In mathematics, the error function (also called the Gauss error function), often denoted by <math>\mathbf{erf}</math>, is the function
<math display="block">\operatorname{erf}(z) = \frac{2}{\sqrt\pi}\int_0^z e^{-t^2}\,dt.</math>
{\sqrt\pi} + C</math>
| taylor_series = <math display="block">\begin{align}
\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{align}</math>
The integral here is a complex contour integral which is path-independent because <math>\exp(-t^2)</math> is holomorphic on the whole complex plane <math>\mathbb{C}</math>. In many applications, the function argument is a real number, in which case the function value is also real.
In some older texts, the error function is defined without the factor of <math>2/\sqrt{\pi}</math>.
This nonelementary integral is a sigmoid function that occurs often in probability, statistics, and partial differential equations.
In statistics, for non-negative real values of <math>X</math>, the error function has the following interpretation: for a real random variable <math>Y</math> that is normally distributed with mean 0 and standard deviation <math>1/\sqrt{2}</math>, <math>\operatorname{erf}(x)</math> is the probability that <math>Y</math> falls in the range <math>[-x,x]</math>.
Two closely related functions are the complementary error function
:<math>\operatorname{erfc}(z) = 1 - \operatorname{erf}(z)</math>
and the imaginary error function
:<math>\operatorname{erfi}(z) = -i\operatorname{erf}(iz),</math>
where <math>i</math> is the imaginary unit.
Name
The name "error function" and its abbreviation <math>\operatorname{erf}</math> were proposed by J. W. L. Glaisher in 1871 on account of its connection with "the theory of probability, and notably the theory of errors". The complementary error function was also discussed by Glaisher in a separate publication in the same year. For the "law of facility" of errors whose density is given by
:<math>f(x) = \left(\frac{c}{\pi}\right)^{1/2} e^{-c x^2}</math>
(the normal distribution), Glaisher calculates the probability of an error lying between <math>p</math> and <math>q</math> as
:<math>\left(\frac{c}{\pi}\right)^\frac{1}{2} \int_p^qe^{-cx^2}\,dx = \frac{1}{2} \big(\operatorname{erf}(q\sqrt{c}) - \operatorname{erf}(p\sqrt{c})\big).</math>
Applications
When the results of a series of measurements are described by a normal distribution with standard deviation <math>\sigma</math> and expected value zero, then
:<math>\operatorname{erf}\bigg(\frac{a}{\sigma\sqrt{2\bigg)</math>
is the probability that the error of a single measurement lies between <math>-a</math> and <math>a</math>. This is useful, for example, in determining the bit error rate of a digital communication system.
The error and complementary error functions occur, for example, in solutions of the heat equation when boundary conditions are given by the Heaviside step function.
The error function and its approximations can be used to estimate results that hold with high probability or with low probability. Given a normally distributed random variable <math>X</math> with mean <math>\mu</math> and standard deviation <math>\sigma</math> and a constant <math>L>\mu</math>, it can be shown (via integration by substitution) that
:<math>\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)</math>
where <math>A</math> and <math>B</math> are certain numeric constants. If <math>L</math> is sufficiently far from the mean, specifically, <math>\mu-L\geq \sigma\sqrt{\log(k)}</math>, then
:<math>\Pr[X\leq L] \leq A \exp (-B \log(k))=\frac{A}{k^B}</math>
and so the probability goes to 0 as <math>k\to\infty</math>.
The probability for <math>X</math> being in the interval <math>[L_a,L_b]</math> can be derived as
<math display="block">\begin{align}
\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 \\[4pt]
&= \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{align}</math>
Properties
The error function is an odd function. This directly results from the fact that the integrand <math>e^{-t^2}</math> is an even function (since the antiderivative of an even function which is zero at the origin is an odd function, and vice versa).
Since the error function is an entire function which maps real numbers to real numbers, for any complex number <math>z</math>,
:<math>\operatorname{erf}(\bar{z}) = \overline{\operatorname{erf}(z)}</math>
where <math>\bar{z}</math> denotes the complex conjugate of <math>z</math>.
