ERFC

What it does

ERFC(x)

ERFC(x) is the normalised integral of exp(-t*t) from x to infinity. Mathematically it is 1-erf(x), but a direct tail calculation retains information when erf is close to one.

Try it

=ERFC(0.25) is about 0.723673609831763; =ERFC(0) is 1. A large positive argument has a small positive tail, whereas a large negative argument approaches two.

Quirks

Nonnumeric input gives #VALUE!. Subtracting a value rounded to one from one can erase a representable tail. The PRECISE spelling is not a whole-domain accuracy guarantee. The current kernel flushes a final subnormal tail to zero.

How to implement

Let D denote ordinary binary64 storage and R a 64-bit-significand operation followed by a binary64 store. The current finite-input branch table is:

For the positive upper stage, store z=R(x*x), obtain the stored public exponential E=EXP(-z), and form R(E*F(x)). Excel’s rational factor F is still open. The current code uses published Cody coefficients with stored arithmetic as a stand-in; copying that table does not complete an identified Excel tail recipe.

Worked below-half case: the stated ERF reconstruction at 0.25 is 0.276326390168237, so subtracting it from one gives about 0.723673609831763. In the upper stage at x=1, z is one and E is about 0.36787944117144233; the still unresolved F must supply the remaining factor.

Apply the current final subnormal-to-zero rule after the chosen branch. In particular, the negative half interval is not routed through the positive small-rational complement by this implementation. The exact half-point ownership in private Excel code is unseparated by the retained equal-answer control. Keep the reconstructed outer stages, current numerical substitutes and unmeasured transfer cases distinct.

Rust code

92 erfc "ERFC" (1, 1) [V "x"] |s, a, sv| {
    reject_logical(&a[0])?;
    out(s, sd::erfc_kernel(num(s, a, 0)?))
}

No implementation has been published with a Handbook test suite for this function.

Version differences

The recorded environments below distinguish a reading from a same-input comparison. Counts describe captured inputs in each environment; overlapping captures and comparisons are not totals of unique worksheet calls.

The record contains no same-input comparison between these environments. Reading different inputs on different builds does not establish either a version change or cross-build agreement.

Recorded environments and comparison details.

Alternatives

ERF computes the central integral; NORM.S.DIST expresses normal probabilities. Use a scaled complementary error function internally to avoid overflow when designing a mathematical tail algorithm.

Named variants

No named variants are recorded yet.