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Error function

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From Wikipedia, the free encyclopedia

Sigmoid shape special function

Not to be confused withLoss function.

In mathematics, theerror function (also called theGauss error function), often denoted byerf, is a function $\mathrm {erf} :\mathbb {C} \to \mathbb {C}$ defined as:^[1] $\operatorname {erf} (z)={\frac {2}{\sqrt {\pi }}}\int _{0}^{z}e^{-t^{2}}\,dt.$

Error function
Plot of the error function over real numbers
General information
General definition	$\operatorname {erf} (z)={\frac {2}{\sqrt {\pi }}}\int _{0}^{z}e^{-t^{2}}\,\mathrm {d} t$
Fields of application	Probability, thermodynamics, digital communications
Domain, codomain and image
Domain	$\mathbb {C}$
Image	$\left(-1,1\right)$
Basic features
Parity	Odd
Specific features
Root	0
Derivative	${\frac {d}{dz}}\operatorname {erf} (z)={\frac {2}{\sqrt {\pi }}}e^{-z^{2}}$
Antiderivative	$\int \operatorname {erf} (z)\,dz=z\operatorname {erf} (z)+{\frac {e^{-z^{2}}}{\sqrt {\pi }}}+C$
Series definition
Taylor series	$\operatorname {erf} (z)={\frac {2}{\sqrt {\pi }}}\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{2n+1}}{\frac {z^{2n+1}}{n!}}$

The integral here is a complexcontour integral which is path-independent because $\exp(-t^{2})$ isholomorphic on the whole complex plane $\mathbb {C}$ . In many applications, the function argument is areal number, in which case the function value is also real.

In some old texts,^[2]the error function is defined without the factor of ${\tfrac {2}{\sqrt {\pi }}}$ .Thisnonelementary integral is asigmoid function that occurs often inprobability,statistics, andpartial differential equations.

In statistics, for non-negative real values ofx, the error function has the following interpretation: for a realrandom variableY that isnormally distributed withmean 0 andstandard deviation ${\tfrac {1}{\sqrt {2}}}$ ,erf(x) is the probability thatY falls in the range[−x,x].

Two closely related functions are thecomplementary error function erfc: $\mathbb {C} \to \mathbb {C} ,$ defined as

$\operatorname {erfc} (z)=1-\operatorname {erf} (z),$

and theimaginary error function erfi: $\mathbb {C} \to \mathbb {C} ,$ defined as

$\operatorname {erfi} (z)=-i\operatorname {erf} (iz),$

wherei is theimaginary unit.

Name

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The name "error function" and its abbreviationerf were proposed byJ. W. L. Glaisher in 1871 on account of its connection with "the theory of probability, and notably the theory oferrors".^[3] The error function complement was also discussed by Glaisher in a separate publication in the same year.^[4]For the "law of facility" of errors whosedensity is given by $f(x)=\left({\frac {c}{\pi }}\right)^{1/2}e^{-cx^{2}}$ (thenormal distribution), Glaisher calculates the probability of an error lying betweenp andq as $\left({\frac {c}{\pi }}\right)^{\frac {1}{2}}\int _{p}^{q}e^{-cx^{2}}\,dx={\frac {1}{2}}{\big (}\operatorname {erf} (q{\sqrt {c}})-\operatorname {erf} (p{\sqrt {c}}){\big )}.$

Applications

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When the results of a series of measurements are described by anormal distribution withstandard deviationσ andexpected value 0, thenerf (⁠a/σ√2⁠) is the probability that the error of a single measurement lies between−a and+a, for positivea. This is useful, for example, in determining thebit error rate of a digital communication system.

The error and complementary error functions occur, for example, in solutions of theheat equation whenboundary conditions are given by theHeaviside step function.

The error function and its approximations can be used to estimate results that holdwith high probability or with low probability. Given a random variableX ~ Norm[μ,σ] (a normal distribution with meanμ and standard deviationσ) and a constantL >μ, it can be shown viaintegration by substitution: ${\begin{aligned}\Pr[X\leq L]&={\frac {1}{2}}+{\frac {1}{2}}\operatorname {erf} \left({\frac {L-\mu }{{\sqrt {2}}\sigma }}\right)\\&\approx A\exp \left(-B\left({\frac {L-\mu }{\sigma }}\right)^{2}\right)\end{aligned}}$

whereA andB are certain numeric constants. IfL is sufficiently far from the mean, specificallyμ −L ≥σ√ln(k), then:

$\Pr[X\leq L]\leq A\exp(-B\ln(k))={\frac {A}{k^{B}}}$

so the probability goes to 0 ask → ∞.

