Water keeps puzzling scientists because of its numerous properties which behave oppositely to those of usual liquids: for instance, water expands upon cooling, and liquid water is denser than ice. To explain this anomalous behavior, several theories have been proposed, with different predictions for the properties of supercooled water (liquid at conditions where ice is stable). However, discriminating between those theories with experiments has remained elusive because of spontaneous ice nucleation. Here we measure the sound velocity in liquid water stretched to negative pressure and derive an experimental equation of state, which reveals compressibility anomalies. We show by rigorous thermodynamic relations how these anomalies are intricately linked with the density anomaly. Some features we observe are necessary conditions for the validity of two theories of water.