Expected maximum of two absolute standard normal variables ($\mathbb{E}\max (|N_1|,|N_2|)=2/\sqrt{\pi }$)

For two independent standard normal random variables, the maximum of their absolute values has distribution function $(2\Phi (x)-1{)}^2$ for $x\ge 0$. Differentiation and integration by parts, using ${\phi }^{\prime }(x)=-x\phi (x)$, give its expectation as $8{\int }_0^{\infty }\phi (x{)}^{2}dx=2/\sqrt{\pi }$. This exact value can detect missing factors in two-sided Gaussian-maximum estimates.