Any ideas how I can obtain a tight lower, upper bound, or approximation on
$$f(x)=\frac{1-\mathrm{erf}(x)}{1+\mathrm{erf}(x)}$$ for almost any $x$.
This function $f(x)$ grows as $e^{x^2}$ on $x\ll 0$ and decays as $e^{-x^2}$ on $x\gg 0$. I need a tight lower, upper bound, or approximation that captures both the exponential growth for $x<0$ and the exponential decay for $x>0$.
Thank you for you efforts!
In terms of approximation, instead of looking at $$f(x)=\frac{1-\text{erf}(x)}{1+\text{erf}(x)}$$ it could probably be better to consider $$g(x)=\log \left(\frac{1-\text{erf}(x)}{1+\text{erf}(x)}\right)$$ Expanded as series around $x=0$ $$g(x)=-\frac{4 x}{\sqrt{\pi }}\sum_{n=0}^\infty a_n\, x^{2n}$$ Even if all coefficients are known, the series solution or its corresponding $[2n+1,2n]$ Padé approximants are acceptable for a limited range $(|x]< \pi)$.
So, the best I could do is a curve fit by a rational approximation $$g(x)=-\frac{4 x}{\sqrt{\pi }}\, \frac {1+\sum_{n=1}^p b_n\, x^{2n} }{1+\sum_{n=1}^p c_n\, x^{2n} }$$
Generating the data for $0 \leq x \leq 10$, using $p=3$, the regression leads to $R^2=0.999999967$ and the rationalized coefficients are $$b_1=\frac{3235}{2114} \qquad \qquad b_2= \frac{806}{3745}\qquad \qquad b_3=\frac{1}{357} $$ $$c_1=\frac{3307}{2291} \qquad \qquad c_2= \frac{309}{4049}\qquad \qquad c_3=\frac{1}{4885} $$