On the Project Nayuki article on Barrett reduction, one important lemma is the inequality

$$$\displaystyle\frac{x}{n} - 1 < \frac{xr}{4^k} ≤ \frac{x}{n} \implies \frac{x}{n} - 2 < \left\lfloor \frac{x}{n} - 1 \right\rfloor ≤ \left\lfloor \frac{xr}{4^k} \right\rfloor ≤ \frac{x}{n}$$$

We shall show that this inequality holds even if $$$\dfrac{xr}{4^k}$$$ is replaced by $$$\dfrac{(x-\delta) r}{4^k}$$$, with $$$0 ≤ \delta < 2^{62}$$$.

First we need to obtain a tighter bound on $$$r$$$. Given that $$$r$$$, $$$k$$$, and $$$n$$$ are fixed in our algorithm, we can verify that $$$\dfrac {4^k} n - r < 2^{-9}$$$. Let this value be $$$\varepsilon$$$. We thus obtain $$$\displaystyle \frac{4^k}{n} - \varepsilon < r < \frac{4^k}{n}$$$

Multiply by $$$x-\delta \geq 0$$$: $$$\displaystyle (x-\delta)\left(\frac{4^k}{n} - \varepsilon\right) < (x-\delta)r < (x-\delta)\frac{4^k}{n}$$$

Given that $$$\delta$$$ is nonnegative we can relax the right bound to $$$x\dfrac{4^k}{n}$$$.

Divide by $$$4^k$$$: $$$\displaystyle (x-\delta)\left(\frac{1}{n} - \frac \varepsilon {4^k}\right) < \frac {(x-\delta)r}{4^k} < \frac{x}{n}$$$

Recompose the leftmost expression: $$$\displaystyle \frac x n - \left(\frac {\varepsilon x} {4^k} + \frac \delta n\right) + \frac{\delta\varepsilon} {4^k} < \frac {(x-\delta)r}{4^k} < \frac{x}{n}$$$

$$$\displaystyle\frac{\delta\varepsilon} {4^k} \geq 0$$$, so relax the bound: $$$\displaystyle \frac x n - \left(\frac {\varepsilon x} {4^k} + \frac \delta n\right) < \frac {(x-\delta)r}{4^k} < \frac{x}{n}$$$

$$$x < n^2 < 4^k \implies \dfrac{x}{4^k} < 1$$$, so further relax the bound: $$$\displaystyle \frac x n - \left({\varepsilon} + \frac \delta n\right) < \frac {(x-\delta)r}{4^k} < \frac{x}{n}$$$

$$$\delta < 2^{62}$$$ while $$$n$$$ is very slightly below $$$2^{63}$$$, so it can be verified that $$$\dfrac \delta n$$$ must be $$$< \dfrac34$$$.

$$$\dfrac34 + \varepsilon < 1$$$, therefore $$$\displaystyle \frac x n - 1 < \frac {(x-\delta)r}{4^k} < \frac{x}{n}$$$

We can thus follow the rest of the proof in the Nayuki article to prove that our $$$t$$$ is in $$$[-n, n)$$$.