2Sum

Given two floating-point numbers $a$ and $b$, 2Sum computes the floating-point sum $s\; :=\; a\; \backslash oplus\; b$ rounded to nearest and the floating-point error $t\; :=\; a\; +\; b\; -\; (a\; \backslash oplus\; b)$ so that $s\; +\; t\; =\; a\; +\; b$, where $\backslash oplus$ and $\backslash ominus$ respectively denote the addition and subtraction rounded to nearest.

The error $t$ is itself a floating-point number.

:Inputs floating-point numbers $a,\; b$

:Outputs rounded sum $s\; =\; a\; \backslash oplus\; b$ and exact error $t\; =\; a\; +\; b\; -\; (a\; \backslash oplus\; b)$

:# $s\; :=\; a\; \backslash oplus\; b$

:# $a\text{'}\; :=\; s\; \backslash ominus\; b$

:# $b\text{'}\; :=\; s\; \backslash ominus\; a\text{'}$

:# $\backslash delta\_a\; :=\; a\; \backslash ominus\; a\text{'}$

:# $\backslash delta\_b\; :=\; b\; \backslash ominus\; b\text{'}$

:# $t\; :=\; \backslash delta\_a\; \backslash oplus\; \backslash delta\_b$

:# return $(s,\; t)$

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 {{nowrap|$s\; +\; t\; =\; a\; +\; b$.}}{{cite book

|last1=Knuth

|first1=Donald E.

|author-link=Donald Knuth

|title=The Art of Computer Programming, Volume II: Seminumerical Algorithms

|date=1998

|publisher=Addison–Wesley

|isbn=978-0-201-89684-8

|page=236

|edition=3rd

|url=https://www-cs-faculty.stanford.edu/~knuth/taocp.html

|access-date=2020-09-20

|archive-date=2017-07-16

|archive-url=https://web.archive.org/web/20170716132045/http://cs.stanford.edu/~uno/taocp.html

|url-status=live

}}

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 $a$ is at least as large as the exponent of $b$, such as when {{nowrap|$\backslash left|a\backslash right|\; \backslash geq\; \backslash left|b\backslash right|$:}}{{cite book

|last1=Sterbenz

|first1=Pat H.

|title=Floating-Point Computation

|date=1974

|publisher=Prentice-Hall

|location=Englewood Cliffs, NJ, United States

|isbn=0-13-322495-3

|pages=138–143

}}{{cite journal

|last1=Dekker

|first1=T.J.

|author-link=Theodorus Dekker

|title=A floating-point technique for extending the available precision

|journal=Numerische Mathematik

|date=June 1971

|volume=18

|issue=3

|pages=224–242

|doi=10.1007/BF01397083

|doi-access=free

|s2cid=63218464

|url=https://ir.cwi.nl/pub/9159

|access-date=2020-09-24

|archive-date=2020-07-19

|archive-url=https://web.archive.org/web/20200719135521/https://ir.cwi.nl/pub/9159/

|url-status=live

}}

:Inputs radix-2 or radix-3 floating-point numbers $a$ and $b$, of which at least one is zero, or which respectively have normalized exponents $e\_a\; \backslash geq\; e\_b$

:Outputs rounded sum $s\; =\; a\; \backslash oplus\; b$ and exact error $t\; =\; a\; +\; b\; -\; (a\; \backslash oplus\; b)$

:# $s\; :=\; a\; \backslash oplus\; b$

:# $z\; =\; s\; \backslash ominus\; a$

:# $t\; =\; b\; \backslash ominus\; z$

:# return $(s,\; t)$

Even if the conditions are not satisfied, 2Sum and Fast2Sum often provide reasonable approximations to the error, i.e. $s\; +\; t\; \backslash approx\; a\; +\; b$, 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.

More complicated variants of 2Sum and Fast2Sum also exist for rounding modes other than round-to-nearest.

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

{{reflist}}

Category:Computer arithmetic

Category:Floating point

Category:Numerical analysis