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Recently proposed circuit-level models of carbon nanotube transistor (CNT) for SPICE-like simulators suffer from numerical complexities as they rely on numerical evaluation of integrals or internal Newton-Raphson iterations to find solutions of non-linear dependencies or both. Recently an approach has been proposed which eliminates the need for numerical integration when calculating the charge densities in CNT through the use of piece-wise linear approximation. This paper extends the effective employment of numerical piece-wise approximation to the solution of the Fermi-Dirac integral to accelerate the CNT model speed when evaluating the source-drain current while maintaining high modeling accuracy. Our results show a speed up of more than three orders of magnitude compared with the theoretical CNT model implemented in FETToy, used as a reference for verifying newer models. Comparisons of drain-source current characteristics of the new model against that in FETToy are presented, confirming that the accuracy of the proposed approach is maintained within less than 5% in terms of RMS error.