$f(x) = \begin{cases} \dfrac{\sin (\lfloor x^2 \rfloor \pi)}{x^2-3x-18} +ax^3+b & \text{for } 0 \le x \le 1 \\ 2 \cos (\pi x)+ \tan^{-1} x & \text{for } 1 \le x \le 2 \end{cases}$

If $f(x)$ is differentiable in $[0,2]$, find the value of $\left| \dfrac{\pi}{4} - b - a \right|$.

**Note:**

- $\lfloor \cdot \rfloor$ denotes the floor function
- $|\cdot |$ denotes the modulus function.