Distribution of the Ratio (Quotient) of a Normal Distribution: Fieller-Hinkley
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When the means of X1 and X2 are both 0, the distribution is a Cauchy distribution; otherwise, it becomes quite complex.
(Note) It appears there was a typographical error in the formula in the original paper, so I have corrected it to match the formula in another paper (see the last link) (June 2, 2023).