Let \( f : (0, \infty) \to \mathbb{R} \) be a twice differentiable function. If for some \( a \neq 0 \),
\[
\int_0^a f(x) \, dx = f(a),
\]
and given that
\[
f(1) = 1, \quad f(16) = \frac{1}{8},
\]
then
\[
16 - f^{-1}\left( \frac{1}{16} \right)
\]
is equal to: