Beating Floating Point at its Own Game: Posit Arithmetic

Abstract

A new data type called a posit is designed as a direct drop-in replacement for IEEE Standard 754 floating-point numbers (floats). Unlike earlier forms of universal number (unum) arithmetic, posits do not require interval arithmetic or variable size operands; like floats, they round if an answer is inexact. However, they provide compelling advantages over floats, including larger dynamic range, higher accuracy, better closure, bitwise identical results across systems, simpler hardware, and simpler exception handling. Posits never overflow to infinity or underflow to zero, and "Nota-Number" (NaN) indicates an action instead of a bit pattern. A posit processing unit takes less circuitry than an IEEE float FPU. With lower power use and smaller silicon footprint, the posit operations per second (POPS) supported by a chip can be significantly higher than the FLOPS using similar hardware resources. GPU accelerators and Deep Learning processors, in particular, can do more per watt and per dollar with posits, yet deliver superior answer quality.A comprehensive series of benchmarks compares floats and posits for decimals of accuracy produced for a set precision. Low precision posits provide a better solution than "approximate computing" methods that try to tolerate decreased answer quality. High precision posits provide more correct decimals than floats of the same size; in some cases, a 32-bit posit may safely replace a 64-bit float. In other words, posits beat floats at their own game.Keywords: computer arithmetic, energy-efficient computing, floating point, posits, LINPACK, linear algebra, neural networks, unum computing, valid arithmetic. Background: Type I and Type II UnumsThe unum (universal number) arithmetic framework has several forms. The original "Type I" unum is a superset of IEEE 754 Standard floating-point format [2,7]; it uses a "ubit" at the end of the fraction to indicate whether a real number is an exact float or lies in the open interval between adjacent floats. While the sign, exponent, and fraction bit fields take their definition from IEEE 754, the exponent and fraction field lengths vary automatically, from a single bit up to some maximum set by the user. Type I unums provide a compact way to express interval arithmetic, but their variable length demands extra management. They can duplicate IEEE float behavior, via an explicit rounding function.The "Type II" unum [4] abandons compatibility with IEEE floats, permitting a clean, mathematical design based on the projective reals. The key observation is that signed (two's complement) integers map elegantly to the projective reals, with the same wraparound of positive numbers to negative numbers, and the same ordering. To quote William Kahan [5]:"They typically save storage space because what you're manipulating are not the numbers, but pointers to the values. And so, it's possible to run this arithmetic very, very fast."The structure for 5-bit Type II unums is shown in fig. 1. With n bits per unum, the "u-lattice" populates the upper right quadrant of t...

Posits as an alternative to floats for weather and climate models

“…A resulting higher precision around one is traded against a gradually lower precision for very large or very small numbers. A positive posit number p is decoded as [15,16] (negative posit numbers are converted first to their two's complement, see Eq. 3)…”

“…1. They fill gaps of powers of 2 spanned by useed = 4, 16, 256, ... for e s = 1, 2, 3, ..., and every posit number can be written as p = ±2 n · (1 + f ) with a given integer n [7,16]. Throughout this article we will use a notation where Posit(n,e s ) defines the posit numbers with n bits including e s exponent bits.…”

“…Posit numbers are a recently proposed alternative to floats and claim to provide more precision in arithmetic calculations with fewer bits in algorithms of linear algebra or machine learning [16]. However, posits remain untested for weather and climate simulations.…”

An approach to generate correctly rounded math libraries for new floating point variants

“…There is an ever-increasing need for improved FP performance in domains such as machine learning and high performance computing (HPC). Hence, several new variants and alternatives to FP have been proposed recently such as Bfloat16 [Tagliavini et al 2018], posits [Gustafson 2017;Gustafson and Yonemoto 2017], and TensorFloat32 [NVIDIA 2020].…”

An Accelerator for Posit Arithmetic Targeting Posit Level 1 BLAS Routines and Pair-HMM

“…https://doi.org /10.1145/3316279.3316284 claim to be a better alternative to the IEEE 754 floating point standard. One of the recent and most promising alternatives is the posit number format, as presented by John L. Gustafson [6].…”