Using a novel approach , the distribution of fluences of long gamma-ray bursts derived from the Swift-BAT catalog was reproduced by a jet-model characterized by the distribution of the total radiated energy in \gamma -rays and the distribution of the aperture angle of the emission cone .
The best fit between simulated and observed fluence distributions permits to estimate the parameters of the model .
An evolution of the median energy of the bursts is required to adequately reproduce the observed redshift distribution of the events when the formation rate of \gamma -ray bursts follows the cosmic star formation rate .
For our preferred model , the median jet energy evolves as E _ { J } \propto e ^ { 0.5 ( 1 + z ) } and the mean expected jet energy is 3.0 \times 10 ^ { 49 } erg , which agrees with the mean value derived from afterglow data .
The estimated local formation rate is R _ { grb } = 290 ~ { } Gpc ^ { -3 } yr ^ { -1 } , representing less than 9 % of the local formation rate of type Ibc supernovae .
This result also suggests that the progenitors of long gamma-ray bursts have masses \geq 90 M _ { \odot } when a Miller-Scalo initial mass function is assumed .