# inverse complementary error function values near 0

$\newcommand{\erf}{\operatorname{erf}}\newcommand{\erfc}{\operatorname{erfc}}$Let's define for each $x>0$ $$\erf(x)=\frac {2}{\sqrt{\pi}}\int_0^xe^{-t^2} dt$$ and $$\erfc(x)=\frac {2}{\sqrt{\pi}}\int_x^\infty e^{-t^2} dt=1-\erf(x)$$

I want to approximate the inverse functions $\erf^{-1}$ and $\erfc^{-1}$ for very small values of $x$.

I know that $\erf\sim f(x)=\frac{2x}{\sqrt{\pi}}$ now since the inverse function of $f$ is close to $0$ when $x$ is close to $0$ I can use that as an approximation for $\erf^{-1}$, in fact $\erf(f(x))\sim x$ when $x<<1$.

Now this doesn't work with erfc since the inverse function of $1-f(x)$ is not small for small values of $x$

Though I know that $\erfc(x) \sim g(x)=\frac{1}{\pi}\frac{e^{-x^2}}x$ for $x\gg1$. Since $g^{-1}$ is an increasing function is should be true that $g^{-1}(x)\gg1$ when $x\gg1$ but is there any simple way to compute the inverse of this function?

Thank you!

This is only a partial answer.

For the inverse error function, for small arguments, Taylor series seem to be quite good $$\text{erf}^{-1}(x)=\frac{\sqrt{\pi } }{2}x\Big(1+\frac{\pi }{12}x^2+\frac{7 \pi ^2 }{480}x^4+\frac{127 \pi ^3 }{40320}x^6+O\left(x^8\right)\Big)$$ Pade approximants $$\text{erf}^{-1}(x)=\frac{\sqrt{\pi } }{2}x\frac{1-\frac{11 \pi }{120}x^2}{1-\frac{7 \pi }{40}x^2}$$ $$\text{erf}^{-1}(x)=\frac{\sqrt{\pi } }{2}x\frac{1-\frac{4397 \pi }{17352}x^2+\frac{111547 \pi ^2 }{14575680}x^4}{1-\frac{5843 \pi }{17352}x^2+\frac{20533 \pi ^2 }{971712}x^4}$$ are even much better.

For the inverse complementary error function, I really did not find anything which could be satisfactory except the fact that $$x e^{x^2} \text{erfc}^{-1}(x)$$ could probably be fitted as $x(x-1)P_n(x)$ but it will not work well close to $x=0$.

Edit

If you are not looking for a very high accuracy and using really small values of $x$, it seems that $$\text{erfc}^{-1}(x)\approx \sqrt{\frac{1}{2} \left(\log \left(\frac{2}{\pi x^2}\right)-\log \left(\log \left(\frac{2}{\pi x^2}\right)\right)\right)}$$ could be acceptable $$\left( \begin{array}{ccc} x & \text{exact} & \text{approximation} \\ 0.0001 & 2.75106 & 2.74595 \\ 0.0002 & 2.62974 & 2.62435 \\ 0.0003 & 2.55640 & 2.55085 \\ 0.0004 & 2.50322 & 2.49754 \\ 0.0005 & 2.46127 & 2.45550 \\ 0.0006 & 2.42652 & 2.42068 \\ 0.0007 & 2.39679 & 2.39089 \\ 0.0008 & 2.37078 & 2.36483 \\ 0.0009 & 2.34763 & 2.34163 \\ 0.0010 & 2.32675 & 2.32071 \\ 0.0020 & 2.18512 & 2.17883 \\ 0.0030 & 2.09851 & 2.09210 \\ 0.0040 & 2.03517 & 2.02871 \\ 0.0050 & 1.98487 & 1.97839 \\ 0.0060 & 1.94297 & 1.93650 \\ 0.0070 & 1.90696 & 1.90049 \\ 0.0080 & 1.87530 & 1.86886 \\ 0.0090 & 1.84700 & 1.84060 \\ 0.0100 & 1.82139 & 1.81503 \end{array} \right)$$

For inverse erf, I inverted a low order the pade approximant (it is valid up to $\pm 0.97$) Orange part is the approximation:

$$\frac{3-\sqrt{9-3 \pi x^2}}{\sqrt{\pi } x}$$

For inverse erfc, I inverted the asmyptotic form of erfc:

$$\frac{\sqrt{W\left(\frac{2}{\pi x^2}\right)}}{\sqrt{2}}$$

This one is even better(just subtract $0.5x$) (but this sacrifices accuracy near x=0, keep that in mind):