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Both f(x) and g(x) could be calc'ed to proper machine precision (say, GSL double prec ~ 1E-15). Would this imply, that beyond the machine precision the values are effectively fixed to resp. zero or infinity, instead of carrying around the extreme orders of magnitude?

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I'm not exactly sure what you're asking here, but the point is that "to machine precision" is relative: if f(x) and g(x) are O(1e200), then the absolute error of each is still O(1e185). However f(x)/g(x) will still be very accurate (with absolute error O(1e-15)).

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