From 44ab256961029c9e104e70fb804617fd582412cd Mon Sep 17 00:00:00 2001 From: Gilles Date: Sun, 14 May 2017 23:30:00 +0200 Subject: [PATCH] MATH-1416: Delete functionality now in "Commons Numbers". --- .../apache/commons/math4/special/Beta.java | 484 --------- .../apache/commons/math4/special/Gamma.java | 746 ------------- .../commons/math4/special/BetaTest.java | 979 ----------------- .../commons/math4/special/GammaTest.java | 998 ------------------ 4 files changed, 3207 deletions(-) delete mode 100644 src/main/java/org/apache/commons/math4/special/Beta.java delete mode 100644 src/main/java/org/apache/commons/math4/special/Gamma.java delete mode 100644 src/test/java/org/apache/commons/math4/special/BetaTest.java delete mode 100644 src/test/java/org/apache/commons/math4/special/GammaTest.java diff --git a/src/main/java/org/apache/commons/math4/special/Beta.java b/src/main/java/org/apache/commons/math4/special/Beta.java deleted file mode 100644 index 9355099a3..000000000 --- a/src/main/java/org/apache/commons/math4/special/Beta.java +++ /dev/null @@ -1,484 +0,0 @@ -/* - * Licensed to the Apache Software Foundation (ASF) under one or more - * contributor license agreements. See the NOTICE file distributed with - * this work for additional information regarding copyright ownership. - * The ASF licenses this file to You under the Apache License, Version 2.0 - * (the "License"); you may not use this file except in compliance with - * the License. You may obtain a copy of the License at - * - * http://www.apache.org/licenses/LICENSE-2.0 - * - * Unless required by applicable law or agreed to in writing, software - * distributed under the License is distributed on an "AS IS" BASIS, - * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. - * See the License for the specific language governing permissions and - * limitations under the License. - */ -package org.apache.commons.math4.special; - -import org.apache.commons.numbers.gamma.Gamma; -import org.apache.commons.numbers.gamma.LogGamma; -import org.apache.commons.math4.exception.NumberIsTooSmallException; -import org.apache.commons.math4.exception.OutOfRangeException; -import org.apache.commons.math4.util.ContinuedFraction; -import org.apache.commons.math4.util.FastMath; - -/** - *

- * This is a utility class that provides computation methods related to the - * Beta family of functions. - *

- *

- * Implementation of {@link #logBeta(double, double)} is based on the - * algorithms described in - *

- * and implemented in the - * NSWC Library of Mathematical Functions, - * available - * here. - * This library is "approved for public release", and the - * Copyright guidance - * indicates that unless otherwise stated in the code, all FORTRAN functions in - * this library are license free. Since no such notice appears in the code these - * functions can safely be ported to Commons-Math. - */ -public class Beta { - /** Maximum allowed numerical error. */ - private static final double DEFAULT_EPSILON = 1E-14; - - /** The constant value of ½log 2π. */ - private static final double HALF_LOG_TWO_PI = .9189385332046727; - - /** - *

- * The coefficients of the series expansion of the Δ function. This function - * is defined as follows - *

- *
Δ(x) = log Γ(x) - (x - 0.5) log a + a - 0.5 log 2π,
- *

- * see equation (23) in Didonato and Morris (1992). The series expansion, - * which applies for x ≥ 10, reads - *

- * 
-     *                 14
-     *                ====
-     *             1  \                2 n
-     *     Δ(x) = ---  >    d  (10 / x)
-     *             x  /      n
-     *                ====
-     *                n = 0
-     * 
-     */
-    private static final double[] DELTA = {
-        .833333333333333333333333333333E-01,
-        -.277777777777777777777777752282E-04,
-        .793650793650793650791732130419E-07,
-        -.595238095238095232389839236182E-09,
-        .841750841750832853294451671990E-11,
-        -.191752691751854612334149171243E-12,
-        .641025640510325475730918472625E-14,
-        -.295506514125338232839867823991E-15,
-        .179643716359402238723287696452E-16,
-        -.139228964661627791231203060395E-17,
-        .133802855014020915603275339093E-18,
-        -.154246009867966094273710216533E-19,
-        .197701992980957427278370133333E-20,
-        -.234065664793997056856992426667E-21,
-        .171348014966398575409015466667E-22
-    };
-
-    /**
-     * Default constructor.  Prohibit instantiation.
-     */
-    private Beta() {}
-
-    /**
-     * Returns the
-     * 
-     * regularized beta function I(x, a, b).
-     *
-     * @param x Value.
-     * @param a Parameter {@code a}.
-     * @param b Parameter {@code b}.
-     * @return the regularized beta function I(x, a, b).
-     * @throws org.apache.commons.math4.exception.MaxCountExceededException
-     * if the algorithm fails to converge.
-     */
-    public static double regularizedBeta(double x, double a, double b) {
-        return regularizedBeta(x, a, b, DEFAULT_EPSILON, Integer.MAX_VALUE);
-    }
-
-    /**
-     * Returns the
-     * 
-     * regularized beta function I(x, a, b).
-     *
-     * @param x Value.
-     * @param a Parameter {@code a}.
-     * @param b Parameter {@code b}.
-     * @param epsilon When the absolute value of the nth item in the
-     * series is less than epsilon the approximation ceases to calculate
-     * further elements in the series.
-     * @return the regularized beta function I(x, a, b)
-     * @throws org.apache.commons.math4.exception.MaxCountExceededException
-     * if the algorithm fails to converge.
-     */
-    public static double regularizedBeta(double x,
-                                         double a, double b,
-                                         double epsilon) {
-        return regularizedBeta(x, a, b, epsilon, Integer.MAX_VALUE);
-    }
-
-    /**
-     * Returns the regularized beta function I(x, a, b).
-     *
-     * @param x the value.
-     * @param a Parameter {@code a}.
-     * @param b Parameter {@code b}.
-     * @param maxIterations Maximum number of "iterations" to complete.
-     * @return the regularized beta function I(x, a, b)
-     * @throws org.apache.commons.math4.exception.MaxCountExceededException
-     * if the algorithm fails to converge.
-     */
-    public static double regularizedBeta(double x,
-                                         double a, double b,
-                                         int maxIterations) {
-        return regularizedBeta(x, a, b, DEFAULT_EPSILON, maxIterations);
-    }
-
-    /**
-     * Returns the regularized beta function I(x, a, b).
-     *
-     * The implementation of this method is based on:
-     * 
-     *
-     * @param x the value.
-     * @param a Parameter {@code a}.
-     * @param b Parameter {@code b}.
-     * @param epsilon When the absolute value of the nth item in the
-     * series is less than epsilon the approximation ceases to calculate
-     * further elements in the series.
-     * @param maxIterations Maximum number of "iterations" to complete.
-     * @return the regularized beta function I(x, a, b)
-     * @throws org.apache.commons.math4.exception.MaxCountExceededException
-     * if the algorithm fails to converge.
-     */
-    public static double regularizedBeta(double x,
-                                         final double a, final double b,
-                                         double epsilon, int maxIterations) {
-        double ret;
-
-        if (Double.isNaN(x) ||
-            Double.isNaN(a) ||
-            Double.isNaN(b) ||
-            x < 0 ||
-            x > 1 ||
-            a <= 0 ||
-            b <= 0) {
-            ret = Double.NaN;
-        } else if (x > (a + 1) / (2 + b + a) &&
-                   1 - x <= (b + 1) / (2 + b + a)) {
-            ret = 1 - regularizedBeta(1 - x, b, a, epsilon, maxIterations);
-        } else {
-            ContinuedFraction fraction = new ContinuedFraction() {
-
-                /** {@inheritDoc} */
-                @Override
-                protected double getB(int n, double x) {
-                    double ret;
-                    double m;
-                    if (n % 2 == 0) { // even
-                        m = n / 2.0;
-                        ret = (m * (b - m) * x) /
-                            ((a + (2 * m) - 1) * (a + (2 * m)));
-                    } else {
-                        m = (n - 1.0) / 2.0;
-                        ret = -((a + m) * (a + b + m) * x) /
-                                ((a + (2 * m)) * (a + (2 * m) + 1.0));
-                    }
-                    return ret;
-                }
-
-                /** {@inheritDoc} */
-                @Override
-                protected double getA(int n, double x) {
-                    return 1.0;
-                }
-            };
-            ret = FastMath.exp((a * FastMath.log(x)) + (b * FastMath.log1p(-x)) -
-                FastMath.log(a) - logBeta(a, b)) *
-                1.0 / fraction.evaluate(x, epsilon, maxIterations);
-        }
-
-        return ret;
-    }
-
-    /**
-     * Returns the value of log Γ(a + b) for 1 ≤ a, b ≤ 2. Based on the
-     * NSWC Library of Mathematics Subroutines double precision
-     * implementation, {@code DGSMLN}. In {@code BetaTest.testLogGammaSum()},
-     * this private method is accessed through reflection.
-     *
-     * @param a First argument.
-     * @param b Second argument.
-     * @return the value of {@code log(Gamma(a + b))}.
-     * @throws OutOfRangeException if {@code a} or {@code b} is lower than
-     * {@code 1.0} or greater than {@code 2.0}.
-     */
-    private static double logGammaSum(final double a, final double b)
-        throws OutOfRangeException {
-
-        if ((a < 1.0) || (a > 2.0)) {
-            throw new OutOfRangeException(a, 1.0, 2.0);
-        }
-        if ((b < 1.0) || (b > 2.0)) {
-            throw new OutOfRangeException(b, 1.0, 2.0);
-        }
-
-        final double x = (a - 1.0) + (b - 1.0);
-        if (x <= 0.5) {
-            return org.apache.commons.math4.special.Gamma.logGamma1p(1.0 + x);
-        } else if (x <= 1.5) {
-            return org.apache.commons.math4.special.Gamma.logGamma1p(x) + FastMath.log1p(x);
-        } else {
-            return org.apache.commons.math4.special.Gamma.logGamma1p(x - 1.0) + FastMath.log(x * (1.0 + x));
-        }
-    }
-
-    /**
-     * Returns the value of log[Γ(b) / Γ(a + b)] for a ≥ 0 and b ≥ 10. Based on
-     * the NSWC Library of Mathematics Subroutines double precision
-     * implementation, {@code DLGDIV}. In
-     * {@code BetaTest.testLogGammaMinusLogGammaSum()}, this private method is
-     * accessed through reflection.
-     *
-     * @param a First argument.
-     * @param b Second argument.
-     * @return the value of {@code log(Gamma(b) / Gamma(a + b))}.
-     * @throws NumberIsTooSmallException if {@code a < 0.0} or {@code b < 10.0}.
-     */
-    private static double logGammaMinusLogGammaSum(final double a,
-                                                   final double b)
-        throws NumberIsTooSmallException {
-
-        if (a < 0.0) {
-            throw new NumberIsTooSmallException(a, 0.0, true);
-        }
-        if (b < 10.0) {
-            throw new NumberIsTooSmallException(b, 10.0, true);
-        }
-
-        /*
-         * d = a + b - 0.5
-         */
-        final double d;
-        final double w;
-        if (a <= b) {
-            d = b + (a - 0.5);
-            w = deltaMinusDeltaSum(a, b);
-        } else {
-            d = a + (b - 0.5);
-            w = deltaMinusDeltaSum(b, a);
-        }
-
-        final double u = d * FastMath.log1p(a / b);
-        final double v = a * (FastMath.log(b) - 1.0);
-
-        return u <= v ? (w - u) - v : (w - v) - u;
-    }
-
-    /**
-     * Returns the value of Δ(b) - Δ(a + b), with 0 ≤ a ≤ b and b ≥ 10. Based
-     * on equations (26), (27) and (28) in Didonato and Morris (1992).
-     *
-     * @param a First argument.
-     * @param b Second argument.
-     * @return the value of {@code Delta(b) - Delta(a + b)}
-     * @throws OutOfRangeException if {@code a < 0} or {@code a > b}
-     * @throws NumberIsTooSmallException if {@code b < 10}
-     */
-    private static double deltaMinusDeltaSum(final double a,
-                                             final double b)
-        throws OutOfRangeException, NumberIsTooSmallException {
-
-        if ((a < 0) || (a > b)) {
-            throw new OutOfRangeException(a, 0, b);
-        }
-        if (b < 10) {
-            throw new NumberIsTooSmallException(b, 10, true);
-        }
-
-        final double h = a / b;
-        final double p = h / (1.0 + h);
-        final double q = 1.0 / (1.0 + h);
-        final double q2 = q * q;
-        /*
-         * s[i] = 1 + q + ... - q**(2 * i)
-         */
-        final double[] s = new double[DELTA.length];
-        s[0] = 1.0;
-        for (int i = 1; i < s.length; i++) {
-            s[i] = 1.0 + (q + q2 * s[i - 1]);
-        }
-        /*
-         * w = Delta(b) - Delta(a + b)
-         */
-        final double sqrtT = 10.0 / b;
-        final double t = sqrtT * sqrtT;
-        double w = DELTA[DELTA.length - 1] * s[s.length - 1];
-        for (int i = DELTA.length - 2; i >= 0; i--) {
-            w = t * w + DELTA[i] * s[i];
-        }
-        return w * p / b;
-    }
-
-    /**
-     * Returns the value of Δ(p) + Δ(q) - Δ(p + q), with p, q ≥ 10. Based on
-     * the NSWC Library of Mathematics Subroutines double precision
-     * implementation, {@code DBCORR}. In
-     * {@code BetaTest.testSumDeltaMinusDeltaSum()}, this private method is
-     * accessed through reflection.
-     *
-     * @param p First argument.
-     * @param q Second argument.
-     * @return the value of {@code Delta(p) + Delta(q) - Delta(p + q)}.
-     * @throws NumberIsTooSmallException if {@code p < 10.0} or {@code q < 10.0}.
-     */
-    private static double sumDeltaMinusDeltaSum(final double p,
-                                                final double q) {
-
-        if (p < 10.0) {
-            throw new NumberIsTooSmallException(p, 10.0, true);
-        }
-        if (q < 10.0) {
-            throw new NumberIsTooSmallException(q, 10.0, true);
-        }
-
-        final double a = FastMath.min(p, q);
-        final double b = FastMath.max(p, q);
-        final double sqrtT = 10.0 / a;
-        final double t = sqrtT * sqrtT;
-        double z = DELTA[DELTA.length - 1];
-        for (int i = DELTA.length - 2; i >= 0; i--) {
-            z = t * z + DELTA[i];
-        }
-        return z / a + deltaMinusDeltaSum(a, b);
-    }
-
-    /**
-     * Returns the value of {@code log B(p, q)} for {@code 0 ≤ x ≤ 1} and {@code p, q > 0}. Based on the
-     * NSWC Library of Mathematics Subroutines implementation,
-     * {@code DBETLN}.
-     *
-     * @param p First argument.
-     * @param q Second argument.
-     * @return the value of {@code log(Beta(p, q))}, {@code NaN} if
-     * {@code p <= 0} or {@code q <= 0}.
-     */
-    public static double logBeta(final double p, final double q) {
-        if (Double.isNaN(p) || Double.isNaN(q) || (p <= 0.0) || (q <= 0.0)) {
-            return Double.NaN;
-        }
-
-        final double a = FastMath.min(p, q);
-        final double b = FastMath.max(p, q);
-        if (a >= 10.0) {
-            final double w = sumDeltaMinusDeltaSum(a, b);
-            final double h = a / b;
-            final double c = h / (1.0 + h);
-            final double u = -(a - 0.5) * FastMath.log(c);
-            final double v = b * FastMath.log1p(h);
-            if (u <= v) {
-                return (((-0.5 * FastMath.log(b) + HALF_LOG_TWO_PI) + w) - u) - v;
-            } else {
-                return (((-0.5 * FastMath.log(b) + HALF_LOG_TWO_PI) + w) - v) - u;
-            }
-        } else if (a > 2.0) {
-            if (b > 1000.0) {
-                final int n = (int) FastMath.floor(a - 1.0);
-                double prod = 1.0;
-                double ared = a;
-                for (int i = 0; i < n; i++) {
-                    ared -= 1.0;
-                    prod *= ared / (1.0 + ared / b);
-                }
-                return (FastMath.log(prod) - n * FastMath.log(b)) +
-                        (LogGamma.value(ared) +
-                         logGammaMinusLogGammaSum(ared, b));
-            } else {
-                double prod1 = 1.0;
-                double ared = a;
-                while (ared > 2.0) {
-                    ared -= 1.0;
-                    final double h = ared / b;
-                    prod1 *= h / (1.0 + h);
-                }
-                if (b < 10.0) {
-                    double prod2 = 1.0;
-                    double bred = b;
-                    while (bred > 2.0) {
-                        bred -= 1.0;
-                        prod2 *= bred / (ared + bred);
-                    }
-                    return FastMath.log(prod1) +
-                           FastMath.log(prod2) +
-                           (LogGamma.value(ared) +
-                           (LogGamma.value(bred) -
-                            logGammaSum(ared, bred)));
-                } else {
-                    return FastMath.log(prod1) +
-                           LogGamma.value(ared) +
-                           logGammaMinusLogGammaSum(ared, b);
-                }
-            }
-        } else if (a >= 1.0) {
-            if (b > 2.0) {
-                if (b < 10.0) {
-                    double prod = 1.0;
-                    double bred = b;
-                    while (bred > 2.0) {
-                        bred -= 1.0;
-                        prod *= bred / (a + bred);
-                    }
-                    return FastMath.log(prod) +
-                           (LogGamma.value(a) +
-                            (LogGamma.value(bred) -
-                             logGammaSum(a, bred)));
-                } else {
-                    return LogGamma.value(a) +
-                           logGammaMinusLogGammaSum(a, b);
-                }
-            } else {
-                return LogGamma.value(a) +
-                       LogGamma.value(b) -
-                       logGammaSum(a, b);
-            }
-        } else {
-            if (b >= 10.0) {
-                return LogGamma.value(a) +
-                       logGammaMinusLogGammaSum(a, b);
-            } else {
-                // The following command is the original NSWC implementation.
-                // return LogGamma.value(a) +
-                // (LogGamma.value(b) - LogGamma.value(a + b));
-                // The following command turns out to be more accurate.
-                return FastMath.log(Gamma.value(a) * Gamma.value(b) /
-                                    Gamma.value(a + b));
-            }
-        }
-    }
-}
diff --git a/src/main/java/org/apache/commons/math4/special/Gamma.java b/src/main/java/org/apache/commons/math4/special/Gamma.java
deleted file mode 100644
index 3c4ff97e1..000000000
--- a/src/main/java/org/apache/commons/math4/special/Gamma.java
+++ /dev/null
@@ -1,746 +0,0 @@
-/*
- * Licensed to the Apache Software Foundation (ASF) under one or more
- * contributor license agreements.  See the NOTICE file distributed with
- * this work for additional information regarding copyright ownership.
- * The ASF licenses this file to You under the Apache License, Version 2.0
- * (the "License"); you may not use this file except in compliance with
- * the License.  You may obtain a copy of the License at
- *
- *      http://www.apache.org/licenses/LICENSE-2.0
- *
- * Unless required by applicable law or agreed to in writing, software
- * distributed under the License is distributed on an "AS IS" BASIS,
- * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
- * See the License for the specific language governing permissions and
- * limitations under the License.
- */
-package org.apache.commons.math4.special;
-
-import org.apache.commons.math4.exception.MaxCountExceededException;
-import org.apache.commons.math4.exception.NumberIsTooLargeException;
-import org.apache.commons.math4.exception.NumberIsTooSmallException;
-import org.apache.commons.math4.util.ContinuedFraction;
-import org.apache.commons.math4.util.FastMath;
-
-/**
- * 

