On the number of segments needed in a piecewise linear approximation

The introduction of high-speed circuits to realize an arithmetic function ff as a piecewise linear approximation has created a need to understand how the number of segments depends on the interval a≤x≤ba≤x≤b and the desired approximation error εε. For the case of optimum non-uniform segments, we show that the number of segments is given as s(ε)∼cε, (ε→0+ε→0+), where c=14∫ab|f″(x)|dx. Experimental data shows that this approximation is close to the exact number of segments for a set of 14 benchmark functions. We also show that, if the segments have the same width (to reduce circuit complexity), then the number of segments is given by s(ε)∼cε, (ε→0+ε→0+), where c=(b−a)|f″|max4.