= Interval square range
{title2=$\operatorname{sqr}(X)=\{x^2:x\in X\}$}
= Square range operation
{synonym}
The <interval square range> treats squaring as one <function> of one input. For $X=[a,b]$ it is exactly
$$
\operatorname{sqr}([a,b])=
\begin{cases}
[0,\max\{a^2,b^2\}],&a\leq0\leq b,\\
[\min\{a^2,b^2\},\max\{a^2,b^2\}],&\text{otherwise}.
\end{cases}
$$
It retains the relationship that ordinary interval <multiplication> loses: for $Z=[-1/2,1/2]$, the <interval square range> is $[0,1/4]$, whereas the independent-input <product> $ZZ$ is $[-1/4,1/4]$.
Consequently, <completing the square> for $x(1-x)$ and using the <interval square range> gives
$$
Z=X-[1/2,1/2]=[-1/2,1/2],\qquad [1/4,1/4]-\operatorname{sqr}(Z)=[0,1/4].
$$
This is the exact <image of a function> on $X=[0,1]$. Merely rewriting the expression and evaluating $[1/4,1/4]-ZZ$ gives $[0,1/2]$, so it does not achieve the same sharpening. A mathematical $Z^2$ must be interpreted explicitly here: a <square range operation> and an independent-input interval <product> have different meanings. Choosing an operation that preserves known structure can save many steps of <interval subdivision>.
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