<s>[INST] Unifying the sum of two error functions. Let's propose you have the sum of two error functions:
$f(x) = \text{erf}(ax)+\text{erf(bx)}$
If you wanted to solve for x, you'd first unify them into a third function
$f(x) = \text{erf}(cx)$
Or something similar. How could you approach this problem?
 [/INST] As already said in comments and answer, you will need some numerical method for solving for $x$ the nonlinear equation
$$y=\text{erf}(a x)+\text{erf}(b x)$$
I shall consider the case where $a$ and $b$ are positive (the other situations being more complex).
The first point is to notice that $y$ is bounded by $2\text{erf}(a x)$ and $2\text{erf}(b x)$.
Now, to have bounds, we could use the simple approximation
$$\text{erf}(t)\approx \sqrt{1-\exp\Big(-\frac 4 {\pi}t^2 \Big)}$$ which makes the solution to be between
$$\frac{\sqrt{\pi }}{2}  \sqrt{-\frac{\log \left(1-\frac{y^2}{4}\right)}{a^2}}\qquad \text{and} \qquad \frac{\sqrt{\pi }}{2}  \sqrt{-\frac{\log \left(1-\frac{y^2}{4}\right)}{b^2}}$$
Start at the midpoint and use Newton method. Using for example $a=0.5$, $b=1.5$ and $y=1.2345$, the iterates would be
$$\left(
\begin{array}{cc}
 n & x_n \\
 0 & 0.81835949 \\
 1 & 0.67741900 \\
 2 & 0.69292373 \\
 3 & 0.69314665 \\
 4 & 0.69314670
\end{array}
\right)$$
What we could also do is to expand as Taylor series
$$y=\frac{2 x (a+b)}{\sqrt{\pi }}-\frac{2 x^3 \left(a^3+b^3\right)}{3 \sqrt{\pi
   }}+\frac{x^5 \left(a^5+b^5\right)}{5 \sqrt{\pi }}-\frac{x^7
   \left(a^7+b^7\right)}{21 \sqrt{\pi }}+O\left(x^9\right)$$ and use series reversion to get
$$x=t+\alpha t^3+\beta t^5+\gamma t^7+O\left(t^9\right)\qquad \text{and} \qquad t=\frac{\sqrt{\pi } }{2 (a+b)}y$$
$$\alpha=\frac{1}{3} \left(a^2-a b+b^2\right)$$
$$\beta=\frac{1}{30} \left(7 a^4-17 a^3 b+27 a^2 b^2-17 a b^3+7 b^4\right)$$
$$\gamma=\frac{1}{630} \left(127 a^6-519 a^5 b+1191 a^4 b^2-1471 a^3 b^3+1191 a^2 b^4-519 a b^5+127 b^6\right)$$
For the worked example, this would give, as an estimate to start with, $x=0.685654$.
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