Given $a,b,n\in\mathbb{R}$ such that $n=ab$, suppose I want to find a numerical approximation of $n$ in the form of an interval that the exact value of $n$ must fall in, that I want the length of the interval to be less than some given $\epsilon\in\mathbb{Q_{>0}}$, and that in order to calculate the approximation, I must approximate $a$ and $b$ separately and multiply the results. It isn't possible to know a priori how small the error intervals for $a$ and $b$ must be in order to hit the desired accuracy for $n$, because the error for $a$ gets scaled up in the product by the magnitude of $b$ and vice versa. For example, suppose I approximate each to within an interval of length 1, and get that $a\in[1,2]$ and $b\in[9,10]$. Then $ab\in[9,20]$, which would suffice for $\epsilon>11$. On the other hand, suppose $b$ were larger, say, $b\in[99,100]$. Then despite the fact that the accuracy of the estimates for $a$ and $b$ is just as good, $n$ is known only to within an interval of size $101$, namely, $[99,200]$.
Of course, I could always just keep halving the error bounds of the approximations for $a$ and $b$ until the interval for $n$ ends up as small as desired. However, since the magnitudes of $a$ and $b$ more or less determine how small the error bounds need to be, and a single round of approximation establishes bounds on what those magnitudes can be, it seems like it should be possible to derive error bounds based on the first round that guarantee that the approximation for $n$ will meet the desired accuracy in the second round. Is that right? I find it tricky to reason about the details.
Apologies if there is vocabulary for these concepts that I'm not taking advantage of. I've been exploring them independently.
Impose some sane limits, such as $\bar δ,ϵ<1$. Then if $a,b$ are the exact values and $a+\delta_a,b+δ_b$ values from the intervals with $|δ_a|,|δ_b|\le\bar δ$, then $$ |(a+\delta_a)(b+δ_b)-ab|\le(|a|+|b|+1)\bar δ $$ so that you can prescribe $$\bar δ=\min(1,\fracϵ{|a|+|b|+1})$$ as the maximal interval length for the factors.
For $a∈[1,2]$ and $b∈[9,10]$ this would result in $\bar δ=\min(1,\fracϵ{13})$.