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apache / commons-rng / 5f640304da601dc45748c5f11c9e9e37f3b4180e / . / commons-rng-examples / examples-jmh / src / main / java / org / apache / commons / rng / examples / jmh / sampling / distribution / ZigguratSamplerPerformance.java
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/*
* 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.rng.examples.jmh.sampling.distribution;
import org.apache.commons.math3.util.FastMath;
import org.apache.commons.rng.UniformRandomProvider;
import org.apache.commons.rng.sampling.distribution.ContinuousSampler;
import org.apache.commons.rng.sampling.distribution.DiscreteSampler;
import org.apache.commons.rng.sampling.distribution.LongSampler;
import org.apache.commons.rng.sampling.distribution.ZigguratSampler;
import org.apache.commons.rng.sampling.distribution.ZigguratNormalizedGaussianSampler;
import org.apache.commons.rng.simple.RandomSource;
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;
import java.util.Arrays;
import java.util.concurrent.ThreadLocalRandom;
import java.util.concurrent.TimeUnit;
/**
* Executes a benchmark to compare the speed of generation of random numbers
* using variations of the ziggurat method.
*/
@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", "-Xms128M", "-Xmx128M"})
public class ZigguratSamplerPerformance {
// Production versions
/** The name for a copy of the {@link ZigguratNormalizedGaussianSampler} with a table of size 128.
* This matches the version in Commons RNG release v1.1 to 1.3. */
static final String GAUSSIAN_128 = "Gaussian128";
/** The name for the {@link ZigguratNormalizedGaussianSampler} (table of size 256).
* This is the version in Commons RNG release v1.4+. */
static final String GAUSSIAN_256 = "Gaussian256";
/** The name for the {@link org.apache.commons.rng.sampling.distribution.ZigguratSampler.NormalizedGaussian}. */
static final String MOD_GAUSSIAN = "ModGaussian";
/** The name for the {@link org.apache.commons.rng.sampling.distribution.ZigguratSampler.Exponential}. */
static final String MOD_EXPONENTIAL = "ModExponential";
// Testing versions
/** The name for the {@link ZigguratExponentialSampler} with a table of size 256.
* This is an exponential sampler using Marsaglia's ziggurat method. */
static final String EXPONENTIAL = "Exponential";
/** The name for the {@link ModifiedZigguratNormalizedGaussianSampler}.
* This is a base implementation of McFarland's ziggurat method. */
static final String MOD_GAUSSIAN2 = "ModGaussian2";
/** The name for the {@link ModifiedZigguratNormalizedGaussianSamplerSimpleOverhangs}. */
static final String MOD_GAUSSIAN_SIMPLE_OVERHANGS = "ModGaussianSimpleOverhangs";
/** The name for the {@link ModifiedZigguratNormalizedGaussianSamplerInlining}. */
static final String MOD_GAUSSIAN_INLINING = "ModGaussianInlining";
/** The name for the {@link ModifiedZigguratNormalizedGaussianSamplerInliningShift}. */
static final String MOD_GAUSSIAN_INLINING_SHIFT = "ModGaussianInliningShift";
/** The name for the {@link ModifiedZigguratNormalizedGaussianSamplerInliningSimpleOverhangs}. */
static final String MOD_GAUSSIAN_INLINING_SIMPLE_OVERHANGS = "ModGaussianInliningSimpleOverhangs";
/** The name for the {@link ModifiedZigguratNormalizedGaussianSamplerIntMap}. */
static final String MOD_GAUSSIAN_INT_MAP = "ModGaussianIntMap";
/** The name for the {@link ModifiedZigguratNormalizedGaussianSamplerEMaxTable}. */
static final String MOD_GAUSSIAN_E_MAX_TABLE = "ModGaussianEMaxTable";
/** The name for the {@link ModifiedZigguratNormalizedGaussianSamplerEMax2}. */
static final String MOD_GAUSSIAN_E_MAX_2 = "ModGaussianEMax2";
/** The name for the {@link ModifiedZigguratNormalizedGaussianSampler512} using a table size of 512. */
static final String MOD_GAUSSIAN_512 = "ModGaussian512";
/** The name for the {@link ModifiedZigguratExponentialSampler}.
* This is a base implementation of McFarland's ziggurat method. */
static final String MOD_EXPONENTIAL2 = "ModExponential2";
/** The name for the {@link ModifiedZigguratExponentialSamplerSimpleOverhangs}. */
static final String MOD_EXPONENTIAL_SIMPLE_OVERHANGS = "ModExponentialSimpleOverhangs";
/** The name for the {@link ModifiedZigguratExponentialSamplerInlining}. */
static final String MOD_EXPONENTIAL_INLINING = "ModExponentialInlining";
/** The name for the {@link ModifiedZigguratExponentialSamplerLoop}. */
static final String MOD_EXPONENTIAL_LOOP = "ModExponentialLoop";
/** The name for the {@link ModifiedZigguratExponentialSamplerLoop2}. */
static final String MOD_EXPONENTIAL_LOOP2 = "ModExponentialLoop2";
/** The name for the {@link ModifiedZigguratExponentialSamplerRecursion}. */
static final String MOD_EXPONENTIAL_RECURSION = "ModExponentialRecursion";
/** The name for the {@link ModifiedZigguratExponentialSamplerIntMap}. */
static final String MOD_EXPONENTIAL_INT_MAP = "ModExponentialIntMap";
/** The name for the {@link ModifiedZigguratExponentialSamplerEMaxTable}. */
static final String MOD_EXPONENTIAL_E_MAX_TABLE = "ModExponentialEmaxTable";
/** The name for the {@link ModifiedZigguratExponentialSamplerEMax2}. */
static final String MOD_EXPONENTIAL_E_MAX_2 = "ModExponentialEmax2";
/** The name for the {@link ModifiedZigguratExponentialSampler512} using a table size of 512. */
static final String MOD_EXPONENTIAL_512 = "ModExponential512";
/** Mask to create an unsigned long from a signed long. */
private static final long MAX_INT64 = 0x7fffffffffffffffL;
/** 2^53. */
private static final double TWO_POW_63 = 0x1.0p63;
/**
* The value.
