Execution of accurate eye movements depends critically on the cerebellum, suggesting that the major output neurons of the cerebellum, Purkinje cells, may predict motion of the eye. However, this encoding of action for rapid eye movements (saccades) has remained unclear: Purkinje cells show little consistent modulation with respect to saccade amplitude or direction, and critically, their discharge lasts longer than the duration of a saccade. Here we analysed Purkinje-cell discharge in the oculomotor vermis of behaving rhesus monkeys (Macaca mulatta) and found neurons that increased or decreased their activity during saccades. We estimated the combined effect of these two populations via their projections to the caudal fastigial nucleus, and uncovered a simple-spike population response that precisely predicted the real-time motion of the eye. When we organized the Purkinje cells according to each cell's complex-spike directional tuning, the simple-spike population response predicted both the real-time speed and direction of saccade multiplicatively via a gain field. This suggests that the cerebellum predicts the real-time motion of the eye during saccades via the combined inputs of Purkinje cells onto individual nucleus neurons. A gain-field encoding of simple spikes emerges if the Purkinje cells that project onto a nucleus neuron are not selected at random but share a common complex-spike property.