The widespread application of artificial neural networks has prompted researchers to experiment with field-programmable gate array and customized ASIC designs to speed up their computation. These implementation efforts have generally focused on weight multiplication and signal summation operations, and less on activation functions used in these applications. Yet, efficient hardware implementations of nonlinear activation functions like exponential linear units (ELU), scaled ELU (SELU), and hyperbolic tangent (tanh), are central to designing effective neural network accelerators, since these functions require lots of resources. In this paper, we explore efficient hardware implementations of activation functions using purely combinational circuits, with a focus on two widely used nonlinear activation functions, i.e., SELU and tanh. Our experiments demonstrate that neural networks are generally insensitive to the precision of the activation function. The results also prove that the proposed combinational circuit-based approach is very efficient in terms of speed and area, with negligible accuracy loss on the MNIST, CIFAR-10, and IMAGE NET benchmarks. Synopsys design compiler synthesis results show that circuit designs for tanh and SELU can save between ${\times 3.13\sim \times 7.69}$ and ${ {\times 4.45\sim \times 8.45}}$ area compared to the look-up table/memory-based implementations, and can operate at 5.14 GHz and 4.52 GHz using the 28-nm SVT library, respectively. The implementation is available at: https://github.com/ThomasMrY/ActivationFunctionDemo.