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Posted to commits@commons.apache.org by ah...@apache.org on 2022/01/21 15:04:09 UTC
[commons-statistics] branch master updated: STATISTICS-52: Add a high precision PDF to the normal distribution
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-statistics.git
The following commit(s) were added to refs/heads/master by this push:
new f931dd5 STATISTICS-52: Add a high precision PDF to the normal distribution
f931dd5 is described below
commit f931dd5c6946f505495262beb5ad36603b4ab5bc
Author: aherbert <ah...@apache.org>
AuthorDate: Fri Jan 21 15:01:00 2022 +0000
STATISTICS-52: Add a high precision PDF to the normal distribution
Exploit information in the round-off from x*x to increase the precision
of exp(-0.5*x*x) when x is large.
Add a benchmark to demonstrate this has minor impact on the runtime
performance. Accuracy is increased to within 3 ULP (down from hundreds)
for large x values.
---
.../statistics/distribution/ExtendedPrecision.java | 90 ++
.../distribution/NormalDistribution.java | 10 +-
.../distribution/ExtendedPrecisionTest.java | 44 +
.../distribution/NormalDistributionTest.java | 32 +-
.../commons/statistics/distribution/expmhxx.csv | 1038 +++++++++++++++++++
.../commons/statistics/distribution/normpdf2.csv | 1039 ++++++++++++++++++++
commons-statistics-examples/examples-jmh/pom.xml | 5 +
.../jmh/distribution/NormalPDFPerformance.java | 204 ++++
8 files changed, 2454 insertions(+), 8 deletions(-)
diff --git a/commons-statistics-distribution/src/main/java/org/apache/commons/statistics/distribution/ExtendedPrecision.java b/commons-statistics-distribution/src/main/java/org/apache/commons/statistics/distribution/ExtendedPrecision.java
index 4fdbec2..fd7c401 100644
--- a/commons-statistics-distribution/src/main/java/org/apache/commons/statistics/distribution/ExtendedPrecision.java
+++ b/commons-statistics-distribution/src/main/java/org/apache/commons/statistics/distribution/ExtendedPrecision.java
@@ -54,6 +54,12 @@ final class ExtendedPrecision {
private static final double SQRT2PI_L;
/** Round-off from sqrt(2 pi) as a double. */
private static final double SQRT2PI_R;
+ /** X-value where {@code exp(-0.5*x*x)} cannot increase accuracy using the round-off
+ * from x squared. */
+ private static final int EXP_M_HALF_XX_MIN_VALUE = 2;
+ /** Approximate x-value where {@code exp(-0.5*x*x) == 0}. This is above
+ * {@code -2 * ln(2^-1074)} due to rounding performed within the exp function. */
+ private static final int EXP_M_HALF_XX_MAX_VALUE = 1491;
static {
// Initialise constants
@@ -180,6 +186,72 @@ final class ExtendedPrecision {
}
/**
+ * Compute {@code exp(-0.5*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(-0.5*x*x) is close to sub-normal. For very
+ * small exp(-0.5*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(-0.5*x*x). This occurs at
+ * {@code x <= sqrt(2)}.
+ *
+ * @param x Value
+ * @return exp(-0.5*x*x)
+ * @see <a href="https://issues.apache.org/jira/browse/STATISTICS-52">STATISTICS-52</a>
+ */
+ static double expmhxx(double x) {
+ final double z = x * x;
+ if (z <= EXP_M_HALF_XX_MIN_VALUE) {
+ return Math.exp(-0.5 * z);
+ } else if (z >= EXP_M_HALF_XX_MAX_VALUE) {
+ // exp(-745.5) == 0
+ return 0;
+ }
+ // Split the number
+ final double hx = highPartUnscaled(x);
+ final double lx = x - hx;
+ // Compute the round-off
+ final double zz = squareLow(hx, lx, z);
+ return expxx(-0.5 * z, -0.5 * zz);
+ }
+
+ /**
+ * 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)
+ * @see <a href="https://issues.apache.org/jira/projects/NUMBERS/issues/NUMBERS-177">
+ * Numbers-177: Accurate scaling by exp(z*z)</a>
+ */
+ 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;
+ }
+
+ /**
* Implement Dekker's method to split a value into two parts. Multiplying by (2^s + 1) creates
* a big value from which to derive the two split parts.