- * This is a utility class that provides computation methods related to the
- * Γ (Gamma) family of functions.
- * 

- * 

- * Implementation of {@link #invGamma1pm1(double)} and
- * {@link #logGamma1p(double)} is based on the algorithms described in
- * 

- * and implemented in the
- * NSWC Library of Mathematical Functions,
- * available
- * here.
- * This library is "approved for public release", and the
- * Copyright guidance
- * indicates that unless otherwise stated in the code, all FORTRAN functions in
- * this library are license free. Since no such notice appears in the code these
- * functions can safely be ported to Commons-Math.
- *
- */
-public class Gamma {
-    /**
-     * Euler-Mascheroni constant
-     *
-     * @since 2.0
-     */
-    public static final double GAMMA = 0.577215664901532860606512090082;
-
-    /**
-     * Constant \( g = \frac{607}{128} \) in the {@link #lanczos(double) Lanczos approximation}.
-     *
-     * @since 3.1
-     */
-    public static final double LANCZOS_G = 607.0 / 128.0;
-
-    /** Maximum allowed numerical error. */
-    private static final double DEFAULT_EPSILON = 10e-15;
-
-    /** Lanczos coefficients */
-    private static final double[] LANCZOS = {
-        0.99999999999999709182,
-        57.156235665862923517,
-        -59.597960355475491248,
-        14.136097974741747174,
-        -0.49191381609762019978,
-        .33994649984811888699e-4,
-        .46523628927048575665e-4,
-        -.98374475304879564677e-4,
-        .15808870322491248884e-3,
-        -.21026444172410488319e-3,
-        .21743961811521264320e-3,
-        -.16431810653676389022e-3,
-        .84418223983852743293e-4,
-        -.26190838401581408670e-4,
-        .36899182659531622704e-5,
-    };
-
-    /** Avoid repeated computation of log of 2 PI in logGamma */
-    private static final double HALF_LOG_2_PI = 0.5 * FastMath.log(2.0 * FastMath.PI);
-
-    /** The constant value of √(2π). */
-    private static final double SQRT_TWO_PI = 2.506628274631000502;
-
-    // limits for switching algorithm in digamma
-    /** C limit. */
-    private static final double C_LIMIT = 49;
-
-    /** S limit. */
-    private static final double S_LIMIT = 1e-5;
-
-    /*
-     * Constants for the computation of double invGamma1pm1(double).
-     * Copied from DGAM1 in the NSWC library.
-     */
-
-    /** The constant {@code A0} defined in {@code DGAM1}. */
-    private static final double INV_GAMMA1P_M1_A0 = .611609510448141581788E-08;
-
-    /** The constant {@code A1} defined in {@code DGAM1}. */
-    private static final double INV_GAMMA1P_M1_A1 = .624730830116465516210E-08;
-
-    /** The constant {@code B1} defined in {@code DGAM1}. */
-    private static final double INV_GAMMA1P_M1_B1 = .203610414066806987300E+00;
-
-    /** The constant {@code B2} defined in {@code DGAM1}. */
-    private static final double INV_GAMMA1P_M1_B2 = .266205348428949217746E-01;
-
-    /** The constant {@code B3} defined in {@code DGAM1}. */
-    private static final double INV_GAMMA1P_M1_B3 = .493944979382446875238E-03;
-
-    /** The constant {@code B4} defined in {@code DGAM1}. */
-    private static final double INV_GAMMA1P_M1_B4 = -.851419432440314906588E-05;
-
-    /** The constant {@code B5} defined in {@code DGAM1}. */
-    private static final double INV_GAMMA1P_M1_B5 = -.643045481779353022248E-05;
-
-    /** The constant {@code B6} defined in {@code DGAM1}. */
-    private static final double INV_GAMMA1P_M1_B6 = .992641840672773722196E-06;
-
-    /** The constant {@code B7} defined in {@code DGAM1}. */
-    private static final double INV_GAMMA1P_M1_B7 = -.607761895722825260739E-07;
-
-    /** The constant {@code B8} defined in {@code DGAM1}. */
-    private static final double INV_GAMMA1P_M1_B8 = .195755836614639731882E-09;
-
-    /** The constant {@code P0} defined in {@code DGAM1}. */
-    private static final double INV_GAMMA1P_M1_P0 = .6116095104481415817861E-08;
-
-    /** The constant {@code P1} defined in {@code DGAM1}. */
-    private static final double INV_GAMMA1P_M1_P1 = .6871674113067198736152E-08;
-
-    /** The constant {@code P2} defined in {@code DGAM1}. */
-    private static final double INV_GAMMA1P_M1_P2 = .6820161668496170657918E-09;
-
-    /** The constant {@code P3} defined in {@code DGAM1}. */
-    private static final double INV_GAMMA1P_M1_P3 = .4686843322948848031080E-10;
-
-    /** The constant {@code P4} defined in {@code DGAM1}. */
-    private static final double INV_GAMMA1P_M1_P4 = .1572833027710446286995E-11;
-
-    /** The constant {@code P5} defined in {@code DGAM1}. */
-    private static final double INV_GAMMA1P_M1_P5 = -.1249441572276366213222E-12;
-
-    /** The constant {@code P6} defined in {@code DGAM1}. */
-    private static final double INV_GAMMA1P_M1_P6 = .4343529937408594255178E-14;
-
-    /** The constant {@code Q1} defined in {@code DGAM1}. */
-    private static final double INV_GAMMA1P_M1_Q1 = .3056961078365221025009E+00;
-
-    /** The constant {@code Q2} defined in {@code DGAM1}. */
-    private static final double INV_GAMMA1P_M1_Q2 = .5464213086042296536016E-01;
-
-    /** The constant {@code Q3} defined in {@code DGAM1}. */
-    private static final double INV_GAMMA1P_M1_Q3 = .4956830093825887312020E-02;
-
-    /** The constant {@code Q4} defined in {@code DGAM1}. */
-    private static final double INV_GAMMA1P_M1_Q4 = .2692369466186361192876E-03;
-
-    /** The constant {@code C} defined in {@code DGAM1}. */
-    private static final double INV_GAMMA1P_M1_C = -.422784335098467139393487909917598E+00;
-
-    /** The constant {@code C0} defined in {@code DGAM1}. */
-    private static final double INV_GAMMA1P_M1_C0 = .577215664901532860606512090082402E+00;
-
-    /** The constant {@code C1} defined in {@code DGAM1}. */
-    private static final double INV_GAMMA1P_M1_C1 = -.655878071520253881077019515145390E+00;
-
-    /** The constant {@code C2} defined in {@code DGAM1}. */
-    private static final double INV_GAMMA1P_M1_C2 = -.420026350340952355290039348754298E-01;
-
-    /** The constant {@code C3} defined in {@code DGAM1}. */
-    private static final double INV_GAMMA1P_M1_C3 = .166538611382291489501700795102105E+00;
-
-    /** The constant {@code C4} defined in {@code DGAM1}. */
-    private static final double INV_GAMMA1P_M1_C4 = -.421977345555443367482083012891874E-01;
-
-    /** The constant {@code C5} defined in {@code DGAM1}. */
-    private static final double INV_GAMMA1P_M1_C5 = -.962197152787697356211492167234820E-02;
-
-    /** The constant {@code C6} defined in {@code DGAM1}. */
-    private static final double INV_GAMMA1P_M1_C6 = .721894324666309954239501034044657E-02;
-
-    /** The constant {@code C7} defined in {@code DGAM1}. */
-    private static final double INV_GAMMA1P_M1_C7 = -.116516759185906511211397108401839E-02;
-
-    /** The constant {@code C8} defined in {@code DGAM1}. */
-    private static final double INV_GAMMA1P_M1_C8 = -.215241674114950972815729963053648E-03;
-
-    /** The constant {@code C9} defined in {@code DGAM1}. */
-    private static final double INV_GAMMA1P_M1_C9 = .128050282388116186153198626328164E-03;
-
-    /** The constant {@code C10} defined in {@code DGAM1}. */
-    private static final double INV_GAMMA1P_M1_C10 = -.201348547807882386556893914210218E-04;
-
-    /** The constant {@code C11} defined in {@code DGAM1}. */
-    private static final double INV_GAMMA1P_M1_C11 = -.125049348214267065734535947383309E-05;
-
-    /** The constant {@code C12} defined in {@code DGAM1}. */
-    private static final double INV_GAMMA1P_M1_C12 = .113302723198169588237412962033074E-05;
-
-    /** The constant {@code C13} defined in {@code DGAM1}. */
-    private static final double INV_GAMMA1P_M1_C13 = -.205633841697760710345015413002057E-06;
-
-    /**
-     * Class contains only static methods.
-     */
-    private Gamma() {}
-
-    /**
-     * Computes the function \( \ln \Gamma(x) \) for \( x \gt 0 \).
-     *
-     * 

-     * For \( x \leq 8 \), the implementation is based on the double precision
-     * implementation in the NSWC Library of Mathematics Subroutines,
-     * {@code DGAMLN}. For \( x \geq 8 \), the implementation is based on
-     * 

-     *
-     * 
-     *
-     * @param x Argument.
-     * @return \( \ln \Gamma(x) \), or {@code NaN} if {@code x <= 0}.
-     */
-    public static double logGamma(double x) {
-        double ret;
-
-        if (Double.isNaN(x) || (x <= 0.0)) {
-            ret = Double.NaN;
-        } else if (x < 0.5) {
-            return logGamma1p(x) - FastMath.log(x);
-        } else if (x <= 2.5) {
-            return logGamma1p((x - 0.5) - 0.5);
-        } else if (x <= 8.0) {
-            final int n = (int) FastMath.floor(x - 1.5);
-            double prod = 1.0;
-            for (int i = 1; i <= n; i++) {
-                prod *= x - i;
-            }
-            return logGamma1p(x - (n + 1)) + FastMath.log(prod);
-        } else {
-            double sum = lanczos(x);
-            double tmp = x + LANCZOS_G + .5;
-            ret = ((x + .5) * FastMath.log(tmp)) - tmp +
-                HALF_LOG_2_PI + FastMath.log(sum / x);
-        }
-
-        return ret;
-    }
-
-    /**
-     * Computes the regularized gamma function \( P(a, x) \).
-     *
-     * @param a Parameter \( a \).
-     * @param x Value.
-     * @return \( P(a, x) \)
-     * @throws MaxCountExceededException if the algorithm fails to converge.
-     */
-    public static double regularizedGammaP(double a, double x) {
-        return regularizedGammaP(a, x, DEFAULT_EPSILON, Integer.MAX_VALUE);
-    }
-
-    /**
-     * Computes the regularized gamma function \( P(a, x) \).
-     *
-     * The implementation of this method is based on:
-     * 
-     *
-     * @param a Parameter \( a \).
-     * @param x Argument.
-     * @param epsilon When the absolute value of the nth item in the
-     * series is less than epsilon the approximation ceases to calculate
-     * further elements in the series.
-     * @param maxIterations Maximum number of "iterations" to complete.
-     * @return \( P(a, x) \)
-     * @throws MaxCountExceededException if the algorithm fails to converge.
-     */
-    public static double regularizedGammaP(double a,
-                                           double x,
-                                           double epsilon,
-                                           int maxIterations) {
-        double ret;
-
-        if (Double.isNaN(a) || Double.isNaN(x) || (a <= 0.0) || (x < 0.0)) {
-            ret = Double.NaN;
-        } else if (x == 0.0) {
-            ret = 0.0;
-        } else if (x >= a + 1) {
-            // use regularizedGammaQ because it should converge faster in this
-            // case.
-            ret = 1.0 - regularizedGammaQ(a, x, epsilon, maxIterations);
-        } else {
-            // calculate series
-            double n = 0.0; // current element index
-            double an = 1.0 / a; // n-th element in the series
-            double sum = an; // partial sum
-            while (FastMath.abs(an/sum) > epsilon &&
-                   n < maxIterations &&
-                   sum < Double.POSITIVE_INFINITY) {
-                // compute next element in the series
-                n += 1.0;
-                an *= x / (a + n);
-
-                // update partial sum
-                sum += an;
-            }
-            if (n >= maxIterations) {
-                throw new MaxCountExceededException(maxIterations);
-            } else if (Double.isInfinite(sum)) {
-                ret = 1.0;
-            } else {
-                ret = FastMath.exp(-x + (a * FastMath.log(x)) - logGamma(a)) * sum;
-            }
-        }
-
-        return ret;
-    }
-
-    /**
-     * Computes the regularized gamma function \( Q(a, x) = 1 - P(a, x) \).
-     *
-     * @param a Parameter \( a \).
-     * @param x Argument.
-     * @return \( Q(a, x) \)
-     * @throws MaxCountExceededException if the algorithm fails to converge.
-     */
-    public static double regularizedGammaQ(double a, double x) {
-        return regularizedGammaQ(a, x, DEFAULT_EPSILON, Integer.MAX_VALUE);
-    }
-
-    /**
-     * Computes the regularized gamma function \( Q(a, x) = 1 - P(a, x) \).
-     *
-     * The implementation of this method is based on:
-     * 
-     *
-     * @param a Parameter \( a \).
-     * @param x Argument.
-     * @param epsilon When the absolute value of the nth item in the
-     * series is less than epsilon the approximation ceases to calculate
-     * further elements in the series.
-     * @param maxIterations Maximum number of "iterations" to complete.
-     * @return \( Q(a, x) \)
-     * @throws MaxCountExceededException if the algorithm fails to converge.
-     */
-    public static double regularizedGammaQ(final double a,
-                                           double x,
-                                           double epsilon,
-                                           int maxIterations) {
-        double ret;
-
-        if (Double.isNaN(a) || Double.isNaN(x) || (a <= 0.0) || (x < 0.0)) {
-            ret = Double.NaN;
-        } else if (x == 0.0) {
-            ret = 1.0;
-        } else if (x < a + 1.0) {
-            // use regularizedGammaP because it should converge faster in this
-            // case.
-            ret = 1.0 - regularizedGammaP(a, x, epsilon, maxIterations);
-        } else {
-            // create continued fraction
-            ContinuedFraction cf = new ContinuedFraction() {
-
-                /** {@inheritDoc} */
-                @Override
-                protected double getA(int n, double x) {
-                    return ((2.0 * n) + 1.0) - a + x;
-                }
-
-                /** {@inheritDoc} */
-                @Override
-                protected double getB(int n, double x) {
-                    return n * (a - n);
-                }
-            };
-
-            ret = 1.0 / cf.evaluate(x, epsilon, maxIterations);
-            ret = FastMath.exp(-x + (a * FastMath.log(x)) - logGamma(a)) * ret;
-        }
-
-        return ret;
-    }
-
-
-    /**
-     * Computes the digamma function, defined as the logarithmic derivative
-     * of the \( \Gamma \) function:
-     * \( \frac{d}{dx}(\ln \Gamma(x)) = \frac{\Gamma^\prime(x)}{\Gamma(x)} \).
-     *
-     * 
This is an independently written implementation of the algorithm described in
-     * Jose Bernardo, Algorithm AS 103: Psi (Digamma) Function, Applied Statistics, 1976.
-     * A 
-     * reflection formula is incorporated to improve performance on negative values.

-     *
-     * 
Some of the constants have been changed to increase accuracy at the moderate
-     * expense of run-time.  The result should be accurate to within \( 10^{-8} \)
-     * relative tolerance for \( 0 \le x \le 10^{-5} \) and within \( 10^{-8} \) absolute
-     * tolerance otherwise.

