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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 forn-bits, our key idea is to generate a polynomial approximation for a representation withn+2-bits using theround-to-oddmode. We prove that the resulting polynomial approximation will produce correctly rounded results for all five rounding modes in the standard and for multiple representations withk-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 withn+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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This content will become publicly available on June 10, 2026
Random Variate Generation with Formal Guarantees
Generating random variates is a fundamental operation in diverse areas of computer science and is supported in almost all modern programming languages. Traditional software libraries for random variate generation are grounded in the idealized Real-RAM model of computation, where algorithms are assumed to be able to access uniformly distributed real numbers from the unit interval and compute with infinite-precision real arithmetic. These assumptions are unrealistic, as any software implementation of a Real-RAM algorithm on a physical computer can instead access a stream of individual random bits and computes with finite-precision arithmetic. As a result, existing libraries have few theoretical guarantees in practice. For example, the actual distribution of a random variate generator is generally unknown, intractable to quantify, and arbitrarily different from the desired distribution; causing runtime errors, unexpected behavior, and inconsistent APIs. This article introduces a new approach to principled and practical random variate generation with formal guarantees. The key idea is to first specify the desired probability distribution in terms of a finite-precision numerical program that defines its cumulative distribution function (CDF), and then generate exact random variates according to this CDF. We present a universal and fully automated method to synthesize exact random variate generators given any numerical CDF implemented in any binary number format, such as floating-point, fixed-point, and posits. The method is guaranteed to operate with the same precision used to specify the CDF, does not overflow, avoids expensive arbitrary-precision arithmetic, and exposes a consistent API. The method rests on a novel space-time optimal implementation for the class of generators that attain the information-theoretically optimal Knuth and Yao entropy rate, consuming the least possible number of input random bits per output variate. We develop a random variate generation library using our method in C and evaluate it on a diverse set of continuous and discrete distributions, showing competitive runtime with the state-of-the-art GNU Scientific Library while delivering higher accuracy, entropy efficiency, and automation.
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- Award ID(s):
- 2311983
- PAR ID:
- 10638716
- Publisher / Repository:
- ACM
- Date Published:
- Journal Name:
- Proceedings of the ACM on Programming Languages
- Volume:
- 9
- Issue:
- PLDI
- ISSN:
- 2475-1421
- Page Range / eLocation ID:
- 125 to 149
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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