The error function at <math>\infty</math> is exactly <math>1</math> (see Gaussian integral). At the real axis, <math>\operatorname{erf}(z)</math> approaches <math>1</math> at <math>z\to\infty</math> and <math>-1</math> at <math>z\to-\infty</math>. At the imaginary axis, it tends to <math>\pm i\infty</math>.
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Taylor series
The error function is an entire function; it has no singularities (except at infinity) and its Taylor expansion always converges. For <math>x\gg 1</math>, however, cancellation of leading terms makes the Taylor expansion impractical.
The defining integral cannot be evaluated in closed form in terms of elementary functions (see Liouville's theorem), but by expanding the integrand <math>e^{-z^2}</math> into its Maclaurin series, integrating term by term, and using the fact that <math>\operatorname{erf}(0)=0</math>, one obtains the error function's Maclaurin series as:
<math display="block">\begin{align}
\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{align}</math>
which holds for every complex number <math>z</math>. The denominator terms form sequence A007680 in the OEIS. This is a special case of Kummer's function:
:<math>\operatorname{erf}(z) = \frac{2z}{\sqrt\pi}\,{}_1F_1\bigg(\frac{1}{2}, \frac{3}{2}, -z^2\bigg).</math>
For iterative calculation of the above series, the following alternative formulation may be useful:
<math display="block">\begin{align}
\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{align}</math>
because
:<math>\frac{-(2k-1)z^2}{k(2k+1)}</math>
expresses the multiplier to turn the <math>k</math>-th term into the <math>(k+1)</math>-th term (considering <math>z</math> as the first term).
The imaginary error function has a similar Maclaurin series:
<math display="block">\begin{align}
\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{align}</math>
which holds for every complex number <math>z</math>.
Derivative and integral
The derivative of the error function follows immediately from its definition:
<math display="block">\frac{d}{dz}\operatorname{erf}(z) =\frac{2}{\sqrt\pi} e^{-z^2}.</math>
From this, the derivative of the imaginary error function is also immediate:
<math display="block">\frac{d}{dz}\operatorname{erfi}(z) =\frac{2}{\sqrt\pi} e^{z^2}.</math>Higher order derivatives are given by
<math display="block">\operatorname{erf}^{(k)}(z) = \frac{2 (-1)^{k-1{\sqrt\pi} \mathit{H}_{k-1}(z) e^{-z^2} = \frac{2}{\sqrt\pi} \frac{d^{k-1{dz^{k-1 \big(e^{-z^2}\big),</math>
where the <math>H_k</math> are the physicists' Hermite polynomials.
An antiderivative of the error function, obtainable by integration by parts, is
<math display="block">\int \operatorname{erf}(z) dz = z\operatorname{erf}(z) + \frac{e^{-z^2{\sqrt\pi}+C.</math>
An antiderivative of the imaginary error function, also obtainable by integration by parts, is
<math display="block">\int \operatorname{erfi}(z) dz = z\operatorname{erfi}(z) - \frac{e^{z^2{\sqrt\pi}+C.</math>
Bürmann series
An expansion which converges more rapidly for all real values of <math>x</math> than a Taylor expansion is obtained by using Bürmann's theorem:
