Erf

The Gauss error function $\operatorname{erf}(z) = \frac{2}{\sqrt\pi}\int_0^z e^{-t^2}\,dt$, the shape behind the normal distribution's CDF.

✓ checked against that system · ! checked, with disagreements (hover a chip)WikipediaError function↗MathWorldErf↗WikidataQ579262↗DLMF7.2.E1↗FungrimErf✓↗Fungrim entryWolfram LanguageErf!↗

Erf(z)the Gauss error function $\operatorname{erf}(z)$.

SageMatherf(z)SymPyerf(z)mpmatherf(z)Juliaenumeratio_ball(erf, z)Rusterf(z)

Typed signatures

Type	Package	Applies
(complex | signed_infinity) -> complex	compute-engine	always
(number, number?) -> number	analyticreplaces compute-engine	always

Domain: Special functions

Details

• Odd function: $\operatorname{erf}(-z) = -\operatorname{erf}(z)$.
• $\operatorname{erf}(0) = 0$, and $\operatorname{erf}(\pm\infty) = \pm 1$ -- the total (signed) area under the Gaussian.
• Complementary with Erfc: $\operatorname{erf}(z) + \operatorname{erfc}(z) = 1$.
• Builds the normal CDF: $\Phi(x) = \frac{1}{2}\left(1 + \operatorname{erf}(x/\sqrt{2})\right)$.
• compute-engine requires an inexact (floating-point) argument to produce a numeric value under plain evaluation; an exact integer or rational argument is left symbolic outside the special points 0 and $\pm\infty$.

Plot

Examples

Basic #

live ↗

In$\operatorname{erf}(0)$Out$0$

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In$\operatorname{erf}(∞)$Out$1$

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In$\operatorname{erf}(-∞)$Out$-1$

erf as z -> ComplexInfinity has no limit -- the approach direction decides it (1 real, i imaginary)

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In$\operatorname{erf}(\tilde∞)$Out$\operatorname{Indeterminate}$

A floating-point argument evaluates directly to a high-precision decimal

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In$\operatorname{erf}(0.5)$Out$0.520\,499\,877\,813\,046\,537\,683$

A machine-precision argument

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In$\operatorname{erf}(0.95)$Out$0.820\,890\,807\,273\,277\,941\,908$

Scope #

Threads element-wise over a list, as Wolfram's Listable heads do

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In$\operatorname{erf}(\bigl\lbrack0, 1\bigr\rbrack)$Out$\bigl\lbrack0, \operatorname{erf}(1)\bigr\rbrack$

Complex arguments evaluate directly

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In$\operatorname{erf}(1.5-i)$Out$1.078\,399\,207\,498\,933\,5+0.027\,963\,711\,238\,655\,833i$

Two arguments: the generalized $\operatorname{erf}(z_0, z_1) = \operatorname{erf}(z_1) - \operatorname{erf}(z_0)$

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In$\operatorname{erf}(0.5, 1.5)$Out$0.445\,605\,268\,662\,264\,2$

...which with exact arguments is the difference $\operatorname{erf}(2) - \operatorname{erf}(1)$

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In$\operatorname{erf}(1, 2)$Out$\operatorname{erf}(2)-\operatorname{erf}(1)$

Interval arithmetic: Erf is increasing, so the image of an interval is the interval of the endpoint images

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In$\operatorname{erf}(\mathrm{Interval}(-2.1, -1.9))$Out$\mathrm{Interval}(-0.997\,020\,533\,343\,667, -0.992\,790\,429\,235\,257\,5)$

Uncertainty propagation, with slope $\frac{2}{\sqrt\pi}e^{-4}$

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In$\operatorname{erf}(\mathrm{Around}(2, 0.01))$Out$\mathrm{Around}(0.995\,322\,265\,018\,952\,7, 0.000\,206\,669\,853\,540\,920\,54)$