-     *
-     * @param x Argument.
-     * @return digamma(x) to within \( 10^{-8} \) relative or absolute error whichever is larger.
-     *
-     * @see Digamma
-     * @see Bernardo's original article
-     *
-     * @since 2.0
-     */
-    public static double digamma(double x) {
-        if (Double.isNaN(x) || Double.isInfinite(x)) {
-            return x;
-        }
-
-        double digamma = 0.0;
-        if (x < 0) {
-            // use reflection formula to fall back into positive values
-            digamma -= FastMath.PI / FastMath.tan(FastMath.PI * x);
-            x =  1 - x;
-        }
-
-        if (x > 0 && x <= S_LIMIT) {
-            // use method 5 from Bernardo AS103
-            // accurate to O(x)
-            return digamma -GAMMA - 1 / x;
-        }
-
-        while (x < C_LIMIT) {
-            digamma -= 1.0 / x;
-            x += 1.0;
-        }
-
-        // use method 4 (accurate to O(1/x^8)
-        double inv = 1 / (x * x);
-        //            1       1        1         1
-        // log(x) -  --- - ------ + ------- - -------
-        //           2 x   12 x^2   120 x^4   252 x^6
-        digamma += FastMath.log(x) - 0.5 / x - inv * ((1.0 / 12) + inv * (1.0 / 120 - inv / 252));
-
-        return digamma;
-    }
-
-    /**
-     * Computes the trigamma function \( \psi_1(x) = \frac{d^2}{dx^2} (\ln \Gamma(x)) \).
-     * 

-     * This function is the derivative of the {@link #digamma(double) digamma function}.
-     * 

-     *
-     * @param x Argument.
-     * @return \( \psi_1(x) \) to within \( 10^{-8} \) relative or absolute
-     * error whichever is smaller
-     *
-     * @see Trigamma
-     * @see #digamma(double)
-     *
-     * @since 2.0
-     */
-    public static double trigamma(double x) {
-        if (Double.isNaN(x) || Double.isInfinite(x)) {
-            return x;
-        }
-
-        if (x > 0 && x <= S_LIMIT) {
-            return 1 / (x * x);
-        }
-
-        if (x >= C_LIMIT) {
-            double inv = 1 / (x * x);
-            //  1    1      1       1       1
-            //  - + ---- + ---- - ----- + -----
-            //  x      2      3       5       7
-            //      2 x    6 x    30 x    42 x
-            return 1 / x + inv / 2 + inv / x * (1.0 / 6 - inv * (1.0 / 30 + inv / 42));
-        }
-
-        return trigamma(x + 1) + 1 / (x * x);
-    }
-
-    /**
-     * Computes the Lanczos approximation used to compute the gamma function.
-     *
-     * 

-     * The Lanczos approximation is related to the Gamma function by the
-     * following equation
-     * \[
-     * \Gamma(x) = \sqrt{2\pi} \, \frac{(g + x + \frac{1}{2})^{x + \frac{1}{2}} \, e^{-(g + x + \frac{1}{2})} \, \mathrm{lanczos}(x)}
-     *                                 {x}
-     * \]
-     * where \(g\) is the Lanczos constant.
-     * 

-     *
-     * @param x Argument.
-     * @return The Lanczos approximation.
-     *
-     * @see Lanczos Approximation
-     * equations (1) through (5), and Paul Godfrey's
-     * Note on the computation
-     * of the convergent Lanczos complex Gamma approximation
-     *
-     * @since 3.1
-     */
-    public static double lanczos(final double x) {
-        double sum = 0.0;
-        for (int i = LANCZOS.length - 1; i > 0; --i) {
-            sum += LANCZOS[i] / (x + i);
-        }
-        return sum + LANCZOS[0];
-    }
-
-    /**
-     * Computes the function \( \frac{1}{\Gamma(1 + x)} - 1 \) for \( -0.5 \leq x \leq 1.5 \).
-     * 

-     * This implementation is based on the double precision implementation in
-     * the NSWC Library of Mathematics Subroutines, {@code DGAM1}.
-     * 

-     *
-     * @param x Argument.
-     * @return \( \frac{1}{\Gamma(1 + x)} - 1 \)
-     * @throws NumberIsTooSmallException if {@code x < -0.5}
-     * @throws NumberIsTooLargeException if {@code x > 1.5}
-     *
-     * @since 3.1
-     */
-    public static double invGamma1pm1(final double x) {
-
-        if (x < -0.5) {
-            throw new NumberIsTooSmallException(x, -0.5, true);
-        }
-        if (x > 1.5) {
-            throw new NumberIsTooLargeException(x, 1.5, true);
-        }
-
-        final double ret;
-        final double t = x <= 0.5 ? x : (x - 0.5) - 0.5;
-        if (t < 0.0) {
-            final double a = INV_GAMMA1P_M1_A0 + t * INV_GAMMA1P_M1_A1;
-            double b = INV_GAMMA1P_M1_B8;
-            b = INV_GAMMA1P_M1_B7 + t * b;
-            b = INV_GAMMA1P_M1_B6 + t * b;
-            b = INV_GAMMA1P_M1_B5 + t * b;
-            b = INV_GAMMA1P_M1_B4 + t * b;
-            b = INV_GAMMA1P_M1_B3 + t * b;
-            b = INV_GAMMA1P_M1_B2 + t * b;
-            b = INV_GAMMA1P_M1_B1 + t * b;
-            b = 1.0 + t * b;
-
-            double c = INV_GAMMA1P_M1_C13 + t * (a / b);
-            c = INV_GAMMA1P_M1_C12 + t * c;
-            c = INV_GAMMA1P_M1_C11 + t * c;
-            c = INV_GAMMA1P_M1_C10 + t * c;
-            c = INV_GAMMA1P_M1_C9 + t * c;
-            c = INV_GAMMA1P_M1_C8 + t * c;
-            c = INV_GAMMA1P_M1_C7 + t * c;
-            c = INV_GAMMA1P_M1_C6 + t * c;
-            c = INV_GAMMA1P_M1_C5 + t * c;
-            c = INV_GAMMA1P_M1_C4 + t * c;
-            c = INV_GAMMA1P_M1_C3 + t * c;
-            c = INV_GAMMA1P_M1_C2 + t * c;
-            c = INV_GAMMA1P_M1_C1 + t * c;
-            c = INV_GAMMA1P_M1_C + t * c;
-            if (x > 0.5) {
-                ret = t * c / x;
-            } else {
-                ret = x * ((c + 0.5) + 0.5);
-            }
-        } else {
-            double p = INV_GAMMA1P_M1_P6;
-            p = INV_GAMMA1P_M1_P5 + t * p;
-            p = INV_GAMMA1P_M1_P4 + t * p;
-            p = INV_GAMMA1P_M1_P3 + t * p;
-            p = INV_GAMMA1P_M1_P2 + t * p;
-            p = INV_GAMMA1P_M1_P1 + t * p;
-            p = INV_GAMMA1P_M1_P0 + t * p;
-
-            double q = INV_GAMMA1P_M1_Q4;
-            q = INV_GAMMA1P_M1_Q3 + t * q;
-            q = INV_GAMMA1P_M1_Q2 + t * q;
-            q = INV_GAMMA1P_M1_Q1 + t * q;
-            q = 1.0 + t * q;
-
-            double c = INV_GAMMA1P_M1_C13 + (p / q) * t;
-            c = INV_GAMMA1P_M1_C12 + t * c;
-            c = INV_GAMMA1P_M1_C11 + t * c;
-            c = INV_GAMMA1P_M1_C10 + t * c;
-            c = INV_GAMMA1P_M1_C9 + t * c;
-            c = INV_GAMMA1P_M1_C8 + t * c;
-            c = INV_GAMMA1P_M1_C7 + t * c;
-            c = INV_GAMMA1P_M1_C6 + t * c;
-            c = INV_GAMMA1P_M1_C5 + t * c;
-            c = INV_GAMMA1P_M1_C4 + t * c;
-            c = INV_GAMMA1P_M1_C3 + t * c;
-            c = INV_GAMMA1P_M1_C2 + t * c;
-            c = INV_GAMMA1P_M1_C1 + t * c;
-            c = INV_GAMMA1P_M1_C0 + t * c;
-
-            if (x > 0.5) {
-                ret = (t / x) * ((c - 0.5) - 0.5);
-            } else {
-                ret = x * c;
-            }
-        }
-
-        return ret;
-    }
-
-    /**
-     * Computes the function \( \ln \Gamma(1 + x) \) for \( -0.5 \leq x \leq 1.5 \).
-     * 

-     * This implementation is based on the double precision implementation in
-     * the NSWC Library of Mathematics Subroutines, {@code DGMLN1}.
-     * 

-     *
-     * @param x Argument.
-     * @return \( \ln \Gamma(1 + x) \)
-     * @throws NumberIsTooSmallException if {@code x < -0.5}.
-     * @throws NumberIsTooLargeException if {@code x > 1.5}.
-     * @since 3.1
-     */
-    public static double logGamma1p(final double x)
-        throws NumberIsTooSmallException, NumberIsTooLargeException {
-
-        if (x < -0.5) {
-            throw new NumberIsTooSmallException(x, -0.5, true);
-        }
-        if (x > 1.5) {
-            throw new NumberIsTooLargeException(x, 1.5, true);
-        }
-
-        return -FastMath.log1p(invGamma1pm1(x));
-    }
-
-
-    /**
-     * Computes the value of \( \Gamma(x) \).
-     * 

-     * Based on the NSWC Library of Mathematics Subroutines double
-     * precision implementation, {@code DGAMMA}.
-     * 

-     *
-     * @param x Argument.
-     * @return \( \Gamma(x) \)
-     *
-     * @since 3.1
-     */
-    public static double gamma(final double x) {
-
-        if ((x == FastMath.rint(x)) && (x <= 0.0)) {
-            return Double.NaN;
-        }
-
-        final double ret;
-        final double absX = FastMath.abs(x);
-        if (absX <= 20.0) {
-            if (x >= 1.0) {
-                /*
-                 * From the recurrence relation
-                 * Gamma(x) = (x - 1) * ... * (x - n) * Gamma(x - n),
-                 * then
-                 * Gamma(t) = 1 / [1 + invGamma1pm1(t - 1)],
-                 * where t = x - n. This means that t must satisfy
-                 * -0.5 <= t - 1 <= 1.5.
-                 */
-                double prod = 1.0;
-                double t = x;
-                while (t > 2.5) {
-                    t -= 1.0;
-                    prod *= t;
-                }
-                ret = prod / (1.0 + invGamma1pm1(t - 1.0));
-            } else {
-                /*
-                 * From the recurrence relation
-                 * Gamma(x) = Gamma(x + n + 1) / [x * (x + 1) * ... * (x + n)]
-                 * then
-                 * Gamma(x + n + 1) = 1 / [1 + invGamma1pm1(x + n)],
-                 * which requires -0.5 <= x + n <= 1.5.
-                 */
-                double prod = x;
-                double t = x;
-                while (t < -0.5) {
-                    t += 1.0;
-                    prod *= t;
-                }
-                ret = 1.0 / (prod * (1.0 + invGamma1pm1(t)));
-            }
-        } else {
-            final double y = absX + LANCZOS_G + 0.5;
-            final double gammaAbs = SQRT_TWO_PI / absX *
-                                    FastMath.pow(y, absX + 0.5) *
-                                    FastMath.exp(-y) * lanczos(absX);
-            if (x > 0.0) {
-                ret = gammaAbs;
-            } else {
-                /*
-                 * From the reflection formula
-                 * Gamma(x) * Gamma(1 - x) * sin(pi * x) = pi,
-                 * and the recurrence relation
-                 * Gamma(1 - x) = -x * Gamma(-x),
-                 * it is found
-                 * Gamma(x) = -pi / [x * sin(pi * x) * Gamma(-x)].
-                 */
-                ret = -FastMath.PI /
-                      (x * FastMath.sin(FastMath.PI * x) * gammaAbs);
-            }
-        }
-        return ret;
-    }
-}
diff --git a/src/test/java/org/apache/commons/math4/special/BetaTest.java b/src/test/java/org/apache/commons/math4/special/BetaTest.java
deleted file mode 100644
index 5a42dbc0c..000000000
--- a/src/test/java/org/apache/commons/math4/special/BetaTest.java
+++ /dev/null
@@ -1,979 +0,0 @@
-/*
- * Licensed to the Apache Software Foundation (ASF) under one or more
- * contributor license agreements.  See the NOTICE file distributed with
- * this work for additional information regarding copyright ownership.
- * The ASF licenses this file to You under the Apache License, Version 2.0
- * (the "License"); you may not use this file except in compliance with
- * the License.  You may obtain a copy of the License at
- *
- *      http://www.apache.org/licenses/LICENSE-2.0
- *
- * Unless required by applicable law or agreed to in writing, software
- * distributed under the License is distributed on an "AS IS" BASIS,
- * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
- * See the License for the specific language governing permissions and
- * limitations under the License.
- */
-package org.apache.commons.math4.special;
-
-import java.lang.reflect.InvocationTargetException;
-import java.lang.reflect.Method;
-
-import org.apache.commons.math4.TestUtils;
-import org.apache.commons.math4.exception.MathIllegalArgumentException;
-import org.apache.commons.math4.exception.NumberIsTooSmallException;
-import org.apache.commons.math4.exception.OutOfRangeException;
-import org.apache.commons.math4.special.Beta;
-import org.apache.commons.math4.util.FastMath;
-import org.junit.Assert;
-import org.junit.Test;
-
-/**
- */
-public class BetaTest {
-
-    /*
-     * Use reflection to test private methods.
-     */
-    private static final Method LOG_GAMMA_SUM_METHOD;
-
-    private static final Method LOG_GAMMA_MINUS_LOG_GAMMA_SUM_METHOD;
-
-    private static final Method SUM_DELTA_MINUS_DELTA_SUM_METHOD;
-
-    static {
-        final Class b;
-        final Class d = Double.TYPE;
-        b = Beta.class;
-        Method m = null;
-        try {
-            m = b.getDeclaredMethod("logGammaSum", d, d);
-        } catch (NoSuchMethodException e) {
-            Assert.fail(e.getMessage());
-        }
-        LOG_GAMMA_SUM_METHOD = m;
-        LOG_GAMMA_SUM_METHOD.setAccessible(true);
-
-        m = null;
-        try {
-            m = b.getDeclaredMethod("logGammaMinusLogGammaSum",d, d);
-        } catch (NoSuchMethodException e) {
-            Assert.fail(e.getMessage());
-        }
-        LOG_GAMMA_MINUS_LOG_GAMMA_SUM_METHOD = m;
-        LOG_GAMMA_MINUS_LOG_GAMMA_SUM_METHOD.setAccessible(true);
-
-        m = null;
-        try {
-            m = b.getDeclaredMethod("sumDeltaMinusDeltaSum",d, d);
-        } catch (NoSuchMethodException e) {
-            Assert.fail(e.getMessage());
-        }
-        SUM_DELTA_MINUS_DELTA_SUM_METHOD = m;
-        SUM_DELTA_MINUS_DELTA_SUM_METHOD.setAccessible(true);
-    }
-
-    private void testRegularizedBeta(double expected, double x,
-                                     double a, double b) {
-        double actual = Beta.regularizedBeta(x, a, b);
-        TestUtils.assertEquals(expected, actual, 10e-15);
-    }
-
-    private void testLogBeta(double expected, double a, double b) {
-        double actual = Beta.logBeta(a, b);
-        TestUtils.assertEquals(expected, actual, 10e-15);
-    }
-
-    @Test
-    public void testRegularizedBetaNanPositivePositive() {
-        testRegularizedBeta(Double.NaN, Double.NaN, 1.0, 1.0);
-    }
-
-    @Test
-    public void testRegularizedBetaPositiveNanPositive() {
-        testRegularizedBeta(Double.NaN, 0.5, Double.NaN, 1.0);
-    }
-
-    @Test
-    public void testRegularizedBetaPositivePositiveNan() {
-        testRegularizedBeta(Double.NaN, 0.5, 1.0, Double.NaN);
-    }
-
-    @Test
-    public void testRegularizedBetaNegativePositivePositive() {
-        testRegularizedBeta(Double.NaN, -0.5, 1.0, 2.0);
-    }
-
-    @Test
-    public void testRegularizedBetaPositiveNegativePositive() {
-        testRegularizedBeta(Double.NaN, 0.5, -1.0, 2.0);
-    }
-
-    @Test
-    public void testRegularizedBetaPositivePositiveNegative() {
-        testRegularizedBeta(Double.NaN, 0.5, 1.0, -2.0);
-    }
-
-    @Test
-    public void testRegularizedBetaZeroPositivePositive() {
-        testRegularizedBeta(0.0, 0.0, 1.0, 2.0);
-    }
-
-    @Test
-    public void testRegularizedBetaPositiveZeroPositive() {
-        testRegularizedBeta(Double.NaN, 0.5, 0.0, 2.0);
-    }
-
-    @Test
-    public void testRegularizedBetaPositivePositiveZero() {
-        testRegularizedBeta(Double.NaN, 0.5, 1.0, 0.0);
-    }
-
-    @Test
-    public void testRegularizedBetaPositivePositivePositive() {
-        testRegularizedBeta(0.75, 0.5, 1.0, 2.0);
-    }
-
-    @Test
-    public void testRegularizedBetaTinyArgument() {
-        double actual = Beta.regularizedBeta(1e-17, 1.0, 1e12);
-        // This value is from R: pbeta(1e-17,1,1e12)
-        TestUtils.assertEquals(9.999950000166648e-6, actual, 1e-16);
-    }
-
-    @Test
-    public void testMath1067() {
-        final double x = 0.22580645161290325;
-        final double a = 64.33333333333334;
-        final double b = 223;
-
-        try {
-            Beta.regularizedBeta(x, a, b, 1e-14, 10000);
-        } catch (StackOverflowError error) {
-            Assert.fail("Infinite recursion");
-        }
-    }
-
-    @Test
-    public void testLogBetaNanPositive() {
-        testLogBeta(Double.NaN, Double.NaN, 2.0);
-    }
-
-    @Test
-    public void testLogBetaPositiveNan() {
-        testLogBeta(Double.NaN, 1.0, Double.NaN);
-    }
-
-    @Test
-    public void testLogBetaNegativePositive() {
-        testLogBeta(Double.NaN, -1.0, 2.0);
-    }
-
-    @Test
-    public void testLogBetaPositiveNegative() {
-        testLogBeta(Double.NaN, 1.0, -2.0);
-    }
-
-    @Test
-    public void testLogBetaZeroPositive() {
-        testLogBeta(Double.NaN, 0.0, 2.0);
-    }
-
-    @Test
-    public void testLogBetaPositiveZero() {
-        testLogBeta(Double.NaN, 1.0, 0.0);
-    }
-
-    @Test
-    public void testLogBetaPositivePositive() {
-        testLogBeta(-0.693147180559945, 1.0, 2.0);
-    }
-
-    /**
-     * Reference data for the {@link #logGammaSum(double, double)}
-     * function. This data was generated with the following
-     * Maxima script.
-     *
-     * 
-     * kill(all);
-     *
-     * fpprec : 64;
-     * gsumln(a, b) := log(gamma(a + b));
-     *
-     * x : [1.0b0, 1.125b0, 1.25b0, 1.375b0, 1.5b0, 1.625b0, 1.75b0, 1.875b0, 2.0b0];
-     *
-     * for i : 1 while i <= length(x) do
-     *   for j : 1 while j <= length(x) do block(
-     *     a : x[i],
-     *     b : x[j],
-     *     print("{", float(a), ",", float(b), ",", float(gsumln(a, b)), "},")
-     *   );
-     * 

