Working on an approximation of the error function I found this (true?)inequality :
let $40/9\leq x\leq 90/4$ then it seems we have :
$$f\left(x\right)+f\left(\phi\right)<0$$
Where :
$$f\left(x\right)=\operatorname{erf}\left(x\right)-\tanh\left(x\right)+\frac{\left(-\tanh\left(\frac{x}{4}+\frac{1}{2x}\right)-\frac{x^{\phi}}{x^{e}+1}+1\right)}{2}$$
And $e$ is the euler constant and $\phi$ the golden ratio.
Using the expansion found on wikipedia for the error function or simply using a second derivative I can show for $x\geq 0$:
$$\operatorname{erf}(x)-\tanh(x)\geq 0$$
But we have a big problem around $x=40/9$ and the expansion becomes not praticable .
How to (dis)prove it ?