We show that for deterministic polynomial time computations, oracle access to
$\mathbf{APP}$, the class of real functions
approximable by probabilistic Turing machines, is the same as having oracle access to
promise-$\mathbf{BPP}$. First,
we construct a mapping that maps every function in $\mathbf{APP}$ to a promise problem
in $\mathbf{prBPP}$, and that preserves completeness, i.e. maps complete functions to complete promise problems.
Next, we construct
a mapping from
$\mathbf{prBPP}$ into $\mathbf{APP}$, that maps every promise problem to a function
in $\mathbf{APP}$, and which preserves completeness.
Then, we prove that $\mathbf{P^{\mathbf{APP}}} = \mathbf{P^{\mathbf{prBPP}}}$.
Finally, we use our results to simplify proofs of important results on $\mathbf{APP}$, such as
the $\mathbf{APP}$-completeness of the function $f_{\mathrm{CAPP}}$,
which approximates the acceptance
probability of a Boolean circuit, the error probability reduction Theorem
for $\mathbf{APP}$ functions, and the conditional derandomization result
$\mathbf{APP} = \mathbf{AP} \mbox{ iff } \mathbf{prBPP}$ is easy.

We show that for determinictic polynomial time computation, oracle access to
$\mathbf{APP}$, the class of real functions
approximable by probabilistic Turing machines, is the same as having oracle access to
promise-$\mathbf{BPP}$. First
we construct a mapping that maps every function in $\mathbf{APP}$ to a promise problem
in $\mathbf{prBPP}$, and that maps complete functions to complete promise problems.
Then we show an analogue result in the opposite direction, by constructing a mapping from
$\mathbf{prBPP}$ into $\mathbf{APP}$, that maps every promise problem to a function
in $\mathbf{APP}$, and mapping complete promise problems to complete functions.
Second we prove that $\mathbf{P^{\mathbf{APP}}} = \mathbf{P^{\mathbf{prBPP}}}$.
Finally we use our results to simplify proofs of important results on $\mathbf{APP}$, such as
the $\mathbf{APP}$-completeness of the function $f_{\mathrm{CAPP}}$ that approximates the acceptance
probability of a Boolean circuit, or the possibility (similarily
to the case of $\mathbf{BPP}$) to reduce the
error probability for $\mathbf{APP}$ functions, or the conditionnal derandomization result
$\mathbf{APP} = \mathbf{AP} \mbox{ iff } \mathbf{prBPP}$ is easy.