Let $\pi:X\rightarrow Z$ be a Fano type fibration with $\dim X-\dim Z=d$ and let $(X,B)$ be an $\epsilon$-lc pair with $K_X+B\sim_{R} 0/Z$. The canonical bundle formula gives $(Z,B_Z+M_Z)$ where $B_Z$ is the discriminant part and $M_Z$ is the moduli part which is determined up to R-linear equivalence. Shokurov conjectured that one can choose $M_Z\geq 0$ such that $(Z,B_Z+M_Z)$ is $\delta$-lc where $\delta$ only depends on $d,\epsilon$. Very recently, this conjecture was proved by Birkar. For $d=1$ and $\epsilon=1$, Han, Jiang and Luo gived the optimal value of $\delta=1/2$. In this paper, we prove that for $d=1$ and arbitrary $0<\epsilon\leq 1$, one can take $$\delta= \sup\{(\epsilon-1/n)/(n-1)\mid n\in \mathbb N_{\geq 2}\}.$$ In particular, one can take $\delta = \epsilon^2/4$ in this case. An example indicates that one can not take $\delta> \epsilon^2$ and hence the order $O(\epsilon^2)$ is sharp.
Comment: Add an example to show that the order O(\epsilon^2) is sharp