To 40 significant digits, every one correctly rounded

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In$N(\operatorname{erf}(1), 40)$Out$0.842\,700\,792\,949\,714\,869\,341\,220\,635\,082\,609\,259\,296\,1$

Applications #

$\operatorname{erf}(1/\sqrt2) \approx 0.6827$: the '68' in the empirical 68-95-99.7 rule for one standard deviation

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In$\operatorname{erf}(0.707\,106\,781\,186\,547\,6)$Out$0.682\,689\,492\,137\,085\,948\,91$

The standard normal CDF $\Phi(x) = \frac12(1 + \operatorname{erf}(x/\sqrt2))$ at $x = 1.96$: the familiar 97.5%

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In$0.5×(1+\operatorname{erf}(\frac{1.96}{\sqrt{2}}))$Out$0.975\,002\,104\,851\,779\,565\,863\,5$

Properties #

Odd function: $\operatorname{erf}(-z) = -\operatorname{erf}(z)$, here at $z = 2$

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In$\operatorname{erf}(2)+\operatorname{erf}(-2)=0$Out$\mathrm{True}$

Complementary with Erfc: $\operatorname{erf}(z) + \operatorname{erfc}(z) = 1$. See Erfc

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In$\operatorname{erf}(2)+\operatorname{erfc}(2)=1$Out$\mathrm{True}$

Oddness applied symbolically: $\operatorname{erf}(-x) = -\operatorname{erf}(x)$

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In$\operatorname{erf}(-x)$Out$-\operatorname{erf}(x)$

$\operatorname{erf}(\operatorname{erfinv}(x)) = x$ symbolically. See ErfInv

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In$\operatorname{erf}(\mathrm{ErfInv}(x))$Out$x$

On the imaginary axis: $\operatorname{erf}(i) = i\,\operatorname{erfi}(1)$

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In$\operatorname{erf}(i)$Out$i\mathrm{Erfi}(1)$

$\frac{d}{dx}\operatorname{erf}(x) = \frac{2}{\sqrt\pi}e^{-x^2}$, the Gaussian itself

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In$\frac{\mathrm{d}\operatorname{erf}(x)}{\mathrm{d}x}$Out$(2e^{-x^2})/\sqrt{π}$

Possible issues #

An exact integer argument is left unevaluated under plain evaluation -- pair with N() or use an inexact argument like 0.5 for a decimal

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In$\operatorname{erf}(1)$Out$\operatorname{erf}(1)$

compute-engine's complex Erf runs in doubles, so past a double's 15 or so figures N leaves it unevaluated rather than print a double as 25.

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In$N(\operatorname{erf}(1+i), 25)$Out$\operatorname{erf}(1+i)$

Neat examples #

By $z=3.5$, erf is already within $10^{-6}$ of its limit of 1

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In$\operatorname{erf}(3.5)$Out$0.999\,999\,256\,901\,627\,658\,587$

About

Examples	24
Checked against	Wolfram Language 21 agree3 differSageMath 17 agree2 differ1 unevaluatedSymPy 21 agree2 differmpmath 16 agree1 differJulia 15 agree1 differ1 errorRust 11 agree
Crosswalk	WikipediaMathWorldWikidataDLMFFungrimFungrim entry ×9Wolfram LanguageSageMathSymPympmathJulia ×2Rust

Implementation

mappedsympypackages/oracle/src/mappings.ts

mappedsympypackages/oracle/src/mappings.ts

SymPy's erf2(x, y) is the same difference but stays unevaluated even at float arguments, so the difference is written out: erf(z1) - erf(z0), each term SymPy's own erf.

mappedmpmathpackages/oracle/src/mappings.ts

mappedmpmathpackages/oracle/src/mappings.ts

mappedsagepackages/oracle/src/mappings.ts

mappedjuliapackages/oracle/src/mappings.ts

mappedjuliapackages/oracle/src/mappings.ts

mappedrustpackages/oracle/src/mappings.ts

See also: Erfc, ErfInv