Distributions

The approximations in this section, all have the format of:

\(\Phi\left(z\right) \approx \frac{1}{2}\times\left(1 + \sqrt{1 - \alpha}\right)\)

They come from the relation between the sncdf and the error function. The error function is defined as:

\( \text{Erf}\left(x\right) = \frac{2}{\sqrt{\pi}}\int_{i=0}^x e^{-i^2} \)

This can then be converted to the sncdf by using:

\( \Phi\left(z\right) = \frac{1 + \text{Erf}\left(\frac{z}{\sqrt{2}}\right)}{2}\)

One that is often referenced is Polya (1949), although this article was from a conference that was held in 1946. The same approximation was also published by Williams (1946) who gets far less often referenced. The approximation:

\(\Phi\left(z\right) \approx \frac{1}{2}\times\left(1 + \sqrt{1 - e^{-\frac{2\times z^2}{\pi}}}\right)\)

Cadwell (1951) improved on this using:

\(\Phi\left(z\right) \approx \frac{1}{2}\times\left(1 + \sqrt{1 - e^{-\frac{2\times z^2}{\pi}+ \frac{2\times\left(\pi - 3\right)\times z^4}{3\times\pi^2}} }\right)\)

Cadwell also noted that since the above formula was not asymptotically correct another adjustment for Polya/Williams:

\(\Phi\left(z\right) \approx \frac{1}{2}\times\left(1 + \sqrt{1 - e^{-\frac{2\times z^2}{\pi}} - 0.0005\times z^6 + 0.00002\times z^8}\right)\)

Strecok (1968) uses the error function as well, but with a summation and sinus operator. See the Single 'simple' equations - With Sums section for this approach

Hamaker (1978) continued on the work from Page (1977) and Zelen and Severo, but then actually came up with a formula very similar to the ones above.

\(\Phi\left(z\right) \approx \frac{1}{2}\times\left(1 + \text{sign}\left(z\right)\times\sqrt{1 - e^{-\left(0.806\times \left|z\right|\times\left(1 -\times0.018\times\left|z\right|\right)\right)^2}}\right)\)

Lin (1988) further improved on Hamaker's equation:

\(\Phi\left(z\right) \approx \frac{1}{2}\times\left(1 + \sqrt{1 - e^{-\left(\frac{z}{1.237+0.0249\times z}\right)^2}}\right)\)

Aludaat and Alodat (2008) looked at Polya, Chokri, Bailey and Johnson, but found that the equations were either too complicated or that the inverse was complicated or impossible. They proposed:

\(\Phi\left(z\right) \approx \frac{1}{2}\times\left(1 + \sqrt{1 - e^{-\sqrt{\frac{\pi}{8}}\times z^2}}\right)\)

Winitzki (2008) proposes:

\(\Phi\left(z\right) \approx \frac{1}{2}\times\left(1 + \sqrt{1 - e^{-\frac{z^2}{2}\times\frac{\frac{8}{\pi}+0.147\times z^2}{2+0.147\times z^2}}}\right)\)

Eidous and Al-Salman (2016) noted that Aludaat and Alodat's improvement on Polya was not fully proven to be an improvement as an upper bound and improved on it themselves by adjusting it to:

\(\Phi\left(z\right) \approx \frac{1}{2}\times\left(1 + \sqrt{1 - e^{-\frac{5}{8}\times z^2}}\right)\)

Note that they also shown that this is an improvement over Bowling et. al () and Lin (), which will be discussed later. The maximum absolute error was stated to be 0.00180786.

Abderrahmane and Boukhetala (2016) also looked at Aludaat and Alodat's improvement of Polya and improved on it themselves to:

\(\Phi\left(z\right) \approx \frac{1}{2}\times\left(1 + \sqrt{1 - e^{-0.62306179\times z^2}}\right)\)

They don't seem to have considered Eidos and Al-Salman's version, however they state that their maximum absolute error is 0.001622801925711, slightly less than the other. Note that they also made another approximation based on a different one, to be discussed later below.