*
* <p>This must NOT be final!</p>
*/
private double value;
/**
* Defines method to use for creating {@code int} index values from a random long.
*/
@State(Scope.Benchmark)
public static class IndexSources {
/** The method to obtain the index. */
@Param({"CastMask", "MaskCast"})
private String method;
/** The sampler. */
private DiscreteSampler sampler;
/**
* @return the sampler.
*/
public DiscreteSampler getSampler() {
return sampler;
}
/** Instantiates sampler. */
@Setup
public void setup() {
// Use a fast generator
final UniformRandomProvider rng = RandomSource.XO_RO_SHI_RO_128_PP.create();
if ("CastMask".equals(method)) {
sampler = () -> {
final long x = rng.nextLong();
return ((int) x) & 0xff;
};
} else if ("MaskCast".equals(method)) {
sampler = () -> {
final long x = rng.nextLong();
return (int) (x & 0xff);
};
} else {
throwIllegalStateException(method);
}
}
}
/**
* Defines method to use for creating an index values from a random long and comparing
* it to an {@code int} limit.
*/
@State(Scope.Benchmark)
public static class IndexCompareSources {
/** The method to compare the index against the limit. */
@Param({"CastMaskIntCompare", "MaskCastIntCompare", "MaskLongCompareCast"})
private String method;
/** The sampler. */
private DiscreteSampler sampler;
/**
* @return the sampler.
*/
public DiscreteSampler getSampler() {
return sampler;
}
/** Instantiates sampler. */
@Setup
public void setup() {
// The limit:
// exponential = 252
// gaussian = 253
final int limit = 253;
// Use a fast generator:
final UniformRandomProvider rng = RandomSource.XO_RO_SHI_RO_128_PP.create();
if ("CastMaskIntCompare".equals(method)) {
sampler = () -> {
final long x = rng.nextLong();
final int i = ((int) x) & 0xff;
if (i < limit) {
return i;
}
return 0;
};
} else if ("MaskCastIntCompare".equals(method)) {
sampler = () -> {
final long x = rng.nextLong();
final int i = (int) (x & 0xff);
if (i < limit) {
return i;
}
return 0;
};
} else if ("MaskLongCompareCast".equals(method)) {
sampler = () -> {
final long x = rng.nextLong();
final long i = x & 0xff;
if (i < limit) {
return (int) i;
}
return 0;
};
} else {
throwIllegalStateException(method);
}
}
}
/**
* Defines method to use for creating unsigned {@code long} values.
*/
@State(Scope.Benchmark)
public static class LongSources {
/** The method to obtain the long. */
@Param({"Mask", "Shift"})
private String method;
/** The sampler. */
private LongSampler sampler;
/**
* @return the sampler.
*/
public LongSampler getSampler() {
return sampler;
}
/** Instantiates sampler. */
@Setup
public void setup() {
// Use a fast generator:
// Here we use a simple linear congruential generator
// which should have constant speed and a random upper bit.
final long[] s = {ThreadLocalRandom.current().nextLong()};
if ("Mask".equals(method)) {
sampler = () -> {
final long x = s[0];
s[0] = updateLCG(x);
return x & Long.MAX_VALUE;
};
} else if ("Shift".equals(method)) {
sampler = () -> {
final long x = s[0];
s[0] = updateLCG(x);
return x >>> 1;
};
} else {
throwIllegalStateException(method);
}
}
}
/**
* Defines method to use for interpolating the X or Y tables from unsigned {@code long} values.
*/
@State(Scope.Benchmark)
public static class InterpolationSources {
/** The method to perform interpolation. */
@Param({"U1", "1minusU2", "U_1minusU"})
private String method;
/** The sampler. */
private ContinuousSampler sampler;
/**
* @return the sampler.
*/
public ContinuousSampler getSampler() {
return sampler;
}
/** Instantiates sampler. */
@Setup
public void setup() {
// Use a fast generator:
final UniformRandomProvider rng = RandomSource.XO_RO_SHI_RO_128_PP.create();
// Get an x table. This is length 254.
// We will sample from this internally to avoid index out-of-bounds issues.
final int start = 42;
final double[] x = Arrays.copyOfRange(
ModifiedZigguratNormalizedGaussianSampler.getX(), start, start + 129);
final int mask = 127;
if ("U1".equals(method)) {
sampler = () -> {
final long u = rng.nextLong();
final int j = 1 + (((int) u) & mask);
// double multiply
// double add
// double subtract
// long convert to double
// double multiply
// int subtract
// three index loads
// Same as sampleX(x, j, u)
return x[j] * TWO_POW_63 + (x[j - 1] - x[j]) * u;
};
} else if ("1minusU2".equals(method)) {
sampler = () -> {
final long u = rng.nextLong();
final int j = 1 + (((int) u) & mask);
// Since u is in [0, 2^63) to create (1 - u) using Long.MIN_VALUE
// as an unsigned integer of 2^63.
// double multiply
// double add
// double subtract
// long subtract
// long convert to double
// double multiply
// int subtract * 2
// three index loads
// Same as sampleY(x, j, u)
return x[j - 1] * TWO_POW_63 + (x[j] - x[j - 1]) * (Long.MIN_VALUE - u);
};
} else if ("U_1minusU".equals(method)) {
sampler = () -> {
final long u = rng.nextLong();
final int j = 1 + (((int) u) & mask);
// Interpolation between bounds a and b using:
// a * u + b * (1 - u) == b + u * (a - b)
// long convert to double
// double multiply
// double add
// long subtract
// long convert to double
// double multiply
// int subtract
// two index loads
return x[j - 1] * u + x[j] * (Long.MIN_VALUE - u);
};
} else {
throwIllegalStateException(method);
}
}
}
/**
* Defines method to extract a sign bit from a {@code long} value.
