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by LegionMammal978
88 days ago
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Just looking at the formula in the code (and the book it came from), we see that the approximation is of form arcsin(x) = π/2 - P(x)*sqrt(1-x). It is called a minimax solution in both, and the simplest form of minimax optimization is for polynomials. So we look at P(x) = (π/2 - arcsin(x))/sqrt(1-x): plotting out its error function with the original coefficients, it has the clear equioscillations that you'd expect from an optimized polynomial, i.e., each local peak has the exact same height, which is the max error. But if we look at the error curve in terms of arcsin(x), then its oscillations no longer have the same height, which indicates that the approximation can be improved. |
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