Problem 1931

Created attachment 436 [details]
Improved numerical precision using differences in radius and hypot to avoid numerical cancellations between large squared numbers

The calculation of transverse distances called “pt" or “it" (often using pt2, it2 for distance squared) is done in G4Torus.cc in several places using code like

rho2 = p.x()*p.x() + p.y()*p.y();
rho = std::sqrt(rho2);
pt2 = rho2+p.z()*p.z() +fRtor * (fRtor-2*rho);
pt2 = std::max(pt2, 0.0);
pt = std::sqrt(pt2) ;

resulting in poor accuracy (and in some cases incorrect pt=0) when fRtor and rho are similar in magnitude.

A much better precision is obtained using instead :

rho = std::hypot(p.x(),p.y());
pt  = std::hypot(p.z(),rho-fRtor);

or basically using (a-b)^2 instead of $a^2+b^2-2ab$ avoiding cancellations between large squared numbers.

Improved G4Torus.cc attached.