Editing Error function (section)

===Approximation with elementary functions===
<ul>
<li>
[[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: {{val|5e-4}})
{{pb}}
where {{math|''a''<sub>1</sub> {{=}} 0.278393}}, {{math|''a''<sub>2</sub> {{=}} 0.230389}}, {{math|''a''<sub>3</sub> {{=}} 0.000972}}, {{math|''a''<sub>4</sub> {{=}} 0.078108}}

<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: {{val|2.5e-5}})
{{pb}}
where {{math|''p'' {{=}} 0.47047}}, {{math|''a''<sub>1</sub> {{=}} 0.3480242}}, {{math|''a''<sub>2</sub> {{=}} −0.0958798}}, {{math|''a''<sub>3</sub> {{=}} 0.7478556}}

<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: {{val|3e-7}})
{{pb}}
where {{math|''a''<sub>1</sub> {{=}} 0.0705230784}}, {{math|''a''<sub>2</sub> {{=}} 0.0422820123}}, {{math|''a''<sub>3</sub> {{=}} 0.0092705272}}, {{math|''a''<sub>4</sub> {{=}} 0.0001520143}}, {{math|''a''<sub>5</sub> {{=}} 0.0002765672}}, {{math|''a''<sub>6</sub> {{=}} 0.0000430638}}

<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: {{val|1.5e-7}})
{{pb}}
where {{math|''p'' {{=}} 0.3275911}}, {{math|''a''<sub>1</sub> {{=}} 0.254829592}}, {{math|''a''<sub>2</sub> {{=}} −0.284496736}}, {{math|''a''<sub>3</sub> {{=}} 1.421413741}}, {{math|''a''<sub>4</sub> {{=}} −1.453152027}}, {{math|''a''<sub>5</sub> {{=}} 1.061405429}}
{{pb}}
All of these approximations are valid for {{math|''x'' ≥ 0}}.  To use these approximations for negative {{mvar|x}}, use the fact that {{math|erf ''x''}} is an odd function, so {{math|erf ''x'' {{=}} −erf(−''x'')}}.
</li>

<li>
Exponential bounds and a pure exponential approximation for the complementary error function are given by<ref>{{cite journal |url = http://campus.unibo.it/85943/1/mcddmsTranWIR2003.pdf |last1= Chiani|first1= M.|last2= Dardari|first2= D. |last3=Simon |first3= M.K.|date = 2003 |title = New Exponential Bounds and Approximations for the Computation of Error Probability in Fading Channels|journal = IEEE Transactions on Wireless Communications|volume = 2|number=4|pages = 840–845| doi=10.1109/TWC.2003.814350 | citeseerx= 10.1.1.190.6761}}</ref>
<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}, &\quad x &> 0 \\[1.5ex]
  \operatorname{erfc} x &\approx \frac{1}{6}e^{-x^2} + \frac{1}{2}e^{-\frac{4}{3} x^2}, &\quad x &> 0 .
\end{align}</math>
</li>

<li>
The above have been generalized to sums of {{mvar|N}} exponentials<ref>{{cite journal |doi=10.1109/TCOMM.2020.3006902 |title=Global minimax approximations and bounds for the Gaussian Q-function by sums of exponentials|journal=IEEE Transactions on Communications |year=2020 |last1=Tanash |first1=I.M. |last2=Riihonen |first2=T. |volume=68 |issue=10 |pages=6514–6524 |arxiv=2007.06939 |s2cid=220514754}}</ref> with increasing accuracy in terms of {{mvar|N}} so that {{math|erfc ''x''}} can be accurately approximated or bounded by {{math|2''Q̃''({{sqrt|2}}''x'')}}, 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 {{math|{(''a<sub>n</sub>'',''b<sub>n</sub>'')}{{su|b=''n'' {{=}} 1|p=''N''}}}} that yield a [[minimax approximation algorithm|minimax]] approximation or bound for the closely related [[Q-function]]: {{math|''Q''(''x'') ≈ ''Q̃''(''x'')}}, {{math|''Q''(''x'') ≤ ''Q̃''(''x'')}}, or {{math|''Q''(''x'') ≥ ''Q̃''(''x'')}} for {{math|''x'' ≥ 0}}. The coefficients {{math|{(''a<sub>n</sub>'',''b<sub>n</sub>'')}{{su|b=''n'' {{=}} 1|p=''N''}}}} for many variations of the exponential approximations and bounds up to {{math|''N'' {{=}} 25}} have been released to open access as a comprehensive dataset.<ref>{{cite journal | doi=10.5281/zenodo.4112978 | title=Coefficients for Global Minimax Approximations and Bounds for the Gaussian Q-Function by Sums of Exponentials [Data set] | url=https://zenodo.org/record/4112978 | website=Zenodo | year=2020 | last1=Tanash | first1=I.M. | last2=Riihonen | first2=T.}}</ref></li>

