To address the time consuming nature of Monte Carlo simulations, the obtained MI values are approximated through curve fitting processes The formula used for the approximation is presented in [2], and is defined for a constellation diagram with $N$ symbols as follows: $$ I(\text{SNR}) = \log_2(N) \cdot \left(1 - \sum_{i=1}^{M} a_i \cdot \exp(-b_i \cdot \text{SNR})\right), \quad i \in \{1, 2, \ldots, M\} $$

Based on the constellation order, employing an higher number $M$ of fitting coefficients $a$_$i$ and $b$_$i$ can lead to an increased accuracy of the approximation. This is expecially true for higher order constellations.

The value of the root mean square error (RMSE) can be used to measure the accuracy of the approximation between the simulated MI value, denoted as $I$_s, and the approximated MI value, denoted as $I$_a, respectively. For $R$ values of MI, it is defined as: $$ \text{RMSE} = \sqrt{\frac{1}{R} \sum_{i=1}^{R} \left(I_s(\text{SNR}) - I_a(\text{SNR})\right)^2} $$

By transmitting symbols with $M$_b bits per symbol, the RMSE only improves with an increasing number of coefficients up to
$M$ = $M$_b - 3. The addition of extra fitting coefficients results in minimal to no further improvement in the RMSE.

[2] C. Ouyang, S. Wu, and H. Yang, Mutual information approximation, IEEE Transactions on Information Theory, pp. 2-3, 2019. DOI: 10.48550/arXiv.1908.09622