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Let $f(x,1)=\sin(x)$ and $f(x,i)=f(\sin(x),i-1)$ ($f$ is the iterated sine function). For arbitrary $N$,$x_0$, how quickly can $f(x_0,N)$ be computed?

Answer to this question discusses Carleman-Matrix method. Is this method precise for very big $N$?

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