NAD(+) hydrolase (NADase) that catalyzes cleavage of NAD(+) into ADP-D-ribose (ADPR) and nicotinamide (PubMed:29395922). In addition to ADPR, also generates a cyclization variant of cyclic ADPR (cADPR), termed v-cADPR, for which the cyclizing bond is unknown (PubMed:29395922). H2O + NAD(+) = ADP-D-ribose + H(+) + nicotinamide The TIR domain mediates NAD(+) hydrolase (NADase) activity. Self-association of TIR domains is required for NADase activity.