Execution traces, such as are used to study and analyze program behavior, are often so large that they need to be stored in compressed form. This paper describes the design and implementation of four value prediction-based compression (VPC) algorithms for traces that record the PC as well as other information about executed instructions. VPC1 directly compresses traces using value predictors, VPC2 adds a second compression stage, and VPC3 utilizes value predictors to convert traces into streams that can be compressed better and more quickly than the original traces. VPC4 introduces further algorithmic enhancements and is automatically synthesized. Of the 55 SPECcpu2000 traces we evaluate, VPC4 compresses 36 better, decompresses 26 faster, and compresses 53 faster than BZIP2, MACHE, PDATS II, SBC, and SEQUITUR. It delivers the highest geometric-mean compression rate, decompression speed, and compression speed because of the predictors' simplicity and their ability to exploit local value locality. Most other compression algorithms can only exploit global value locality.