Show this function is always $\geq 0$

Question

I'm trying to show that:

$$ 1 + \frac{xy}{e^{\frac{1}{2}(x^2+y^2-2)}} \ge 0, \forall~x,y \in \mathbb{R^2} $$

But I end up with the following inequality:

$$ \ln{xy} - \frac{1}{2}(x^2+y^2-2) \ge \ln{(-1)} $$

Which looks like nonsense to me since $xy$ could easily be negative (say $x=-2, y=+2 \rightarrow xy = -4$).

So I'm guessing I'm approaching this in a wrong way. I've also plotted the function and it's indeed $\ge 0, \forall~x,y \in \mathbb{R^2}$ converging to 1 when $x,y \rightarrow \pm \infty$.

Any tips or sources on how to approach the problem (analytically) would be appreciated. Also, why is it that I end up with such a nonsense expression following the initial approach?

Edit:

1. I missed a $\frac{1}{2}$ in the denominator, just added it.
2. $\ln{(-1)}$ is not defined in the real numbers, that's what I mean by "nonsense".
3. $\ln{(xy)}$ causes the same issue with $xy<0$.