ERF Function (Gnumeric)

• Introduced: First release not established.
• Supported in: Gnumeric 1.12.62, with no retirement established.
• Scope: Desktop Gnumeric with the Engineering plugin; English comma-separated formulas are shown.
• Checked: October 14, 2026. Reference and boundary tables plus the explicit advanced-check formula were recalculated through 1.12.57 ssconvert. Current 1.12.62 source was inspected; that later release was not executed.

ERF evaluates the Gauss error function. With one argument x it integrates from zero to x, scaled by 2/SQRT(PI()). With two arguments it integrates from the first limit to the second. The order of those limits controls the sign. Versioned function help.

=ERF(lower[,upper])

One or two numeric arguments are accepted. In the one-argument form, the supplied value is the endpoint: =ERF(1) means the integral from 0 to 1. In the two-argument form, =ERF(lower,upper) uses both limits. Do not interpret the optional second position as a cumulative flag. Brackets mark optional arguments; do not type the brackets into a formula. Function implementation.

Ordinary results are rounded to twelve decimal places; tiny nonzero values use scientific notation. The eight reference rows were independently checked.

For the interval from 1 to 2, =ERF(1,2) is approximately 0.152621472069. Reversing the limits gives approximately −0.152621472069, while equal limits give zero. Put the lower and upper limits in separately labeled cells before filling a table.

The result is a signed scaled integral. For example, =ERF(-1,1) is approximately 1.685401585899, which exceeds 1. That is valid for this function and should not be described as a probability without the appropriate transformation.

For a standard normal variable Z, P(a<Z≤b)=ERF(a/SQRT(2),b/SQRT(2))/2. The factor 1/2 and the SQRT(2) scaling are essential. As a simpler check, the executed identity =ERF(1)+ERFC(1) returned 1.

The native =ERF(6,10) returned a small nonzero result, approximately 2.151973671250E-17. Computing ERF(10)−ERF(6) separately can lose that tail difference when both single-argument values round to 1. The two-limit implementation uses a normal-interval helper rather than requiring the user to subtract two near-one displayed values.

Negative arguments are accepted and ERF is odd: ERF(−x)=−ERF(x). A negative interval result need not indicate an error; inspect the order of the limits. The function does not silently sort them.

The listed coercion tests accepted TRUE as 1 and an empty F5 as 0, while text “abc” failed. Such observations apply to these exact formulas and import path; they are not a rule that all text or missing data should be treated numerically. Very narrow intervals and extreme tails still require accuracy checks beyond the ordinary examples.

F5 below is an actually empty cell. The listed errors are native error results, not mathematical zeroes.

• Worksheet entry
• Numeric or argument-handling source

The retained workbook results include a deliberately stale SUM cache and an uncached SUM formula, both recalculated correctly. Native evidence covers the named 1.12.57 build and listed cases; cell-reference layouts and other explanatory compositions are instructions or independent mathematical checks unless explicitly identified as executed. Other locales, direct GUI entry and later-release output were not tested.