Post-training quantization (PTQ) is a powerful technique for model compression, reducing the numerical precision in neural networks without additional training overhead. Recent works have investigated adopting 8-bit floating-point formats (\code{FP8}) in the context of PTQ for model inference. However, floating-point formats smaller than 8 bits and their relative comparison in terms of accuracy-hardware cost with integers remains unexplored on FPGAs. In this work, we present minifloats, which are reduced-precision floating-point formats capable of further reducing the memory footprint, latency, and energy cost of a model while approaching full-precision model accuracy. We implement a custom FPGA-based multiply-accumulate operator library and explore the vast design space, comparing minifloat and integer representations across 3 to 8 bits for both weights and activations. We also examine the applicability of various integer-based quantization techniques to minifloats. Our experiments show that minifloats offer a promising alternative for emerging workloads such as vision transformers.