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Posted to commits@commons.apache.org by ah...@apache.org on 2022/01/22 13:13:58 UTC
[commons-numbers] 03/05: NUMBERS-181: Port the Boost beta functions
This is an automated email from the ASF dual-hosted git repository.
aherbert pushed a commit to branch master
in repository https://gitbox.apache.org/repos/asf/commons-numbers.git
commit ad04a0700e282b0b7f303a5fc7d2436555f040c6
Author: Alex Herbert <ah...@apache.org>
AuthorDate: Wed Jan 5 23:49:59 2022 +0000
NUMBERS-181: Port the Boost beta functions
---
.../apache/commons/numbers/gamma/BoostBeta.java | 1356 ++++++++++++++
.../apache/commons/numbers/gamma/BoostGamma.java | 15 +-
.../commons/numbers/gamma/BoostBetaTest.java | 1236 +++++++++++++
.../apache/commons/numbers/gamma/beta_exp_data.csv | 373 ++++
.../apache/commons/numbers/gamma/beta_med_data.csv | 1852 ++++++++++++++++++++
.../commons/numbers/gamma/beta_small_data.csv | 45 +
.../numbers/gamma/binomial_extra_large_data.csv | 531 ++++++
.../commons/numbers/gamma/binomial_huge_data.csv | 530 ++++++
.../commons/numbers/gamma/binomial_large_data.csv | 252 +++
.../commons/numbers/gamma/binomial_small_data.csv | 181 ++
.../numbers/gamma/ibeta_derivative_large_data.csv | 1233 +++++++++++++
.../numbers/gamma/ibeta_derivative_med_data.csv | 523 ++++++
.../numbers/gamma/ibeta_derivative_small_data.csv | 523 ++++++
.../gamma/ibeta_derivative_small_int_data.csv | 1023 +++++++++++
.../commons/numbers/gamma/ibeta_large_data.csv | 1233 +++++++++++++
.../commons/numbers/gamma/ibeta_med_data.csv | 523 ++++++
.../commons/numbers/gamma/ibeta_small_data.csv | 523 ++++++
.../commons/numbers/gamma/ibeta_small_int_data.csv | 1023 +++++++++++
.../checkstyle/checkstyle-suppressions.xml | 3 +
src/main/resources/pmd/pmd-ruleset.xml | 20 +-
20 files changed, 12991 insertions(+), 7 deletions(-)
diff --git a/commons-numbers-gamma/src/main/java/org/apache/commons/numbers/gamma/BoostBeta.java b/commons-numbers-gamma/src/main/java/org/apache/commons/numbers/gamma/BoostBeta.java
new file mode 100644
index 0000000..f468d04
--- /dev/null
+++ b/commons-numbers-gamma/src/main/java/org/apache/commons/numbers/gamma/BoostBeta.java
@@ -0,0 +1,1356 @@
+/*
+ * 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.
+ */
+
+// Copyright John Maddock 2006.
+// Use, modification and distribution are subject to the
+// Boost Software License, Version 1.0. (See accompanying file
+// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)
+
+package org.apache.commons.numbers.gamma;
+
+import java.util.function.DoubleSupplier;
+import java.util.function.Supplier;
+import org.apache.commons.numbers.fraction.GeneralizedContinuedFraction;
+import org.apache.commons.numbers.fraction.GeneralizedContinuedFraction.Coefficient;
+
+/**
+ * Implementation of the
+ * <a href="https://mathworld.wolfram.com/RegularizedBetaFunction.html">regularized beta functions</a> and
+ * <a href="https://mathworld.wolfram.com/IncompleteBetaFunction.html">incomplete beta functions</a>.
+ *
+ * <p>This code has been adapted from the <a href="https://www.boost.org/">Boost</a>
+ * {@code c++} implementation {@code <boost/math/special_functions/beta.hpp>}.
+ * All work is copyright to the original authors and subject to the Boost Software License.
+ *
+ * @see
+ * <a href="https://www.boost.org/doc/libs/1_77_0/libs/math/doc/html/math_toolkit/sf_beta.html">
+ * Boost C++ beta functions</a>
+ */
+final class BoostBeta {
+ //
+ // Code ported from Boost 1.77.0
+ //
+ // boost/math/special_functions/beta.hpp
+ //
+ // Original code comments are preserved.
+ //
+ // Changes to the Boost implementation:
+ // - Update method names to replace underscores with camel case
+ // - Remove checks for under/overflow. In this implementation no error is raised
+ // for overflow (infinity is returned) or underflow (sub-normal or zero is returned).
+ // This follows the conventions in java.lang.Math for the same conditions.
+ // - Removed the pointer p_derivative in the betaIncompleteImp. This is used
+ // in the Boost code for the beta_inv functions for a derivative
+ // based inverse function. This is currently not supported.
+ // - Altered series generators to use integer counters added to the double term
+ // replacing directly incrementing a double term. When the term is large it cannot
+ // be incremented: 1e16 + 1 == 1e16.
+ // - Added the use of the classic continued fraction representation for cases
+ // where the Boost implementation detects sub-normal terms and does not evaluate.
+ // - Updated the method used to compute the binomialCoefficient. This can use a
+ // series evaluation when n > max factorial given that n - k < 40.
+ // - Changed convergence criteria for betaSmallBLargeASeries to stop when r has
+ // no effect on the sum. The Boost code uses machine epsilon (ignoring the policy eps).
+
+ /** Default epsilon value for relative error.
+ * This is equal to the Boost constant {@code boost::math::EPSILON}. */
+ private static final double EPSILON = 0x1.0p-52;
+ /** Approximate value for ln(Double.MAX_VALUE).
+ * This is equal to the Boost constant {@code boost::math::tools::log_max_value<double>()}.
+ * No term {@code x} should be used in {@code exp(x)} if {@code x > LOG_MAX_VALUE} to avoid
+ * overflow. */
+ private static final int LOG_MAX_VALUE = 709;
+ /** Approximate value for ln(Double.MIN_VALUE).
+ * This is equal to the Boost constant {@code boost::math::tools::log_min_value<double>()}.
+ * No term {@code x} should be used in {@code exp(x)} if {@code x < LOG_MIN_VALUE} to avoid
+ * underflow to sub-normal or zero. */
+ private static final int LOG_MIN_VALUE = -708;
+ /** pi/2. */
+ private static final double HALF_PI = Math.PI / 2;
+ /** The largest factorial that can be represented as a double.
+ * This is equal to the Boost constant {@code boost::math::max_factorial<double>::value}. */
+ private static final int MAX_FACTORIAL = 170;
+ /** Size of the table of Pn's.
+ * Equal to {@code Pn_size<double>} suitable for 16-20 digit accuracy. */
+ private static final int PN_SIZE = 30;
+ /** 2^53. Used to scale sub-normal values. */
+ private static final double TWO_POW_53 = 0x1.0p53;
+ /** 2^-53. Used to rescale values. */
+ private static final double TWO_POW_M53 = 0x1.0p-53;
+
+ /** Private constructor. */
+ private BoostBeta() {
+ // intentionally empty.
+ }
+
+ /**
+ * Beta function.
+ * <p>\[ B(p, q) = \frac{\Gamma(p) \Gamma(q)}{\Gamma(p+q)} \]
+ *
+ * @param p Argument p
+ * @param q Argument q
+ * @return beta value
+ */
+ static double beta(double p, double q) {
+ if (!(p > 0 && q > 0)) {
+ // Domain error
+ return Double.NaN;
+ }
+
+ final double c = p + q;
+
+ // Special cases:
+ if (c == p && q < EPSILON) {
+ return 1 / q;
+ } else if (c == q && p < EPSILON) {
+ return 1 / p;
+ }
+ if (q == 1) {
+ return 1 / p;
+ } else if (p == 1) {
+ return 1 / q;
+ } else if (c < EPSILON) {
+ return (c / p) / q;
+ }
+
+ // Create input a > b
+ final double a = p < q ? q : p;
+ final double b = p < q ? p : q;
+
+ // Lanczos calculation:
+ final double agh = a + BoostGamma.Lanczos.gmh();
+ final double bgh = b + BoostGamma.Lanczos.gmh();
+ final double cgh = c + BoostGamma.Lanczos.gmh();
+ double result = BoostGamma.Lanczos.lanczosSumExpGScaled(a) *
+ (BoostGamma.Lanczos.lanczosSumExpGScaled(b) / BoostGamma.Lanczos.lanczosSumExpGScaled(c));
+ final double ambh = a - 0.5f - b;
+ if (Math.abs(b * ambh) < cgh * 100 && a > 100) {
+ // Special case where the base of the power term is close to 1
+ // compute (1+x)^y instead:
+ result *= Math.exp(ambh * Math.log1p(-b / cgh));
+ } else {
+ result *= Math.pow(agh / cgh, ambh);
+ }
+
+ if (cgh > 1e10f) {
+ // this avoids possible overflow, but appears to be marginally less accurate:
+ result *= Math.pow((agh / cgh) * (bgh / cgh), b);
+ } else {
+ result *= Math.pow((agh * bgh) / (cgh * cgh), b);
+ }
+ result *= Math.sqrt(Math.E / bgh);
+
+ return result;
+ }
+
+ /**
+ * Derivative of the regularised incomplete beta.
+ * <p>\[ \frac{\delta}{\delta x} I_x(a, b) = \frac{(1-x)^{b-1} x^{a-1}}{\B(a, b)} \]
+ *
+ * <p>Adapted from {@code boost::math::ibeta_derivative}.
+ *
+ * @param a Argument a
+ * @param b Argument b
+ * @param x Argument x
+ * @return ibeta derivative
+ */
+ static double ibetaDerivative(double a, double b, double x) {
+ //
+ // start with the usual error checks:
+ //
+ if (!(a > 0 && b > 0) || !(x >= 0 && x <= 1)) {
+ // Domain error
+ return Double.NaN;
+ }
+ //
+ // Now the corner cases:
+ //
+ if (x == 0) {
+ if (a > 1) {
+ return 0;
+ }
+ // a == 1 : return 1 / beta(a, b) == b
+ return a == 1 ? b : Double.POSITIVE_INFINITY;
+ } else if (x == 1) {
+ if (b > 1) {
+ return 0;
+ }
+ // b == 1 : return 1 / beta(a, b) == a
+ return b == 1 ? a : Double.POSITIVE_INFINITY;
+ }
+
+ // Update with extra edge cases
+ if (b == 1) {
+ // ibeta = x^a
+ return a * Math.pow(x, a - 1);
+ }
+ if (a == 1) {
+ // ibeta = 1 - (1-x)^b
+ if (x >= 0.5) {
+ return b * Math.pow(1 - x, b - 1);
+ }
+ return b * Math.exp(Math.log1p(-x) * (b - 1));
+ }
+
+ //
+ // Now the regular cases:
+ //
+ final double y = (1 - x) * x;
+ return ibetaPowerTerms(a, b, x, 1 - x, true, 1 / y);
+ }
+
+ /**
+ * Compute the leading power terms in the incomplete Beta.
+ *
+ * <p>Utility function to call
+ * {@link #ibetaPowerTerms(double, double, double, double, boolean, double)}
+ * using a multiplication prefix of {@code 1.0}.
+ *
+ * @param a Argument a
+ * @param b Argument b
+ * @param x Argument x
+ * @param y Argument 1-x
+ * @param normalised true to divide by beta(a, b)
+ * @return incomplete beta power terms
+ */
+ private static double ibetaPowerTerms(double a, double b, double x,
+ double y, boolean normalised) {
+ return ibetaPowerTerms(a, b, x, y, normalised, 1);
+ }
+
+ /**
+ * Compute the leading power terms in the incomplete Beta.
+ *
+ * <pre>
+ * (x^a)(y^b)/Beta(a,b) when normalised, and
+ * (x^a)(y^b) otherwise.
+ * </pre>
+ *
+ * <p>Almost all of the error in the incomplete beta comes from this function:
+ * particularly when a and b are large. Computing large powers are *hard*
+ * though, and using logarithms just leads to horrendous cancellation errors.
+ *
+ * @param a Argument a
+ * @param b Argument b
+ * @param x Argument x
+ * @param y Argument 1-x
+ * @param normalised true to divide by beta(a, b)
+ * @param prefix Prefix to multiply by the result
+ * @return incomplete beta power terms
+ */
+ private static double ibetaPowerTerms(double a, double b, double x,
+ double y, boolean normalised, double prefix) {
+ if (!normalised) {
+ // can we do better here?
+ return Math.pow(x, a) * Math.pow(y, b);
+ }
+
+ double result;
+
+ final double c = a + b;
+
+ // combine power terms with Lanczos approximation:
+ final double agh = a + BoostGamma.Lanczos.gmh();
+ final double bgh = b + BoostGamma.Lanczos.gmh();
+ final double cgh = c + BoostGamma.Lanczos.gmh();
+ result = BoostGamma.Lanczos.lanczosSumExpGScaled(c) /
+ (BoostGamma.Lanczos.lanczosSumExpGScaled(a) * BoostGamma.Lanczos.lanczosSumExpGScaled(b));
+ result *= prefix;
+ // combine with the leftover terms from the Lanczos approximation:
+ result *= Math.sqrt(bgh / Math.E);
+ result *= Math.sqrt(agh / cgh);
+
+ // l1 and l2 are the base of the exponents minus one:
+ double l1 = (x * b - y * agh) / agh;
+ double l2 = (y * a - x * bgh) / bgh;
+ if (Math.min(Math.abs(l1), Math.abs(l2)) < 0.2) {
+ // when the base of the exponent is very near 1 we get really
+ // gross errors unless extra care is taken:
+ if (l1 * l2 > 0 || Math.min(a, b) < 1) {
+ //
+ // This first branch handles the simple cases where either:
+ //
+ // * The two power terms both go in the same direction
+ // (towards zero or towards infinity). In this case if either
+ // term overflows or underflows, then the product of the two must
+ // do so also.
+ // * Alternatively if one exponent is less than one, then we
+ // can't productively use it to eliminate overflow or underflow
+ // from the other term. Problems with spurious overflow/underflow
+ // can't be ruled out in this case, but it is *very* unlikely
+ // since one of the power terms will evaluate to a number close to 1.
+ //
+ if (Math.abs(l1) < 0.1) {
+ result *= Math.exp(a * Math.log1p(l1));
+ } else {
+ result *= Math.pow((x * cgh) / agh, a);
+ }
+ if (Math.abs(l2) < 0.1) {
+ result *= Math.exp(b * Math.log1p(l2));
+ } else {
+ result *= Math.pow((y * cgh) / bgh, b);
+ }
+ } else if (Math.max(Math.abs(l1), Math.abs(l2)) < 0.5) {
+ //
+ // Both exponents are near one and both the exponents are
+ // greater than one and further these two
+ // power terms tend in opposite directions (one towards zero,
+ // the other towards infinity), so we have to combine the terms
+ // to avoid any risk of overflow or underflow.
+ //
+ // We do this by moving one power term inside the other, we have:
+ //
+ // (1 + l1)^a * (1 + l2)^b
+ // = ((1 + l1)*(1 + l2)^(b/a))^a
+ // = (1 + l1 + l3 + l1*l3)^a ; l3 = (1 + l2)^(b/a) - 1
+ // = exp((b/a) * log(1 + l2)) - 1
+ //
+ // The tricky bit is deciding which term to move inside :-)
+ // By preference we move the larger term inside, so that the
+ // size of the largest exponent is reduced. However, that can
+ // only be done as long as l3 (see above) is also small.
+ //
+ final boolean smallA = a < b;
+ final double ratio = b / a;
+ if ((smallA && ratio * l2 < 0.1) || (!smallA && l1 / ratio > 0.1)) {
+ double l3 = Math.expm1(ratio * Math.log1p(l2));
+ l3 = l1 + l3 + l3 * l1;
+ l3 = a * Math.log1p(l3);
+ result *= Math.exp(l3);
+ } else {
+ double l3 = Math.expm1(Math.log1p(l1) / ratio);
+ l3 = l2 + l3 + l3 * l2;
+ l3 = b * Math.log1p(l3);
+ result *= Math.exp(l3);
+ }
+ } else if (Math.abs(l1) < Math.abs(l2)) {
+ // First base near 1 only:
+ double l = a * Math.log1p(l1) + b * Math.log((y * cgh) / bgh);
+ if (l <= LOG_MIN_VALUE || l >= LOG_MAX_VALUE) {
+ l += Math.log(result);
+ // Update: Allow overflow to infinity
+ result = Math.exp(l);
+ } else {
+ result *= Math.exp(l);
+ }
+ } else {
+ // Second base near 1 only:
+ double l = b * Math.log1p(l2) + a * Math.log((x * cgh) / agh);
+ if (l <= LOG_MIN_VALUE || l >= LOG_MAX_VALUE) {
+ l += Math.log(result);
+ // Update: Allow overflow to infinity
+ result = Math.exp(l);
+ } else {
+ result *= Math.exp(l);
+ }
+ }
+ } else {
+ // general case:
+ final double b1 = (x * cgh) / agh;
+ final double b2 = (y * cgh) / bgh;
+ l1 = a * Math.log(b1);
+ l2 = b * Math.log(b2);
+ if (l1 >= LOG_MAX_VALUE || l1 <= LOG_MIN_VALUE || l2 >= LOG_MAX_VALUE || l2 <= LOG_MIN_VALUE) {
+ // Oops, under/overflow, sidestep if we can:
+ if (a < b) {
+ final double p1 = Math.pow(b2, b / a);
+ final double l3 = a * (Math.log(b1) + Math.log(p1));
+ if (l3 < LOG_MAX_VALUE && l3 > LOG_MIN_VALUE) {
+ result *= Math.pow(p1 * b1, a);
+ } else {
+ l2 += l1 + Math.log(result);
+ // Update: Allow overflow to infinity
+ result = Math.exp(l2);
+ }
+ } else {
+ final double p1 = Math.pow(b1, a / b);
+ final double l3 = (Math.log(p1) + Math.log(b2)) * b;
+ if (l3 < LOG_MAX_VALUE && l3 > LOG_MIN_VALUE) {
+ result *= Math.pow(p1 * b2, b);
+ } else {
+ l2 += l1 + Math.log(result);
+ // Update: Allow overflow to infinity
+ result = Math.exp(l2);
+ }
+ }
+ } else {
+ // finally the normal case:
+ result *= Math.pow(b1, a) * Math.pow(b2, b);
+ }
+ }
+
+ return result;
+ }
+
+ /**
+ * Full incomplete beta.
+ * <p>\[ B_x(a,b) = \int_0^x t^{a-1}\,(1-t)^{b-1}\,dt \]
+ *
+ * @param a Argument a
+ * @param b Argument b
+ * @param x Argument x
+ * @return lower beta value
+ */
+ static double beta(double a, double b, double x) {
+ return betaIncompleteImp(a, b, x, Policy.getDefault(), false, false);
+ }
+
+ /**
+ * Full incomplete beta.
+ * <p>\[ B_x(a,b) = \int_0^x t^{a-1}\,(1-t)^{b-1}\,dt \]
+ *
+ * @param a Argument a
+ * @param b Argument b
+ * @param x Argument x
+ * @param policy Function evaluation policy
+ * @return lower beta value
+ */
+ static double beta(double a, double b, double x, Policy policy) {
+ return betaIncompleteImp(a, b, x, policy, false, false);
+ }
+
+ /**
+ * Complement of the incomplete beta.
