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It has been conjectured that the relative phase between strong andelectromagnetic amplitudes is universally $-90^{\circ}$ in charmonium decays.$\psi^{\prime}$ decaying into pseudoscalar pair provides a possibility to testthis conjecture. However, the experimentally observed cross section for such aprocess is depicted by the two-fold integral which takes into account theinitial state radiative (ISR) correction and energy spread effect. Using thegeneralized linear regression approach, a complex energy-dependent factor isapproximated by a linear function of energy. Taking advantage of thissimplification, the integration of ISR correction can be performed and ananalytical expression with accuracy at the level of 1% is obtained. Then, theoriginal two-fold integral is simplified into a one-fold integral, whichreduces the total computing time by two orders of magnitude. Such a simplifiedexpression for the observed cross section usually plays an indispensable rolein the optimization of scan data taking, the determination of systematicuncertainty, and the analysis of data correlation.

Fast computation of observed cross section for $\psi^{\prime} \to PP$ decays