-     */
-    private static final double[][] LOG_GAMMA_SUM_REF = {
-        { 1.0 , 1.0 , 0.0 },
-        { 1.0 , 1.125 , .05775985153034387 },
-        { 1.0 , 1.25 , .1248717148923966 },
-        { 1.0 , 1.375 , .2006984603774558 },
-        { 1.0 , 1.5 , .2846828704729192 },
-        { 1.0 , 1.625 , .3763336820249054 },
-        { 1.0 , 1.75 , .4752146669149371 },
-        { 1.0 , 1.875 , .5809359740231859 },
-        { 1.0 , 2.0 , .6931471805599453 },
-        { 1.125 , 1.0 , .05775985153034387 },
-        { 1.125 , 1.125 , .1248717148923966 },
-        { 1.125 , 1.25 , .2006984603774558 },
-        { 1.125 , 1.375 , .2846828704729192 },
-        { 1.125 , 1.5 , .3763336820249054 },
-        { 1.125 , 1.625 , .4752146669149371 },
-        { 1.125 , 1.75 , .5809359740231859 },
-        { 1.125 , 1.875 , .6931471805599453 },
-        { 1.125 , 2.0 , 0.811531653906724 },
-        { 1.25 , 1.0 , .1248717148923966 },
-        { 1.25 , 1.125 , .2006984603774558 },
-        { 1.25 , 1.25 , .2846828704729192 },
-        { 1.25 , 1.375 , .3763336820249054 },
-        { 1.25 , 1.5 , .4752146669149371 },
-        { 1.25 , 1.625 , .5809359740231859 },
-        { 1.25 , 1.75 , .6931471805599453 },
-        { 1.25 , 1.875 , 0.811531653906724 },
-        { 1.25 , 2.0 , .9358019311087253 },
-        { 1.375 , 1.0 , .2006984603774558 },
-        { 1.375 , 1.125 , .2846828704729192 },
-        { 1.375 , 1.25 , .3763336820249054 },
-        { 1.375 , 1.375 , .4752146669149371 },
-        { 1.375 , 1.5 , .5809359740231859 },
-        { 1.375 , 1.625 , .6931471805599453 },
-        { 1.375 , 1.75 , 0.811531653906724 },
-        { 1.375 , 1.875 , .9358019311087253 },
-        { 1.375 , 2.0 , 1.06569589786406 },
-        { 1.5 , 1.0 , .2846828704729192 },
-        { 1.5 , 1.125 , .3763336820249054 },
-        { 1.5 , 1.25 , .4752146669149371 },
-        { 1.5 , 1.375 , .5809359740231859 },
-        { 1.5 , 1.5 , .6931471805599453 },
-        { 1.5 , 1.625 , 0.811531653906724 },
-        { 1.5 , 1.75 , .9358019311087253 },
-        { 1.5 , 1.875 , 1.06569589786406 },
-        { 1.5 , 2.0 , 1.200973602347074 },
-        { 1.625 , 1.0 , .3763336820249054 },
-        { 1.625 , 1.125 , .4752146669149371 },
-        { 1.625 , 1.25 , .5809359740231859 },
-        { 1.625 , 1.375 , .6931471805599453 },
-        { 1.625 , 1.5 , 0.811531653906724 },
-        { 1.625 , 1.625 , .9358019311087253 },
-        { 1.625 , 1.75 , 1.06569589786406 },
-        { 1.625 , 1.875 , 1.200973602347074 },
-        { 1.625 , 2.0 , 1.341414578068493 },
-        { 1.75 , 1.0 , .4752146669149371 },
-        { 1.75 , 1.125 , .5809359740231859 },
-        { 1.75 , 1.25 , .6931471805599453 },
-        { 1.75 , 1.375 , 0.811531653906724 },
-        { 1.75 , 1.5 , .9358019311087253 },
-        { 1.75 , 1.625 , 1.06569589786406 },
-        { 1.75 , 1.75 , 1.200973602347074 },
-        { 1.75 , 1.875 , 1.341414578068493 },
-        { 1.75 , 2.0 , 1.486815578593417 },
-        { 1.875 , 1.0 , .5809359740231859 },
-        { 1.875 , 1.125 , .6931471805599453 },
-        { 1.875 , 1.25 , 0.811531653906724 },
-        { 1.875 , 1.375 , .9358019311087253 },
-        { 1.875 , 1.5 , 1.06569589786406 },
-        { 1.875 , 1.625 , 1.200973602347074 },
-        { 1.875 , 1.75 , 1.341414578068493 },
-        { 1.875 , 1.875 , 1.486815578593417 },
-        { 1.875 , 2.0 , 1.6369886482725 },
-        { 2.0 , 1.0 , .6931471805599453 },
-        { 2.0 , 1.125 , 0.811531653906724 },
-        { 2.0 , 1.25 , .9358019311087253 },
-        { 2.0 , 1.375 , 1.06569589786406 },
-        { 2.0 , 1.5 , 1.200973602347074 },
-        { 2.0 , 1.625 , 1.341414578068493 },
-        { 2.0 , 1.75 , 1.486815578593417 },
-        { 2.0 , 1.875 , 1.6369886482725 },
-        { 2.0 , 2.0 , 1.791759469228055 },
-    };
-
-    private static double logGammaSum(final double a, final double b) {
-
-        /*
-         * Use reflection to access private method.
-         */
-        try {
-            return ((Double) LOG_GAMMA_SUM_METHOD.invoke(null, a, b)).doubleValue();
-        } catch (final IllegalAccessException e) {
-            Assert.fail(e.getMessage());
-        } catch (final IllegalArgumentException e) {
-            Assert.fail(e.getMessage());
-        } catch (final InvocationTargetException e) {
-            final Throwable te = e.getTargetException();
-            if (te instanceof MathIllegalArgumentException) {
-                throw (MathIllegalArgumentException) te;
-            }
-            Assert.fail(e.getMessage());
-        }
-        return Double.NaN;
-    }
-
-    @Test
-    public void testLogGammaSum() {
-        final int ulps = 2;
-        for (int i = 0; i < LOG_GAMMA_SUM_REF.length; i++) {
-            final double[] ref = LOG_GAMMA_SUM_REF[i];
-            final double a = ref[0];
-            final double b = ref[1];
-            final double expected = ref[2];
-            final double actual = logGammaSum(a, b);
-            final double tol = ulps * FastMath.ulp(expected);
-            final StringBuilder builder = new StringBuilder();
-            builder.append(a).append(", ").append(b);
-            Assert.assertEquals(builder.toString(), expected, actual, tol);
-        }
-    }
-
-    @Test(expected = OutOfRangeException.class)
-    public void testLogGammaSumPrecondition1() {
-
-        logGammaSum(0.0, 1.0);
-    }
-
-    @Test(expected = OutOfRangeException.class)
-    public void testLogGammaSumPrecondition2() {
-
-        logGammaSum(3.0, 1.0);
-    }
-
-    @Test(expected = OutOfRangeException.class)
-    public void testLogGammaSumPrecondition3() {
-
-        logGammaSum(1.0, 0.0);
-    }
-
-    @Test(expected = OutOfRangeException.class)
-    public void testLogGammaSumPrecondition4() {
-
-        logGammaSum(1.0, 3.0);
-    }
-
-    private static final double[][] LOG_GAMMA_MINUS_LOG_GAMMA_SUM_REF = {
-//        { 0.0 , 8.0 , 0.0 },
-//        { 0.0 , 9.0 , 0.0 },
-        { 0.0 , 10.0 , 0.0 },
-        { 0.0 , 11.0 , 0.0 },
-        { 0.0 , 12.0 , 0.0 },
-        { 0.0 , 13.0 , 0.0 },
-        { 0.0 , 14.0 , 0.0 },
-        { 0.0 , 15.0 , 0.0 },
-        { 0.0 , 16.0 , 0.0 },
-        { 0.0 , 17.0 , 0.0 },
-        { 0.0 , 18.0 , 0.0 },
-//        { 1.0 , 8.0 , - 2.079441541679836 },
-//        { 1.0 , 9.0 , - 2.19722457733622 },
-        { 1.0 , 10.0 , - 2.302585092994046 },
-        { 1.0 , 11.0 , - 2.397895272798371 },
-        { 1.0 , 12.0 , - 2.484906649788 },
-        { 1.0 , 13.0 , - 2.564949357461537 },
-        { 1.0 , 14.0 , - 2.639057329615258 },
-        { 1.0 , 15.0 , - 2.70805020110221 },
-        { 1.0 , 16.0 , - 2.772588722239781 },
-        { 1.0 , 17.0 , - 2.833213344056216 },
-        { 1.0 , 18.0 , - 2.890371757896165 },
-//        { 2.0 , 8.0 , - 4.276666119016055 },
-//        { 2.0 , 9.0 , - 4.499809670330265 },
-        { 2.0 , 10.0 , - 4.700480365792417 },
-        { 2.0 , 11.0 , - 4.882801922586371 },
-        { 2.0 , 12.0 , - 5.049856007249537 },
-        { 2.0 , 13.0 , - 5.204006687076795 },
-        { 2.0 , 14.0 , - 5.347107530717468 },
-        { 2.0 , 15.0 , - 5.480638923341991 },
-        { 2.0 , 16.0 , - 5.605802066295998 },
-        { 2.0 , 17.0 , - 5.723585101952381 },
-        { 2.0 , 18.0 , - 5.834810737062605 },
-//        { 3.0 , 8.0 , - 6.579251212010101 },
-//        { 3.0 , 9.0 , - 6.897704943128636 },
-        { 3.0 , 10.0 , - 7.185387015580416 },
-        { 3.0 , 11.0 , - 7.447751280047908 },
-        { 3.0 , 12.0 , - 7.688913336864796 },
-        { 3.0 , 13.0 , - 7.912056888179006 },
-        { 3.0 , 14.0 , - 8.11969625295725 },
-        { 3.0 , 15.0 , - 8.313852267398207 },
-        { 3.0 , 16.0 , - 8.496173824192162 },
-        { 3.0 , 17.0 , - 8.668024081118821 },
-        { 3.0 , 18.0 , - 8.830543010616596 },
-//        { 4.0 , 8.0 , - 8.977146484808472 },
-//        { 4.0 , 9.0 , - 9.382611592916636 },
-        { 4.0 , 10.0 , - 9.750336373041954 },
-        { 4.0 , 11.0 , - 10.08680860966317 },
-        { 4.0 , 12.0 , - 10.39696353796701 },
-        { 4.0 , 13.0 , - 10.68464561041879 },
-        { 4.0 , 14.0 , - 10.95290959701347 },
-        { 4.0 , 15.0 , - 11.20422402529437 },
-        { 4.0 , 16.0 , - 11.4406128033586 },
-        { 4.0 , 17.0 , - 11.66375635467281 },
-        { 4.0 , 18.0 , - 11.87506544834002 },
-//        { 5.0 , 8.0 , - 11.46205313459647 },
-//        { 5.0 , 9.0 , - 11.94756095037817 },
-        { 5.0 , 10.0 , - 12.38939370265721 },
-        { 5.0 , 11.0 , - 12.79485881076538 },
-        { 5.0 , 12.0 , - 13.16955226020679 },
-        { 5.0 , 13.0 , - 13.517858954475 },
-        { 5.0 , 14.0 , - 13.84328135490963 },
-        { 5.0 , 15.0 , - 14.14866300446081 },
-        { 5.0 , 16.0 , - 14.43634507691259 },
-        { 5.0 , 17.0 , - 14.70827879239624 },
-        { 5.0 , 18.0 , - 14.96610790169833 },
-//        { 6.0 , 8.0 , - 14.02700249205801 },
-//        { 6.0 , 9.0 , - 14.58661827999343 },
-        { 6.0 , 10.0 , - 15.09744390375942 },
-        { 6.0 , 11.0 , - 15.56744753300516 },
-        { 6.0 , 12.0 , - 16.002765604263 },
-        { 6.0 , 13.0 , - 16.40823071237117 },
-        { 6.0 , 14.0 , - 16.78772033407607 },
-        { 6.0 , 15.0 , - 17.14439527801481 },
-        { 6.0 , 16.0 , - 17.48086751463602 },
-        { 6.0 , 17.0 , - 17.79932124575455 },
-        { 6.0 , 18.0 , - 18.10160211762749 },
-//        { 7.0 , 8.0 , - 16.66605982167327 },
-//        { 7.0 , 9.0 , - 17.29466848109564 },
-        { 7.0 , 10.0 , - 17.8700326259992 },
-        { 7.0 , 11.0 , - 18.40066087706137 },
-        { 7.0 , 12.0 , - 18.89313736215917 },
-        { 7.0 , 13.0 , - 19.35266969153761 },
-        { 7.0 , 14.0 , - 19.78345260763006 },
-        { 7.0 , 15.0 , - 20.18891771573823 },
-        { 7.0 , 16.0 , - 20.57190996799433 },
-        { 7.0 , 17.0 , - 20.9348154616837 },
-        { 7.0 , 18.0 , - 21.27965594797543 },
-//        { 8.0 , 8.0 , - 19.37411002277548 },
-//        { 8.0 , 9.0 , - 20.06725720333542 },
-        { 8.0 , 10.0 , - 20.70324597005542 },
-        { 8.0 , 11.0 , - 21.29103263495754 },
-        { 8.0 , 12.0 , - 21.83757634132561 },
-        { 8.0 , 13.0 , - 22.3484019650916 },
-        { 8.0 , 14.0 , - 22.82797504535349 },
-        { 8.0 , 15.0 , - 23.27996016909654 },
-        { 8.0 , 16.0 , - 23.70740418392348 },
-        { 8.0 , 17.0 , - 24.11286929203165 },
-        { 8.0 , 18.0 , - 24.49853177284363 },
-//        { 9.0 , 8.0 , - 22.14669874501526 },
-//        { 9.0 , 9.0 , - 22.90047054739164 },
-        { 9.0 , 10.0 , - 23.59361772795159 },
-        { 9.0 , 11.0 , - 24.23547161412398 },
-        { 9.0 , 12.0 , - 24.8333086148796 },
-        { 9.0 , 13.0 , - 25.39292440281502 },
-        { 9.0 , 14.0 , - 25.9190174987118 },
-        { 9.0 , 15.0 , - 26.41545438502569 },
-        { 9.0 , 16.0 , - 26.88545801427143 },
-        { 9.0 , 17.0 , - 27.33174511689985 },
-        { 9.0 , 18.0 , - 27.75662831086511 },
-//        { 10.0 , 8.0 , - 24.97991208907148 },
-//        { 10.0 , 9.0 , - 25.7908423052878 },
-        { 10.0 , 10.0 , - 26.53805670711802 },
-        { 10.0 , 11.0 , - 27.23120388767797 },
-        { 10.0 , 12.0 , - 27.87783105260302 },
-        { 10.0 , 13.0 , - 28.48396685617334 },
-        { 10.0 , 14.0 , - 29.05451171464095 },
-        { 10.0 , 15.0 , - 29.59350821537364 },
-        { 10.0 , 16.0 , - 30.10433383913963 },
-        { 10.0 , 17.0 , - 30.58984165492133 },
-        { 10.0 , 18.0 , - 31.05246517686944 },
-    };
-
-    private static double logGammaMinusLogGammaSum(final double a, final double b) {
-
-        /*
-         * Use reflection to access private method.
-         */
-        try {
-            final Method m = LOG_GAMMA_MINUS_LOG_GAMMA_SUM_METHOD;
-            return ((Double) m.invoke(null, a, b)).doubleValue();
-        } catch (final IllegalAccessException e) {
-            Assert.fail(e.getMessage());
-        } catch (final IllegalArgumentException e) {
-            Assert.fail(e.getMessage());
-        } catch (final InvocationTargetException e) {
-            final Throwable te = e.getTargetException();
-            if (te instanceof MathIllegalArgumentException) {
-                throw (MathIllegalArgumentException) te;
-            }
-            Assert.fail(e.getMessage());
-        }
-        return Double.NaN;
-    }
-
-    @Test
-    public void testLogGammaMinusLogGammaSum() {
-        final int ulps = 4;
-        for (int i = 0; i < LOG_GAMMA_MINUS_LOG_GAMMA_SUM_REF.length; i++) {
-            final double[] ref = LOG_GAMMA_MINUS_LOG_GAMMA_SUM_REF[i];
-            final double a = ref[0];
-            final double b = ref[1];
-            final double expected = ref[2];
-            final double actual = logGammaMinusLogGammaSum(a, b);
-            final double tol = ulps * FastMath.ulp(expected);
-            final StringBuilder builder = new StringBuilder();
-            builder.append(a).append(", ").append(b);
-            Assert.assertEquals(builder.toString(), expected, actual, tol);
-        }
-    }
-
-    @Test(expected = NumberIsTooSmallException.class)
-    public void testLogGammaMinusLogGammaSumPrecondition1() {
-        logGammaMinusLogGammaSum(-1.0, 8.0);
-    }
-
-    @Test(expected = NumberIsTooSmallException.class)
-    public void testLogGammaMinusLogGammaSumPrecondition2() {
-        logGammaMinusLogGammaSum(1.0, 7.0);
-    }
-
-    private static final double[][] SUM_DELTA_MINUS_DELTA_SUM_REF = {
-        { 10.0 , 10.0 , .01249480717472882 },
-        { 10.0 , 11.0 , .01193628470267385 },
-        { 10.0 , 12.0 , .01148578547212797 },
-        { 10.0 , 13.0 , .01111659739668398 },
-        { 10.0 , 14.0 , .01080991216314295 },
-        { 10.0 , 15.0 , .01055214134859758 },
-        { 10.0 , 16.0 , .01033324912491747 },
-        { 10.0 , 17.0 , .01014568069918883 },
-        { 10.0 , 18.0 , .009983653199146491 },
-        { 10.0 , 19.0 , .009842674320242729 },
-        { 10.0 , 20.0 , 0.0097192081956071 },
-        { 11.0 , 10.0 , .01193628470267385 },
-        { 11.0 , 11.0 , .01135973290745925 },
-        { 11.0 , 12.0 , .01089355537047828 },
-        { 11.0 , 13.0 , .01051064829297728 },
-        { 11.0 , 14.0 , 0.0101918899639826 },
-        { 11.0 , 15.0 , .009923438811859604 },
-        { 11.0 , 16.0 , .009695052724952705 },
-        { 11.0 , 17.0 , 0.00949900745283617 },
-        { 11.0 , 18.0 , .009329379874933402 },
-        { 11.0 , 19.0 , 0.00918156080743147 },
-        { 11.0 , 20.0 , 0.00905191635141762 },
-        { 12.0 , 10.0 , .01148578547212797 },
-        { 12.0 , 11.0 , .01089355537047828 },
-        { 12.0 , 12.0 , .01041365883144029 },
-        { 12.0 , 13.0 , .01001867865848564 },
-        { 12.0 , 14.0 , 0.00968923999191334 },
-        { 12.0 , 15.0 , .009411294976563555 },
-        { 12.0 , 16.0 , .009174432043268762 },
-        { 12.0 , 17.0 , .008970786693291802 },
-        { 12.0 , 18.0 , .008794318926790865 },
-        { 12.0 , 19.0 , .008640321527910711 },
-        { 12.0 , 20.0 , .008505077879954796 },
-        { 13.0 , 10.0 , .01111659739668398 },
-        { 13.0 , 11.0 , .01051064829297728 },
-        { 13.0 , 12.0 , .01001867865848564 },
-        { 13.0 , 13.0 , .009613018147953376 },
-        { 13.0 , 14.0 , .009274085618154277 },
-        { 13.0 , 15.0 , 0.0089876637564166 },
-        { 13.0 , 16.0 , .008743200745261382 },
-        { 13.0 , 17.0 , .008532715206686251 },
-        { 13.0 , 18.0 , .008350069108807093 },
-        { 13.0 , 19.0 , .008190472517984874 },
-        { 13.0 , 20.0 , .008050138630244345 },
-        { 14.0 , 10.0 , .01080991216314295 },
-        { 14.0 , 11.0 , 0.0101918899639826 },
-        { 14.0 , 12.0 , 0.00968923999191334 },
-        { 14.0 , 13.0 , .009274085618154277 },
-        { 14.0 , 14.0 , .008926676241967286 },
-        { 14.0 , 15.0 , .008632654302369184 },
-        { 14.0 , 16.0 , .008381351102615795 },
-        { 14.0 , 17.0 , .008164687232662443 },
-        { 14.0 , 18.0 , .007976441942841219 },
-        { 14.0 , 19.0 , .007811755112234388 },
-        { 14.0 , 20.0 , .007666780069317652 },
-        { 15.0 , 10.0 , .01055214134859758 },
-        { 15.0 , 11.0 , .009923438811859604 },
-        { 15.0 , 12.0 , .009411294976563555 },
-        { 15.0 , 13.0 , 0.0089876637564166 },
-        { 15.0 , 14.0 , .008632654302369184 },
-        { 15.0 , 15.0 , 0.00833179217417291 },
-        { 15.0 , 16.0 , .008074310643041299 },
-        { 15.0 , 17.0 , .007852047581145882 },
-        { 15.0 , 18.0 , .007658712051540045 },
-        { 15.0 , 19.0 , .007489384065757007 },
-        { 15.0 , 20.0 , .007340165635725612 },
-        { 16.0 , 10.0 , .01033324912491747 },
-        { 16.0 , 11.0 , .009695052724952705 },
-        { 16.0 , 12.0 , .009174432043268762 },
-        { 16.0 , 13.0 , .008743200745261382 },
-        { 16.0 , 14.0 , .008381351102615795 },
-        { 16.0 , 15.0 , .008074310643041299 },
-        { 16.0 , 16.0 , .007811229919967624 },
-        { 16.0 , 17.0 , .007583876618287594 },
-        { 16.0 , 18.0 , .007385899933505551 },
-        { 16.0 , 19.0 , .007212328560607852 },
-        { 16.0 , 20.0 , .007059220321091879 },
-        { 17.0 , 10.0 , .01014568069918883 },
-        { 17.0 , 11.0 , 0.00949900745283617 },
-        { 17.0 , 12.0 , .008970786693291802 },
-        { 17.0 , 13.0 , .008532715206686251 },
-        { 17.0 , 14.0 , .008164687232662443 },
-        { 17.0 , 15.0 , .007852047581145882 },
-        { 17.0 , 16.0 , .007583876618287594 },
-        { 17.0 , 17.0 , .007351882161431358 },
-        { 17.0 , 18.0 , .007149662089534654 },
-        { 17.0 , 19.0 , .006972200907152378 },
-        { 17.0 , 20.0 , .006815518216094137 },
-        { 18.0 , 10.0 , .009983653199146491 },
-        { 18.0 , 11.0 , .009329379874933402 },
-        { 18.0 , 12.0 , .008794318926790865 },
-        { 18.0 , 13.0 , .008350069108807093 },
-        { 18.0 , 14.0 , .007976441942841219 },
-        { 18.0 , 15.0 , .007658712051540045 },
-        { 18.0 , 16.0 , .007385899933505551 },
-        { 18.0 , 17.0 , .007149662089534654 },
-        { 18.0 , 18.0 , .006943552208153373 },
-        { 18.0 , 19.0 , .006762516574228829 },
-        { 18.0 , 20.0 , .006602541598043117 },
-        { 19.0 , 10.0 , .009842674320242729 },
-        { 19.0 , 11.0 , 0.00918156080743147 },
-        { 19.0 , 12.0 , .008640321527910711 },
-        { 19.0 , 13.0 , .008190472517984874 },
-        { 19.0 , 14.0 , .007811755112234388 },
-        { 19.0 , 15.0 , .007489384065757007 },
-        { 19.0 , 16.0 , .007212328560607852 },
-        { 19.0 , 17.0 , .006972200907152378 },
-        { 19.0 , 18.0 , .006762516574228829 },
-        { 19.0 , 19.0 , .006578188655176814 },
-        { 19.0 , 20.0 , .006415174623476747 },
-        { 20.0 , 10.0 , 0.0097192081956071 },
-        { 20.0 , 11.0 , 0.00905191635141762 },
-        { 20.0 , 12.0 , .008505077879954796 },
-        { 20.0 , 13.0 , .008050138630244345 },
-        { 20.0 , 14.0 , .007666780069317652 },
-        { 20.0 , 15.0 , .007340165635725612 },
-        { 20.0 , 16.0 , .007059220321091879 },
-        { 20.0 , 17.0 , .006815518216094137 },
-        { 20.0 , 18.0 , .006602541598043117 },
-        { 20.0 , 19.0 , .006415174623476747 },
-        { 20.0 , 20.0 , .006249349445691423 },
-    };
-
-    private static double sumDeltaMinusDeltaSum(final double a,
-                                                final double b) {
-
-        /*
-         * Use reflection to access private method.
-         */
-        try {
-            final Method m = SUM_DELTA_MINUS_DELTA_SUM_METHOD;
-            return ((Double) m.invoke(null, a, b)).doubleValue();
-        } catch (final IllegalAccessException e) {
-            Assert.fail(e.getMessage());
-        } catch (final IllegalArgumentException e) {
-            Assert.fail(e.getMessage());
-        } catch (final InvocationTargetException e) {
-            final Throwable te = e.getTargetException();
-            if (te instanceof MathIllegalArgumentException) {
-                throw (MathIllegalArgumentException) te;
-            }
-            Assert.fail(e.getMessage());
-        }
-        return Double.NaN;
-    }
-
-    @Test
-    public void testSumDeltaMinusDeltaSum() {
-
-        final int ulps = 3;
-        for (int i = 0; i < SUM_DELTA_MINUS_DELTA_SUM_REF.length; i++) {
-            final double[] ref = SUM_DELTA_MINUS_DELTA_SUM_REF[i];
-            final double a = ref[0];
-            final double b = ref[1];
-            final double expected = ref[2];
-            final double actual = sumDeltaMinusDeltaSum(a, b);
-            final double tol = ulps * FastMath.ulp(expected);
-            final StringBuilder builder = new StringBuilder();
-            builder.append(a).append(", ").append(b);
-            Assert.assertEquals(builder.toString(), expected, actual, tol);
