Let me share one approach that makes sense to use in some cases.
In my case, the signal follows roughly the inverse square law in magnitude, but also goes above and below zero, crossing zero at various points. I am interested in the relative error (i.e. far away, where the signal is microvolts, I need precision down to the nanovolt, but near the source, where the signal is a few volts, I need millivolt precision, and would like to ignore deviations in the nanovolt range; so using absolute error doesn't make sense).
But, if I simply divide, either by the true signal, the approximation, or various combinations of the two, the relative error shoots to infinity near the zero-crossings.
The solution is to weigh the absolute error by the inverse of a yardstick signal, that has a similar fall-off properties to the signals of interest, and is positive everywhere. In the formula for relative error, the true signal itself is used for that, but it doesn't have to be, to produce the behaviour you expect from the relative error.
In fact, the normalising signal could be wrong by a multiplicative factor (e.g. if your space is anisotropic, but you still use
1/r^2 as the denominator), and the ratio would still work well as a relative error. Thinking in terms of a log scale helps somewhat, because the relative error becomes a subtraction, rather than division.
To quote an article (1) with 600+ citations reported by Google Scholar, from an authority in these numerical issues:
$\epsilon = (f_2 - f_1) / f_1\;\;\;$ (7)
$E_1$ may of course be expressed as a percent, and like any
relative error indicator it will become meaningless when $f_1$ [...] is zero or small relative to $(f_2 - f_1)$, in which case
the denominator of Eq. (7) should be replaced with some
suitable normalizing value for the problem at hand [...]
(Note that $E_1$ is defined to be a multiple of $\epsilon$ in the article, but these details are irrelevant in the present context.)
I take this as a strong indicator that up until at least 1994, there was no better analogue of relative error for signals that cross zero, than the idea being proposed here, namely, dividing by a normalising signal (that I call the "yardstick" signal above).
(1) PJ Roache, "Perspective: a method for uniform reporting of grid refinement studies"
- Journal of Fluids Engineering, 1994