Error Function Calculator

No elementary antiderivative — and that is a theorem

erf(x) = (2/√π) ∫₀ˣ e^(−t²) dt. The 2/√π is there to make erf(∞) come out at exactly 1, since ∫₀^∞ e^(−t²) dt = √π/2. No elementary function differentiates to e^(−t²), so this integral cannot be evaluated in closed form — erf is the name given to the answer, not a shorthand for one.

This is worth stating precisely because it sounds like an admission of defeat and is not. Liouville’s theorem proves that no combination of polynomials, exponentials, logarithms and trigonometric functions differentiates to e⁻ᵗ². Nobody is going to find one.

So erf is a genuinely new function, defined by its integral, exactly as sin and log are defined by their own properties rather than reduced to something simpler. Every value has to be computed numerically, and the interesting question becomes how well.

Why erfc has to be its own function

Computing erfc as 1 − erf(x) gives 1.53743684450 × 10^−12 against the true 1.53745979443 × 10^−12. erf(x) is so close to 1 here that the subtraction throws most of the precision away: about 4 correct significant figures survive out of the 16 a double carries, and the gap widens as x grows — by x = 6 there is nothing left at all, because erf(6) rounds to exactly 1 and the subtraction returns a flat zero where the true erfc is 2.15197 × 10^−17. This is why erfc exists as its own function rather than as a convenience name for 1 − erf.

The problem is catastrophic cancellation. erf(5) is 0.999999999998, and subtracting that from 1 leaves a result made mostly of the digits that were rounded away. The subtraction is exact; the input to it has already lost the information.

At this size 1 − erf(x) = 2.20904969986 × 10^−5 still agrees with erfc(x) = 2.20904969986 × 10^−5. The two only diverge once erf gets close to 1, past about x = 3.

So erfc is computed directly from its own series rather than by subtracting, which is why it stays meaningful out to erfc(5) = 1.53745979443 × 10^−12 and beyond. Any library that defines erfc as 1 − erf is unusable in exactly the regime where erfc is the quantity you wanted.

The normal-distribution link, and the √2

Φ(z) = ½(1 + erf(z/√2)), so erf and the normal CDF carry the same information on different scales — the √2 is the whole conversion. Read this erf value as a probability and it says: a standard normal lies within ±1.4142136 standard deviations with probability 0.8427007929. That is the TWO-sided probability. The one-sided upper tail beyond that same z is 0.07864960353, which is a different number and the one people substitute by mistake.

The two functions carry the same information on different scales. erf integrates e⁻ᵗ²; the normal density has e⁻ᵗ²⃗² instead, and that factor of 2 in the exponent becomes a √2 in the argument.

For x = 1, Φ(x) = 0.841344746069 while erf(x) = 0.84270079295 — different numbers describing the same distribution. The equivalent z is 1.414213562, and it is the conversion that gets dropped when a formula is copied between the two conventions.

One tail or two, and which one you asked for

erf(x) is the TWO-sided probability: the chance a standard normal lands within ±x√2 of the mean. For x = 1 that is 0.84270079295.

The single upper tail beyond the same point is 0.0786496035251 — a different number, and roughly half of what is left over. Substituting one for the other is the commonest error in applying either function, and it is the kind that produces a plausible answer.

The rule: if the question is “within” or “between ±”, it is two-sided and erf gives it directly. If it is “greater than” or “more extreme than” in one direction, it is one-sided and you want half of erfc.

The inverse, and the range it is defined on

Read the other way: erf⁻¹(0.5) = 0.4769362762 is the upper limit that makes the integral come to 0.5.

erf only ever returns values strictly between −1 and 1 — it approaches both without reaching them — so erf⁻¹ is defined on (−1, 1) and nowhere else. Asking for erf⁻¹(1) is asking for infinity.

erf only ever returns values strictly between −1 and 1, so erf⁻¹(2) does not exist. The inverse is defined on (−1, 1) and nowhere else.

Practically the inverse is how a confidence level becomes a bound: given the probability you want to capture, it returns how many standard units wide the interval has to be. That is the direction most applied questions run.

How this page computes it

The usual implementation is the Abramowitz & Stegun rational approximation, which is accurate to about 1.5×10⁻⁷. That is fine for a table and visibly wrong in the seventh decimal.

This page uses the relationship erf(x) = P(½, x²) with the regularised incomplete gamma function, computed by its series for small arguments and by a continued fraction for large ones. The result holds full double precision — about 10⁻¹⁶ relative — across the whole range.

Middle range: this is where erf does almost all of its moving — from about 0.52 at x = 0.5 to 0.9999779 at x = 3. Neither the small-x series nor the large-x expansion is comfortable here, which is why the incomplete-gamma route is used throughout.

erf is an odd function: erf(−x) = −erf(x). So erf(1.5) = 0.966105146475, and erf(0) = 0 exactly — the integral from 0 to 0.

Sources and methodology

The error function and its relationship to the incomplete gamma function are standard; the references below carry the canonical definitions.

• NIST Digital Library of Mathematical Functions — §7.2, Error Functions
• NIST DLMF — §8.2, Incomplete Gamma Functions
• How we calculate — methodology

Method. Every figure on this page comes from src/lib/error-function.ts over src/lib/algebra/special.ts. erf is computed as P(½, x²) through the regularised incomplete gamma function — series below the crossover, modified Lentz continued fraction above it — rather than through the A&S rational fit. erfc is computed from Q(½, x²) directly, never by subtracting erf from 1. That engine is verified on every change against 76 hand-written assertions, including agreement with published values to better than 1e-15 relative, and that erf(erf⁻¹(y)) returns y. The count and the per-case breakdown are published on the formula verification page.

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Read the guide

The error function is the incomplete gamma function in disguise, and the Gamma Function Calculator covers the family it belongs to. For summarising a dataset rather than a distribution, the Standard Deviation Calculator is the tool.