-        }
-    }
-
-    @Test(expected = NumberIsTooSmallException.class)
-    public void testSumDeltaMinusDeltaSumPrecondition1() {
-
-        sumDeltaMinusDeltaSum(9.0, 10.0);
-    }
-
-    @Test(expected = NumberIsTooSmallException.class)
-    public void testSumDeltaMinusDeltaSumPrecondition2() {
-
-        sumDeltaMinusDeltaSum(10.0, 9.0);
-    }
-
-    private static final double[][] LOG_BETA_REF = {
-        { 0.125 , 0.125 , 2.750814190409515 },
-        { 0.125 , 0.25 , 2.444366899981226 },
-        { 0.125 , 0.5 , 2.230953804989556 },
-        { 0.125 , 1.0 , 2.079441541679836 },
-        { 0.125 , 2.0 , 1.961658506023452 },
-        { 0.125 , 3.0 , 1.901033884207018 },
-        { 0.125 , 4.0 , 1.860211889686763 },
-        { 0.125 , 5.0 , 1.829440231020009 },
-        { 0.125 , 6.0 , 1.804747618429637 },
-        { 0.125 , 7.0 , 1.784128331226902 },
-        { 0.125 , 8.0 , 1.766428754127501 },
-        { 0.125 , 9.0 , 1.750924567591535 },
-        { 0.125 , 10.0 , 1.7371312454592 },
-        { 0.125 , 1000.0 , 1.156003642015969 },
-        { 0.125 , 1001.0 , 1.155878649827818 },
-        { 0.125 , 10000.0 , .8681312798751318 },
-        { 0.25 , 0.125 , 2.444366899981226 },
-        { 0.25 , 0.25 , 2.003680106471455 },
-        { 0.25 , 0.5 , 1.657106516191482 },
-        { 0.25 , 1.0 , 1.386294361119891 },
-        { 0.25 , 2.0 , 1.163150809805681 },
-        { 0.25 , 3.0 , 1.045367774149297 },
-        { 0.25 , 4.0 , 0.965325066475761 },
-        { 0.25 , 5.0 , .9047004446593261 },
-        { 0.25 , 6.0 , .8559102804898941 },
-        { 0.25 , 7.0 , 0.815088285969639 },
-        { 0.25 , 8.0 , .7799969661583689 },
-        { 0.25 , 9.0 , .7492253074916152 },
-        { 0.25 , 10.0 , .7218263333035008 },
-        { 0.25 , 1000.0 , - .4388225372378877 },
-        { 0.25 , 1001.0 , - .4390725059930951 },
-        { 0.25 , 10000.0 , - 1.014553193217846 },
-        { 0.5 , 0.125 , 2.230953804989556 },
-        { 0.5 , 0.25 , 1.657106516191482 },
-        { 0.5 , 0.5 , 1.1447298858494 },
-        { 0.5 , 1.0 , .6931471805599453 },
-        { 0.5 , 2.0 , .2876820724517809 },
-        { 0.5 , 3.0 , .06453852113757118 },
-//        { 0.5 , 4.0 , - .08961215868968714 },
-        { 0.5 , 5.0 , - .2073951943460706 },
-        { 0.5 , 6.0 , - .3027053741503954 },
-        { 0.5 , 7.0 , - .3827480818239319 },
-        { 0.5 , 8.0 , - .4517409533108833 },
-        { 0.5 , 9.0 , - .5123655751273182 },
-        { 0.5 , 10.0 , - .5664327963975939 },
-        { 0.5 , 1000.0 , - 2.881387696571577 },
-        { 0.5 , 1001.0 , - 2.881887571613228 },
-        { 0.5 , 10000.0 , - 4.032792743063396 },
-        { 1.0 , 0.125 , 2.079441541679836 },
-        { 1.0 , 0.25 , 1.386294361119891 },
-        { 1.0 , 0.5 , .6931471805599453 },
-        { 1.0 , 1.0 , 0.0 },
-        { 1.0 , 2.0 , - .6931471805599453 },
-        { 1.0 , 3.0 , - 1.09861228866811 },
-        { 1.0 , 4.0 , - 1.386294361119891 },
-        { 1.0 , 5.0 , - 1.6094379124341 },
-        { 1.0 , 6.0 , - 1.791759469228055 },
-        { 1.0 , 7.0 , - 1.945910149055313 },
-        { 1.0 , 8.0 , - 2.079441541679836 },
-        { 1.0 , 9.0 , - 2.19722457733622 },
-        { 1.0 , 10.0 , - 2.302585092994046 },
-        { 1.0 , 1000.0 , - 6.907755278982137 },
-        { 1.0 , 1001.0 , - 6.90875477931522 },
-        { 1.0 , 10000.0 , - 9.210340371976184 },
-        { 2.0 , 0.125 , 1.961658506023452 },
-        { 2.0 , 0.25 , 1.163150809805681 },
-        { 2.0 , 0.5 , .2876820724517809 },
-        { 2.0 , 1.0 , - .6931471805599453 },
-        { 2.0 , 2.0 , - 1.791759469228055 },
-        { 2.0 , 3.0 , - 2.484906649788 },
-        { 2.0 , 4.0 , - 2.995732273553991 },
-        { 2.0 , 5.0 , - 3.401197381662155 },
-        { 2.0 , 6.0 , - 3.737669618283368 },
-        { 2.0 , 7.0 , - 4.02535169073515 },
-        { 2.0 , 8.0 , - 4.276666119016055 },
-        { 2.0 , 9.0 , - 4.499809670330265 },
-        { 2.0 , 10.0 , - 4.700480365792417 },
-        { 2.0 , 1000.0 , - 13.81651005829736 },
-        { 2.0 , 1001.0 , - 13.81850806096003 },
-        { 2.0 , 10000.0 , - 18.4207807389527 },
-        { 3.0 , 0.125 , 1.901033884207018 },
-        { 3.0 , 0.25 , 1.045367774149297 },
-        { 3.0 , 0.5 , .06453852113757118 },
-        { 3.0 , 1.0 , - 1.09861228866811 },
-        { 3.0 , 2.0 , - 2.484906649788 },
-        { 3.0 , 3.0 , - 3.401197381662155 },
-        { 3.0 , 4.0 , - 4.0943445622221 },
-        { 3.0 , 5.0 , - 4.653960350157523 },
-        { 3.0 , 6.0 , - 5.123963979403259 },
-        { 3.0 , 7.0 , - 5.529429087511423 },
-        { 3.0 , 8.0 , - 5.886104031450156 },
-        { 3.0 , 9.0 , - 6.20455776256869 },
-        { 3.0 , 10.0 , - 6.492239835020471 },
-        { 3.0 , 1000.0 , - 20.03311615938222 },
-        { 3.0 , 1001.0 , - 20.03611166836202 },
-        { 3.0 , 10000.0 , - 26.9381739103716 },
-        { 4.0 , 0.125 , 1.860211889686763 },
-        { 4.0 , 0.25 , 0.965325066475761 },
-//        { 4.0 , 0.5 , - .08961215868968714 },
-        { 4.0 , 1.0 , - 1.386294361119891 },
-        { 4.0 , 2.0 , - 2.995732273553991 },
-        { 4.0 , 3.0 , - 4.0943445622221 },
-        { 4.0 , 4.0 , - 4.941642422609304 },
-        { 4.0 , 5.0 , - 5.634789603169249 },
-        { 4.0 , 6.0 , - 6.222576268071369 },
-        { 4.0 , 7.0 , - 6.733401891837359 },
-        { 4.0 , 8.0 , - 7.185387015580416 },
-        { 4.0 , 9.0 , - 7.590852123688581 },
-        { 4.0 , 10.0 , - 7.958576903813898 },
-        { 4.0 , 1000.0 , - 25.84525465867605 },
-        { 4.0 , 1001.0 , - 25.84924667994559 },
-        { 4.0 , 10000.0 , - 35.05020194868867 },
-        { 5.0 , 0.125 , 1.829440231020009 },
-        { 5.0 , 0.25 , .9047004446593261 },
-        { 5.0 , 0.5 , - .2073951943460706 },
-        { 5.0 , 1.0 , - 1.6094379124341 },
-        { 5.0 , 2.0 , - 3.401197381662155 },
-        { 5.0 , 3.0 , - 4.653960350157523 },
-        { 5.0 , 4.0 , - 5.634789603169249 },
-        { 5.0 , 5.0 , - 6.445719819385578 },
-        { 5.0 , 6.0 , - 7.138866999945524 },
-        { 5.0 , 7.0 , - 7.745002803515839 },
-        { 5.0 , 8.0 , - 8.283999304248526 },
-        { 5.0 , 9.0 , - 8.769507120030227 },
-        { 5.0 , 10.0 , - 9.211339872309265 },
-        { 5.0 , 1000.0 , - 31.37070759780783 },
-        { 5.0 , 1001.0 , - 31.37569513931887 },
-        { 5.0 , 10000.0 , - 42.87464787956629 },
-        { 6.0 , 0.125 , 1.804747618429637 },
-        { 6.0 , 0.25 , .8559102804898941 },
-        { 6.0 , 0.5 , - .3027053741503954 },
-        { 6.0 , 1.0 , - 1.791759469228055 },
-        { 6.0 , 2.0 , - 3.737669618283368 },
-        { 6.0 , 3.0 , - 5.123963979403259 },
-        { 6.0 , 4.0 , - 6.222576268071369 },
-        { 6.0 , 5.0 , - 7.138866999945524 },
-        { 6.0 , 6.0 , - 7.927324360309794 },
-        { 6.0 , 7.0 , - 8.620471540869739 },
-        { 6.0 , 8.0 , - 9.239510749275963 },
-        { 6.0 , 9.0 , - 9.799126537211386 },
-        { 6.0 , 10.0 , - 10.30995216097738 },
-        { 6.0 , 1000.0 , - 36.67401250586691 },
-        { 6.0 , 1001.0 , - 36.67999457754446 },
-        { 6.0 , 10000.0 , - 50.47605021415003 },
-        { 7.0 , 0.125 , 1.784128331226902 },
-        { 7.0 , 0.25 , 0.815088285969639 },
-        { 7.0 , 0.5 , - .3827480818239319 },
-        { 7.0 , 1.0 , - 1.945910149055313 },
-        { 7.0 , 2.0 , - 4.02535169073515 },
-        { 7.0 , 3.0 , - 5.529429087511423 },
-        { 7.0 , 4.0 , - 6.733401891837359 },
-        { 7.0 , 5.0 , - 7.745002803515839 },
-        { 7.0 , 6.0 , - 8.620471540869739 },
-        { 7.0 , 7.0 , - 9.39366142910322 },
-        { 7.0 , 8.0 , - 10.08680860966317 },
-        { 7.0 , 9.0 , - 10.71541726908554 },
-        { 7.0 , 10.0 , - 11.2907814139891 },
-        { 7.0 , 1000.0 , - 41.79599038729854 },
-        { 7.0 , 1001.0 , - 41.80296600103496 },
-        { 7.0 , 10000.0 , - 57.89523093697012 },
-        { 8.0 , 0.125 , 1.766428754127501 },
-        { 8.0 , 0.25 , .7799969661583689 },
-        { 8.0 , 0.5 , - .4517409533108833 },
-        { 8.0 , 1.0 , - 2.079441541679836 },
-        { 8.0 , 2.0 , - 4.276666119016055 },
-        { 8.0 , 3.0 , - 5.886104031450156 },
-        { 8.0 , 4.0 , - 7.185387015580416 },
-        { 8.0 , 5.0 , - 8.283999304248526 },
-        { 8.0 , 6.0 , - 9.239510749275963 },
-        { 8.0 , 7.0 , - 10.08680860966317 },
-        { 8.0 , 8.0 , - 10.84894866171006 },
-        { 8.0 , 9.0 , - 11.54209584227001 },
-        { 8.0 , 10.0 , - 12.17808460899001 },
-        { 8.0 , 1000.0 , - 46.76481113096179 },
-        { 8.0 , 1001.0 , - 46.77277930061096 },
-        { 8.0 , 10000.0 , - 65.16036091500527 },
-        { 9.0 , 0.125 , 1.750924567591535 },
-        { 9.0 , 0.25 , .7492253074916152 },
-        { 9.0 , 0.5 , - .5123655751273182 },
-        { 9.0 , 1.0 , - 2.19722457733622 },
-        { 9.0 , 2.0 , - 4.499809670330265 },
-        { 9.0 , 3.0 , - 6.20455776256869 },
-        { 9.0 , 4.0 , - 7.590852123688581 },
-        { 9.0 , 5.0 , - 8.769507120030227 },
-        { 9.0 , 6.0 , - 9.799126537211386 },
-        { 9.0 , 7.0 , - 10.71541726908554 },
-        { 9.0 , 8.0 , - 11.54209584227001 },
-        { 9.0 , 9.0 , - 12.29586764464639 },
-        { 9.0 , 10.0 , - 12.98901482520633 },
-        { 9.0 , 1000.0 , - 51.60109303791327 },
-        { 9.0 , 1001.0 , - 51.61005277928474 },
-        { 9.0 , 10000.0 , - 72.29205942547217 },
-        { 10.0 , 0.125 , 1.7371312454592 },
-        { 10.0 , 0.25 , .7218263333035008 },
-        { 10.0 , 0.5 , - .5664327963975939 },
-        { 10.0 , 1.0 , - 2.302585092994046 },
-        { 10.0 , 2.0 , - 4.700480365792417 },
-        { 10.0 , 3.0 , - 6.492239835020471 },
-        { 10.0 , 4.0 , - 7.958576903813898 },
-        { 10.0 , 5.0 , - 9.211339872309265 },
-        { 10.0 , 6.0 , - 10.30995216097738 },
-        { 10.0 , 7.0 , - 11.2907814139891 },
-        { 10.0 , 8.0 , - 12.17808460899001 },
-        { 10.0 , 9.0 , - 12.98901482520633 },
-        { 10.0 , 10.0 , - 13.73622922703655 },
-        { 10.0 , 1000.0 , - 56.32058348093065 },
-        { 10.0 , 1001.0 , - 56.33053381178382 },
-        { 10.0 , 10000.0 , - 79.30607481535498 },
-        { 1000.0 , 0.125 , 1.156003642015969 },
-        { 1000.0 , 0.25 , - .4388225372378877 },
-        { 1000.0 , 0.5 , - 2.881387696571577 },
-        { 1000.0 , 1.0 , - 6.907755278982137 },
-        { 1000.0 , 2.0 , - 13.81651005829736 },
-        { 1000.0 , 3.0 , - 20.03311615938222 },
-        { 1000.0 , 4.0 , - 25.84525465867605 },
-        { 1000.0 , 5.0 , - 31.37070759780783 },
-        { 1000.0 , 6.0 , - 36.67401250586691 },
-        { 1000.0 , 7.0 , - 41.79599038729854 },
-        { 1000.0 , 8.0 , - 46.76481113096179 },
-        { 1000.0 , 9.0 , - 51.60109303791327 },
-        { 1000.0 , 10.0 , - 56.32058348093065 },
-        { 1000.0 , 1000.0 , - 1388.482601635902 },
-        { 1000.0 , 1001.0 , - 1389.175748816462 },
-        { 1000.0 , 10000.0 , - 3353.484270767097 },
-        { 1001.0 , 0.125 , 1.155878649827818 },
-        { 1001.0 , 0.25 , - .4390725059930951 },
-        { 1001.0 , 0.5 , - 2.881887571613228 },
-        { 1001.0 , 1.0 , - 6.90875477931522 },
-        { 1001.0 , 2.0 , - 13.81850806096003 },
-        { 1001.0 , 3.0 , - 20.03611166836202 },
-        { 1001.0 , 4.0 , - 25.84924667994559 },
-        { 1001.0 , 5.0 , - 31.37569513931887 },
-        { 1001.0 , 6.0 , - 36.67999457754446 },
-        { 1001.0 , 7.0 , - 41.80296600103496 },
-        { 1001.0 , 8.0 , - 46.77277930061096 },
-        { 1001.0 , 9.0 , - 51.61005277928474 },
-        { 1001.0 , 10.0 , - 56.33053381178382 },
-        { 1001.0 , 1000.0 , - 1389.175748816462 },
-        { 1001.0 , 1001.0 , - 1389.869395872064 },
-        { 1001.0 , 10000.0 , - 3355.882166039895 },
-        { 10000.0 , 0.125 , .8681312798751318 },
-        { 10000.0 , 0.25 , - 1.014553193217846 },
-        { 10000.0 , 0.5 , - 4.032792743063396 },
-        { 10000.0 , 1.0 , - 9.210340371976184 },
-        { 10000.0 , 2.0 , - 18.4207807389527 },
-        { 10000.0 , 3.0 , - 26.9381739103716 },
-        { 10000.0 , 4.0 , - 35.05020194868867 },
-        { 10000.0 , 5.0 , - 42.87464787956629 },
-        { 10000.0 , 6.0 , - 50.47605021415003 },
-        { 10000.0 , 7.0 , - 57.89523093697012 },
-        { 10000.0 , 8.0 , - 65.16036091500527 },
-        { 10000.0 , 9.0 , - 72.29205942547217 },
-        { 10000.0 , 10.0 , - 79.30607481535498 },
-        { 10000.0 , 1000.0 , - 3353.484270767097 },
-        { 10000.0 , 1001.0 , - 3355.882166039895 },
-        { 10000.0 , 10000.0 , - 13866.28325676141 },
-    };
-
-    @Test
-    public void testLogBeta() {
-        final int ulps = 3;
-        for (int i = 0; i < LOG_BETA_REF.length; i++) {
-            final double[] ref = LOG_BETA_REF[i];
-            final double a = ref[0];
-            final double b = ref[1];
-            final double expected = ref[2];
-            final double actual = Beta.logBeta(a, b);
-            final double tol = ulps * FastMath.ulp(expected);
-            final StringBuilder builder = new StringBuilder();
-            builder.append(a).append(", ").append(b);
-            Assert.assertEquals(builder.toString(), expected, actual, tol);
-        }
-    }}
diff --git a/src/test/java/org/apache/commons/math4/special/GammaTest.java b/src/test/java/org/apache/commons/math4/special/GammaTest.java
deleted file mode 100644
index c3205267b..000000000
--- a/src/test/java/org/apache/commons/math4/special/GammaTest.java
+++ /dev/null
@@ -1,998 +0,0 @@
-/*
- * Licensed to the Apache Software Foundation (ASF) under one or more
- * contributor license agreements.  See the NOTICE file distributed with
- * this work for additional information regarding copyright ownership.
- * The ASF licenses this file to You under the Apache License, Version 2.0
- * (the "License"); you may not use this file except in compliance with
- * the License.  You may obtain a copy of the License at
- *
- *      http://www.apache.org/licenses/LICENSE-2.0
- *
- * Unless required by applicable law or agreed to in writing, software
- * distributed under the License is distributed on an "AS IS" BASIS,
- * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
- * See the License for the specific language governing permissions and
- * limitations under the License.
- */
-package org.apache.commons.math4.special;
-
-import org.apache.commons.math4.TestUtils;
-import org.apache.commons.math4.exception.NumberIsTooLargeException;
-import org.apache.commons.math4.exception.NumberIsTooSmallException;
-import org.apache.commons.math4.util.FastMath;
-import org.junit.Assert;
-import org.junit.Test;
-
-/**
- */
-public class GammaTest {
-    private void testRegularizedGamma(double expected, double a, double x) {
-        double actualP = Gamma.regularizedGammaP(a, x);
-        double actualQ = Gamma.regularizedGammaQ(a, x);
-        TestUtils.assertEquals(expected, actualP, 10e-15);
-        TestUtils.assertEquals(actualP, 1.0 - actualQ, 10e-15);
-    }
-
-    private void testLogGamma(double expected, double x) {
-        double actual = Gamma.logGamma(x);
-        TestUtils.assertEquals(expected, actual, 10e-15);
-    }
-
-    @Test
-    public void testRegularizedGammaNanPositive() {
-        testRegularizedGamma(Double.NaN, Double.NaN, 1.0);
-    }
-
-    @Test
-    public void testRegularizedGammaPositiveNan() {
-        testRegularizedGamma(Double.NaN, 1.0, Double.NaN);
-    }
-
-    @Test
-    public void testRegularizedGammaNegativePositive() {
-        testRegularizedGamma(Double.NaN, -1.5, 1.0);
-    }
-
-    @Test
-    public void testRegularizedGammaPositiveNegative() {
-        testRegularizedGamma(Double.NaN, 1.0, -1.0);
-    }
-
-    @Test
-    public void testRegularizedGammaZeroPositive() {
-        testRegularizedGamma(Double.NaN, 0.0, 1.0);
-    }
-
-    @Test
-    public void testRegularizedGammaPositiveZero() {
-        testRegularizedGamma(0.0, 1.0, 0.0);
-    }
-
-    @Test
-    public void testRegularizedGammaPositivePositive() {
-        testRegularizedGamma(0.632120558828558, 1.0, 1.0);
-    }
-
-    @Test
-    public void testLogGammaNan() {
-        testLogGamma(Double.NaN, Double.NaN);
-    }
-
-    @Test
-    public void testLogGammaNegative() {
-        testLogGamma(Double.NaN, -1.0);
-    }
-
-    @Test
-    public void testLogGammaZero() {
-        testLogGamma(Double.NaN, 0.0);
-    }
-
-    @Test
-    public void testLogGammaPositive() {
-        testLogGamma(0.6931471805599457, 3.0);
-    }
-
-    @Test
-    public void testDigammaLargeArgs() {
-        double eps = 1e-8;
-        Assert.assertEquals(4.6001618527380874002, Gamma.digamma(100), eps);
-        Assert.assertEquals(3.9019896734278921970, Gamma.digamma(50), eps);
-        Assert.assertEquals(2.9705239922421490509, Gamma.digamma(20), eps);
-        Assert.assertEquals(2.9958363947076465821, Gamma.digamma(20.5), eps);
-        Assert.assertEquals(2.2622143570941481605, Gamma.digamma(10.1), eps);
-        Assert.assertEquals(2.1168588189004379233, Gamma.digamma(8.8), eps);
-        Assert.assertEquals(1.8727843350984671394, Gamma.digamma(7), eps);
-        Assert.assertEquals(0.42278433509846713939, Gamma.digamma(2), eps);
-        Assert.assertEquals(-100.56088545786867450, Gamma.digamma(0.01), eps);
-        Assert.assertEquals(-4.0390398965921882955, Gamma.digamma(-0.8), eps);
-        Assert.assertEquals(4.2003210041401844726, Gamma.digamma(-6.3), eps);
-        Assert.assertEquals(-3.110625123035E-5, Gamma.digamma(1.4616), eps);
-    }
-
-    @Test
-    public void testDigammaSmallArgs() {
-        // values for negative powers of 10 from 1 to 30 as computed by webMathematica with 20 digits
-        // see functions.wolfram.com
-        double[] expected = {-10.423754940411076795, -100.56088545786867450, -1000.5755719318103005,
-                -10000.577051183514335, -100000.57719921568107, -1.0000005772140199687e6, -1.0000000577215500408e7,
-                -1.0000000057721564845e8, -1.0000000005772156633e9, -1.0000000000577215665e10, -1.0000000000057721566e11,
-                -1.0000000000005772157e12, -1.0000000000000577216e13, -1.0000000000000057722e14, -1.0000000000000005772e15, -1e+16,
-                -1e+17, -1e+18, -1e+19, -1e+20, -1e+21, -1e+22, -1e+23, -1e+24, -1e+25, -1e+26,
-                -1e+27, -1e+28, -1e+29, -1e+30};
-        for (double n = 1; n < 30; n++) {
-            checkRelativeError(String.format("Test %.0f: ", n), expected[(int) (n - 1)], Gamma.digamma(FastMath.pow(10.0, -n)), 1e-8);
-        }
-    }
-
-    @Test
-    public void testDigammaNonRealArgs() {
-        Assert.assertTrue(Double.isNaN(Gamma.digamma(Double.NaN)));
-        Assert.assertTrue(Double.isInfinite(Gamma.digamma(Double.POSITIVE_INFINITY)));
-        Assert.assertTrue(Double.isInfinite(Gamma.digamma(Double.NEGATIVE_INFINITY)));
-    }
-
-    @Test
-    public void testTrigamma() {
-        double eps = 1e-8;
-        // computed using webMathematica.  For example, to compute trigamma($i) = Polygamma(1, $i), use
-        //
-        // http://functions.wolfram.com/webMathematica/Evaluated.jsp?name=PolyGamma2&plottype=0&vars={%221%22,%22$i%22}&digits=20
-        double[] data = {
-                1e-4, 1.0000000164469368793e8,
-                1e-3, 1.0000016425331958690e6,
-                1e-2, 10001.621213528313220,
-                1e-1, 101.43329915079275882,
-                1, 1.6449340668482264365,
-                2, 0.64493406684822643647,
-                3, 0.39493406684822643647,
-                4, 0.28382295573711532536,
-                5, 0.22132295573711532536,
-                10, 0.10516633568168574612,
-                20, 0.051270822935203119832,
-                50, 0.020201333226697125806,
-                100, 0.010050166663333571395
-        };
-        for (int i = data.length - 2; i >= 0; i -= 2) {
-            Assert.assertEquals(String.format("trigamma %.0f", data[i]), data[i + 1], Gamma.trigamma(data[i]), eps);
-        }
-    }
-
-    @Test
-    public void testTrigammaNonRealArgs() {
-        Assert.assertTrue(Double.isNaN(Gamma.trigamma(Double.NaN)));
-        Assert.assertTrue(Double.isInfinite(Gamma.trigamma(Double.POSITIVE_INFINITY)));
-        Assert.assertTrue(Double.isInfinite(Gamma.trigamma(Double.NEGATIVE_INFINITY)));
-    }
-
-    /**
-     * Reference data for the {@link Gamma#logGamma(double)} function. This data
-     * was generated with the following Maxima script.
-     *
-     * 
-     * kill(all);
-     *
-     * fpprec : 64;
-     * gamln(x) := log(gamma(x));
-     * x : append(makelist(bfloat(i / 8), i, 1, 80),
-     *     [0.8b0, 1b2, 1b3, 1b4, 1b5, 1b6, 1b7, 1b8, 1b9, 1b10]);
-     *
-     * for i : 1 while i <= length(x) do
-     *     print("{", float(x[i]), ",", float(gamln(x[i])), "},");
-     * 

