You can click "derivation" to see the exact process it was derived. It's the first term of a taylor series expansion around x = 0, which is indeed f(0) = 1/sqrt(9*0-6) = sqrt(-1/6). I'm not sure why x was chosen outside of its input domain though.
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ADDED LATER: I think I have the answer. Herbie seems to always assume x = 0, but there are a fixed set of transformations to get around that fact. Currently it only tries f(x), 1/f(x) and 1/-f(x) [1].
> Maybe there aren’t any value options that provide a speedup, but the tool allows increasingly reduced accuracy (down to zero) until it finds a speedup?
AFAIK Herbie works by trying a bunch of transformations to get candidate expressions and evaluate them to get the pareto frontier. So if there is a single constant expression generated, it will always be available as the worst alternative.
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ADDED LATER: I think I have the answer. Herbie seems to always assume x = 0, but there are a fixed set of transformations to get around that fact. Currently it only tries f(x), 1/f(x) and 1/-f(x) [1].
[1] https://github.com/herbie-fp/herbie/blob/b4a4bb4c61749912a64...
> Maybe there aren’t any value options that provide a speedup, but the tool allows increasingly reduced accuracy (down to zero) until it finds a speedup?
AFAIK Herbie works by trying a bunch of transformations to get candidate expressions and evaluate them to get the pareto frontier. So if there is a single constant expression generated, it will always be available as the worst alternative.