In software and hardware testing, generating multiple inputs which satisfy a given set of constraints is an important problem with applications in fuzz testing and stimulus generation. However, it is a challenge to perform the sampling efficiently, while generating a diverse set of inputs which satisfy the constraints. We developed a new algorithm QuickSampler which requires a small number of solver calls to produce millions of samples which satisfy the constraints with high probability. We evaluate QuickSampler on large real-world benchmarks and show that it can produce unique valid solutions orders of magnitude faster than other state-of-the-art sampling tools, with a distribution which is reasonably close to uniform in practice.