The probability forX being in the interval[L_a,L_b] can be derived as ${\begin{aligned}\Pr[L_{a}\leq X\leq L_{b}]&=\int _{L_{a}}^{L_{b}}{\frac {1}{{\sqrt {2\pi }}\sigma }}\exp \left(-{\frac {(x-\mu )^{2}}{2\sigma ^{2}}}\right)\,dx\\&={\frac {1}{2}}\left(\operatorname {erf} \left({\frac {L_{b}-\mu }{{\sqrt {2}}\sigma }}\right)-\operatorname {erf} \left({\frac {L_{a}-\mu }{{\sqrt {2}}\sigma }}\right)\right).\end{aligned}}$

Properties

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Plots in the complex plane

Integrandexp(−z²)

erf(z)

The propertyerf (−z) = −erf(z) means that the error function is anodd function. This directly results from the fact that the integrande^−t² is aneven function (the antiderivative of an even function which is zero at the origin is an odd function and vice versa).

Since the error function is anentire function which maps real numbers to real numbers, for anycomplex numberz: $\operatorname {erf} ({\overline {z}})={\overline {\operatorname {erf} (z)}}$ where ${\overline {z}}$ denotes thecomplex conjugate of $z {\displaystyle z}$ .

The integrandf = exp(−z²) andf = erf(z) are shown in the complexz-plane in the figures at right withdomain coloring.

The error function at+∞ is exactly 1 (seeGaussian integral). At the real axis,erfz approaches unity atz → +∞ and −1 atz → −∞. At the imaginary axis, it tends to±i∞.

Taylor series

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The error function is anentire function; it has no singularities (except that at infinity) and itsTaylor expansion always converges. Forx >> 1, however, cancellation of leading terms makes the Taylor expansion unpractical.

The defining integral cannot be evaluated inclosed form in terms ofelementary functions (seeLiouville's theorem), but by expanding theintegrande^−z² into itsMaclaurin series and integrating term by term, one obtains the error function's Maclaurin series as: ${\begin{aligned}\operatorname {erf} (z)&={\frac {2}{\sqrt {\pi }}}\sum _{n=0}^{\infty }{\frac {(-1)^{n}z^{2n+1}}{n!(2n+1)}}\\[6pt]&={\frac {2}{\sqrt {\pi }}}\left(z-{\frac {z^{3}}{3}}+{\frac {z^{5}}{10}}-{\frac {z^{7}}{42}}+{\frac {z^{9}}{216}}-\cdots \right)\end{aligned}}$ which holds for everycomplex number z. The denominator terms are sequenceA007680 in theOEIS.

It is a special case ofKummer's function:

$\operatorname {erf} (z)={\frac {2z}{\sqrt {\pi }}}\,{}_{1}F_{1}(1/2;3/2;-z^{2}).$

For iterative calculation of the above series, the following alternative formulation may be useful: ${\begin{aligned}\operatorname {erf} (z)&={\frac {2}{\sqrt {\pi }}}\sum _{n=0}^{\infty }\left(z\prod _{k=1}^{n}{\frac {-(2k-1)z^{2}}{k(2k+1)}}\right)\\[6pt]&={\frac {2}{\sqrt {\pi }}}\sum _{n=0}^{\infty }{\frac {z}{2n+1}}\prod _{k=1}^{n}{\frac {-z^{2}}{k}}\end{aligned}}$ because⁠−(2k − 1)z²/k(2k + 1)⁠ expresses the multiplier to turn thekth term into the(k + 1)th term (consideringz as the first term).

The imaginary error function has a very similar Maclaurin series, which is: ${\begin{aligned}\operatorname {erfi} (z)&={\frac {2}{\sqrt {\pi }}}\sum _{n=0}^{\infty }{\frac {z^{2n+1}}{n!(2n+1)}}\\[6pt]&={\frac {2}{\sqrt {\pi }}}\left(z+{\frac {z^{3}}{3}}+{\frac {z^{5}}{10}}+{\frac {z^{7}}{42}}+{\frac {z^{9}}{216}}+\cdots \right)\end{aligned}}$ which holds for everycomplex number z.

Given a complex numberz, there is not aunique complex numberw satisfyingerf(w) =z, so a true inverse function would be multivalued. However, for−1 <x < 1, there is a uniquereal number denotederf⁻¹(x) satisfying $\operatorname {erf} \left(\operatorname {erf} ^{-1}(x)\right)=x.$

Theinverse error function is usually defined with domain(−1,1), and it is restricted to this domain in manycomputer algebra systems. However, it can be extended to the disk|z| < 1 of the complex plane, using the Maclaurin series^[8] $\operatorname {erf} ^{-1}(z)=\sum _{k=0}^{\infty }{\frac {c_{k}}{2k+1}}\left({\frac {\sqrt {\pi }}{2}}z\right)^{2k+1},$ wherec₀ = 1 and ${\begin{aligned}c_{k}&=\sum _{m=0}^{k-1}{\frac {c_{m}c_{k-1-m}}{(m+1)(2m+1)}}\\[1ex]&=\left\{1,1,{\frac {7}{6}},{\frac {127}{90}},{\frac {4369}{2520}},{\frac {34807}{16200}},\ldots \right\}.\end{aligned}}$

So we have the series expansion (common factors have been canceled from numerators and denominators): $\operatorname {erf} ^{-1}(z)={\frac {\sqrt {\pi }}{2}}\left(z+{\frac {\pi }{12}}z^{3}+{\frac {7\pi ^{2}}{480}}z^{5}+{\frac {127\pi ^{3}}{40320}}z^{7}+{\frac {4369\pi ^{4}}{5806080}}z^{9}+{\frac {34807\pi ^{5}}{182476800}}z^{11}+\cdots \right).$ (After cancellation the numerator and denominator values in (sequenceA092676 in theOEIS) and (sequenceA092677 in theOEIS) respectively; without cancellation the numerator terms are values in (sequenceA002067 in theOEIS).) The error function's value at ±∞ is equal to ±1.

