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package org.apache.commons.math3.special;

import org.apache.commons.math3.util.FastMath;

This is a utility class that provides computation methods related to the error functions.
/** * This is a utility class that provides computation methods related to the * error functions. * */
public class Erf {
The number X_CRIT is used by erf(double, double) internally. This number solves erf(x)=0.5 within 1ulp. More precisely, the current implementations of erf(double) and erfc(double) satisfy:
erf(X_CRIT) < 0.5,
erf(Math.nextUp(X_CRIT) > 0.5,
erfc(X_CRIT) = 0.5, and
erfc(Math.nextUp(X_CRIT) < 0.5
/** * The number {@code X_CRIT} is used by {@link #erf(double, double)} internally. * This number solves {@code erf(x)=0.5} within 1ulp. * More precisely, the current implementations of * {@link #erf(double)} and {@link #erfc(double)} satisfy:<br/> * {@code erf(X_CRIT) < 0.5},<br/> * {@code erf(Math.nextUp(X_CRIT) > 0.5},<br/> * {@code erfc(X_CRIT) = 0.5}, and<br/> * {@code erfc(Math.nextUp(X_CRIT) < 0.5} */
private static final double X_CRIT = 0.4769362762044697;
Default constructor. Prohibit instantiation.
/** * Default constructor. Prohibit instantiation. */
private Erf() {}
Returns the error function.

erf(x) = 2/√π 0x e-t2dt

This implementation computes erf(x) using the regularized gamma function, following Erf, equation (3)

The value returned is always between -1 and 1 (inclusive). If abs(x) > 40, then erf(x) is indistinguishable from either 1 or -1 as a double, so the appropriate extreme value is returned.

Params:
  • x – the value.
Throws:
See Also:
Returns:the error function erf(x)
/** * Returns the error function. * * <p>erf(x) = 2/&radic;&pi; <sub>0</sub>&int;<sup>x</sup> e<sup>-t<sup>2</sup></sup>dt </p> * * <p>This implementation computes erf(x) using the * {@link Gamma#regularizedGammaP(double, double, double, int) regularized gamma function}, * following <a href="http://mathworld.wolfram.com/Erf.html"> Erf</a>, equation (3)</p> * * <p>The value returned is always between -1 and 1 (inclusive). * If {@code abs(x) > 40}, then {@code erf(x)} is indistinguishable from * either 1 or -1 as a double, so the appropriate extreme value is returned. * </p> * * @param x the value. * @return the error function erf(x) * @throws org.apache.commons.math3.exception.MaxCountExceededException * if the algorithm fails to converge. * @see Gamma#regularizedGammaP(double, double, double, int) */
public static double erf(double x) { if (FastMath.abs(x) > 40) { return x > 0 ? 1 : -1; } final double ret = Gamma.regularizedGammaP(0.5, x * x, 1.0e-15, 10000); return x < 0 ? -ret : ret; }
Returns the complementary error function.

erfc(x) = 2/√π x e-t2dt
= 1 - erf(x)

This implementation computes erfc(x) using the regularized gamma function, following Erf, equation (3).

The value returned is always between 0 and 2 (inclusive). If abs(x) > 40, then erf(x) is indistinguishable from either 0 or 2 as a double, so the appropriate extreme value is returned.

Params:
  • x – the value
Throws:
See Also:
Returns:the complementary error function erfc(x)
Since:2.2
/** * Returns the complementary error function. * * <p>erfc(x) = 2/&radic;&pi; <sub>x</sub>&int;<sup>&infin;</sup> e<sup>-t<sup>2</sup></sup>dt * <br/> * = 1 - {@link #erf(double) erf(x)} </p> * * <p>This implementation computes erfc(x) using the * {@link Gamma#regularizedGammaQ(double, double, double, int) regularized gamma function}, * following <a href="http://mathworld.wolfram.com/Erf.html"> Erf</a>, equation (3).</p> * * <p>The value returned is always between 0 and 2 (inclusive). * If {@code abs(x) > 40}, then {@code erf(x)} is indistinguishable from * either 0 or 2 as a double, so the appropriate extreme value is returned. * </p> * * @param x the value * @return the complementary error function erfc(x) * @throws org.apache.commons.math3.exception.MaxCountExceededException * if the algorithm fails to converge. * @see Gamma#regularizedGammaQ(double, double, double, int) * @since 2.2 */
public static double erfc(double x) { 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; }
Returns the difference between erf(x1) and erf(x2). The implementation uses either erf(double) or erfc(double) depending on which provides the most precise result.
Params:
  • x1 – the first value
  • x2 – the second value
Returns:erf(x2) - erf(x1)
/** * Returns the difference between erf(x1) and erf(x2). * * The implementation uses either erf(double) or erfc(double) * depending on which provides the most precise result. * * @param x1 the first value * @param x2 the second value * @return erf(x2) - erf(x1) */
public static double erf(double x1, double x2) { if(x1 > x2) { return -erf(x2, x1); } return x1 < -X_CRIT ? x2 < 0.0 ? erfc(-x2) - erfc(-x1) : erf(x2) - erf(x1) : x2 > X_CRIT && x1 > 0.0 ? erfc(x1) - erfc(x2) : erf(x2) - erf(x1); }
Returns the inverse erf.