*/
@State(Scope.Benchmark)
public static class SignBitSources {
/** The method to obtain the sign bit. */
@Param({"ifNegative", "ifSignBit", "ifBit", "bitSubtract", "signBitSubtract"})
private String method;
/** The sampler. */
private ContinuousSampler sampler;
/**
* @return the sampler.
*/
public ContinuousSampler getSampler() {
return sampler;
}
/** Instantiates sampler. */
@Setup
public void setup() {
// Use a fast generator:
final UniformRandomProvider rng = RandomSource.XO_RO_SHI_RO_128_PP.create();
if ("ifNegative".equals(method)) {
sampler = () -> {
final long x = rng.nextLong();
return x < 0 ? -1.0 : 1.0;
};
} else if ("ifSignBit".equals(method)) {
sampler = () -> {
final long x = rng.nextLong();
return (x >>> 63) == 0 ? -1.0 : 1.0;
};
} else if ("ifBit".equals(method)) {
sampler = () -> {
final long x = rng.nextLong();
return (x & 0x100) == 0 ? -1.0 : 1.0;
};
} else if ("bitSubtract".equals(method)) {
sampler = () -> {
final long x = rng.nextLong();
return ((x >>> 7) & 0x2) - 1.0;
};
} else if ("signBitSubtract".equals(method)) {
sampler = () -> {
final long x = rng.nextLong();
return ((x >>> 62) & 0x2) - 1.0;
};
} else {
throwIllegalStateException(method);
}
}
}
/**
* Defines method to use for computing {@code exp(x)} when {@code -8 <= x <= 0}.
*/
@State(Scope.Benchmark)
public static class ExpSources {
/** The method to compute exp(x). */
@Param({"noop", "Math.exp", "FastMath.exp"})
private String method;
/** The sampler. */
private ContinuousSampler sampler;
/**
* @return the sampler.
*/
public ContinuousSampler getSampler() {
return sampler;
}
/** Instantiates sampler. */
@Setup
public void setup() {
// Use a fast generator:
// Here we use a simple linear congruential generator
// which should have constant speed and a random upper bit.
final long[] s = {ThreadLocalRandom.current().nextLong()};
// From a random long we create a value in (-8, 0] by using the
// method to generate [0, 1) and then multiplying by -8:
// (x >>> 11) * 0x1.0p-53 * -0x1.0p3
if ("noop".equals(method)) {
sampler = () -> {
final long x = s[0];
s[0] = updateLCG(x);
return (x >>> 11) * -0x1.0p-50;
};
} else if ("Math.exp".equals(method)) {
sampler = () -> {
final long x = s[0];
s[0] = updateLCG(x);
return Math.exp((x >>> 11) * -0x1.0p-50);
};
} else if ("FastMath.exp".equals(method)) {
sampler = () -> {
final long x = s[0];
s[0] = updateLCG(x);
return FastMath.exp((x >>> 11) * -0x1.0p-50);
};
} else {
throwIllegalStateException(method);
}
}
}
/**
* Update the state of the linear congruential generator.
* <pre>
* s = m*s + a
* </pre>
*
* <p>This can be used when the upper bits of the long are important.
* The lower bits will not be very random. Each bit has a period of
* 2^p where p is the bit significance.
*
* @param state the state
* @return the new state
*/
private static long updateLCG(long state) {
// m is the multiplier used for the LCG component of the JDK 17 generators.
// a can be any odd number.
// Use the golden ratio from a SplitMix64 generator.
return 0xd1342543de82ef95L * state + 0x9e3779b97f4a7c15L;
}
/**
* The samplers to use for testing the ziggurat method.
*/
@State(Scope.Benchmark)
public abstract static class Sources {
/**
* The sampler type.
*/
@Param({// Production versions
GAUSSIAN_128, GAUSSIAN_256, MOD_GAUSSIAN, MOD_EXPONENTIAL,
// Experimental Marsaglia exponential ziggurat sampler
EXPONENTIAL,
// Experimental McFarland Gaussian ziggurat samplers
MOD_GAUSSIAN2, MOD_GAUSSIAN_SIMPLE_OVERHANGS,
MOD_GAUSSIAN_INLINING, MOD_GAUSSIAN_INLINING_SHIFT,
MOD_GAUSSIAN_INLINING_SIMPLE_OVERHANGS, MOD_GAUSSIAN_INT_MAP,
MOD_GAUSSIAN_E_MAX_TABLE, MOD_GAUSSIAN_E_MAX_2, MOD_GAUSSIAN_512,
// Experimental McFarland Gaussian ziggurat samplers
MOD_EXPONENTIAL2, MOD_EXPONENTIAL_SIMPLE_OVERHANGS, MOD_EXPONENTIAL_INLINING,
MOD_EXPONENTIAL_LOOP, MOD_EXPONENTIAL_LOOP2,
MOD_EXPONENTIAL_RECURSION, MOD_EXPONENTIAL_INT_MAP,
MOD_EXPONENTIAL_E_MAX_TABLE, MOD_EXPONENTIAL_E_MAX_2, MOD_EXPONENTIAL_512})
protected String type;
/**
* Creates the sampler.
