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https://issues.apache.org/jira/browse/NUMBERS-177?page=com.atlassian.jira.plugin.system.issuetabpanels:all-tabpanel
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Alex Herbert updated NUMBERS-177:
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Attachment: expn.png
> Scaled complementary error function
> -----------------------------------
>
> Key: NUMBERS-177
> URL: https://issues.apache.org/jira/browse/NUMBERS-177
> Project: Commons Numbers
> Issue Type: New Feature
> Components: gamma
> Affects Versions: 1.1
> Reporter: Alex Herbert
> Priority: Minor
> Fix For: 1.1
>
> Attachments: erfc.png, erfcx.large.png, erfcx.medium.png, expn.png
>
>
> Add a scaled complementary error function:
> {noformat}
> erfcx(x) = exp(x^2) * erfc(x)
> {noformat}
> For small x this can use the current rational function approximations for erf
> or erfc and remove the scaling down by exp(-x^2). For large x this can use
> the [continued fraction expansion of
> erfc|https://en.wikipedia.org/wiki/Error_function#Continued_fraction_expansion]
> and remove the scaling by e^-z:
> {noformat}
> z 1
> erfc z = -------- e^-z^2 -----------------------------
> sqrt(pi) z^2 + 1/2
> -----------------------
> 1 + 1
> -------------------
> z^2 + 3/2
> -------------
> 1 + 2
> ---------
> z^2 + ...
> {noformat}
> At very large x the rational function cannot be evaluated as {{z^2 + 0.5 ~
> z^2}} and the following approximation can be used:
> {noformat}
> 1 1
> erfcx z = -------- * -
> sqrt(pi) z
> {noformat}
> This occurs at 2^26 or approximately 6.71e7.
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