Large language models (LLMs) are impressive feats of engineering, capable of generating human-like text that's both creative and informative. However, these powerful tools can sometimes go off the rails, producing unwanted outputs like buggy code, fabricated personal information, or even offensive language. Researchers are constantly working on ways to prevent these issues, but current methods often involve a trade-off: either they require massive computational resources or they skew the output in undesirable ways. A new technique called "Approximately Aligned Decoding" (AprAD) offers a clever compromise. Imagine trying to write a poem without using the letter 'E'. Traditional methods might simply forbid the LLM from ever generating 'E', which often leads to awkward phrasing and unnatural language. AprAD, on the other hand, takes a more nuanced approach. It allows the LLM to explore different possibilities, but gently nudges it away from generating unwanted outputs. This method draws inspiration from a technique called "speculative decoding", which uses a smaller, faster model to preview possible outputs and then checks them against the desired constraints. AprAD cleverly adapts this idea to create a system that's both efficient and effective. The key innovation lies in how it handles errors. Instead of starting from scratch every time an error is detected, AprAD cleverly reuses part of the generated text, reducing wasted computation. Experiments show that AprAD achieves a sweet spot between quality and efficiency. In tests involving generating “lipograms” (text avoiding specific letters) and preventing code hallucinations, AprAD produced high-quality outputs while requiring significantly less computation than other methods. While not perfect, AprAD represents a significant step towards generating high-quality, error-free text from LLMs. Future research will likely explore more refined variations of this method, potentially leading to even more efficient and controllable AI text generation.