Matić, Radoičić, and Stefanica (2018) compared many different approximations, and gave the following of their own:

\(\Phi\left(z\right) \approx \frac{1}{2}\times\left(1 + \sqrt{1-e^{-\frac{2\times z^2}{\pi}\times\left(1+\sum_{k=1}^5 a_{2\times k}\times z^{2\times k}\right)}}\right)\)

The approximations in this section all have a format of:

\(\Phi\left(z\right) \approx \frac{1}{1+e^{\alpha}}\)

Lin (1990) uses a logistic approximation on the previous section, specifically from Hamaker and Lin (1988). The approximation proposed is:

\(\Phi\left(z\right) \approx \frac{1}{1+e^{-\frac{4.2\times\pi\times z}{9-z}}}\)

Vedder (1993) improved on this with:

\(\Phi\left(z\right) \approx \frac{1}{1+e^{-\sqrt{\frac{8}{\pi}}\times z - \frac{4-\pi}{3\times\pi}\times\sqrt{\frac{2}{\pi}}\times z^3}}\)

Waissi and Rossin (1996) gave a three parameter approximation:

\(\Phi\left(z\right) \approx \frac{1}{1+e^{-\sqrt{\pi}\times\left(-0.0004406\times z^5+0.0418198\times z^3+0.9\times z\right)}}\)

Bowling et al. (2009) continued the work and gave two approximations. One with one parameter, and one with two. Bowling does not reference Waissi and Rossin nor Vedder.

\(\Phi\left(z\right) \approx \frac{1}{1+e^{-1.702\times z}}\)

\(\Phi\left(z\right) \approx \frac{1}{1+e^{-\left(0.07056\times z^3+1.5976\times z\right)}}\)

Mill's ratio is another function that is approximated from which the normal cdf can be approximated. These have the general form of:

\(\Phi\left(z\right) \approx 1-\phi\left(z\right)\times f\left(z\right)\)

Mill's ratio for the normal distribution can be defined as:

\(R\left(z\right) =e^{-z^2}\times\int_{i=z}^\infty e^{-\frac{i^2}{2}}\)

This can then be converted to the sncdf by using:

\( \Phi\left(z\right) = 1 -R\left(z\right)\times\phi\left(z\right)\)

Hastings (1955) looks at the error function but this can be converted to the Mill's ratio. Hasting proposes to use:

\(\text{erf}\left(x\right) \approx 1 - \frac{2}{\sqrt{\pi}}\times e^{-x^2}\times\sum_{i=1}^k a_i\times t^i\)

This can be re-written to the form of:

\(\Phi\left(z\right) \approx 1-\phi\left(z\right)\times \sqrt{2} \times \sum_{i=1}^k a_i\times t^i\)

With:

\(t = \frac{1}{p\times x}\)

\(x = \frac{z}{\sqrt{2}}\)

And Hastings gives three different sets for \(p\) and \(a_i\):

Version 1 - Sheet 43

\(p = 0.47047, a_{1\dots 3} = \left\{0.3084284,-0.0849713,0.6627698\right\}\)

Version 2 - Sheet 44

\(p = 0.381965, a_{1\dots 4} = \left\{0.12771538,0.54107939,-0.53859539,0.75602755\right\}\)

Version 3 - Sheet 45

\(p = 0.3275911, a_{1\dots 4} = \left\{0.225836846,-0.252128668,1.25969513,-1.287822453,0.94064607\right\}\)

Note: Version 1 and 3 can be found as well in Abramowitz and Stegun’s book, in chapter 7 written by Gautschi (1970) as Algorithm 7.1.25 and Algorithm 7.1.26 resp.. The values for a_i are multiplied by 2/√π, and the equation adjusted accordingly

Hart (1957) uses the Mill's Ratio, and proposed:

\(\Phi\left(z\right) \approx 1-\frac{\phi\left(z\right)}{z+0.8\times e^{-0.4\times z}}\)

Hart (1966) proposed:

\(\Phi\left(z\right) \approx 1 - \frac{\phi\left(z\right)}{z}\times\left(1 - \frac{\frac{\sqrt{1+c\times z^2}}{1+b\times z^2}}{a\times z+\sqrt{a^2\times z^2+e^{-\frac{z^2}{2}}}\times\frac{\sqrt{1+c\times z^2}}{1+b\times z^2}}\right)\)

with:

\(a = \sqrt{\frac{\pi}{2}} \)

\(b = \frac{1+\sqrt{1-2\times\pi^2+6\times\pi}}{2\times\pi} \)

\(c = 2\times\pi\times b^2\)

Schucany and Gray (1968) improved on Hart (1966), and proposed for z>2:

