Due to its ability to significantly enhance communication reliability, the fluid antenna system (FAS) has emerged as a promising technology for ultra-reliable communications (URC). However, existing expressions to evaluate the outage probability of FAS are either computationally complex or simplified at the cost of accuracy. To address this issue, we propose an accuracy-controllable approximation method under Rayleigh fading channels. Specifically, we first approximate the fully correlated Rayleigh fading model as a block-correlation Rayleigh model, where the channel gains of ports in different blocks are assumed to be mutually independent, while the channel gains of ports in the same block are considered equally correlated, to simplify the analysis while maintaining accuracy. Then, we propose a novel method to approximate the Bessel function I₀, introduced by the correlated Rayleigh fading, using an M -term series expansion, where the parameter M provides a tunable trade-off between approximation accuracy and computational complexity. Based on this approximation, a closed-form expression for the outage probability of FAS is derived. Simulation results demonstrate that our derived expression offers low computational complexity for evaluating the FAS outage probability while maintaining excellent approximation accuracy in the medium-to-high signal-to-noise ratio (SNR) range.