## 3 Proposed Algorithm for Fractional-Bit Compensation  
This section discusses the proposed eADP-efficient antilogarithmic converters using the fractional-bit compensation schemes. To design an efficient antilogarithmic converter, we first compare [PERSON]'s [3] straight-line \(out=1+f\) with the accurate antilogarithmic curve \(out=2^{\prime}\) and analyze the percentage conversion error, as shown in Figures 3 and 4, respectively. The percentage conversion error is defined as 100% \(\times\) (conversion error divided by \(2^{\prime}\)):  
\[\text{Percentage Conversion Error }=\frac{\text{Conversion Error}}{2^{ \prime}} \tag{8}\] \[=\frac{((1+f)-2^{\prime})}{2^{\prime}}\times 100\%\,\ 0\leq f<1\]  
The maximum percentage error is defined as 100% \(\times\) (sum of absolute value of maximum positive percentage error and minimum negative percentage error):  
\[=(\frac{\text{Maximum Percentage Error}}{2^{\prime}})\times 100\%\,\ 0\leq f<1 \tag{9}\]  
For the fractional-bit compensation method, we found that the first region and the last region could not be compensated by Figure 3. The compensated region is divided by the \(2^{n}\) region, where \(n\) denotes the first most significant bits (MSB). For example, \(n=4\) is divided by 16 compensated regions. That is, \(f_{-1}\), \(f_{-2}\), \(f_{-3}\), and \(f_{-4}\) bits in the fraction part are used to partition 16 regions. Table 2 shows the maximum conversion error and percentage error for the uncompensated first region and last region, where \(n\) is set to 3 to 8. The partition region is considered to be a uniform partition. Table 2 presents the local errors and local percentage errors for the first and last regions. The larger values of the first region and the last region are considered to be the absolute maximum errors and absolute maximum percentage errors. Table 2 presents the important indexes for deciding the partition and compensation values. Taking \(n=4\) for example, the partition number is 16 regions. The upper bound of the maximum percentage conversion error is 1.7327%. That is, a designed maximum percent  
Figure 3: Comparison of [PERSON]’s [3] straight-line \(out=1+f\) and the accurate antilogarithmic curve \(out=2^{\prime}\). age conversion error below 1.7327% cannot be achieved for 16 uniform regions, even if a full-precision fractional-bit compensation value is used. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{\(f_{-1}\)-\(f_{-n}\)} & **Uniform** & \multicolumn{2}{c}{**Key**} & \multicolumn{2}{c}{**Local**} & \multicolumn{2}{c}{**Absolute**} & \multicolumn{2}{c}{**Maximum**} \\  & **Partition** & **Uncompensated** & **Error** & **Error** & **Error** & **Error** & **Percentage** \\ \hline \multirow{2}{*}{\(n=3\)} & \multirow{2}{*}{8} & First region & \multirow{2}{*}{0.0342} & \multirow{2}{*}{3.1427\%} & \multirow{2}{*}{0.0410} & \multirow{2}{*}{3.1427\%} \\  & & Last region & & & & \\ \hline \multirow{2}{*}{\(n=4\)} & \multirow{2}{*}{16} & First region & \multirow{2}{*}{0.0181} & \multirow{2}{*}{1.7327\%} & \multirow{2}{*}{0.0221} & \multirow{2}{*}{1.7327\%} \\  & & Last region & & & & \\ \hline \multirow{2}{*}{\(n=5\)} & \multirow{2}{*}{32} & First region & \multirow{2}{*}{0.0093} & \multirow{2}{*}{0.9083\%} & \multirow{2}{*}{0.0115} & \multirow{2}{*}{0.9083\%} \\  & & Last region & & & & \\ \hline \multirow{2}{*}{\(n=6\)} & \multirow{2}{*}{64} & First region & \multirow{2}{*}{0.0045} & \multirow{2}{*}{0.4502\%} & \multirow{2}{*}{0.0057} & \multirow{2}{*}{0.4502\%} \\  & & Last region & & & & \\ \hline \multirow{2}{*}{\(n=7\)} & \multirow{2}{*}{128} & First region & \multirow{2}{*}{0.0021} & \multirow{2}{*}{0.2126\%} & \multirow{2}{*}{0.0027} & \multirow{2}{*}{0.2126\%} \\  & & Last region & & & & \\ \hline \multirow{2}{*}{\(n=8\)} & \multirow{2}{*}{256} & First region & \multirow{2}{*}{0.0009} & \multirow{2}{*}{0.0916\%} & \multirow{2}{*}{0.0012} & \multirow{2}{*}{0.0578\%} \\  & & Last region & & & & \\ \hline \hline \end{tabular}
\end{table}