  We study properties of Euclidean affine functions (EAFs), namely those of the
form $f(r) = (\alpha\cdot r + \beta)/\delta$, and their closely related
expression $\mathring{f}(r) = (\alpha\cdot r + \beta)\%\delta$, where $r$,
$\alpha$, $\beta$ and $\delta$ are integers, and where $/$ and $\%$
respectively denote the quotient and remainder of Euclidean division. We derive
algebraic relations and numerical approximations that are important for the
efficient evaluation of these expressions in modern CPUs. Since simple division
and remainder are particular cases of EAFs (when $\alpha = 1$ and $\beta = 0$),
the optimisations proposed in this paper can also be appplied to them. Such
expressions appear in some of the most common tasks in any computer system,
such as printing numbers, times and dates. We use calendar calculations as the
main application example because it is richer with respect to the number of
EAFs employed. Specifically, the main application presented in this article
relates to Gregorian calendar algorithms. We will show how they can be
implemented substantially more efficiently than is currently the case in widely
used C, C++, C# and Java open source libraries. Gains in speed of a factor of
two or more are common.
