  We investigate the evolution of the galaxy Star Formation Rate Function
(SFRF) and Cosmic Star Formation Rate Density (CSFRD) of $z\sim 0-8 $ galaxies
in the Evolution and Assembly of GaLaxies and their Environments (EAGLE)
simulations. In addition, we present a compilation of UV, IR and H$\alpha$
SFRFs and compare these with the predictions from the EAGLE suite of
cosmological hydrodynamic simulations. We find that the constraints implied by
different indicators are inconsistent with each other for the highest
star-forming objects at z < 2, a problem that is possibly related to selection
biases and the uncertainties of dust attenuation effects. EAGLE's feedback
parameters were calibrated to reproduce realistic galaxy sizes and stellar
masses at z = 0.1. In this work we test if and why those choices yield
realistic Star Formation Rates (SFRs) for $z \sim 0-8$ as well. We demonstrate
that SNe feedback plays a major role at setting the abundance of galaxies at
all star-forming regimes, especially at high redshifts. On the contrary, Active
Galactic Nuclei (AGN) feedback becomes more prominent at lower redshifts and is
a major mechanism that affects only the highest star-forming systems.
Furthermore, we find that galaxies with SFR $\sim 1-10 \, {\rm M_{\odot} \,
yr^{-1}}$ dominate the CSFRD at redshifts z < 5, while rare high star-forming
galaxies (SFR $\sim 10-100 \,{\rm M_{\odot} \, yr^{-1}}$) contribute
significantly only briefly around the peak era ($z \sim 2$) and then are
quenched by AGN feedback. In the absence of this prescription objects with SFR
$\sim 10-100 \,{\rm M_{\odot} \, yr^{-1}}$ would dominate the CSFRD, while the
cosmic budget of star formation would be extremely high. Finally, we
demonstrate that the majority of the cosmic star formation occurs in relatively
rare high mass halos ($ {\rm M_{Halo}} \sim 10^{11-13} \, {\rm M_{\odot}}$)
even at the earliest epochs.