*
* @param type Type of sampler
* @param rng RNG
* @return the sampler
*/
static ContinuousSampler createSampler(String type, UniformRandomProvider rng) {
if (GAUSSIAN_128.equals(type)) {
return new ZigguratNormalizedGaussianSampler128(rng);
} else if (GAUSSIAN_256.equals(type)) {
return ZigguratNormalizedGaussianSampler.of(rng);
} else if (MOD_GAUSSIAN.equals(type)) {
return ZigguratSampler.NormalizedGaussian.of(rng);
} else if (MOD_EXPONENTIAL.equals(type)) {
return ZigguratSampler.Exponential.of(rng);
} else if (EXPONENTIAL.equals(type)) {
return new ZigguratExponentialSampler(rng);
} else if (MOD_GAUSSIAN2.equals(type)) {
return new ModifiedZigguratNormalizedGaussianSampler(rng);
} else if (MOD_GAUSSIAN_SIMPLE_OVERHANGS.equals(type)) {
return new ModifiedZigguratNormalizedGaussianSamplerSimpleOverhangs(rng);
} else if (MOD_GAUSSIAN_INLINING.equals(type)) {
return new ModifiedZigguratNormalizedGaussianSamplerInlining(rng);
} else if (MOD_GAUSSIAN_INLINING_SHIFT.equals(type)) {
return new ModifiedZigguratNormalizedGaussianSamplerInliningShift(rng);
} else if (MOD_GAUSSIAN_INLINING_SIMPLE_OVERHANGS.equals(type)) {
return new ModifiedZigguratNormalizedGaussianSamplerInliningSimpleOverhangs(rng);
} else if (MOD_GAUSSIAN_INT_MAP.equals(type)) {
return new ModifiedZigguratNormalizedGaussianSamplerIntMap(rng);
} else if (MOD_GAUSSIAN_E_MAX_TABLE.equals(type)) {
return new ModifiedZigguratNormalizedGaussianSamplerEMaxTable(rng);
} else if (MOD_GAUSSIAN_E_MAX_2.equals(type)) {
return new ModifiedZigguratNormalizedGaussianSamplerEMax2(rng);
} else if (MOD_GAUSSIAN_512.equals(type)) {
return new ModifiedZigguratNormalizedGaussianSampler512(rng);
} else if (MOD_EXPONENTIAL2.equals(type)) {
return new ModifiedZigguratExponentialSampler(rng);
} else if (MOD_EXPONENTIAL_SIMPLE_OVERHANGS.equals(type)) {
return new ModifiedZigguratExponentialSamplerSimpleOverhangs(rng);
} else if (MOD_EXPONENTIAL_INLINING.equals(type)) {
return new ModifiedZigguratExponentialSamplerInlining(rng);
} else if (MOD_EXPONENTIAL_LOOP.equals(type)) {
return new ModifiedZigguratExponentialSamplerLoop(rng);
} else if (MOD_EXPONENTIAL_LOOP2.equals(type)) {
return new ModifiedZigguratExponentialSamplerLoop2(rng);
} else if (MOD_EXPONENTIAL_RECURSION.equals(type)) {
return new ModifiedZigguratExponentialSamplerRecursion(rng);
} else if (MOD_EXPONENTIAL_INT_MAP.equals(type)) {
return new ModifiedZigguratExponentialSamplerIntMap(rng);
} else if (MOD_EXPONENTIAL_E_MAX_TABLE.equals(type)) {
return new ModifiedZigguratExponentialSamplerEMaxTable(rng);
} else if (MOD_EXPONENTIAL_E_MAX_2.equals(type)) {
return new ModifiedZigguratExponentialSamplerEMax2(rng);
} else if (MOD_EXPONENTIAL_512.equals(type)) {
return new ModifiedZigguratExponentialSampler512(rng);
} else {
throw new IllegalStateException("Unknown type: " + type);
}
}
}
/**
* The samplers to use for testing the ziggurat method with single sample generation.
* Defines the RandomSource and the sampler type.
*/
@State(Scope.Benchmark)
public static class SingleSources extends Sources {
/**
* RNG providers.
*
* <p>Use different speeds.</p>
*
* @see <a href="https://commons.apache.org/proper/commons-rng/userguide/rng.html">
* Commons RNG user guide</a>
*/
@Param({"XO_RO_SHI_RO_128_PP",
"MWC_256",
"JDK"})
private String randomSourceName;
/** The sampler. */
private ContinuousSampler sampler;
/**
* @return the sampler.
*/
public ContinuousSampler getSampler() {
return sampler;
}
/** Instantiates sampler. */
@Setup
public void setup() {
final RandomSource randomSource = RandomSource.valueOf(randomSourceName);
final UniformRandomProvider rng = randomSource.create();
sampler = createSampler(type, rng);
}
}
/**
* The samplers to use for testing the ziggurat method with sequential sample generation.
* Defines the RandomSource and the sampler type.
*
* <p>This specifically targets repeat calls to the same sampler.
* Performance should scale linearly with the size. A plot of size against time
* can identify outliers and should allow ranking of different methods.
*
* <p>Note: Testing using a single call to the sampler may return different relative
* performance of the samplers than when testing using multiple calls. This is due to the
* single calls being performed multiple times by the JMH framework rather than a
* single block of code. Rankings should be consistent and the optimal sampler method
* chosen using both sets of results.
*/
@State(Scope.Benchmark)
public static class SequentialSources extends Sources {
/**
* RNG providers.
*
* <p>Use different speeds.</p>
*
* @see <a href="https://commons.apache.org/proper/commons-rng/userguide/rng.html">
* Commons RNG user guide</a>
*/
@Param({"XO_RO_SHI_RO_128_PP",
//"MWC_256",
//"JDK"
})
private String randomSourceName;
/** The size. */
@Param({"1", "2", "4", "8", "16", "32", "64"})
private int size;
/** The sampler. */
private ContinuousSampler sampler;
/**
* @return the sampler.
*/
public ContinuousSampler getSampler() {
return sampler;
}
/** Instantiates sampler. */
@Setup
public void setup() {
final RandomSource randomSource = RandomSource.valueOf(randomSourceName);
final UniformRandomProvider rng = randomSource.create();
final ContinuousSampler s = createSampler(type, rng);
sampler = createSequentialSampler(size, s);
}
/**
* Creates the sampler for the specified number of samples.
*
* @param size the size
* @param s the sampler to create the samples
* @return the sampler
*/
private static ContinuousSampler createSequentialSampler(int size, ContinuousSampler s) {
// Create size samples
switch (size) {
case 1:
return new Size1Sampler(s);
case 2:
return new Size2Sampler(s);
case 3:
return new Size3Sampler(s);
case 4:
return new Size4Sampler(s);
case 5:
return new Size5Sampler(s);
default:
return new SizeNSampler(s, size);
}
}
/**
* Create a specified number of samples from an underlying sampler.
*/
abstract static class SizeSampler implements ContinuousSampler {
/** The sampler. */
protected ContinuousSampler delegate;
/**
* @param delegate the sampler to create the samples
*/
SizeSampler(ContinuousSampler delegate) {
this.delegate = delegate;
}
}
/**
* Create 1 sample from the sampler.