-     */
-    private static final double[][] LOG_GAMMA_REF = {
-        { 0.125 , 2.019418357553796 },
-        { 0.25 , 1.288022524698077 },
-        { 0.375 , .8630739822706475 },
-        { 0.5 , .5723649429247001 },
-        { 0.625 , .3608294954889402 },
-        { 0.75 , .2032809514312954 },
-        { 0.875 , .08585870722533433 },
-        { 0.890625 , .07353860936979656 },
-        { 0.90625 , .06169536624059108 },
-        { 0.921875 , .05031670080005688 },
-        { 0.9375 , 0.0393909017345823 },
-        { 0.953125 , .02890678734595923 },
-        { 0.96875 , .01885367233441289 },
-        { 0.984375 , .009221337197578781 },
-        { 1.0 , 0.0 },
-        { 1.015625 , - 0.00881970970573307 },
-        { 1.03125 , - .01724677500176807 },
-        { 1.046875 , - .02528981394675729 },
-        { 1.0625 , - .03295710029357782 },
-        { 1.078125 , - .04025658272400143 },
-        { 1.09375 , - .04719590272716985 },
-        { 1.109375 , - .05378241123619192 },
-        { 1.125 , - .06002318412603958 },
-        { 1.25 , - .09827183642181316 },
-        { 1.375 , - .1177552707410788 },
-        { 1.5 , - .1207822376352452 },
-        { 1.625 , - .1091741337567954 },
-        { 1.75 , - .08440112102048555 },
-        { 1.875 , - 0.0476726853991883 },
-        { 1.890625 , - .04229320615532515 },
-        { 1.90625 , - .03674470657266143 },
-        { 1.921875 , - .03102893865389552 },
-        { 1.9375 , - .02514761940298887 },
-        { 1.953125 , - .01910243184040138 },
-        { 1.96875 , - .01289502598016741 },
-        { 1.984375 , - .006527019770560387 },
-        { 2.0 , 0.0 },
-        { 2.015625 , .006684476830232185 },
-        { 2.03125 , .01352488366498562 },
-        { 2.046875 , .02051972208453692 },
-        { 2.0625 , .02766752152285702 },
-        { 2.078125 , 0.0349668385135861 },
-        { 2.09375 , .04241625596251728 },
-        { 2.109375 , .05001438244545164 },
-        { 2.125 , .05775985153034387 },
-        { 2.25 , .1248717148923966 },
-        { 2.375 , .2006984603774558 },
-        { 2.5 , .2846828704729192 },
-        { 2.625 , .3763336820249054 },
-        { 2.75 , .4752146669149371 },
-        { 2.875 , .5809359740231859 },
-        { 2.890625 , .5946142560817441 },
-        { 2.90625 , .6083932548009232 },
-        { 2.921875 , .6222723333588501 },
-        { 2.9375 , .6362508628423761 },
-        { 2.953125 , .6503282221022278 },
-        { 2.96875 , .6645037976116387 },
-        { 2.984375 , 0.678776983328359 },
-        { 3.0 , .6931471805599453 },
-        { 3.015625 , .7076137978322324 },
-        { 3.03125 , .7221762507608962 },
-        { 3.046875 , .7368339619260166 },
-        { 3.0625 , 0.751586360749556 },
-        { 3.078125 , .7664328833756681 },
-        { 3.09375 , .7813729725537568 },
-        { 3.109375 , .7964060775242092 },
-        { 3.125 , 0.811531653906724 },
-        { 3.25 , .9358019311087253 },
-        { 3.375 , 1.06569589786406 },
-        { 3.5 , 1.200973602347074 },
-        { 3.625 , 1.341414578068493 },
-        { 3.75 , 1.486815578593417 },
-        { 3.875 , 1.6369886482725 },
-        { 4.0 , 1.791759469228055 },
-        { 4.125 , 1.950965937095089 },
-        { 4.25 , 2.114456927450371 },
-        { 4.375 , 2.282091222188554 },
-        { 4.5 , 2.453736570842442 },
-        { 4.625 , 2.62926886637513 },
-        { 4.75 , 2.808571418575736 },
-        { 4.875 , 2.99153431107781 },
-        { 5.0 , 3.178053830347946 },
-        { 5.125 , 3.368031956881733 },
-        { 5.25 , 3.561375910386697 },
-        { 5.375 , 3.757997741998131 },
-        { 5.5 , 3.957813967618717 },
-        { 5.625 , 4.160745237339519 },
-        { 5.75 , 4.366716036622286 },
-        { 5.875 , 4.57565441552762 },
-        { 6.0 , 4.787491742782046 },
-        { 6.125 , 5.002162481906205 },
-        { 6.25 , 5.219603986990229 },
-        { 6.375 , 5.439756316011858 },
-        { 6.5 , 5.662562059857142 },
-        { 6.625 , 5.887966185430003 },
-        { 6.75 , 6.115915891431546 },
-        { 6.875 , 6.346360475557843 },
-        { 7.0 , 6.579251212010101 },
-        { 7.125 , 6.814541238336996 },
-        { 7.25 , 7.05218545073854 },
-        { 7.375 , 7.292140407056348 },
-        { 7.5 , 7.534364236758733 },
-        { 7.625 , 7.778816557302289 },
-        { 7.75 , 8.025458396315983 },
-        { 7.875 , 8.274252119110479 },
-        { 8.0 , 8.525161361065415 },
-        { 8.125 , 8.77815096449171 },
-        { 8.25 , 9.033186919605123 },
-        { 8.375 , 9.290236309282232 },
-        { 8.5 , 9.549267257300997 },
-        { 8.625 , 9.810248879795765 },
-        { 8.75 , 10.07315123968124 },
-        { 8.875 , 10.33794530382217 },
-        { 9.0 , 10.60460290274525 },
-        { 9.125 , 10.87309669270751 },
-        { 9.25 , 11.14340011995171 },
-        { 9.375 , 11.41548738699336 },
-        { 9.5 , 11.68933342079727 },
-        { 9.625 , 11.96491384271319 },
-        { 9.75 , 12.24220494005076 },
-        { 9.875 , 12.52118363918365 },
-        { 10.0 , 12.80182748008147 },
-        { 0.8 , .1520596783998376 },
-        { 100.0 , 359.1342053695754 },
-        { 1000.0 , 5905.220423209181 },
-        { 10000.0 , 82099.71749644238 },
-        { 100000.0 , 1051287.708973657 },
-        { 1000000.0 , 1.2815504569147612e+7 },
-        { 10000000.0 , 1.511809493694739e+8 },
-        { 1.e+8 , 1.7420680661038346e+9 },
-        { 1.e+9 , 1.972326582750371e+10 },
-        { 1.e+10 , 2.202585092888106e+11 },
-    };
-
-    @Test
-    public void testLogGamma() {
-        final int ulps = 3;
-        for (int i = 0; i < LOG_GAMMA_REF.length; i++) {
-            final double[] data = LOG_GAMMA_REF[i];
-            final double x = data[0];
-            final double expected = data[1];
-            final double actual = Gamma.logGamma(x);
-            final double tol;
-            if (expected == 0.0) {
-                tol = 1E-15;
-            } else {
-                tol = ulps * FastMath.ulp(expected);
-            }
-            Assert.assertEquals(Double.toString(x), expected, actual, tol);
-        }
-    }
-
-    @Test
-    public void testLogGammaPrecondition1() {
-        Assert.assertTrue(Double.isNaN(Gamma.logGamma(0.0)));
-    }
-
-    @Test
-    public void testLogGammaPrecondition2() {
-        Assert.assertTrue(Double.isNaN(Gamma.logGamma(-1.0)));
-    }
-
-    /**
-     * 

