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 ∫ a b | f ″ ( x ) | d x . 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 ) ∼ c ε , ( ε → 0 + ), where c = ( b − a ) | f ″ | max 4 .