+ * <p>\[ betac(a, b, x) = beta(b, a, 1-x) = B_{1-x}(b,a) \]
+ *
+ * @param a Argument a
+ * @param b Argument b
+ * @param x Argument x
+ * @return upper beta value
+ */
+ static double betac(double a, double b, double x) {
+ return betaIncompleteImp(a, b, x, Policy.getDefault(), false, true);
+ }
+
+ /**
+ * Complement of the incomplete beta.
+ * <p>\[ betac(a, b, x) = beta(b, a, 1-x) = B_{1-x}(b,a) \]
+ *
+ * @param a Argument a
+ * @param b Argument b
+ * @param x Argument x
+ * @param policy Function evaluation policy
+ * @return upper beta value
+ */
+ static double betac(double a, double b, double x, Policy policy) {
+ return betaIncompleteImp(a, b, x, policy, false, true);
+ }
+
+ /**
+ * Regularised incomplete beta.
+ * <p>\[ I_x(a,b) = \frac{1}{B(a, b)} \int_0^x t^{a-1}\,(1-t)^{b-1}\,dt \]
+ *
+ * @param a Argument a
+ * @param b Argument b
+ * @param x Argument x
+ * @return p
+ */
+ static double ibeta(double a, double b, double x) {
+ return betaIncompleteImp(a, b, x, Policy.getDefault(), true, false);
+ }
+
+ /**
+ * Regularised incomplete beta.
+ * <p>\[ I_x(a,b) = \frac{1}{B(a, b)} \int_0^x t^{a-1}\,(1-t)^{b-1}\,dt \]
+ *
+ * @param a Argument a
+ * @param b Argument b
+ * @param x Argument x
+ * @param policy Function evaluation policy
+ * @return p
+ */
+ static double ibeta(double a, double b, double x, Policy policy) {
+ return betaIncompleteImp(a, b, x, policy, true, false);
+ }
+
+ /**
+ * Complement of the regularised incomplete beta.
+ * <p>\[ ibetac(a, b, x) = 1 - I_x(a,b) = I_{1-x}(b, a) \]
+ *
+ * @param a Argument a
+ * @param b Argument b
+ * @param x Argument x
+ * @return q
+ */
+ static double ibetac(double a, double b, double x) {
+ return betaIncompleteImp(a, b, x, Policy.getDefault(), true, true);
+ }
+
+ /**
+ * Complement of the regularised incomplete beta.
+ * <p>\[ ibetac(a, b, x) = 1 - I_x(a,b) = I_{1-x}(b, a) \]
+ *
+ * @param a Argument a
+ * @param b Argument b
+ * @param x Argument x
+ * @param policy Function evaluation policy
+ * @return q
+ */
+ static double ibetac(double a, double b, double x, Policy policy) {
+ return betaIncompleteImp(a, b, x, policy, true, true);
+ }
+
+ /**
+ * Main incomplete beta entry point, handles all four incomplete betas.
+ * Adapted from {@code boost::math::detail::ibeta_imp}.
+ *
+ * <p>The Boost code has a pointer {@code p_derivative} that can be set to the
+ * value of the derivative. This is used for the inverse incomplete
+ * beta functions. It is not required for the forward evaluation functions.
+ *
+ * @param a Argument a
+ * @param b Argument b
+ * @param x Argument x
+ * @param pol Function evaluation policy
+ * @param normalised true to compute the regularised value
+ * @param inv true to compute the complement value
+ * @return incomplete beta value
+ */
+ private static double betaIncompleteImp(double a, double b, double x,
+ Policy pol, boolean normalised, boolean inv) {
+ //
+ // The incomplete beta function implementation:
+ // This is just a big bunch of spaghetti code to divide up the
+ // input range and select the right implementation method for
+ // each domain:
+ //
+
+ if (!(x >= 0 && x <= 1)) {
+ // Domain error
+ return Double.NaN;
+ }
+
+ if (normalised) {
+ if (!(a >= 0 && b >= 0)) {
+ // Domain error
+ return Double.NaN;
+ }
+ // extend to a few very special cases:
+ if (a == 0) {
+ if (b == 0) {
+ // a and b cannot both be zero
+ return Double.NaN;
+ }
+ // Assume b > 0
+ return inv ? 0 : 1;
+ } else if (b == 0) {
+ // assume a > 0
+ return inv ? 1 : 0;
+ }
+ } else {
+ if (!(a > 0 && b > 0)) {
+ // Domain error
+ return Double.NaN;
+ }
+ }
+
+ if (x == 0) {
+ if (inv) {
+ return normalised ? 1 : beta(a, b);
+ }
+ return 0;
+ }
+ if (x == 1) {
+ if (!inv) {
+ return normalised ? 1 : beta(a, b);
+ }
+ return 0;
+ }
+
+ if (a == 0.5f && b == 0.5f) {
+ // We have an arcsine distribution:
+ final double z = inv ? 1 - x : x;
+ final double asin = Math.asin(Math.sqrt(z));
+ return normalised ? asin / HALF_PI : 2 * asin;
+ }
+
+ boolean invert = inv;
+ double y = 1 - x;
+ if (a == 1) {
+ // swap(a, b)
+ double tmp = a;
+ a = b;
+ b = tmp;
+ // swap(x, y)
+ tmp = x;
+ x = y;
+ y = tmp;
+ invert = !invert;
+ }
+ if (b == 1) {
+ //
+ // Special case see:
+ // http://functions.wolfram.com/GammaBetaErf/BetaRegularized/03/01/01/
+ //
+ if (a == 1) {
+ return invert ? y : x;
+ }
+
+ double p;
+ if (y < 0.5) {
+ p = invert ? -Math.expm1(a * Math.log1p(-y)) : Math.exp(a * Math.log1p(-y));
+ } else {
+ p = invert ? -BoostMath.powm1(x, a) : Math.pow(x, a);
+ }
+ if (!normalised) {
+ p /= a;
+ }
+ return p;
+ }
+
+ double fract;
+ if (Math.min(a, b) <= 1) {
+ if (x > 0.5) {
+ // swap(a, b)
+ double tmp = a;
+ a = b;
+ b = tmp;
+ // swap(x, y)
+ tmp = x;
+ x = y;
+ y = tmp;
+ invert = !invert;
+ }
+ if (Math.max(a, b) <= 1) {
+ // Both a,b < 1:
+ if (a >= Math.min(0.2, b) || Math.pow(x, a) <= 0.9) {
+ if (invert) {
+ fract = -(normalised ? 1 : beta(a, b));
+ invert = false;
+ fract = -ibetaSeries(a, b, x, fract, normalised, pol);
+ } else {
+ fract = ibetaSeries(a, b, x, 0, normalised, pol);
+ }
+ } else {
+ // swap(a, b)
+ double tmp = a;
+ a = b;
+ b = tmp;
+ // swap(x, y)
+ tmp = x;
+ x = y;
+ y = tmp;
+ invert = !invert;
+ if (y >= 0.3) {
+ if (invert) {
+ fract = -(normalised ? 1 : beta(a, b));
+ invert = false;
+ fract = -ibetaSeries(a, b, x, fract, normalised, pol);
+ } else {
+ fract = ibetaSeries(a, b, x, 0, normalised, pol);
+ }
+ } else {
+ // Sidestep on a, and then use the series representation:
+ double prefix;
+ if (normalised) {
+ prefix = 1;
+ } else {
+ prefix = risingFactorialRatio(a + b, a, 20);
+ }
+ fract = ibetaAStep(a, b, x, y, 20, normalised);
+ if (invert) {
+ fract -= normalised ? 1 : beta(a, b);
+ invert = false;
+ fract = -betaSmallBLargeASeries(a + 20, b, x, y, fract, prefix, pol, normalised);
+ } else {
+ fract = betaSmallBLargeASeries(a + 20, b, x, y, fract, prefix, pol, normalised);
+ }
+ }
+ }
+ } else {
+ // One of a, b < 1 only:
+ if (b <= 1 || (x < 0.1 && Math.pow(b * x, a) <= 0.7)) {
+ if (invert) {
+ fract = -(normalised ? 1 : beta(a, b));
+ invert = false;
+ fract = -ibetaSeries(a, b, x, fract, normalised, pol);
+ } else {
+ fract = ibetaSeries(a, b, x, 0, normalised, pol);
+ }
+ } else {
+ // swap(a, b)
+ double tmp = a;
+ a = b;
+ b = tmp;
+ // swap(x, y)
+ tmp = x;
+ x = y;
+ y = tmp;
+ invert = !invert;
+
+ if (y >= 0.3) {
+ if (invert) {
+ fract = -(normalised ? 1 : beta(a, b));
+ invert = false;
+ fract = -ibetaSeries(a, b, x, fract, normalised, pol);
+ } else {
+ fract = ibetaSeries(a, b, x, 0, normalised, pol);
+ }
+ } else if (a >= 15) {
+ if (invert) {
+ fract = -(normalised ? 1 : beta(a, b));
+ invert = false;
+ fract = -betaSmallBLargeASeries(a, b, x, y, fract, 1, pol, normalised);
+ } else {
+ fract = betaSmallBLargeASeries(a, b, x, y, 0, 1, pol, normalised);
+ }
+ } else {
+ // Sidestep to improve errors:
+ double prefix;
+ if (normalised) {
+ prefix = 1;
+ } else {
+ prefix = risingFactorialRatio(a + b, a, 20);
+ }
+ fract = ibetaAStep(a, b, x, y, 20, normalised);
+ if (invert) {
+ fract -= normalised ? 1 : beta(a, b);
+ invert = false;
+ fract = -betaSmallBLargeASeries(a + 20, b, x, y, fract, prefix, pol, normalised);
+ } else {
+ fract = betaSmallBLargeASeries(a + 20, b, x, y, fract, prefix, pol, normalised);
+ }
+ }
+ }
+ }
+ } else {
+ // Both a,b >= 1:
+ // Note:
+ // median ~ (a - 1/3) / (a + b - 2/3) ~ a / (a + b)
+ // if x > a / (a + b) => a - (a + b) * x < 0
+ double lambda;
+ if (a < b) {
+ lambda = a - (a + b) * x;
+ } else {
+ lambda = (a + b) * y - b;
+ }
+ if (lambda < 0) {
+ // swap(a, b)
+ double tmp = a;
+ a = b;
+ b = tmp;
+ // swap(x, y)
+ tmp = x;
+ x = y;
+ y = tmp;
+ invert = !invert;
+ }
+
+ if (b < 40) {
+ // Note: y != 1 check is required for non-zero x < epsilon
+ if (Math.rint(a) == a && Math.rint(b) == b && a < (Integer.MAX_VALUE - 100) && y != 1) {
+ // Here: a in [2, 2^31 - 102] && b in [2, 39]
+
+ // relate to the binomial distribution and use a finite sum:
+ final int k = (int) (a - 1);
+ final int n = (int) (b + k);
+ fract = binomialCCdf(n, k, x, y);
+ if (!normalised) {
+ fract *= beta(a, b);
+ }
+ } else if (b * x <= 0.7) {
+ if (invert) {
+ fract = -(normalised ? 1 : beta(a, b));
+ invert = false;
+ fract = -ibetaSeries(a, b, x, fract, normalised, pol);
+ } else {
+ fract = ibetaSeries(a, b, x, 0, normalised, pol);
+ }
+ } else if (a > 15) {
+ // sidestep so we can use the series representation:
+ int n = (int) b;
+ if (n == b) {
+ --n;
+ }
+ final double bbar = b - n;
+ double prefix;
+ if (normalised) {
+ prefix = 1;
+ } else {
+ prefix = risingFactorialRatio(a + bbar, bbar, n);
+ }
+ fract = ibetaAStep(bbar, a, y, x, n, normalised);
+ fract = betaSmallBLargeASeries(a, bbar, x, y, fract, 1, pol, normalised);
+ fract /= prefix;
+ } else if (normalised) {
+ // The formula here for the non-normalised case is tricky to figure
+ // out (for me!!), and requires two pochhammer calculations rather
+ // than one, so leave it for now and only use this in the normalized case....
+ int n = (int) Math.floor(b);
+ double bbar = b - n;
+ if (bbar <= 0) {
+ --n;
+ bbar += 1;
+ }
+ fract = ibetaAStep(bbar, a, y, x, n, normalised);
+ fract += ibetaAStep(a, bbar, x, y, 20, normalised);
+ if (invert) {
+ // Note this line would need changing if we ever enable this branch in
+ // non-normalized case
+ fract -= 1;
+ }
+ fract = betaSmallBLargeASeries(a + 20, bbar, x, y, fract, 1, pol, normalised);
+ if (invert) {
+ fract = -fract;
+ invert = false;
+ }
+ } else {
+ fract = ibetaFraction2(a, b, x, y, pol, normalised);
+ }
+ } else {
+ fract = ibetaFraction2(a, b, x, y, pol, normalised);
+ }
+ }
+ if (invert) {
+ return (normalised ? 1 : beta(a, b)) - fract;
+ }
+ return fract;
+ }
+
+ /**
+ * Series approximation to the incomplete beta.
+ *
+ * @param a Argument a
+ * @param b Argument b
+ * @param x Argument x
+ * @param s0 Initial sum for the series
+ * @param normalised true to compute the regularised value
+ * @param pol Function evaluation policy
+ * @return incomplete beta series
+ */
+ private static double ibetaSeries(double a, double b, double x, double s0, boolean normalised, Policy pol) {
+ double result;
+
+ if (normalised) {
+ final double c = a + b;
+
+ // incomplete beta power term, combined with the Lanczos approximation:
+ final double agh = a + BoostGamma.Lanczos.gmh();
+ final double bgh = b + BoostGamma.Lanczos.gmh();
+ final double cgh = c + BoostGamma.Lanczos.gmh();
+ result = BoostGamma.Lanczos.lanczosSumExpGScaled(c) /
+ (BoostGamma.Lanczos.lanczosSumExpGScaled(a) * BoostGamma.Lanczos.lanczosSumExpGScaled(b));
+
+ final double l1 = Math.log(cgh / bgh) * (b - 0.5f);
+ final double l2 = Math.log(x * cgh / agh) * a;
+ //
+ // Check for over/underflow in the power terms:
+ //
+ if (l1 > LOG_MIN_VALUE && l1 < LOG_MAX_VALUE && l2 > LOG_MIN_VALUE && l2 < LOG_MAX_VALUE) {
+ if (a * b < bgh * 10) {
+ result *= Math.exp((b - 0.5f) * Math.log1p(a / bgh));
+ } else {
+ result *= Math.pow(cgh / bgh, b - 0.5f);
+ }
+ result *= Math.pow(x * cgh / agh, a);
+ result *= Math.sqrt(agh / Math.E);
+ } else {
+ //
+ // Oh dear, we need logs, and this *will* cancel:
+ //
+ result = Math.log(result) + l1 + l2 + (Math.log(agh) - 1) / 2;
+ result = Math.exp(result);
+ }
+ } else {
+ // Non-normalised, just compute the power:
+ result = Math.pow(x, a);
+ }
+ double rescale = 1.0;
+ if (result < Double.MIN_NORMAL) {
+ // Safeguard: series can't cope with denorms.
+
+ // Update:
+ // The entire series is only based on the magnitude of 'result'.
+ // If the first term can be added to s0 (e.g. if s0 == 0) then
+ // scale s0 and result, compute the series and then rescale.
+
+ // Intentional floating-point equality check.
+ if (s0 + result / a == s0) {
+ return s0;
+ }
+ s0 *= TWO_POW_53;
+ result *= TWO_POW_53;
+ rescale = TWO_POW_M53;
+ }
+
+ final double eps = pol.getEps();
+ final int maxIterations = pol.getMaxIterations();
+
+ // Create effectively final 'result' for initialisation
+ final double result1 = result;
+ final DoubleSupplier gen = new DoubleSupplier() {
+ /** Next result term. */
+ private double result = result1;
+ /** Pochhammer term. */
+ private final double poch = -b;
+ /** Iteration. */
+ private int n;
+
+ @Override
+ public double getAsDouble() {
+ final double r = result / (a + n);
+ n++;
+ result *= (n + poch) * x / n;
+ return r;
+ }
+ };
+
+ return BoostTools.sumSeries(gen, eps, maxIterations, s0) * rescale;
+ }
+
+ /**
+ * Rising factorial ratio.
+ * This function is only needed for the non-regular incomplete beta,
+ * it computes the delta in:
+ * <pre>
+ * beta(a,b,x) = prefix + delta * beta(a+k,b,x)
+ * </pre>
+ * <p>It is currently only called for small k.
+ *
+ * @param a Argument a
+ * @param b Argument b
+ * @param k Argument k
+ * @return the rising factorial ratio
+ */
+ private static double risingFactorialRatio(double a, double b, int k) {
+ // calculate:
+ // (a)(a+1)(a+2)...(a+k-1)
+ // _______________________
+ // (b)(b+1)(b+2)...(b+k-1)
+
+ // This is only called with small k, for large k
+ // it is grossly inefficient, do not use outside it's
+ // intended purpose!!!
+ double result = 1;
+ for (int i = 0; i < k; ++i) {
+ result *= (a + i) / (b + i);
+ }
+ return result;
+ }
+
+ /**
+ * Binomial complementary cdf.
+ * For integer arguments we can relate the incomplete beta to the
+ * complement of the binomial distribution cdf and use this finite sum.
+ *
+ * @param n Argument n (called with {@code [2, 39] + k})
+ * @param k Argument k (called with {@code k in [1, Integer.MAX_VALUE - 102]})
+ * @param x Argument x
+ * @param y Argument 1-x
+ * @return Binomial complementary cdf
+ */
+ private static double binomialCCdf(int n, int k, double x, double y) {
+ double result = Math.pow(x, n);
+
+ if (result > Double.MIN_NORMAL) {
+ double term = result;
+ for (int i = n - 1; i > k; --i) {
+ term *= ((i + 1) * y) / ((n - i) * x);
+ result += term;
+ }
+ } else {
+ // First term underflows so we need to start at the mode of the
+ // distribution and work outwards:
+ int start = (int) (n * x);
+ if (start <= k + 1) {
+ start = k + 2;
+ }
+ // Update:
+ // Carefully compute this term to guard against extreme parameterisation
+ result = binomialTerm(n, start, x, y);
+ if (result == 0) {
+ // OK, starting slightly above the mode didn't work,
+ // we'll have to sum the terms the old fashioned way:
+ for (int i = start - 1; i > k; --i) {
+ result += binomialTerm(n, i, x, y);
+ }
+ } else {
+ double term = result;
+ final double startTerm = result;
+ for (int i = start - 1; i > k; --i) {
+ term *= ((i + 1) * y) / ((n - i) * x);
+ result += term;
+ }
+ term = startTerm;
+ for (int i = start + 1; i <= n; ++i) {
+ term *= (n - i + 1) * x / (i * y);
+ result += term;
+ }
+ }
+ }
+
+ return result;
+ }
+
+ /**
+ * Compute the binomial term.
