How useful is this when you are using numbers in a reasonable range, like 10^-12 to 10^12? Generally I try to scale my numbers to be in this range, whether by picking the right units or scaling constraints and objectives when doing nonlinear programming/ optimization.

Like looking at this example,

https://herbie.uwplse.org/demo/b070b371a661191752fe37ce0321c...

It is claimed that for the function f(x) =sqrt(x+1) -1

Accuracy is increased by from 8.5% accuracy to 98% for alternative 5 Which has f(x) = 0.5x

Ok so x=99, the right answer is sqrt(100) -1 = 9

But 0.5 * 99 = 49.5 which doesn't seem too accurate to me.

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yossi_peti 4 hours ago | parent | next [-]

The precondition on the link you shared has -1 <= x && x <= 1, so 99 is way outside of that range. But even so, testing for x=1, which is supposed to be inside that range, 0.5 doesn't seem tolerably close to 0.4142.

	▲	LiamPowell 7 minutes ago \| parent [-]
		I have a suspicion that the accuracy number is the mean of accuracies over all valid floats in the range (or something approximating that), which is going to be weighted towards zero where the accuracy is higher, and perhaps where sqrt near 1 has some artefacts.

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hmpc 4 hours ago | parent | prev [-]

Check the specification at the top. The range for x is [-1, 1]. For the range you provided the accuracy of the 0.5x alternative is reported as only 33%: https://herbie.uwplse.org/demo/570b973df0f1f4a78fe791858038a...

	▲	fluorinerocket 4 hours ago \| parent [-]
		You're right I misread the graph. That said though I have played around with Herbie before, trying it out on a few of the more gnarly expressions I had in my code (analytical partial derivatives if equations of motion if launch vehicle in rotating spherical frame) and didn't see much appreciable improvement over the expected range of values, but then again I didn't check every single one. What would be cool is if you could some how have this kind of analysis done automatically for your whole program where it finds the needle in the haystack expression that can be improved, assuming you gave expected ranges for your variables