<math display="block">\begin{align}
\operatorname{erf}(x)
&= \frac{2}{\sqrt\pi} \sgn(x) \cdot \sqrt{1-e^{-x^2 \left( 1-\frac{1}{12} \left (1-e^{-x^2} \right ) -\frac{7}{480} \left (1-e^{-x^2} \right )^2 -\frac{5}{896} \left (1-e^{-x^2} \right )^3 - \cdots \right) \\[10pt]
&= \frac{2}{\sqrt\pi} \sgn(x) \cdot \sqrt{1-e^{-x^2 \left(\frac{\sqrt\pi}{2} + \sum_{k=1}^\infty c_k e^{-kx^2} \right)
\end{align}</math>
where <math>\operatorname{sgn}</math> is the sign function. By keeping only the first two coefficients and choosing <math>c_1=31/200</math> and <math>c_2=-341/8000</math>, the resulting approximation shows its largest relative error at <math>x=\pm 1.40587</math>, where it is less than <math>0.0034361</math>:
<math display="block">\operatorname{erf}(x) \approx \frac{2}{\sqrt\pi}\sgn(x) \cdot \sqrt{1-e^{-x^2 \left(\frac{\sqrt{\pi{2} + \frac{31}{200}e^{-x^2}-\frac{341}{8000} e^{-2x^2}\right). </math>
Inverse functions
thumb|300px|Inverse error function
Given a complex number <math>z</math>, there is not a unique complex number <math>w</math> satisfying <math>\operatorname{erf}(w)=z</math>, so a true inverse function would be multivalued. However, for <math>-1<x<1</math>, there is a unique real number denoted <math>\operatorname{erf}^{-1}(x)</math> satisfying
:<math>\operatorname{erf}\left(\operatorname{erf}^{-1}(x)\right) =x.</math>
The inverse error function is usually defined with domain <math>(-1,1)</math>, and it is restricted to this domain in many computer algebra systems. However, it can be extended to the disk <math>|z|<1</math> of the complex plane, using the Maclaurin series
<math display="block">\operatorname{erf}^{-1}(z)=\sum_{k=0}^\infty\frac{c_k}{2k+1}\left (\frac{\sqrt\pi}{2}z\right )^{2k+1},</math>
where <math>c_0=1</math> and
<math display="block">\begin{align}
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{align}</math>
So we have the series expansion (common factors have been canceled from numerators and denominators):
:<math>\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 ).</math>
(After cancellation the numerator and denominator values in and respectively; without cancellation the numerator terms are values in .) The error function's value at <math>\pm\infty</math> is equal to <math>\pm 1</math>.
For <math>|z|<1</math>, we have <math>\operatorname{erf}(\operatorname{erf}^{-1}(z))=z</math>.
The inverse complementary error function is defined as
<math display="block">\operatorname{erfc}^{-1}(1-z) = \operatorname{erf}^{-1}(z).</math>
For real <math>x</math>, there is a unique real number <math>\operatorname{erfi}^{-1}(x)</math> satisfying <math>\operatorname{erfi}(\operatorname{erfi}^{-1}(x))=x</math>. The inverse imaginary error function is defined as <math>\operatorname{erfi}^{-1}(x)</math>.
For any real <math>x</math>, Newton's method can be used to compute <math>\operatorname{erfi}^{-1}(x)</math>, and for <math>-1 \leq x \leq 1</math>, the following Maclaurin series converges:
:<math>\operatorname{erfi}^{-1}(z) =\sum_{k=0}^\infty\frac{(-1)^k c_k}{2k+1} \left( \frac{\sqrt\pi}{2} z \right)^{2k+1},</math>
where <math>c_k</math> is defined as above.
