ERFC 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.

ERFC evaluates the complementary Gauss error function, erfc(x). Mathematically it equals 1−erf(x), but evaluating the complement directly can retain a tiny positive tail that subtraction from 1 would lose. It takes one endpoint, not a lower/upper pair. Versioned function help.

=ERFC(x)

Exactly one numeric x is required. Negative x is accepted; the mathematical range over finite real arguments is between 0 and 2. =ERFC(0) is 1. A second argument is an arity error in Gnumeric and must not be copied from a different product’s function contract. Function implementation.

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

A direct tail calculation at x=6 gives =ERFC(6) approximately 2.151973671250E-17 in the native run. This is tiny but nonzero. Display it in scientific notation rather than with a small fixed number of decimal places that would make it appear to be exactly zero.

For a standard normal right-tail probability at z, the mathematical relationship is ERFC(z/SQRT(2))/2. ERFC alone is not already that normal-tail probability: both the input scaling and final division matter.

The executed comparison =ERFC(6)-(1-ERF(6)) returned approximately 2.151973671250E-17. In this case the subtraction-based complement rounded to zero, while direct ERFC retained the tail. This illustrates cancellation; it does not establish perfect relative accuracy for every extreme argument.

The symmetry erfc(−x)=2−erfc(x) explains why negative inputs can return values above 1. In particular ERFC(−1) is approximately 1.842700792950, whereas ERFC(1) is approximately 0.157299207050.

The native result at x=30 was 0, and at −30 it was 2. Those are finite-precision endpoints: the mathematical positive tail at 30 is nonzero but smaller than ordinary binary64 can represent. A displayed or stored zero does not prove a mathematically impossible event.

An actually empty F5 became the zero endpoint and returned 1 in this fixture. Text “abc” failed. Preserve the distinction between an absent measurement and a deliberate zero; successful numeric coercion alone does not validate an input column.

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.