Increase precision of custom function

Question

I hope the title is not misleading:

Suppose I have a function that is quite complicated, e.g.

f[u_] := Exp[-Exp[- Abs[c.u]^a] Sin[d.u] Sin[(Abs[c.u]^a) ...

I want to use this function for numerical integration NIntegrate[f, {s, bound1, bound2}] that is prone to numerical error. Mathematica keeps telling me, that the function evaluates not to the desired precision, I have chosen for the numerical integration. Is there a way to improve the precision of the function evaluation, e.g. using arbitrary precision packages/functions? What could I do?

Thanks for your help, it is highly appreciated.

EDIT:

To be a bit more precise: I am trying to do an oscillatory Hilbert transform for a function (King 2002: "Numerical Evaluation of Hilbert transforms for oscillatory functions"). So the error is scaled up with $t^{-1}$ around the singularity. So far the values I obtain are far from being precise, so maybe the technique won't work at all. But I wanted to give it a try. I now use exact number constants like a := 1/10 and T := N[Tan[Pi*a/2], 100]. I hope that Pi will also have the required number of digits then.