Taking the tangent of both sides, we get $\tan (\tan^{-1} x + \tan^{-1} y) = \tan \frac{\pi}{4} = 1.$  Then from the tangent addition formula,
\[\frac{x + y}{1 - xy} = 1.\]Then $x + y = 1 - xy,$ so $xy + x + y = \boxed{1}.$
If $\tan^{-1} x + \tan^{-1} y = \frac{\pi}{4},$ then compute $xy + x + y.$
{'answer': '1', 'subject': 'Precalculus', 'level': 2, 'unique_id': 'test/precalculus/439.json'}