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Posted to commits@commons.apache.org by ah...@apache.org on 2021/12/08 15:24:24 UTC
[commons-numbers] branch master updated: NUMBERS-177: Add scaled complementary error function Erfcx
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
The following commit(s) were added to refs/heads/master by this push:
new 797eba3 NUMBERS-177: Add scaled complementary error function Erfcx
797eba3 is described below
commit 797eba323327f8bc232213c25a5b1ccbfadbc2bb
Author: Alex Herbert <ah...@apache.org>
AuthorDate: Mon Dec 6 08:34:26 2021 +0000
NUMBERS-177: Add scaled complementary error function Erfcx
Updates the rational function approximation for erfc(x > 4) to a
function valid for the approximation limit of erfcx (x ~ 6.71e7).
Improves the scaling by exp(-x*x) and exp(x*x).
---
.../org/apache/commons/numbers/gamma/BoostErf.java | 298 +++++++++++++++++----
.../org/apache/commons/numbers/gamma/Erfcx.java | 55 ++++
.../apache/commons/numbers/gamma/BoostErfTest.java | 268 +++++++++++++++++-
.../apache/commons/numbers/gamma/ErfcxTest.java | 78 ++++++
.../commons/numbers/gamma/erfcx_huge_data.csv | 159 +++++++++++
.../commons/numbers/gamma/erfcx_large_data.csv | 105 ++++++++
.../commons/numbers/gamma/erfcx_medium_data.csv | 190 +++++++++++++
.../commons/numbers/gamma/erfcx_neg_data.csv | 224 ++++++++++++++++
.../numbers/gamma/erfcx_neg_medium_data.csv | 188 +++++++++++++
.../commons/numbers/gamma/erfcx_neg_small_data.csv | 158 +++++++++++
.../commons/numbers/gamma/erfcx_small_data.csv | 158 +++++++++++
src/changes/changes.xml | 3 +
12 files changed, 1833 insertions(+), 51 deletions(-)
diff --git a/commons-numbers-gamma/src/main/java/org/apache/commons/numbers/gamma/BoostErf.java b/commons-numbers-gamma/src/main/java/org/apache/commons/numbers/gamma/BoostErf.java
index e6a51c3..e8d3790 100644
--- a/commons-numbers-gamma/src/main/java/org/apache/commons/numbers/gamma/BoostErf.java
+++ b/commons-numbers-gamma/src/main/java/org/apache/commons/numbers/gamma/BoostErf.java
@@ -28,7 +28,11 @@ package org.apache.commons.numbers.gamma;
*
* <p>This code has been adapted from the <a href="https://www.boost.org/">Boost</a>
* {@code c++} implementation {@code <boost/math/special_functions/erf.hpp>}.
- * All work is copyright John Maddock 2006 and subject to the Boost Software License.
+ * The erf/erfc functions and their inverses are copyright John Maddock 2006 and subject to
+ * the Boost Software License.
+ *
+ * <p>Additions made to support the erfcx function are original work under the Apache software
+ * license.
*
* @see
* <a href="https://www.boost.org/doc/libs/1_77_0/libs/math/doc/html/math_toolkit/sf_erf/error_function.html">
@@ -42,6 +46,23 @@ final class BoostErf {
* and the multiplier is {@code 2^27 + 1}.
*/
private static final double MULTIPLIER = 1.0 + 0x1.0p27;
+ /** 1 / sqrt(pi). Used for the scaled complementary error function erfcx. */
+ private static final double ONE_OVER_ROOT_PI = 0.5641895835477562869480794515607725858;
+ /** Threshold for the scaled complementary error function erfcx
+ * where the approximation {@code (1 / sqrt(pi)) / x} can be used. */
+ private static final double ERFCX_APPROX = 6.71e7;
+ /** Threshold for the erf implementation for |x| where the computation
+ * uses {@code erf(x)}; otherwise {@code erfc(x)} is computed. The final result is
+ * achieved by suitable application of symmetry. */
+ private static final double COMPUTE_ERF = 0.5;
+ /** Threshold for the scaled complementary error function erfcx for negative x
+ * where {@code 2 * exp(x*x)} will overflow. Value is 26.62873571375149. */
+ private static final double ERFCX_NEG_X_MAX = Math.sqrt(Math.log(Double.MAX_VALUE / 2));
+ /** Threshold for the scaled complementary error function erfcx for x
+ * where {@code exp(x*x) == 1; x <= t}. Value is (1 + 5/16) * 2^-27 = 9.778887033462524E-9.
+ * <p>Note: This is used for performance. If set to 0 then the result is computed
+ * using expm1(x*x) with the same final result. */
+ private static final double EXP_XX_1 = 0x1.5p-27;
/** Private constructor. */
private BoostErf() {
@@ -60,14 +81,21 @@ final class BoostErf {
// - Explicitly inline the polynomial function evaluation
// using Horner's method (https://en.wikipedia.org/wiki/Horner%27s_method)
// - Support odd function for f(0.0) = -f(-0.0)
+ // - Support the scaled complementary error function erfcx
// Erf:
// - Change extended precision z*z to compute the square round-off
// using Dekker's method
- // - Change the erf threshold for z when p=1 from
+ // - Change extended precision exp(-z*z) to compute using a
+ // round-off addition to the standard exp result (see NUMBERS-177)
+ // - Change the erf threshold for z when erf(z)=1 from
// z > 5.8f to z > 5.930664
- // - Change edge case detection for integer z
+ // - Change the erfc threshold for z when erfc(z)=0 from
+ // z < 28 to z < 27.3
+ // - Change rational function approximation for z > 4 to a function
+ // suitable for erfcx (see NUMBERS-177)
// Inverse erf:
// - Change inverse erf edge case detection to include NaN
+ // - Change edge case detection for integer z
//
// Note:
// Constants using the 'f' suffix are machine
@@ -82,7 +110,7 @@ final class BoostErf {
* @return the complementary error function.
*/
static double erfc(double x) {
- return erfImp(x, true);
+ return erfImp(x, true, false);
}
/**
@@ -92,28 +120,34 @@ final class BoostErf {
* @return the error function.
*/
static double erf(double x) {
- return erfImp(x, false);
+ return erfImp(x, false, false);
}
/**
* 53-bit implementation for the error function.
*
+ * <p>Note: The {@code scaled} flag only applies when
+ * {@code z >= 0.5} and {@code invert == true}.
+ * This functionality is used to compute erfcx(z) for positive z.
