Why Computers Lie Badly At Alarming Speed and the unum Promise

The translation of arithmetic to physical hardware with using the IEEE standard employed numerical representation is fraught with difficulty. As is well known by any who have used even a pocket calculator, computer processors are imprecise with dangerous rounding errors, which vary on different systems. Further, the standard representation method, IEEE 754 "Standard for Floating-Point Arithmetic" (1985, revised 2008), is extremely inefficient from an engineering perspective with increasing physical cost when additional precision is sought.

The basic issue is the limitations in converting decimal or floating point notation into binary form. The IEEE standard suggests that when a calculation overflows the value +inf should be used instead, and when a number is too small the standard says to use 0 instead. Inserting infinity to represent "a very big number" or 0 to represent a "very small number" will certainly cause computational issues. Floating point operations have additional issues when employed in parallel, breaking the logic of associative properties. The equation (a + b) + (c + d) in parallel will not equal the equation ((a + b) + c) + d when run in serial.

These issues have been known in computer science for some decades (Goldberg, 1991). In recent years an attempt has been made to reconstruct the physical implementation of arithmetic to physical hardware by providing a superset to IEEE's 754 standard and IEEE 1788, Standard for Interval
Arithmetic. This number format, the Unum (Gustafson, 2015), consists of a bit string of variable length with six sub-fields: a sign bit, exponent, fraction, uncertainty bit, exponent size, and fraction size. The uncertainty bit, or ubit, specifies whether or not there are additional bits after fraction, instead of rounding, in other words a precise interval. This means that numbers that are close to
zero or infinity are treated as such and are never represented as zero or infinity. To date, Unums have not been translated into hardware as they require more logic than floating-point numbers, but software logic has been provided.

Why Computers Lie Badly At Alarming Speed and the unum Promise
Challenges in High Performance Computing Conference
2-6 Sept, 2019 Mathematical Sciences Institute, Australian National University