\(\Phi\left(z\right) \approx 1 - \frac{z\times\phi\left(z\right)}{z^2+2}\times\frac{z^6+6\times z^4-14\times z^2 - 28}{z^6+5\times z^4-20\times z^2 - 4}\)

Zelen and Severo (1970) for version 1 and 2:

\(\Phi\left(z\right) \approx 1 - \phi\left(z\right)\times\sum_{i=1}^k a_i\times\left(\frac{1}{1+p\times z}\right)^i\)

With for version 1:

\( p=0.33267, a_{k=3,1\dots3}\left\{0.4361836,-0.1201676,0.937298\right\} \)

and for version 2:

\( p=0.2316419, a_{k=5,1\dots5}\left\{0.319381530,-0.356563782,1.781477937,-1.821255978,1.330274429\right\} \)

Note that version 1 is also sometimes referenced as from Texas Instruments (1980, as cited in McConnell (1981))

Divgi (1979) also use the Mill's ratio with a summation over 11 values:

\(\Phi\left(z\right) \approx 1 - \phi\left(z\right)\times\sum_{i=0}^n a_i\times z^i\)

With:

\( a_{i=0\dots 10}\left\{1.253313402, -0.9999673043,0.6262972801, -0.3316218430, 0.1522723563, -5.982834993\times 10^{-2}, 1.915649350\times 10^{-2}, -4.644960579\times 10^{-3}, 7.771088713\times 10^{-4}, -7.830823677\times 10^{-5}, 3.534244658\times 10^{-6}\right\} \)

Bryc (2002) gave a more precise version of this as:

\(\Phi\left(z\right) \approx 1-\frac{\left(4-\pi\right)\times z+\sqrt{2\times\pi}\times\left(\pi-2\right)}{\left(4-\pi\right)\times z^2 + 2\times\left(\pi-2\right)+\sqrt{2\times\pi}\times z} \times\phi\left(z\right)\)

which got approximated to:

\(\Phi\left(z\right) \approx 1-\frac{z+0.333}{\sqrt{2\times\pi}\times z^2+7.32\times z + 6.666} \times e^{-\frac{z^2}{2}}\)

The author also proposed a second iteration for Mill's ratio and approximated that with:

\(\Phi\left(z\right) \approx 1-\frac{z^2+5.575192695\times z+12.77436324}{\sqrt{2\times\pi}\times z^3+14.38718147\times z^2 + 31.53531977\times z +25.54872648} \times e^{-\frac{z^2}{2}}\)

Choudhury (2014) compared various approximations, including Bryc's (but doesn't reference Hart) and derived:

\(\Phi\left(z\right) \approx 1-\frac{\phi\left(z\right)}{0.226+0.64\times z+0.33\times\sqrt{z^2+3}}\)

Yerukala and Boiroju (2015) second approximation, does not fit the general format of this section, but is a combination of Hart and Byrc's approximations:

\(\Phi\left(z\right) \approx 0.268\times\Phi_{Hart}\left(z\right) + 0.732\times\Phi_{Byrc}\left(z\right)\)

Yerukala and Boiroju (2015) third approximation is a modification of Choudhury (2014) (see error function):

\(\Phi\left(z\right) \approx 1 - \phi\left(z\right)\times\frac{\sqrt{2\times\pi}}{\frac{44}{79}+\frac{8}{5}\times z+\frac{5}{6}\times\sqrt{z^2+3}}\)

Abderrahmane and Boukhetala (2016) besides an approximation based on the error function, also gave an improvement on Hart's approximation as:

\(\Phi\left(z\right) \approx 1-\frac{\phi\left(z\right)}{z+0.79758\times e^{-0.4446\times z}}\)

A few approximation use trigonometry, with almost all using the hyperbolic tangent function (tanh).

The tanh approximations use:

\(\frac{e^{2\times\alpha}}{1+e^{2\times\alpha}} = \frac{1}{2}\times\left(1+\text{tanh}\left(\alpha\right)\right)\)

Raab and Green (1961) proposed:

\(\Phi\left(z\right) \approx = 1 - \frac{1+\cos\left(z\right)}{\pi}\times\frac{\pi - z -\sin\left(z\right)}{2\times\pi}\)

This approximation seems to be the worst of all of them, but is interestingly the only one with sinus and cosine.