*/
static class Size1Sampler extends SizeSampler {
/**
* @param delegate the sampler to create the samples
*/
Size1Sampler(ContinuousSampler delegate) {
super(delegate);
}
@Override
public double sample() {
return delegate.sample();
}
}
/**
* Create 2 samples from the sampler.
*/
static class Size2Sampler extends SizeSampler {
/**
* @param delegate the sampler to create the samples
*/
Size2Sampler(ContinuousSampler delegate) {
super(delegate);
}
@Override
public double sample() {
delegate.sample();
return delegate.sample();
}
}
/**
* Create 3 samples from the sampler.
*/
static class Size3Sampler extends SizeSampler {
/**
* @param delegate the sampler to create the samples
*/
Size3Sampler(ContinuousSampler delegate) {
super(delegate);
}
@Override
public double sample() {
delegate.sample();
delegate.sample();
return delegate.sample();
}
}
/**
* Create 4 samples from the sampler.
*/
static class Size4Sampler extends SizeSampler {
/**
* @param delegate the sampler to create the samples
*/
Size4Sampler(ContinuousSampler delegate) {
super(delegate);
}
@Override
public double sample() {
delegate.sample();
delegate.sample();
delegate.sample();
return delegate.sample();
}
}
/**
* Create 5 samples from the sampler.
*/
static class Size5Sampler extends SizeSampler {
/**
* @param delegate the sampler to create the samples
*/
Size5Sampler(ContinuousSampler delegate) {
super(delegate);
}
@Override
public double sample() {
delegate.sample();
delegate.sample();
delegate.sample();
delegate.sample();
return delegate.sample();
}
}
/**
* Create N samples from the sampler.
*/
static class SizeNSampler extends SizeSampler {
/** The number of samples minus 1. */
private final int sizeM1;
/**
* @param delegate the sampler to create the samples
* @param size the size
*/
SizeNSampler(ContinuousSampler delegate, int size) {
super(delegate);
if (size < 1) {
throw new IllegalArgumentException("Size must be above zero: " + size);
}
this.sizeM1 = size - 1;
}
@Override
public double sample() {
for (int i = sizeM1; i != 0; i--) {
delegate.sample();
}
return delegate.sample();
}
}
}
/**
* <a href="https://en.wikipedia.org/wiki/Ziggurat_algorithm">
* Marsaglia and Tsang "Ziggurat" method</a> for sampling from a NormalizedGaussian
* distribution with mean 0 and standard deviation 1.
*
* <p>This is a copy of {@link ZigguratNormalizedGaussianSampler} using a table size of 256.
*/
static class ZigguratNormalizedGaussianSampler128 implements ContinuousSampler {
/** Start of tail. */
private static final double R = 3.442619855899;
/** Inverse of R. */
private static final double ONE_OVER_R = 1 / R;
/** Index of last entry in the tables (which have a size that is a power of 2). */
private static final int LAST = 127;
/** Auxiliary table. */
private static final long[] K;
/** Auxiliary table. */
private static final double[] W;
/** Auxiliary table. */
private static final double[] F;
/**
* The multiplier to convert the least significant 53-bits of a {@code long} to a {@code double}.
* Taken from org.apache.commons.rng.core.util.NumberFactory.
*/
private static final double DOUBLE_MULTIPLIER = 0x1.0p-53d;
/** Underlying source of randomness. */
private final UniformRandomProvider rng;
static {
// Filling the tables.
// Rectangle area.
final double v = 9.91256303526217e-3;
// Direction support uses the sign bit so the maximum magnitude from the long is 2^63
final double max = Math.pow(2, 63);
final double oneOverMax = 1d / max;
K = new long[LAST + 1];
W = new double[LAST + 1];
F = new double[LAST + 1];
double d = R;
double t = d;
double fd = pdf(d);
final double q = v / fd;
K[0] = (long) ((d / q) * max);
K[1] = 0;
W[0] = q * oneOverMax;
W[LAST] = d * oneOverMax;
F[0] = 1;
F[LAST] = fd;
for (int i = LAST - 1; i >= 1; i--) {
d = Math.sqrt(-2 * Math.log(v / d + fd));
fd = pdf(d);
K[i + 1] = (long) ((d / t) * max);
t = d;
F[i] = fd;
W[i] = d * oneOverMax;
}
}
/**
* @param rng Generator of uniformly distributed random numbers.
*/
ZigguratNormalizedGaussianSampler128(UniformRandomProvider rng) {
this.rng = rng;
}
/** {@inheritDoc} */
@Override
public double sample() {
final long j = rng.nextLong();
final int i = ((int) j) & LAST;
if (Math.abs(j) < K[i]) {
// This branch is called about 0.972101 times per sample.
return j * W[i];
}
return fix(j, i);
}
/**
* Gets the value from the tail of the distribution.
*
* @param hz Start random integer.
* @param iz Index of cell corresponding to {@code hz}.
* @return the requested random value.
*/
private double fix(long hz,
int iz) {
if (iz == 0) {
// Base strip.
// This branch is called about 5.7624515E-4 times per sample.
double y;
double x;
do {
// Avoid infinity by creating a non-zero double.
y = -Math.log(makeNonZeroDouble(rng.nextLong()));
x = -Math.log(makeNonZeroDouble(rng.nextLong())) * ONE_OVER_R;
} while (y + y < x * x);
final double out = R + x;
return hz > 0 ? out : -out;
}
// Wedge of other strips.
// This branch is called about 0.027323 times per sample.
final double x = hz * W[iz];
if (F[iz] + rng.nextDouble() * (F[iz - 1] - F[iz]) < pdf(x)) {
// This branch is called about 0.014961 times per sample.
return x;
}
// Try again.
// This branch is called about 0.012362 times per sample.
return sample();
}
/**
* Creates a {@code double} in the interval {@code (0, 1]} from a {@code long} value.
*
* @param v Number.
* @return a {@code double} value in the interval {@code (0, 1]}.
