I'm really peeved that I cannot seem to find an algorithm for computing arcsine that works like what I have. This algorithm is based on my work from years ago to compute sine and I have no idea why I never reversed it before now. Anyways, the algorithm:
ArcSine(x), ##x\in[-1,1]##
    x=2x
    s=z=sign(x)
iterate n times
    x=x*x-2
    if x<0
        s=-s
    z=2z+s
return pi*z*2^(-n-2)
This algorithm extracts one bit each iteration, so for 16 bit of accuracy, you would iterate 16 times. I think it is a pretty cute and compact algorithm, so where is it? @_@

As a note, at the endpoints {-1,1} it may not give the expected answer. In both cases, if allowed to run infinitely, it would return (in binary): ##\frac{\pi}{2}.1111111111111..._{2}## which of course is equivalent to ##\frac{\pi}{2}##.