###### Abstract
This method contributes to the delay of the critical path. In addition to the improvements mentioned above, we point out the limitations of the method for finding the MAE in advanced segmentation methods. The minimizations of MAE are found in [25,26] by parallel shifting, which is not suitable for all nonlinear unary functions. Take the sine function as an example, as shown in Figure 2.  
Figure 1: SAA parts of ML-PLAC (**a**) and OML-PLAC (**b**). Assume the segment is \(\sin(x)\), where \(x\)\(\in\)[0, 2\(\pi\)]. If the shifting method of [25] or [26] is used, then the final approximate line is the x-axis, and the MAE is 1. However, it is easy to prove mathematically that Line L is the optimal approximate line. L1 and L2 are tangent lines to \(\sin(x)\), and the tangent points are P1 and P2, respectively. L1 is parallel to L2, and L is the midline of L1 and L2. Line L passes through A. The MAE in this case is the distance between point P1 or P2 and line L, and it is obvious that the value is less than 1. In fact, the above problem may occur when the line on which the two endpoints lie intersects the function itself. This problem does not arise when the first derivative and the second derivative of the function are both greater than zero. This can be proved by the Cauchy mean value theorem and proof by contradiction. This issue is not the focus of this paper and will not be further explored. We just offer one solution here to this problem, traverse the line composed of any two actual points between two endpoints, take them as the target line for parallel shifting, and finally find the line with the minimum MAE. The cost of this approach is an increase in the number of iterations. We use MATLAB to model the proposed improved segmentation and MAE adaptation method. Then Verilog HDL is used to model the hardware architecture. As for showing the superiority of OML-PLAC over other methods, we synthesize with TSMC 90-nm, TSMC 65-nm and TSMC 40-nm technologies. The synthesized results prove that OML-PLAC has increased performance over ML-PLAC. The contributions of this paper are summarized as follows:  
1. By controlling the slope property of the piecewise approximation function, the multi-plexers in the SAA part can be fully used. As a result, fewer segments are required under the same limitation of MAE. Shift-and-add operations can be reduced in some cases as well, and these contribute to less area and delay;
2. The proposed MAE adaptation method can reduce the actual MAE as much as possible on the premise of ensuring the predefined MAE and the maximum number of segments, and the percentage of the final MAE to the limit MAE is controllable;
3. The tree-cascaded method is used to implement multi-input multiplexers with the nodes of 2-input and 3-input multiplexers in hardware architecture, trying to flatten the delay from input to output. This method is beneficial to reduce the length of critical path;
4. The limitations of the method for finding the MAE in advanced segmentation methods has been presented and a corresponding solution is given. The rest of this paper is organized as follows: Section 2 introduces state-of-the-art research on the PWL so far. Section 3 introduces the proposed optimization method in detail, including the optimized segmentation method, the error adaptation and the hardware improvement. In Section 4, the proposed optimization method is applied to typical nonlinear functions, and the experimental results are compared and analyzed with those of the existing state-of-the-art methods and some other studies. Section 5 provides the conclusions of this paper. Figure 2: The graph of the sine function.