text: This is not a complete analysis, and I am not an expert in this area, and just the result of a few random thoughts. You can flesh this out as needed. \n\n\nLet f(n,x)=the nth application of function g to x. In your case g(x)=sin(x). In general, a function g(x) will have some values x at which f(n,x) converges to a fixed point, and some values x at which it does not, but diverges, or exhibits periodic or chaotic behavior (wanders around in some bounded region). \n\nFor the case of g(x)=sin(x), \n\nf(1e1,1)=.463\nf(1e2,1)=.1689\nf(1e3,1)=.0546\nf(1e4,1)=.0173\nf(1e5,1)=.0055\nf(1e6,1)=.0017\n\nPlotting log(f(1en,1)) versus n shows an approximately straight line, so that if a0=-1.1274 and a1=0.4303, then y(n)=a0*n+a1 is a reasonable approximation for log(f(1en,1)), and log(f(1en,1))-y(n) is \n\n(-0.0729, 0.0460, 0.0441, 0.0222, 0.0036, -0.0431) \n\nfor n=1,2,3,4,5,6 respectively. \n\nI notice that there appears to be some oscillatory behavior in this residual, which you might be able to model fairly accurately as well for more variance reduction in your convergence prediction. \n\nMaybe this will get you started and you can dig into this further, depending on how much depth you need.