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Comment by raphlinus | Hacker News Reader

A bit OT but my favorite way to approximate tanh is to do g(f(x)) where f(x) is an odd polynomial and g(x) is 1/sqrt(1 + x^2). The recip sqrt can be computed very quickly using Newton approximation, and many SIMD instructions have an instruction for it. Highly recommended for audio applications.

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Something is definitely off about your recollection here. g(x) = 1/√(1+x²) is always positive, whereas tanh(x) is negative for negative values of x. So regardless of your f, g(f(x)) can’t ever be a correct approximation for negative x.

Looks like grandparent probably meant g(x) = x/√(1+x²)

Looks like f(x)=x+x^3/5 gets you two decimal digits everywhere [1], and you could improve that with more terms in the polynomial.

Edit: you can derive f(x) by doing a series expansion of tanh(x)/√(1-tanh(x)²) about x=0, and Mathematica tells me the first few terms are

f(x) = x + x^3/6 + x^5/120 + x^7/5040 + O(x^9)

[1] https://www.desmos.com/calculator/zbyll5oeqk

Oops, yes, you're correct, I was typing from memory and didn't double-check.

And yes, the Taylor series is pretty good but you can usually squeeze a bit more accuracy-over-range by, eg, using an optimizer for the coefficients. Also, note that in your notebook you have x^3/5 but x^3/6 is probably what you wanted.

Could be a typo, f(g(x)) with g(x)=x/√(1+x²) and f a polynomial makes more sense to me.

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