Pulsation of bubbles at their natural frequency leaves their signature on various natural and man-made processes such as cavitation, sound of falling water drops, diagnostic and therapeutic ultrasound with bubbles as contrast agents, and gas seepage from underwater natural gas sources. For small-amplitude adiabatic pulsations, the well-known formula for "Minnaert frequency" asserts that the product of frequency and the equilibrium bubble radius is independent of the oscillation amplitude, and is in fact a constant for a given identity of the bubble gas. In practice, the oscillation amplitude cannot be ruled out to be small, being dependent on impulsive events or additional forces that exist before the initiation of bubble pulsation. The nonlinear nature of such large amplitude oscillations renders the 'Minnaert formula' inapplicable. Here, we propose simple closed-form corrections to the Minnaert formula, through systematic singular perturbation analysis of the effect of finite but small oscillation amplitudes. Through symbolic computations, an extended version of the classical Poincare-Lindstedt method is applied to the Rayleigh Plesset equation. Finite amplitude oscillations shift the bubble natural frequency downwards. The predicted time-resolved bubble growth curves and the amplitude-dependence of the nonlinear oscillation frequency show good agreement with the predictions from the direct numerical solution of the Rayleigh Plesset equation.