Given that $\displaystyle X,Y$ are independant Normal distributed variables, with $\displaystyle N(1,\sigma^2), \sigma = \frac{1}{2}$

How do I calculate $\displaystyle P(X+Y\leq t)$

I'm not quite sure how to calculate this since the cdf of the normal distribution doesn't have a closed form. And approximating with an error-function seems quite a bother. Any hints?