Let $\phi(t) := \frac{1}{\sqrt{2\pi}}\exp\{-t^2/2\}$ be the standard Gaussian pdf function and $\Phi(t) := \int_{-\infty}^t \phi(u)du$ be the Gaussian CDF function. Consider equation $$ \Phi(x) + \phi(x) = 1. $$
I'm wondering whether such an equation has simple closed-form solutions $x$?
There is one single zero for the equation $$f(x)=\frac{1}{2}\text{erf}\left(\frac{x}{\sqrt{2}}\right)+\frac{e^{-\frac{x^2}{2}}}{\sqrt{2 \pi}}-\frac{1}{2}$$ and it is close to $0.3$.
For a good approximation, we can write the $[n,n]$ Padé approximant $P_n$ and solve for $x$ $$P_n=\frac 12$$ leading to explicit formulae if $n\leq 4$.
Form example $$P_1=\frac{3 x+2}{\sqrt{2 \pi } (x+2)} \qquad \implies x_1=0.290051$$ $$P_2=\frac{-11 x^2+66 x+60}{\sqrt{2 \pi } \left(13 x^2+6 x+60\right)}\qquad \implies x_2=0.302645$$ $$P_3=\frac{-707 x^3+4332 x^2+3180 x-600}{3 \sqrt{2 \pi } \left(277 x^3+84 x^2+1260 x-200\right)}\qquad \implies x_3=0.302651$$ $$P_4=\frac{326537 x^4+1877380 x^3-5065740 x^2+14346360 x+15622320}{3 \sqrt{2 \pi } \left(116679 x^4-59420 x^3+1340460 x^2-425320 x+5207440\right)}$$ would give $x_4=0.302631$ which is the "exact" solution for six significant figures.
Another solution would consist in a Taylor expansion followed by a power series reversion. This would give $$x=\sum_{n=1}^\infty a_n\,t^n \qquad \text{where} \qquad t=\sqrt{\frac{\pi }{2}}-1$$
SInce we know all coefficients of the initial series, we know all the $a_n$, the first of them making the sequence $$\left\{1,\frac{1}{2},\frac{2}{3},\frac{11}{12},\frac{43}{30},\frac{851}{360},\frac{10259}{2520},\frac{145667}{20160}, \frac{1190911}{90720},\frac{22036891}{907200},\frac{455267359}{9979200}\right\}$$