+ *
* @param z Point to evaluate
* @param invert true to invert the result (for the complementary error function)
+ * @param scaled true to compute the scaled complementary error function
* @return the error function result
*/
- private static double erfImp(double z, boolean invert) {
+ private static double erfImp(double z, boolean invert, boolean scaled) {
if (Double.isNaN(z)) {
return Double.NaN;
}
if (z < 0) {
+ // Here the scaled flag is ignored.
if (!invert) {
- return -erfImp(-z, invert);
+ return -erfImp(-z, invert, false);
} else if (z < -0.5) {
- return 2 - erfImp(-z, invert);
+ return 2 - erfImp(-z, invert, false);
} else {
- return 1 + erfImp(-z, false);
+ return 1 + erfImp(-z, false, false);
}
}
@@ -124,10 +158,11 @@ final class BoostErf {
// which implementation to use,
// try to put most likely options first:
//
- if (z < 0.5) {
+ if (z < COMPUTE_ERF) {
//
// We're going to calculate erf:
//
+ // Here the scaled flag is ignored.
if (z < 1e-10) {
if (z == 0) {
result = z;
@@ -157,11 +192,13 @@ final class BoostErf {
Q = 1.0 + Q * zz;
result = z * (Y + P / Q);
}
- // Note: Boost threshold of 5.8f has been raised to approximately 5.93 (6073 / 1024)
- } else if (invert ? (z < 28) : (z < 5.9306640625f)) {
+ // Note: Boost threshold of 5.8f has been raised to approximately 5.93 (6073 / 1024);
+ // threshold of 28 has been lowered to approximately 27.3 (6989/256) where exp(-z*z) = 0.
+ } else if (scaled || (invert ? (z < 27.300781f) : (z < 5.9306640625f))) {
//
// We'll be calculating erfc:
//
+ // Here the scaled flag is used.
invert = !invert;
if (z < 1.5f) {
// Maximum Deviation Found: 3.702e-17
@@ -186,7 +223,11 @@ final class BoostErf {
Q = 1.84759070983002217845 + Q * zm;
Q = 1.0 + Q * zm;
result = Y + P / Q;
- result *= Math.exp(-z * z) / z;
+ if (scaled) {
+ result /= z;
+ } else {
+ result *= expmxx(z) / z;
+ }
} else if (z < 2.5f) {
// Max Error found at double precision = 6.599585e-18
// Maximum Deviation Found: 3.909e-18
@@ -209,10 +250,14 @@ final class BoostErf {
Q = 1.53991494948552447182 + Q * zm;
Q = 1.0 + Q * zm;
result = Y + P / Q;
- final double sq = z * z;
- final double errSqr = squareLowUnscaled(z, sq);
- result *= Math.exp(-sq) * Math.exp(-errSqr) / z;
- } else if (z < 4.5f) {
+ if (scaled) {
+ result /= z;
+ } else {
+ result *= expmxx(z) / z;
+ }
+ // Lowered Boost threshold from 4.5 to 4.0 as this is the limit
+ // for the Cody erfc approximation
+ } else if (z < 4.0f) {
// Maximum Deviation Found: 1.512e-17
// Expected Error Term: 1.512e-17
// Maximum Relative Change in Control Points: 2.222e-04
@@ -234,46 +279,62 @@ final class BoostErf {
Q = 1.04217814166938418171 + Q * zm;
Q = 1.0 + Q * zm;
result = Y + P / Q;
- final double sq = z * z;
- final double errSqr = squareLowUnscaled(z, sq);
- result *= Math.exp(-sq) * Math.exp(-errSqr) / z;
+ if (scaled) {
+ result /= z;
+ } else {
+ result *= expmxx(z) / z;
+ }
} else {
- // Max Error found at double precision = 2.997958e-17
- // Maximum Deviation Found: 2.860e-17
- // Expected Error Term: 2.859e-17
- // Maximum Relative Change in Control Points: 1.357e-05
- final double Y = 0.5579090118408203125f;
- final double iz = 1 / z;
- double P;
- P = -2.8175401114513378771;
- P = -3.22729451764143718517 + P * iz;
- P = -2.5518551727311523996 + P * iz;
- P = -0.687717681153649930619 + P * iz;
- P = -0.212652252872804219852 + P * iz;
- P = 0.0175389834052493308818 + P * iz;
- P = 0.00628057170626964891937 + P * iz;
- double Q;
- Q = 5.48409182238641741584;
- Q = 13.5064170191802889145 + Q * iz;
- Q = 22.9367376522880577224 + Q * iz;
- Q = 15.930646027911794143 + Q * iz;
- Q = 11.0567237927800161565 + Q * iz;
- Q = 2.79257750980575282228 + Q * iz;
- Q = 1.0 + Q * iz;
- result = Y + P / Q;
- final double sq = z * z;
- final double errSqr = squareLowUnscaled(z, sq);
- result *= Math.exp(-sq) * Math.exp(-errSqr) / z;
+ // Rational function approximation for erfc(x > 4.0)
+ //
+ // This approximation is not the Boost implementation.
+ // The Boost function is suitable for [4.5 < z < 28].
+ //
+ // This function is suitable for erfcx(z) as it asymptotes
+ // to (1 / sqrt(pi)) / z at large z.
+ //
+ // Taken from "Rational Chebyshev approximations for the error function"
+ // by W. J. Cody, Math. Comp., 1969, PP. 631-638.
+ //
+ // See NUMBERS-177.
+
+ final double izz = 1 / (z * z);
+ double p;
+ p = 1.63153871373020978498e-2;
+ p = 3.05326634961232344035e-1 + p * izz;
+ p = 3.60344899949804439429e-1 + p * izz;
+ p = 1.25781726111229246204e-1 + p * izz;
+ p = 1.60837851487422766278e-2 + p * izz;
+ p = 6.58749161529837803157e-4 + p * izz;
+ double q;
+ q = 1;
+ q = 2.56852019228982242072e00 + q * izz;
+ q = 1.87295284992346047209e00 + q * izz;
+ q = 5.27905102951428412248e-1 + q * izz;
+ q = 6.05183413124413191178e-2 + q * izz;
+ q = 2.33520497626869185443e-3 + q * izz;
+
+ result = izz * p / q;
+ result = (ONE_OVER_ROOT_PI - result) / z;
+
+ if (!scaled) {
+ // exp(-z*z) can be sub-normal so
+ // multiply by any sub-normal after divide by z
+ result *= expmxx(z);
+ }
}
} else {
//
- // Any value of z larger than 28 will underflow to zero:
+ // Any value of z larger than 27.3 will underflow to zero:
//
result = 0;
invert = !invert;
}
if (invert) {
+ // Note: If 0.5 <= z < 28 and the scaled flag is true then
+ // invert will have been flipped to false and the
+ // the result is unchanged as erfcx(z)
result = 1 - result;
}
@@ -281,6 +342,70 @@ final class BoostErf {
}
/**
+ * Returns the scaled complementary error function.
+ * <pre>
+ * erfcx(x) = exp(x^2) * erfc(x)
+ * </pre>
+ *
+ * @param x the value.
+ * @return the scaled complementary error function.
+ */
+ static double erfcx(double x) {
+ if (Double.isNaN(x)) {
+ return Double.NaN;
+ }
+
+ // For |z| < 0.5 erfc is computed using erf
+ final double ax = Math.abs(x);
+ if (ax < COMPUTE_ERF) {
+ // Use the erf(x) result.
+ // (1 - erf(x)) * exp(x*x)
+
+ final double erfx = erf(x);
+ if (ax < EXP_XX_1) {
+ // No exponential required
+ return 1 - erfx;
+ }
+
+ // exp(x*x) - exp(x*x) * erf(x)
+ // Avoid use of exp(x*x) with expm1:
+ // exp(x*x) - 1 - (erf(x) * (exp(x*x) - 1)) - erf(x) + 1
+
+ // Sum small to large: |erf(x)| > expm1(x*x)
+ // -erf(x) * expm1(x*x) + expm1(x*x) - erf(x) + 1
+ // Negative x: erf(x) < 0, summed terms are positive, no cancellation occurs.
+ // Positive x: erf(x) > 0 so cancellation can occur.
+ // When terms are ordered by absolute magnitude the magnitude of the next term
+ // is above the round-off from adding the previous term to the sum. Thus
+ // cancellation is negligible compared to errors in the largest computed term (erf(x)).
+
+ final double em1 = Math.expm1(x * x);
+ return -erfx * em1 + em1 - erfx + 1;
+ }
+
+ // Handle negative arguments
+ if (x < 0) {
+ // erfcx(x) = 2*exp(x*x) - erfcx(-x)
+
+ if (x < -ERFCX_NEG_X_MAX) {
+ // Overflow
+ return Double.POSITIVE_INFINITY;
+ }
+
+ final double e = expxx(x);
+ return e - erfImp(-x, true, true) + e;
+ }
+
+ // Approximation for large positive x
+ if (x > ERFCX_APPROX) {
+ return ONE_OVER_ROOT_PI / x;
+ }
+
+ // Compute erfc scaled
+ return erfImp(x, true, true);
+ }
+
+ /**
* Returns the inverse complementary error function.
*
* @param z Value (in {@code [0, 2]}).
@@ -589,6 +714,85 @@ final class BoostErf {
return result;
}
+ /**
+ * Compute {@code exp(x*x)} with high accuracy. This is performed using
+ * information in the round-off from {@code x*x}.
+ *
+ * <p>This is accurate at large x to 1 ulp.
+ *
+ * <p>At small x the accuracy cannot be improved over using exp(x*x).
+ * This occurs at {@code x <= 1}.
+ *
+ * <p>Warning: This has no checks for overflow. The method is never called
+ * when {@code x*x > log(MAX_VALUE/2)}.
+ *
+ * @param x Value
+ * @return exp(x*x)
+ */
+ static double expxx(double x) {
+ // Note: If exp(a) overflows this can create NaN if the
+ // round-off b is negative or zero:
+ // exp(a) * exp1m(b) + exp(a)
+ // inf * 0 + inf or inf * -b + inf
+ final double a = x * x;
+ final double b = squareLowUnscaled(x, a);
+ return expxx(a, b);
+ }
+
+ /**
+ * Compute {@code exp(-x*x)} with high accuracy. This is performed using
+ * information in the round-off from {@code x*x}.
+ *
+ * <p>This is accurate at large x to 1 ulp until exp(-x*x) is close to
+ * sub-normal. For very small exp(-x*x) the adjustment is sub-normal and
+ * bits can be lost in the adjustment for a max observed error of {@code < 2} ulp.
+ *
+ * <p>At small x the accuracy cannot be improved over using exp(-x*x).
+ * This occurs at {@code x <= 1}.
+ *
+ * @param x Value
+ * @return exp(-x*x)
+ */
+ static double expmxx(double x) {
+ final double a = x * x;
+ final double b = squareLowUnscaled(x, a);
+ return expxx(-a, -b);
+ }
+
+ /**
+ * Compute {@code exp(a+b)} with high accuracy assuming {@code a+b = a}.
+ *
+ * <p>This is accurate at large positive a to 1 ulp. If a is negative and exp(a) is
+ * close to sub-normal a bit of precision may be lost when adjusting result
+ * as the adjustment is sub-normal (max observed error {@code < 2} ulp).
+ * For the use case of multiplication of a number less than 1 by exp(-x*x), a = -x*x,
+ * the result will be sub-normal and the rounding error is lost.
+ *
+ * <p>At small |a| the accuracy cannot be improved over using exp(a) as the
+ * round-off is too small to create terms that can adjust the standard result by
+ * more than 0.5 ulp. This occurs at {@code |a| <= 1}.
+ *
+ * @param a High bits of a split number
+ * @param b Low bits of a split number
+ * @return exp(a+b)
+ */
+ private static double expxx(double a, double b) {
+ // exp(a+b) = exp(a) * exp(b)
+ // = exp(a) * (exp(b) - 1) + exp(a)
+ // Assuming:
+ // 1. -746 < a < 710 for no under/overflow of exp(a)
+ // 2. a+b = a
+ // As b -> 0 then exp(b) -> 1; expm1(b) -> b
+ // The round-off b is limited to ~ 0.5 * ulp(746) ~ 5.68e-14
+ // and we can use an approximation for expm1 (x/1! + x^2/2! + ...)
+ // The second term is required for the expm1 result but the
+ // bits are not significant to change the product with exp(a)
+
+ final double ea = Math.exp(a);
+ // b ~ expm1(b)
+ return ea * b + ea;
+ }
+
// Extended precision multiplication specialised for the square adapted from:
// org.apache.commons.numbers.core.ExtendedPrecision
diff --git a/commons-numbers-gamma/src/main/java/org/apache/commons/numbers/gamma/Erfcx.java b/commons-numbers-gamma/src/main/java/org/apache/commons/numbers/gamma/Erfcx.java
new file mode 100644
index 0000000..ca8b907
--- /dev/null
+++ b/commons-numbers-gamma/src/main/java/org/apache/commons/numbers/gamma/Erfcx.java
@@ -0,0 +1,55 @@
+/*
+ * 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;
+
+/**
+ * Scaled complementary error function.
+ *
+ * <p>\[ \operatorname{erfcx}(z) = \operatorname{erfc}(z)\ e^{z^2} \]
+ *
+ * <p>For large z the value is approximately:
+ *
+ * <p>\[ \operatorname{erfcx}(z) = \frac{1}{z \sqrt{\pi}} \]
+ *
+ * @see Erfc
+ * @since 1.1
+ */
+public final class Erfcx {
+ /** Private constructor. */
+ private Erfcx() {
+ // intentionally empty.
+ }
+
+ /**
+ * Returns the scaled complementary error function.
+ *
+ * <p>Special cases:
+ * <ul>
+ * <li>If the argument is 0, then the result is 1.
+ * <li>If the argument is +infinity, then the result is 0.
+ * <li>If the argument is negative and {@code exp(z*z)} is infinite, then the result is +infinity.
+ * <li>If the argument is nan, then the result is nan.
+ * </ul>
+ *
+ * @param x Value.
+ * @return the scaled complementary error function.
+ */
+ public static double value(double x) {
+ return BoostErf.erfcx(x);
+ }
+}
+
diff --git a/commons-numbers-gamma/src/test/java/org/apache/commons/numbers/gamma/BoostErfTest.java b/commons-numbers-gamma/src/test/java/org/apache/commons/numbers/gamma/BoostErfTest.java
index 5fd4e78..da7bd0e 100644
--- a/commons-numbers-gamma/src/test/java/org/apache/commons/numbers/gamma/BoostErfTest.java
+++ b/commons-numbers-gamma/src/test/java/org/apache/commons/numbers/gamma/BoostErfTest.java
@@ -49,7 +49,8 @@ class BoostErfTest {
ERF(BoostErf::erf, true),
ERFC(BoostErf::erfc, false),
ERF_INV(BoostErf::erfInv, true),
- ERFC_INV(BoostErf::erfcInv, false);
+ ERFC_INV(BoostErf::erfcInv, false),
+ ERFCX(BoostErf::erfcx, false);
/** The function. */
private final DoubleUnaryOperator fun;
@@ -112,13 +113,13 @@ class BoostErfTest {
*/
private enum TestCase {
/** Erf Boost data. */
- ERF(TestFunction.ERF, 1.4, 0.25),
+ ERF(TestFunction.ERF, 1.3, 0.2),
/** Erfc Boost data. */
ERFC(TestFunction.ERFC, 2.2, 0.5),
/** Erf Boost large data (simple case where all z>8, p=1.0). */
ERF_LARGE(TestFunction.ERF, 0, 0.0),
/** Erfc Boost large data. */
- ERFC_LARGE(TestFunction.ERFC, 1.99, 0.75),
+ ERFC_LARGE(TestFunction.ERFC, 1.75, 0.7),
/** Erf Boost small data (no exponentiation required). */
ERF_SMALL(TestFunction.ERF, 1.2, 0.25),
/** Erfc Boost small data (no exponentiation required: ulp=0). */
@@ -132,7 +133,19 @@ class BoostErfTest {
/** Inverse Erf limit data. */
ERF_INV_LIMIT(TestFunction.ERF_INV, 1.4, 0.5),
/** Inverse Erfc limit data. */
- ERFC_INV_LIMIT(TestFunction.ERFC_INV, 1.2, 0.5);
+ ERFC_INV_LIMIT(TestFunction.ERFC_INV, 1.2, 0.5),
+ /** Erfcx negative medium data. */
+ ERFCX_NEG_MEDIUM(TestFunction.ERFCX, 1.7, 0.55),
+ /** Erfcx negative small data. */
+ ERFCX_NEG_SMALL(TestFunction.ERFCX, 0.7, 0.061),
+ /** Erfcx small data. */
+ ERFCX_SMALL(TestFunction.ERFCX, 0.6, 0.073),
+ /** Erfcx medium data. */
+ ERFCX_MED(TestFunction.ERFCX, 1.2, 0.5),
+ /** Erfcx large data. */
+ ERFCX_LARGE(TestFunction.ERFCX, 1.1, 0.45),
+ /** Erfcx huge data. */
+ ERFCX_HUGE(TestFunction.ERFCX, 1.25, 0.45);
/** Sum of the squared ULP error and count n. */
private final TestUtils.ErrorStatistics stats = new TestUtils.ErrorStatistics();
@@ -209,6 +222,111 @@ class BoostErfTest {
@ParameterizedTest
@CsvSource({
+ "0, 1",
+ // Reference data from GCC libquadmath expq(z*z)
+ // Cases from known max ulp errors for:
+ // a = z*z; b = z*z round-off
+ // exp(a+b) = exp(a) * exp(b) = exp(a) * expm1(b) + exp(a)
+ "0.042876182518572857, 1.00184005785999452616485960307584103",
+ "0.12332761808971203, 1.01532595755115384513226892929687518",
+ "0.23492402237866017, 1.05674063282543772769083554593703693",
+ "0.5005007866910639, 1.28466892274154983265904587935363446",
+ "0.6079329058333609, 1.44713019294681595655835657289687865",
+ "0.70256943723915932, 1.6382093968128949474570450121807659",
+ "0.82241395271131823, 1.96671514042639717092581737392007089",
+ "1.1676941591314409, 3.90989159781783456283136476225348036",
+ "1.2872328693421708, 5.24339117555116565213868516013500793",
+ "1.763493621168525, 22.4190210354395450203564424019491017",
+ "2.2911777554159483, 190.470153340296021320752950876036293",
+ "2.6219566826446719, 967.443326246348231629933262201707317",
+ "3.0037511703473512, 8287.64469163961117832508854092931197",
+ "4.9612493700241904, 48946578790.7631709834358118976268355",
+ "9.3076562180806182, 4.20727779143532487527563444483221012e+37",
+ "14.323050579907544, 1.24570874136900582743589942668130243e+89",
+ "19.789244604728378, 1.19093202729437857229577485425178318e+170",
+ "25.264725343479071, 1.63276667418921707391087152900497326e+277",
+ "26.471453083170552, 2.12115354204783587764203142231215649e+304",
+ "26.628235718509234, 8.7522773171090885282464768100291983e+307",
+ // Limit.
+ // The next double up (26.64174755704633) has
+ // exp(x*x) = 1.79769313486244732925408679719502907e+308 = Infinity
+ // This is computed as NaN due to lack of overflow checks.
+ "26.641747557046327, 1.79769313486210702414246567679413232e+308",
+ })
+ void testExpxx(double z, BigDecimal expected) {
+ final double actual = BoostErf.expxx(z);
+ TestUtils.assertEquals(expected, actual, 1.0);
+ }
+
+ @ParameterizedTest
+ @CsvSource({
+ // Known cases where exp(x*x) is infinite and the round-off is zero or negative
+ "27, 3.98728526204259656354686104733435016e+316",
+ "27.45162436, 1.79769313486244732925408679719502907e+308",
+ // Next double after 26.641747557046327
+ "26.64174755704633, 1.79769313486244732925408679719502907e+308",
+ })
+ void testExpxxOverflow(double z, double expected) {
+ final double e = Math.exp(z * z);
+ Assertions.assertEquals(Double.POSITIVE_INFINITY, e);
+ Assertions.assertEquals(Double.POSITIVE_INFINITY, expected);
+ Assertions.assertEquals(Double.NaN, BoostErf.expxx(z), "No overflow checks");
+ }
+
+ @ParameterizedTest
+ @CsvSource({
+ "0, 1",
+ // Reference data from GCC libquadmath expq(-z*z)
+ // Cases from known max ulp errors for:
+ // a = z*z; b = z*z round-off
+ // exp(a+b) = exp(a) * exp(b) = exp(a) * expm1(b) + exp(a)
+ "0.018888546570790723, 0.999643286445856926713199201695735708",
+ "0.044040879540028978, 0.998062280736064566626960146268218869",
+ "0.14252706610502067, 0.979890973955195922461144945194213951",
+ "0.21644880196960639, 0.954230441398827896658935167789523746",
+ "0.38611178653477424, 0.861498200598095608584361373395644347",
+ "0.48918821106359167, 0.78717467413707227187574762653633039",
+ "0.69762286071744906, 0.614665134758267412066486574992456239",
+ "0.74212965000887676, 0.576513560506426682539710563714000483",
+ "0.83549504832885912, 0.497553606822144444881962542058093636",
+ "0.93551572137564831, 0.416782963065511995785537520046193978",
+ "1.1557980830683314, 0.262929535430060880774643925549236367",
+ "1.4043799106718902, 0.139138848619574074162424456941838826",
+ "1.76928217755289, 0.0437020868591733321664690441540917611",
+ "3.030043930000375, 0.000102960388874918024753918581478636755",
+ "6.8941333932059292, 2.28236391243884653067874490191517352e-21",
+ "12.150966601450184, 7.55373161692552909052687379129491314e-65",
+ "18.145618649215113, 1.00621134630738394627962504745954853e-143",
+ "22.58677686154909, 2.74945208846365015258084878726170577e-222",
+ "25.307715003447811, 6.96433586576952601048539830338695099e-279",
+ "26.314662520770881, 1.85270995749302588464727454552962812e-301",
+ "26.550566379026709, 7.1067746279803211201902681713374704e-307",
+ // Limit
+ "27.297128403953796, 2.47032822920651974943004172315065178e-324",
+ // exp(-z*z) = 0
+ "27.2971284039538, 2.47032822920604061009296400001395168e-324",
+ })
+ void testExpmxx(double z, BigDecimal expected) {
+ final double actual = BoostErf.expmxx(z);
+ TestUtils.assertEquals(expected, actual, 1.0);
+ }
+
+ @ParameterizedTest
+ @CsvSource({
+ // Reference data from GCC libquadmath expq(z)
+ // exp(z) is nearly sub-normal
+ "26.589967471163597, 8.75686990774305433076998380040492953e-308",
+ "26.592726991055095, 7.56157741629803260538530412232841961e-308",
+ "26.593224983876986, 7.36392883240998076191982506525092463e-308",
+ "26.596608821043414, 6.15095812619805721349676447899566968e-308",
+ })
+ void testExpmxxCloseToSubNormal(double z, BigDecimal expected) {
+ final double actual = BoostErf.expmxx(z);
+ TestUtils.assertEquals(expected, actual, 1.3);
+ }
+
+ @ParameterizedTest
+ @CsvSource({
"-Infinity, -1",
"Infinity, 1",
"0, 0",
@@ -475,6 +593,148 @@ class BoostErfTest {
assertRms(name, data, rmsUlp);
}
+ @ParameterizedTest
+ @CsvSource({
+ "NaN, NaN",
+ "0, 1",
+ "-0.0, 1",
+ "1e-100, 1",
+ // expected 1.0000000000000002
+ "-2.220446049250313E-16, 1.0000000000000002505505",
+ // expected 1.0
+ "-1.1102230246251565E-16, 1.000000000000000125275",
+ "1.734723475976807e-17, 0.99999999999999998042574169",
+ // expected 0.9999999999999999
+ "5.551115123125783e-17, 0.99999999999999993736237340916",
+ // max value
+ "1.7976931348623157e308, 3.13840873398544321279297017922274491e-309",
+ "Infinity, 0",
+ "-27, Infinity",
+ "-26.62873571375149, 1.7976931348622485388617592502115433e+308",
+ // This is infinite as a double: matlab erfcx = Inf
+ "-26.628735713751492, 1.79769313486258867776818776259144527e+308",
+ })
+ void testErfcxEdgeCases(double z, double expected) {
+ final double actual = BoostErf.erfcx(z);
+ // Workaround for NaN not having a valid ulp
+ if (Double.isFinite(expected)) {
+ Assertions.assertEquals(expected, actual, Math.ulp(expected));
+ } else {
+ Assertions.assertEquals(expected, actual);
+ }
+ }
+
+ @ParameterizedTest
+ @CsvSource({
+ // matlab: fprintf("%.17g\n, erfcx(z))
+ "-24.356, 8.5293595881160216e+257",
+ "-12.34, 2.7132062210034015e+66",
+ "-6.89, 8.2775358436372447e+20",
+ "-1.11134, 6.4783098090861095",
+ "-0.67868, 2.6356650381821858",
+ "-0.1234, 1.156008270595601",
+ "0.1234, 0.87467990946395457",
+ "0.2836, 0.74601996202714793",
+ "0.7521364, 0.5061525663103037",
+ "1.678, 0.2947001506216585",
+ "2.67868, 0.19827490572168827",
+ "5.6788, 0.09787639472753934",
+ "10.67182, 0.052638121464397732",
+ "15.678, 0.035913308816213706",
+ "23.975, 0.023511995366729203",
+ "26.8989, 0.020959983993738648",
+ "27.1678, 0.020752808901245822",
+ "27.789, 0.020289502724156101",
+ "28.4567, 0.019814028635674531",
+ "33.67868, 0.016744753846685077",
+ "56.567, 0.0099722712078122132",
+ "101.101, 0.0055801820870247853",
+ "234.7, 0.0024038536962965006",
+ "678658.678, 8.3133039013043281e-07",
+ })
+ void testErfcxSpotTests(double z, double expected) {
+ final double actual = BoostErf.erfcx(z);
+ Assertions.assertEquals(expected, actual, Math.ulp(expected));
+ }
+
+ @ParameterizedTest
+ @Order(1)
+ @CsvFileSource(resources = "erfcx_neg_medium_data.csv")
+ void testErfcxNegMedium(double z, BigDecimal expected) {
+ assertErf(TestCase.ERFCX_NEG_MEDIUM, z, expected);
+ }
+
+ @Test
+ @Order(3010)
+ void testErfcxNegMediumRMS() {
+ assertRms(TestCase.ERFCX_NEG_MEDIUM);
+ }
+
+ @ParameterizedTest
+ @Order(1)
+ @CsvFileSource(resources = "erfcx_neg_small_data.csv")
+ void testErfcxNegSmall(double z, BigDecimal expected) {
+ assertErf(TestCase.ERFCX_NEG_SMALL, z, expected);
+ }
+
+ @Test
+ @Order(3020)
+ void testErfcxNegSmallRMS() {
+ assertRms(TestCase.ERFCX_NEG_SMALL);
+ }
+
+ @ParameterizedTest
+ @Order(1)
+ @CsvFileSource(resources = "erfcx_small_data.csv")
+ void testErfcxSmall(double z, BigDecimal expected) {
+ assertErf(TestCase.ERFCX_SMALL, z, expected);
+ }
+
+ @Test
+ @Order(3030)
+ void testErfcxSmallRMS() {
+ assertRms(TestCase.ERFCX_SMALL);
+ }
+
+ @ParameterizedTest
+ @Order(1)
+ @CsvFileSource(resources = "erfcx_medium_data.csv")
+ void testErfcxMedium(double z, BigDecimal expected) {
+ assertErf(TestCase.ERFCX_MED, z, expected);
+ }
+
+ @Test
+ @Order(3040)
+ void testErfcxMediumRMS() {
+ assertRms(TestCase.ERFCX_MED);
+ }
+
+ @ParameterizedTest
+ @Order(1)
+ @CsvFileSource(resources = "erfcx_large_data.csv")
+ void testErfcxLarge(double z, BigDecimal expected) {
+ assertErf(TestCase.ERFCX_LARGE, z, expected);
+ }
+
+ @Test
+ @Order(3050)
+ void testErfcxLargeRMS() {
+ assertRms(TestCase.ERFCX_LARGE);
+ }
+
+ @ParameterizedTest
+ @Order(1)
+ @CsvFileSource(resources = "erfcx_huge_data.csv")
+ void testErfcxHuge(double z, BigDecimal expected) {
+ assertErf(TestCase.ERFCX_HUGE, z, expected);
+ }
+
+ @Test
+ @Order(3060)
+ void testErfcxHugeRMS() {
+ assertRms(TestCase.ERFCX_HUGE);
+ }
+
/**
* Assert the function using extended precision.
*
diff --git a/commons-numbers-gamma/src/test/java/org/apache/commons/numbers/gamma/ErfcxTest.java b/commons-numbers-gamma/src/test/java/org/apache/commons/numbers/gamma/ErfcxTest.java
new file mode 100644
index 0000000..a877255
--- /dev/null
+++ b/commons-numbers-gamma/src/test/java/org/apache/commons/numbers/gamma/ErfcxTest.java
@@ -0,0 +1,78 @@
+/*
+ * 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 org.junit.jupiter.api.Assertions;
+import org.junit.jupiter.params.ParameterizedTest;
+import org.junit.jupiter.params.provider.CsvSource;
+
+/**
+ * Tests for {@link Erfcx}.
+ */
+class ErfcxTest {
+ @ParameterizedTest
+ @CsvSource({
+ "NaN, NaN",
+ "0, 1",
+ "Infinity, 0",
+ "-27, Infinity",
+ })
+ void testErfcxEdgeCases(double z, double expected) {
+ Assertions.assertEquals(expected, Erfcx.value(z));
+ }
+
+ /**
+ * Compare erfcx against reference values computed using matlab.
+ */
+ @ParameterizedTest
+ @CsvSource({
+ // Positives
+ "0.123, 0.87504503469728478",
+ "0.356, 0.69768949677109249",
+ "0.598, 0.56869994421241077",
+ "1.12, 0.39689157691768223",
+ "2.54, 0.20787186512579933",
+ "4.78, 0.11560177456745414",
+ "8.89, 0.06306928075747556",
+ "16.133, 0.034904353289554449",
+ "25.677, 0.021955940669412959",
+ "32.123, 0.017554917112870515",
+ "64.156, 0.0087929589162697615",
+ "128.1256, 0.00440327616930610247",
+ "256.245, 0.0022017416435368476",
+ "512.123, 0.0011016660215700644",
+ "1024.1211, 0.0005509009770282079",
+ "1.87e5, 3.0170565964689267e-06",
+ "1.23e6, 4.5869071833135764e-07",
+ "6.23e7, 9.0560125770105324e-09",
+ "2.34e8, 2.4110665963579326e-09",
+ "1.7976931348623157E308, 3.1384087339854447e-309",
+ // Negatives require accurate computation of exp(z^2)
+ "-0.123, 1.1554430105980469",
+ "-0.356, 1.5725451177335525",
+ "-0.598, 2.2910985922662546",
+ "-1.12, 6.6145770282875613",
+ "-2.54, 1267.2229816784452",
+ "-4.78, 16748041793.844347",
+ "-8.89, 4.2095426970353675e+34",
+ "-16.133, 2.1699859052022138e+113",
+ "-25.677, 4.3151823545771249e+286",
+ })
+ void testErfcx(double z, double expected) {
+ Assertions.assertEquals(expected, Erfcx.value(z), Math.ulp(expected));
+ }
+}
diff --git a/commons-numbers-gamma/src/test/resources/org/apache/commons/numbers/gamma/erfcx_huge_data.csv b/commons-numbers-gamma/src/test/resources/org/apache/commons/numbers/gamma/erfcx_huge_data.csv
new file mode 100755
index 0000000..fb1ff0e
--- /dev/null
+++ b/commons-numbers-gamma/src/test/resources/org/apache/commons/numbers/gamma/erfcx_huge_data.csv
@@ -0,0 +1,159 @@
+# 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.
+
+# Random data using 52-bit mantissa scaled with 2^b.
+# Generated in blocks of 10 with b:
+# {-50:5:-5} {-4:1:-1} {0,0,0} {1,1,1} {2,2,2} {3,3,3}
+# {4,4,4,4} {5,5,5,5} {6,6,6,6} {8:2:28}
+# An additional 10 positive test points generated in the range
+# [26.3, 27.3] where erfc(x) transitions to sub-normal.
+# Reference values from customized Boost C++ 128-bit implementation
+# of erfc modifed to remove scaling by exp(-x^2). Values for large x
+# generated using the upper_gamma_fraction(...) continued fraction
+# See: https://mathworld.wolfram.com/Erfc.html Eq.3
+
+# Partition: 100 <= z
+# z, erfcx(z)
+102.29952077160934,0.00551481206974884676446875286320606304
+103.50017201936511,0.00545084363111614798636261825463792064
+103.58396304142204,0.00544643475187555024591078495502437022
+105.4230044810421,0.00535143356841716843901531292236915805
+105.5609746650351,0.00534443977181417466332969500524315112
+106.42113607755707,0.00530124653443092598862333303403629019
+108.03229628904964,0.00522219220926894822590339118923021224
+108.16026285278699,0.00521601425786995434907993750413885907
+109.06091689498882,0.00517294266142791156920201386171246745
+109.14121820310775,0.00516913695730079816350658506310362864
+110.72131298270907,0.00509537468728679300565610449495526792
+113.26841264500692,0.00498080255000802037240193343782011139
+114.89406925601759,0.00491033366525260497335914091987985151
+115.63063949639751,0.00487905705603983364464237577891180739
+115.68588725972278,0.00487672715346130108939230992266420234
+117.18136887657732,0.00481449438195476545482630095240549111
+119.14209812058073,0.00473526756280483156886008696843067396
+120.38323500797867,0.00468645090105367705171643131620436972
+120.41023568103655,0.00468540008838105233260995906005558465
+126.88534637716943,0.00444631377660281965906859967330572902
+127.72124544528185,0.00441721571249154135488633281918770819
+277.00614398390172,0.00203672705291104777583204844776911297
+340.41023941371702,0.00165737420285223247808389825662612108
+353.60693843856711,0.00159552108901264796377973576891841113
+379.24468349357852,0.00148766125609616685305692882930632355
+389.67535179416376,0.00144784042205956146381612909598924147
+407.31665019254052,0.00138513336731049367042784898180518142
+455.26473141354859,0.00123925308419539458183963546878165528
+482.62337451604969,0.00116900341394750798114964738099450841
+486.77026438101694,0.00115904449037064664459835973344161781
+491.68385507128755,0.00114746174978904342735468544513088594
+1139.2998051302197,0.000495207112015481365513324664372789488
+1433.3824404997233,0.000393607058595642220192282045823871305
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+1496.6721775606327,0.000376962614841793930739264081108515307
+1536.2975918474183,0.000367239698233429860653400719789006797
+1726.0550697923725,0.000326866447505449488076128110891920101
+1758.6148735991735,0.000320814693885077616602102952253879833
+1809.0879619294565,0.000311864049303789868356183099204901648
+1976.5423481793755,0.000285442662971467749121888067767341569
+2018.5584390666806,0.000279501204124545022599098887774946884
+4739.2536190637038,0.000119046081163227144232931266337319176
+5082.1008438065819,0.000111015029013672563225258479926713095
+5188.4834818492718,0.000108738820320539123015098528862024766
+5403.1147245551574,0.000104419321566658875168734509678447818
+5713.8075191740199,9.87414387015816107023827069759622627e-05
+5948.2910540726043,9.48490197346140614087319767193547019e-05
+5970.4037762536664,9.44977252422832637455927654346081355e-05
+7770.7716332808805,7.26040611539550605245329046399135673e-05
+7810.5121853103692,7.22346455056638593185054972008764896e-05
+8185.2763156086721,6.89273712435880705461739677780665729e-05
+19581.670153982923,2.88121277896874661118282250082911149e-05
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+27106.067107807456,2.08141439671014442936845785790601098e-05
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+28385.407440818399,1.98760431525931806992901782528061484e-05
+29886.805942506533,1.88775469790005163573019478179647374e-05
+29990.650587052041,1.88121822031330501019607004076527426e-05
+77190.222301186426,7.30908095197621938022384568077022062e-06
+89335.328909363438,6.31541396220544530883900204608017677e-06
+91081.63525676656,6.19432865827733064419669562106847442e-06
+101242.60563416444,5.57264977512440217825430688792505877e-06
+103530.10912244421,5.44952177007922771623901997517984162e-06
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diff --git a/commons-numbers-gamma/src/test/resources/org/apache/commons/numbers/gamma/erfcx_large_data.csv b/commons-numbers-gamma/src/test/resources/org/apache/commons/numbers/gamma/erfcx_large_data.csv
new file mode 100644
index 0000000..7743234
--- /dev/null
+++ b/commons-numbers-gamma/src/test/resources/org/apache/commons/numbers/gamma/erfcx_large_data.csv
@@ -0,0 +1,105 @@
+# 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.
+
+# Random data using 52-bit mantissa scaled with 2^b.
+# Generated in blocks of 10 with b:
+# {-50:5:-5} {-4:1:-1} {0,0,0} {1,1,1} {2,2,2} {3,3,3}
+# {4,4,4,4} {5,5,5,5} {6,6,6,6} {8:2:28}
+# An additional 10 positive test points generated in the range
+# [26.3, 27.3] where erfc(x) transitions to sub-normal.
+# Reference values from customized Boost C++ 128-bit implementation
+# of erfc modifed to remove scaling by exp(-x^2). Values for large x
+# generated using the upper_gamma_fraction(...) continued fraction
+# See: https://mathworld.wolfram.com/Erfc.html Eq.3
+
+# Partition: 26.543258454251 <= z < 100
+# z, erfcx(z)
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diff --git a/commons-numbers-gamma/src/test/resources/org/apache/commons/numbers/gamma/erfcx_medium_data.csv b/commons-numbers-gamma/src/test/resources/org/apache/commons/numbers/gamma/erfcx_medium_data.csv
new file mode 100644
index 0000000..db08dfc
--- /dev/null
+++ b/commons-numbers-gamma/src/test/resources/org/apache/commons/numbers/gamma/erfcx_medium_data.csv
@@ -0,0 +1,190 @@
+# 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.
+
+# Random data using 52-bit mantissa scaled with 2^b.
+# Generated in blocks of 10 with b:
+# {-50:5:-5} {-4:1:-1} {0,0,0} {1,1,1} {2,2,2} {3,3,3}
+# {4,4,4,4} {5,5,5,5} {6,6,6,6} {8:2:28}
+# An additional 10 positive test points generated in the range
+# [26.3, 27.3] where erfc(x) transitions to sub-normal.
+# Reference values from customized Boost C++ 128-bit implementation
+# of erfc modifed to remove scaling by exp(-x^2). Values for large x
+# generated using the upper_gamma_fraction(...) continued fraction
+# See: https://mathworld.wolfram.com/Erfc.html Eq.3
+
+# Partition: 0.5 <= z < 26.543258454251
+# z, erfcx(z)
+0.55076013518044176,0.590563844804729867935975631890725524
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diff --git a/commons-numbers-gamma/src/test/resources/org/apache/commons/numbers/gamma/erfcx_neg_data.csv b/commons-numbers-gamma/src/test/resources/org/apache/commons/numbers/gamma/erfcx_neg_data.csv
new file mode 100644
index 0000000..9722731
--- /dev/null
+++ b/commons-numbers-gamma/src/test/resources/org/apache/commons/numbers/gamma/erfcx_neg_data.csv
@@ -0,0 +1,224 @@
+# 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.
+
+# Random data using 52-bit mantissa scaled with 2^b.
+# Generated in blocks of 10 with b:
+# {} and 26.6287357137515 <= z < 0
+# z, erfcx(z)
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diff --git a/commons-numbers-gamma/src/test/resources/org/apache/commons/numbers/gamma/erfcx_neg_medium_data.csv b/commons-numbers-gamma/src/test/resources/org/apache/commons/numbers/gamma/erfcx_neg_medium_data.csv
new file mode 100755
index 0000000..5822867
--- /dev/null
+++ b/commons-numbers-gamma/src/test/resources/org/apache/commons/numbers/gamma/erfcx_neg_medium_data.csv
@@ -0,0 +1,188 @@
+# 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.
+
+# Random data using 52-bit mantissa scaled with 2^b.
+# Generated in blocks of 10 with b:
+# {-50:5:-5} {-4:1:-1} {0,0,0} {1,1,1} {2,2,2} {3,3,3}
+# {4,4,4,4} {5,5,5,5} {6,6,6,6} {8:2:28}
+# An additional 10 positive test points generated in the range
+# [26.3, 27.3] where erfc(x) transitions to sub-normal.
+# Reference values from customized Boost C++ 128-bit implementation
+# of erfc modifed to remove scaling by exp(-x^2). Values for large x
+# generated using the upper_gamma_fraction(...) continued fraction
+# See: https://mathworld.wolfram.com/Erfc.html Eq.3
+
+# Partition: z < -0.5
+# z, erfcx(z)
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diff --git a/commons-numbers-gamma/src/test/resources/org/apache/commons/numbers/gamma/erfcx_neg_small_data.csv b/commons-numbers-gamma/src/test/resources/org/apache/commons/numbers/gamma/erfcx_neg_small_data.csv
new file mode 100755
index 0000000..c7d5a4a
--- /dev/null
+++ b/commons-numbers-gamma/src/test/resources/org/apache/commons/numbers/gamma/erfcx_neg_small_data.csv
@@ -0,0 +1,158 @@
+# 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.
+
+# Random data using 52-bit mantissa scaled with 2^b.
+# Generated in blocks of 10 with b:
+# {-50:5:-5} {-4:1:-1} {0,0,0} {1,1,1} {2,2,2} {3,3,3}
+# {4,4,4,4} {5,5,5,5} {6,6,6,6} {8:2:28}
+# An additional 10 positive test points generated in the range
+# [26.3, 27.3] where erfc(x) transitions to sub-normal.
+# Reference values from customized Boost C++ 128-bit implementation
+# of erfc modifed to remove scaling by exp(-x^2). Values for large x
+# generated using the upper_gamma_fraction(...) continued fraction
+# See: https://mathworld.wolfram.com/Erfc.html Eq.3
+
+# Partition: -0.5 <= z < 0
+# z, erfcx(z)
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diff --git a/commons-numbers-gamma/src/test/resources/org/apache/commons/numbers/gamma/erfcx_small_data.csv b/commons-numbers-gamma/src/test/resources/org/apache/commons/numbers/gamma/erfcx_small_data.csv
new file mode 100644
index 0000000..59b9987
--- /dev/null
+++ b/commons-numbers-gamma/src/test/resources/org/apache/commons/numbers/gamma/erfcx_small_data.csv
@@ -0,0 +1,158 @@
+# 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.
+
+# Random data using 52-bit mantissa scaled with 2^b.
+# Generated in blocks of 10 with b:
+# {-50:5:-5} {-4:1:-1} {0,0,0} {1,1,1} {2,2,2} {3,3,3}
+# {4,4,4,4} {5,5,5,5} {6,6,6,6} {8:2:28}
+# An additional 10 positive test points generated in the range
+# [26.3, 27.3] where erfc(x) transitions to sub-normal.
+# Reference values from customized Boost C++ 128-bit implementation
+# of erfc modifed to remove scaling by exp(-x^2). Values for large x
+# generated using the upper_gamma_fraction(...) continued fraction
+# See: https://mathworld.wolfram.com/Erfc.html Eq.3
+
+# Partition: 0 <= z < 0.5
+# z, erfcx(z)
+9.7130154481995625e-16,0.99999999999999890400357185747403679
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diff --git a/src/changes/changes.xml b/src/changes/changes.xml
index 4b7d6e7..3f6fbb6 100644
--- a/src/changes/changes.xml
+++ b/src/changes/changes.xml
@@ -72,6 +72,9 @@ N.B. the Performance testing module requires Java 9+.
(The unit tests require Java 8+)
">
+ <action dev="aherbert" type="add" issue="NUMBERS-177">
+ "Erfcx": Compute a scaled complementary error function: erfcx(z) = erfc(z) * exp(z*z).
+ </action>
<action dev="aherbert" type="update" issue="NUMBERS-178">
"Factorial/FactorialDouble": Tabulate all factorials with exact 64-bit double
representations of n! up to n=170. This change deprecates the FactorialDouble