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  1. Posit is a recently proposed alternative to the floating point representation (FP). It provides tapered accuracy. Given a fixed number of bits, the posit representation can provide better precision for some numbers compared to FP, which has generated significant interest in numerous domains. Being a representation with tapered accuracy, it can introduce high rounding errors for numbers outside the above golden zone. Programmers currently lack tools to detect and debug errors while programming with posits. This paper presents PositDebug, a compile-time instrumentation that performs shadow execution with high pre- cision values to detect various errors in computation using posits. To assist the programmer in debugging the reported error, PositDebug also provides directed acyclic graphs of instructions, which are likely responsible for the error. A contribution of this paper is the design of the metadata per memory location for shadow execution that enables productive debugging of errors with long-running programs. We have used PositDebug to detect and debug errors in various numerical applications written using posits. To demonstrate that these ideas are applicable even for FP programs, we have built a shadow execution framework for FP programs that is an order of magnitude faster than Herbgrind. 
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  2. Posit is a recently proposed representation for approximating real numbers using a finite number of bits. In contrast to the floating point (FP) representation, posit provides variable precision with a fixed number of total bits (i.e., tapered accuracy). Posit can represent a set of numbers with higher precision than FP and has garnered significant interest in various domains. The posit ecosystem currently does not have a native general-purpose math library. This paper presents our results in developing a math library for posits using the CORDIC method. CORDIC is an iterative algorithm to approximate trigonometric functions by rotating a vector with different angles in each iteration. This paper proposes two extensions to the CORDIC algorithm to account for tapered accuracy with posits that improves precision: (1) fast-forwarding of iterations to start the CORDIC algorithm at a later iteration and (2) the use of a wide accumulator (i.e., the quire data type) to minimize precision loss with accumulation. Our results show that a 32-bit posit implementation of trigonometric functions with our extensions is more accurate than a 32-bit FP implementation. 
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