= Solution
Expand $|b\rangle=\sum_i b_i|\lambda_i\rangle$ in the nondegenerate eigenbasis of $A$. Apply <quantum phase estimation> to $U_+=e^{2\pi iA}$, using $U_-$ when inverse powers are needed, to coherently attach the eigenvalue:
$$
\sum_i b_i|\lambda_i\rangle|\lambda_i\rangle_{\rm val}.
$$
On an ancillary qubit perform the eigenvalue-controlled rotation
$$
|0\rangle\longmapsto
\sqrt{1-e^{2(\lambda_i-1)}}|0\rangle
+e^{\lambda_i-1}|1\rangle.
$$
Uncompute the value register and measure the ancilla. Conditional on outcome one, the system is proportional to
$$
\sum_i b_ie^{\lambda_i}|\lambda_i\rangle=e^A|b\rangle,
$$
and normalization gives the desired $|\psi\rangle$.
The success probability is
$$
\boxed{P_S=e^{-2}\langle b|e^{2A}|b\rangle}.
$$
Since every $\lambda_i>0$,
$$
\boxed{P_S>e^{-2}},
$$
a positive lower bound independent of $|b\rangle$. The promised precision assumptions permit the eigenvalue-controlled arithmetic and rotation.
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