Tao Theorem 5.1 as its proof gives it

Bound for minor arc sums, with the constants its own proof yields. Let $4\alpha=\frac aq+\beta$ for a natural number $q\ge4$ with $(a,q)=1$ and $|\beta|\le q^{-2}$. Let $1<U,V<x$ with $UV\le\frac x4$, $UV^2\ge x$ and $U,V\ge40$. Then

This is the source's Theorem 5.1 with each of its four terms replaced by what its own chain of estimates delivers. The source prints $0.5$, $0.89$ and $0.78$ where this statement has $1$ (with an extra additive $4$), $1.78$ and $1.1$. It is still strong enough for everything the source does with it: the exponential sum estimate of Theorem 1.3 follows from this form with its printed constants, at $U=V=\frac1{10}x^{2/5}$ rather than the source's $U=\frac14x^{2/5}$, $V=\frac12x^{2/5}$.

Where the three changes come from

The factor $2$ in the first two terms. The Type I argument bounds each block $2jq+\frac q2<d\le2(j+1)q+\frac q2$ by Corollary 3.5. That block has length exactly $2q$, so the corollary's prefactor is $\lfloor\frac{2q}{2q}\rfloor+1=2$, giving $2\bigl(2A_j+\frac2\pi Cq\log4q\bigr)$ with $A_j=\frac12\frac{x}{2jq+q/2}\log x+C$ and $C=4(\log2)\log2x$. The source's display uses $2A_j+\frac2\pi Cq\log 4q$, that is prefactor $1$. Carrying the correct prefactor doubles both the harmonic term and the block bookkeeping, and $\frac{4\log2}\pi\le0.89$ becomes $\frac{8\log2}\pi\le1.78$.

The additive $4$. The integral test $\sum_{0\le j\le\frac{UV}{2q}-\frac14}\frac{x}{2jq+\frac q2}\le\frac1{2q}\int_{q/2}^{UV+2q}\frac xy\,dy$ compares a decreasing summand with the integral over the preceding block of length $2q$, which covers $j\ge1$ but not $j=0$; the uncovered term is $\frac{x}{q/2}=\frac{x}{2q}\cdot4$. At $q=4$, $UV=40$ the sum is $0.7235x$ and the printed bound $0.3973x$.

The last constant. In the square-root expansion of the Type II envelope the cross term is $\sqrt{2q}\cdot\sqrt{\frac{x}{2Wq}}\cdot\sqrt x=x\sqrt{\frac{2q}{2Wq}}=\frac{x}{\sqrt W}$, coefficient $1$ and not $\frac1{\sqrt2}$; integrating against $\frac{4dW}{W}$ turns $\frac{1.1}{\sqrt2}\le0.78$ into $1.1$.

Formalization Note The alternative form of the first term available when $a=\pm1$ and $UV<q-1$ is omitted, since the derivation of Theorem 1.3 does not use it. The smoothed sum is the platform definition.