1. A method of processing a network input through a neural network having one or more initial neural network layers followed by a softmax output layer to generate a neural network output for the network input, the method comprising: obtaining a layer output generated by processing the network input through the one or more initial neural network layers, wherein the initial neural network layers are implemented by a processing system that performs computations specified by the one or more initial neural network layers using quantized arithmetic such that output values generated by the processing system can take only values from a predetermined finite set of values, and wherein: the layer output has a plurality of layer output values, and each layer output value is a respective one of a predetermined finite set of possible output values; and processing the layer output through the softmax output layer to generate the neural network output for the network input, comprising: determining, for each possible output value in the predetermined finite set of possible output values, a count of occurrences of the possible output value among the plurality of layer output values that are in the layer output; for each possible output value that occurs at least once in the plurality of layer output values, determining a respective exponentiation measure of the possible output value; determining a normalization factor for the layer output by combining the exponentiation measures in accordance with the counts of occurrences of the possible output values; and determining, for each of the plurality of layer output values, a softmax probability value from the respective exponentiation measure for the layer output value and the normalization factor for the layer output.