Any good approximation for $\Phi^{-1}(1-\Phi(a))$?

Question

Let $\Phi$ be the standard Gaussian CDF and $a > 0$.

Question

Is there any good approximation for $\Phi^{-1}(1-\Phi(a))$ ?

Answer 1

For $a \in \mathbb{R}$ and not just $a > 0$, the exact solution is

$$\Phi^{-1}(1-\Phi(a)) = -a \\ \because 1 - \Phi(a) = \Phi(-a) $$ due to the symmetry with respect to zero. This applies to any symmetric density, or equivalently, any CDF that are rotationally symmetric with respect to $(0,\frac12)$.