* <pre>
@@ -238,4 +310,22 @@ final class ExtendedPrecision {
// low = lx * ly - err3
return lx * ly - (((xy - hx * hy) - lx * hy) - hx * ly);
}
+
+ /**
+ * Compute the low part of the double length number {@code (z,zz)} for the exact
+ * square of {@code x} using Dekker's mult12 algorithm. The standard precision product
+ * {@code x*x} must be provided. The number {@code x} should already be split into low
+ * and high parts.
+ *
+ * <p>Note: This is a specialisation of
+ * {@link #productLow(double, double, double, double, double)}.
+ *
+ * @param hx High part of factor.
+ * @param lx Low part of factor.
+ * @param xx Square of the factor.
+ * @return <code>lx * lx - (((xx - hx * hx) - lx * hx) - hx * lx)</code>
+ */
+ private static double squareLow(double hx, double lx, double xx) {
+ return lx * lx - ((xx - hx * hx) - 2 * lx * hx);
+ }
}
diff --git a/commons-statistics-distribution/src/main/java/org/apache/commons/statistics/distribution/NormalDistribution.java b/commons-statistics-distribution/src/main/java/org/apache/commons/statistics/distribution/NormalDistribution.java
index af8dae7..883757e 100644
--- a/commons-statistics-distribution/src/main/java/org/apache/commons/statistics/distribution/NormalDistribution.java
+++ b/commons-statistics-distribution/src/main/java/org/apache/commons/statistics/distribution/NormalDistribution.java
@@ -110,9 +110,8 @@ public final class NormalDistribution extends AbstractContinuousDistribution {
/** {@inheritDoc} */
@Override
public double density(double x) {
- final double x0 = x - mean;
- final double x1 = x0 / standardDeviation;
- return Math.exp(-0.5 * x1 * x1) / sdSqrt2pi;
+ final double z = (x - mean) / standardDeviation;
+ return ExtendedPrecision.expmhxx(z) / sdSqrt2pi;
}
/** {@inheritDoc} */
@@ -131,9 +130,8 @@ public final class NormalDistribution extends AbstractContinuousDistribution {
/** {@inheritDoc} */
@Override
public double logDensity(double x) {
- final double x0 = x - mean;
- final double x1 = x0 / standardDeviation;
- return -0.5 * x1 * x1 - logStandardDeviationPlusHalfLog2Pi;
+ final double z = (x - mean) / standardDeviation;
+ return -0.5 * z * z - logStandardDeviationPlusHalfLog2Pi;
}
/** {@inheritDoc} */
diff --git a/commons-statistics-distribution/src/test/java/org/apache/commons/statistics/distribution/ExtendedPrecisionTest.java b/commons-statistics-distribution/src/test/java/org/apache/commons/statistics/distribution/ExtendedPrecisionTest.java
index 9378cdf..6784249 100644
--- a/commons-statistics-distribution/src/test/java/org/apache/commons/statistics/distribution/ExtendedPrecisionTest.java
+++ b/commons-statistics-distribution/src/test/java/org/apache/commons/statistics/distribution/ExtendedPrecisionTest.java
@@ -42,6 +42,10 @@ class ExtendedPrecisionTest {
private static final RMS SQRT2XX_RMS2 = new RMS();
/** The sum of the squared ULP error for the first computation for x * sqrt(2 pi). */
private static final RMS XSQRT2PI_RMS = new RMS();
+ /** The sum of the squared ULP error for the first computation for exp(-0.5*x*x). */
+ private static final RMS EXPMHXX_RMS1 = new RMS();
+ /** The sum of the squared ULP error for the second computation for exp(-0.5*x*x). */
+ private static final RMS EXPMHXX_RMS2 = new RMS();
/**
* Class to compute the root mean squared error (RMS).