For|z| < 1, we haveerf(erf⁻¹(z)) =z.

Theinverse complementary error function is defined as $\operatorname {erfc} ^{-1}(1-z)=\operatorname {erf} ^{-1}(z).$ For realx, there is a uniquereal numbererfi⁻¹(x) satisfyingerfi(erfi⁻¹(x)) =x. Theinverse imaginary error function is defined aserfi⁻¹(x).^[9]

For any realx,Newton's method can be used to computeerfi⁻¹(x), and for−1 ≤x ≤ 1, the following Maclaurin series converges: $\operatorname {erfi} ^{-1}(z)=\sum _{k=0}^{\infty }{\frac {(-1)^{k}c_{k}}{2k+1}}\left({\frac {\sqrt {\pi }}{2}}z\right)^{2k+1},$ wherec_k is defined as above.

Asymptotic expansion

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A usefulasymptotic expansion of the complementary error function (and therefore also of the error function) for large realx is ${\begin{aligned}\operatorname {erfc} (x)&={\frac {e^{-x^{2}}}{x{\sqrt {\pi }}}}\left(1+\sum _{n=1}^{\infty }(-1)^{n}{\frac {1\cdot 3\cdot 5\cdots (2n-1)}{\left(2x^{2}\right)^{n}}}\right)\\[6pt]&={\frac {e^{-x^{2}}}{x{\sqrt {\pi }}}}\sum _{n=0}^{\infty }(-1)^{n}{\frac {(2n-1)!!}{\left(2x^{2}\right)^{n}}},\end{aligned}}$ where(2n − 1)!! is thedouble factorial of(2n − 1), which is the product of all odd numbers up to(2n − 1). This series diverges for every finitex, and its meaning as asymptotic expansion is that for any integerN ≥ 1 one has $\operatorname {erfc} (x)={\frac {e^{-x^{2}}}{x{\sqrt {\pi }}}}\sum _{n=0}^{N-1}(-1)^{n}{\frac {(2n-1)!!}{\left(2x^{2}\right)^{n}}}+R_{N}(x)$ where the remainder is $R_{N}(x):={\frac {(-1)^{N}\,(2N-1)!!}{{\sqrt {\pi }}\cdot 2^{N-1}}}\int _{x}^{\infty }t^{-2N}e^{-t^{2}}\,\mathrm {d} t,$ which follows easily by induction, writing $e^{-t^{2}}=-{\frac {1}{2t}}\,{\frac {\mathrm {d} }{\mathrm {d} t}}e^{-t^{2}}$ and integrating by parts.

The asymptotic behavior of the remainder term, inLandau notation, is $R_{N}(x)=O\left(x^{-(1+2N)}e^{-x^{2}}\right)$ asx → ∞. This can be found by $R_{N}(x)\propto \int _{x}^{\infty }t^{-2N}e^{-t^{2}}\,\mathrm {d} t=e^{-x^{2}}\int _{0}^{\infty }(t+x)^{-2N}e^{-t^{2}-2tx}\,\mathrm {d} t\leq e^{-x^{2}}\int _{0}^{\infty }x^{-2N}e^{-2tx}\,\mathrm {d} t\propto x^{-(1+2N)}e^{-x^{2}}.$ For large enough values ofx, only the first few terms of this asymptotic expansion are needed to obtain a good approximation oferfcx (while for not too large values ofx, the above Taylor expansion at 0 provides a very fast convergence).

Continued fraction expansion

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Acontinued fraction expansion of the complementary error function was found byLaplace:^[10]^[11] $\operatorname {erfc} (z)={\frac {z}{\sqrt {\pi }}}e^{-z^{2}}{\cfrac {1}{z^{2}+{\cfrac {a_{1}}{1+{\cfrac {a_{2}}{z^{2}+{\cfrac {a_{3}}{1+\dotsb }}}}}}}},\qquad a_{m}={\frac {m}{2}}.$

