In previous work, we presented a tight approximate response-time analysis for tasks with offsets. While providing a tight bound on response times, the tight analysis exhibits similarly long execution times as does the traditional methods for calculating response-times for tasks with offsets. The existing method for fast analysis of tasks with offsets is not applicable to the tight analysis. In this paper we extend the fast analysis to handle the distinguishing trait of the tight analysis; continuously increasing interference functions. Furthermore, we provide another speedup; by introducing pessimism in the modelling of interference at certain points, we speed up the convergence of the numerical solving for response-times without increasing the pessimism of the resulting responsetimes. The presented fast-and-tight analysis is guaranteed to calculate the same response-times as the tight analysis, and in a simulation study we obtain speedups of more than two orders of magnitude for realistically sized tasks sets compared to the tight analysis. We also demonstrate that the fast-and-tight analysis has comparable execution time to that of the fast analysis. Hence, we conclude that the fast-and-tight analysis is the preferred analysis technique when tight estimates of response-times are needed, and that we do not need to sacrifice tightness for analysis speed; both are obtained with the fast-and-tight analysis.