*/
private static double makeNonZeroDouble(long v) {
// This matches the method in o.a.c.rng.core.util.NumberFactory.makeDouble(long)
// but shifts the range from [0, 1) to (0, 1].
return ((v >>> 11) + 1L) * DOUBLE_MULTIPLIER;
}
/**
* Compute the Gaussian probability density function {@code f(x) = e^-0.5x^2}.
*
* @param x Argument.
* @return \( e^{-\frac{x^2}{2}} \)
*/
private static double pdf(double x) {
return Math.exp(-0.5 * x * x);
}
}
/**
* <a href="https://en.wikipedia.org/wiki/Ziggurat_algorithm">
* Marsaglia and Tsang "Ziggurat" method</a> for sampling from an exponential
* distribution.
*
* <p>The algorithm is explained in this
* <a href="http://www.jstatsoft.org/article/view/v005i08/ziggurat.pdf">paper</a>
* and this implementation has been adapted from the C code provided therein.</p>
*/
static class ZigguratExponentialSampler implements ContinuousSampler {
/** Start of tail. */
private static final double R = 7.69711747013104972;
/** Index of last entry in the tables (which have a size that is a power of 2). */
private static final int LAST = 255;
/** Auxiliary table. */
private static final long[] K;
/** Auxiliary table. */
private static final double[] W;
/** Auxiliary table. */
private static final double[] F;
/** Underlying source of randomness. */
private final UniformRandomProvider rng;
static {
// Filling the tables.
// Rectangle area.
final double v = 0.0039496598225815571993;
// No support for unsigned long so the upper bound is 2^63
final double max = Math.pow(2, 63);
final double oneOverMax = 1d / max;
K = new long[LAST + 1];
W = new double[LAST + 1];
F = new double[LAST + 1];
double d = R;
double t = d;
double fd = pdf(d);
final double q = v / fd;
K[0] = (long) ((d / q) * max);
K[1] = 0;
W[0] = q * oneOverMax;
W[LAST] = d * oneOverMax;
F[0] = 1;
F[LAST] = fd;
for (int i = LAST - 1; i >= 1; i--) {
d = -Math.log(v / d + fd);
fd = pdf(d);
K[i + 1] = (long) ((d / t) * max);
t = d;
F[i] = fd;
W[i] = d * oneOverMax;
}
}
/**
* @param rng Generator of uniformly distributed random numbers.
*/
ZigguratExponentialSampler(UniformRandomProvider rng) {
this.rng = rng;
}
/** {@inheritDoc} */
@Override
public double sample() {
// An unsigned long in [0, 2^63)
final long j = rng.nextLong() >>> 1;
final int i = ((int) j) & LAST;
if (j < K[i]) {
// This branch is called about 0.977777 times per call into createSample.
// Note: Frequencies have been empirically measured for the first call to
// createSample; recursion due to retries have been ignored. Frequencies sum to 1.
return j * W[i];
}
return fix(j, i);
}
/**
* Gets the value from the tail of the distribution.
*
* @param jz Start random integer.
* @param iz Index of cell corresponding to {@code jz}.
* @return the requested random value.
*/
private double fix(long jz,
int iz) {
if (iz == 0) {
// Base strip.
// This branch is called about 0.000448867 times per call into createSample.
return R - Math.log(rng.nextDouble());
}
// Wedge of other strips.
final double x = jz * W[iz];
if (F[iz] + rng.nextDouble() * (F[iz - 1] - F[iz]) < pdf(x)) {
// This branch is called about 0.0107820 times per call into createSample.
return x;
}
// Try again.
// This branch is called about 0.0109920 times per call into createSample
// i.e. this is the recursion frequency.
return sample();
}
/**
* Compute the exponential probability density function {@code f(x) = e^-x}.
*
* @param x Argument.
* @return {@code e^-x}
*/
private static double pdf(double x) {
return Math.exp(-x);
}
}
/**
* Modified Ziggurat method for sampling from a Gaussian distribution with mean 0 and standard deviation 1.
*
* <p>Uses the algorithm from:
*
* <blockquote>
* McFarland, C.D. (2016)<br>
* "A modified ziggurat algorithm for generating exponentially and normally distributed pseudorandom numbers".<br>
* <i>Journal of Statistical Computation and Simulation</i> <b>86</b>, 1281-1294.
* </blockquote>
*
* <p>This class uses the same tables as the production version
* {@link org.apache.commons.rng.sampling.distribution.ZigguratSampler.NormalizedGaussian}
* with the overhang sampling matching the reference c implementation. Methods and members
* are protected to allow the implementation to be modified in sub-classes.
*
* @see <a href="https://www.tandfonline.com/doi/abs/10.1080/00949655.2015.1060234">
* McFarland (2016) JSCS 86, 1281-1294</a>
*/
static class ModifiedZigguratNormalizedGaussianSampler implements ContinuousSampler {
// Ziggurat volumes:
// Inside the layers = 98.8281% (253/256)
// Fraction outside the layers:
// concave overhangs = 76.1941%
// inflection overhang = 0.1358%
// convex overhangs = 21.3072%
// tail = 2.3629%
/** The number of layers in the ziggurat. Maximum i value for early exit. */
protected static final int I_MAX = 253;
/** The point where the Gaussian switches from convex to concave.
* This is the largest value of X[j] below 1. */
protected static final int J_INFLECTION = 204;
/** Maximum epsilon distance of convex pdf(x) above the hypotenuse value for early rejection.
* Equal to approximately 0.2460 scaled by 2^63. This is negated on purpose as the
* distance for a point (x,y) above the hypotenuse is negative:
* {@code (|d| < max) == (d >= -max)}. */
protected static final long CONVEX_E_MAX = -2269182951627976004L;
/** Maximum distance of concave pdf(x) below the hypotenuse value for early exit.
