Claimed Quadrature Results seem Impossible We've been preparing a preprint that shows that the convergence bounds proved for tanh-sinh quadrature for numerical integration, cannot possibly hold, and an error must exist - since they imply a P time algorithm to a #P problem. The preprint can be accessed at https://github.com/naturalog/prodcos/blob/master/prodcos.pdf
Maybe someone here have any idea about what is going on here?
 A: I don't know the solution, but the thing that most bothers me about your preprint is the non-explicit nature of Theorem 2 and your inferences from it.  Each $O(\,)$ has some "constant" implied in it, but what do the "constants" depend on? Theorem 2 as you state it suggests to me that the "constants" may depend on $f$, which potentially kills your argument. To make your case convincing you need a form of Theorem 2 where each $O(\,)$ is replaced by an explicit bound on the quantity that is being estimated.
Answering first comment: I don't think the bound can possibly be independent of $f$.  For given $N$ we can modify $f$ any way we like except at the $2N+1$ points we are evaluating it at. I don't believe all such modifications have the same integral within the given error; why should they?  Second, looking back at the definition of $H^2$, I also suspect that the integral error depends on the actual value of the sup and not just on whether it is finite or infinite. Third, another way that the bound can depend on $f$ is that the "for large enough $N$" implicit in the $O(\,)$ notation can depend on $f$.
Further comments: (A) I cannot find [5] in "Mathematics of Computation" or any other journal. If that is true, I suggest you communicate with the authors as to why their paper did not appear.  (B) The presence of $O(\,)$ expressions in Thm 2 is still enough to doubt your claims. You need to bound every quantity with an explicit formula. (C) The number of function evaluations is not enough. You have to consider the cost of very high precision evaluation of the transform.
