Sanitizing and disinfecting compositions with relatively low levels of free available chlorine, with good effectiveness against microbes. The composition may include less than 0.5% by weight hypochlorite or other free available chlorine level, at least one nonionic or zwitterionic surfactant, with a pH from 8 to 12.5. The composition is characterized by an R value of greater than 0 (e.g., greater than 0.5, or equal to 1), where R is defined as the sum of the concentration of any nonionic, zwitterionic, and cationic surfactants divided by the total surfactant concentration (and total surfactant concentration may include any surfactant aids). The compositions may exhibit at least a 3-log reduction in M. bovis or C. diff population within 10 minutes (e.g., 4-6 minutes).