 Solution

Squaring a variable with the standard normal distribution gives the required chi-squared distribution. Its two inverse images at $y>0$ are $\pm \sqrt{y}$, so the change-of-variable formula givesDraw the branch-selection uniform variable independently of the randomness used for the normal draw. Then the pair $(Y,U)$ has the product law: conditioning on $Y$ leaves $U$ uniform. This independence is exactly what justifies interpreting the subsequent threshold as the conditional probability of selecting the smaller root.