At first this was pretty surprising to me, and I was pretty sure it must be a bug in the code. Well, I tried it myself and got the same results. :-)
After a few false starts, here's what I think is the probable explanation. Note that we're trying to approximate √3 by a fraction of the form (ad-bc)/(ac+bd). The closest fractions we get (among a, b, c, d below a given magnitude) could be either below or above √3.
Consider an angle of (π/3+ε). If you look tan(π/3+ε)-tan(π/3), it works out to be (tan(ε) + 3ε) / (1 - ε√3).
When ε is replaced by -ε, if you look at tan(π/3)-tan(π/3-ε), it works out to be (tan(ε) + 3ε) / (1 + ε√3).
For small ε, the former, namely (tan(ε) + 3ε) / (1 - ε√3), is larger than the latter, namely (tan(ε) + 3ε) / (1 + ε√3). (More generally, this boils down to the fact that the second derivative of tan(x) is positive.)
This means that it's easier (you're allowed a larger difference in the angle) to achieve a given closeness of the fraction (ad-bc)/(ac+bd) to √3 by picking the angle to be greater than π/3 (=60°) than if the angle is less than π/3.
For larger denominators, √3 becomes about as easy to approximate to a given closeness from below as from above (by fractions of the form (ad-bc)/(ac+bd)), but among roughly equally distant approximations, the ones from above are closer in angle than the ones from below.
Can't edit the above post, but when trying to write this up more carefully I realized it's not correct:
(1) For one thing, the difference in epsilon is very small: the ratio (1 + ε√3)/(1 - ε√3) is 1+2√3ε+O(ε²), that is, we're comparing something like 4ε+4√3ε² versus 4ε-4√3ε², which is probably too small to matter.
(2) The argument looks like it applies to all angles, but in fact experimentation shows the same phenomenon does not appear, and finally,