Thocher (1963) proposed (re-written in tanh form):

\(\Phi\left(z\right) \approx \frac{1}{2}\times\left(1+\text{tanh}\left(\sqrt{\frac{2}{\pi}}\times z\right)\right)\)

Strecok (1968) uses the error function as well, but with a summation and sinus operator. See the Single 'simple' equations - With Sums section for this approach

Page (1977) improved on this with:

\(\Phi\left(z\right) \approx \frac{1}{2}\times\left(1+\text{tanh}\left(\sqrt{\frac{2}{\pi}}\times z\times\left(1+0.044715\times z^2\right)\right)\right)\)

Moran (1980) improved on the work from Strecok, also with a summation and sinus operator. See the Single 'simple' equations - With Sums section for this approach

Vedder (1993) also gave a tanh version of the approximation shown in the logistic section:

\(\Phi\left(z\right) \approx \frac{1}{2}\times\left(1+\text{tanh}\left(\frac{1}{2}\times z\times\left(\sqrt{\frac{8}{\pi}}+\frac{4-\pi}{3\times\pi}\times\sqrt{\frac{2}{\pi}}\times z^2\right)\right)\right)\)

Yun (2009) had the following approximation:

\(\Phi\left(z\right) \approx \frac{1}{2}\times\left(1 + \tanh\left(\frac{18.445}{20}\times\left(\frac{1}{\left(1-\frac{z}{\sqrt{\frac{\pi}{2}}\times18.445}\right)^{10}} - \frac{1}{\left(1+\frac{z}{\sqrt{\frac{\pi}{2}}\times18.445}\right)^{10}}\right)\right)\right)\)

Yun (2009) also simplified the approximation using the inverse of tanh

\(\Phi\left(z\right) \approx \frac{1}{2}\times\left(1+\text{tanh}\left( 2.5673\times\text{atanh}\left(\frac{z}{\sqrt{\frac{\pi}{2}}\times2.5673}\right)\right)\right)\)

Yerukala et al. (2011) used a neural network approach and found:

\(\Phi\left(z\right) \approx 0.5 - 1.136\times\text{tanh}\left(-0.2695\times z\right) + 2.47\times\text{tanh}\left(0.5416\times z\right) - 3.013\times\text{tanh}\left(0.4134\times z\right)\)

They tested this against Polya, Lin, Aludaat and Alodat, and Bowling, and found this outperforms those. The max absolute error was 0.0013 for z-values between 0 and 3.

Vazquez et al. (2012) derived an approximation which then got converted into fractions, and then to exponential:

\(\Phi\left(z\right) \approx \frac{1}{2} \times \left(1 + \text{tanh}\left(\frac{39\times z}{2\times\sqrt{2\times\pi}} - \frac{111}{2} \times \text{atan}\left(\frac{35\times z}{111\times\sqrt{2\times\pi}}\right)\right)\right)\)

\(\Phi\left(z\right) \approx \frac{1}{2} \times \left(1 + \text{tanh}\left(\frac{179\times z}{23} - \frac{111}{2} \times \text{atan}\left(\frac{37\times z}{294}\right)\right)\right)\)

\(\Phi\left(z\right) \approx \frac{1}{1+e^{-\frac{358\times z}{23}+111\times\text{tanh}\left(\frac{37\times z}{294}\right)}}\)

Yerukala and Boiroju (2015) propose:

\(\Phi\left(z\right) \approx \frac{1}{1+e^{-0.125-3.611\times\tanh\left(0.043+0.2624\times z\right)+4.658\times\tanh\left(-1.687-0.519\times z\right)-4.982\times\tanh\left(-1.654+0.5044\times z\right)}} \)

The approximations in this section all use a summation.

Hastings (1955) proposed

\(\Phi\left(z\right) \approx 1 - \frac{1}{2\times\left(1+\sum_{i=1}^k a_i\times\left(\frac{z}{\sqrt{2}}\right)^i\right)^{2^k -z}}\)

Hastings gave three different suggestions for \(a_i\):

\(a_{k=4, 1\dots4} = \left\{0.278393,0.230389,0.000972,0.078108\right\}\)

\(a_{k=5, 1\dots5} = \left\{0.14112821,0.08864027,0.02743349,0.00039446,0.00328975\right\}\)

\(a_{k=6, 1\dots6} = \left\{0.0705230784,0.0422820123,0.0092705272,0.0001520143,0.0002765672,0.000430638\right\}\)

