As the complexity of FPGA-based systems scales, the importance of efficiently handling irregular code increases. Recent work has proposed Irregular Code Energy Reducers (ICERs), a high-level synthesis approach for FPGAs that offers significant energy reduction for irregular code compared to a soft core processor. ICERs target the hot-spots of programs, and are seamlessly connected via a shared L1 cache with a soft processor that executes the cold code. This paper evaluates the application of the selective depipelining (SDP) technique to ICERs, which greatly reduces both the execution time and energy of irregular computations. SDP enables irregular computations to be expressed as large, fast, low-power combinational blocks. SDP maintains high memory bandwidth by scheduling the many potentially dependent memory operations within these blocks onto a high-frequency, highly-multiplexed coherent memory while scheduling combinational operations at a much lower frequency. SDP is a key enabler for improving the execution properties of irregular computations that are difficult to parallelize. We show that applying SDP to ICERs reduces energy-delay by 2.62× relative to ICERs. ICERs with SDP are up to 2.38× faster than a soft core processor and reduce energy consumption by up to 15.83× for a variety of irregular applications.