@@ -186,6 +190,46 @@ class ExtendedPrecisionTest {
assertPrecision(XSQRT2PI_RMS, 1.2, 0.6);
}
+ @ParameterizedTest
+ @ValueSource(doubles = {0, 0.5, 1, 2, 3, 4, 5, 38.5, Double.MAX_VALUE, Double.POSITIVE_INFINITY, Double.NaN})
+ void testExpmhxxEdgeCases(double x) {
+ final double expected = Math.exp(-0.5 * x * x);
+ Assertions.assertEquals(expected, ExtendedPrecision.expmhxx(x));
+ Assertions.assertEquals(expected, ExtendedPrecision.expmhxx(-x));
+ }
+
+ /**
+ * Test the extended precision {@code exp(-0.5 * x * x)}. The expected result
+ * is an extended precision computation. For comparison ulp errors are collected for
+ * the standard precision computation.
+ *
+ * @param x Value x
+ * @param expected Expected result of {@code exp(-0.5 * x * x)}.
+ */
+ @ParameterizedTest
+ @Order(1)
+ @CsvFileSource(resources = "expmhxx.csv")
+ void testExpmhxx(double x, BigDecimal expected) {
+ final double e = expected.doubleValue();
+ final double actual = ExtendedPrecision.expmhxx(x);
+ Assertions.assertEquals(e, actual, Math.ulp(e) * 2);
+ // Compute errors
+ addError(actual, expected, e, EXPMHXX_RMS1);
+ addError(Math.exp(-0.5 * x * x), expected, e, EXPMHXX_RMS2);
+ }
+
+ @Test
+ void testExpmhxxHighPrecision() {
+ // Typical result: max 0.9727 rms 0.3481
+ assertPrecision(EXPMHXX_RMS1, 1.5, 0.5);
+ }
+
+ @Test
+ void testExpmhxxStandardPrecision() {
+ // Typical result: max 385.7193 rms 50.7769
+ assertPrecision(EXPMHXX_RMS2, 400, 60);
+ }
+
private static void assertPrecision(RMS rms, double maxError, double rmsError) {
Assertions.assertTrue(rms.getMax() < maxError, () -> "max error: " + rms.getMax());
Assertions.assertTrue(rms.getRMS() < rmsError, () -> "rms error: " + rms.getRMS());
diff --git a/commons-statistics-distribution/src/test/java/org/apache/commons/statistics/distribution/NormalDistributionTest.java b/commons-statistics-distribution/src/test/java/org/apache/commons/statistics/distribution/NormalDistributionTest.java
index 4f8e42b..cf5c608 100644
--- a/commons-statistics-distribution/src/test/java/org/apache/commons/statistics/distribution/NormalDistributionTest.java
+++ b/commons-statistics-distribution/src/test/java/org/apache/commons/statistics/distribution/NormalDistributionTest.java
@@ -177,13 +177,41 @@ class NormalDistributionTest extends BaseContinuousDistributionTest {
}
}
+ /**
+ * Test the PDF using high-accuracy uniform x data.
+ *
+ * <p>This dataset uses uniformly spaced machine representable x values that have no
+ * round-off component when squared. If the density is implemented using
+ * {@code exp(logDensity(x))} the test will fail. Using the log density requires a
+ * tolerance of approximately 53 ULP to pass the test of larger x values.
+ */
@ParameterizedTest
@CsvFileSource(resources = "normpdf.csv")
void testPDF(double x, BigDecimal expected) {
+ assertPDF(x, expected, 2);
+ }
+
+ /**
+ * Test the PDF using high-accuracy random x data.
+ *
+ * <p>This dataset uses random x values with full usage of the 52-bit mantissa to ensure
+ * that there is a round-off component when squared. It requires a high precision exponential
+ * function using the round-off to compute {@code exp(-0.5*x*x)} accurately.