Asymptotic expansion
A useful asymptotic expansion of the complementary error function (and therefore also of the error function) for large real <math>x</math> is
<math display="block">\begin{align}
\operatorname{erfc}(x) &= \frac{e^{-x^2{x\sqrt{\pi\left(1 + \sum_{n=1}^\infty (-1)^n \frac{1\cdot3\cdot5\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{align}</math>
where <math>(2n-1)!!</math> is the double factorial of <math>2n-1</math>, i.e. the product of all odd numbers up to <math>2n-1</math>. This series diverges for every finite <math>x</math>, and its meaning as asymptotic expansion is that for any integer <math>N\geq 1</math> one has
:<math>\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),</math>
where the remainder is
:<math>R_N(x) := \frac{(-1)^N \, (2 N - 1)!!}{\sqrt{\pi} \cdot 2^{N - 1 \int_x^\infty t^{-2N}e^{-t^2}\, dt,</math>
which follows easily by induction, writing
:<math>e^{-t^2} = -\frac{1}{2 t} \, \frac{d}{dt} e^{-t^2}</math>
and integrating by parts. The asymptotic behavior of the remainder term is
:<math>R_N(x) = O\Big(x^{- (1 + 2N)} e^{-x^2}\Big)</math>
as <math>x\to\infty</math>. This can be found by
:<math>R_N(x) \propto \int_x^\infty t^{-2N}e^{-t^2}\, dt = e^{-x^2} \int_0^\infty (t+x)^{-2N}e^{-t^2-2tx}\,dt\leq e^{-x^2} \int_0^\infty x^{-2N} e^{-2tx}\,dt \propto x^{-(1+2N)}e^{-x^2}.</math>
For large enough values of <math>x</math>, only the first few terms of this asymptotic expansion are needed to obtain a good approximation of <math>\operatorname{erfc}(x)</math> (while for not too large values of <math>x</math>, the above Taylor expansion at 0 provides a very fast convergence).
Continued fraction expansion
A continued fraction expansion of the complementary error function was found by Laplace:
<math display="block">\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</math>
where <math>a_m = \frac{m}{2}</math>.
Factorial series
The inverse factorial series
:<math>\begin{align}
\operatorname{erfc}(z)
&= \frac{e^{-z^2{\sqrt{\pi}\,z} \sum_{n=0}^\infty \frac{\left(-1\right)^n Q_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{align}</math>
converges for <math>\operatorname{Re}(z^2)>0</math>. Here
:<math>\begin{align} Q_n
&=
\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{align}</math>
where <math>z^{\bar{n</math> denotes the rising factorial, and <math>s(n,k)</math> denotes a signed Stirling number of the first kind.
The Taylor series can be written in terms of the double factorial:
:<math>\operatorname{erf}(z) = \frac{2}{\sqrt\pi} \sum_{n=0}^\infty \frac{(-2)^n(2n-1)!!}{(2n+1)!}z^{2n+1}.</math>
Bounds and numerical approximations
Approximation with elementary functions
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:
<math display="block">\operatorname{erf}(x) \approx 1 - \frac{1}{\left(1 + a_1x + a_2x^2 + a_3x^3 + a_4x^4\right)^4}, \qquad x \geq 0</math>
(maximum error: )
where , , ,
<math display="block">\operatorname{erf}(x) \approx 1 - \left(a_1t + a_2t^2 + a_3t^3\right)e^{-x^2},\quad t=\frac{1}{1 + px}, \qquad x \geq 0</math>
(maximum error: )
where , , ,
<math display="block">\operatorname{erf}(x) \approx 1 - \frac{1}{\left(1 + a_1x + a_2x^2 + \cdots + a_6x^6\right)^{16, \qquad x \geq 0</math>
(maximum error: )
where , , , , ,
<math display="block">\operatorname{erf}(x) \approx 1 - \left(a_1t + a_2t^2 + \cdots + a_5t^5\right)e^{-x^2},\quad t = \frac{1}{1 + px}</math>
(maximum error: )
where , , , , ,
One can improve the accuracy of the A&S approximation by extending it with three extra parameters,
<math display="block">\operatorname{erf}(x) \approx 1 - \left(a_1t + a_2t^2 + \cdots + a_5t^5+a_6t^6+a_7t^7\right)e^{-x^2},\quad t = \frac{1}{1 + p_1x+p_2x^2}</math>
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 about . The parameters are obtained by fitting the extended approximation to the accurate values of the error function using the following Python code.
