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