Factorial series

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The inversefactorial series: ${\begin{aligned}\operatorname {erfc} (z)&={\frac {e^{-z^{2}}}{{\sqrt {\pi }}\,z}}\sum _{n=0}^{\infty }{\frac {\left(-1\right)^{n}Q_{n}}{{\left(z^{2}+1\right)}^{\bar {n}}}}\\[1ex]&={\frac {e^{-z^{2}}}{{\sqrt {\pi }}\,z}}\left[1-{\frac {1}{2}}{\frac {1}{(z^{2}+1)}}+{\frac {1}{4}}{\frac {1}{\left(z^{2}+1\right)\left(z^{2}+2\right)}}-\cdots \right]\end{aligned}}$ converges forRe(z²) > 0. Here ${\begin{aligned}Q_{n}&{\overset {\text{def}}{{}={}}}{\frac {1}{\Gamma {\left({\frac {1}{2}}\right)}}}\int _{0}^{\infty }\tau (\tau -1)\cdots (\tau -n+1)\tau ^{-{\frac {1}{2}}}e^{-\tau }\,d\tau \\[1ex]&=\sum _{k=0}^{n}{\frac {s(n,k)}{2^{\bar {k}}}},\end{aligned}}$ zⁿ denotes therising factorial, ands(n,k) denotes a signedStirling number of the first kind.^[12]^[13]The Taylor series can be written in terms of thedouble factorial: $\operatorname {erf} (z)={\frac {2}{\sqrt {\pi }}}\sum _{n=0}^{\infty }{\frac {(-2)^{n}(2n-1)!!}{(2n+1)!}}z^{2n+1}$

Bounds and numerical approximations

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Approximation with elementary functions

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Abramowitz and Stegun give several approximations of varying accuracy (equations 7.1.25–28). This allows one to choose the fastest approximation suitable for a given application. In order of increasing accuracy, they are: $\operatorname {erf} (x)\approx 1-{\frac {1}{\left(1+a_{1}x+a_{2}x^{2}+a_{3}x^{3}+a_{4}x^{4}\right)^{4}}},\qquad x\geq 0$ (maximum error:5×10⁻⁴)

wherea₁ = 0.278393,a₂ = 0.230389,a₃ = 0.000972,a₄ = 0.078108

$\operatorname {erf} (x)\approx 1-\left(a_{1}t+a_{2}t^{2}+a_{3}t^{3}\right)e^{-x^{2}},\quad t={\frac {1}{1+px}},\qquad x\geq 0$ (maximum error:2.5×10⁻⁵)

wherep = 0.47047,a₁ = 0.3480242,a₂ = −0.0958798,a₃ = 0.7478556

$\operatorname {erf} (x)\approx 1-{\frac {1}{\left(1+a_{1}x+a_{2}x^{2}+\cdots +a_{6}x^{6}\right)^{16}}},\qquad x\geq 0$ (maximum error:3×10⁻⁷)

wherea₁ = 0.0705230784,a₂ = 0.0422820123,a₃ = 0.0092705272,a₄ = 0.0001520143,a₅ = 0.0002765672,a₆ = 0.0000430638

$\operatorname {erf} (x)\approx 1-\left(a_{1}t+a_{2}t^{2}+\cdots +a_{5}t^{5}\right)e^{-x^{2}},\quad t={\frac {1}{1+px}}$ (maximum error:1.5×10⁻⁷)

wherep = 0.3275911,a₁ = 0.254829592,a₂ = −0.284496736,a₃ = 1.421413741,a₄ = −1.453152027,a₅ = 1.061405429

One can improve the accuracy of the A&S approximation by extending it with three extra parameters, $\operatorname {erf} (x)\approx 1-\left(a_{1}t+a_{2}t^{2}+\cdots +a_{5}t^{5}+a_{6}t^{6}+a_{7}t^{7}\right)e^{-x^{2}},\quad t={\frac {1}{1+p_{1}x+p_{2}x^{2}}}$ where p1 = 0.406742016006509,p2 = 0.0072279182302319,a1 = 0.316879890481381,a2 = -0.138329314150635,a3 = 1.08680830347054,a4 = -1.11694155120396,a5 = 1.20644903073232,a6 = -0.393127715207728,a7 = 0.0382613542530727.The maximum error of this approximation is about2×10⁻⁹. The parameters are obtained by fitting the extended approximation to the accurate values of the error function using the following Python code.

