Bit error rate (BER) is an important figure of merit to evaluate the performance of a communication system. Analyzing the BER of a linear-time-invariant system has been extensively studied. However, analyzing the BER of nonlinear circuits and systems is challenging because it cannot rely on linear time invariant (LTI) principles, while an exhaustive nonlinear simulation is computationally prohibitive. To find BER, an exhaustive approach requires nonlinear simulations of $2^m$ bit patterns for a channel of m-bit memory, where m can be larger than 100 in today's high-speed and high-performance design. In this work, we develop a fast and accurate method to analyze the BER of large-scale nonlinear circuits. Only O(k) nonlinear simulations are required, with k much less than 2^m and independent of $2^m$ . While performing O(k) nonlinear simulations only, we find a method to determine the probability density function of the nonlinear responses, from which accurate BER results can be obtained. An error assessment method is also developed to evaluate the true error of the nonlinear signaling analysis without the need for knowing the entire nonlinear responses of the channel. Simulations of large-scale real-world nonlinear circuits have demonstrated the accuracy, efficiency, and capacity of the proposed method. A BER as low as $10^{-56}$ is accurately predicted.