+ * Using a standard precision computation requires a tolerance of approximately 383 ULP
+ * to pass the test of larger x values.
+ *
+ * <p>See STATISTICS-52.
+ */
+ @ParameterizedTest
+ @CsvFileSource(resources = "normpdf2.csv")
+ void testPDF2(double x, BigDecimal expected) {
+ assertPDF(x, expected, 3);
+ }
+
+ private static void assertPDF(double x, BigDecimal expected, int ulpTolerance) {
final double e = expected.doubleValue();
final double a = STANDARD_NORMAL.density(x);
- // Require high precision. Currently this does not work at 1 ULP.
- Assertions.assertEquals(e, a, Math.ulp(e) * 2,
+ Assertions.assertEquals(e, a, Math.ulp(e) * ulpTolerance,
() -> "ULP error: " + expected.subtract(new BigDecimal(a)).doubleValue() / Math.ulp(e));
}
}
diff --git a/commons-statistics-distribution/src/test/resources/org/apache/commons/statistics/distribution/expmhxx.csv b/commons-statistics-distribution/src/test/resources/org/apache/commons/statistics/distribution/expmhxx.csv
new file mode 100644
index 0000000..1e99112
--- /dev/null
+++ b/commons-statistics-distribution/src/test/resources/org/apache/commons/statistics/distribution/expmhxx.csv
@@ -0,0 +1,1038 @@
+# 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.
+
+# High-precision test data for the normal distribution PDF.
+# Computed using using gcc 128-bit quadmath using Boost multiprecision.
+#
+# #include <boost/multiprecision/float128.hpp>
+# #include <iostream>
+# #include <limits>
+#
+# template <class T>
+# static T expmhxx(T x) {
+# return exp(-0.5*x*x);
+# }
+#
+# // for x in ...
+# std::cout.precision(std::numeric_limits<double>::max_digits10);
+# std::cout << x << ',';
+# std::cout.precision(std::numeric_limits<boost::multiprecision::float128>::max_digits10);
+# std::cout << expmhxx((boost::multiprecision::float128) x) << std::endl;
+#
+# Random x values are approximately log-uniformly distributed in [1, 39).
+# This was done by mapping 1.0 and 39.0 to their raw long bits, sampling uniformly
+# in the range and mapping the bits back to a double.
+# Randomness uses the entire 52-bit mantissa and ensures x*x has a round-off component.
+
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diff --git a/commons-statistics-distribution/src/test/resources/org/apache/commons/statistics/distribution/normpdf2.csv b/commons-statistics-distribution/src/test/resources/org/apache/commons/statistics/distribution/normpdf2.csv
new file mode 100644
index 0000000..444b1ff
--- /dev/null
+++ b/commons-statistics-distribution/src/test/resources/org/apache/commons/statistics/distribution/normpdf2.csv
@@ -0,0 +1,1039 @@
+# 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.
+
+# High-precision test data for the normal distribution PDF.
+# Computed using using gcc 128-bit quadmath using Boost multiprecision.
+#
+# #include <boost/multiprecision/float128.hpp>
+# #include <boost/math/constants/constants.hpp>
+# #include <iostream>
+# #include <limits>
+#
+# template <class T>
+# static T normpdf(T x) {
+# return exp(-0.5*x*x)/boost::math::constants::root_two_pi<T>();
+# }
+#
+# // for x in ...
+# std::cout.precision(std::numeric_limits<double>::max_digits10);
+# std::cout << x << ',';
+# std::cout.precision(std::numeric_limits<boost::multiprecision::float128>::max_digits10);
+# std::cout << normpdf((boost::multiprecision::float128) x) << std::endl;
+#
+# Random x values are approximately log-uniformly distributed in [1, 39).
+# This was done by mapping 1.0 and 39.0 to their raw long bits, sampling uniformly
+# in the range and mapping the bits back to a double.