<syntaxhighlight lang="python">
import numpy as np
from math import erf, exp, sqrt
from scipy.optimize import least_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].
def approx_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^6
for ak in a:
poly += ak * tt
tt *= t
return 1.0 - t * np.exp(-x * x) * poly
def residuals(params, xs, ys):
return approx_erf(params, xs) - ys
- Prepare data for fitting
N = 300
xmin = 0
xmax = 10
xs = np.linspace(xmin, xmax, N)
ys = np.array([erf(x) for x in xs], dtype=float)
- Initial guess for parameters
- Start from original A&S values and extend them conservatively
p1_0 = 0.3275911 # original A&S p
p2_0 = 0.0 # new denominator parameter
- original A&S 5 coefficients, add two => 7 in total
a0 = [
0.254829592,
-0.284496736,
1.421413741,
-1.453152027,
1.061405429,
0.0, # new term
0.0, # 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.x
p1_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},")
for i, ai in enumerate(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.")
</syntaxhighlight>
All of these approximations are valid for . To use these approximations for negative , use the fact that is an odd function, so .
Exponential bounds and a pure exponential approximation for the complementary error function are given by
<math display="block">\begin{align}
\operatorname{erfc}(x) &\leq \frac{1}{2}e^{-2 x^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{align}</math>
The above have been generalized to sums of exponentials with increasing accuracy in terms of so that can be accurately approximated or bounded by , where
<math display="block">\tilde{Q}(x) = \sum_{n=1}^N a_n e^{-b_n x^2}.</math>
In particular, there is a systematic methodology to solve the numerical coefficients that yield a minimax approximation or bound for the closely related Q-function: , , or for . The coefficients for many variations of the exponential approximations and bounds up to have been released to open access as a comprehensive dataset.
A tight approximation of the complementary error function for is given by Karagiannidis & Lioumpas (2007), who showed for the appropriate choice of parameters that
<math display="block">\operatorname{erfc}(x) \approx \frac{\left(1 - e^{-Ax}\right)e^{-x^2{B\sqrt{\pi} x}.</math>
They determined , which gave a good approximation for all . Alternative coefficients are also available for tailoring accuracy for a specific application or transforming the expression into a tight bound.
A single-term lower bound is
<math display="block" display="block">\operatorname{erfc}(x) \geq \sqrt{\frac{2 e}{\pi \frac{\sqrt{\beta - 1{\beta} e^{- \beta x^2}, \qquad x \ge 0,\quad \beta > 1,</math>
where the parameter can be picked to minimize error on the desired interval of approximation.
Another approximation is given by Sergei Winitzki using his "global Padé approximations":
<math display="block">\operatorname{erf}(x) \approx \sgn x \cdot \sqrt{1 - \exp\left(-x^2\frac{\frac{4}{\pi} + ax^2}{1 + ax^2}\right)}</math>
where
<math display="block">a = \frac{8(\pi - 3)}{3\pi(4 - \pi)} \approx 0.140012.</math>
This is designed to be very accurate in the neighborhoods of 0 and infinity, and the relative error is less than 0.00035 for all real . Using the alternate value reduces the maximum relative error to about 0.00013.
The extended "global Pade" approximation,
<math display="block">\operatorname{erf}(x) \approx \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)}\,,</math>
provides a maximum error of about , as demonstrated by the following Python script.
<syntaxhighlight lang="python">
import numpy,math
from scipy.optimize import least_squares
- approximation to erf(x)
def approx_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)
return numpy.sign(x)*numpy.sqrt(
1-numpy.exp(-x*x*frac))
def residuals(params, xs, ys):
return approx_erf(params, xs) - ys
- data for fitting
N = 200
xmin = 0
xmax = 9
xs = numpy.linspace(xmin, xmax, N)
ys = numpy.array([math.erf(x) for x in xs], dtype=float)
params0 = numpy.array([0.9,0.1,0.008,0.8,0.1,0.008], dtype=float)
- fitting
result = 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 parameters
print("\nFitted parameters:")
for i, pi in enumerate(params, 0):
print(f"p{i} = {pi:.15g},")
- evaluate approximation error
approx_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.")
