erf(x) - erf(y) = erfc(y) - erfc(x) about herbie HOT 7 CLOSED

Ok, so there are actually a few things going on.

First of all, the original program is only inaccurate for a relatively small range, for x and y between 5 and 28. For larger values, erf(x) - erf(y) returns 0, which is actually the closest floating-point result for the correct answer. Since the range is small, Herbie doesn't sample many points in it (only one in (2048/3)^2 ≈ 466k points falls into this range) and ends up basically ignoring it. So Herbie was "correctly" determining that the original expression had very little error and didn't need any changes. The best way to fix this is to add a precondition, either using :pre (if on the command line) or by clicking the "Additional options" link (if on the web version).

Second of all, one of Herbie's rewrite rules (erfc-erf) was less general than it should have been. I've tweaked that in the last commit.

from herbie.

Great question, I'll look into it. I think I know why but give me a moment to prove out my theory.

from herbie.

Awesome! Thanks for taking a look. Actually I'm trying to find a robust expression to compute the variance of a truncated normal (https://en.wikipedia.org/wiki/Truncated_normal_distribution#Two_sided_truncation[2]).
My manual attempts have failed so far.
Will you release a new version of Herbie with the erf rules fix so I can try it? Thanks!
Edit: I'll try compiling from master.

from herbie.

I tried erf(x) - erf(y) with precondition x > y on Herbie master but I still don't get a transformation to erfc. Maybe I need to do something else?

from herbie.

Hi @cossio, I got good results when I specified the precondition 5 < x < 28 and 5 < y < 28. I bet reasonably similar bounds on x and y would work.

from herbie.

Thanks. I get

Don't know why it doesn't simplify the square root.

In fact, it seems that Herbie gets to (−erfc(x))+erfc(y) as an intermediate expression, but then applies a couple of rules that introduce the square roots (which seem superfluous to me).

from herbie.

Yeah, Herbie does that sometimes, and I don't know why. I've been meaning to look into it. In some cases, that may twiddle some low-order bit in just the right way to minutely increase accuracy, but it might also be totally useless. I'll add it to my list of things to look into.

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