We present algorithms for performing the five elementary arithmetic operations ( $+$ , $-$ , ×, $\div$ , and $\sqrt{\phantom{x}}$ ) in floating point arithmetic with stochastic rounding, and demonstrate the value of these algorithms by discussing various applications where stochastic rounding is beneficial. The algorithms require that the hardware be compliant with the IEEE 754 floating-point standard and that a floating-point pseudorandom number generator be available. The goal of these techniques is to emulate stochastic rounding when the underlying hardware does not support this rounding mode, as is the case for most existing CPUs and GPUs. By simulating stochastic rounding in software, one has the possibility to explore the behavior of this rounding mode and develop new algorithms even without having access to hardware implementing stochastic rounding—once such hardware becomes available, it suffices to replace the proposed algorithms by calls to the corresponding hardware routines. When stochastically rounding double precision operations, the algorithms we propose are between 7.3 and 19 times faster than the implementations that use the GNU MPFR library to simulate extended precision. We test our algorithms on various tasks, including summation algorithms and solvers for ordinary differential equations, where stochastic rounding is expected to bring advantages.

中文翻译：

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IEEE 754 浮点算术中随机舍入基本算术运算的算法

我们提出了执行五种基本算术运算的算法（ $+$ , $-$ , ×, $\div$ ， 和 $\sqrt{\phantom{x}}$ ) 在具有随机舍入的浮点运算中，并通过讨论随机舍入有益的各种应用来证明这些算法的价值。这些算法要求硬件符合 IEEE 754 浮点标准并且可以使用浮点伪随机数生成器。这些技术的目标是在底层硬件不支持这种舍入模式时模拟随机舍入，就像大多数现有 CPU 和 GPU 的情况一样。通过在软件中模拟随机舍入，即使无法访问实现随机舍入的硬件，也可以探索这种舍入模式的行为并开发新算法——一旦这种硬件可用，通过调用相应的硬件例程来替换所提出的算法就足够了。当随机舍入双精度运算时，我们提出的算法比使用 GNU MPFR 库来模拟扩展精度的实现快 7.3 到 19 倍。我们在各种任务上测试我们的算法，包括求和算法和常微分方程求解器，其中随机舍入有望带来优势。