Algorithms for stochastically rounded elementary arithmetic operations in IEEE 754 floating-point arithmetic

We present algorithms for performing the five elementary arithmetic operations (+, -, ×, ÷, and √) 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. Simulating stochastically rounded floating-point operations can be used to explore the behavior of this rounding, as well as to develop applications before hardware with stochastic rounding is available-once such hardware becomes available, the proposed algorithms can be replaced by calls to the relevant 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 problems where stochastic rounding is expected to bring advantages, which includes summation algorithms and ordinary differential equation solvers