  The question how much star formation is occurring at low metallicity
throughout the cosmic history appears crucial for the discussion of the origin
of various energetic transients, and possibly - double black hole mergers. We
revisit the observation-based distribution of birth metallicities of stars
(f$_{\rm SFR}$(Z,z)), focusing on several factors that strongly affect its low
metallicity part: (i) the method used to describe the metallicity distribution
of galaxies (redshift-dependent mass metallicity relation - MZR, or
redshift-invariant fundamental metallicity relation - FMR), (ii) the
contribution of starburst galaxies and (iii) the slope of the MZR. We
empirically construct the FMR based on the low-redshift scaling relations,
which allows us to capture the systematic differences in the relation caused by
the choice of metallicity and star formation rate (SFR) determination
techniques and discuss the related f$_{\rm SFR}$(Z,z) uncertainty. We indicate
factors that dominate the f$_{\rm SFR}$(Z,z) uncertainty in different
metallicity and redshift regimes. The low metallicity part of the distribution
is poorly constrained even at low redshifts (even a factor of $\sim$200
difference between the model variations) The non-evolving FMR implies a much
shallower metallicity evolution than the extrapolated MZR, however, its effect
on the low metallicity part of the f$_{\rm SFR}$(Z,z) is counterbalanced by the
contribution of starbursts (assuming that they follow the FMR). A
non-negligible fraction of starbursts in our model may be necessary to satisfy
the recent high-redshift SFR density constraints.
