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I wonder if it makes sense to treat floating points not as single points on the number line but as a range. This way when you subtract two similar numbers the result is a range that is wide in relation to the distance from 0, explicitly encoding the bigger relative rounding error.

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That exists and is called interval arithmetic.

There are even more improvements (in some regard) to this, e.g. Affine Arithmetic[0, 1]. Here, error terms from different operations are accumulated symbolically so that in the case where they would cancel out (as could be useful for numerical algorithms), tighter bounds can be obtained. There are several research groups working on improving such analyses.

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[0] - https://en.wikipedia.org/wiki/Affine_arithmetic

[1] - http://users.ece.cmu.edu/~rutenbar/pdf/rutenbar-intervalicas...

edit: formatting

And the problem with it is that it can lead to [-inf;+inf] intervals on sane computation such as Newton's method (refinements have been introduced but nothing that fully solves the problem).

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