Let $C$ be a curve in a smooth $3$-fold $X$ with an ordinary node $p\in X$. Blow-up $p$ let $E$ be the exceptional divisor, and $\widetilde{C}$ the strict transform of $C$. Furthermore let $L$ be the line in $E$ through the two points $E\cap \widetilde{C}$.

Now, let us blow-up $\widetilde{C}$ (with exceptional divisor $E_{C}$) and then flop the strict transform of $L$. Let $Z$ be the resulting variety. The strict transform of $E$ in $Z$ is isomorphic to $\mathbb{P}^1\times\mathbb{P}^1$, and can be blow-down to another variety $W$. We get then a divisorial contraction $f:W\rightarrow X$ contracting the strict transform of $E_C$ to $C$.

Is $f$ the blow-up of $X$ along $C$ ?

To me it is clear that it must be the blow-up over any point $q\in C\setminus \{p\}$ but I am quite confused on what happens over $p$.