In the International Commission on Illumination (CIE) color appearance model CIECAM02, a modified hyperbolic function is used to represent luminance adaptation. The same nonlinear function is also used in the new color appearance model CAM16 [Color Res Appl., 2017;42:703-718]. Although the modified hyperbolic function seems reasonable based on physiological evidence, it has an infinite slope at the origin, which causes instability for both the forward and inverse modes of the CIECAM02/CAM16 models. In this article, various possible extensions to the nonlinear luminance adaptation function in CIECAM02/CAM16 are reviewed and evaluated. Based on these investigations, the Gill extension to the hyperbolic function that is used to represent luminance adaptation [Proceedings of 16th Color and Imaging Conference, pp. 327-331, 2008], is recommended at both the lower end () and the upper end (), whereresponse. In addition, the new recommended function can be readily inverted for use in the appropriate inverse appearance model. From an extensive analysis using available experimental data sets, we also propose that, for the lower and upper limits of the luminance range in the extended model, the values (L)=0.26 and q(U)=150 be used, respectively.