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  1. Mainstream math libraries for floating point (FP) do not produce correctly rounded results for all inputs. In contrast, CR-LIBM and RLIBM provide correctly rounded implementations for a specific FP representation with one rounding mode. Using such libraries for a representation with a new rounding mode or with different precision will result in wrong results due to double rounding. This paper proposes a novel method to generate a single polynomial approximation that produces correctly rounded results for all inputs for multiple rounding modes and multiple precision configurations. To generate a correctly rounded library for n -bits, our key idea is to generate a polynomial approximation for a representation with n +2-bits using the round-to-odd mode. We prove that the resulting polynomial approximation will produce correctly rounded results for all five rounding modes in the standard and for multiple representations with k -bits such that | E | +1 < k ≤ n , where | E | is the number of exponent bits in the representation. Similar to our prior work in the RLIBM project, we approximate the correctly rounded result when we generate the library with n +2-bits using the round-to-odd mode. We also generate polynomial approximations by structuring it as a linear programming problem but propose enhancements to polynomial generation to handle the round-to-odd mode. Our prototype is the first 32-bit float library that produces correctly rounded results with all rounding modes in the IEEE standard for all inputs with a single polynomial approximation. It also produces correctly rounded results for any FP configuration ranging from 10-bits to 32-bits while also being faster than mainstream libraries. 
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  2. null (Ed.)
    This paper proposes a new approach for debugging errors in floating point computation by performing shadow execution with higher precision in parallel. The programmer specifies parts of the program that need to be debugged for errors. Our compiler creates shadow execution tasks, which execute on different cores and perform the computation with higher precision. We propose a novel method to execute a shadow execution task from an arbitrary memory state, which is necessary because we are creating a parallel shadow execution from a sequential program. Our approach also ensures that the shadow execution follows the same control flow path as the original program. Our runtime automatically distributes the shadow execution tasks to balance the load on the cores. Our prototype for parallel shadow execution, PFPSanitizer, provides comprehensive detection of errors while having lower performance overheads than prior approaches. 
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  3. null (Ed.)
    This paper proposes a set of techniques to develop correctly rounded math libraries for 32-bit float and posit types. It enhances our RLIBM approach that frames the problem of generating correctly rounded libraries as a linear programming problem in the context of 16-bit types to scale to 32-bit types. Specifically, this paper proposes new algorithms to (1) generate polynomials that produce correctly rounded outputs for all inputs using counterexample guided polynomial generation, (2) generate efficient piecewise polynomials with bit-pattern based domain splitting, and (3) deduce the amount of freedom available to produce correct results when range reduction involves multiple elementary functions. The resultant math library for the 32-bit float type is faster than state-of-the-art math libraries while producing the correct output for all inputs. We have also developed a set of correctly rounded elementary functions for 32-bit posits. 
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  4. null (Ed.)
    Given the importance of floating point (FP) performance in numerous domains, several new variants of FP and its alternatives have been proposed (e.g., Bfloat16, TensorFloat32, and posits). These representations do not have correctly rounded math libraries. Further, the use of existing FP libraries for these new representations can produce incorrect results. This paper proposes a novel approach for generating polynomial approximations that can be used to implement correctly rounded math libraries. Existing methods generate polynomials that approximate the real value of an elementary function 𝑓 (𝑥) and produce wrong results due to approximation errors and rounding errors in the implementation. In contrast, our approach generates polynomials that approximate the correctly rounded value of 𝑓 (𝑥) (i.e., the value of 𝑓 (𝑥) rounded to the target representation). It provides more margin to identify efficient polynomials that produce correctly rounded results for all inputs. We frame the problem of generating efficient polynomials that produce correctly rounded results as a linear programming problem. Using our approach, we have developed correctly rounded, yet faster, implementations of elementary functions for multiple target representations. 
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  5. 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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  6. 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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