Strecok (1968) uses the sinus to approximate the error function with:

\(\text{erf}\left(x\right) \approx \frac{2}{\pi}\times\left(\frac{x}{5}+\sum_{n=1}^{37} \frac{1}{n}\times e^{-\left(\frac{n}{5}\right)^2}\times\sin\left(\frac{2\times n\times x}{5}\right)\right)\)

Zelen and Severo (1970) for version 3 and 4:

\(\Phi\left(z\right) \approx 1 - \frac{1}{2\times\left(1+\sum_{i=1}^k a_i\times z^i\right)^{2^{k-2}}}\)

With for version 3:

\( a_{k=3,1\dots3}\left\{0.196854,0.115194,0.000344,0.019527\right\} \)

and for version 4:

\( a_{k=6,1\dots6}\left\{0.0498673470,0.0211410061,0.0032776263,0.000380036,0.0000488906,0.0000053830\right\} \)

Kerridge and Cook (1976) use the following infinite series:

\(\Phi\left(z\right) \approx 1 - \frac{1}{2} + \phi\left(z\right)\times z\times\sum_{i=0}^{\infty}\frac{a_{2\times i}}{2\times i + 1}\)

With the recursive function:

\( a_i = \frac{z^2}{4}\times\frac{a_i-a_{i-1}}{n+1} \)

\( a_0 = 1, a_1 = \frac{z^2}{4} \)

Moran (1980) continues on the work from Strecock (1968) and suggests two similar approximations. Version 1:

\(\Phi\left(z\right) \approx \frac{1}{2} + \frac{1}{\pi}\times\left(\frac{z}{3\times\sqrt{2}} + \sum_{n-1}^{\infty} \frac{e^{\frac{-n^2}{9}}}{n}\times\sin\left(\frac{n\times z\times\sqrt{2}}{3}\right)\right)\)

and version 2:

\(\Phi\left(z\right) \approx \frac{1}{2} + \frac{1}{\pi}\times\left(\sum_{n-1}^{\infty} \frac{e^{\frac{\left(n+\frac{1}{2}\right)^2}{9}}}{n+\frac{1}{2}}\times\sin\left(\frac{\left(n+\frac{1}{2}\right)\times z\times\sqrt{2}}{3}\right)\right)\)

Moran mentions that version 1 when n = 12 will be accurate to nine decimal places while z ≤ 7, and for version 2 when n = 7.

Abernathy (1988) uses a series expansion of \(e^x\) very similar to Mill's ratio:

\(\Phi\left(z\right) \approx \frac{1}{2} + \frac{1}{\sqrt{2\times\pi}}\times\sum_{n=0}^\infty \frac{\left(-1\right)^n\times z^{2\times n+1}}{2^n\times n!\times\left(2\times n + 1\right)}\)

Patry and Keller (1964) proposed (using the error function):

\(\Phi\left(z\right) \approx 1 - \frac{e^{-\frac{z^2}{2}}\times t}{2\times 1.77245385\times\left(\frac{z}{\sqrt{2}}\times t+1\right)}\)

with:

\(t = 1.77245385+\frac{z}{\sqrt{2}}\times\left(2-q\right) \)

\(q = \frac{0.858407657+\frac{z}{\sqrt{2}}\times\left(0.307818193+\frac{z}{\sqrt{2}}\times\left(0.0638323891-\frac{z}{\sqrt{2}}\times0.000182405075\right)\right)}{1+\frac{z}{\sqrt{2}}\times\left(0.650974265+\frac{z}{\sqrt{2}}\times\left(0.229484819+\frac{z}{\sqrt{2}}\times 0.0340301823\right)\right)}\)

Burr (1967) proposed:

\(\Phi_{\text{Burr}}\left(z\right) \approx 1 - \left(1 + \left(0.644693 + 0.161984\times z\right)^{4.874}\right)^{-6.158}\)

Version 2 was a modification using:

\(\Phi\left(z\right) \approx \frac{\Phi_{\text{Burr}}\left(z\right)+1-\Phi_{\text{Burr}}\left(-z\right)}{2}\)

Lin (1989) proposed:

\(\Phi\left(z\right) \approx 1 - \frac{e^{-0.717\times z - 0.416\times z^2}}{2}\)

This came from using an ordinary least square to find the optimal solution for values using:

\( 1 - \Phi\left(z\right) \approx \frac{e^{b\times z + a\times z^2}}{2} \)

Bagby (1995) proposed:

\(\Phi\left(z\right) \approx \frac{1}{2}\times\left(1+\sqrt{1-\frac{1}{30}\times\left(7\times e^{-\frac{z^2}{2}}+16\times e^{-z^2\times\left(2-\sqrt{2}\right)}+ \left(7+\frac{\pi}{4}\times z^2\right)\times e^{-z^2}\right)}\right)\)

Note that this resembles a lot format of the error function based equations.

