  We provide the most accurate estimate yet of the bright end of the infrared
(IR) luminosity functions (LFs) and the abundance of hyperluminous IR galaxies
(HLIRGs) with IR luminosities > 10^13 L_solar, thanks to the combination of the
high sensitivity, angular resolution, and large area of the LOFAR Deep Fields,
which probes an unprecedented dynamic range of luminosity and volume. We
cross-match Herschel sources and LOFAR sources in Bootes (8.63 deg^2), Lockman
Hole (10.28 deg^2), and ELAIS-N1 (6.74 deg^2) with rms sensitivities of around
32, 22, and 20 mJy per beam, respectively. We divide the matched samples into
unique and multiple categories. For the multiple matches, we de-blend the
Herschel fluxes using the LOFAR positions and the 150-MHz flux densities as
priors. We perform spectral energy distribution (SED) fitting, combined with
multi-wavelength counterpart identifications and photometric redshift
estimates, to derive IR luminosities. The depth of the LOFAR data allows us to
identify highly complete (around 92% completeness) samples of bright Herschel
sources with a simple selection based on the 250 micron flux density (45, 40,
and 35 mJy in Bootes, Lockman Hole, and ELAIS-N1, respectively). Most of the
bright Herschel sources fall into the unique category (i.e. a single LOFAR
counterpart). For the multiple matches, there is excellent correspondence
between the radio emission and the far-IR emission. We find a good agreement in
the IR LFs with a previous study out to z around 6 which used de-blended
Herschel data. Our sample gives the strongest and cleanest indication to date
that the population of HLIRGs has surface densities of around 5 to 18 / deg^2
(with variations due to a combination of the applied flux limit and cosmic
variance) and an uncertainty of a factor of 2. In comparison, the GALFORM
semi-analytic model significantly under-predicts the abundance of HLIRGs.
