Applications with large amounts of data, real-time constraints, ultra-low power requirements, and heavy computational complexity present significant challenges for modern computing systems, and often fall within the category of high performance computing (HPC). As such, computer architects have looked to high performance single instruction multiple data (SIMD) architectures, such as accelerator-rich platforms, for handling these workloads. However, since the results of these applications do not always require exact precision, approximate computing may also be leveraged. In this work, we introduce BRAINIAC, a heterogeneous platform that combines precise accelerators with neural-network-based approximate accelerators. These reconfigurable accelerators are leveraged in a multi-stage flow that begins with simple approximations and resorts to more complex ones as needed. We employ high-level, applicationspecific light-weight checks (LWCs) to throttle this multi-stage acceleration flow and reliably ensure user-specified accuracy at runtime. Evaluation of the performance and energy of our heterogeneous platform for error tolerance thresholds of 5%-25% demonstrates an average of 3× gain over computation that only includes precise acceleration, and 15×-35× gain over softwarebased computation.