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Let $\pi:X\rightarrow Z$ be a Fano type fibration with $\dim X-\dim Z=d$ andlet $(X,B)$ be an $\epsilon$-lc pair with $K_X+B\sim_{R} 0/Z$. The canonicalbundle 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, thisconjecture was proved by Birkar. For $d=1$ and $\epsilon=1$, Han, Jiang and Luogived 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, onecan take $\delta = \epsilon^2/4$ in this case. An example indicates that onecan not take $\delta> \epsilon^2$ and hence the order $O(\epsilon^2)$ is sharp.

Explicit bound for singularities on the base of a Fano type fibration of relative dimension one