As it turns out, a good approximation for the error function \(\textrm{erf}(x)\) is the function \(\tanh\left(\dfrac{2x}{\sqrt{\pi}}\right)\).

How good of an approximation? Tell me yourself by finding the total area bounded by these two functions. That is, find the value of:

\[ \int_{-\infty}^{\infty} \left|\textrm{erf}(x) -\tanh\left(\frac{2x}{\sqrt{\pi}}\right)\right| dx\]

Please round to 3 decimal places.

Note: \(\displaystyle \textrm{erf}(x)= \dfrac{2}{\sqrt{\pi}}\int_{0}^{x} e^{-t^2} dt\) and \(\tanh(x)= \dfrac{\sinh(x)}{\cosh(x)}\).

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