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With the proliferation of embedded systems requiring intelligent behavior, custom number systems to optimize performance per Watt of the entire system become essential components for successful commercial products. We present the Universal Number Library, a high-performance number systems library that includes arbitrary integer, decimal, fixed-point, floating-point, and introduces two tapered floating-point types, posit and valid, that support reproducible arithmetic computation in arbitrary concurrency environments. We discuss the design of the Universal library as a run-time for application development, and as a platform for application-driven hardware validation. The library implementation is described, and examples are provided to show educational examples to elucidate the number system properties, and how specialization is used to yield very high-performance emulation on existing x86, ARM, and POWER processors. We will highlight the integration of the library in larger application environments in computational science and engineering to enable multi-precision and adaptive precision algorithms to improve performance and efficiency of large scale and real-time applications. We will demonstrate the integration of the Universal library into a high-performance reproducible linear algebra run-time. We will conclude with the roadmap of additional functionality of the library as we are targeting new application domains, such as Software Defined Radio, instrumentation, sensor fusion, and model-predictive control.