ evaluates to one. Exact arithmetic makes both one: changing summation order changes the accumulated <roundoff error>. MIT's summation lecture explains why <numerical stability> and the <condition number> must be assessed separately.https://ocw.mit.edu/courses/18-335j-introduction-to-numerical-methods-spring-2019/pages/week-2/[Steven G. Johnson, MIT 18.335J, Lecture 3: Floating-Point Summation and Backwards Stability, 2019]{ref}
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