Cheap algorithm to find measure of angle between vectors

Question

Finding the angle between two vectors is not hard using the cosine rule. However, because I am programming for a platform with very limited resources, I would like to avoid

Answer 1

Here on SO I still don't have the privilege to comment (though I have at math.se) so this is actually a reply to Timo's post on diamond angles.

The whole concept of diamond angles based on the L1 norm is most interesting and if it were merely a comparison of which vector has a greater/lesser w.r.t. the positive X axis it would be sufficient. However, the OP did mention angle between two generic vectors, and I presume the OP wants to compare it to some tolerance for finding smoothness/corner status or sth like that, but unfortunately, it seems that with only the formulae provided on jsperf.com or freesteel.co.uk (links above) it seems it is not possible to do this using diamond angles.

Observe the following output from my Asymptote implementation of the formulae:

Vectors : 50,20 and -40,40
Angle diff found by acos      : 113.199
Diff of angles found by atan2 : 113.199
Diamond minus diamond         : 1.21429
Convert that to degrees       : 105.255
Rotate same vectors by 30 deg.
Vectors : 33.3013,42.3205 and -54.641,14.641
Angle diff found by acos      : 113.199
Diff of angles found by atan2 : 113.199
Diamond minus diamond         : 1.22904
Convert that to degrees       : 106.546
Rotate same vectors by 30 deg.
Vectors : 7.67949,53.3013 and -54.641,-14.641
Angle diff found by acos      : 113.199
Diff of angles found by atan2 : 113.199
Diamond minus diamond         : 1.33726
Convert that to degrees       : 116.971

So the point is you can't do diamond(alpha)-diamond(beta) and compare it to some tolerance unlike you can do with the output of atan2. If all you want to do is diamond(alpha)>diamond(beta) then I suppose diamond is fine.