I don't know much about computing functions in general but I would like to understand how Mathematica computes the hyperbolic tangent function for large values of x.
How do you compute the hyperbolic tangent function for high values of x?
2026-03-29 10:47:48.1774781268
How do you compute the hyperbolic tangent function for high values of x?
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Disclaimer: I have no idea how Mathematica does it. Wolfram like to keep their algorithms quiet, or at least quieter than other mathematics software providers.
We may write $$ \tanh{x} = \frac{e^x-e^{-x}}{e^x+e^{-x}} = \frac{e^{2x}-1}{e^{2x}+1} = 1 - \frac{2e^{-2x}}{1+e^{-2x}}. $$ This then has an expansion in terms of an alternating series of decaying exponentials, which decrease very rapidly to zero: $$ 1 - \frac{2e^{-2x}}{1+e^{-2x}} = 1 + 2\sum_{k=1}^{\infty} (-1)^k(e^{-2x})^k $$ So calculate $e^{-2x}$ accurately and it's easy.