+# Randomness uses the entire 52-bit mantissa and ensures x*x has a round-off component.
+
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diff --git a/commons-statistics-examples/examples-jmh/pom.xml b/commons-statistics-examples/examples-jmh/pom.xml
index 32be83e..d2f8ec4 100644
--- a/commons-statistics-examples/examples-jmh/pom.xml
+++ b/commons-statistics-examples/examples-jmh/pom.xml
@@ -37,6 +37,11 @@
</dependency>
<dependency>
+ <groupId>org.apache.commons</groupId>
+ <artifactId>commons-rng-simple</artifactId>
+ </dependency>
+
+ <dependency>
<groupId>org.openjdk.jmh</groupId>
<artifactId>jmh-core</artifactId>
<version>${jmh.version}</version>
diff --git a/commons-statistics-examples/examples-jmh/src/main/java/org/apache/commons/statistics/examples/jmh/distribution/NormalPDFPerformance.java b/commons-statistics-examples/examples-jmh/src/main/java/org/apache/commons/statistics/examples/jmh/distribution/NormalPDFPerformance.java
new file mode 100644
index 0000000..c290df5
--- /dev/null
+++ b/commons-statistics-examples/examples-jmh/src/main/java/org/apache/commons/statistics/examples/jmh/distribution/NormalPDFPerformance.java
@@ -0,0 +1,204 @@
+/*
+ * 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.statistics.examples.jmh.distribution;
+
+import java.util.concurrent.TimeUnit;
+import java.util.function.DoubleSupplier;
+import java.util.function.DoubleUnaryOperator;
+import org.apache.commons.rng.sampling.distribution.ContinuousUniformSampler;
+import org.apache.commons.rng.sampling.distribution.ZigguratSampler;
+import org.apache.commons.rng.simple.RandomSource;
+import org.apache.commons.statistics.distribution.NormalDistribution;
+import org.openjdk.jmh.annotations.Benchmark;
+import org.openjdk.jmh.annotations.BenchmarkMode;
+import org.openjdk.jmh.annotations.Fork;
+import org.openjdk.jmh.annotations.Measurement;
+import org.openjdk.jmh.annotations.Mode;
+import org.openjdk.jmh.annotations.OutputTimeUnit;
+import org.openjdk.jmh.annotations.Param;
+import org.openjdk.jmh.annotations.Scope;
+import org.openjdk.jmh.annotations.Setup;
+import org.openjdk.jmh.annotations.State;
+import org.openjdk.jmh.annotations.Warmup;
+
+/**
+ * Executes a benchmark of the probability density function for
+ * the normal distribution.
+ */
+@BenchmarkMode(Mode.AverageTime)
+@OutputTimeUnit(TimeUnit.NANOSECONDS)
+@Warmup(iterations = 5, time = 1, timeUnit = TimeUnit.SECONDS)
+@Measurement(iterations = 5, time = 1, timeUnit = TimeUnit.SECONDS)
+@State(Scope.Benchmark)
+@Fork(value = 1, jvmArgs = {"-server", "-Xms512M", "-Xmx512M"})
+public class NormalPDFPerformance {
+
+ /**
+ * Source of a function to compute the normal distribution PDF.
+ * The source of the random X deviate is implemented in sub-classes.
+ */
+ @State(Scope.Benchmark)
+ public abstract static class Source {
+ /** The method. */
+ @Param({"baseline", "std", "hp"})
+ private String method;
+
+ /** The generator to supply the next density value. */
+ private DoubleSupplier gen;
+
+ /**
+ * @return the next value
+ */
+ public double next() {
+ return gen.getAsDouble();
+ }
+
+ /**
+ * Create the normal distribution PDF and a supplier of random X deviates.
+ * This is used to create a supplier of density values.
+ */
+ @Setup
+ public void setup() {
+ // Note:
+ // These exist to exercise the full the density function.