* Equal to approximately 0.08244 scaled by 2^63. */
protected static final long CONCAVE_E_MAX = 760463704284035184L;
/** Beginning of tail. Equal to X[0] * 2^63. */
protected static final double X_0 = 3.6360066255009455861;
/** 1/X_0. Used for tail sampling. */
protected static final double ONE_OVER_X_0 = 1.0 / X_0;
/** The alias map. An integer in [0, 255] stored as a byte to save space. */
protected static final byte[] MAP = {
/* [ 0] */ (byte) 0, (byte) 0, (byte) 239, (byte) 2, (byte) 0, (byte) 0, (byte) 0, (byte) 0,
/* [ 8] */ (byte) 0, (byte) 0, (byte) 0, (byte) 0, (byte) 1, (byte) 1, (byte) 1, (byte) 253,
/* [ 16] */ (byte) 253, (byte) 253, (byte) 253, (byte) 253, (byte) 253, (byte) 253, (byte) 253, (byte) 253,
/* [ 24] */ (byte) 253, (byte) 253, (byte) 253, (byte) 253, (byte) 253, (byte) 253, (byte) 253, (byte) 253,
/* [ 32] */ (byte) 253, (byte) 253, (byte) 253, (byte) 253, (byte) 253, (byte) 253, (byte) 253, (byte) 253,
/* [ 40] */ (byte) 253, (byte) 253, (byte) 253, (byte) 253, (byte) 253, (byte) 253, (byte) 253, (byte) 253,
/* [ 48] */ (byte) 253, (byte) 253, (byte) 253, (byte) 253, (byte) 253, (byte) 253, (byte) 253, (byte) 253,
/* [ 56] */ (byte) 253, (byte) 253, (byte) 253, (byte) 253, (byte) 253, (byte) 253, (byte) 253, (byte) 253,
/* [ 64] */ (byte) 253, (byte) 253, (byte) 253, (byte) 253, (byte) 253, (byte) 253, (byte) 253, (byte) 253,
/* [ 72] */ (byte) 253, (byte) 253, (byte) 253, (byte) 253, (byte) 253, (byte) 253, (byte) 253, (byte) 253,
/* [ 80] */ (byte) 253, (byte) 253, (byte) 253, (byte) 253, (byte) 253, (byte) 253, (byte) 253, (byte) 253,
/* [ 88] */ (byte) 253, (byte) 253, (byte) 253, (byte) 253, (byte) 253, (byte) 253, (byte) 253, (byte) 253,
/* [ 96] */ (byte) 253, (byte) 253, (byte) 253, (byte) 253, (byte) 253, (byte) 253, (byte) 253, (byte) 253,
/* [104] */ (byte) 253, (byte) 253, (byte) 253, (byte) 253, (byte) 253, (byte) 253, (byte) 253, (byte) 253,
/* [112] */ (byte) 253, (byte) 253, (byte) 253, (byte) 253, (byte) 253, (byte) 253, (byte) 253, (byte) 253,
/* [120] */ (byte) 253, (byte) 253, (byte) 253, (byte) 253, (byte) 253, (byte) 253, (byte) 253, (byte) 253,
/* [128] */ (byte) 253, (byte) 253, (byte) 253, (byte) 253, (byte) 253, (byte) 253, (byte) 253, (byte) 253,
/* [136] */ (byte) 253, (byte) 253, (byte) 253, (byte) 253, (byte) 253, (byte) 253, (byte) 253, (byte) 253,
/* [144] */ (byte) 253, (byte) 253, (byte) 253, (byte) 253, (byte) 253, (byte) 253, (byte) 253, (byte) 253,
/* [152] */ (byte) 253, (byte) 253, (byte) 253, (byte) 253, (byte) 253, (byte) 253, (byte) 253, (byte) 253,
/* [160] */ (byte) 253, (byte) 253, (byte) 253, (byte) 253, (byte) 253, (byte) 253, (byte) 253, (byte) 253,
/* [168] */ (byte) 253, (byte) 253, (byte) 253, (byte) 253, (byte) 253, (byte) 253, (byte) 253, (byte) 253,
/* [176] */ (byte) 253, (byte) 253, (byte) 253, (byte) 253, (byte) 253, (byte) 253, (byte) 253, (byte) 253,
/* [184] */ (byte) 253, (byte) 253, (byte) 253, (byte) 253, (byte) 253, (byte) 253, (byte) 253, (byte) 253,
/* [192] */ (byte) 253, (byte) 253, (byte) 252, (byte) 252, (byte) 252, (byte) 252, (byte) 252, (byte) 252,
/* [200] */ (byte) 252, (byte) 252, (byte) 252, (byte) 252, (byte) 251, (byte) 251, (byte) 251, (byte) 251,
/* [208] */ (byte) 251, (byte) 251, (byte) 251, (byte) 250, (byte) 250, (byte) 250, (byte) 250, (byte) 250,
/* [216] */ (byte) 249, (byte) 249, (byte) 249, (byte) 248, (byte) 248, (byte) 248, (byte) 247, (byte) 247,
/* [224] */ (byte) 247, (byte) 246, (byte) 246, (byte) 245, (byte) 244, (byte) 244, (byte) 243, (byte) 242,
/* [232] */ (byte) 240, (byte) 2, (byte) 2, (byte) 3, (byte) 3, (byte) 0, (byte) 0, (byte) 240,
/* [240] */ (byte) 241, (byte) 242, (byte) 243, (byte) 244, (byte) 245, (byte) 246, (byte) 247, (byte) 248,
/* [248] */ (byte) 249, (byte) 250, (byte) 251, (byte) 252, (byte) 253, (byte) 1, (byte) 0, (byte) 0,
};
/** The alias inverse PMF. */
protected static final long[] IPMF = {
/* [ 0] */ 9223372036854775296L, 1100243796534090752L, 7866600928998383104L, 6788754710675124736L,
/* [ 4] */ 9022865200181688320L, 6522434035205502208L, 4723064097360024576L, 3360495653216416000L,
/* [ 8] */ 2289663232373870848L, 1423968905551920384L, 708364817827798016L, 106102487305601280L,
/* [ 12] */ -408333464665794560L, -853239722779025152L, -1242095211825521408L, -1585059631105762048L,
/* [ 16] */ -1889943050287169024L, -2162852901990669824L, -2408637386594511104L, -2631196530262954496L,
/* [ 20] */ -2833704942520925696L, -3018774289025787392L, -3188573753472222208L, -3344920681707410944L,
/* [ 24] */ -3489349705062150656L, -3623166100042179584L, -3747487436868335360L, -3863276422712173824L,
/* [ 28] */ -3971367044063130880L, -4072485557029824000L, -4167267476830916608L, -4256271432240159744L,
