The present invention involves a rotary peristaltic pump and associated method employing the relaxation of pressure on flexible tubing between the rotor and stator of the pump to restore the rotor to a start angle while a valve on the output of the pump is closed to avoid progress of fluid through the system. Three different implementations of the pump and method are presented including reciprocating the stator, reciprocating the rotor, and retracting the idler rollers of the rotor into the rotor to relieve the pressure on the flexible tubing. A controller ensures the appropriate switching and timing of the valve and rotor of the pump. The pump and associated method are directed to the precision dispensing of pharmaceutical fluids into containers, including containers held in container nests.