+ * <pre>
+ * x^k * (1-x)^(n-k) * binomial(n, k)
+ * </pre>
+ * <p>This is a helper function used to guard against extreme values generated
+ * in the term which can produce NaN from zero multiplied by infinity.
+ *
+ * @param n Argument n
+ * @param k Argument k
+ * @param x Argument x
+ * @param y Argument 1-x
+ * @return Binomial term
+ */
+ private static double binomialTerm(int n, int k, double x, double y) {
+ // This function can be called with the following extreme that will overflow:
+ // binomial(2147483545 + 39, 38) ~ 7.84899e309
+ // Guard this as the power functions are very likely to be zero with large n and k.
+
+ final double binom = binomialCoefficient(n, k);
+ if (!Double.isFinite(binom)) {
+ // Product of the power functions will be zero with large n and k
+ return 0;
+ }
+
+ // The power terms are below 1.
+ // Multiply by the largest so that any sub-normal term is used last
+ // This method is called where x^n is sub-normal so assume the other term is larger.
+ return binom * Math.pow(y, n - k) * Math.pow(x, k);
+ }
+
+ /**
+ * Computes the binomial coefficient.
+ *
+ * <p>Adapted from
+ * {@code org.apache.commons.numbers.combinatorics.BinomialCoefficientDouble}.
+ *
+ * <p>Note: This does not use {@code BinomialCoefficientDouble}
+ * to avoid a circular dependency as the combinatorics depends on the gamma package.
+ * No checks are made on the arguments.
+ *
+ * @param n Size of the set (must be positive).
+ * @param k Size of the subsets to be counted (must be in [0, n]).
+ * @return {@code n choose k}.
+ */
+ static double binomialCoefficient(int n, int k) {
+ // Assume: n >= 0; 0 <= k <= n
+ // Use symmetry
+ final int m = Math.min(k, n - k);
+
+ // This function is typically called with m <= 3 so handle these special cases
+ if (m == 0) {
+ return 1;
+ }
+ if (m == 1) {
+ return n;
+ }
+ if (m == 2) {
+ return 0.5 * n * (n - 1);
+ }
+ if (m == 3) {
+ // Divide by 3 at the end to avoid using an 1/6 inexact initial multiplier.
+ return 0.5 * n * (n - 1) * (n - 2) / 3;
+ }
+
+ double result;
+ if (n <= MAX_FACTORIAL) {
+ // Use fast table lookup:
+ result = BoostGamma.uncheckedFactorial(n);
+ // Smaller m will have a more accurate factorial
+ result /= BoostGamma.uncheckedFactorial(m);
+ result /= BoostGamma.uncheckedFactorial(n - m);
+ } else {
+ // Updated:
+ // Do not use the beta function as per <boost/math/special_functions/binomial.hpp>,
+ // e.g. result = 1 / (m * beta(m, n - m + 1)) == gamma(n+1) / (m * gamma(m) * gamma(n-m+1))
+
+ // Use a summation as per BinomialCoefficientDouble which is more accurate
+ // than using the beta function. This is only used up to m = 39 so
+ // may overflow *only* on the final term (i.e. m << n when overflow can occur).
+
+ result = 1;
+ for (int i = 1; i < m; i++) {
+ result *= n - m + i;
+ result /= i;
+ }
+ // Final term may cause overflow.
+ if (result * n > Double.MAX_VALUE) {
+ result /= m;
+ result *= n;
+ } else {
+ result *= n;
+ result /= m;
+ }
+ }
+ // convert to nearest integer:
+ return Math.ceil(result - 0.5f);
+ }
+
+ /**
+ * Computes the difference between ibeta(a,b,x) and ibeta(a+k,b,x).
+ *
+ * @param a Argument a
+ * @param b Argument b
+ * @param x Argument x
+ * @param y Argument 1-x
+ * @param k Argument k
+ * @param normalised true to compute the regularised value
+ * @return ibeta difference
+ */
+ private static double ibetaAStep(double a, double b, double x, double y, int k, boolean normalised) {
+ double prefix = ibetaPowerTerms(a, b, x, y, normalised);
+ prefix /= a;
+ if (prefix == 0) {
+ return prefix;
+ }
+ double sum = 1;
+ double term = 1;
+ // series summation from 0 to k-1:
+ for (int i = 0; i < k - 1; ++i) {
+ term *= (a + b + i) * x / (a + i + 1);
+ sum += term;
+ }
+ prefix *= sum;
+
+ return prefix;
+ }
+
+ /**
+ * Computes beta(a, b, x) using small b and large a.
+ *
+ * @param a Argument a
+ * @param b Argument b
+ * @param x Argument x
+ * @param y Argument 1-x
+ * @param s0 Initial sum for the series
+ * @param mult Multiplication prefix factor
+ * @param pol Function evaluation policy
+ * @param normalised true to compute the regularised value
+ * @return beta series
+ */
+ private static double betaSmallBLargeASeries(double a, double b, double x, double y, double s0, double mult,
+ Policy pol, boolean normalised) {
+ //
+ // This is DiDonato and Morris's BGRAT routine, see Eq's 9 through 9.6.
+ //
+ // Some values we'll need later, these are Eq 9.1:
+ //
+ final double bm1 = b - 1;
+ final double t = a + bm1 / 2;
+ double lx;
+ double u;
+ if (y < 0.35) {
+ lx = Math.log1p(-y);
+ } else {
+ lx = Math.log(x);
+ }
+ u = -t * lx;
+ // and from from 9.2:
+ final double h = BoostGamma.regularisedGammaPrefix(b, u);
+ if (h <= Double.MIN_NORMAL) {
+ // Update:
+ // Boost returns s0.
+ // If this is zero then compute an expected sub-normal value
+ // using the classic continued fraction representation.
+ if (s0 == 0) {
+ return ibetaFraction(a, b, x, y, pol, normalised);
+ }
+ return s0;
+ }
+ double prefix;
+ if (normalised) {
+ prefix = h / GammaRatio.delta(a, b);
+ prefix /= Math.pow(t, b);
+ } else {
+ prefix = BoostGamma.fullIgammaPrefix(b, u) / Math.pow(t, b);
+ }
+ prefix *= mult;
+ //
+ // now we need the quantity Pn, unfortunately this is computed
+ // recursively, and requires a full history of all the previous values
+ // so no choice but to declare a big table and hope it's big enough...
+ //
+ final double[] p = new double[PN_SIZE];
+ // see 9.3.
+ p[0] = 1;
+ //
+ // Now an initial value for J, see 9.6:
+ //
+ double j = BoostGamma.gammaQ(b, u, pol) / h;
+ //
+ // Now we can start to pull things together and evaluate the sum in Eq 9:
+ //
+ // Value at N = 0
+ double sum = s0 + prefix * j;
+ // some variables we'll need:
+ // 2*N+1
+ int tnp1 = 1;
+ double lx2 = lx / 2;
+ lx2 *= lx2;
+ double lxp = 1;
+ final double t4 = 4 * t * t;
+ double b2n = b;
+
+ for (int n = 1; n < PN_SIZE; ++n) {
+ //
+ // begin by evaluating the next Pn from Eq 9.4:
+ //
+ tnp1 += 2;
+ p[n] = 0;
+ double mbn = b - n;
+ int tmp1 = 3;
+ for (int m = 1; m < n; ++m) {
+ mbn = m * b - n;
+ p[n] += mbn * p[n - m] / BoostGamma.uncheckedFactorial(tmp1);
+ tmp1 += 2;
+ }
+ p[n] /= n;
+ p[n] += bm1 / BoostGamma.uncheckedFactorial(tnp1);
+ //
+ // Now we want Jn from Jn-1 using Eq 9.6:
+ //
+ j = (b2n * (b2n + 1) * j + (u + b2n + 1) * lxp) / t4;
+ lxp *= lx2;
+ b2n += 2;
+ //
+ // pull it together with Eq 9:
+ //
+ final double r = prefix * p[n] * j;
+ final double previous = sum;
+ sum += r;
+ // Update:
+ // This continues until convergence at machine epsilon
+ // |r| < eps * |sum|; r < 1
+ // |r| / eps < |sum|; r > 1
+ if (sum == previous) {
+ break;
+ }
+ }
+ return sum;
+ }
+
+ /**
+ * Evaluate the incomplete beta via a continued fraction representation.
+ *
+ * <p>Note: This is not a generic continued fraction. The formula is from <a
+ * href="https://dl.acm.org/doi/10.1145/131766.131776">Didonato and Morris</a> and is
+ * only used when a and b are above 1. See <a
+ * href="https://www.boost.org/doc/libs/1_77_0/libs/math/doc/html/math_toolkit/sf_beta/ibeta_function.html">Incomplete
+ * Beta Function: Implementation</a>.
+ *
+ * @param a Argument a
+ * @param b Argument b
+ * @param x Argument x
+ * @param y Argument 1-x
+ * @param pol Function evaluation policy
+ * @param normalised true to compute the regularised value
+ * @return incomplete beta
+ */
+ static double ibetaFraction2(double a, double b, double x, double y, Policy pol, boolean normalised) {
+ final double result = ibetaPowerTerms(a, b, x, y, normalised);
+ if (result == 0) {
+ return result;
+ }
+
+ final double eps = pol.getEps();
+ final int maxIterations = pol.getMaxIterations();
+
+ final Supplier<Coefficient> gen = new Supplier<Coefficient>() {
+ /** Iteration. */
+ private int m;
+
+ @Override
+ public Coefficient get() {
+ double aN = (a + m - 1) * (a + b + m - 1) * m * (b - m) * x * x;
+ final double denom = a + 2 * m - 1;
+ aN /= denom * denom;
+
+ double bN = m;
+ bN += (m * (b - m) * x) / (a + 2 * m - 1);
+ bN += ((a + m) * (a * y - b * x + 1 + m * (2 - x))) / (a + 2 * m + 1);
+
+ ++m;
+ return Coefficient.of(aN, bN);
+ }
+ };
+
+ // Note: The first generated term a0 is discarded
+ final double fract = GeneralizedContinuedFraction.value(gen, eps, maxIterations);
+ return result / fract;
+ }
+
+ /**
+ * Evaluate the incomplete beta via the classic continued fraction representation.
+ *
+ * <p>Note: This is not part of the Boost C++ implementation.
+ * This is a generic continued fraction applicable to all arguments.
+ * It is used when alternative methods are unsuitable due to the presence of sub-normal
+ * terms and the result is expected to be sub-normal. In this case the original Boost
+ * implementation would return zero.
+ *
+ * <p>This continued fraction was the evaluation method used in Commons Numbers 1.0.
+ * This method has been updated to use the {@code ibetaPowerTerms} function to compute
+ * the power terms. Reversal of the arguments to call {@code 1 - ibetaFraction(b, a, 1 - x)}
+ * is not performed as the result is expected to be very small and this optimisation
+ * for accuracy is not required.
+ *
+ * @param a Argument a
+ * @param b Argument b
+ * @param x Argument x
+ * @param y Argument 1-x
+ * @param pol Function evaluation policy
+ * @param normalised true to compute the regularised value
+ * @return incomplete beta
+ */
+ static double ibetaFraction(double a, double b, double x, double y, Policy pol, boolean normalised) {
+ final double result = ibetaPowerTerms(a, b, x, y, normalised);
+ if (result == 0) {
+ return result;
+ }
+
+ final double eps = pol.getEps();
+ final int maxIterations = pol.getMaxIterations();
+
+ final Supplier<Coefficient> gen = new Supplier<Coefficient>() {
+ /** Iteration. */
+ private int n;
+
+ @Override
+ public Coefficient get() {
+ // https://functions.wolfram.com/GammaBetaErf/Beta3/10/0001/
+
+ final int m = n;
+ final int k = m / 2;
+ double aN;
+ if ((m & 0x1) == 0) {
+ // even
+ // m = 2k
+ aN = (k * (b - k) * x) /
+ ((a + m - 1) * (a + m));
+ } else {
+ // odd
+ // k = (m - 1) / 2 due to integer truncation
+ // m - 1 = 2k
+ aN = -((a + k) * (a + b + k) * x) /
+ ((a + m - 1) * (a + m));
+ }
+
+ n = m + 1;
+ return Coefficient.of(aN, 1);
+ }
+ };
+
+ // Note: The first generated term a0 is discarded
+ final double fract = GeneralizedContinuedFraction.value(gen, eps, maxIterations);
+ return (result / a) / fract;
+ }
+}
diff --git a/commons-numbers-gamma/src/main/java/org/apache/commons/numbers/gamma/BoostGamma.java b/commons-numbers-gamma/src/main/java/org/apache/commons/numbers/gamma/BoostGamma.java
index ea79786..07747fe 100644
--- a/commons-numbers-gamma/src/main/java/org/apache/commons/numbers/gamma/BoostGamma.java
+++ b/commons-numbers-gamma/src/main/java/org/apache/commons/numbers/gamma/BoostGamma.java
@@ -567,6 +567,19 @@ final class BoostGamma {
}
/**
+ * Returns the factorial of n.
+ * This is unchecked as an index out of bound exception will occur
+ * if the value n is not within the range [0, 170].
+ * This function is exposed for use in {@link BoostBeta}.
+ *
+ * @param n Argument n (must be in [0, 170])
+ * @return n!
+ */
+ static double uncheckedFactorial(int n) {
+ return FACTORIAL[n];
+ }
+
+ /**
* Gamma function.
*
* <p>For small {@code z} this is based on the <em>NSWC Library of Mathematics
@@ -1665,7 +1678,7 @@ final class BoostGamma {
* @param z Argument z
* @return incomplete gamma prefix
*/
- private static double fullIgammaPrefix(double a, double z) {
+ static double fullIgammaPrefix(double a, double z) {
if (z > Double.MAX_VALUE) {
return 0;
}
diff --git a/commons-numbers-gamma/src/test/java/org/apache/commons/numbers/gamma/BoostBetaTest.java b/commons-numbers-gamma/src/test/java/org/apache/commons/numbers/gamma/BoostBetaTest.java
new file mode 100644
index 0000000..d4b613a
--- /dev/null
+++ b/commons-numbers-gamma/src/test/java/org/apache/commons/numbers/gamma/BoostBetaTest.java
@@ -0,0 +1,1236 @@
+/*
+ * 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.numbers.gamma;
+
+import java.io.IOException;
+import java.math.BigDecimal;
+import java.util.Arrays;
+import java.util.function.DoubleBinaryOperator;
+import org.junit.jupiter.api.Assertions;
+import org.junit.jupiter.api.MethodOrderer;
+import org.junit.jupiter.api.Test;
+import org.junit.jupiter.api.TestMethodOrder;
+import org.junit.jupiter.params.ParameterizedTest;
+import org.junit.jupiter.params.provider.CsvSource;
+import org.junit.jupiter.params.provider.EnumSource;
+import org.junit.jupiter.params.provider.EnumSource.Mode;
+
+/**
+ * Tests for {@link BoostBeta}.
+ *
+ * <p>Note: Some resource data files used in these tests have been extracted
+ * from the Boost test files for the beta functions.
+ */
+@TestMethodOrder(MethodOrderer.OrderAnnotation.class)
+class BoostBetaTest {
+ /**
+ * Represents an operation upon three {@code double}-valued operands and producing a
+ * {@code double}-valued result.
+ */
+ interface DoubleTernaryOperator {
+ /**
+ * Applies this operator to the given operands.
+ *
+ * @param x the first operand
+ * @param y the second operand
+ * @param z the third operand
+ * @return the operator result
+ */
+ double applyAsDouble(double x, double y, double z);
+ }
+
+ /** Define the expected error for a test. */
+ private interface TestError {
+ /**
+ * @return maximum allowed error
+ */
+ double getTolerance();
+
+ /**
+ * @return maximum allowed RMS error
+ */
+ double getRmsTolerance();
+ }
+
+ /**
+ * Define the test cases for each resource file for two argument functions.
+ * This encapsulates the function to test, the expected maximum and RMS error, and
+ * the resource file containing the data.
+ *
+ * <h2>Note on accuracy</h2>
+ *
+ * <p>The Boost functions use the default policy of internal promotion
+ * of double to long double if it offers more precision. Java does not
+ * support long double computation. Tolerances have been set to allow tests to
+ * pass. Spot checks on larger errors have been verified against the reference
+ * implementation compiled with promotion of double <strong>disabled</strong>.
+ *
+ * @see <a href="https://www.boost.org/doc/libs/1_77_0/libs/math/doc/html/math_toolkit/relative_error.html">Relative error</a>
+ * @see <a href="https://www.boost.org/doc/libs/1_77_0/libs/math/doc/html/math_toolkit/pol_tutorial/policy_tut_defaults.html">Policy defaults</a>
+ */
+ private enum BiTestCase implements TestError {
+ // beta(a, b)
+ // Note that the worst errors occur when a or b are large, and that
+ // when this is the case the result is very close to zero, so absolute
+ // errors will be very small.
+ /** beta Boost small data. */
+ BETA_SMALL(BoostBeta::beta, "beta_small_data.csv", 4, 1.7),
+ /** beta Boost medium data. */
+ BETA_MED(BoostBeta::beta, "beta_med_data.csv", 200, 35),
+ /** beta Boost divergent data. */
+ BETA_EXP(BoostBeta::beta, "beta_exp_data.csv", 17, 3.6),
+ // LogBeta based implementation is worse
+ /** beta Boost small data. */
+ BETA1_SMALL(BoostBetaTest::beta, "beta_small_data.csv", 110, 28),
+ /** beta Boost medium data. */
+ BETA1_MED(BoostBetaTest::beta, "beta_med_data.csv", 280, 46),
+ /** beta Boost divergent data. */
+ BETA1_EXP(BoostBetaTest::beta, "beta_exp_data.csv", 28, 4.5),
+ /** binomial coefficient Boost small argument data. */
+ BINOMIAL_SMALL(BoostBetaTest::binomialCoefficient, "binomial_small_data.csv", -2, 0.5),
+ /** binomial coefficient Boost large argument data. */
+ BINOMIAL_LARGE(BoostBetaTest::binomialCoefficient, "binomial_large_data.csv", 5, 1.1),
+ /** binomial coefficient extra large argument data. */
+ BINOMIAL_XLARGE(BoostBetaTest::binomialCoefficient, "binomial_extra_large_data.csv", 9, 2),
+ /** binomial coefficient huge argument data. */
+ BINOMIAL_HUGE(BoostBetaTest::binomialCoefficient, "binomial_huge_data.csv", 9, 2),
+ // Using the beta function is worse
+ /** binomial coefficient Boost large argument data computed using the beta function. */
+ BINOMIAL1_LARGE(BoostBetaTest::binomialCoefficient1, "binomial_large_data.csv", 31, 9),
+ /** binomial coefficient huge argument data computed using the beta function. */
+ BINOMIAL1_HUGE(BoostBetaTest::binomialCoefficient1, "binomial_huge_data.csv", 70, 19);
+
+ /** The function. */
+ private final DoubleBinaryOperator fun;
+
+ /** The filename containing the test data. */
+ private final String filename;
+
+ /** The field containing the expected value. */
+ private final int expected;
+
+ /** The maximum allowed ulp. */
+ private final double maxUlp;
+
+ /** The maximum allowed RMS ulp. */
+ private final double rmsUlp;
+
+ /**
+ * Instantiates a new test case.
+ *
+ * @param fun function to test
+ * @param filename Filename of the test data
+ * @param maxUlp maximum allowed ulp
+ * @param rmsUlp maximum allowed RMS ulp
+ */
+ BiTestCase(DoubleBinaryOperator fun, String filename, double maxUlp, double rmsUlp) {
+ this(fun, filename, 2, maxUlp, rmsUlp);
+ }
+
+ /**
+ * Instantiates a new test case.
+ *
+ * @param fun function to test
+ * @param filename Filename of the test data
+ * @param expected Expected result field index
+ * @param maxUlp maximum allowed ulp
+ * @param rmsUlp maximum allowed RMS ulp
+ */
+ BiTestCase(DoubleBinaryOperator fun, String filename, int expected, double maxUlp, double rmsUlp) {
+ this.fun = fun;
+ this.filename = filename;
+ this.expected = expected;
+ this.maxUlp = maxUlp;
+ this.rmsUlp = rmsUlp;
+ }
+
+ /**
+ * @return function to test
+ */
+ DoubleBinaryOperator getFunction() {
+ return fun;
+ }
+
+ /**
+ * @return Filename of the test data
+ */
+ String getFilename() {
+ return filename;
+ }
+
+ /**
+ * @return Expected result field index
+ */
+ int getExpectedField() {
+ return expected;
+ }
+
+ @Override
+ public double getTolerance() {
+ return maxUlp;
+ }
+
+ @Override
+ public double getRmsTolerance() {
+ return rmsUlp;
+ }
+ }
+
+
+ /**
+ * Define the test cases for each resource file for three argument functions.
+ * This encapsulates the function to test, the expected maximum and RMS error, and
+ * the resource file containing the data.
+ */
+ private enum TriTestCase implements TestError {
+ /** ibeta derivative Boost small integer data. */
+ IBETA_DERIV_SMALL_INT(BoostBeta::ibetaDerivative, "ibeta_derivative_small_int_data.csv", 60, 13),
+ /** ibeta derivative Boost small data. */
+ IBETA_DERIV_SMALL(BoostBeta::ibetaDerivative, "ibeta_derivative_small_data.csv", 22, 4),
+ /** ibeta derivative Boost medium data. */
+ IBETA_DERIV_MED(BoostBeta::ibetaDerivative, "ibeta_derivative_med_data.csv", 150, 33),
+ /** ibeta derivative Boost large and diverse data. */
+ IBETA_DERIV_LARGE(BoostBeta::ibetaDerivative, "ibeta_derivative_large_data.csv", 3900, 260),
+ // LogGamma based implementation is worse
+ /** ibeta derivative Boost small integer data. */
+ IBETA_DERIV1_SMALL_INT(BoostBetaTest::ibetaDerivative1, "ibeta_derivative_small_int_data.csv", 220, 55),
+ /** ibeta derivative Boost small data. */
+ IBETA_DERIV1_SMALL(BoostBetaTest::ibetaDerivative1, "ibeta_derivative_small_data.csv", 75, 10.5),
+ /** ibeta derivative Boost medium data. */
+ IBETA_DERIV1_MED(BoostBetaTest::ibetaDerivative1, "ibeta_derivative_med_data.csv", 1500, 300),
+ /** ibeta derivative Boost large and diverse data. */
+ IBETA_DERIV1_LARGE(BoostBetaTest::ibetaDerivative1, "ibeta_derivative_large_data.csv", 9e7, 250000),
+ // LogBeta based implementation is worse
+ /** ibeta derivative Boost small integer data. */
+ IBETA_DERIV2_SMALL_INT(BoostBetaTest::ibetaDerivative2, "ibeta_derivative_small_int_data.csv", 180, 31),
+ /** ibeta derivative Boost small data. */
+ IBETA_DERIV2_SMALL(BoostBetaTest::ibetaDerivative2, "ibeta_derivative_small_data.csv", 75, 8.5),
+ /** ibeta derivative Boost medium data. */
+ IBETA_DERIV2_MED(BoostBetaTest::ibetaDerivative2, "ibeta_derivative_med_data.csv", 500, 85),
+ /** ibeta derivative Boost large and diverse data. */
+ IBETA_DERIV2_LARGE(BoostBetaTest::ibetaDerivative2, "ibeta_derivative_large_data.csv", 28000, 1200),
+
+ /** ibeta Boost small integer data. */
+ IBETA_SMALL_INT(BoostBeta::beta, "ibeta_small_int_data.csv", 48, 11),
+ /** ibeta Boost small data. */
+ IBETA_SMALL(BoostBeta::beta, "ibeta_small_data.csv", 17, 3.3),
+ /** ibeta Boost medium data. */
+ IBETA_MED(BoostBeta::beta, "ibeta_med_data.csv", 190, 20),
+ /** ibeta Boost large and diverse data. */
+ IBETA_LARGE(BoostBeta::beta, "ibeta_large_data.csv", 1300, 50),
+ /** ibetac Boost small integer data. */
+ IBETAC_SMALL_INT(BoostBeta::betac, "ibeta_small_int_data.csv", 4, 57, 11),
+ /** ibetac Boost small data. */
+ IBETAC_SMALL(BoostBeta::betac, "ibeta_small_data.csv", 4, 14, 3.2),
+ /** ibetac Boost medium data. */
+ IBETAC_MED(BoostBeta::betac, "ibeta_med_data.csv", 4, 130, 24),
+ /** ibetac Boost large and diverse data. */
+ IBETAC_LARGE(BoostBeta::betac, "ibeta_large_data.csv", 4, 7000, 220),
+ /** regularised ibeta Boost small integer data. */
+ RBETA_SMALL_INT(BoostBeta::ibeta, "ibeta_small_int_data.csv", 5, 7.5, 1.2),
+ /** regularised ibeta Boost small data. */
+ RBETA_SMALL(BoostBeta::ibeta, "ibeta_small_data.csv", 5, 14, 3.3),
+ /** regularised ibeta Boost medium data. */
+ RBETA_MED(BoostBeta::ibeta, "ibeta_med_data.csv", 5, 200, 28),
+ /** regularised ibeta Boost large and diverse data. */
+ RBETA_LARGE(BoostBeta::ibeta, "ibeta_large_data.csv", 5, 2400, 100),
+ /** regularised ibetac Boost small integer data. */
+ RBETAC_SMALL_INT(BoostBeta::ibetac, "ibeta_small_int_data.csv", 6, 8, 1.6),
+ /** regularised ibetac Boost small data. */
+ RBETAC_SMALL(BoostBeta::ibetac, "ibeta_small_data.csv", 6, 11, 2.75),
+ /** regularised ibetac Boost medium data. */
+ RBETAC_MED(BoostBeta::ibetac, "ibeta_med_data.csv", 6, 105, 23),
+ /** regularised ibetac Boost large and diverse data. */
+ RBETAC_LARGE(BoostBeta::ibetac, "ibeta_large_data.csv", 6, 4000, 180),
+
+ // Classic continued fraction representation is:
+ // - worse on small data
+ // - comparable (or better) on medium data
+ // - much worse on large data
+ /** regularised ibeta Boost small data using the classic continued fraction evaluation. */
+ RBETA1_SMALL(BoostBetaTest::ibeta, "ibeta_small_data.csv", 5, 45, 5),
+ /** regularised ibeta Boost small data using the classic continued fraction evaluation. */
+ RBETA1_MED(BoostBetaTest::ibeta, "ibeta_med_data.csv", 5, 200, 26),
+ /** regularised ibeta Boost large and diverse data. */
+ RBETA1_LARGE(BoostBetaTest::ibeta, "ibeta_large_data.csv", 5, 150000, 7500),
+ /** regularised ibeta Boost small data using the classic continued fraction evaluation. */
+ RBETAC1_SMALL(BoostBetaTest::ibetac, "ibeta_small_data.csv", 6, 30, 4.5),
+ /** regularised ibeta Boost small data using the classic continued fraction evaluation. */
+ RBETAC1_MED(BoostBetaTest::ibetac, "ibeta_med_data.csv", 6, 100, 22),
+ /** regularised ibetac Boost large and diverse data. */
+ RBETAC1_LARGE(BoostBetaTest::ibetac, "ibeta_large_data.csv", 6, 370000, 11000);
+
+ /** The function. */
+ private final DoubleTernaryOperator fun;
+
+ /** The filename containing the test data. */
+ private final String filename;
+
+ /** The field containing the expected value. */
+ private final int expected;
+
+ /** The maximum allowed ulp. */
+ private final double maxUlp;
+
+ /** The maximum allowed RMS ulp. */
+ private final double rmsUlp;
+
+ /**
+ * Instantiates a new test case.
+ *
+ * @param fun function to test
+ * @param filename Filename of the test data
+ * @param maxUlp maximum allowed ulp
+ * @param rmsUlp maximum allowed RMS ulp
+ */
+ TriTestCase(DoubleTernaryOperator fun, String filename, double maxUlp, double rmsUlp) {
+ this(fun, filename, 3, maxUlp, rmsUlp);
+ }
+
+ /**
+ * Instantiates a new test case.
+ *
+ * @param fun function to test
+ * @param filename Filename of the test data
+ * @param expected Expected result field index
+ * @param maxUlp maximum allowed ulp
+ * @param rmsUlp maximum allowed RMS ulp
+ */
+ TriTestCase(DoubleTernaryOperator fun, String filename, int expected, double maxUlp, double rmsUlp) {
+ this.fun = fun;
+ this.filename = filename;
+ this.expected = expected;
+ this.maxUlp = maxUlp;
+ this.rmsUlp = rmsUlp;
+ }
+
+ /**
+ * @return function to test
+ */
+ DoubleTernaryOperator getFunction() {
+ return fun;
+ }
+
+ /**
+ * @return Filename of the test data
+ */
+ String getFilename() {
+ return filename;
+ }
+
+ /**
+ * @return Expected result field index
+ */
+ int getExpectedField() {
+ return expected;
+ }
+
+ @Override
+ public double getTolerance() {
+ return maxUlp;
+ }
+
+ @Override
+ public double getRmsTolerance() {
+ return rmsUlp;
+ }
+ }
+
+ @ParameterizedTest
+ @CsvSource({
+ // Argument a > 0
+ "NaN, 1, NaN",
+ "0, 1, NaN",
+ "-1, 1, NaN",
+ // Argument b > 0
+ "1, NaN, NaN",
+ "1, 0, NaN",
+ "1, -1, NaN",
+ })
+ void testBetaEdgeCases(double a, double b, double expected) {
+ Assertions.assertEquals(expected, BoostBeta.beta(a, b), "beta");
+ }
+
+ /**
+ * beta spot tests extracted from
+ * {@code boost/libs/math/test/test_beta.hpp}.
+ */
+ @Test
+ void testBetaSpotTests() {
+ final int tolerance = 20;
+ // small = epsilon / 1024
+ final double small = Math.ulp(1.0) / 1024;
+ assertClose(BoostBeta::beta, 1, 1, 1, 0);
+ assertClose(BoostBeta::beta, 1, 4, 0.25, 0);
+ assertClose(BoostBeta::beta, 4, 1, 0.25, 0);
+ assertClose(BoostBeta::beta, small, 4, 1 / small, 0);
+ assertClose(BoostBeta::beta, 4, small, 1 / small, 0);
+ assertClose(BoostBeta::beta, 4, 20, 0.00002823263692828910220214568040654997176736, tolerance);
+ assertClose(BoostBeta::beta, 0.0125, 0.000023, 43558.24045647538375006349016083320744662, tolerance * 2);
+
+ // Additional tests
+
+ // a + b > 1e10
+ assertClose(BoostBeta::beta, 1e6, 1e6, 0, 0);
+ assertClose(BoostBeta::beta, 1e10, 100, 0, 0);
+ assertClose(BoostBeta::beta, 1e11, 1, 1e-11, 0);
+ assertClose(BoostBeta::beta, 1e11, 2, 1 / (1e11 * (1e11 + 1)), 5);
+
+ // a + b == a; b > epsilon
+ assertClose(BoostBeta::beta, 5, 0x1.0p-51, 2.2517998136852459167e15, 5);
+ // a + b == b; a > epsilon
+ assertClose(BoostBeta::beta, 0x1.0p-50, 11, 1.125899906842621071e15, 5);
+ }
+
+ @ParameterizedTest
+ @EnumSource(value = BiTestCase.class, mode = Mode.MATCH_ANY, names = {"BETA_.*", "BETA1_.*"})
+ void testBeta(BiTestCase tc) {
+ assertFunction(tc);
+ }
+
+ @ParameterizedTest
+ @CsvSource({
+ // k=0 or k=n
+ "1, 0, 1",
+ "42, 0, 1",
+ "1, 1, 1",
+ "42, 42, 1",
+ // k=1 or k=n-1
+ "6, 1, 6",
+ "42, 1, 42",
+ "6, 5, 6",
+ "42, 41, 42",
+ })
+ void testBinomialCoefficientEdgeCases(int n, int k, double expected) {
+ Assertions.assertEquals(expected, BoostBeta.binomialCoefficient(n, k));
+ }
+
+ /**
+ * Binomial coefficient spot tests extracted from
+ * {@code boost/libs/math/test/test_binomial_coeff.cpp}.
+ */
+ @Test
+ void testBinomialCoefficientSpotTests() {
+ Assertions.assertEquals(1, BoostBeta.binomialCoefficient(20, 0));
+ Assertions.assertEquals(20, BoostBeta.binomialCoefficient(20, 1));
+ Assertions.assertEquals(190, BoostBeta.binomialCoefficient(20, 2));
+ Assertions.assertEquals(1140, BoostBeta.binomialCoefficient(20, 3));
+ Assertions.assertEquals(1, BoostBeta.binomialCoefficient(20, 20));
+ Assertions.assertEquals(20, BoostBeta.binomialCoefficient(20, 19));
+ Assertions.assertEquals(190, BoostBeta.binomialCoefficient(20, 18));
+ Assertions.assertEquals(1140, BoostBeta.binomialCoefficient(20, 17));
+ Assertions.assertEquals(184756, BoostBeta.binomialCoefficient(20, 10));
+
+ // Requires tolerance 50 if using the beta function approximation
+ // in <boost/math/special_functions/binomial.hpp>.
+ // Lower if using BinomialCoefficientDouble (see below).
+ int tolerance = 2;
+
+ // 0 ulp
+ assertClose(BoostBetaTest::binomialCoefficient, 100, 5, 7.528752e7, tolerance);
+ // 1 ulp
+ assertClose(BoostBetaTest::binomialCoefficient, 100, 81, 1.323415729392122674e20, tolerance);
+
+ // 0 ulp
+ assertClose(BoostBetaTest::binomialCoefficient, 300, 3, 4.45510e6, tolerance);
+ // 0 ulp
+ assertClose(BoostBetaTest::binomialCoefficient, 300, 7, 4.04385595614e13, tolerance);
+ // -3 ulp / 1 ulp
+ assertClose(BoostBetaTest::binomialCoefficient, 300, 290, 1.39832023324170177e18, tolerance);
+ // 36 ulp / 2 ulp
+ assertClose(BoostBetaTest::binomialCoefficient, 300, 275, 1.953265141442868389822364184842211512e36, tolerance);
+
+ // Additional tests
+
+ // Overflow if computed using
+ // result = 1
+ // for i in [1, m]:
+ // result *= n - m + i
+ // result /= i
+ // The final term must use divide by i first.
+ tolerance = 5;
+ assertClose(BoostBetaTest::binomialCoefficient, 1786388282, 38, 7.187239013254065384599502085053593e306, tolerance);
+ assertClose(BoostBetaTest::binomialCoefficient, 1914878305, 38, 100.6570419073661447979173868523364e306, tolerance);
+ assertClose(BoostBetaTest::binomialCoefficient, 1179067476, 39, 30.22890249420109200962786203300876e306, tolerance);
+ // n == 2^31 - 1
+ assertClose(BoostBetaTest::binomialCoefficient, 2147483647, 37, 1.388890512412231479281222156415993e302, tolerance);
+ }
+
+ @ParameterizedTest
+ @EnumSource(value = BiTestCase.class, mode = Mode.MATCH_ANY, names = {"BINOM.*"})
+ void testBinomialCoefficient(BiTestCase tc) {
+ assertFunction(tc);
+ }
+
+ /**
+ * Demonstrate the binomial coefficient is better without the beta function at large k.
+ */
+ @ParameterizedTest
+ @CsvSource({
+ "500, 225, 9.5909622148251792594e147",
+ "500, 250, 1.1674431578827768292e149",
+ "600, 300, 1.3510794199619426851e179",
+ "700, 350, 1.5857433585316795488e209",
+ "800, 400, 1.8804244186835312701e239",
+ "1000, 500, 2.7028824094543656952e299",
+ "100000, 50, 3.2479111644852887358e185",
+ "100000, 70, 8.1490000781382598363e249",
+ "100000, 80, 1.3537701492763430639e281",
+ "100000, 85, 3.4252195975122556458e296",
+ "1030, 515, 2.8596413729978081638e308",
+ })
+ void testBinomialCoefficientLargeK(int n, int k, double nCk) {
+ final double bc1 = BoostBeta.binomialCoefficient(n, k);
+ final double bc2 = BoostBetaTest.binomialCoefficient1(n, k);
+ assertCloser("nCk", nCk, bc1, bc2);
+ }
+
+ @ParameterizedTest
+ @CsvSource({
+ // No protection against overflow when k ~ n/2, only for k << n
+ "1040, 450, 2.3101613255412135615e307",
+ "1029, 514, 1.4298206864989040819e308",
+ })
+ void testBinomialCoefficientOverflowAtLargeK(int n, int k, double nCk) {
+ // Note: This could be made to work correctly but is not required for the
+ // current use cases.
+ Assertions.assertEquals(Double.POSITIVE_INFINITY, BoostBeta.binomialCoefficient(n, k));
+
+ // Using the beta function is not very accurate but it does work
+ Assertions.assertEquals(nCk, BoostBetaTest.binomialCoefficient1(n, k), nCk * 1e-12);
+ }
+
+ @ParameterizedTest
+ @CsvSource({
+ // Argument a > 0
+ "NaN, 1, 0.5, NaN",
+ "0, 1, 0.5, NaN",
+ "-1, 1, 0.5, NaN",
+ // Argument b > 0
+ "1, NaN, 0.5, NaN",
+ "1, 0, 0.5, NaN",
+ "1, -1, 0.5, NaN",
+ // Argument x in [0, 1]
+ "1, 1, NaN, NaN",
+ "1, 1, -1, NaN",
+ "1, 1, 2, NaN",
+ })
+ void testIBetaEdgeCases(double a, double b, double x, double expected) {
+ // Note: Despite being named 'beta' this function is the full incomplete beta
+ Assertions.assertEquals(expected, BoostBeta.beta(a, b, x), "beta");
+ Assertions.assertEquals(expected, BoostBeta.betac(a, b, x), "betac");
+ final Policy pol = Policy.getDefault();
+ Assertions.assertEquals(expected, BoostBeta.beta(a, b, x, pol), "beta");
+ Assertions.assertEquals(expected, BoostBeta.betac(a, b, x, pol), "betac");
+
+ // The ibetaDerivative has the same invalid domain for a, b and x
+ // as beta. This is despite the ibeta being defined when a or b is 0.
+ // In that case the value x is ignored in ibeta (it is a constant
+ // function) and there is no derivative.
+ Assertions.assertEquals(expected, BoostBeta.ibetaDerivative(a, b, x), "ibetaDerivative");
+ }
+
+ @ParameterizedTest
+ @CsvSource({
+ // Argument a >= 0
+ "NaN, 1, 0.5, NaN",
+ "-1, 1, 0.5, NaN",
+ // Argument b >= 0
+ "1, NaN, 0.5, NaN",
+ "1, -1, 0.5, NaN",
+ // a and b cannot both be 0
+ "0, 0, 0.5, NaN",
+ // a or b are allowed to be zero, x is ignored
+ "0, 1, 0.25, 1",
+ "0, 1, 0.5, 1",
+ "0, 2, 0.5, 1",
+ "1, 0, 0.25, 0",
+ "1, 0, 0.5, 0",
+ "2, 0, 0.5, 0",
+ // Argument x in [0, 1]
+ "1, 1, NaN, NaN",
+ "1, 1, -1, NaN",
+ "1, 1, 2, NaN",
+ })
+ void testRegularisedIBetaEdgeCases(double a, double b, double x, double expected) {
+ Assertions.assertEquals(expected, BoostBeta.ibeta(a, b, x), "ibeta");
+ Assertions.assertEquals(1 - expected, BoostBeta.ibetac(a, b, x), "ibetac");
+ final Policy pol = Policy.getDefault();
+ Assertions.assertEquals(expected, BoostBeta.ibeta(a, b, x, pol), "ibeta");
+ Assertions.assertEquals(1 - expected, BoostBeta.ibetac(a, b, x, pol), "ibetac");
+ }
+
+ /**
+ * ibeta derivative spot tests extracted from
+ * {@code boost/libs/math/test/test_ibeta_derivative.hpp}.
+ */
+ @Test
+ void testIBetaDerivativeSpotTests() {
+ final int tolerance = 400;
+ assertClose(BoostBeta::ibetaDerivative, 2, 4, Math.scalb(1.0, -557), 4.23957586190238472641508753637420672781472122471791800210e-167, tolerance * 4);
+ assertClose(BoostBeta::ibetaDerivative, 2, 4.5, Math.scalb(1.0, -557), 5.24647512910420109893867082626308082567071751558842352760e-167, tolerance * 4);
+
+ // Additions
+ for (final double p : new double[] {3, 13, 42}) {
+ // x==0
+ assertClose(BoostBeta::ibetaDerivative, 2, p, 0, 0.0, 0);
+ assertClose(BoostBeta::ibetaDerivative, 1, p, 0, p, 0);
+ assertClose(BoostBeta::ibetaDerivative, 0.5, p, 0, Double.POSITIVE_INFINITY, 0);
+ // x==1
+ assertClose(BoostBeta::ibetaDerivative, p, 2, 1, 0.0, 0);
+ assertClose(BoostBeta::ibetaDerivative, p, 1, 1, p, 0);
+ assertClose(BoostBeta::ibetaDerivative, p, 0.5, 1, Double.POSITIVE_INFINITY, 0);
+ }
+ }
+
+ @ParameterizedTest
+ @EnumSource(value = TriTestCase.class, mode = Mode.MATCH_ANY, names = {"IBETA_DERIV.*"})
+ void testIBetaDerivative(TriTestCase tc) {
+ assertFunction(tc);
+ }
+
+ /**
+ * ibeta spot tests extracted from
+ * {@code boost/libs/math/test/test_ibeta.hpp}.
+ */
+ @Test
+ void testIBetaSpotTests() {
+ int tolerance = 30;
+
+ // Spot values are from http://functions.wolfram.com/webMathematica/FunctionEvaluation.jsp?name=BetaRegularized
+ // using precision of 50 decimal digits.
+ assertClose(BoostBeta::ibeta, 159.0 / 10000, 1184.0 / 1000000000L, 6917.0 / 10000,
+ 0.000075393541456247525676062058821484095548666733251733, tolerance);
+ assertClose(BoostBeta::ibeta, 4243.0 / 100, 3001.0 / 10000, 9157.0 / 10000,
+ 0.0028387319012616013434124297160711532419664289474798, tolerance);
+ assertClose(BoostBeta::ibeta, 9713.0 / 1000, 9940.0 / 100, 8391.0 / 100000,
+ 0.46116895440368248909937863372410093344466819447476, tolerance * 2);
+ // extra tolerance needed on linux X86EM64
+ assertClose(BoostBeta::ibeta, 72.5, 1.125, 0.75, 1.3423066982487051710597194786268004978931316494920e-9,
+ tolerance * 3);
+ assertClose(BoostBeta::ibeta, 4985.0 / 1000, 1066.0 / 1000, 7599.0 / 10000,
+ 0.27533431334486812211032939156910472371928659321347, tolerance);
+ assertClose(BoostBeta::ibeta, 6813.0 / 1000, 1056.0 / 1000, 1741.0 / 10000,
+ 7.6736128722762245852815040810349072461658078840945e-6, tolerance);
+ assertClose(BoostBeta::ibeta, 4898.0 / 10000, 2251.0 / 10000, 2003.0 / 10000,
+ 0.17089223868046209692215231702890838878342349377008, tolerance);
+ assertClose(BoostBeta::ibeta, 4049.0 / 1000, 1540.0 / 10000, 6537.0 / 10000,
+ 0.017273988301528087878279199511703371301647583919670, tolerance);
+ assertClose(BoostBeta::ibeta, 7269.0 / 1000, 1190.0 / 10000, 8003.0 / 10000,
+ 0.013334694467796052900138431733772122625376753696347, tolerance);
+ assertClose(BoostBeta::ibeta, 2726.0 / 1000, 1151.0 / 100000, 8665.0 / 100000,
+ 5.8218877068298586420691288375690562915515260230173e-6, tolerance);
+ assertClose(BoostBeta::ibeta, 3431.0 / 10000, 4634.0 / 100000, 7582.0 / 10000,
+ 0.15132819929418661038699397753916091907278005695387, tolerance);
+
+ assertClose(BoostBeta::ibeta, 0.34317314624786377, 0.046342257410287857, 0, 0, tolerance);
+ assertClose(BoostBeta::ibetac, 0.34317314624786377, 0.046342257410287857, 0, 1, tolerance);
+ assertClose(BoostBeta::ibeta, 0.34317314624786377, 0.046342257410287857, 1, 1, tolerance);
+ assertClose(BoostBeta::ibetac, 0.34317314624786377, 0.046342257410287857, 1, 0, tolerance);
+ assertClose(BoostBeta::ibeta, 1, 4634.0 / 100000,
+ 32.0 / 100, 0.017712849440718489999419956301675684844663359595318, tolerance);
+ assertClose(BoostBeta::ibeta, 4634.0 / 100000,
+ 1, 32.0 / 100, 0.94856839398626914764591440181367780660208493234722, tolerance);
+
+ // try with some integer arguments:
+ assertClose(BoostBeta::ibeta, 3, 8, 0.25, 0.474407196044921875, tolerance);
+ assertClose(BoostBeta::ibeta, 6, 8, 0.25, 0.08021259307861328125, tolerance);
+ assertClose(BoostBeta::ibeta, 12, 1, 0.25, 5.9604644775390625e-8, tolerance);
+ assertClose(BoostBeta::ibeta, 1, 8, 0.25, 0.8998870849609375, tolerance);
+
+ // very naive check on derivative:
+ tolerance = 100;
+ assertClose(BoostBeta::ibetaDerivative, 2, 3, 0.5,
+ Math.pow(0.5, 2) * Math.pow(0.5, 1) / BoostBeta.beta(2, 3), tolerance);
+
+ //
+ // Special cases and error handling:
+ //
+ Assertions.assertEquals(1, BoostBeta.ibeta(0, 2, 0.5));
+ Assertions.assertEquals(0, BoostBeta.ibeta(3, 0, 0.5));
+ Assertions.assertEquals(0, BoostBeta.ibetac(0, 2, 0.5));
+ Assertions.assertEquals(1, BoostBeta.ibetac(4, 0, 0.5));
+
+ // Domain errors
+ Assertions.assertEquals(Double.NaN, BoostBeta.beta(0, 2, 0.5));
+ Assertions.assertEquals(Double.NaN, BoostBeta.beta(3, 0, 0.5));
+ Assertions.assertEquals(Double.NaN, BoostBeta.betac(0, 2, 0.5));
+ Assertions.assertEquals(Double.NaN, BoostBeta.betac(4, 0, 0.5));
+
+ Assertions.assertEquals(Double.NaN, BoostBeta.ibetac(0, 0, 0.5));
+ Assertions.assertEquals(Double.NaN, BoostBeta.ibetac(-1, 2, 0.5));
+ Assertions.assertEquals(Double.NaN, BoostBeta.ibetac(2, -2, 0.5));
+ Assertions.assertEquals(Double.NaN, BoostBeta.ibetac(2, 2, -0.5));
+ Assertions.assertEquals(Double.NaN, BoostBeta.ibetac(2, 2, 1.5));
+
+ //
+ // a = b = 0.5 is a special case:
+ //
+ assertClose(BoostBeta::ibeta, 0.5, 0.5, 0.25, 1.0 / 3, tolerance);
+ assertClose(BoostBeta::ibetac, 0.5, 0.5, 0.25, 2.0 / 3, tolerance);
+ assertClose(BoostBeta::ibeta, 0.5, 0.5, 0.125, 0.230053456162615885213780567705142893009911395270714102055874, tolerance);
+ assertClose(BoostBeta::ibetac, 0.5, 0.5, 0.125, 0.769946543837384114786219432294857106990088604729285897944125, tolerance);
+ assertClose(BoostBeta::ibeta, 0.5, 0.5, 0.825, 0.725231121519469565327291851560156562956885802608457839260161, tolerance);
+ assertClose(BoostBeta::ibetac, 0.5, 0.5, 0.825, 0.274768878480530434672708148439843437043114197391542160739838, tolerance);
+ // beta(0.5, 0.5) = pi. Multiply by pi for full incomplete beta
+ assertClose(BoostBeta::beta, 0.5, 0.5, 0.25, Math.PI / 3, tolerance);
+ assertClose(BoostBeta::betac, 0.5, 0.5, 0.25, Math.PI * 2 / 3, tolerance);
+ //
+ // Second argument is 1 is a special case, see http://functions.wolfram.com/GammaBetaErf/BetaRegularized/03/01/01/
+ //
+ assertClose(BoostBeta::ibeta, 0.5, 1, 0.825, 0.908295106229247499626759842915458109758420750043003849691665, tolerance);
+ assertClose(BoostBeta::ibetac, 0.5, 1, 0.825, 0.091704893770752500373240157084541890241579249956996150308334, tolerance);
+ assertClose(BoostBeta::ibeta, 30, 1, 0.825, 0.003116150729395132012981654047222541793435357905008020740211, tolerance);
+ assertClose(BoostBeta::ibetac, 30, 1, 0.825, 0.996883849270604867987018345952777458206564642094991979259788, tolerance);
+
+ //
+ // Bug cases from Rocco Romeo:
+ //
+ assertClose(BoostBeta::beta, 2, 24, Math.scalb(1.0, -52),
+ 2.46519032881565349871772482100516780410072110983579277754743e-32, tolerance);
+ assertClose(BoostBeta::ibeta, 2, 24, Math.scalb(1.0, -52),
+ 1.47911419728939209923063489260310068246043266590147566652846e-29, tolerance);
+ assertClose(BoostBeta::beta, 3, 2, Math.scalb(1.0, -270),
+ 4.88182556606650701438035298707052523938789614661168065734809e-245, tolerance);
+ assertClose(BoostBeta::beta, 2, 31, Math.scalb(1.0, -373),
+ 1.35080680244581673116149460571129957689952846520037541640260e-225, tolerance);
+ assertClose(BoostBeta::ibeta, 3, 2, Math.scalb(1.0, -270),
+ 5.85819067927980841725642358448463028726547537593401678881771e-244, tolerance);
+ assertClose(BoostBeta::ibeta, 2, 31, Math.scalb(1.0, -373),
+ 1.34000034802625019731220264886560918028433223747877241307138e-222, tolerance);
+ //
+ // Bug cases from Rocco Romeo:
+ //
+ assertClose(BoostBeta::beta, 2, 4, Math.scalb(1 + 1.0 / 1024, -351),
+ 2.381008060978474962211278613067275529112106932635520021e-212, tolerance);
+ assertClose(BoostBeta::beta, 2, 4, Math.scalb(1 + 1.0 / 2048, -351),
+ 2.378685692854274898232669682422430136513931911501225435e-212, tolerance);
+ assertClose(BoostBeta::ibeta, 3, 5, Math.scalb(1 + 15.0 / 16, -268),
+ 2.386034198603463687323052353589201848077110231388968865e-240, tolerance);
+ assertClose(BoostBeta::ibetaDerivative, 2, 4, Math.scalb(1.0, -557),
+ 4.23957586190238472641508753637420672781472122471791800210e-167, tolerance * 4);
+ assertClose(BoostBeta::ibetaDerivative, 2, 4.5, Math.scalb(1.0, -557),
+ 5.24647512910420109893867082626308082567071751558842352760e-167, tolerance * 20);
+
+ // Additional tests
+
+ // Underflow of binomialTerm(n, n, x, y) but binomialTerm(n, n - 1, x, y) is non-zero.
+ // Result computed using Maxima: beta_incomplete_regularized with 64 digits precision.
+ // Note this is moderately accurate until the missing first term is a significant.
+ tolerance = 5;
+ assertClose(BoostBeta::ibeta, 20, 2, Math.scalb(1.0, -52),
+ 1.78247646441082836775138741451452643650923455144048771440215391e-312, tolerance);
+ assertClose(BoostBeta::ibeta, 33, 2, Math.scalb(1.0, -32),
+ 4.403555717560735620179800985566548469434721409459606451475482401e-317, tolerance);
+ assertClose(BoostBeta::ibeta, 759, 2, 0.375,
+ 2.327049191912271223071639014409322663839431670313469640444508565e-321, tolerance);
+
+ // With integer a and/or b just above 1
+ // ibeta_small_int_data uses 5 as the smallest parameter above 1
+ tolerance = 1;
+ assertClose(BoostBeta::ibeta, 1, 2, 0.125, 0.234375, tolerance);
+ assertClose(BoostBeta::ibeta, 1, 2, 0.5, 0.75, tolerance);
+ assertClose(BoostBeta::ibeta, 1, 2, 0.75, 0.9375, tolerance);
+ assertClose(BoostBeta::ibeta, 2, 1, 0.125, 0.015625, tolerance);
+ assertClose(BoostBeta::ibeta, 2, 1, 0.5, 0.25, tolerance);
+ assertClose(BoostBeta::ibeta, 2, 1, 0.75, 0.5625, tolerance);
+ assertClose(BoostBeta::ibeta, 2, 2, 0.125, 0.04296875, tolerance);
+ assertClose(BoostBeta::ibeta, 2, 2, 0.5, 0.5, tolerance);
+ assertClose(BoostBeta::ibeta, 2, 2, 0.75, 0.84375, tolerance);
+ assertClose(BoostBeta::ibeta, 2, 3, 0.125, 0.078857421875, tolerance);
+ assertClose(BoostBeta::ibeta, 2, 3, 0.5, 0.6875, tolerance);
+ assertClose(BoostBeta::ibeta, 2, 3, 0.75, 0.94921875, tolerance);
+
+ // Extreme integer arguments. Creates an overflow in the binomial coefficient
+ // binomial(2147483545 + 39, 38) ~ 7.84899e309
+ Assertions.assertEquals(1, BoostBeta.ibeta(39, 2147483546, Math.nextDown(1.0)));
+ Assertions.assertEquals(0, BoostBeta.ibetac(39, 2147483546, Math.nextDown(1.0)));
+ // Executes a different code path not using the binomial
+ Assertions.assertEquals(1, BoostBeta.ibeta(39, Integer.MAX_VALUE, Math.nextDown(1.0)));
+ Assertions.assertEquals(0, BoostBeta.ibetac(39, Integer.MAX_VALUE, Math.nextDown(1.0)));
+
+ // x==0 or 1
+ Assertions.assertEquals(0, BoostBeta.ibeta(2, 4, 0));
+ Assertions.assertEquals(1, BoostBeta.ibeta(2, 4, 1));
+ Assertions.assertEquals(1, BoostBeta.ibetac(2, 4, 0));
+ Assertions.assertEquals(0, BoostBeta.ibetac(2, 4, 1));
+ Assertions.assertEquals(0, BoostBeta.beta(2, 4, 0));
+ TestUtils.assertEquals(0.05, BoostBeta.beta(2, 4, 1), 1);
+ TestUtils.assertEquals(0.05, BoostBeta.betac(2, 4, 0), 1);
+ Assertions.assertEquals(0, BoostBeta.betac(2, 4, 1));
+
+ tolerance = 30;
+
+ // a > 15, integer b < 40, b * x >= 0.7
+ assertClose(BoostBeta::ibeta, 72.5, 2, 0.75, 1.673181444858556263e-8, tolerance * 2);
+
+ // a < 15, integer b < 40, b * x >= 0.7, regularized
+ assertClose(BoostBeta::ibeta, 14.5, 2, 0.75, 7.1367429756558048437e-2, tolerance);
+ }
+
+ @ParameterizedTest
+ @EnumSource(value = TriTestCase.class, mode = Mode.MATCH_ANY,
+ names = {"IBETA_[SML].*", "IBETAC_[SML].*", "RBETA.*"})
+ void testIBeta(TriTestCase tc) {
+ assertFunction(tc);
+ }
+
+ /**
+ * Test the incomplete beta function uses the policy containing the epsilon and
+ * maximum iterations for series evaluations. The data targets methods computed
+ * using a series component to check the policy is not ignored.
+ *
+ * <p>This does not target
+ * {@link BoostBeta#ibetaFraction(double, double, double, double, Policy, boolean)}.
+ * This is used when the result is sub-normal and the continued
+ * fraction converges without iteration. The policy has no effect.
+ *
+ * <p>Running the policy tests on their own should hit the code paths
+ * using the policy for function evaluation:
+ * <pre>
+ * mvn clean test -Dtest=BoostBetaTest#testIBetaPolicy* jacoco:report
+ * </pre>
+ */
+ @ParameterizedTest
+ @CsvSource(value = {
+ // Data extracted from the resource files and formatted to double precision
+
+ // Target ibetaSeries
+ "4.201121919322759E-5,2.1881177963223308E-4,0.6323960423469543,23803.707603529016,4569.595256369948,0.838947362634037,0.16105263736596298",
+ "0.22512593865394592,0.4898320138454437,0.7996731996536255,4.764013990849158,0.9820281524501243,0.8290948573018541,0.17090514269814597",
+ "4.623167842510156E-5,4.340034502092749E-5,0.135563462972641,21628.337679706918,23043.143905699715,0.48416432390665337,0.5158356760933467",
+ "2.9415024982881732E-5,4.1924233664758503E-4,0.3082362115383148,33995.42298781772,2386.0630988783787,0.9344154580933705,0.0655845419066295",
+ "1.184685606858693E-5,0.015964560210704803,0.3082362115383148,84409.7658651171,63.42758275377908,0.9992491395179357,7.508604820642986E-4",
+ "3.529437162796967E-5,2.2326681573758833E-5,0.6323960423469543,28333.671885777032,44788.91695876111,0.3874817937042091,0.6125182062957909",
+ "0.06715317070484161,2.306319236755371,0.9084427952766418,13.787272071732604,0.001859217475218142,0.999865167903893,1.3483209610697333E-4",
+ "0.3183284401893616,3.165504217147827,0.07764927297830582,1.3374998709679642,0.6794195418585712,0.6631399660602084,0.3368600339397915",
+ "0.15403440594673157,4.049813747406006,0.34629878401756287,4.872033861103044,0.08561968850485947,0.9827297959310547,0.01727020406894529",
+ "1.3317101001739502,0.7650398015975952,0.6445860862731934,0.47144799136487586,0.5594135526519237,0.4573339592510717,0.5426660407489283",
+ "0.11902070045471191,7.269547462463379,0.19963125884532928,6.225047194692518,0.08420413075357451,0.9866538632858122,0.013346136714187858",
+ "2.664715051651001,0.6914005279541016,0.8443243503570557,0.3338388990912521,0.3587830340198169,0.4819929648946269,0.518007035105373",
+ "1.0562920570373535,6.812713623046875,0.8258343935012817,0.12732498812245932,9.807107058749213E-7,0.9999922976379849,7.702362015088557E-6",
+ "1.7118667364120483,3.0191311836242676,0.07594671100378036,0.0064151684204504875,0.10850933283432233,0.05582072012850319,0.9441792798714969",
+ // Target betaSmallBLargeASeries
+ "0.04634225741028786,0.34317314624786377,0.24176712334156036,20.363670894714268,3.6307737402387885,0.8486827348798152,0.15131726512018484",
+ "2.113992923113983E-5,1.7535277947899885E-5,0.8350250720977783,47305.46963235176,57026.27394012006,0.4534139659948463,0.5465860340051537",
+ "4.005068331025541E-5,42.84983825683594,0.12707412242889404,24964.03974453176,4.764518849491958E-4,0.9999999809144722,1.9085527852527327E-8",
+ "67.90167999267578,0.8324270844459534,0.9676981568336487,0.002669338211636283,0.031098353139101195,0.0790500654578503,0.9209499345421497",
+ "0.395370751619339,0.004023698624223471,0.9058013558387756,4.307485901473997,246.2348442100959,0.017192647244702382,0.9828073527552976",
+ "3.444607973098755,66.36054992675781,0.09654488414525986,1.4741027361579568E-6,8.307589573110104E-8,0.9466497330301025,0.05335026696989754",
+ "1.0665277242660522,4.985442161560059,0.2400285303592682,0.12523918055824373,0.0476465037156954,0.7244045745268357,0.2755954254731643",
+ // Target ibetaFraction2
+ "1.319732904434204,4.903014659881592,0.33251503109931946,0.0837704419861451,0.021604794441302123,0.7949727547593459,0.20502724524065405",
+ "485.7690734863281,190.16734313964844,0.6323960423469543,7.8023253024461E-182,7.885435919806278E-176,9.894592590329194E-7,0.9999990105407409",
+ })
+ void testIBetaPolicy(double a, double b, double x, double beta, double betac, double ibeta, double ibetac) {
+ // Low iterations should fail to converge
+ final Policy pol1 = new Policy(0x1.0p-52, 1);
+ Assertions.assertThrows(ArithmeticException.class, () -> BoostBeta.beta(a, b, x, pol1), "beta");
+ Assertions.assertThrows(ArithmeticException.class, () -> BoostBeta.betac(a, b, x, pol1), "betac");
+ Assertions.assertThrows(ArithmeticException.class, () -> BoostBeta.ibeta(a, b, x, pol1), "ibeta");
+ Assertions.assertThrows(ArithmeticException.class, () -> BoostBeta.ibetac(a, b, x, pol1), "ibetac");
+
+ // Low epsilon should not be as accurate
+ final Policy pol2 = new Policy(1e-3, Integer.MAX_VALUE);
+
+ // Ignore infinite
+ if (Double.isFinite(beta)) {
+ final double u1 = BoostBeta.beta(a, b, x);
+ final double u2 = BoostBeta.beta(a, b, x, pol2);
+ assertCloser("beta", beta, u1, u2);
+ }
+ if (Double.isFinite(betac)) {
+ final double l1 = BoostBeta.betac(a, b, x);
+ final double l2 = BoostBeta.betac(a, b, x, pol2);
+ assertCloser("betac", betac, l1, l2);
+ }
+
+ // Ignore 0 or 1
+ if ((int) ibeta != ibeta) {
+ final double p1 = BoostBeta.ibeta(a, b, x);
+ final double p2 = BoostBeta.ibeta(a, b, x, pol2);
+ assertCloser("ibeta", ibeta, p1, p2);
+ }
+ if ((int) ibetac != ibetac) {
+ final double q1 = BoostBeta.ibetac(a, b, x);
+ final double q2 = BoostBeta.ibetac(a, b, x, pol2);
+ assertCloser("ibetac", ibetac, q1, q2);
+ }
+ }
+
+ /**
+ * Assert x is closer to the expected result than y.
+ */
+ private static void assertCloser(String msg, double expected, double x, double y) {
+ if (!Double.isFinite(expected)) {
+ // Test both answers are correct
+ Assertions.assertEquals(expected, x);
+ Assertions.assertEquals(expected, y);
+ } else {
+ final double dx = Math.abs(expected - x);
+ final double dy = Math.abs(expected - y);
+ Assertions.assertTrue(dx < dy,
+ () -> String.format("%s %s : %s (%s) : %s (%s)", msg, expected,
+ x, dx / Math.ulp(expected), y, dy / Math.ulp(expected)));
+ }
+ }
+
+ /**
+ * Computes the binomial coefficient.
+ *
+ * <p>Wrapper to convert double arguments to integer and call the
+ * {@link BoostBeta#binomialCoefficient(int, int)} method.
+ *
+ * @param n Size of the set.
+ * @param k Size of the subsets to be counted.
+ * @return {@code n choose k}.
+ */
+ private static double binomialCoefficient(double n, double k) {
+ return BoostBeta.binomialCoefficient((int) n, (int) k);
+ }
+
+ /**
+ * Computes the binomial coefficient using the beta function.
+ *
+ * <p>Adapted from {@code <boost/math/special_functions/binomial.hpp>}.
+ *
+ * @param n1 Size of the set.
+ * @param k1 Size of the subsets to be counted.
+ * @return {@code n choose k}.
+ */
+ private static double binomialCoefficient1(double n1, double k1) {
+ // Use symmetry
+ int n = (int) n1;
+ int k = (int) k1;
+ final int m = Math.min(k, n - k);
+ if (m == 0) {
+ return 1;
+ }
+ if (m == 1) {
+ return n;
+ }
+ if (m == 2) {
+ // Cannot overflow a long
+ return (n - 1L) * n / 2;
+ }
+
+ double result;
+ if (n <= 170) {
+ // Use fast table lookup:
+ // Note: This has lower error on test data than calling BinomialCoefficientDouble.
+ result = BoostGamma.uncheckedFactorial(n);
+ // Smaller m will have a more accurate factorial
+ result /= BoostGamma.uncheckedFactorial(m);
+ result /= BoostGamma.uncheckedFactorial(n - m);
+ } else {
+ if (k < n - k) {
+ result = k * BoostBeta.beta(k, n - k + 1);
+ } else {
+ result = (n - k) * BoostBeta.beta(k + 1, n - k);
+ }
+ result = 1 / result;
+ }
+ return Math.ceil(result - 0.5f);
+ }
+
+ /**
+ * Beta function.
+ * <p>\[ B(p, q) = \frac{\Gamma(p) \Gamma(q)}}{\Gamma(p+q)} \]
+ *
+ * <p>Computed using {@link LogBeta}.
+ *
+ * @param p Argument p
+ * @param q Argument q
+ * @return beta value
+ */
+ private static double beta(double p, double q) {
+ return Math.exp(LogBeta.value(p, q));
+ }
+
+ /**
+ * Derivative of the regularised incomplete beta.
+ * <p>\[ \frac{\delta}{\delta x} I_x(a, b) = \frac{(1-x)^{b-1} x^{a-1}}{\B(a, b)} \]
+ *
+ * <p>Computed using {@link LogGamma}.
+ *
+ * @param a Argument a
+ * @param b Argument b
+ * @param x Argument x
+ * @return ibeta derivative
+ */
+ private static double ibetaDerivative1(double a, double b, double x) {
+ if (x < 0 || x > 1) {
+ return 0;
+ } else if (x == 0) {
+ if (a < 1) {
+ // Distribution is not valid when x=0, alpha<1
+ // due to a divide by zero error.
+ // Do not raise an exception and return the limit.
+ return Double.POSITIVE_INFINITY;
+ }
+ // Special case of cancellation: x^(a-1) (1-x)^(b-1) / B(a, b)
+ if (a == 1) {
+ return b;
+ }
+ return 0;
+ } else if (x == 1) {
+ if (b < 1) {
+ // Distribution is not valid when x=1, beta<1
+ // due to a divide by zero error.
+ // Do not raise an exception and return the limit.
+ return Double.POSITIVE_INFINITY;
+ }
+ // Special case of cancellation: x^(a-1) (1-x)^(b-1) / B(a, b)
+ if (b == 1) {
+ return a;
+ }
+ return 0;
+ } else {
+ final double z = LogGamma.value(a) + LogGamma.value(b) - LogGamma.value(a + b);
+ final double logX = Math.log(x);
+ final double log1mX = Math.log1p(-x);
+ return Math.exp((a - 1) * logX + (b - 1) * log1mX - z);
+ }
+ }
+
+ /**
+ * Derivative of the regularised incomplete beta.
+ * <p>\[ \frac{\delta}{\delta x} I_x(a, b) = \frac{(1-x)^{b-1} x^{a-1}}{\B(a, b)} \]
+ *
+ * <p>Computed using {@link LogBeta}.
+ *
+ * @param a Argument a
+ * @param b Argument b
+ * @param x Argument x
+ * @return ibeta derivative
+ */
+ private static double ibetaDerivative2(double a, double b, double x) {
+ if (x < 0 || x > 1) {
+ return 0;
+ } else if (x == 0) {
+ if (a < 1) {
+ // Distribution is not valid when x=0, alpha<1
+ // due to a divide by zero error.
+ // Do not raise an exception and return the limit.
+ return Double.POSITIVE_INFINITY;
+ }
+ // Special case of cancellation: x^(a-1) (1-x)^(b-1) / B(a, b)
+ if (a == 1) {
+ return b;
+ }
+ return 0;
+ } else if (x == 1) {
+ if (b < 1) {
+ // Distribution is not valid when x=1, beta<1
+ // due to a divide by zero error.
+ // Do not raise an exception and return the limit.
+ return Double.POSITIVE_INFINITY;
+ }
+ // Special case of cancellation: x^(a-1) (1-x)^(b-1) / B(a, b)
+ if (b == 1) {
+ return a;
+ }
+ return 0;
+ } else {
+ final double z = LogBeta.value(a, b);
+ final double logX = Math.log(x);
+ final double log1mX = Math.log1p(-x);
+ return Math.exp((a - 1) * logX + (b - 1) * log1mX - z);
+ }
+ }
+
+ /**
+ * Regularised incomplete beta.
+ * Helper method to call the classic continued fraction representation.
+ *
+ * @param a Argument a
+ * @param b Argument b
+ * @param x Argument x
+ * @return p
+ */
+ private static double ibeta(double a, double b, double x) {
+ return ibetaImp(a, b, x, false);
+ }
+
+ /**
+ * Complement of the regularised incomplete beta.
+ * Helper method to call the classic continued fraction representation.
+ *
+ * @param a Argument a
+ * @param b Argument b
+ * @param x Argument x
+ * @return q
+ */
+ private static double ibetac(double a, double b, double x) {
+ return ibetaImp(a, b, x, true);
+ }
+
+ /**
+ * Regularised incomplete beta implementation using
+ * the classic continued fraction representation.
+ *
+ * <p>This is a partial implementation with no special case handling.
+ *
+ * @param a Argument a
+ * @param b Argument b
+ * @param x Argument x
+ * @param inv true to compute the complement value
+ * @return p
+ */
+ private static double ibetaImp(double a, double b, double x, boolean inv) {
+ // This logic is from Commons Numbers 1.0 for use of the
+ // complement of the regularised beta function
+ double result;
+ if (x > (a + 1) / (2 + b + a)) {
+ result = BoostBeta.ibetaFraction(b, a, 1 - x, x, Policy.getDefault(), true);
+ inv = !inv;
+ } else {
+ result = BoostBeta.ibetaFraction(a, b, x, 1 - x, Policy.getDefault(), true);
+ }
+ return inv ? 1 - result : result;
+ }
+
+ /**
+ * Assert the function is close to the expected value.
+ *
+ * @param fun Function
+ * @param x Input value
+ * @param y Input value
+ * @param expected Expected value
+ * @param tolerance the tolerance
+ */
+ private static void assertClose(DoubleBinaryOperator fun, double x, double y, double expected, int tolerance) {
+ final double actual = fun.applyAsDouble(x, y);
+ TestUtils.assertEquals(expected, actual, tolerance, null, () -> x + ", " + y);
+ }
+
+ /**
+ * Assert the function is close to the expected value.
+ *
+ * @param fun Function
+ * @param x Input value
+ * @param y Input value
+ * @param expected Expected value
+ * @param eps Relative tolerance
+ */
+ private static void assertWithinEps(DoubleBinaryOperator fun, double x, double y, double expected, double eps) {
+ final double actual = fun.applyAsDouble(x, y);
+ Assertions.assertEquals(expected, actual, Math.abs(expected) * eps, () -> x + ", " + y);
+ }
+
+ /**
+ * Assert the function is close to the expected value.
+ *
+ * @param fun Function
+ * @param x Input value
+ * @param y Input value
+ * @param z Input value
+ * @param expected Expected value
+ * @param tolerance the tolerance
+ */
+ private static void assertClose(DoubleTernaryOperator fun, double x, double y, double z, double expected, int tolerance) {
+ final double actual = fun.applyAsDouble(x, y, z);
+ TestUtils.assertEquals(expected, actual, tolerance, null, () -> x + ", " + y + ", " + z);
+ }
+
+ /**
+ * Assert the function using extended precision.
+ *
+ * @param tc Test case
+ */
+ private static void assertFunction(BiTestCase tc) {
+ final TestUtils.ErrorStatistics stats = new TestUtils.ErrorStatistics();
+ try (DataReader in = new DataReader(tc.getFilename())) {
+ while (in.next()) {
+ try {
+ final double x = in.getDouble(0);
+ final double y = in.getDouble(1);
+ final BigDecimal expected = in.getBigDecimal(tc.getExpectedField());
+ final double actual = tc.getFunction().applyAsDouble(x, y);
+ TestUtils.assertEquals(expected, actual, tc.getTolerance(), stats::add,
+ () -> tc + " x=" + x + ", y=" + y);
+ } catch (final NumberFormatException ex) {
+ Assertions.fail("Failed to load data: " + Arrays.toString(in.getFields()), ex);
+ }
+ }
+ } catch (final IOException ex) {
+ Assertions.fail("Failed to load data: " + tc.getFilename(), ex);
+ }
+
+ assertRms(tc, stats);
+ }
+
+ /**
+ * Assert the function using extended precision.
+ *
+ * @param tc Test case
+ */
+ private static void assertFunction(TriTestCase tc) {
+ final TestUtils.ErrorStatistics stats = new TestUtils.ErrorStatistics();
+ try (DataReader in = new DataReader(tc.getFilename())) {
+ while (in.next()) {
+ try {
+ final double x = in.getDouble(0);
+ final double y = in.getDouble(1);
+ final double z = in.getDouble(2);
+ final BigDecimal expected = in.getBigDecimal(tc.getExpectedField());
+ final double actual = tc.getFunction().applyAsDouble(x, y, z);
+ TestUtils.assertEquals(expected, actual, tc.getTolerance(), stats::add,
+ () -> tc + " x=" + x + ", y=" + y + ", z=" + z);
+ } catch (final NumberFormatException ex) {
+ Assertions.fail("Failed to load data: " + Arrays.toString(in.getFields()), ex);
+ }
+ }
+ } catch (final IOException ex) {
+ Assertions.fail("Failed to load data: " + tc.getFilename(), ex);
+ }
+
+ assertRms(tc, stats);
+ }
+
+ /**
+ * Assert the Root Mean Square (RMS) error of the function is below the allowed
+ * maximum for the specified TestError.
+ *
+ * @param te Test error
+ * @param stats Error statistics
+ */
+ private static void assertRms(TestError te, TestUtils.ErrorStatistics stats) {
+ final double rms = stats.getRMS();
+ //debugRms(te.toString(), stats.getMaxAbs(), rms, stats.getMean(), stats.size());
+ Assertions.assertTrue(rms <= te.getRmsTolerance(),
+ () -> String.format("%s RMS %s < %s", te, rms, te.getRmsTolerance()));
+ }
+
+ /**
+ * Output the maximum and RMS ulp for the named test. Used for reporting the
+ * errors and setting appropriate test tolerances. This is relevant across
+ * different JDK implementations where the java.util.Math functions used in
+ * BoostGamma may compute to different accuracy.
+ *
+ * @param name Test name
+ * @param maxAbsUlp Maximum |ulp|
+ * @param rmsUlp RMS ulp
+ * @param meanUlp Mean ulp
+ * @param size Number of measurements
+ */
+ private static void debugRms(String name, double maxAbsUlp, double rmsUlp, double meanUlp, int size) {
+ // CHECKSTYLE: stop regexp
+ System.out.printf("%-35s max %10.6g RMS %10.6g mean %14.6g n %4d%n",
+ name, maxAbsUlp, rmsUlp, meanUlp, size);
+ }
+}
diff --git a/commons-numbers-gamma/src/test/resources/org/apache/commons/numbers/gamma/beta_exp_data.csv b/commons-numbers-gamma/src/test/resources/org/apache/commons/numbers/gamma/beta_exp_data.csv
new file mode 100644
index 0000000..e6f0f3c
--- /dev/null
+++ b/commons-numbers-gamma/src/test/resources/org/apache/commons/numbers/gamma/beta_exp_data.csv
@@ -0,0 +1,373 @@
+# 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.
+
+# (C) Copyright John Maddock 2006.
+# Use, modification and distribution are subject to the
+# Boost Software License, Version 1.0. (See accompanying file
+# LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)
+
+# Extracted from: boost/libs/math/test/beta_exp_data.ipp
+# a, b, beta(a, b)
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diff --git a/commons-numbers-gamma/src/test/resources/org/apache/commons/numbers/gamma/beta_med_data.csv b/commons-numbers-gamma/src/test/resources/org/apache/commons/numbers/gamma/beta_med_data.csv
new file mode 100644
index 0000000..3fa0da1
--- /dev/null
+++ b/commons-numbers-gamma/src/test/resources/org/apache/commons/numbers/gamma/beta_med_data.csv
@@ -0,0 +1,1852 @@
+# 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.
+
+# (C) Copyright John Maddock 2006.
+# Use, modification and distribution are subject to the
+# Boost Software License, Version 1.0. (See accompanying file
+# LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)
+
+# Extracted from: boost/libs/math/test/beta_med_data.ipp
+# a, b, beta(a, b)
+0.4883005917072296142578125,0.4883005917072296142578125,3.245912809500479157065104747353807392371
+3.5808107852935791015625,0.4883005917072296142578125,1.007653173802923954909901438393379243537
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diff --git a/commons-numbers-gamma/src/test/resources/org/apache/commons/numbers/gamma/beta_small_data.csv b/commons-numbers-gamma/src/test/resources/org/apache/commons/numbers/gamma/beta_small_data.csv
new file mode 100644
index 0000000..58c5fe3
--- /dev/null
+++ b/commons-numbers-gamma/src/test/resources/org/apache/commons/numbers/gamma/beta_small_data.csv
@@ -0,0 +1,45 @@
+# 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.
+
+# (C) Copyright John Maddock 2006.
+# Use, modification and distribution are subject to the
+# Boost Software License, Version 1.0. (See accompanying file
+# LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)
+
+# Extracted from: boost/libs/math/test/beta_small_data.ipp
+# a, b, beta(a, b)
+0.1730655412757187150418758392333984375e-5,0.1730655412757187150418758392333984375e-5,1155631.551635027016649268884796909927277
+0.216575062950141727924346923828125e-5,0.1730655412757187150418758392333984375e-5,1039549.452063747329381617654200841254652
+0.216575062950141727924346923828125e-5,0.216575062950141727924346923828125e-5,923467.3524924676425690820378921903570447
+0.72700195232755504548549652099609375e-5,0.1730655412757187150418758392333984375e-5,715366.9882608199489156088500706918884474
+0.72700195232755504548549652099609375e-5,0.216575062950141727924346923828125e-5,599284.8886895402674421924477675861806454
+0.72700195232755504548549652099609375e-5,0.72700195232755504548549652099609375e-5,275102.4248866129549503632384850636365051
+0.14000004739500582218170166015625e-4,0.1730655412757187150418758392333984375e-5,649244.3230419389091055494874477610228104
+0.14000004739500582218170166015625e-4,0.216575062950141727924346923828125e-5,533162.2234706592346717218748633348329255
+0.14000004739500582218170166015625e-4,0.72700195232755504548549652099609375e-5,208979.7596677320047637517666971035319141
+0.14000004739500582218170166015625e-4,0.14000004739500582218170166015625e-4,142857.0944488511634633414470885024340382
+0.17196454791701398789882659912109375e-4,0.1730655412757187150418758392333984375e-5,635967.2966761120408405544283252854891051
+0.17196454791701398789882659912109375e-4,0.216575062950141727924346923828125e-5,519885.1971048323697502064131013612391462
+0.17196454791701398789882659912109375e-4,0.72700195232755504548549652099609375e-5,195702.7333019051790657557774869592595619
+0.17196454791701398789882659912109375e-4,0.14000004739500582218170166015625e-4,129580.0680830243894812627999923649303964
+0.17196454791701398789882659912109375e-4,0.17196454791701398789882659912109375e-4,116303.0417171976400618567245671670493297
+0.60085076256655156612396240234375e-4,0.1730655412757187150418758392333984375e-5,594458.8435535046961781833034439822035691
+0.60085076256655156612396240234375e-4,0.216575062950141727924346923828125e-5,478376.7439822250699480739314338811953296
+0.60085076256655156612396240234375e-4,0.72700195232755504548549652099609375e-5,154194.2801792984055346510144704788292969
+0.60085076256655156612396240234375e-4,0.14000004739500582218170166015625e-4,88071.61496041830983457563965247905141874
+0.60085076256655156612396240234375e-4,0.17196454791701398789882659912109375e-4,74794.58859459188997822531683455967562115
+0.60085076256655156612396240234375e-4,0.60085076256655156612396240234375e-4,33286.13547199056171829186221434157408122
+2.9833362924800826973163861261851735349505886138402e-154,3.7291703656001033716454826577314669186882357673002e-155,3.0167567842370843474041556245963153786828573096333e154
+2.9833362924800826973163861261851735349505886138402e-154,1.8645851828000516858227413288657334593441178836501e-155,5.6983183702256037673189606242374846041787304737518e154
diff --git a/commons-numbers-gamma/src/test/resources/org/apache/commons/numbers/gamma/binomial_extra_large_data.csv b/commons-numbers-gamma/src/test/resources/org/apache/commons/numbers/gamma/binomial_extra_large_data.csv
new file mode 100644
index 0000000..2e777ec
--- /dev/null
+++ b/commons-numbers-gamma/src/test/resources/org/apache/commons/numbers/gamma/binomial_extra_large_data.csv
@@ -0,0 +1,531 @@
+# 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.
+
+# Generated using the following maxima script:
+# https://maxima.sourceforge.io/
+#
+# /* Formatting function to convert big float exponent to e. */
+# str(x) := ssubst("e","b",string(x));
+#
+# fpprec : 20;
+# /* 2^16 - 1 */
+# low : 65535;
+# range : 65536;
+# for i : 1 step 1 thru 500 do
+# (n : low + random(range),
+# k : 3 + random(40 - 3),
+# printf(true, "~d,~d,~a~%", n, k, str(bfloat(binomial(n, k)))));
+
+# n, k, binom(n, k)
+83054,23,5.3885911169764183093e90
+71537,13,2.0613205382191374626e53
+120323,16,9.2156438924733450329e67
+98958,16,4.0370886626967470753e66
+85750,3,1.050838417265e14
+124078,14,2.3496333826719206799e60
+117873,4,8.04310983397714338e18
+117612,22,3.150041964993879019e90
+72801,24,7.8875831533180831083e92
+86130,18,1.0608830322707442375e73
+65713,20,9.2394356945403991745e77
+128452,24,6.5485027724279123717e98
+72186,15,5.7499396974672586986e60
+122239,36,3.6878965811886522957e141
+111097,28,6.223671832590949927e111
+98123,21,1.3119392733579376484e85
+98585,28,2.1922867997123553876e110
+123328,9,1.8182993749751390747e40
+87222,36,1.944755499080514648e136
+97101,38,6.206340232275634454e144
+69632,24,2.7101017063891064298e92
+101749,34,6.0738730210052009257e131
+71181,34,3.2121138938456030679e126
+68447,14,5.6751333244774206782e56
+95707,31,3.1054372066810163323e120
+103163,26,5.554381649182754217e103
+100441,17,3.0257349403031532927e70
+66126,16,6.3759993965447551013e63
+98026,30,2.0637208370232392108e117
+96330,5,6.9116457495077692216e22
+88602,21,1.5379446179306371529e84
+96565,13,1.0186513183499753733e55
+130691,33,7.8644266172309236522e131
+66765,30,2.0416228162195863746e112
+85320,27,1.257775794193431467e105
+119242,32,1.0562279424722963374e127
+85824,24,4.0973928557690470772e94
+71839,7,1.9586843568894074649e30
+72099,31,4.7641271672681815552e116
+116633,20,8.9042386475545034504e82
+98499,39,2.6976671878123600342e148
+86289,22,3.4601129404582725729e87
+73147,3,6.5225962207165e13
+110676,15,3.4985667907933082442e63
+97701,17,1.8906453153041203804e70
+124335,27,3.2795394938856867187e109
+72682,15,6.3720031819728966148e60
+126150,21,2.5678250881451306898e87
+79831,27,2.0880450901420743528e104
+82142,22,1.1707063663413173301e87
+91707,36,1.1829612278092587847e137
+69200,12,2.514907030427952146e49
+102043,8,2.9149160441956506072e35
+98620,38,1.1195751664782463759e145
+92522,3,1.3199871517404e14
+66841,16,7.5733438240762662136e63
+80369,34,1.9944110148976864025e128
+103174,16,7.8704497736431134073e66
+93737,33,1.3549784757802358118e127
+97533,36,1.0867164232814021882e138
+118612,17,5.1123454608786895021e71
+67883,22,1.7639975677530050699e85
+117006,15,8.0580996100662627927e63
+104692,38,1.0846154573821817126e146
+111583,33,4.2657380461776737403e129
+91366,11,9.2728571405573868197e46
+95219,22,3.0206422399771023021e88
+69383,35,2.6662446406310088728e129
+91414,35,4.153222377910206417e133
+80971,38,6.2262916606399520637e141
+122505,31,6.5478371447273970586e123
+83396,11,3.3974672443481260896e46
+104146,32,1.3877866894462058283e125
+72547,11,7.3337998378259946702e45
+100838,6,1.4599844824930570206e27
+99516,23,3.4509474202480706631e92
+73011,28,4.8807069547168688695e106
+126796,17,1.5894982325438978811e72
+98015,18,1.0870432121053496808e74
+107325,13,4.0226820214727683359e55
+68711,12,2.3097292993641297517e49
+106756,17,8.5319278721302668396e70
+114699,8,7.4276342090481540609e35
+103676,7,2.5540484098160723133e31
+66493,16,6.9664482694581155184e63
+120982,34,2.1894104863898304505e134
+66459,16,6.9096652532796266565e63
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+129865,20,7.6398063515796468562e83
+95610,37,1.3704408862478088058e141
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+117037,27,6.403840800611511466e108
+85385,8,7.0047028695171188503e34
+78900,6,3.3500165719061273412e26
+67680,23,4.8581393908231702211e88
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+130119,39,1.4035549678089922179e153
+93923,24,3.5689869360948714551e95
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+96109,29,3.562739545493827488e113
+129239,9,2.7711388742924068012e40
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+81261,16,1.7252563403657014826e65
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+125255,6,5.3627298701263443402e27
+93187,11,1.152116995584867467e47
+102591,11,3.3175183882519413555e47
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+93979,38,1.79226220229002579e144
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+94895,27,2.2232105456066626829e106
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+80062,9,3.7228847574645417898e38
+114056,24,3.7764053340214200565e97
+103875,12,3.2924525566982830783e51
+79912,16,1.3198367197436398786e65
+85340,17,1.8959819233205704028e69
+105617,13,3.2654173288162460891e55
+96358,36,7.0240082636012256384e137
+130386,3,3.6942956078104e14
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+90638,19,1.2675991404294848422e77
+90276,9,1.0970200011160756868e39
+78063,38,1.5505538955490543342e141
+101668,7,2.2272538477049390432e31
+119406,20,1.4246470844030597473e83
+121466,38,3.0778360006651540377e148
+109345,27,1.0213958331458526752e108
+96400,22,3.9617529605866064775e88
+108500,35,1.6727318287390625167e136
+106328,36,2.4333611081740052637e139
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+130936,36,4.3802767245868981656e142
+105227,30,1.7312511962902776714e118
+84163,21,5.2253781602887299722e83
+108631,15,2.6447660031811876066e63
+94615,13,7.8134269544350854138e54
+75540,3,7.183935812338e13
+106385,28,1.8491284500677821335e111
+75945,26,1.929750167583858096e100
+110220,39,2.1659614260719573545e150
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+93201,21,4.4515318145032866471e84
+94930,22,2.825225431774064856e88
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diff --git a/commons-numbers-gamma/src/test/resources/org/apache/commons/numbers/gamma/binomial_huge_data.csv b/commons-numbers-gamma/src/test/resources/org/apache/commons/numbers/gamma/binomial_huge_data.csv
new file mode 100644
index 0000000..121cb61
--- /dev/null
+++ b/commons-numbers-gamma/src/test/resources/org/apache/commons/numbers/gamma/binomial_huge_data.csv
@@ -0,0 +1,530 @@
+# 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.
+
+# Generated using the following maxima script:
+# https://maxima.sourceforge.io/
+#
+# /* Formatting function to convert big float exponent to e. */
+# str(x) := ssubst("e","b",string(x));
+#
+# fpprec : 20;
+# /* 2^30 - 1 */
+# low : 1073741823;
+# range : 1073741824;
+# for i : 1 step 1 thru 500 do
+# (n : low + random(range),
+# k : 3 + random(40 - 3),
+# printf(true, "~d,~d,~a~%", n, k, str(bfloat(binomial(n, k)))));
+
+# n, k, binom(n, k)
+1875541801,11,2.5306873612606207317e94
+1547005290,12,3.9225149066625458446e101
+1706082374,8,1.7802436234953121403e69
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diff --git a/commons-numbers-gamma/src/test/resources/org/apache/commons/numbers/gamma/binomial_large_data.csv b/commons-numbers-gamma/src/test/resources/org/apache/commons/numbers/gamma/binomial_large_data.csv
new file mode 100644
index 0000000..b777e1f
--- /dev/null
+++ b/commons-numbers-gamma/src/test/resources/org/apache/commons/numbers/gamma/binomial_large_data.csv
@@ -0,0 +1,252 @@
+# 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.
+
+# (C) Copyright John Maddock 2006.
+# Use, modification and distribution are subject to the
+# Boost Software License, Version 1.0. (See accompanying file
+# LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)
+
+# Extracted from: boost/libs/math/test/binomial_large_data.ipp
+# n, k, binom(n, k)
+0.174e3,0.4e1,0.36890001e8
+0.174e3,0.5e1,0.1254260034e10
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+0.174e3,0.22e2,0.4359346248249975839021623926e28
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+0.174e3,0.36e2,0.24968911018132735303785934846085665809e38
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+0.398e3,0.11e2,0.864983200770704352428e21
+0.398e3,0.13e2,0.828287951907239471630766e24
+0.398e3,0.22e2,0.776122173564640399865233597134960648e36
+0.398e3,0.374e3,0.198248598682272276052532494920343209e39
+0.398e3,0.375e3,0.12687910315665425667362079674901965376e38
+0.398e3,0.376e3,0.776122173564640399865233597134960648e36
+0.398e3,0.378e3,0.2516163840026833008699548944439896e34
+0.398e3,0.379e3,0.13277909446051889227965957490448e33
+0.398e3,0.386e3,0.27895708224855215365803e23
+0.398e3,0.388e3,0.24522719609478731641e20
+0.398e3,0.394e3,0.1029804105e10
+0.398e3,0.395e3,0.10428396e8
+0.398e3,0.397e3,0.398e3
+0.398e3,0.398e3,0.1e1
+0.399e3,0.4e1,0.1040232501e10
+0.399e3,0.5e1,0.82178367579e11
+0.399e3,0.6e1,0.5396379471021e13
+0.399e3,0.8e1,0.14845439924778771e17
+0.399e3,0.9e1,0.644951890065388829e18
+0.399e3,0.11e2,0.889505920380183084069e21
+0.399e3,0.13e2,0.856183660132094686996569e24
+0.399e3,0.22e2,0.821413122685123394021825478134878776e36
+0.399e3,0.375e3,0.210936508997937701719894574595245174376e39
+0.399e3,0.376e3,0.13464032489230066067227313272036926024e38
+0.399e3,0.378e3,0.47807112960509827165291429944358024e35
+0.399e3,0.379e3,0.2648942934487351900979208519344376e34
+0.399e3,0.386e3,0.856183660132094686996569e24
+0.399e3,0.388e3,0.889505920380183084069e21
+0.399e3,0.394e3,0.82178367579e11
+0.399e3,0.395e3,0.1040232501e10
+0.399e3,0.397e3,0.79401e5
+0.399e3,0.398e3,0.399e3
+0.399e3,0.399e3,0.1e1
diff --git a/commons-numbers-gamma/src/test/resources/org/apache/commons/numbers/gamma/binomial_small_data.csv b/commons-numbers-gamma/src/test/resources/org/apache/commons/numbers/gamma/binomial_small_data.csv
new file mode 100644
index 0000000..dcc74a0
--- /dev/null
+++ b/commons-numbers-gamma/src/test/resources/org/apache/commons/numbers/gamma/binomial_small_data.csv
@@ -0,0 +1,181 @@
+# 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.
+
+# (C) Copyright John Maddock 2006.