</syntaxhighlight>
Winitzki's approximation can be inverted to obtain an approximation for the inverse error function:
<math display="block">\operatorname{erf}^{-1}(x) \approx \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)}.</math>
An approximation with a maximal error of for any real argument is:
<math display="block">\begin{align}
\operatorname{erf}(x) &= \begin{cases}
1-\tau, & x\ge 0\\
\tau-1, & x < 0
\end{cases}\\
\tau &= t\cdot\exp\left(-x^2-1.26551223+1.00002368 t+0.37409196 t^2+0.09678418 t^3 -0.18628806 t^4\right.\\
&\left. \qquad\qquad\qquad +0.27886807 t^5-1.13520398 t^6+1.48851587 t^7 -0.82215223 t^8+0.17087277 t^9\right)\\
t &= \frac{1}{1 + \frac{1}{2}|x|}
\end{align}</math>
An approximation of <math>\operatorname{erfc}</math> with a maximum relative error less than <math>2^{-53}</math> <math>\left(\approx 1.1 \times 10^{-16}\right)</math> in absolute value is:
for
<math display="block">\begin{aligned}
\operatorname{erfc} \left(x\right)
& =
\left(\frac{0.56418958354775629}{x+2.06955023132914151}\right) \left(\frac{x^2+2.71078540045147805 x+5.80755613130301624}{x^2+3.47954057099518960 x+12.06166887286239555}\right) \\ & \left(\frac{x^2+3.47469513777439592 x+12.07402036406381411}{x^2+3.72068443960225092 x+8.44319781003968454}\right)
\left(\frac{x^2+4.00561509202259545 x+9.30596659485887898}{x^2+3.90225704029924078 x+6.36161630953880464}\right) \\
& \left(\frac{x^2+5.16722705817812584 x+9.12661617673673262}{x^2+4.03296893109262491 x+5.13578530585681539}\right)
\left(\frac{x^2+5.95908795446633271 x+9.19435612886969243}{x^2+4.11240942957450885 x+4.48640329523408675}\right) e^{-x^2} \\
\end{aligned}</math>
and for <math>x<0</math>
<math display="block">\operatorname{erfc} \left(x\right) = 2 - \operatorname{erfc} \left(-x\right)</math>
A simple approximation for real-valued arguments can be done through hyperbolic functions:
<math display="block">\operatorname{erf} \left(x\right) \approx z(x) = \tanh\left(\frac{2}{\sqrt{\pi\left(x+\frac{11}{123}x^3\right)\right)</math>
which keeps the absolute difference
Since the error function and the Gaussian Q-function are closely related through the identity <math>\operatorname{erfc}(x) = 2 Q(\sqrt{2} x)</math> or equivalently <math>Q(x) = \frac{1}{2} \operatorname{erfc}\left(\frac{x}{\sqrt{2\right)</math>, 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 <math>x \in [0, \infty)</math> was introduced by Abreu (2012) based on a simple algebraic expression with only two exponential terms:
<math display="block">\begin{align}
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{align}</math>
These bounds stem from a unified form <math display="block">Q_{\mathrm{B(x; a, b) = \frac{\exp(-x^2)}{a} + \frac{\exp(-x^2 / 2)}{b (x + 1)},</math> where the parameters <math>a</math> and <math>b</math> are selected to ensure the bounding properties: for the lower bound, <math>a_{\mathrm{L = 12</math> and <math>b_{\mathrm{L = \sqrt{2\pi}</math>, and for the upper bound, <math>a_{\mathrm{U = 50</math> and <math>b_{\mathrm{U = 2</math>.
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 <math>Q^n(x)</math> for positive integers <math>n</math> via the binomial theorem, suggesting potential adaptability for powers of <math>\operatorname{erfc}(x)</math>, though this is less commonly required in error function applications.