-     * Reference values for the {@link Gamma#invGamma1pm1(double)} method.
-     * These values were generated with the following Maxima script
-     * 

-     *
-     * 
-     * kill(all);
-     *
-     * fpprec : 64;
-     * gam1(x) := 1 / gamma(1 + x) - 1;
-     * x : makelist(bfloat(i / 8), i, -4, 12);
-     *
-     * for i : 1 while i <= length(x) do print("{",
-     *                                         float(x[i]),
-     *                                         ",",
-     *                                         float(gam1(x[i])),
-     *                                         "},");
-     * 

-     */
-    private static final double[][] INV_GAMMA1P_M1_REF = {
-        { -0.5 , -.4358104164522437 },
-        { -0.375 , -.3029021533379859 },
-        { -0.25 , -0.183951060901737 },
-        { -0.125 , -.08227611018520711 },
-        { 0.0 , 0.0 },
-        { 0.125 , .06186116458306091 },
-        { 0.25 , .1032626513208373 },
-        { 0.375 , .1249687649039041 },
-        { 0.5 , .1283791670955126 },
-        { 0.625 , .1153565546592225 },
-        { 0.75 , 0.0880652521310173 },
-        { 0.875 , .04882730264547758 },
-        { 1.0 , 0.0 },
-        { 1.125 , -.05612340925950141 },
-        { 1.25 , -.1173898789433302 },
-        { 1.375 , -.1818408982517061 },
-        { 1.5 , -0.247747221936325 },
-    };
-
-    @Test
-    public void testInvGamma1pm1() {
-
-        final int ulps = 3;
-        for (int i = 0; i < INV_GAMMA1P_M1_REF.length; i++) {
-            final double[] ref = INV_GAMMA1P_M1_REF[i];
-            final double x = ref[0];
-            final double expected = ref[1];
-            final double actual = Gamma.invGamma1pm1(x);
-            final double tol = ulps * FastMath.ulp(expected);
-            Assert.assertEquals(Double.toString(x), expected, actual, tol);
-        }
-    }
-
-    @Test(expected = NumberIsTooSmallException.class)
-    public void testInvGamma1pm1Precondition1() {
-
-        Gamma.invGamma1pm1(-0.51);
-    }
-
-    @Test(expected = NumberIsTooLargeException.class)
-    public void testInvGamma1pm1Precondition2() {
-
-        Gamma.invGamma1pm1(1.51);
-    }
-
-    private static final double[][] LOG_GAMMA1P_REF = {
-        { - 0.5 , .5723649429247001 },
-        { - 0.375 , .3608294954889402 },
-        { - 0.25 , .2032809514312954 },
-        { - 0.125 , .08585870722533433 },
-        { 0.0 , 0.0 },
-        { 0.125 , - .06002318412603958 },
-        { 0.25 , - .09827183642181316 },
-        { 0.375 , - .1177552707410788 },
-        { 0.5 , - .1207822376352452 },
-        { 0.625 , - .1091741337567954 },
-        { 0.75 , - .08440112102048555 },
-        { 0.875 , - 0.0476726853991883 },
-        { 1.0 , 0.0 },
-        { 1.125 , .05775985153034387 },
-        { 1.25 , .1248717148923966 },
-        { 1.375 , .2006984603774558 },
-        { 1.5 , .2846828704729192 },
-    };
-
-    @Test
-    public void testLogGamma1p() {
-
-        final int ulps = 3;
-        for (int i = 0; i < LOG_GAMMA1P_REF.length; i++) {
-            final double[] ref = LOG_GAMMA1P_REF[i];
-            final double x = ref[0];
-            final double expected = ref[1];
-            final double actual = Gamma.logGamma1p(x);
-            final double tol = ulps * FastMath.ulp(expected);
-            Assert.assertEquals(Double.toString(x), expected, actual, tol);
-        }
-    }
-
-    @Test(expected = NumberIsTooSmallException.class)
-    public void testLogGamma1pPrecondition1() {
-
-        Gamma.logGamma1p(-0.51);
-    }
-
-    @Test(expected = NumberIsTooLargeException.class)
-    public void testLogGamma1pPrecondition2() {
-
-        Gamma.logGamma1p(1.51);
-    }
-
-    /**
-     * Reference data for the {@link Gamma#gamma(double)} function. This
-     * data was generated with the following Maxima script.
-     *
-     * 
-     * kill(all);
-     *
-     * fpprec : 64;
-     *
-     * EPSILON : 10**(-fpprec + 1);
-     * isInteger(x) := abs(x - floor(x)) <= EPSILON * abs(x);
-     *
-     * x : makelist(bfloat(i / 8), i, -160, 160);
-     * x : append(x, makelist(bfloat(i / 2), i, 41, 200));
-     *
-     * for i : 1 while i <= length(x) do if not(isInteger(x[i])) then
-     *     print("{", float(x[i]), ",", float(gamma(x[i])), "},");
-     * 

