[jira] Commented: (MATH-364) Make Erf more precise in the tails by providing erfc

["Phil Steitz (JIRA)" <ji...@apache.org>]

    [ https://issues.apache.org/jira/browse/MATH-364?page=com.atlassian.jira.plugin.system.issuetabpanels:comment-tabpanel&focusedCommentId=12975677#action_12975677 ] 

Phil Steitz commented on MATH-364:
----------------------------------

To maintain the identity erfc(x) = 1 - erf(x) and to top code for values not distinguishable from 0/2, I think erfc should be
{code}
public static double erfc(double x) throws MathException {
        if (FastMath.abs(x) > 40) {
            return x > 0 ? 0 : 2;
        }
        final double ret = Gamma.regularizedGammaQ(0.5, x * x, 1.0e-15, 10000);
        return x < 0 ? 2 - ret : ret;
    }
{code}

I understand the intent of erf(x1, x2) but am unable to derive myself or find a reference verifying the superior numerics of the implementation above.  Can anyone else provide a reference or explain why the impl given is optimal or at least better than erf(x2) = erf(x1)?

> Make Erf more precise in the tails by providing erfc
> ----------------------------------------------------
>
>                 Key: MATH-364
>                 URL: https://issues.apache.org/jira/browse/MATH-364
>             Project: Commons Math
>          Issue Type: Improvement
>    Affects Versions: 1.1, 1.2, 2.0, 2.1
>            Reporter: Christian Winter
>            Priority: Minor
>             Fix For: 2.2
>
>
> First I want to thank Phil Steitz for making Erf stable in the tails through adjusting the choices in calculating the regularized gamma functions, see [Math-282|https://issues.apache.org/jira/browse/MATH-282]. However, the precision of Erf in the tails is limitted to fixed point precision because of the closeness to +/-1.0, although the Gamma class could provide much more accuracy. Thus I propose to add the methods erfc(double) and erf(double, double) to the class Erf:
> {code:borderStyle=solid}
> /**
>  * Returns the complementary error function erfc(x).
>  * @param x the value
>  * @return the complementary error function erfc(x)
>  * @throws MathException if the algorithm fails to converge
>  */
> public static double erfc(double x) throws MathException {
> double ret = Gamma.regularizedGammaQ(0.5, x * x, 1.0e-15, 10000);
> 	if (x < 0) {
> 		ret = -ret;
> 	}
> 	return ret;
> }
> /**
>  * Returns the difference of the error function values of x1 and x2.
>  * @param x1 the first bound
>  * @param x2 the second bound
>  * @return erf(x2) - erf(x1)
>  * @throws MathException
>  */
> public static double erf(double x1, double x2) throws MathException {
> 	if(x1>x2)
> 		return erf(x2, x1);
> 	if(x1==x2)
> 		return 0.0;
>     	
> 	double f1 = erf(x1);
> 	double f2 = erf(x2);
> 	
> 	if(f2 > 0.5)
> 		if(f1 > 0.5)
> 			return erfc(x1) - erfc(x2);
> 		else
> 			return (0.5-erfc(x2)) + (0.5-f1);
> 	else
> 		if(f1 < -0.5)
> 			if(f2 < -0.5)
> 				return erfc(-x2) - erfc(-x1);
> 			else
> 				return (0.5-erfc(-x1)) + (0.5+f2);
> 		else
> 			return f2 - f1;
> }
> {code} 
> Further this can be used to improve the NormalDistributionImpl through
> {code:borderStyle=solid}
> @Override
> public double cumulativeProbability(double x0, double x1) throws MathException {
> 	return 0.5 * Erf.erf(
> 			(x0 - getMean()) / (getStandardDeviation() * sqrt2),
> 			(x1 - getMean()) / (getStandardDeviation() * sqrt2) );
> }
> {code}

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