牛顿引力N体问题直接求解器的验证：IEEE浮点数与Posits的系统比较
Validating direct solvers for Newton's gravitational N-body problem, and the systematic comparison between IEEE floating point and Posits
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中文总结 AI 辅助
本研究系统比较了IEEE浮点数、两种Posits实现及任意精度算术求解牛顿混沌N体问题的精度与速度，发现Posits暂非fp64的理想替代方案。
中文摘要 AI 辅助
我们对任意精度算术与积分、符合IEEE-754标准的浮点算术（fp16、bfp16、fp32、双精度fp64、四精度fp128），以及两种Posits（III型unum）实现进行了系统比较，以求解牛顿混沌N体问题。每种实现均以任意精度计算为基准，客观评估其精度与速度性能。fp64依赖硬件与编译器实现，任意精度算术与Posits则采用软件实现。半精度算术（fp16、bfp16及Posits<16,1>）求解牛顿运动方程的精度不足；单精度（fp32及Posits<32,2>）可用于统计系综计算，但在任何单独强相遇中会产生较大误差。所有64位实现（fp64及Posits<64,3>）在我们的测试中均存在困难。其中一种Posits实现（Universal）的精度与fp64相当，但速度较慢（校正后者更高效的硬件支持后，至少比fp64慢一个数量级）；另一种实现（CPPPosits）速度与fp64相当，但系统误差更大（比fp64约大一个数量级，超出值超过两个数量级），导致结果空间出现系统漂移，难以分辨近距离相遇。在运动参考系中积分动力学系统时，Posits与fp64均存在困难，无法测试伽利略不变性。就当前实现而言，在积分牛顿运动方程等混沌或刚性常微分方程时，Posits似乎并非fp64的理想替代方案。
英文摘要
We present a systematic comparison between arbitrary precise arithmetic and integration, IEEE-754 compliant floating point arithmetic (fp16, bfp16, fp32, double precision fp64, and quadruple precision fp128), and two implementations of Posits (type III unum) for solving Newton's chaotic N-body problem. Each implementation is benchmarked with arbitrary precise calculations to objectively evaluate their performance in precision as well as speed. We rely on hardware and compiler implementations for fp64, and software implementations for arbitrary-precision arithmetic and Posits. Half precision arithmetic (fp16, bfp16, and Posits$<16,1>$) are insufficiently precise for solving Newton's equations of motion. Single precision (fp32, and Posits$<32,2>$) could be used for statistical ensemble calculations, but lead to relatively large errors in any individual strong encounter. All 64-bit implementations fp64 as well as Posits (Posits$<64,3>$) experience difficulty in our tests. One of the implementations of Posits (Universal) gives precision comparable to fp64 but is slow (by at least an orders of magnitude compared to fp64 after correcting for the more efficient hardware support for the latter). The other (CPPPosits) has a speed comparable to fp64 but has systematically larger errors (by about an order of magnitude compared to fp64 with excesses exceeding two orders of magnitude). As a consequence, this implementation leads to a systematic drift in the result space and has difficulty resolving close encounters. Posits and fp64 have difficulty when integrating a dynamical system in a moving reference frame; testing Galileo invariancy. In their current implementation, Posits do not seem to be the ideal alternative for fp64 when integrating chaotic or stiff ordinary differential equations, such as Newton's equations of motion.
