Inspired by an undisclosed old problem

Geometry Level 5

\[ \large 199 x^5+800 x^4+750 x^3+18 x^2-83 x-12 = 0 \]

Let \(x_1, x_2,x_3,x_4,x_5\) be roots of the equation above.

If the value of \( \displaystyle \tan\left( \sum_{m=1}^5 \tan^{-1}(x_m) \right) \) is equals to \( \frac ab\) for coprime positive integers \(a\) and \(b\), what is the value of \(a - b \)?

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