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Sorry to be overly harsh! I think I was initially excited by the suggestion, then disappointed when I thought it through and found it less compelling that it sounded. :)

Re small angles, I don’t quite follow; if I have a couple of small angles, say 0.01 and 0.015, isn’t it useful to have good resolution around 0 so I can represent them reasonably accurately? The Farey sequence seems to have its biggest gap right at that point, between 0 and 1/N.

I don't follow either. When sin(0.01rad) = 0,0099998333... and you do need the 0,00000026 precision you just choose representation with enough bits, same as with floating point.

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