As machine learning gets deployed more and more widely, and model sizes continue to grow, improving computational efficiency during model inference has become a key challenge. In many commonly used model architectures, including Transformers, a significant portion of the inference computation is comprised of exponential non-linearities such as Softmax. In this work, we develop QuAKE, a collection of novel operators that leverage certain properties of IEEE-754 floating point representations to quickly approximate the exponential function without requiring specialized hardware, extra memory, or precomputation. We propose optimizations that enhance the efficiency of QuAKE in commonly used exponential non-linearities such as Softmax, GELU, and the Logistic function. Our benchmarks demonstrate substantial inference speed improvements between 10% and 35% on server CPUs, and 5% and 45% on embedded and mobile-scale CPUs for a variety of model architectures and sizes. Evaluations of model performance on standard datasets and tasks from various domains show that QuAKE operators are able to provide sizable speed benefits with little to no loss of performance on downstream tasks.