  Let $\mathcal{T}$ be any of the three canonical truth theories
$\textsf{CT}^-$ (Compositional truth without extra induction), $\textsf{FS}^-$
(Friedman--Sheard truth without extra induction), and $\textsf{KF}^-$
(Kripke--Feferman truth without extra induction), where the base theory of
$\mathcal{T}$ is $\textsf{PA}$ (Peano arithmetic). We show that $\mathcal{T}$
is \textit{feasibly reducible to} $\textsf{PA}$, i.e., there is a polynomial
time computable function $f$ such that for any proof $\pi $ of an arithmetical
sentence $\phi $ in $\mathcal{T}$, $f(\pi )$ is a proof of $\phi $ in
$\textsf{PA}$. In particular, $\mathcal{T}$ has at most polynomial speed-up
over $\textsf{PA}$, in sharp contrast to the situation for
$\mathcal{T}[\textsf{B}]$ for \textit{finitely axiomatizable} base theories
$\textsf{B}$.