<li>
A tight approximation of the complementary error function for {{math|''x'' ∈ [0,∞)}} is given by [[George Karagiannidis|Karagiannidis]] & Lioumpas (2007)<ref>{{cite journal|last1=Karagiannidis |first1=G. K. |last2=Lioumpas |first2=A. S. |url=http://users.auth.gr/users/9/3/028239/public_html/pdf/Q_Approxim.pdf |title=An improved approximation for the Gaussian Q-function |date=2007 |journal=IEEE Communications Letters |volume=11 |issue=8 |pages=644–646|doi=10.1109/LCOMM.2007.070470 |s2cid=4043576 }}</ref> who showed for the appropriate choice of parameters {{math|{''A'',''B''}<nowiki/>}} that
<math display="block">\operatorname{erfc} x \approx \frac{\left(1 - e^{-Ax}\right)e^{-x^2}}{B\sqrt{\pi} x}.</math>
They determined {{math|{''A'',''B''} {{=}} {1.98,1.135}<nowiki/>}}, which gave a good approximation for all {{math|''x'' ≥ 0}}. Alternative coefficients are also available for tailoring accuracy for a specific application or transforming the expression into a tight bound.<ref>{{cite journal |doi=10.1109/LCOMM.2021.3052257|title=Improved coefficients for the Karagiannidis–Lioumpas approximations and bounds to the Gaussian Q-function|journal=IEEE Communications Letters | year=2021 | last1=Tanash | first1=I.M.|last2=Riihonen|first2=T.|volume=25|issue=5|pages=1468–1471|arxiv=2101.07631|s2cid=231639206}}</ref>
</li>

<li>
A single-term lower bound is<ref>{{cite journal |last1=Chang |first1=Seok-Ho |last2=Cosman |first2=Pamela C. |author-link2 = Pamela Cosman |last3=Milstein |first3=Laurence B. |date=November 2011 |title=Chernoff-Type Bounds for the Gaussian Error Function |url=http://escholarship.org/uc/item/6hw4v7pg |journal=IEEE Transactions on Communications |volume=59 |issue=11 |pages=2939–2944 |doi=10.1109/TCOMM.2011.072011.100049 |s2cid=13636638}}</ref>
<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 {{mvar|β}} can be picked to minimize error on the desired interval of approximation.
</li>

<li>
Another approximation is given by Sergei Winitzki using his "global Padé approximations":<ref>{{cite book |last=Winitzki |first=Sergei |title=Computational Science and Its Applications – ICCSA 2003 |date=2003  |volume=2667 |chapter=Uniform approximations for transcendental functions |publisher=Springer, Berlin |pages=[https://archive.org/details/computationalsci0000iccs_a2w6/page/780 780–789] |isbn=978-3-540-40155-1 |doi=10.1007/3-540-44839-X_82 |chapter-url-access=registration |chapter-url=https://archive.org/details/computationalsci0000iccs_a2w6 |series=Lecture Notes in Computer Science }}</ref><ref>{{cite journal|last1=Zeng |first1=Caibin |last2=Chen |first2=Yang Cuan |title=Global Padé approximations of the generalized Mittag-Leffler function and its inverse |journal=Fractional Calculus and Applied Analysis |date=2015 |volume=18 |issue=6 | pages=1492–1506 |doi= 10.1515/fca-2015-0086 |quote=Indeed, Winitzki [32] provided the so-called global Padé approximation | arxiv=1310.5592 |s2cid=118148950 }}</ref>{{rp|2–3}}
<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 a neighborhood of 0 and a neighborhood of infinity, and the ''relative'' error is less than 0.00035 for all real {{mvar|x}}. Using the alternate value {{math|''a'' ≈ 0.147}} reduces the maximum relative error to about 0.00013.<ref>{{Cite web <!-- Deny Citation Bot--> |url=https://www.academia.edu/9730974/A_handy_approximation_for_the_error_function_and_its_inverse |last=Winitzki |first=Sergei |date=6 February 2008 |title=A handy approximation for the error function and its inverse }}</ref>
{{pb}}
This 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>
</li>

<li>
An approximation with a maximal error of {{val|1.2e-7}} for any real argument is:<ref>{{cite book | last = Press | first = William H. | title = Numerical Recipes in Fortran 77: The Art of Scientific Computing | isbn = 0-521-43064-X | year = 1992 | page = 214 | publisher = Cambridge University Press }}</ref>
<math display="block">\operatorname{erf} x = \begin{cases}
1-\tau &  x\ge 0\\
\tau-1 &  x < 0
\end{cases}</math>
with
<math display="block">\begin{align}
\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)
\end{align}</math>
and
<math display="block">t = \frac{1}{1 + \frac{1}{2}|x|}.</math>
</li>

<li>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:<ref>{{Cite journal | last = Dia | first = Yaya D. |date = 2023 | title = Approximate Incomplete Integrals, Application to Complementary Error Function | url = https://www.ssrn.com/abstract=4487559 | journal = SSRN Electronic Journal | language = en | doi = 10.2139/ssrn.4487559 | issn = 1556-5068}}</ref>
for {{nowrap|<math>x\ge 0</math>,}}
<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>
</li>
<li>
A simple approximation for real-valued arguments could 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 {{nowrap|<math>\left|\operatorname{erf} \left(x\right)-z(x)\right| < 0.000358,\, \forall x</math>.}}
</li>
<li>
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)<ref>{{cite journal |doi=10.1109/TCOMM.2012.080612.110075 |title=Very Simple Tight Bounds on the Q-Function |journal=IEEE Transactions on Communications |volume=60 |issue=9 |pages=2415–2420 |year=2012 |last=Abreu |first=Giuseppe}}</ref> based on a simple algebraic expression with only two exponential terms:
<math display="block">Q(x) \geq \frac{1}{12} e^{-x^2} + \frac{1}{\sqrt{2\pi} (x + 1)} e^{-x^2 / 2}, \qquad x \geq 0,</math>
and
<math display="block">Q(x) \leq \frac{1}{50} e^{-x^2} + \frac{1}{2 (x + 1)} e^{-x^2 / 2}, \qquad x \geq 0.</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.
</li>
</ul>