Shore (2005) created three variations using the RMM error distribution. Version 1 looks as follows:

\(\Phi\left(z\right) \approx 1 - e^{-\ln\left(2\right)\times e^{b\times\left(e^{c\times z}-1\right) + d\times z}}\)

With:

\(b=-1.81280,c=-0.472245,d=0.294549\)

For version 2 and 3 the following is used:

\(\Phi\left(z\right) \approx \frac{1+g\left(-z\right)-g\left(z\right)}{2}\)

With for version 2:

\(g\left(z\right) = e^{-\ln\left(2\right)\times e^{b\times\left(e^{a+c\times z}-1\right)+d\times z}}\)

\(a=0.072880383,b=-2.36699428,c=-0.40639929,d=0.19417768\)

and for version 3:

\(g\left(z\right) = e^{-\ln\left(2\right)\times e^{\frac{a\times b}{c}\times\left(\left(1+b\times z\right)^{\frac{c}{b}}-1\right)+d\times z}}\)

\(a=-6.37309208,b=-0.11105481,c=-0.61228883,d=0.44334159\)

Soranzo and Epure (2014) gave a very different formula as an approximation, compared to the others:

\(\Phi\left(z\right) \approx 2^{-22^{1-41^{z/10}}}\)

Shchigolev (2020) used a Variational Iteration Method to derive the following approximation:

\(\Phi\left(z\right) \approx \frac{1}{2} + \frac{1}{\sqrt{2}}\times\left(7-\left(7+3\times\sqrt{2}\times z+\frac{5}{4}\times z^2 + \frac{\sqrt{2}}{4}\times z^3 + \frac{z^4}{8}\right)\times e^{-\frac{z}{\sqrt{2}}}\right)\)

Cody (1969) gives the following set of equations for the error function:

\(\text{erf}\left(z\right) \approx \begin{cases} x\times R\left(x^2\right) & \text{if } \left|x\right|≤0.5 \\ 1 - e^{-x^2}\times R\left(x\right) & \text{if } 0.46875≤x≤4 \\ 1 - \frac{e^{-x^2}}{x}\times\left(\frac{1}{\sqrt{\pi}} + \frac{1}{x^2}\times R\left(\frac{1}{x^2}\right)\right) & \text{if } x≥4 \end{cases}\)

With:

\(R\left(x\right) = \frac{\sum_{i=0}^n p_i\times x^i}{\sum_{i=0}^n q_i\times x^i}\)

The values for \(p_i\) and \(q_i\) are given by Cody in various tables.

Badhe (1976) continued on the results from Patry and Keller (1964) and gave the following three approximations:

For \(z≥2\):

\(\Phi\left(z\right) \approx 1 - \phi\left(z\right) \times \frac{1}{z}\times\left(1-\frac{1}{y}\times\left(1+\frac{1}{y}\times\left(7+1/y\times\left(55+\frac{1}{y}\times\left(445+37450\times \frac{q}{y}\right)\right)\right)\right)\right)\)

With:

\(q=1+\frac{8.5}{z^2-\frac{0.4284639753}{z^2}+1.240964109}\)

\(y = z^2+10\)

For \(z<2\):

\(\Phi\left(z\right) \approx \frac{1}{2}+z\times\left(a_1+y\times\left(a_2+y\times\left(a_3+y\times\left(a_4+y\times\left(a_5+y\times\left(a_6+y\times\left(a_7+a_8\times y\right)\right)\right)\right)\right)\right)\right)\)

With:

\(y=\frac{z^2}{32}\)

\(a_{i=1\dots 8} = \left\{0.3989422784,-2.127690079,10.2125662121,-38.8830314909,120.2836370787,-303.2973153419,575.073131917,-603.9068092058\right\}\)

