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."
It depends on how much accuracy you need and over what interval.  It seems that you are happy with a few percent.  There is an approximation in Abromowitz &amp; Stegun that gives $\text{erf}$ in terms of a rational polynomial times a Gaussian over $[0,\infty)$ out to $\sim 10^{-5}$ accuracy.  

In case you care, in the next column, there is a series for erf of a complex number that is accurate to $10^{-16}$ relative error!  I have used this in my work and got incredible accuracy with just one term in the sum.
And another one not so nice but reversible by $\ln(x)$  to depressed quartic equation solvable by Ferrari's formulas and more accurate $\operatorname{erf}(x)= \operatorname{sgn}(x) \tanh[1.152|x| + 0.064|x|^4]$ (error below 0.5%).