  The EXtreme PREcision Spectrograph (EXPRES) is an environmentally stabilized,
fiber-fed, $R=137,500$, optical spectrograph. It was recently commissioned at
the 4.3-m Lowell Discovery Telescope (LDT) near Flagstaff, Arizona. The
spectrograph was designed with a target radial-velocity (RV) precision of
30$\mathrm{~cm~s^{-1}}$. In addition to instrumental innovations, the EXPRES
pipeline, presented here, is the first for an on-sky, optical, fiber-fed
spectrograph to employ many novel techniques---including an "extended flat"
fiber used for wavelength-dependent quantum efficiency characterization of the
CCD, a flat-relative optimal extraction algorithm, chromatic barycentric
corrections, chromatic calibration offsets, and an ultra-precise laser
frequency comb for wavelength calibration. We describe the reduction,
calibration, and radial-velocity analysis pipeline used for EXPRES and present
an example of our current sub-meter-per-second RV measurement precision, which
reaches a formal, single-measurement error of 0.3$\mathrm{~m~s^{-1}}$ for an
observation with a per-pixel signal-to-noise ratio of 250. These velocities
yield an orbital solution on the known exoplanet host 51 Peg that matches
literature values with a residual RMS of 0.895$\mathrm{~m~s^{-1}}$.
