I'm integrating two standard normal distributions and, for numerical problems, I would like to simplify the expresion below, since the only difference between the numerator and denominator is the upper limit, where $\theta$, $\beta \in \mathbb{R}$.
$$ \cfrac{\int _{-\infty }^{\theta}\!{\frac {{{\rm e}^{-{x}^{2}/2\,}}}{ \sqrt{2\,\pi }}}{dx}}{ \int _{-\infty }^{\beta}\!{\frac {{{\rm e}^{-{x}^{2}/2\,}}}{ \sqrt{2\,\pi }}}{dx}}. $$
Any hint on how can I do this simplification?