2Sum

Given two floating-point numbers ${\displaystyle a}$ and ${\displaystyle b}$, 2Sum computes the floating-point sum ${\displaystyle s:=\operatorname {fl} (a+b)}$ and the floating-point error ${\displaystyle t:=a+b-\operatorname {fl} (a+b)}$ so that ${\displaystyle s+t=a+b}$. The error ${\displaystyle t}$ is itself a floating-point number.

Inputs floating-point numbers ${\displaystyle a,b}$

Outputs sum ${\displaystyle s=\operatorname {fl} (a+b)}$ and error ${\displaystyle t=a+b-\operatorname {fl} (a+b)}$

1. ${\displaystyle s:=a\oplus b}$
2. ${\displaystyle a':=s\ominus b}$
3. ${\displaystyle b':=s\ominus a'}$
4. ${\displaystyle \delta _{a}:=a\ominus a'}$
5. ${\displaystyle \delta _{b}:=b\ominus b'}$
6. ${\displaystyle t:=\delta _{a}\oplus \delta _{b}}$
7. return ${\displaystyle (s,t)}$

Provided the floating-point arithmetic is correctly rounded to nearest (with ties resolved any way), as is the default in IEEE 754, and provided the sum does not overflow and, if it underflows, underflows gradually, it can be proven that ${\displaystyle s+t=a+b}$.[1][6][2]

A variant of 2Sum called Fast2Sum uses only three floating-point operations, for floating-point arithmetic in radix 2 or radix 3, under the assumption that the exponent of ${\displaystyle a}$ is at least as large as the exponent of ${\displaystyle b}$, such as when ${\displaystyle \left|a\right|\geq \left|b\right|}$:[1][6][7][4]

Inputs radix-2 or radix-3 floating-point numbers ${\displaystyle a}$ and ${\displaystyle b}$, of which at least one is zero, or which respectively have normalized exponents ${\displaystyle e_{a}\geq e_{b}}$

Outputs sum ${\displaystyle s=\operatorname {fl} (a+b)}$ and error ${\displaystyle t=a+b-\operatorname {fl} (a+b)}$

1. ${\displaystyle s:=a\oplus b}$
2. ${\displaystyle z=s\ominus a}$
3. ${\displaystyle t=b\ominus z}$
4. return ${\displaystyle (s,t)}$

Even if the conditions are not satisfied, 2Sum and Fast2Sum often provide reasonable approximations to the error so that ${\displaystyle s+t\approx a+b}$, which enables algorithms for compensated summation, dot-product, etc., to have low error even if the inputs are not sorted or the rounding mode is unusual.[1][2] More complicated variants of 2Sum and Fast2Sum also exist for rounding modes other than round-to-nearest.[1]

• Kahan summation algorithm
• Round-off error
• Double-double arithmetic