Inspired by an undisclosed old problem

Geometry Level 5

199x5+800x4+750x3+18x283x12=0 \large 199 x^5+800 x^4+750 x^3+18 x^2-83 x-12 = 0

Let x1,x2,x3,x4,x5x_1, x_2,x_3,x_4,x_5 be roots of the equation above.

If the value of tan(m=15tan1(xm)) \displaystyle \tan\left( \sum_{m=1}^5 \tan^{-1}(x_m) \right) is equals to ab \frac ab for coprime positive integers aa and bb, what is the value of aba - b ?

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