Python code to fit extended A&S approximation

importnumpyasnpfrommathimporterf,exp,sqrtfromscipy.optimizeimportleast_squares## Extended A&S approximation:# erf(x) ≈ 1 − t * exp(−x^2) * (a1 + a2*t + a3*t^2 + ... + a7*t^6)# where now#      t = 1 / (1 + p1*x + p2*x^2)# We fit parameters p1, p2, a1..a7 over x in [0, 10].#defapprox_erf(params,x):p1=params[0]p2=params[1]a=params[2:]t=1.0/(1.0+p1*x+p2*x*x)poly=np.zeros_like(x)tt=np.ones_like(x)# t^0# polynomial: a1*t^0 + a2*t^1 + ... + a7*t^6forakina:poly+=ak*tttt*=treturn1.0-t*np.exp(-x*x)*polydefresiduals(params,xs,ys):returnapprox_erf(params,xs)-ys## Prepare data for fitting#N=300xmin=0xmax=10xs=np.linspace(xmin,xmax,N)ys=np.array([erf(x)forxinxs],dtype=float)## Initial guess for parameters# Start from original A&S values and extend them conservatively#p1_0=0.3275911# original A&S pp2_0=0.0# new denominator parameter# original A&S 5 coefficients, add two => 7 in totala0=[0.254829592,-0.284496736,1.421413741,-1.453152027,1.061405429,0.0,# new term0.0,# another new term]params0=np.array([p1_0,p2_0]+a0,dtype=float)## Fit using nonlinear least squares (Levenberg–Marquardt)#result=least_squares(residuals,params0,args=(xs,ys),xtol=1e-14,ftol=1e-14,gtol=1e-14,max_nfev=5000)params=result.xp1_fit=params[0]p2_fit=params[1]a_fit=params[2:]## Print fitted parameters#print("\nFitted parameters:")print(f"p1 ={p1_fit:.15g},")print(f"p2 ={p2_fit:.15g},")fori,aiinenumerate(a_fit,1):print(f"a{i} ={ai:.15g},")## Evaluate approximation error#approx_vals=approx_erf(params,xs)abs_err=np.abs(approx_vals-ys)print(f"\nMaximum absolute error on [{xmin},{xmax}]:",np.max(abs_err))print("RMS error:",np.sqrt(np.mean(abs_err**2)))print("Done.")

All of these approximations are valid forx ≥ 0. To use these approximations for negativex, use the fact thaterf(x) is an odd function, soerf(x) = −erf(−x).

Exponential bounds and a pure exponential approximation for the complementary error function are given by^[14] ${\begin{aligned}\operatorname {erfc} (x)&\leq {\frac {1}{2}}e^{-2x^{2}}+{\frac {1}{2}}e^{-x^{2}}\leq e^{-x^{2}},&&x>0\\[1.5ex]\operatorname {erfc} (x)&\approx {\frac {1}{6}}e^{-x^{2}}+{\frac {1}{2}}e^{-{\frac {4}{3}}x^{2}},&&x>0.\end{aligned}}$

The above have been generalized to sums ofN exponentials^[15] with increasing accuracy in terms ofN so thaterfc(x) can be accurately approximated or bounded by2Q̃(√2x), where ${\tilde {Q}}(x)=\sum _{n=1}^{N}a_{n}e^{-b_{n}x^{2}}.$ In particular, there is a systematic methodology to solve the numerical coefficients{(a_n,b_n)}^N
_{n = 1} that yield aminimax approximation or bound for the closely relatedQ-function:Q(x) ≈Q̃(x),Q(x) ≤Q̃(x), orQ(x) ≥Q̃(x) forx ≥ 0. The coefficients{(a_n,b_n)}^N
_{n = 1} for many variations of the exponential approximations and bounds up toN = 25 have been released to open access as a comprehensive dataset.^[16]

A tight approximation of the complementary error function forx ∈ [0,∞) is given byKaragiannidis & Lioumpas (2007),^[17] who showed for the appropriate choice of parameters{A,B} that $\operatorname {erfc} (x)\approx {\frac {\left(1-e^{-Ax}\right)e^{-x^{2}}}{B{\sqrt {\pi }}x}}.$ They determined{A,B} = {1.98,1.135}, which gave a good approximation^[which?] for allx ≥ 0. Alternative coefficients are also available for tailoring accuracy for a specific application or transforming the expression into a tight bound.^[18]

A single-term lower bound is^[19] $\operatorname {erfc} (x)\geq {\sqrt {\frac {2e}{\pi }}}{\frac {\sqrt {\beta -1}}{\beta }}e^{-\beta x^{2}},\qquad x\geq 0,\quad \beta >1,$ where the parameterβ can be picked to minimize error on the desired interval of approximation.

Another approximation is given by Sergei Winitzki using his "global Padé approximations":^[20]^[21]^: 2–3 $\operatorname {erf} (x)\approx \operatorname {sgn} x\cdot {\sqrt {1-\exp \left(-x^{2}{\frac {{\frac {4}{\pi }}+ax^{2}}{1+ax^{2}}}\right)}}$ where $a={\frac {8(\pi -3)}{3\pi (4-\pi )}}\approx 0.140012.$ This is designed to be very accurate in the neighborhoods of 0 and infinity, and therelative error is less than 0.00035 for all realx. Using the alternate valuea ≈ 0.147 reduces the maximum relative error to about 0.00013.^[22]

The extended "global Pade" approximation, $\operatorname {erf} (x)\approx \operatorname {sgn} x\cdot {\sqrt {1-\exp \left(-x^{2}{\frac {4+0.880877880079853x^{2}+0.144026670907584x^{4}+0.0077581300270021x^{6}}{\pi +0.786235558186528x^{2}+0.128368576906837x^{4}+0.00773380006014367x^{6}}}\right)}}\,,$ provides a maximum error of about2×10⁻⁹, as demonstrated by the following Python script.

