  The high-resolution, dual channel, visible and near-infrared spectrograph
CARMENES offers exciting opportunities for stellar and exoplanetary research on
M dwarfs. In this work we address the challenge of reaching the highest radial
velocity precision possible with a complex, actively cooled, cryogenic
instrument, such as the near-infrared channel. We describe the performance of
the instrument and the work flow used to derive precise Doppler measurements
from the spectra. The capability of both CARMENES channels to detect small
exoplanets is demonstrated with the example of the nearby M5.0 V star CD Cet
(GJ 1057), around which we announce a super-Earth ($4.0\pm0.4\,M_\oplus$)
companion on a 2.29 d orbit.
