Pair Arithmetic

3.6 Pair Arithmetic

Pair arithmetic is a numerical representation method designed to extend the precision of floating-point computations by splitting a value into a main part and a remainder part. This approach mitigates rounding errors that arise in standard floating-point arithmetic, particularly for operations involving numbers of vastly different magnitudes.

A pair $〈x,y⟩$ represents a real number $z$ as the sum of two floating-point values:

z= x+ y,

where:

$x$ is the main value, a floating-point number capturing the significant digits of $z$ .
$y$ is the remainder, a floating-point number representing the residual part of $z$ that is too small to be accurately represented in $x$ alone.

The pair $〈x,y⟩$ satisfies the condition:

|y|≤ 1 ulp(x), 2

where $ulp(x)$ (unit in the last place) denotes the spacing between $x$ and the next representable floating-point value. This ensures $y$ is negligible relative to $x$ but still contributes to the overall precision of $z$ .

3.6.1 Not-A-Number Pair

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