The error function

@ac-kit/math-analysisPortable

import { erf, erfc } from "@ac-kit/math-analysis";

Where you meet it. “Three sigma” is a claim about the error function. So is every p-value from a normal test, every process-capability index on a factory floor, and the confidence interval on any measurement assumed to be normally distributed. It also turns up far from statistics — heat diffusion and signal attenuation are described by the same integral.

Formally, erf() computes erf(x) = 2/√π ∫₀ˣ e^(−t²) dt, which is the probability that a standard normal measurement lands within x√2 of its mean.

The complement is not a subtraction

erfc() computes 1 − erf(x) — the area outside the band rather than inside it. That is the quantity you want for a tail probability: “how unlikely is a result at least this extreme?”

It exists as its own function because computing it as a subtraction destroys it. erf(x) reaches exactly 1 in floating point at about x = 6, and 1 - 1 is 0. Everything past that point — every tail probability smaller than about 1e-17 — is gone.

Computed directly, erfc keeps full relative precision all the way down to the underflow limit near x = 26.5, which is what makes a tail probability of 1e-100 a number rather than a zero.

EvidencePast x ≈ 6 the naive 1 − erf(x) returns exactly zero, while erfc(x) stays positive below 1e-100.

Why erfc is not 1 − erf

Both curves are the same mathematical quantity, plotted on a logarithmic axis. The naive 1 − erf(x) dies at x ≈ 6, where erf(x) rounds to exactly 1 and the subtraction returns exactly 0 — there is nothing left to take a logarithm of, so its line simply ends. erfc(x) routes the positive branch through the upper incomplete gamma instead and keeps full relative precision down to the underflow limit near x = 26.5. That gap is what makes a normal tail probability of 1e-100 computable at all.

const xs = linspace(0, upper, samples);

const frame = curveFrame(
  xs,
  [
    { name: "erfc(x)", values: xs.map((x) => erfc(x)) },
    { name: "1 − erf(x)", values: xs.map((x) => 1 - erf(x)) },
  ],
  { x: "x", y: "tail", series: "method" },
);

The rule generalises across this whole package:

Reach for the upper form when the answer is small.

The same pairing appears in the incomplete gamma family, for exactly the same reason.

The error function

The complement is not a subtraction

Further reading