-     */
-    private static final double[][] GAMMA_REF = {
-        { - 19.875 , 4.920331854832504e-18 },
-        { - 19.75 , 3.879938752480031e-18 },
-        { - 19.625 , 4.323498423815027e-18 },
-        { - 19.5 , 5.811045977502237e-18 },
-        { - 19.375 , 9.14330910942125e-18 },
-        { - 19.25 , 1.735229114436739e-17 },
-        { - 19.125 , 4.653521565668223e-17 },
-        { - 18.875 , - 9.779159561479603e-17 },
-        { - 18.75 , - 7.662879036148061e-17 },
-        { - 18.625 , - 8.484865656736991e-17 },
-        { - 18.5 , - 1.133153965612936e-16 },
-        { - 18.375 , - 1.771516139950367e-16 },
-        { - 18.25 , - 3.340316045290721e-16 },
-        { - 18.125 , - 8.899859994340475e-16 },
-        { - 17.875 , 1.845816367229275e-15 },
-        { - 17.75 , 1.436789819277761e-15 },
-        { - 17.625 , 1.580306228567265e-15 },
-        { - 17.5 , 2.096334836383932e-15 },
-        { - 17.375 , 3.255160907158799e-15 },
-        { - 17.25 , 6.096076782655566e-15 },
-        { - 17.125 , 1.613099623974211e-14 },
-        { - 16.875 , - 3.29939675642233e-14 },
-        { - 16.75 , - 2.550301929218027e-14 },
-        { - 16.625 , - 2.785289727849803e-14 },
-        { - 16.5 , - 3.66858596367188e-14 },
-        { - 16.375 , - 5.655842076188414e-14 },
-        { - 16.25 , - 1.051573245008085e-13 },
-        { - 16.125 , - 2.762433106055837e-13 },
-        { - 15.875 , 5.567732026462681e-13 },
-        { - 15.75 , 4.271755731440195e-13 },
-        { - 15.625 , 4.630544172550298e-13 },
-        { - 15.5 , 6.053166840058604e-13 },
-        { - 15.375 , 9.261441399758529e-13 },
-        { - 15.25 , 1.708806523138138e-12 },
-        { - 15.125 , 4.454423383515037e-12 },
-        { - 14.875 , - 8.838774592009505e-12 },
-        { - 14.75 , - 6.728015277018307e-12 },
-        { - 14.625 , - 7.235225269609841e-12 },
-        { - 14.5 , - 9.382408602090835e-12 },
-        { - 14.375 , - 1.423946615212874e-11 },
-        { - 14.25 , - 2.605929947785661e-11 },
-        { - 14.125 , - 6.737315367566492e-11 },
-        { - 13.875 , 1.314767720561414e-10 },
-        { - 13.75 , 9.923822533602004e-11 },
-        { - 13.625 , 1.058151695680439e-10 },
-        { - 13.5 , 1.360449247303171e-10 },
-        { - 13.375 , 2.046923259368506e-10 },
-        { - 13.25 , 3.713450175594567e-10 },
-        { - 13.125 , 9.516457956687671e-10 },
-        { - 12.875 , - 1.8242402122789617e-9 },
-        { - 12.75 , - 1.3645255983702756e-9 },
-        { - 12.625 , - 1.4417316853645984e-9 },
-        { - 12.5 , - 1.836606483859281e-9 },
-        { - 12.375 , - 2.7377598594053765e-9 },
-        { - 12.25 , - 4.9203214826628017e-9 },
-        { - 12.125 , - 1.2490351068152569e-8 },
-        { - 11.875 , 2.3487092733091633e-8 },
-        { - 11.75 , 1.7397701379221012e-8 },
-        { - 11.625 , 1.8201862527728055e-8 },
-        { - 11.5 , 2.295758104824101e-8 },
-        { - 11.375 , 3.3879778260141535e-8 },
-        { - 11.25 , 6.027393816261931e-8 },
-        { - 11.125 , 1.5144550670134987e-7 },
-        { - 10.875 , - 2.7890922620546316e-7 },
-        { - 10.75 , - 2.044229912058469e-7 },
-        { - 10.625 , - 2.1159665188483867e-7 },
-        { - 10.5 , - 2.640121820547716e-7 },
-        { - 10.375 , - 3.8538247770911e-7 },
-        { - 10.25 , - 6.780818043294673e-7 },
-        { - 10.125 , - 1.6848312620525174e-6 },
-        { - 9.875 , 3.0331378349844124e-6 },
-        { - 9.75 , 2.1975471554628537e-6 },
-        { - 9.625 , 2.2482144262764103e-6 },
-        { - 9.5 , 2.772127911575102e-6 },
-        { - 9.375 , 3.998343206232017e-6 },
-        { - 9.25 , 6.95033849437704e-6 },
-        { - 9.125 , 1.7058916528281737e-5 },
-        { - 8.875 , - 2.9952236120471065e-5 },
-        { - 8.75 , - 2.1426084765762826e-5 },
-        { - 8.625 , - 2.163906385291045e-5 },
-        { - 8.5 , - 2.633521515996347e-5 },
-        { - 8.375 , - 3.748446755842515e-5 },
-        { - 8.25 , - 6.429063107298763e-5 },
-        { - 8.125 , - 1.5566261332057085e-4 },
-        { - 7.875 , 2.658260955691807e-4 },
-        { - 7.75 , 1.874782417004247e-4 },
-        { - 7.625 , 1.8663692573135265e-4 },
-        { - 7.5 , 2.238493288596895e-4 },
-        { - 7.375 , 3.1393241580181064e-4 },
-        { - 7.25 , 5.303977063521479e-4 },
-        { - 7.125 , .001264758733229638 },
-        { - 6.875 , - .002093380502607298 },
-        { - 6.75 , - .001452956373178292 },
-        { - 6.625 , - .001423106558701564 },
-        { - 6.5 , - .001678869966447671 },
-        { - 6.375 , - .002315251566538353 },
-        { - 6.25 , - .003845383371053072 },
-        { - 6.125 , - .009011405974261174 },
-        { - 5.875 , .01439199095542518 },
-        { - 5.75 , .009807455518953468 },
-        { - 5.625 , .009428080951397862 },
-        { - 5.5 , .01091265478190986 },
-        { - 5.375 , 0.014759728736682 },
-        { - 5.25 , 0.0240336460690817 },
-        { - 5.125 , .05519486159234969 },
-        { - 4.875 , - 0.0845529468631229 },
-        { - 4.75 , - .05639286923398244 },
-        { - 4.625 , - .05303295535161297 },
-        { - 4.5 , - .06001960130050425 },
-        { - 4.375 , - .07933354195966577 },
-        { - 4.25 , - .1261766418626789 },
-        { - 4.125 , - .2828736656607921 },
-        { - 3.875 , .4121956159577241 },
-        { - 3.75 , .2678661288614166 },
-        { - 3.625 , 0.24527741850121 },
-        { - 3.5 , .2700882058522691 },
-        { - 3.375 , .3470842460735378 },
-        { - 3.25 , .5362507279163854 },
-        { - 3.125 , 1.166853870850768 },
-        { - 2.875 , - 1.597258011836181 },
-        { - 2.75 , - 1.004497983230312 },
-        { - 2.625 , - .8891306420668862 },
-        { - 2.5 , - .9453087204829419 },
-        { - 2.375 , - 1.17140933049819 },
-        { - 2.25 , - 1.742814865728253 },
-        { - 2.125 , - 3.646418346408649 },
-        { - 1.875 , 4.59211678402902 },
-        { - 1.75 , 2.762369453883359 },
-        { - 1.625 , 2.333967935425576 },
-        { - 1.5 , 2.363271801207355 },
-        { - 1.375 , 2.782097159933201 },
-        { - 1.25 , 3.921333447888569 },
-        { - 1.125 , 7.748638986118379 },
-        { - 0.875 , - 8.610218970054413 },
-        { - 0.75 , - 4.834146544295877 },
-        { - 0.625 , - 3.792697895066561 },
-        { - 0.5 , - 3.544907701811032 },
-        { - 0.375 , - 3.825383594908152 },
-        { - 0.25 , - 4.901666809860711 },
-        { - 0.125 , - 8.717218859383175 },
-        { 0.125 , 7.533941598797612 },
-        { 0.25 , 3.625609908221908 },
-        { 0.375 , 2.370436184416601 },
-        { 0.5 , 1.772453850905516 },
-        { 0.625 , 1.434518848090557 },
-        { 0.75 , 1.225416702465178 },
-        { 0.875 , 1.089652357422897 },
-        { 1.0 , 1.0 },
-        { 1.125 , .9417426998497015 },
-        { 1.25 , 0.906402477055477 },
-        { 1.375 , .8889135691562253 },
-        { 1.5 , 0.886226925452758 },
-        { 1.625 , 0.896574280056598 },
-        { 1.75 , .9190625268488832 },
-        { 1.875 , .9534458127450348 },
-        { 2.0 , 1.0 },
-        { 2.125 , 1.059460537330914 },
-        { 2.25 , 1.133003096319346 },
-        { 2.375 , 1.22225615758981 },
-        { 2.5 , 1.329340388179137 },
-        { 2.625 , 1.456933205091972 },
-        { 2.75 , 1.608359421985546 },
-        { 2.875 , 1.78771089889694 },
-        { 3.0 , 2.0 },
-        { 3.125 , 2.251353641828193 },
-        { 3.25 , 2.549256966718529 },
-        { 3.375 , 2.902858374275799 },
-        { 3.5 , 3.323350970447843 },
-        { 3.625 , 3.824449663366426 },
-        { 3.75 , 4.422988410460251 },
-        { 3.875 , 5.139668834328703 },
-        { 4.0 , 6.0 },
-        { 4.125 , 7.035480130713102 },
-        { 4.25 , 8.28508514183522 },
-        { 4.375 , 9.797147013180819 },
-        { 4.5 , 11.63172839656745 },
-        { 4.625 , 13.86363002970329 },
-        { 4.75 , 16.58620653922594 },
-        { 4.875 , 19.91621673302373 },
-        { 5.0 , 24.0 },
-        { 5.125 , 29.02135553919155 },
-        { 5.25 , 35.21161185279968 },
-        { 5.375 , 42.86251818266609 },
-        { 5.5 , 52.34277778455352 },
-        { 5.625 , 64.11928888737773 },
-        { 5.75 , 78.78448106132322 },
-        { 5.875 , 97.09155657349066 },
-        { 6.0 , 120.0 },
-        { 6.125 , 148.7344471383567 },
-        { 6.25 , 184.8609622271983 },
-        { 6.375 , 230.3860352318302 },
-        { 6.5 , 287.8852778150443 },
-        { 6.625 , 360.6709999914997 },
-        { 6.75 , 453.0107661026085 },
-        { 6.875 , 570.4128948692577 },
-        { 7.0 , 720.0 },
-        { 7.125 , 910.9984887224346 },
-        { 7.25 , 1155.38101391999 },
-        { 7.375 , 1468.710974602918 },
-        { 7.5 , 1871.254305797788 },
-        { 7.625 , 2389.445374943686 },
-        { 7.75 , 3057.822671192607 },
-        { 7.875 , 3921.588652226146 },
-        { 8.0 , 5040.0 },
-        { 8.125 , 6490.864232147346 },
-        { 8.25 , 8376.512350919926 },
-        { 8.375 , 10831.74343769652 },
-        { 8.5 , 14034.40729348341 },
-        { 8.625 , 18219.5209839456 },
-        { 8.75 , 23698.12570174271 },
-        { 8.875 , 30882.5106362809 },
-        { 9.0 , 40320.0 },
-        { 9.125 , 52738.27188619719 },
-        { 9.25 , 69106.22689508938 },
-        { 9.375 , 90715.85129070834 },
-        { 9.5 , 119292.461994609 },
-        { 9.625 , 157143.3684865308 },
-        { 9.75 , 207358.5998902487 },
-        { 9.875 , 274082.281896993 },
-        { 10.0 , 362880.0 },
-        { 10.125 , 481236.7309615494 },
-        { 10.25 , 639232.5987795768 },
-        { 10.375 , 850461.1058503906 },
-        { 10.5 , 1133278.388948786 },
-        { 10.625 , 1512504.921682859 },
-        { 10.75 , 2021746.348929925 },
-        { 10.875 , 2706562.533732806 },
-        { 11.0 , 3628800.0 },
-        { 11.125 , 4872521.900985687 },
-        { 11.25 , 6552134.137490662 },
-        { 11.375 , 8823533.973197803 },
-        { 11.5 , 1.1899423083962249e+7 },
-        { 11.625 , 1.6070364792880382e+7 },
-        { 11.75 , 2.1733773250996688e+7 },
-        { 11.875 , 2.943386755434427e+7 },
-        { 12.0 , 3.99168e+7 },
-        { 12.125 , 5.420680614846578e+7 },
-        { 12.25 , 7.371150904676994e+7 },
-        { 12.375 , 1.0036769894512501e+8 },
-        { 12.5 , 1.3684336546556586e+8 },
-        { 12.625 , 1.8681799071723443e+8 },
-        { 12.75 , 2.5537183569921107e+8 },
-        { 12.875 , 3.4952717720783816e+8 },
-        { 13.0 , 4.790016e+8 },
-        { 13.125 , 6.572575245501475e+8 },
-        { 13.25 , 9.029659858229319e+8 },
-        { 13.375 , 1.2420502744459219e+9 },
-        { 13.5 , 1.7105420683195732e+9 },
-        { 13.625 , 2.3585771328050845e+9 },
-        { 13.75 , 3.2559909051649416e+9 },
-        { 13.875 , 4.500162406550916e+9 },
-        { 14.0 , 6.2270208e+9 },
-        { 14.125 , 8.626505009720685e+9 },
-        { 14.25 , 1.196429931215385e+10 },
-        { 14.375 , 1.66124224207142e+10 },
-        { 14.5 , 2.309231792231424e+10 },
-        { 14.625 , 3.213561343446927e+10 },
-        { 14.75 , 4.476987494601794e+10 },
-        { 14.875 , 6.243975339089396e+10 },
-        { 15.0 , 8.71782912e+10 },
-        { 15.125 , 1.218493832623047e+11 },
-        { 15.25 , 1.704912651981923e+11 },
-        { 15.375 , 2.388035722977667e+11 },
-        { 15.5 , 3.348386098735565e+11 },
-        { 15.625 , 4.699833464791132e+11 },
-        { 15.75 , 6.603556554537646e+11 },
-        { 15.875 , 9.287913316895475e+11 },
-        { 16.0 , 1.307674368e+12 },
-        { 16.125 , 1.842971921842358e+12 },
-        { 16.25 , 2.599991794272433e+12 },
-        { 16.375 , 3.671604924078163e+12 },
-        { 16.5 , 5.189998453040126e+12 },
-        { 16.625 , 7.343489788736144e+12 },
-        { 16.75 , 1.040060157339679e+13 },
-        { 16.875 , 1.474456239057157e+13 },
-        { 17.0 , 2.0922789888e+13 },
-        { 17.125 , 2.971792223970803e+13 },
-        { 17.25 , 4.224986665692704e+13 },
-        { 17.375 , 6.012253063177992e+13 },
-        { 17.5 , 8.563497447516206e+13 },
-        { 17.625 , 1.220855177377384e+14 },
-        { 17.75 , 1.742100763543963e+14 },
-        { 17.875 , 2.488144903408952e+14 },
-        { 18.0 , 3.55687428096e+14 },
-        { 18.125 , 5.08919418355e+14 },
-        { 18.25 , 7.288101998319914e+14 },
-        { 18.375 , 1.044628969727176e+15 },
-        { 18.5 , 1.498612053315336e+15 },
-        { 18.625 , 2.151757250127639e+15 },
-        { 18.75 , 3.092228855290534e+15 },
-        { 18.875 , 4.447559014843502e+15 },
-        { 19.0 , 6.402373705728e+15 },
-        { 19.125 , 9.224164457684374e+15 },
-        { 19.25 , 1.330078614693384e+16 },
-        { 19.375 , 1.919505731873686e+16 },
-        { 19.5 , 2.772432298633372e+16 },
-        { 19.625 , 4.007647878362728e+16 },
-        { 19.75 , 5.797929103669752e+16 },
-        { 19.875 , 8.39476764051711e+16 },
-        { 20.0 , 1.21645100408832e+17 },
-        { 20.5 , 5.406242982335075e+17 },
-        { 21.0 , 2.43290200817664e+18 },
-        { 21.5 , 1.10827981137869e+19 },
-        { 22.0 , 5.109094217170944e+19 },
-        { 22.5 , 2.382801594464184e+20 },
-        { 23.0 , 1.124000727777608e+21 },
-        { 23.5 , 5.361303587544415e+21 },
-        { 24.0 , 2.585201673888498e+22 },
-        { 24.5 , 1.259906343072938e+23 },
-        { 25.0 , 6.204484017332395e+23 },
-        { 25.5 , 3.086770540528697e+24 },
-        { 26.0 , 1.551121004333099e+25 },
-        { 26.5 , 7.871264878348176e+25 },
-        { 27.0 , 4.032914611266056e+26 },
-        { 27.5 , 2.085885192762267e+27 },
-        { 28.0 , 1.088886945041835e+28 },
-        { 28.5 , 5.736184280096234e+28 },
-        { 29.0 , 3.048883446117139e+29 },
-        { 29.5 , 1.634812519827427e+30 },
-        { 30.0 , 8.841761993739702e+30 },
-        { 30.5 , 4.822696933490909e+31 },
-        { 31.0 , 2.65252859812191e+32 },
-        { 31.5 , 1.470922564714727e+33 },
-        { 32.0 , 8.222838654177922e+33 },
-        { 32.5 , 4.633406078851391e+34 },
-        { 33.0 , 2.631308369336935e+35 },
-        { 33.5 , 1.505856975626702e+36 },
-        { 34.0 , 8.683317618811885e+36 },
-        { 34.5 , 5.044620868349451e+37 },
-        { 35.0 , 2.952327990396041e+38 },
-        { 35.5 , 1.740394199580561e+39 },
-        { 36.0 , 1.033314796638614e+40 },
-        { 36.5 , 6.178399408510991e+40 },
-        { 37.0 , 3.719933267899013e+41 },
-        { 37.5 , 2.255115784106512e+42 },
-        { 38.0 , 1.376375309122634e+43 },
-        { 38.5 , 8.456684190399419e+43 },
-        { 39.0 , 5.230226174666011e+44 },
-        { 39.5 , 3.255823413303776e+45 },
-        { 40.0 , 2.039788208119745e+46 },
-        { 40.5 , 1.286050248254992e+47 },
-        { 41.0 , 8.159152832478976e+47 },
-        { 41.5 , 5.208503505432716e+48 },
-        { 42.0 , 3.345252661316381e+49 },
-        { 42.5 , 2.161528954754577e+50 },
-        { 43.0 , 1.40500611775288e+51 },
-        { 43.5 , 9.186498057706953e+51 },
-        { 44.0 , 6.041526306337383e+52 },
-        { 44.5 , 3.996126655102524e+53 },
-        { 45.0 , 2.658271574788449e+54 },
-        { 45.5 , 1.778276361520623e+55 },
-        { 46.0 , 1.196222208654802e+56 },
-        { 46.5 , 8.091157444918836e+56 },
-        { 47.0 , 5.502622159812088e+57 },
-        { 47.5 , 3.762388211887259e+58 },
-        { 48.0 , 2.586232415111682e+59 },
-        { 48.5 , 1.787134400646448e+60 },
-        { 49.0 , 1.241391559253607e+61 },
-        { 49.5 , 8.667601843135273e+61 },
-        { 50.0 , 6.082818640342675e+62 },
-        { 50.5 , 4.290462912351959e+63 },
-        { 51.0 , 3.041409320171338e+64 },
-        { 51.5 , 2.16668377073774e+65 },
-        { 52.0 , 1.551118753287382e+66 },
-        { 52.5 , 1.115842141929936e+67 },
-        { 53.0 , 8.065817517094388e+67 },
-        { 53.5 , 5.858171245132164e+68 },
-        { 54.0 , 4.274883284060025e+69 },
-        { 54.5 , 3.134121616145708e+70 },
-        { 55.0 , 2.308436973392413e+71 },
-        { 55.5 , 1.70809628079941e+72 },
-        { 56.0 , 1.269640335365828e+73 },
-        { 56.5 , 9.479934358436728e+73 },
-        { 57.0 , 7.109985878048635e+74 },
-        { 57.5 , 5.356162912516752e+75 },
-        { 58.0 , 4.052691950487721e+76 },
-        { 58.5 , 3.079793674697132e+77 },
-        { 59.0 , 2.350561331282878e+78 },
-        { 59.5 , 1.801679299697822e+79 },
-        { 60.0 , 1.386831185456898e+80 },
-        { 60.5 , 1.071999183320204e+81 },
-        { 61.0 , 8.320987112741391e+81 },
-        { 61.5 , 6.485595059087236e+82 },
-        { 62.0 , 5.075802138772249e+83 },
-        { 62.5 , 3.98864096133865e+84 },
-        { 63.0 , 3.146997326038794e+85 },
-        { 63.5 , 2.492900600836656e+86 },
-        { 64.0 , 1.98260831540444e+87 },
-        { 64.5 , 1.582991881531277e+88 },
-        { 65.0 , 1.268869321858841e+89 },
-        { 65.5 , 1.021029763587673e+90 },
-        { 66.0 , 8.247650592082472e+90 },
-        { 66.5 , 6.687744951499262e+91 },
-        { 67.0 , 5.443449390774431e+92 },
-        { 67.5 , 4.447350392747009e+93 },
-        { 68.0 , 3.647111091818868e+94 },
-        { 68.5 , 3.001961515104231e+95 },
-        { 69.0 , 2.48003554243683e+96 },
-        { 69.5 , 2.056343637846398e+97 },
-        { 70.0 , 1.711224524281413e+98 },
-        { 70.5 , 1.429158828303247e+99 },
-        { 71.0 , 1.19785716699699e+100 },
-        { 71.5 , 1.00755697395379e+101 },
-        { 72.0 , 8.50478588567862e+101 },
-        { 72.5 , 7.20403236376959e+102 },
-        { 73.0 , 6.12344583768861e+103 },
-        { 73.5 , 5.22292346373295e+104 },
-        { 74.0 , 4.47011546151268e+105 },
-        { 74.5 , 3.83884874584372e+106 },
-        { 75.0 , 3.30788544151939e+107 },
-        { 75.5 , 2.85994231565357e+108 },
-        { 76.0 , 2.48091408113954e+109 },
-        { 76.5 , 2.15925644831845e+110 },
-        { 77.0 , 1.88549470166605e+111 },
-        { 77.5 , 1.65183118296361e+112 },
-        { 78.0 , 1.45183092028286e+113 },
-        { 78.5 , 1.2801691667968e+114 },
-        { 79.0 , 1.13242811782063e+115 },
-        { 79.5 , 1.00493279593549e+116 },
-        { 80.0 , 8.94618213078298e+116 },
-        { 80.5 , 7.98921572768712e+117 },
-        { 81.0 , 7.15694570462638e+118 },
-        { 81.5 , 6.43131866078814e+119 },
-        { 82.0 , 5.79712602074737e+120 },
-        { 82.5 , 5.24152470854233e+121 },
-        { 83.0 , 4.75364333701284e+122 },
-        { 83.5 , 4.32425788454742e+123 },
-        { 84.0 , 3.94552396972066e+124 },
-        { 84.5 , 3.6107553335971e+125 },
-        { 85.0 , 3.31424013456535e+126 },
-        { 85.5 , 3.05108825688955e+127 },
-        { 86.0 , 2.81710411438055e+128 },
-        { 86.5 , 2.60868045964056e+129 },
-        { 87.0 , 2.42270953836727e+130 },
-        { 87.5 , 2.25650859758909e+131 },
-        { 88.0 , 2.10775729837953e+132 },
-        { 88.5 , 1.97444502289045e+133 },
-        { 89.0 , 1.85482642257398e+134 },
-        { 89.5 , 1.74738384525805e+135 },
-        { 90.0 , 1.65079551609085e+136 },
-        { 90.5 , 1.56390854150595e+137 },
-        { 91.0 , 1.48571596448176e+138 },
-        { 91.5 , 1.41533723006289e+139 },
-        { 92.0 , 1.3520015276784e+140 },
-        { 92.5 , 1.29503356550754e+141 },
-        { 93.0 , 1.24384140546413e+142 },
-        { 93.5 , 1.19790604809448e+143 },
-        { 94.0 , 1.15677250708164e+144 },
-        { 94.5 , 1.12004215496834e+145 },
-        { 95.0 , 1.08736615665674e+146 },
-        { 95.5 , 1.05843983644508e+147 },
-        { 96.0 , 1.03299784882391e+148 },
-        { 96.5 , 1.01081004380505e+149 },
-        { 97.0 , 9.9167793487095e+149 },
-        { 97.5 , 9.75431692271873e+150 },
-        { 98.0 , 9.61927596824821e+151 },
-        { 98.5 , 9.51045899965076e+152 },
-        { 99.0 , 9.42689044888325e+153 },
-        { 99.5 , 9.367802114656e+154 },
-        { 100.0 , 9.33262154439441e+155 },
-    };
-
-    @Test
-    public void testGamma() {
-
-        for (int i = 0; i < GAMMA_REF.length; i++) {
-            final double[] ref = GAMMA_REF[i];
-            final double x = ref[0];
-            final double expected = ref[1];
-            final double actual = Gamma.gamma(x);
-            final double absX = FastMath.abs(x);
-            final int ulps;
-            if (absX <= 8.0) {
-                ulps = 3;
-            } else if (absX <= 20.0) {
-                ulps = 5;
-            } else if (absX <= 30.0) {
-                ulps = 50;
-            } else if (absX <= 50.0) {
-                ulps = 180;
-            } else {
-                ulps = 500;
-            }
-            final double tol = ulps * FastMath.ulp(expected);
-            Assert.assertEquals(Double.toString(x), expected, actual, tol);
-        }
-    }
-
-    @Test
-    public void testGammaNegativeInteger() {
-
-        for (int i = -100; i <= 0; i++) {
-            Assert.assertTrue(Integer.toString(i), Double.isNaN(Gamma.gamma(i)));
-        }
-    }
-
-    @Test
-    public void testGammaNegativeDouble() {
-        // check that the gamma function properly switches sign
-        // see: https://en.wikipedia.org/wiki/Gamma_function
-
-        double previousGamma = Gamma.gamma(-18.5);
-        for (double x = -19.5; x > -25; x -= 1.0) {
-            double gamma = Gamma.gamma(x);
-            Assert.assertEquals(  (int) FastMath.signum(previousGamma),
-                                - (int) FastMath.signum(gamma));
-
-            previousGamma = gamma;
-        }
-    }
-
-    private void checkRelativeError(String msg, double expected, double actual,
-                                    double tolerance) {
-
-        Assert.assertEquals(msg, expected, actual, FastMath.abs(tolerance * actual));
-    }
-}