Secure hardware design is a challenging task that goes far beyond ensuring functional correctness. Important design properties such as non-interference cannot be verified on functional circuit models due to the lack of essential information (e.g., sensitivity level) for reasoning about security. Hardware information flow tracking (IFT) techniques associate data objects in the hardware design with sensitivity labels for modeling security-related behaviors. They allow the designer to test and verify security properties related to confidentiality, integrity, and logical side channels. However, precisely accounting for each bit of information flow at the hardware level can be expensive. In this work, we focus on the precision of the IFT logic. The key idea is to selectively introduce only one sided errors ( false positives); these provide a conservative and safe information flow response while reducing the complexity of the security logic. We investigate the effect of logic synthesis on the quality and complexity of hardware IFT and reveal how different logic synthesis optimizations affect the amount of false positives and design overheads of IFT logic. We propose novel techniques to further simplify the IFT logic while adding no, or only a minimum number of, false positives. Additionally, we provide a solution to quantitatively introduce false positives in order to accelerate information flow security verification. Experimental results using IWLS benchmarks show that our method can reduce complexity of GLIFT by 14.47% while adding 0.20% of false positives on average. By quantitatively introducing false positives, we can achieve up to a 55.72% speedup in verification time.