## 3 Proposed Method
### Hardware Improvement
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**i** & **1** & **2** & **3** & **4** & **5** & **6** \\ \hline \(\text{k}\text{q}_{\text{i}}\) & 0.75 & 0.75 & 0.75 & 0.75 & 0.75 & 0.75 \\ \(\text{k}\text{q}_{\text{i}}\)\(\text{dcp}\)\({}^{\text{(a)}}\) & \(-1\),\(-2\) & \(-1\),\(-2\) & \(-1\),\(-2\) & \(-1\),\(-2\) & \(-1\),\(-2\) & \(-1\),\(-2\) \\ \(\text{b}\text{q}_{\text{i}}\times 2^{27}\) & 134,064,676 & 133,945,844 & 134,266,135 & 134,585,698 & 134,907,266 & 135,227,824 \\ \(\text{sp}_{\text{i}}\times 2^{27}\) & 0 & 7,405,568 & 27,982,355 & 33,160,625 & 37,123,176 & 40,433,978 \\ \(\text{cp}_{\text{i}}\times 2^{27}\) & 7,405,567 & 27,982,354 & 33,160,624 & 37,123,175 & 40,433,977 & 43,348,127 \\ \hline
**i** & **7** & **8** & **9** & **10** & **11** & **12** \\ \hline \(\text{k}\text{q}_{\text{i}}\) & 0.75 & 1 & 1 & 1 & 1 & 1.0078125 \\ \(\text{k}\text{q}_{\text{i}}\)\(\text{dcp}\)\({}^{\text{(a)}}\) & \(-1\),\(-2\) & 0 & 0 & 0 & 0 & 0,\(-7\) \\ \(\text{b}\text{q}_{\text{i}}\times 2^{27}\) & 135,550,313 & 124,052,312 & 123,723,825 & 123,395,314 & 123,066,929 & 122,272,854 \\ \(\text{sp}_{\text{i}}\times 2^{27}\) & 43,348,128 & 45,969,761 & 48,832,512 & 52,101,120 & 56,033,280 & 61,382,656 \\  & 45,969,760 & 48,832,511 & 52,101,119 & 56,033,279 & 61,382,655 & 83,500,828 \\ \hline
**i** & **13** & **14** & **15** & **16** & **17** & **18** \\ \hline \(\text{k}\text{q}_{\text{i}}\) & 1.125 & 1.25 & 1.25 & 1.25 & 1.25 & 1.25 & 1.5 \\ \(\text{k}\text{q}_{\text{i}}\)\(\text{dcp}\)\({}^{\text{(a)}}\) & 0,\(-3\) & 0,\(-2\) & 0,\(-2\) & 0,\(-2\) & 0,\(-2\) & 0,\(-1\) \\ \(\text{b}\text{q}_{\text{i}}\times 2^{27}\) & 112,508,522 & 99,488,945 & 99,809,534 & 100,128,867 & 100,448,749 & 67,144,445 \\ \(\text{sp}_{\text{i}}\times 2^{27}\) & 83,500,829 & 104,193,313 & 124,058,498 & 128,089,921 & 131,169,714 & 133,757,944 \\  & 104,193,312 & 124,058,497 & 128,089,920 & 131,169,713 & 133,757,943 & 134,217,727 \\ \hline \hline \end{tabular} \({}^{\text{(a)}}\) Decomposition. For example, \(\text{k}\text{q}_{\text{i}}=0.75=0.11_{2}\), then \(\text{k}\text{q}_{\text{i}\text{i}\text{i}\text{i}\text{dcp}}\) includes \(-1\) and \(-2\).  
\end{table} 
Table 4: The result of OML-PLAC for \(2^{\text{x}}\), \(x\in[0,1)\). represents the critical path, including a 27 bit comparator, a multiplexer (MUX2) and two adders. MUX2 contains 2 3:1 multiplexers and 1 2:1 multiplexer; when implemented by 2:1 multiplexer then we need 5. Using the former method, the delay of MUX2 is the sum delay of a 3-input multiplexer and a 2-input multiplexer while the delay is the sum delay of three 2-input multiplexers with the latter method. It is easy to know that the former method has a smaller delay, and as the number of multiplexer inputs increases the advantages of the tree-cascaded structure will become more obvious. In Figure 5, sp\({}_{\text{i}}\) and bq\({}_{\text{j}}\) are the endpoint and the y-intercept of the i\({}^{\text{th}}\) segment, s\({}_{\text{i}}\) is the comparison result between input data and segment starting point. The SAA operations can be represented as the following piecewise function:  
\[ \begin{cases}2^{-1}\times\text{x}_{13}+2^{-2}\times\text{x}_{12}&\text{i}\in[ 1,7]\\ \text{x}_{14}&\text{i}\in[8,11]\\ \text{x}_{14}+2^{-7}\times\text{x}_{7}&\text{i}=12\\ \text{x}_{14}+2^{-3}\times\text{x}_{11}&\text{i}=13\\ \text{x}_{14}+2^{-2}\times\text{x}_{12}&\text{i}\in[14,17]\\ \text{x}_{14}+2^{-1}\times\text{x}_{13}&\text{i}=18\end{cases}  \tag{17}\]  
where I and x\({}_{\text{m}}\) represent the number of segments and the highest m significant bits of input x. We divide the SAA operations into six segments based on the value of slope according to Table 4. Multiplying \(2^{-t}\) by x\({}_{\text{m}}\) is the same thing as shifting x\({}_{\text{m}}\) to the right by t bits, and the sum of m and t is the fractional bit widths of the intermediate data. Next, we give an example to illustrate the case where the piecewise method in this paper reduces the SAA operations and their associated hardware resources.