= Machine epsilon
{title2=$\epsilon_{\rm mach}=\beta^{1-p}$}
{wiki}
= Machine precision
{synonym}
Here <machine epsilon> means the gap from one to the next larger value in normalized <floating-point arithmetic>:
$$
\epsilon_{\rm mach}=\operatorname{next}(1)-1=\beta^{1-p}.
$$
It is a relative resolution scale near one, not the smallest positive representable value and not a fixed absolute error for all magnitudes. In binary64, $p=53$ gives
$$
\epsilon_{\rm mach}=2^{-52}\approx2.220446049250313\times10^{-16},\qquad u=2^{-53}\approx1.1102230246251565\times10^{-16},
$$
where $u$ is <unit roundoff> for rounding to nearest. Some authors instead call $u$ their <machine epsilon>; MIT 18.335J uses this half-gap convention. Always check which definition an error bound uses.https://ocw.mit.edu/courses/18-335j-introduction-to-numerical-methods-spring-2019/2f313023ae3404bc217a81a31b227170_MIT18_335JS19_lec2.pdf[Persson, MIT 18.335J, Lecture 2, slides 3, 8–9: the half-gap convention]{ref}
The gap below one is smaller: the preceding binary64 value is $1-2^{-53}$, whereas the next value above one is $1+2^{-52}$. Under nearest-even rounding, `1.0 + 2.0**-53` is exactly `1.0`: the exact <sum> is a tie and one has the even final <binary digit>. By contrast, `1.0 + 2.0**-52` is the next value. Thus searching for the smallest increment that changes one by trial <addition> also depends on the rounding convention. At $2^{53}$ the next larger binary64 value is $2^{53}+2$, illustrating why a small absolute increment can disappear at large magnitude.
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