Given \[ 0 \le x \le \frac{1}{2}, \] find the value of \[ \tan \left( \sin^{-1}\left( \frac{x}{\sqrt{2}} + \frac{\sqrt{1 - x^2}}{\sqrt{2}} \right) - \sin^{-1} x \right). \]
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When dealing with expressions involving $\sin^{-1}(A \cos B \pm C \sin D)$, try to convert the argument of the inverse sine into a single sine term using trigonometric sum/difference identities. Pay attention to the domain of the inverse function to correctly apply $\sin^{-1}(\sin x) = x$.