Log-Arcsine Algorithm

[Albert Chan]

It may be easier to get recurrence relation, directly from cosines.

Let θ = x/2^(n+1), s(n) = 2^n * sin(x/2^n)

cos(θ) = sin(2θ)/(2*sin(θ)) = s(n)/s(n+1)
cos(2θ) = sin(4θ)/(2*sin(2θ)) = s(n-1)/s(n)

cos(θ) = √((1+cos(2θ))/2)

s(n)/s(n+1) = √((1+s(n-1)/s(n))/2) --> s(n+1) = s(n) * √(2*s(n) / (s(n) + s(n-1)))