# Double-Double

Robert P. Munafo, 2023 Jun 19.

This term refers to a number format for calculation with twice the precision of the "native" hardware-supported double precision. The method was pioneered by T. J. Dekker in 1971 and exploits the fact that the roundoff error in a floating-point addition can be computed exactly to the same precision as the addition itself, and therefore the exact result can be attained by keeping both the rounded result and this precise calculation of the roundoff error. If "a+b" represents an exact addition and "a

`⊕`b" represents a rounded-off calculation e.g. IEEE 754, then

a + b = a `⊕` b + ε

ε = (a `⊖` (a `⊕` b)) `⊕` b

where ε represents the roundoff error. The value of a`⊕`b
is calculated first using a single native operation, and then
ε is calculated from that with two more native operations,
and both are kept for future use.

This is combined with a more traditional multiplication technique (adding together four partial products) and the rest of the operations are built on those to achieve quadruple-precision calculations at about 1/10 the speed of the native double-precision.

Double-double, and the higher-precision variants triple-double, quad-double, etc. serve as a bridge between hardware-supported formats and more generally extensible approaches like arbitrary-precision arithmetic.

From the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo, (c) 1987-2024.

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