3 of ref. [@Frixione:2005vw]), using the following integrals: I\_[[[H]{}]{}]{}\^[()]{}&=&\_[ab]{}dz\_1 dz\_2 d\_2 ([[[S]{}]{}\_]{}[[|\_[ev]{}]{}]{}-[[|\_[[MC]{}]{}]{}]{}), \[Ithreedef\]\
I\_[[[S]{}]{}]{}\^[()]{}&=&\_[ab]{}dz\_1 dz\_2 d\_2 . \[Itwodef\] We remind the reader that all integrals defined in these equations are separately finite, and that \_[tot]{}=I\^[(+)]{}\_[[[S]{}]{}]{}+I\^[(+)]{}\_[[[H]{}]{}]{}+I\^[(-)]{}\_[[[S]{}]{}]{}+I\^[(-)]{}\_[[[H]{}]{}]{} is an exact equation (with $\sigma_{tot}$ the fully-inclusive NLO rate).

Although eqs. (\[Ithreedef\]) and (\[Itwodef\]) solve the problem of the generation of hard-subprocess events, a further simplification is possible. We start by considering the integrals $I_{{{\mathbb H}}}^{(\pm)}$, that are used to generate ${{\mathbb H}}$ events. These two integrals have to be computed separately. However, one observes that, for a given choice of , the kinematics associated with these two contributions are actually identical, and the integrals can therefore be computed together. This is equivalent to using the following integral for the generation of ${{\mathbb H}}$ events: I\_[[[H]{}]{}]{}I\_[[[H]{}]{}]{}\^[(+)]{}+I\_[[[H]{}]{}]{}\^[(-)]{}= \_[ab]{}dz\_1 dz\_2 d\_2 ([[|\_[ev]{}]{}]{}-\_[L=]{}[[|\_[[MC]{}]{}]{}]{}). \[Ithreedef2\] The first term in the integrand has been simplified thanks to eq. (\[Sfunnorm\]). For the computation of the second term in the integrand we stress that, as eq. (\[Ithreedef2\]) explicitly indicates, $z_i$ are the integration variables. Therefore, when we evaluate eqs. (\[dsigMCp\]) and (\[dsigMCm\]), the variables $x_{i\pm}^{(s)}$ have to be computed according to eqs. (\[xtilsp\]) and (\[xtilsm\]). From the physical viewpoint, this implies 