  Close-orbit low-mass X-ray binaries (LMXBs), radio binary millisecond pulsars
(BMSPs) with extremely low-mass helium WDs (ELM He~WDs) and ultra-compact X-ray
binaries (UCXBs) are all part of the same evolutionary sequence. It is
therefore of uttermost importance to understand how these populations evolve
from one specie to another. Moreover, UCXBs are important gravitational wave
(GW) sources and can be detected by future space-borne GW observatories.
However, the formation and evolutionary link between these three different
populations of neutron star (NS) binaries are not fully understood. In
particular, a peculiar fine-tuning problem has previously been demonstrated for
the formation of these systems. In this investigation, we test a newly
suggested magnetic braking prescription and model the formation and evolution
of LMXBs. We compute a grid of binary evolution models and present the initial
parameter space of the progenitor binaries which successfully evolve all the
way to produce UCXBs. We find that the initial orbital period range of LMXBs,
which evolve into detached NS+ELM~He~WD binaries and later UCXBs, becomes
significantly wider compared to evolution with a standard magnetic braking
prescription, and thus helps to relieve the fine-tuning problem. However, we
also find that formation of wide-orbit BMSPs is prohibited for strong versions
of this new magnetic braking prescription, which therefore calls for a revision
of the prescription. Finally, we present examples of the properties of UCXBs as
Galactic GW sources and discuss their detection by the LISA, TianQin and Taiji
observatories.
