Editing Normal distribution (section)

=== Numerical approximations for the normal cumulative distribution function and normal quantile function ===
The standard normal [[cumulative distribution function]] is widely used in scientific and statistical computing.

The values Φ(''x'') may be approximated very accurately by a variety of methods, such as [[numerical integration]], [[Taylor series]], [[asymptotic series]] and [[Gauss's continued fraction#Of Kummer's confluent hypergeometric function|continued fractions]]. Different approximations are used depending on the desired level of accuracy.
* {{harvtxt |Zelen |Severo |1964 }} give the approximation for Φ(''x'') for ''x'' > 0 with the absolute error {{math|{{abs|''ε''(''x'')}} < 7.5·10<sup>−8</sup>}} (algorithm [https://secure.math.ubc.ca/~cbm/aands/page_932.htm 26.2.17]): <math display=block>
    \Phi(x) = 1 - \varphi(x)\left(b_1 t + b_2 t^2 + b_3t^3 + b_4 t^4 + b_5 t^5\right) + \varepsilon(x), \qquad t = \frac{1}{1+b_0x},
</math> where ''ϕ''(''x'') is the standard normal probability density function, and ''b''<sub>0</sub> = 0.2316419, ''b''<sub>1</sub> = 0.319381530, ''b''<sub>2</sub> = −0.356563782, ''b''<sub>3</sub> = 1.781477937, ''b''<sub>4</sub> = −1.821255978, ''b''<sub>5</sub> = 1.330274429.
* {{harvtxt |Hart |1968 }} lists some dozens of approximations – by means of rational functions, with or without exponentials – for the {{mono|erfc()}} function. His algorithms vary in the degree of complexity and the resulting precision, with maximum absolute precision of 24 digits. An algorithm by {{harvtxt |West |2009 }} combines Hart's algorithm 5666 with a [[continued fraction]] approximation in the tail to provide a fast computation algorithm with a 16-digit precision.
* {{harvtxt |Cody |1969 }} after recalling Hart68 solution is not suited for erf, gives a solution for both erf and erfc, with maximal relative error bound, via [[rational function|Rational Chebyshev Approximation]].
* {{harvtxt |Marsaglia |2004 }} suggested a simple algorithm{{NoteTag|For example, this algorithm is given in the article [[Bc programming language#A translated C function|Bc programming language]].}} based on the Taylor series expansion <math display=block>
    \Phi(x) = \frac12 + \varphi(x)\left( x + \frac{x^3} 3 + \frac{x^5}{3 \cdot 5} + \frac{x^7}{3 \cdot 5 \cdot 7} + \frac{x^9}{3 \cdot 5 \cdot 7 \cdot 9} + \cdots \right)
</math> for calculating {{math|Φ(''x'')}} with arbitrary precision. The drawback of this algorithm is comparatively slow calculation time (for example it takes over 300 iterations to calculate the function with 16 digits of precision when {{math|1=''x'' = 10}}).
* The [[GNU Scientific Library]] calculates values of the standard normal cumulative distribution function using Hart's algorithms and approximations with [[Chebyshev polynomial]]s.
* {{harvtxt |Dia|2023 }} proposes the following approximation of <math display=inline>1-\Phi</math> with a maximum relative error less than <math display=inline>2^{-53}</math> <math display=inline>
\left(\approx 1.1 \times 10^{-16}\right)
</math> in absolute value: for <math display=inline>x \ge 0</math><math display=inline>
\begin{aligned}
    1-\Phi\left(x\right) & = \left(\frac{0.39894228040143268}{x+2.92678600515804815}\right)
    \left(\frac{x^2+8.42742300458043240 x+18.38871225773938487}{x^2+5.81582518933527391 x+8.97280659046817350} \right) \\
    & \left(\frac{x^2+7.30756258553673541 x+18.25323235347346525}{x^2+5.70347935898051437 x+10.27157061171363079}\right)
    \left(\frac{x^2+5.66479518878470765 x+18.61193318971775795}{x^2+5.51862483025707963 x+12.72323261907760928}\right) \\
    & \left( \frac{x^2+4.91396098895240075 x+24.14804072812762821}{x^2+5.26184239579604207 x+16.88639562007936908}\right)
    \left( \frac{x^2+3.83362947800146179 x+11.61511226260603247}{x^2+4.92081346632882033 x+24.12333774572479110}\right) e^{-\frac{x^2}{2}}
\end{aligned}
</math> and for <math display=inline>
    x<0
</math>,
<math display=block>
    1-\Phi\left(x\right) = 1-\left(1-\Phi\left(-x\right)\right)
</math>

Shore (1982) introduced simple approximations that may be incorporated in stochastic optimization models of engineering and operations research, like reliability engineering and inventory analysis. Denoting {{math|1=''p'' = Φ(''z'')}}, the simplest approximation for the quantile function is:
<math display=block>z = \Phi^{-1}(p)=5.5556\left[1- \left( \frac{1-p} p \right)^{0.1186}\right],\qquad p\ge 1/2</math>

This approximation delivers for ''z'' a maximum absolute error of 0.026 (for {{math|0.5 ≤ ''p'' ≤ 0.9999}}, corresponding to {{math|0 ≤ ''z'' ≤ 3.719}}). For {{math|''p'' < 1/2}} replace ''p'' by {{math|1 − ''p''}} and change sign. Another approximation, somewhat less accurate, is the single-parameter approximation:
<math display=block> z=-0.4115\left\{ \frac{1-p} p + \log \left[ \frac{1-p} p \right] - 1 \right\}, \qquad p\ge 1/2</math>

The latter had served to derive a simple approximation for the loss integral of the normal distribution, defined by
<math display=block>\begin{align}
L(z) & =\int_z^\infty (u-z)\varphi(u) \, du=\int_z^\infty [1-\Phi (u)] \, du \\[5pt]
L(z) & \approx \begin{cases}
   0.4115\left(\dfrac p {1-p} \right) - z, & p<1/2, \\ \\
   0.4115\left( \dfrac {1-p} p \right), & p\ge 1/2.
\end{cases} \\[5pt]
\text{or, equivalently,} \\
L(z) & \approx \begin{cases}
   0.4115\left\{ 1-\log \left[ \frac p {1-p} \right] \right\}, & p < 1/2, \\ \\
   0.4115 \dfrac{1-p} p, & p\ge 1/2.
\end{cases}
\end{align}</math>

This approximation is particularly accurate for the right far-tail (maximum error of 10<sup>−3</sup> for z≥1.4). Highly accurate approximations for the cumulative distribution function, based on [[Response Modeling Methodology]] (RMM, Shore, 2011, 2012), are shown in Shore (2005).

Some more approximations can be found at: [[Error function#Approximation with elementary functions]]. In particular, small ''relative'' error on the whole domain for the cumulative distribution function {{tmath|\Phi}} and the quantile function <math display=inline>\Phi^{-1}</math> as well, is achieved via an explicitly invertible formula by Sergei Winitzki in 2008.