This implementation is described in the paper: Approximating the erfinv function by Mike Giles, Oxford-Man Institute of Quantitative Finance, which was published in GPU Computing Gems, volume 2, 2010. The source code is available here.

Params:
  • x – the value
Returns:t such that x = erf(t)
Since:3.2
/** * Returns the inverse erf. * <p> * This implementation is described in the paper: * <a href="http://people.maths.ox.ac.uk/gilesm/files/gems_erfinv.pdf">Approximating * the erfinv function</a> by Mike Giles, Oxford-Man Institute of Quantitative Finance, * which was published in GPU Computing Gems, volume 2, 2010. * The source code is available <a href="http://gpucomputing.net/?q=node/1828">here</a>. * </p> * @param x the value * @return t such that x = erf(t) * @since 3.2 */
public static double erfInv(final double x) { // beware that the logarithm argument must be // commputed as (1.0 - x) * (1.0 + x), // it must NOT be simplified as 1.0 - x * x as this // would induce rounding errors near the boundaries +/-1 double w = - FastMath.log((1.0 - x) * (1.0 + x)); double p; if (w < 6.25) { w -= 3.125; p = -3.6444120640178196996e-21; p = -1.685059138182016589e-19 + p * w; p = 1.2858480715256400167e-18 + p * w; p = 1.115787767802518096e-17 + p * w; p = -1.333171662854620906e-16 + p * w; p = 2.0972767875968561637e-17 + p * w; p = 6.6376381343583238325e-15 + p * w; p = -4.0545662729752068639e-14 + p * w; p = -8.1519341976054721522e-14 + p * w; p = 2.6335093153082322977e-12 + p * w; p = -1.2975133253453532498e-11 + p * w; p = -5.4154120542946279317e-11 + p * w; p = 1.051212273321532285e-09 + p * w; p = -4.1126339803469836976e-09 + p * w; p = -2.9070369957882005086e-08 + p * w; p = 4.2347877827932403518e-07 + p * w; p = -1.3654692000834678645e-06 + p * w; p = -1.3882523362786468719e-05 + p * w; p = 0.0001867342080340571352 + p * w; p = -0.00074070253416626697512 + p * w; p = -0.0060336708714301490533 + p * w; p = 0.24015818242558961693 + p * w; p = 1.6536545626831027356 + p * w; } else if (w < 16.0) { w = FastMath.sqrt(w) - 3.25; p = 2.2137376921775787049e-09; p = 9.0756561938885390979e-08 + p * w; p = -2.7517406297064545428e-07 + p * w; p = 1.8239629214389227755e-08 + p * w; p = 1.5027403968909827627e-06 + p * w; p = -4.013867526981545969e-06 + p * w; p = 2.9234449089955446044e-06 + p * w; p = 1.2475304481671778723e-05 + p * w; p = -4.7318229009055733981e-05 + p * w; p = 6.8284851459573175448e-05 + p * w; p = 2.4031110387097893999e-05 + p * w; p = -0.0003550375203628474796 + p * w; p = 0.00095328937973738049703 + p * w; p = -0.0016882755560235047313 + p * w; p = 0.0024914420961078508066 + p * w; p = -0.0037512085075692412107 + p * w; p = 0.005370914553590063617 + p * w; p = 1.0052589676941592334 + p * w; p = 3.0838856104922207635 + p * w; } else if (!Double.isInfinite(w)) { w = FastMath.sqrt(w) - 5.0; p = -2.7109920616438573243e-11; p = -2.5556418169965252055e-10 + p * w; p = 1.5076572693500548083e-09 + p * w; p = -3.7894654401267369937e-09 + p * w; p = 7.6157012080783393804e-09 + p * w; p = -1.4960026627149240478e-08 + p * w; p = 2.9147953450901080826e-08 + p * w; p = -6.7711997758452339498e-08 + p * w; p = 2.2900482228026654717e-07 + p * w; p = -9.9298272942317002539e-07 + p * w; p = 4.5260625972231537039e-06 + p * w; p = -1.9681778105531670567e-05 + p * w; p = 7.5995277030017761139e-05 + p * w; p = -0.00021503011930044477347 + p * w; p = -0.00013871931833623122026 + p * w; p = 1.0103004648645343977 + p * w; p = 4.8499064014085844221 + p * w; } else { // this branch does not appears in the original code, it // was added because the previous branch does not handle // x = +/-1 correctly. In this case, w is positive infinity // and as the first coefficient (-2.71e-11) is negative. // Once the first multiplication is done, p becomes negative // infinity and remains so throughout the polynomial evaluation. // So the branch above incorrectly returns negative infinity // instead of the correct positive infinity. p = Double.POSITIVE_INFINITY; } return p * x; }
Returns the inverse erfc.
Params:
  • x – the value
Returns:t such that x = erfc(t)
Since:3.2
/** * Returns the inverse erfc. * @param x the value * @return t such that x = erfc(t) * @since 3.2 */
public static double erfcInv(final double x) { return erfInv(1 - x); } }