Table of values
{| class="wikitable" style="text-align:left;margin-left:24pt"
! !! !!
|-
|0 || ||
|-
|0.02|| ||
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|0.04|| ||
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|0.06|| ||
|-
|0.08|| ||
|-
|0.1 || ||
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|0.2 || ||
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|0.3 || ||
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|0.4 || ||
|-
|0.5 || ||
|-
|0.6 || ||
|-
|0.7 || ||
|-
|0.8 || ||
|-
|0.9 || ||
|-
|1 || ||
|-
|1.1 || ||
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|1.2 || ||
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|1.3 || ||
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|1.4 || ||
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|1.5 || ||
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|1.6 || ||
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|1.7 || ||
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|1.8 || ||
|-
|1.9 || ||
|-
|2 || ||
|-
|2.1 || ||
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|2.2 || ||
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|2.3 || ||
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|2.4 || ||
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|2.5 || ||
|-
|3 || ||
|-
|3.5 || ||
|}
Related functions
Complementary error function
thumb|Plot of the error function erf(z) in the complex plane from to
The complementary error function, denoted , is defined as
<math display="block">\begin{align}
\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{align}</math>
which also defines , the scaled complementary error function (which can be used instead of to avoid arithmetic underflow). Another form of for is known as Craig's formula, after its discoverer:
<math display="block">
\operatorname{erfc} (x \mid x\ge 0) =
\frac{2}{\pi} \int_0^\frac{\pi}{2} \exp\left(-\frac{x^2}{\sin^2 \theta}\right) \,d\theta.
</math>
This expression is valid only for positive values of , but can be used in conjunction with to obtain for negative values. This form is advantageous in that the range of integration is fixed and finite. An extension of this expression for the of the sum of two non-negative variables is
<math display="block">
\operatorname{erfc}(x + y \mid x, y \ge 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.
</math>
Imaginary error function
thumb|Plot of the imaginary error function erfi(z) in the complex plane from to
The imaginary error function, denoted , is defined as
<math display="block">\begin{align}
\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{align}</math>
where is the Dawson function (which can be used instead of to avoid arithmetic overflow
<math display="block">\begin{align}
i^n\!\operatorname{erfc}(z) &= \int_z^\infty i^{n-1}\!\operatorname{erfc}(\zeta)\,d\zeta \\[6pt]
i^0\!\operatorname{erfc}(z) &= \operatorname{erfc}(z) \\
i^1\!\operatorname{erfc}(z) &= \operatorname{ierfc}(z) = \frac{1}{\sqrt\pi} e^{-z^2} - z \operatorname{erfc}(z) \\
i^2\!\operatorname{erfc}(z) &= \tfrac{1}{4} \left( \operatorname{erfc}(z) -2 z \operatorname{ierfc}(z) \right) \\
\end{align}</math>
The general recurrence formula is
<math display="block">2 n \cdot i^n\!\operatorname{erfc}(z) = i^{n-2}\!\operatorname{erfc}(z) -2 z \cdot i^{n-1}\!\operatorname{erfc}(z)</math>
They have the power series
<math display="block">i^n\!\operatorname{erfc}(z) =\sum_{j=0}^\infty \frac{(-z)^j}{2^{n-j}j! \,\Gamma \left( 1 + \frac{n-j}{2}\right)},</math>
from which follow the symmetry properties
<math display="block">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)!}</math>
and
<math display="block">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)!}.
</math>
Implementations
As real function of a real argument
- In POSIX-compliant operating systems, the header <code>math.h</code> shall declare and the mathematical library <code>libm</code> shall provide the functions <code>erf</code> and <code>erfc</code> (double precision) as well as their single precision and extended precision counterparts <code>erff</code>, <code>erfl</code> and <code>erfcf</code>, <code>erfcl</code>.
- The GNU Scientific Library provides <code>erf</code>, <code>erfc</code>, <code>log(erf)</code>, and scaled error functions.
As complex function of a complex argument
- <code>libcerf</code>, numeric C library for complex error functions, provides the complex functions <code>cerf</code>, <code>cerfc</code>, <code>cerfcx</code> and the real functions <code>erfi</code>, <code>erfcx</code> with approximately 13–14 digits precision, based on the Faddeeva function as implemented in the MIT Faddeeva Package
Notes
References
Further reading
External links
- A Table of Integrals of the Error Functions