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Use the cyclic logarithmic coloringHere the floor function places $x$ in the unique half-open real interval $[(3/2{)}^j,(3/2{)}^{j+1})$. If $y/x\in [1.9,2]$, thenbecause $3/2<1.9\le 2<(3/2{)}^2$. For any real $t$ and $1<u<2$, $\lfloor t+u\rfloor -\lfloor t\rfloor$ is either $1$ or $2$. Applying this to the logarithms of $x$ and $y$ shows that their bin indices differ by $1$ or $2$, and hence have different residues in modular arithmetic modulo $3$. This proves the required separation, including both endpoints of the prescribed ratio interval. The half-open bins remove any ambiguity at their boundaries.

Let the known phase gate on the answer register bewhere the comparison uses ordinary integers, not modular arithmetic. A reversible circuit computes the predicate, applies a Pauli Z gate to its flag, then performs uncomputation.

The compute-phase-uncompute construction now givesUse modular-oracle inversion by negation to realize the last operation with one further $U_f$ query. Exactly two oracle queries implement $I_g$, returning the answer register and comparison workspace to their initial states.