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The standard affine mapping takes a real value x , a scale s and a zero-point z , and produces an integer $x_q = \mathrm{clip}\left(\left\lfloor \frac{x}{s} \right\rceil + z,\ q_{\min},\ q_{\max}\right), \qquad \hat{x} = s\left(x_q - z\right)$. where $\lfloor$ $\cdot$ $\rceil$ is round-to-nearest and $\hat{x}$ is the dequantised approximation actually used in arithmetic. Jacob and colleagues’ integer-arithmetic scheme, still the reference formulation for this mapping, showed that with weights and activations quantized this way, “inference can be carried out using integer-only arithmetic” end to end on ordinary integer hardware, with no floating-point unit required at all, which is what makes the technique valuable on the cheapest edge silicon rather than merely on the largest [ 7 ] .

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