On the number of segments needed in a piecewise linear approximation

The introduction of high-speed circuits to realize an arithmetic function f as a piecewise linear approximation has created a need to understand how the number of segments depends on the interval a ≤ x < b and the desired approximation error ε. For the case of optimum non-uniform segments, we show that the number of segments is given as s(ε) ∼ c √ ε , (ε → 0), where c = 1 4 ∫ b a √ |f ′′(x)|dx. 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) ∼ c √ ε ,(ε → 0), where c = (b−a) √ |f ′′|max 4 .