In this work, we provide energy-efficient architectural support for floating point accuracy. For each floating point addition performed, we "recycle" that operation's rounding error. We make this error architecturally visible such that it can be used, whenever desired, by software. We also design a compiler pass that allows software to automatically use this feature. Experimental results on physical hardware show that software that exploits architecturally recycled error bits can (a) achieve accuracy comparable to a 64-bit FPU with performance and energy that are comparable to a 32-bit FPU, and (b) achieve accuracy comparable to an all-software scheme for 128-bit accuracy with far better performance and energy usage.