Neural networks are increasingly integrated into scientific discovery, where input data reduction and model quantization play a key role in accelerating inference. However, understanding and mitigating the impact of these techniques on output error is critical for ensuring reliable results, particularly in tasks demanding high numerical precision. This paper introduces a comprehensive framework for optimizing neural network inference in scientific computing by combining data reduction and model quantization while maintaining error-controlled outcomes. We develop theoretical analyses to bound error propagation under these techniques and propose a framework that balances computational performance with error constraints. Evaluation on real-world learning-based combustion simulations and satellite image classification shows that our derived error bounds accurately predict observed errors while enabling significant computational speedup under our framework. This work highlights the potential for further leveraging advancements in modern lossy compression algorithms and hardware accelerators that support lower-precision formats.