Negative reciprocal: first step of the Euclidean descent (quotient)

First step of the negative-reciprocal rule of continued fractions (quotient form). If $a>0$ and the leading quotient of the continued fraction of $b/a$ is at least $1$ — equivalently $b\ge a$ — then the Euclidean quotient of $-a$ by $b$ is $-1$:

Combined with the companion remainder identity $(-a)\bmod b=b-a$ this says that the descent of the pair $(b,-a)$ starts with the quotient $-1$ and continues from the pair $(b-a,b)$, which is the mechanism by which the two Euclidean descents used in the $\mathrm{SL}(2,\mathbb Z)$ continued-fraction section merge.