What is the probability $P(X = Y)$ if both $X$ and $Y$ are independent, circularly symmetric Gaussian random variable?

Question

For the random variable x and y mean is zero and variance is $1$.

Efforts

I tried to find its solution by the following technique i.e.

$$\int_{0}^{\infty} \int_{0}^{\infty} \frac{1}{\overline{x}} e^{\frac{-x}{\overline{x}}} \times \frac{1}{\overline{y}} e^{\frac{-y}{\overline{y}}} dx \ dy$$ where $\bar{x}$ and $\bar{y}$ are the respective means of $x,y$.