  Comparing invariants from both topological and geometric perspectives is a
key focus in index theorem. This paper compares higher analytic and topological
torsions and establishes a version of the higher Cheeger-M\"uller/Bismut-Zhang
theorem. In fact, Bismut-Goette achieved this comparison assuming the existence
of fiberwise Morse functions satisfying the fiberwise Thom-Smale transversality
condition (TS condition). To fully generalize the theorem, we should remove
this assumption. Notably, unlike fiberwise Morse functions, fiberwise
generalized Morse functions (GMFs) always exist, we extend Bismut-Goette's
setup by considering a fibration $ M \to S $ with a unitarily flat complex
bundle $ F \to M $ and a fiberwise GMF $ f $, while retaining the TS condition.
  Compared to Bismut-Goette's work, handling birth-death points for a
generalized Morse function poses a key difficulty. To address this, first, by
the work of the author M.P., joint with Zhang and Zhu, we focus on a relative
version of the theorem. Here, analytic and topological torsions are normalized
by subtracting their corresponding torsions for trivial bundles. Next, using
new techniques from by the author J.Y., we excise a small neighborhood around
the locus where $f$ has birth-death points. This reduces the problem to
Bismut-Goette's settings (or its version with boundaries) via a Witten-type
deformation. However, new difficulties arise from very singular critical points
during this deformation.To address these, we extend methods from Bismut-Lebeau,
using Agmon estimates for noncompact manifolds developed by Dai and J.Y.