Python script to fit extended "global Pade" approximation

importnumpy,mathfromscipy.optimizeimportleast_squares# approximation to erf(x)defapprox_erf(p,x):frac=(4+p[0]*x**2+p[1]*x**4+p[2]*x**6)/(math.pi+p[3]*x**2+p[4]*x**4+p[5]*x**6)returnnumpy.sign(x)*numpy.sqrt(1-numpy.exp(-x*x*frac))defresiduals(params,xs,ys):returnapprox_erf(params,xs)-ys# data for fittingN=200xmin=0xmax=9xs=numpy.linspace(xmin,xmax,N)ys=numpy.array([math.erf(x)forxinxs],dtype=float)params0=numpy.array([0.9,0.1,0.008,0.8,0.1,0.008],dtype=float)# fittingresult=least_squares(residuals,params0,args=(xs,ys),xtol=1e-14,ftol=1e-14,gtol=1e-14,max_nfev=5000)params=result.x# print out fitted parametersprint("\nFitted parameters:")fori,piinenumerate(params,0):print(f"p{i} ={pi:.15g},")# evaluate approximation errorapprox_vals=approx_erf(params,xs)abs_err=numpy.abs(approx_vals-ys)print(f"\nMaximum absolute error on [{xmin},{xmax}]:",numpy.max(abs_err))print("RMS error:",numpy.sqrt(numpy.mean(abs_err**2)))print("Done.")

Winitzki's approximation can be inverted to obtain an approximation for the inverse error function: $\operatorname {erf} ^{-1}(x)\approx \operatorname {sgn} x\cdot {\sqrt {{\sqrt {\left({\frac {2}{\pi a}}+{\frac {\ln \left(1-x^{2}\right)}{2}}\right)^{2}-{\frac {\ln \left(1-x^{2}\right)}{a}}}}-\left({\frac {2}{\pi a}}+{\frac {\ln \left(1-x^{2}\right)}{2}}\right)}}.$

An approximation with a maximal error of1.2×10⁻⁷ for any real argument is:^[23] ${\begin{aligned}\operatorname {erf} (x)&={\begin{cases}1-\tau ,&x\geq 0\\\tau -1,&x<0\end{cases}}\\\tau &=t\cdot \exp \left(-x^{2}-1.26551223+1.00002368t+0.37409196t^{2}+0.09678418t^{3}-0.18628806t^{4}\right.\\&\left.\qquad \qquad \qquad +0.27886807t^{5}-1.13520398t^{6}+1.48851587t^{7}-0.82215223t^{8}+0.17087277t^{9}\right)\\t&={\frac {1}{1+{\frac {1}{2}}|x|}}\end{aligned}}$

An approximation of $\operatorname {erfc}$ with a maximum relative error less than $2^{-53}$ $\left(\approx 1.1\times 10^{-16}\right)$ in absolute value is:^[24]for $x\geq 0$ , ${\begin{aligned}\operatorname {erfc} \left(x\right)&=\left({\frac {0.56418958354775629}{x+2.06955023132914151}}\right)\left({\frac {x^{2}+2.71078540045147805x+5.80755613130301624}{x^{2}+3.47954057099518960x+12.06166887286239555}}\right)\\&\left({\frac {x^{2}+3.47469513777439592x+12.07402036406381411}{x^{2}+3.72068443960225092x+8.44319781003968454}}\right)\left({\frac {x^{2}+4.00561509202259545x+9.30596659485887898}{x^{2}+3.90225704029924078x+6.36161630953880464}}\right)\\&\left({\frac {x^{2}+5.16722705817812584x+9.12661617673673262}{x^{2}+4.03296893109262491x+5.13578530585681539}}\right)\left({\frac {x^{2}+5.95908795446633271x+9.19435612886969243}{x^{2}+4.11240942957450885x+4.48640329523408675}}\right)e^{-x^{2}}\\\end{aligned}}$ and for $x<0$ $\operatorname {erfc} \left(x\right)=2-\operatorname {erfc} \left(-x\right)$

A simple approximation for real-valued arguments can be done throughhyperbolic functions: $\operatorname {erf} \left(x\right)\approx z(x)=\tanh \left({\frac {2}{\sqrt {\pi }}}\left(x+{\frac {11}{123}}x^{3}\right)\right)$ which keeps the absolute difference $\left|\operatorname {erf} \left(x\right)-z(x)\right|<0.000358,\,\forall x$ .

