## 3 Results
These metrics play a pivotal role in evaluating the accuracy and performance of the models, providing insights into the degree of variation between predicted and observed values. Their examination enables a comprehensive understanding of the efficacy of the predictive equations. \begin{table}
\begin{tabular}{c c c c} \hline \hline  & **1 st Degree Eq** & **2 nd Degree Eq** & **Rule of Thumb** \\ \hline R\({}^{2}\) & 0.98 & 1 & 0.95 \\ RMSE & 1.82 & 0.71 & 2.67 \\ MSE & 3.33 & 0.51 & 7.13 \\ MAE & 1.44 & 0.53 & 1.45 \\ STD & 1.82 & 0.62 & 2.39 \\ Max Error * & 4.88 & 2.92 & 12.95 \\ Mean Error * & 1.59 & 0.72 & 1.805 \\ \hline \hline \end{tabular}  
* After 10,000 random calculations.  
\end{table} 
Table 2: Error indices. Figure 6: Absolute error rate of the Thumb Eq. vs. Observed data. (**a**) 3D representation. (**b**) Planar representation. The statistical parameters provided for the three equations reveal some interesting insights into their performance. The second-degree equation stands out distinctly, achieving a perfect R-squared value of 1.00, which suggests it perfectly captures the variance in the dependent variable from the predictors. It also records the lowest errors across RMSE, MSE, and MAE, indicating its high accuracy and reliability in prediction. On the other hand, while the first-degree and Thumb equations also exhibit relatively high R-squared values, they fall short on error metrics, with the Thumb equation notably showing the highest errors and thus the least precision. This divergence in performance highlights the superior capability of the second-degree equation in providing reliable and accurate predictions under the conditions tested, while also underscoring the limitations of the more traditionally used Thumb equation in terms of accuracy and consistency. In addition to the accuracy evaluation, it is necessary to study the computational cost of the four equations in terms of time and memory, and these equations are tested with various input sizes (10, 100, 500, 1000, 5000, 10,000, 12,000, 14,000, and 19,000) using random values within the range of 0 to 100. The test is conducted on the Google Colab cloud platform, utilizing a system with 12.67 GB of RAM and two cores of an AMD EPYC 7B12 processor running at 2.25 GHz (AMD, Santa Clara, CA, USA). Execution time is recorded, and memory usage is tracked to measure peak memory consumption. For each input size, random values of temperature and relative humidity are generated, and a mesh grid is created. The execution time and memory usage for each equation are then measured and recorded. Table 3 and Figure 7 capture this test and show the results. In Figure 7 it is possible to observe that for the Magnus-Tetens' equation, for inputs larger than 13,500 iterations, the graphic turns horizontal; this is due to the impossibility to run this simulation because of RAM restrictions. Additionally, equations were tested with the same methodology, replacing the use of RAM for writing intermediate results to a file, as represented in Figure 8. Firstly, data and results are stored one by one (Figure 8a,b), and secondly, in blocks of 1000 entries (Figure 8c,d). The execution time and memory usage for each equation were measured and recorded, ensuring efficient memory management. Figure 8 illustrates the execution time, in the first case, Figure 8a, and in the second case, Figure 8c. Additionally, the memory usage across iterations is shown for the first case in Figure 8b and for the second case in Figure 8d, highlighting the performance differences between the equations. Figure 8a,c displays the averaging of each 100 results to enhance clarity and avoid noise. In the case of RAM usage limitations, the results show a much slower individual behavior. Despite individual and sporadic peaks, they do not represent a significative variance (around 1% over the standard results); likewise in the case of RAM, where just sporadic peaks appear.