Approximating the error function erf by analytical functions The Error function 
$\mathrm{erf}(x)=\frac{2}{\sqrt{\pi}}\int_0^x e^{-t^2}\,dt$
shows up in many contexts, but can't be represented using elementary functions. 
I compared it with another function $f$ which also starts linearly, has $f(0)=0$ and converges against the constant value 1 fast, namely
$\tanh{(x)} = \frac {e^x - e^{-x}} {e^x + e^{-x}}$.
Astoningishly to me, I found that they never differ by more than $|\Delta f|=0.0812$ and converge against each other exponentially fast!
I consider $\tanh{(x)}$ to be the somewhat prettyier function, and so I wanted to find an approximation to $\text{erf}$ with "nice functions" by a short expression. I "naturally" tried 
$f(x)=A\cdot\tanh(k\cdot x^a-d)$
Changing $A=1$ or $d=0$ on it's own makes the approximation go bad and the exponent $a$ is a bit difficult to deal with. However, I found that for $k=\sqrt{\pi}\log{(2)}$ the situation gets "better". I obtained that $k$ value by the requirement that "norm" given by  
$\int_0^\infty\text{erf}(x)-f(x)dx,$
i.e. the difference of the functions areas, should valish. With this value, the maximal value difference even falls under $|\Delta f| = 0.03$. And however you choose the integration bounds for an interval, the area difference is no more than $0.017$.

Numerically speaking and relative to a unit scale, the functions $\text{erf}$ and $\tanh{(\sqrt{\pi}\log{(2)}x)}$ are essentially the same.

My question is if I can find, or if there are known, substitutions for this non-elementary function in terms of elementary ones. In the sense above, i.e. the approximation is compact/rememberable while the values are even better, from a numerical point of view. 
The purpose being for example, that if I see somewhere that for a computation I have to integrate erf, that I can think to myself "oh, yeah that's maybe complicated, but withing the bounds of $10^{-3}$ usign e.g. $\tanh(k\cdot x)$ is an incredible accurate approximation."

I found my own very compact and nice but most importantly readily reversible $\operatorname{erf}(x)= \frac2\pi\arctan[2x(1+x^4)]$ (error below 2%).
