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Define Shannon entropy by $$I(p) = -p \log_2 p$$ Numerical experimentation shows that $\sin(\pi p)^{1-1/e}$ is a good approximation to $I(p) + I(1-p)$ on $[0,1],$ never differing by more than 3.3%. A little more experimentation shows that the $L^{1}$ norm for the difference between $I(p) + I(1-p)$ and $\sin(\pi p)^{x}$ is minimized for $x = 1-1/e.$ Are these known facts? Do they have a simple proof?

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