Since the error function and the Gaussian Q-function are closely related through the identity $\operatorname {erfc} (x)=2Q({\sqrt {2}}x)$ or equivalently $Q(x)={\frac {1}{2}}\operatorname {erfc} \left({\frac {x}{\sqrt {2}}}\right)$ , bounds developed for the Q-function can be adapted to approximate the complementary error function. A pair of tight lower and upper bounds on the Gaussian Q-function for positive arguments $x\in [0,\infty )$ was introduced by Abreu (2012)^[25] based on a simplealgebraic expression with only two exponential terms: ${\begin{aligned}x&\geq 0\\{\frac {1}{2}}\operatorname {erfc} \left({\frac {x}{\sqrt {2}}}\right)&\geq {\frac {1}{12}}e^{-x^{2}}+{\frac {1}{{\sqrt {2\pi }}(x+1)}}e^{-x^{2}/2}\\&\leq {\frac {1}{50}}e^{-x^{2}}+{\frac {1}{2(x+1)}}e^{-x^{2}/2}\\{\frac {1}{25}}e^{-2x^{2}}+{\frac {1}{x+1}}e^{-x^{2}}\geq \operatorname {erfc} (x)&\geq {\frac {1}{6}}e^{-2x^{2}}+{\frac {1}{2{\sqrt {2\pi }}(x+1)}}e^{-x^{2}}\end{aligned}}$

These bounds stem from a unified form $Q_{\mathrm {B} }(x;a,b)={\frac {\exp(-x^{2})}{a}}+{\frac {\exp(-x^{2}/2)}{b(x+1)}},$ where the parameters $a {\displaystyle a}$ and $b {\displaystyle b}$ are selected to ensure the bounding properties: for the lower bound, $a_{\mathrm {L} }=12$ and $b_{\mathrm {L} }={\sqrt {2\pi }}$ , and for the upper bound, $a_{\mathrm {U} }=50$ and $b_{\mathrm {U} }=2$ . These expressions maintain simplicity and tightness, providing a practical trade-off between accuracy and ease of computation. They are particularly valuable in theoretical contexts, such as communication theory over fading channels, where both functions frequently appear. Additionally, the original Q-function bounds can be extended to $Q^{n}(x)$ for positive integers $n {\displaystyle n}$ via thebinomial theorem, suggesting potential adaptability for powers of $\operatorname {erfc} (x)$ , though this is less commonly required in error function applications.

Table of values

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Further information:Interval estimation,Coverage probability, and68–95–99.7 rule

x	erf(x)	1 − erf(x)
0	0	1
0.02	0.022564575	0.977435425
0.04	0.045111106	0.954888894
0.06	0.067621594	0.932378406
0.08	0.090078126	0.909921874
0.1	0.112462916	0.887537084
0.2	0.222702589	0.777297411
0.3	0.328626759	0.671373241
0.4	0.428392355	0.571607645
0.5	0.520499878	0.479500122
0.6	0.603856091	0.396143909
0.7	0.677801194	0.322198806
0.8	0.742100965	0.257899035
0.9	0.796908212	0.203091788
1	0.842700793	0.157299207
1.1	0.880205070	0.119794930
1.2	0.910313978	0.089686022
1.3	0.934007945	0.065992055
1.4	0.952285120	0.047714880
1.5	0.966105146	0.033894854
1.6	0.976348383	0.023651617
1.7	0.983790459	0.016209541
1.8	0.989090502	0.010909498
1.9	0.992790429	0.007209571
2	0.995322265	0.004677735
2.1	0.997020533	0.002979467
2.2	0.998137154	0.001862846
2.3	0.998856823	0.001143177
2.4	0.999311486	0.000688514
2.5	0.999593048	0.000406952
3	0.999977910	0.000022090
3.5	0.999999257	0.000000743

Related functions

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Complementary error function

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Plot of the error function erf(z) in the complex plane from−2 − 2i to2 + 2i

Thecomplementary error function, denotederfc, is defined as ${\begin{aligned}\operatorname {erfc} (x)&=1-\operatorname {erf} (x)\\&={\frac {2}{\sqrt {\pi }}}\int _{x}^{\infty }e^{-t^{2}}\,dt\\&=e^{-x^{2}}\operatorname {erfcx} (x),\end{aligned}}$ which also defineserfcx, thescaled complementary error function^[26] (which can be used instead oferfc to avoidarithmetic underflow^[26]^[27]). Another form oferfcx forx ≥ 0 is known as Craig's formula, after its discoverer:^[28] $\operatorname {erfc} (x\mid x\geq 0)={\frac {2}{\pi }}\int _{0}^{\frac {\pi }{2}}\exp \left(-{\frac {x^{2}}{\sin ^{2}\theta }}\right)\,d\theta .$ This expression is valid only for positive values ofx, but can be used in conjunction witherfc(x) = 2 − erfc(−x) to obtainerfc(x) for negative values. This form is advantageous in that the range of integration is fixed and finite. An extension of this expression for theerfc of the sum of two non-negative variables is^[29] $\operatorname {erfc} (x+y\mid x,y\geq 0)={\frac {2}{\pi }}\int _{0}^{\frac {\pi }{2}}\exp \left(-{\frac {x^{2}}{\sin ^{2}\theta }}-{\frac {y^{2}}{\cos ^{2}\theta }}\right)\,d\theta .$

Imaginary error function

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Theimaginary error function, denotederfi, is defined as ${\begin{aligned}\operatorname {erfi} (x)&=-i\operatorname {erf} (ix)\\&={\frac {2}{\sqrt {\pi }}}\int _{0}^{x}e^{t^{2}}\,dt\\&={\frac {2}{\sqrt {\pi }}}e^{x^{2}}D(x),\end{aligned}}$ whereD(x) is theDawson function (which can be used instead oferfi to avoidarithmetic overflow^[26]).