+ // If these are different from a standard normal distribution N(0, 1)
+ // then the supplier of values may not provide a suitable distribution
+ // of random X deviates.
+ final double mean = 0;
+ final double sd = 1;
+
+ DoubleUnaryOperator fun;
+ if ("baseline".equals(method)) {
+ // No density function. This tests baseline speed of generating the X deviate.
+ fun = x -> x;
+ } else if ("std".equals(method)) {
+ // Standard precision implementation
+ fun = new StandardNormalDistribution(mean, sd)::density;
+ } else if ("hp".equals(method)) {
+ // High-precision implementation in the NormalDistribution class
+ fun = NormalDistribution.of(mean, sd)::density;
+ } else {
+ throw new IllegalStateException("Unknown method: " + method);
+ }
+ final DoubleSupplier nextValue = createValues();
+ gen = () -> fun.applyAsDouble(nextValue.getAsDouble());
+ }
+
+ /**
+ * Creates the generator of random X deviates used to test the normal distribution.
+ *
+ * @return the generator
+ */
+ protected abstract DoubleSupplier createValues();
+
+ /**
+ * Implementation of the normal distribution using standard precision.
+ */
+ private static class StandardNormalDistribution {
+ /** The mean. */
+ private final double mean;
+ /** The standard deviation. */
+ private final double sd;
+ /** Density factor sd * sqrt(2 pi). */
+ private final double sdSqrt2pi;
+
+ /**
+ * @param mean Mean
+ * @param sd Standard deviation
+ */
+ StandardNormalDistribution(double mean, double sd) {
+ this.mean = mean;
+ this.sd = sd;
+ sdSqrt2pi = sd * Math.sqrt(2 * Math.PI);
+ }
+
+ /**
+ * Compute the probability density function.
+ *
+ * @param x Point x
+ * @return pdf(x)
+ */
+ double density(double x) {
+ final double z = (x - mean) / sd;
+ return Math.exp(-0.5 * z * z) / sdSqrt2pi;
+ }
+ }
+ }
+
+ /**
+ * Provides a source of random X deviates from a uniform distribution.
+ */
+ @State(Scope.Benchmark)
+ public static class UniformSource extends Source {
+ /** The lower bound. */
+ @Param({"0"})
+ private double low;
+ /**
+ * The higher bound.
+ *
+ * <p>Note: A standard normal distribution density function is zero above approximately 38.5.
+ *
+ * <p>Note: The high precision normal distribution uses a high precision exponential
+ * function when x^2 is less than 2.0. The worse case scenario for a 50-50 random branch
+ * for the distribution to choose a standard or high precision can be created using a
+ * range of [0, 2 * sqrt(2)], or [0, 2.828].
+ */
+ @Param({"1", "2.828", "4", "16", "64"})
+ private double high;
+
+ /** {@inheritDoc} */
+ @Override
+ protected DoubleSupplier createValues() {
+ return ContinuousUniformSampler.of(RandomSource.XO_RO_SHI_RO_128_PP.create(), low, high)::sample;
+ }
+ }
+
+ /**
+ * Provides a source of random X deviates from a normal distribution.
+ */
+ @State(Scope.Benchmark)
+ public static class NormalSource extends Source {
+ /** {@inheritDoc} */
+ @Override
+ protected DoubleSupplier createValues() {
+ return ZigguratSampler.NormalizedGaussian.of(RandomSource.XO_RO_SHI_RO_128_PP.create())::sample;
+ }
+ }
+
+ /**
+ * Compute the PDF from a uniformally distributed X deviate.
+ *
+ * @param source source of the function
+ * @return the value
+ */
+ @Benchmark
+ public double uniform(UniformSource source) {
+ return source.next();
+ }
+
+ /**
+ * Compute the PDF from a normally distributed X deviate.
+ *
+ * @param source source of the function
+ * @return the value
+ */
+ @Benchmark
+ public double normal(NormalSource source) {
+ return source.next();
+ }
+}