## 5 Results and Discussion  
The mathematical model performances are shown in Figures 12-14 and Table 4 for narrow, intermediate, and deep penetration, respectively. Table 4 shows the R\({}^{2}\), MSE, MAE of proposed model in comparison with NDRC, and Li and Chen model. Figure 12a shows the comparison of the model-predicted values with the actual values of narrow penetration. The coefficient of determination, R\({}^{2}\), is 0.59. Figure 12b illustrates the residual plot of predicted values. The residuals show a constant variance and are evenly spread out. The LOWESS (locally weighted scatterplot smoothing) fit is close to zero but it shows some divergence, as the variance is high at a high predicted value, which is, possibly, an outlier. The frequency distribution of residuals is shown in Figure 12c. Figure 12: Symbolic regression model for narrow penetration. (**a**) Comparison between actual and predicted values for all data, (**b**) residual plot of predicted values, and (**c**) frequency distribution of residuals. Figure 13: Symbolic regression model for intermediate penetration. (**a**) Comparison between actual and predicted values for train and test datasets, (**b**) residual plot of predicted values, and (**c**) frequency distribution of residuals. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Metric**} & \multicolumn{4}{c}{**Comparison of Performance between Different Equations**} \\ \cline{2-5}  & **NDRC** & **Barr** & **Li and Chen** &
\begin{tabular}{c} **Sym. Reg.a** \\ **(Present Study)** \\ \end{tabular} \\ \hline \multicolumn{5}{c}{Narrow penetration (_X/d_ \(<\) 0.5)} \\ \hline R\({}^{2}\) & \(-3.262\) & 0.145 & \(-0.031\) & 0.590 \\ MSE & 0.085 & 0.017 & 0.021 & 0.008 \\ MAE & 0.270 & 0.097 & 0.118 & 0.068 \\ \hline \multicolumn{5}{c}{Intermediate penetration (0.5 \(\leq\) _X/d_ \(<\) 5)} \\ \hline R\({}^{2}\) & 0.746 & 0.650 & 0.746 & 0.884 \\ MSE & 0.103 & 0.142 & 0.231 & 0.106 \\ MAE & 0.222 & 0.261 & 0.330 & 0.216 \\ \hline \multicolumn{5}{c}{Deep penetration (_X/d_ \(\geq\) 5)} \\ \hline R\({}^{2}\) & 0.565 & - & 0.963 & 0.967 \\ MSE & 199.369 & - & 16.877 & 15.087 \\ MAE & 11.295 & - & 2.799 & 2.470 \\ \hline \hline \end{tabular}  
* Equation (1) for narrow penetration, Equation (2) for intermediate penetration, and Equation (3) for deep penetration.  
\end{table} 
Table 4: Statistics of previous studies compared to the proposed study. For the intermediate penetration data, Figure 13a compares model-predicted values with actual values. The prediction of the train and test datasets show that the model performs well on unseen test data. The coefficient of determination, R\({}^{2}\), for all data is 0.88. The residuals of predicted values are evenly distributed, as seen in the Figure 13b,c, which shows constant variance. The LOWESS fit is near zero and does not display diversion at low or high predicted values, as seen in Figure 13b. For deep penetration data, Figure 14a compares model-predicted values with actual values. The model performs well on unseen test datasets. The coefficient of determination, R\({}^{2}\), for all data is 0.97. The residual of predicted values is evenly distributed at low or high predicted values, exhibiting a constant variance (Figure 14b,c). The LOWESS fit is close to zero, showing an absence of divergence at low or high predicted values. Furthermore, Figure 15 shows the prediction of the proposed equation for narrow, medium, and deep penetrations compared to the Barr, NDRC, Li and Chen, and L equations. Figure 15a shows that the NDRC and Li and Chen models are almost not applicable in the narrow penetration depths. However, the present model can be useful, compared to other models with less accuracy, due to the complexity of penetration phenomena. Figure 15b,c shows that the proposed model for the prediction of _X/d_ is more accurate, as compared to the NDRC and Li and Chen models within the range of 0.5 \(\leq\)_X/d_ \(<\) 5.0 and _X/d_ \(\geq\) 5.0. Figure 14: Symbolic regression model for deep penetration.