High-precision spectrographs play a key role in exoplanet searches and Doppler asteroseismology using the radial velocity technique. The 1  m s-1level of precision requires very high stability and uniformity of the illumination of the spectrograph. In fiber-fed spectrographs such as SOPHIE, the fiber-link scrambling properties are one of the main conditions for high precision. To significantly improve the radial velocity precision of the SOPHIE spectrograph, which was limited to 5–6  m s-1, we implemented a piece of octagonal-section fiber in the fiber link. We present here the scientific validation of the upgrade of this instrument, demonstrating a real improvement. The upgraded instrument, renamed SOPHIE+, reaches radial velocity precision in the range of 1–2  m s-1. It is now fully efficient for the detection of low-mass exoplanets down to 5–10 M⊕ and for the identification of acoustic modes down to a few tens of  cm s-1.