Derenzo (1977) approximation:

\(\Phi\left(z\right) \approx \begin{cases} 1-\frac{1}{2}\times e^{-\frac{\left(83\times z+351\right)\times z + 562}{\frac{703}{z}+165}} & \text{ if } z<5.5 \\ 1 - \frac{\sqrt{\frac{2}{\pi}}}{2\times z}\times e^{-\frac{z^2}{2}-\frac{0.94}{z^2}}& \text{ if } z≥5.5 \end{cases}\)

Lew (1981) gives two approximation sets:

version 1:

\(\Phi\left(z\right) \approx \begin{cases} \frac{1}{2}+\frac{z-\frac{1}{7}\times z^3}{\sqrt{2\times\pi}} & \text{ if } 0 ≤z≤1 \\ 1 - \frac{\left(1+z\right)\times\phi\left(z\right)}{z^2+z+1} & \text{ if } z>1 \end{cases}\)

version 2:

\(\Phi\left(z\right) \approx \begin{cases} \frac{1}{2}+\frac{2}{5}\times z & \text{ if } 0 ≤z≤\frac{1}{2} \\ 1 - \frac{\left(1+z\right)\times\phi\left(z\right)}{z^2+z+1} & \text{ if } z>\frac{1}{2} \end{cases}\)

Hawkes (1982):

\(\Phi\left(z\right) \approx \begin{cases} \frac{1}{2}\times\left(1 + \sqrt{1-e^{-2\times\frac{t^2}{\pi}}}\right) & \text{ if } 0 ≤z≤2.2 \\ 1 - \frac{\left(1.184+z\right)\times\phi\left(z\right)}{z^2+1.176\times z+1.209} & \text{ if } z>2.2 \end{cases}\)

With:

\( t=z-\frac{7.5166\times z^3}{10^3}+\frac{3.1737\times z^5}{10^4} - \frac{2.9657\times z^7}{10^6}\)

Shah (1985) gives perhaps a not so accurate, but the perhaps the most simple approximation:

\(\Phi\left(z\right) \approx \begin{cases} \frac{1}{2}+\frac{z\times\left(4.4-z\right)}{10} & \text{ if } 0 ≤z≤2.2 \\ 0.99 & \text{ if } 2.2<z<2.6 \\ 1 & \text{ if } z≥2.6 \end{cases}\)

Norton (1989) builds further on Shah's set, and gives two sets:

version 1:

\(\Phi\left(z\right) \approx \begin{cases} 1-\frac{1}{2}\times e^{-\frac{z^2+z}{2}} & \text{ if } 0 ≤z≤2.6 \\ 1-\frac{\phi\left(z\right)}{z} & \text{ if } z>2.6 \end{cases}\)

version 2:

\(\Phi\left(z\right) \approx \begin{cases} 1-\frac{1}{2}\times e^{-\frac{z^2+1.2\times z^0.8}{2}} & \text{ if } 0 ≤z≤2.7 \\ 1-\frac{\phi\left(z\right)}{z} & \text{ if } z>2.7 \end{cases}\)

Dridi (2003) gives an approximation set for the error function, based on rational fractions. It's a modification of Cody's algorithm

\(\text{erf}\left(x\right) \approx \begin{cases} x\times R_1\left(x^2\right) & \text{ if } 0 ≤x≤1.5 \\ 1 - e^{-x^2}\times\left(\frac{1}{2}\times R_1\left(x^4\right)+\frac{1}{5}\times R_2\left(x^{-4}\right)+\frac{3}{10}\times R_3\left(\frac{1}{x^{-4}}\right)\right) & \text{ if } 0.46875≤x≤4 \\ 1 - \frac{e^{-x^2}}{x}\times\left(\frac{1}{\sqrt{\pi}}+\frac{1}{x^2}\times R\left(\frac{1}{x^2}\right)\right) & \text{ if } x≥4 \end{cases}\)

With:

\(R_i\left(x\right) = \frac{\sum_{i=0}^n p_i\times x^i}{\sum_{i=0}^n q_i\times x^i}\)

The values for \(p_i\) and \(q_i\) are given by Cody in various tables or can be obtained from the example code provided by Dridi.