Despite the name "imaginary error function",erfi(x) is real whenx is real.

When the error function is evaluated for arbitrarycomplex argumentsz, the resultingcomplex error function is usually discussed in scaled form as theFaddeeva function: $w(z)=e^{-z^{2}}\operatorname {erfc} (-iz)=\operatorname {erfcx} (-iz).$

Cumulative distribution function

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The error function is essentially identical to the standardnormal cumulative distribution function, denotedΦ, also namednorm(x) by some software languages^{[citation needed]}, as they differ only by scaling and translation. Indeed, ${\begin{aligned}\Phi (x)&={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{x}e^{\tfrac {-t^{2}}{2}}\,dt\\[6pt]&={\frac {1}{2}}\left(1+\operatorname {erf} \left({\frac {x}{\sqrt {2}}}\right)\right)\\[6pt]&={\frac {1}{2}}\operatorname {erfc} \left(-{\frac {x}{\sqrt {2}}}\right)\end{aligned}}$ or rearranged forerf anderfc: ${\begin{aligned}\operatorname {erf} (x)&=2\Phi {\left(x{\sqrt {2}}\right)}-1\\[6pt]\operatorname {erfc} (x)&=2\Phi {\left(-x{\sqrt {2}}\right)}\\&=2\left(1-\Phi {\left(x{\sqrt {2}}\right)}\right).\end{aligned}}$

Consequently, the error function is also closely related to theQ-function, which is the tail probability of the standard normal distribution. The Q-function can be expressed in terms of the error function as ${\begin{aligned}Q(x)&={\frac {1}{2}}-{\frac {1}{2}}\operatorname {erf} \left({\frac {x}{\sqrt {2}}}\right)\\&={\frac {1}{2}}\operatorname {erfc} \left({\frac {x}{\sqrt {2}}}\right).\end{aligned}}$

Theinverse ofΦ is known as thenormal quantile function, orprobit function and may be expressed in terms of the inverse error function as $\operatorname {probit} (p)=\Phi ^{-1}(p)={\sqrt {2}}\operatorname {erf} ^{-1}(2p-1)=-{\sqrt {2}}\operatorname {erfc} ^{-1}(2p).$

The standard normal cdf is used more often in probability and statistics, and the error function is used more often in other branches of mathematics.

The error function is a special case of theMittag-Leffler function, and can also be expressed as aconfluent hypergeometric function (Kummer's function): $\operatorname {erf} (x)={\frac {2x}{\sqrt {\pi }}}M\left({\tfrac {1}{2}},{\tfrac {3}{2}},-x^{2}\right).$

It has a simple expression in terms of theFresnel integral.^{[further explanation needed]}

In terms of theregularized gamma functionP and theincomplete gamma function, $\operatorname {erf} (x)=\operatorname {sgn}(x)\cdot P\left({\tfrac {1}{2}},x^{2}\right)={\frac {\operatorname {sgn}(x)}{\sqrt {\pi }}}\gamma {\left({\tfrac {1}{2}},x^{2}\right)}.$ sgn(x) is thesign function.

Iterated integrals of the complementary error function

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The general recurrence formula is $2n\cdot i^{n}\!\operatorname {erfc} (z)=i^{n-2}\!\operatorname {erfc} (z)-2z\cdot i^{n-1}\!\operatorname {erfc} (z)$

They have the power series $i^{n}\!\operatorname {erfc} (z)=\sum _{j=0}^{\infty }{\frac {(-z)^{j}}{2^{n-j}j!\,\Gamma \left(1+{\frac {n-j}{2}}\right)}},$ from which follow the symmetry properties $i^{2m}\!\operatorname {erfc} (-z)=-i^{2m}\!\operatorname {erfc} (z)+\sum _{q=0}^{m}{\frac {z^{2q}}{2^{2(m-q)-1}(2q)!(m-q)!}}$ and $i^{2m+1}\!\operatorname {erfc} (-z)=i^{2m+1}\!\operatorname {erfc} (z)+\sum _{q=0}^{m}{\frac {z^{2q+1}}{2^{2(m-q)-1}(2q+1)!(m-q)!}}.$

Implementations

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As real function of a real argument

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InPOSIX-compliant operating systems, the headermath.h shall declare and the mathematical librarylibm shall provide the functionserf anderfc (double precision) as well as theirsingle precision andextended precision counterpartserff,erfl anderfcf,erfcl.^[31]
TheGNU Scientific Library provideserf,erfc,log(erf), and scaled error functions.^[32]

As complex function of a complex argument

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libcerf, numeric C library for complex error functions, provides the complex functionscerf,cerfc,cerfcx and the real functionserfi,erfcx with approximately 13–14 digits precision, based on theFaddeeva function as implemented in theMIT Faddeeva Package