  TOU is an extremely high resolution optical spectrograph (R=$100,000$,
380-900~nm), which is designed to detect low mass exoplanets using the radial
velocity technique.We describe an IDL-based radial velocity (RV) data reduction
pipeline for the TOU spectrograph and its performance with stable stars. This
pipeline uses a least-squares fitting algorithm to match observed stellar
spectra to a high signal-to-noise template created for each star. By carefully
controlling all of the error contributions to RV measurements in both the
hardware and data pipeline, we have achieved $\sim$0.9\ms long-term RV
precision with one of the most RV stable stars, Tau Ceti, similar to what has
been achieved with HARPS. This paper presents steps and details in our data
pipeline on how to reach the \sms RV precision and also all major error sources
which contribute to the final RV measurement uncertainties. The lessons learned
in this pipeline development can be applied to other environmentally
controlled, very high resolution optical spectrographs to improve RV precision.
