The lack of control over the scale to switch between absolute and relative error seems like somewhat of a shortcoming: it only works when you have values in the rough vicinity of 1.
In any case, what I usually want from an error metric is a clear interpretation of what it means, apart from just looking nice. Absolute error is good for measurements where the major error sources are independent of the value, while relative error is good for accuracy loss in floating-point arithmetic (though it gets a bit involved with catastrophic cancellation, where you want to take everything relative to the original input scale). Without a principled reason to do so, I wouldn't want to clump together absolute and relative thresholds and distort their meaning like this.
I used it to pick the constants that minimized error in an approximation of inverse trigonometric functions (-π to +π). Found that it provided a good balance between minimizing either the relative or absolute error.
The lack of control over the scale to switch between absolute and relative error seems like somewhat of a shortcoming: it only works when you have values in the rough vicinity of 1.
In any case, what I usually want from an error metric is a clear interpretation of what it means, apart from just looking nice. Absolute error is good for measurements where the major error sources are independent of the value, while relative error is good for accuracy loss in floating-point arithmetic (though it gets a bit involved with catastrophic cancellation, where you want to take everything relative to the original input scale). Without a principled reason to do so, I wouldn't want to clump together absolute and relative thresholds and distort their meaning like this.
I used it to pick the constants that minimized error in an approximation of inverse trigonometric functions (-π to +π). Found that it provided a good balance between minimizing either the relative or absolute error.