Choudhury, Ray, and Sarkar (2007):

If \(0 < z ≤ 0.7315 \text{ or } 2.2075 < z ≤ 2.7245 \text{ or } 3.056 < z≤4\):

\(\Phi\left(z\right) \approx 1-\frac{z^2+5.575192695\times z+12.77436324}{z^3\times\sqrt{2\times\pi}+14.38718147\times z^2 +31.53531977\times z+25.548726}\times e^{-\frac{z^2}{2}} \)

If \(0.7315 < z ≤ 1.726 \text{ or } 1.8135 < z ≤ 2.2075\):

\(\Phi\left(z\right) \approx 1-\left(0.4361836\times t-0.1201676\times t^2+0.937298\times t^3\right)\times\phi\left(z\right) \)

With \( t = \frac{1}{1+0.33267\times z} \)

If \(1.726 < z ≤ 1.8135 \text{ or } 2.7245 < z ≤ 3.056\):

\(\Phi\left(z\right) \approx \frac{1}{2}\times\left(1 + \sqrt{1 - \frac{1}{30}\times\left(7\times e^{-\frac{z^2}{2}}+16\times e^{-z^2\times\left(2-\sqrt{2}\right)} + \left(7+\frac{\pi\times z^2}{4}\right)\times e^{-z^2}\right)}\right) \)

Kiani, Panaretos, Psarakis, and Saleem (2008):

\( \Phi\left(z\right) \approx \frac{1}{2}\times\left(1-\text{sign}\left(-z\right)\times\sqrt{1-e^{-z^2\times w\left(z\right)}}\right) \)

If \(0 ≤ \left|z\right| < 1.05 \):

\( w\left(z\right) = -\frac{6.62}{10^6}\times\left|z\right|^5 +\frac{4.4166}{10^4}\times z^4 - \frac{1.31}{10^5}\times\left|z\right|^3 - \frac{94661}{9900000}\times z^2 - \frac{4.8}{10^7}\times\left|z\right| + \frac{7073553}{11111111}\)

If \(1.05 ≤ \left|z\right| < 2.29 \):

\( w\left(z\right) = -\frac{1.401663}{10^4}\times\left|z\right|^5 +\frac{1.150811}{10^3}\times z^4 - \frac{1.582555}{10^3}\times\left|z\right|^3 - \frac{7.76126}{10^3}\times z^2 - \frac{1.0608}{10^3}\times\left|z\right| + 0.6368751 \)

If \(2.29 ≤ \left|z\right| < 8 \):

\( w\left(z\right) = \frac{5.8716}{10^5}\times\left|z\right|^5 -\frac{1.221684}{10^3}\times z^4 + \frac{9.841663}{10^3}\times\left|z\right|^3 - \frac{3.55101}{10^2}\times z^2 + \frac{3.29203}{10^2}\times\left|z\right| + 0.62010268 \)

If \(8 ≤ \left|z\right| \):

\( w\left(z\right) = 0.5 \)

Choudhury and Roy (2009):

\(\text{erf}\left(x\right) \approx \begin{cases} \frac{1}{2} + \phi\left(z\right)\times g\left(z\right) & \text{ if } 0 <z<1.8735 \\ a+b\times z + c\times z^2 + \frac{1}{2} + \phi\left(z\right)\times g\left(z\right) & \text{ if } 1.8735≤z≤2.0885 \\ \alpha+\beta\times z + \gamma\times z^2 + 1 - \frac{1}{2}\times \phi\left(z\right)\times h\left(z\right) & \text{ if } 2.0885<z≤2.43 \\ 1 - \frac{1}{2}\times\phi\left(z\right)\times h\left(z\right) & \text{ if } 2.43 <z≤5.75 \\ 1 & \text{ if } z>5.75 \end{cases}\)

With:

\(g\left(z\right) = \frac{654729075\times z+45945900\times z^3+6486480\times z^5+154440\times z^7+4785\times z^9}{654729075-172297125\times z^2+20270250\times z^4-1351350\times z^6+51975\times z^8-945\times z^10} \)

\( a=-0.00007237,b=0.00007680,c=-0.00002041 \)

\( h\left(z\right) = \frac{2\times z^9+88\times z^7+1176\times z^5+5280\times z^3+5790\times z}{z^10+45\times z^8+630\times z^6+3150\times z^4+4725\times z^2+945} \)

\( \alpha=-0.00004861, \beta=0.00004021,\gamma=-0.00000833 \)