+# Use, modification and distribution are subject to the
+# Boost Software License, Version 1.0. (See accompanying file
+# LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)
+
+# Extracted from: boost/libs/math/test/binomial_data.ipp
+# n, k, binom(n, k)
+0.15e2,0.15e2,0.1e1
+0.19e2,0.15e2,0.3876e4
+0.19e2,0.19e2,0.1e1
+0.21e2,0.15e2,0.54264e5
+0.21e2,0.19e2,0.21e3
+0.21e2,0.21e2,0.1e1
+0.29e2,0.15e2,0.7755876e8
+0.29e2,0.19e2,0.2003001e8
+0.29e2,0.21e2,0.4292145e7
+0.29e2,0.29e2,0.1e1
+0.33e2,0.15e2,0.103715832e10
+0.33e2,0.19e2,0.8188092e9
+0.33e2,0.21e2,0.35481732e9
+0.33e2,0.29e2,0.4092e5
+0.33e2,0.33e2,0.1e1
+0.42e2,0.15e2,0.98672427616e11
+0.42e2,0.19e2,0.4467753108e12
+0.42e2,0.21e2,0.53825787444e12
+0.42e2,0.29e2,0.2551873128e11
+0.42e2,0.33e2,0.44589181e9
+0.42e2,0.42e2,0.1e1
+0.46e2,0.15e2,0.511738760544e12
+0.46e2,0.19e2,0.415424667196e13
+0.46e2,0.21e2,0.6943526580276e13
+0.46e2,0.29e2,0.174969502686e13
+0.46e2,0.33e2,0.10176623079e12
+0.46e2,0.42e2,0.163185e6
+0.46e2,0.46e2,0.1e1
+0.82e2,0.15e2,0.996731056598616e16
+0.82e2,0.19e2,0.19710382359693168e19
+0.82e2,0.21e2,0.1833065559451464624e20
+0.82e2,0.29e2,0.1257660762751009389168e23
+0.82e2,0.33e2,0.899986590548788671513e23
+0.82e2,0.42e2,0.41467066225715382349482e24
+0.82e2,0.46e2,0.23223183395337370425708e24
+0.82e2,0.82e2,0.1e1
+0.95e2,0.15e2,0.110375347398090219e18
+0.95e2,0.19e2,0.45038039715653129145e20
+0.95e2,0.21e2,0.611230538998149609825e21
+0.95e2,0.29e2,0.2146280142106099437545685e25
+0.95e2,0.33e2,0.3780222443838484815806271e26
+0.95e2,0.42e2,0.171987522327068244097964157e28
+0.95e2,0.46e2,0.308620560869098008873280965e28
+0.95e2,0.82e2,0.3489735464257595e16
+0.95e2,0.95e2,0.1e1
+0.122e3,0.15e2,0.6156937298378625264e19
+0.122e3,0.19e2,0.819751088529043274604e22
+0.122e3,0.21e2,0.205054879430622110547372e24
+0.122e3,0.29e2,0.965499981454662046766678536e28
+0.122e3,0.33e2,0.68890617994929806723272437813e30
+0.122e3,0.42e2,0.982048893026815201335532502524191e33
+0.122e3,0.46e2,0.9517963588769497111427223674615988e34
+0.122e3,0.82e2,0.254605268562507644790693611765531e33
+0.122e3,0.95e2,0.877923835320476575559398624e27
+0.122e3,0.122e3,0.1e1
+0.125e3,0.15e2,0.90648078331934398e19
+0.125e3,0.19e2,0.1350175763944140060675e23
+0.125e3,0.21e2,0.357796577445197116078875e24
+0.125e3,0.29e2,0.21471697865846785089593512375e29
+0.125e3,0.33e2,0.1743111472200107189546091504625e31
+0.125e3,0.42e2,0.339619764433637564049548277312025e34
+0.125e3,0.46e2,0.38244450869782214079034893241053e35
+0.125e3,0.82e2,0.655545126697486460839825744579025e34
+0.125e3,0.95e2,0.687094331707097122866992396e29
+0.125e3,0.122e3,0.31775e6
+0.125e3,0.125e3,0.1e1
+0.135e3,0.15e2,0.307563739414613748e20
+0.135e3,0.19e2,0.65183278299357679461e23
+0.135e3,0.21e2,0.2070345077412932009547e25
+0.135e3,0.29e2,0.2654598483954587033449048188e30
+0.135e3,0.33e2,0.322268385697858320141605625441e32
+0.135e3,0.42e2,0.165539796634100865734488051471458e36
+0.135e3,0.46e2,0.2961867439565318449706072684149998e37
+0.135e3,0.82e2,0.1323968752490674447090425029339391282e39
+0.135e3,0.95e2,0.319215598884570762368184126129732e35
+0.135e3,0.122e3,0.437531400061434e18
+0.135e3,0.125e3,0.393812684240976e15
+0.135e3,0.135e3,0.1e1
+0.137e3,0.15e2,0.388194525997363728e20
+0.137e3,0.19e2,0.87968625327656981292e23
+0.137e3,0.21e2,0.28916543839848387707556e25
+0.137e3,0.29e2,0.4280069137507949602563401336e30
+0.137e3,0.33e2,0.560540007685072462740701644251e32
+0.137e3,0.42e2,0.3453905364934565879468064249738192e36
+0.137e3,0.46e2,0.67381580139170956477318127290699344e37
+0.137e3,0.95e2,0.3453905364934565879468064249738192e36
+0.137e3,0.122e3,0.388194525997363728e20
+0.137e3,0.125e3,0.55587257066498976e17
+0.137e3,0.135e3,0.9316e4
+0.137e3,0.137e3,0.1e1
+0.143e3,0.15e2,0.76435265515004093841e20
+0.143e3,0.19e2,0.210374461432936964163e24
+0.143e3,0.21e2,0.76395982994646537557478e25
+0.143e3,0.29e2,0.17138496567725350060390313738e31
+0.143e3,0.33e2,0.27947962468928852152878073112114e33
+0.143e3,0.42e2,0.291038363317011408088847346619471914e37
+0.143e3,0.46e2,0.728184459077800535203086713053471437e38
+0.143e3,0.95e2,0.3005697554491346889987208985795179974e39
+0.143e3,0.122e3,0.76395982994646537557478e25
+0.143e3,0.125e3,0.31976918137806418552776e23
+0.143e3,0.135e3,0.3553074951283e13
+0.143e3,0.137e3,0.10679057389e11
+0.143e3,0.143e3,0.1e1
+0.144e3,0.15e2,0.85323087086516197776e20
+0.144e3,0.19e2,0.242351379570743382715776e24
+0.144e3,0.21e2,0.89439199603488629335584e25
+0.144e3,0.29e2,0.214603783108908731190974363328e31
+0.144e3,0.33e2,0.36256816175907700090220202956256e33
+0.144e3,0.42e2,0.410877689388721987890137430521607408e37
+0.144e3,0.46e2,0.1069985327624523235400453945711223336e39
+0.144e3,0.122e3,0.500046434146777336739856e26
+0.144e3,0.125e3,0.242351379570743382715776e24
+0.144e3,0.135e3,0.56849199220528e14
+0.144e3,0.137e3,0.219683466288e12
+0.144e3,0.143e3,0.144e3
+0.144e3,0.144e3,0.1e1
+0.145e3,0.15e2,0.95168058673421912904e20
+0.145e3,0.19e2,0.27889642887109357534752e24
+0.145e3,0.21e2,0.10458616082666009075532e26
+0.145e3,0.29e2,0.26825472888613591398871795416e31
+0.145e3,0.33e2,0.4693962808488050458108865561301e33
+0.145e3,0.42e2,0.57842004816858920625310609151100072e37
+0.145e3,0.46e2,0.156715022732884716296026082957704428e39
+0.145e3,0.122e3,0.315246665005577016640344e27
+0.145e3,0.125e3,0.1757047501887889524689376e25
+0.145e3,0.135e3,0.824313388697656e15
+0.145e3,0.137e3,0.398176282647e13
+0.145e3,0.143e3,0.1044e5
+0.145e3,0.144e3,0.145e3
+0.145e3,0.145e3,0.1e1
+0.148e3,0.15e2,0.131439605927052712464e21
+0.148e3,0.19e2,0.42244624913775365779698e24
+0.148e3,0.21e2,0.16608172537529972375104128e26
+0.148e3,0.29e2,0.51863815313548295889610045344e31
+0.148e3,0.33e2,0.100644585365316213131406856592232e34
+0.148e3,0.42e2,0.1587255130729102485141868921944628536e38
+0.148e3,0.122e3,0.6418858594895863473128136624e29
+0.148e3,0.125e3,0.525225250880542723214261376e27
+0.148e3,0.135e3,0.1525832904625819216e19
+0.148e3,0.137e3,0.12775329171405528e17
+0.148e3,0.143e3,0.552689424e9
+0.148e3,0.144e3,0.19190605e8
+0.148e3,0.145e3,0.529396e6
+0.148e3,0.148e3,0.1e1
+0.149e3,0.15e2,0.146152994650230254904e21
+0.149e3,0.19e2,0.484188393242502269321154e24
+0.149e3,0.21e2,0.19332950844468483467894649e26
+0.149e3,0.29e2,0.643975706809891340629324729688e31
+0.149e3,0.33e2,0.129276234650276859970513979588298e34
+0.149e3,0.42e2,0.2210289854940525890524658592240650952e38
+0.149e3,0.122e3,0.35422590023684579907262679888e30
+0.149e3,0.125e3,0.3260773432550036073288539376e28
+0.149e3,0.135e3,0.16239221627803361656e20
+0.149e3,0.137e3,0.158627003878285306e18
+0.149e3,0.143e3,0.13725120696e11
+0.149e3,0.144e3,0.571880029e9
+0.149e3,0.145e3,0.19720001e8
+0.149e3,0.148e3,0.149e3
+0.149e3,0.149e3,0.1e1
diff --git a/commons-numbers-gamma/src/test/resources/org/apache/commons/numbers/gamma/ibeta_derivative_large_data.csv b/commons-numbers-gamma/src/test/resources/org/apache/commons/numbers/gamma/ibeta_derivative_large_data.csv
new file mode 100644
index 0000000..9ff798f
--- /dev/null
+++ b/commons-numbers-gamma/src/test/resources/org/apache/commons/numbers/gamma/ibeta_derivative_large_data.csv
@@ -0,0 +1,1233 @@
+#
+# 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.
+
+# (C) Copyright John Maddock 2006.
+# Use, modification and distribution are subject to the
+# Boost Software License, Version 1.0. (See accompanying file
+# LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)
+
+# Extracted from: boost/libs/math/test/ibeta_derivative_large_data.ipp
+# a, b, x, ibeta_derivative(a, b, x)
+1.04760829344741068780422210693359375e-5,3.90781875e4,9.13384497165679931640625e-1,1.9269302505055971245108533005908825520385e-41521
+1.127331415773369371891021728515625e-5,2.26620174944400787353515625e-2,1.355634629726409912109375e-1,9.5833311060979905221980727295357737468569e-5
+1.13778432933031581342220306396484375e-5,3.654421865940093994140625e-2,9.688708782196044921875e-1,3.3222023158036735998897099680773504451652e-4
+1.142846667789854109287261962890625e-5,2.44517601095139980316162109375e-3,1.355634629726409912109375e-1,9.7033676664585638381273746677575810050956e-5
+1.184685606858693063259124755859375e-5,1.5964560210704803466796875e-2,3.082362115383148193359375e-1,5.519230686984445546996254585063199948556e-5
+1.215800511999987065792083740234375e-5,2.411049609375e4,1.355634629726409912109375e-1,4.1691304653994518958586532492091054824083e-1530
+1.30341722979210317134857177734375e-5,2.6168341796875e4,1.2707412242889404296875e-1,3.5131369801838022244189263561014529757933e-1549
+1.3885271982871927320957183837890625e-5,4.976274073123931884765625e-2,6.32396042346954345703125e-1,5.6811171490789982191243977143552781827692e-5
+1.39016165121574886143207550048828125e-5,3.188882328686304390430450439453125e-5,8.1474220752716064453125e-1,6.4136860372805759679245829936722379832552e-5
+1.4759907571715302765369415283203125e-5,2.82413125038146972656250,6.32396042346954345703125e-1,3.7609461264452087672100113133375774171817e-6
+1.50794721776037476956844329833984375e-5,4.875471131526865065097808837890625e-5,2.21111953258514404296875e-1,6.6872412074058490069916374670256458887574e-5
+1.519624856882728636264801025390625e-5,1.6177537109375e4,8.1474220752716064453125e-1,2.6910083770664820015271856038619503004517e-11850
+1.554497794131748378276824951171875e-5,4.046924591064453125e1,9.05801355838775634765625e-1,5.5068258519819312527899610527681704563872e-46
+1.5675454051233828067779541015625e-5,1.01913392427377402782440185546875e-4,8.1474220752716064453125e-1,8.9993789667585905658020173941717362428122e-5
+1.5971760149113833904266357421875e-5,1.9206241607666015625e1,9.13384497165679931640625e-1,7.9493607822778359703597009221512935383254e-25
+1.6304524251609109342098236083984375e-5,3.9033559733070433139801025390625e-4,1.2707412242889404296875e-1,1.410795073719112901366482730955481534932e-4
+1.6487292668898589909076690673828125e-5,4.7099715625e5,1.2707412242889404296875e-1,4.7913621881963152427739710649393818331186e-27804
+1.65044693858362734317779541015625e-5,3.575843811035156250,3.082362115383148193359375e-1,2.0724212982706756891833285332465709443072e-5
+1.66259487741626799106597900390625e-5,1.47818875e5,6.32396042346954345703125e-1,6.940975264551968805055110203165413548228e-64250
+1.6847467122715897858142852783203125e-5,2.207652665674686431884765625e-3,9.688708782196044921875e-1,5.5014020851638370971095475825558113712879e-4
+1.747490387060679495334625244140625e-5,2.6349246501922607421875e-1,6.32396042346954345703125e-1,5.7742678064008213233661580897444084068984e-5
+1.7863809262053109705448150634765625e-5,4.3938714599609375e2,8.350250720977783203125e-1,1.8047299302665262153684660799151120674515e-348
+1.895247623906470835208892822265625e-5,1.071095585823059082031250,9.688708782196044921875e-1,1.528521421578502712184688186839025777264e-5
+1.9740999050554819405078887939453125e-5,1.0541565704345703125e2,3.082362115383148193359375e-1,1.2462675909278790257612853184177372341877e-21
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diff --git a/commons-numbers-gamma/src/test/resources/org/apache/commons/numbers/gamma/ibeta_derivative_med_data.csv b/commons-numbers-gamma/src/test/resources/org/apache/commons/numbers/gamma/ibeta_derivative_med_data.csv
new file mode 100644
index 0000000..703b105
--- /dev/null
+++ b/commons-numbers-gamma/src/test/resources/org/apache/commons/numbers/gamma/ibeta_derivative_med_data.csv
@@ -0,0 +1,523 @@
+#
+# 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.
+
+# (C) Copyright John Maddock 2006.
+# Use, modification and distribution are subject to the
+# Boost Software License, Version 1.0. (See accompanying file
+# LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)
+
+# Extracted from: boost/libs/math/test/ibeta_derivative_data.ipp
+# a, b, x, ibeta_derivative(a, b, x)
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diff --git a/commons-numbers-gamma/src/test/resources/org/apache/commons/numbers/gamma/ibeta_derivative_small_data.csv b/commons-numbers-gamma/src/test/resources/org/apache/commons/numbers/gamma/ibeta_derivative_small_data.csv
new file mode 100644
index 0000000..d73297a
--- /dev/null
+++ b/commons-numbers-gamma/src/test/resources/org/apache/commons/numbers/gamma/ibeta_derivative_small_data.csv
@@ -0,0 +1,523 @@
+#
+# 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.
+
+# (C) Copyright John Maddock 2006.
+# Use, modification and distribution are subject to the
+# Boost Software License, Version 1.0. (See accompanying file
+# LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)
+
+# Extracted from: boost/libs/math/test/ibeta_derivative_small_data.ipp
+# a, b, x, ibeta_derivative(a, b, x)
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diff --git a/commons-numbers-gamma/src/test/resources/org/apache/commons/numbers/gamma/ibeta_derivative_small_int_data.csv b/commons-numbers-gamma/src/test/resources/org/apache/commons/numbers/gamma/ibeta_derivative_small_int_data.csv
new file mode 100644
index 0000000..e834d95
--- /dev/null
+++ b/commons-numbers-gamma/src/test/resources/org/apache/commons/numbers/gamma/ibeta_derivative_small_int_data.csv
@@ -0,0 +1,1023 @@
+#
+# 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.
+
+# (C) Copyright John Maddock 2006.
+# Use, modification and distribution are subject to the
+# Boost Software License, Version 1.0. (See accompanying file
+# LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)
+
+# Extracted from: boost/libs/math/test/ibeta_derivative_int_data.ipp
+# a, b, x, ibeta_derivative(a, b, x)
+1.0,1.0,1.2707412242889404296875e-1,1.0
+1.0,1.0,1.355634629726409912109375e-1,1.0
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+1.0,1.0,3.082362115383148193359375e-1,1.0
+1.0,1.0,6.32396042346954345703125e-1,1.0
+1.0,1.0,8.1474220752716064453125e-1,1.0
+1.0,1.0,8.350250720977783203125e-1,1.0
+1.0,1.0,9.05801355838775634765625e-1,1.0
+1.0,1.0,9.13384497165679931640625e-1,1.0
+1.0,1.0,9.688708782196044921875e-1,1.0
+1.0,5.0,1.2707412242889404296875e-1,2.9032168585284850711226834224884111659355
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+1.0,9.0,1.2707412242889404296875e-1,3.0343205259518420518735332481669911487185
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+1.0,9.0,3.082362115383148193359375e-1,4.7195990455409980953587385153068556499237e-1
+1.0,9.0,6.32396042346954345703125e-1,3.0011239767112320752115061811340852668426e-3
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+1.0,9.0,9.688708782196044921875e-1,7.9355577628341500944328871478533028923854e-12
+1.0,1.3e1,1.2707412242889404296875e-1,2.5449061459242510382566663365495531612768
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+1.0,1.3e1,6.32396042346954345703125e-1,7.9159946671589042281490839627098212252494e-5
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+1.0,2.1e1,8.350250720977783203125e-1,4.6835831996654990285225995661517063908955e-15
+1.0,2.1e1,9.05801355838775634765625e-1,6.3549528399482585128034968036488346604573e-20
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+1.0,2.5e1,8.350250720977783203125e-1,4.1301967667394070880700672105908225699136e-18
+1.0,2.5e1,9.05801355838775634765625e-1,5.9567812711150703900877713411857055705582e-24
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+1.0,2.9e1,9.13384497165679931640625e-1,5.1889043468153358488432941691480203476646e-29
+1.0,2.9e1,9.688708782196044921875e-1,1.8666818943878542695166526725683638588841e-41
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diff --git a/commons-numbers-gamma/src/test/resources/org/apache/commons/numbers/gamma/ibeta_large_data.csv b/commons-numbers-gamma/src/test/resources/org/apache/commons/numbers/gamma/ibeta_large_data.csv
new file mode 100644
index 0000000..3c5a2c9
--- /dev/null
+++ b/commons-numbers-gamma/src/test/resources/org/apache/commons/numbers/gamma/ibeta_large_data.csv
@@ -0,0 +1,1233 @@
+#
+# 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.
+
+# (C) Copyright John Maddock 2006.
+# Use, modification and distribution are subject to the
+# Boost Software License, Version 1.0. (See accompanying file
+# LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt
+
+# Extracted from: boost/libs/math/test/ibeta_large_data.ipp
+# a, b, x, beta(a, b, x), betac(a, b, x), ibeta(a, b, x), ibetac(a, b, x
+0.104760829344741068780422210693359375e-4,39078.1875,0.913384497165679931640625,95444.37547576888548779405522478045372688,0.4076397251031275963346153642645211144346e-41521,1.0,0.4270966445860582673748568606264586002504e-41526
+0.1127331415773369371891021728515625e-4,0.0226620174944400787353515625,0.1355634629726409912109375,88703.20318198098901713372585734808194292,45.9460483635769831377505786833842050448,0.9994822930837981780736860587426162687504,0.0005177069162018219263139412573837312496096
+0.113778432933031581342220306396484375e-4,0.03654421865940093994140625,0.9688708782196044921875,87893.29210911967129223056449206866305889,24.13233514927218107101077084629346084654,0.999725511349976252733137998478545702411,0.0002744886500237472668620015214542975889712
+0.1142846667789854109287261962890625e-4,0.00244517601095139980316162109375,0.1355634629726409912109375,87498.94901523223439182946063290895919618,410.8174698800966982396187764410683048418,0.9953268278792474174353255397883295739773,0.004673172120752582564674460211670426022665
+0.1184685606858693063259124755859375e-4,0.015964560210704803466796875,0.3082362115383148193359375,84409.76586511709213323237371074024560834,63.42758275377907514602214367744158071269,0.9992491395179357013663738064942314292026,0.000750860482064298633626193505768570797446
+0.1215800511999987065792083740234375e-4,24110.49609375,0.1355634629726409912109375,82239.66859351932879196086451784571067159,0.1228963536503215267489197519054078145969e-1528,1.0,0.1494368298804234679967895435418141236754e-1533
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+0.13885271982871927320957183837890625e-4,0.04976274073123931884765625,0.632396042346954345703125,72019.23463150248088958543324182892626471,19.53662131349699149968802706905083676178,0.9997288040735046086759949196286322002492,0.0002711959264953913240050803713677997508421
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+0.150794721776037476956844329833984375e-4,0.4875471131526865065097808837890625e-4,0.221111953258514404296875,66314.05927892349341973416366818795267336,20512.09739510432080805623001488366337009,0.763756704421306309320206420222803196485,0.236243295578693690679793579777196803515
+0.1519624856882728636264801025390625e-4,16177.537109375,0.81474220752716064453125,65795.44709267135893551075697872372424105,0.2027537374480409874707652773038782413935e-11849,1.0,0.3081577014933374612265415972232266786926e-11854
+0.1554497794131748378276824951171875e-4,40.46924591064453125,0.905801355838775634765625,64325.19244254056356455961240584532088293,0.822463871000651749828830546797984530393e-43,1.0,0.1278603047686689558334345413308146407234e-47
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