Serverless computing separates function execution from state management. Simple retry-based fault tolerance might corrupt the shared state with duplicate updates. Existing solutions employ log-based fault tolerance to achieve exactly-once semantics, where every single read or write to the external state is associated with a log for deterministic replay. However, logging is not a free lunch, which introduces considerable overhead to stateful serverless applications. We present Halfmoon, a serverless runtime system for fault-tolerant stateful serverless computing. Our key insight is that it is unnecessary to symmetrically log both reads and writes. Instead, it suffices to log either reads or writes, i.e., asymmetrically. We design two logging protocols that enforce exactly-once semantics while providing log-free reads and writes, which are suitable for read- and write-intensive workloads, respectively. We theoretically prove that the two protocols are log-optimal , i.e., no other protocols can achieve lower logging overhead than our protocols. We provide a criterion for choosing the right protocol for a given workload, and a pauseless switching mechanism to switch protocols for dynamic workloads. We implement a prototype of Halfmoon. Experiments show that Halfmoon achieves 20%–40% lower latency and 1.5–4.0× lower logging overhead than the state-of-the-art solution Boki.