/* [ 32] */ -4339990541927306752L, -4418861817133802240L, -4493273980372377088L, -4563574004462246656L,
/* [ 36] */ -4630072609770453760L, -4693048910430964992L, -4752754358862894848L, -4809416110052769536L,
/* [ 40] */ -4863239903586985984L, -4914412541515875840L, -4963104028439161088L, -5009469424769119232L,
/* [ 44] */ -5053650458856559360L, -5095776932695077632L, -5135967952544929024L, -5174333008451230720L,
/* [ 48] */ -5210972924952654336L, -5245980700100460288L, -5279442247516297472L, -5311437055462369280L,
/* [ 52] */ -5342038772315650560L, -5371315728843297024L, -5399331404632512768L, -5426144845448965120L,
/* [ 56] */ -5451811038519422464L, -5476381248265593088L, -5499903320558339072L, -5522421955752311296L,
/* [ 60] */ -5543978956085263616L, -5564613449659060480L, -5584362093436146432L, -5603259257517428736L,
/* [ 64] */ -5621337193070986240L, -5638626184974132224L, -5655154691220933888L, -5670949470294763008L,
/* [ 68] */ -5686035697601807872L, -5700437072199152384L, -5714175914219812352L, -5727273255295220992L,
/* [ 72] */ -5739748920271997440L, -5751621603810412032L, -5762908939773946112L, -5773627565915007744L,
/* [ 76] */ -5783793183152377600L, -5793420610475628544L, -5802523835894661376L, -5811116062947570176L,
/* [ 80] */ -5819209754516120832L, -5826816672854571776L, -5833947916825278208L, -5840613956570608128L,
/* [ 84] */ -5846824665591763456L, -5852589350491075328L, -5857916778480726528L, -5862815203334800384L,
/* [ 88] */ -5867292388935742464L, -5871355631762284032L, -5875011781262890752L, -5878267259039093760L,
/* [ 92] */ -5881128076579883520L, -5883599852028851456L, -5885687825288565248L, -5887396872144963840L,
/* [ 96] */ -5888731517955042304L, -5889695949247728384L, -5890294025706689792L, -5890529289910829568L,
/* [100] */ -5890404977675987456L, -5889924026487208448L, -5889089083913555968L, -5887902514965209344L,
/* [104] */ -5886366408898372096L, -5884482585690639872L, -5882252601321090304L, -5879677752995027712L,
/* [108] */ -5876759083794175232L, -5873497386318840832L, -5869893206505510144L, -5865946846617024256L,
/* [112] */ -5861658367354159104L, -5857027590486131456L, -5852054100063428352L, -5846737243971504640L,
/* [116] */ -5841076134082373632L, -5835069647234580480L, -5828716424754549248L, -5822014871949021952L,
/* [120] */ -5814963157357531648L, -5807559211080072192L, -5799800723447229952L, -5791685142338073344L,
/* [124] */ -5783209670985158912L, -5774371264582489344L, -5765166627072226560L, -5755592207057667840L,
/* [128] */ -5745644193442049280L, -5735318510777133824L, -5724610813433666560L, -5713516480340333056L,
/* [132] */ -5702030608556698112L, -5690148005851018752L, -5677863184109371904L, -5665170350903313408L,
/* [136] */ -5652063400924580608L, -5638535907000141312L, -5624581109999480320L, -5610191908627599872L,
/* [140] */ -5595360848093632768L, -5580080108034218752L, -5564341489875549952L, -5548136403221394688L,
/* [144] */ -5531455851545399296L, -5514290416593586944L, -5496630242226406656L, -5478465016761742848L,
/* [148] */ -5459783954986665216L, -5440575777891777024L, -5420828692432397824L, -5400530368638773504L,
/* [152] */ -5379667916699401728L, -5358227861294116864L, -5336196115274292224L, -5313557951078385920L,
/* [156] */ -5290297970633451520L, -5266400072915222272L, -5241847420214015744L, -5216622401043726592L,
/* [160] */ -5190706591719534080L, -5164080714589203200L, -5136724594099067136L, -5108617109269313024L,
/* [164] */ -5079736143458214912L, -5050058530461741312L, -5019559997031891968L, -4988215100963582976L,
/* [168] */ -4955997165645491968L, -4922878208652041728L, -4888828866780320000L, -4853818314258475776L,
/* [172] */ -4817814175855180032L, -4780782432601701888L, -4742687321746719232L, -4703491227581444608L,
/* [176] */ -4663154564978699264L, -4621635653358766336L, -4578890580370785792L, -4534873055659683584L,
/* [180] */ -4489534251700611840L, -4442822631898829568L, -4394683764809104128L, -4345060121983362560L,
/* [184] */ -4293890858708922880L, -4241111576153830144L, -4186654061692619008L, -4130446006804747776L,
/* [188] */ -4072410698657718784L, -4012466683838401024L, -3950527400305017856L, -3886500774061896704L,
/* [192] */ -3820288777467837184L, -3751786943594897664L, -3680883832433527808L, -3607460442623922176L,
/* [196] */ -3531389562483324160L, -3452535052891361792L, -3370751053395887872L, -3285881101633968128L,
/* [200] */ -3197757155301365504L, -3106198503156485376L, -3011010550911937280L, -2911983463883580928L,
/* [204] */ -2808890647470271744L, -2701487041141149952L, -2589507199690603520L, -2472663129329160192L,
/* [208] */ -2350641842139870464L, -2223102583770035200L, -2089673683684728576L, -1949948966090106880L,
/* [212] */ -1803483646855993856L, -1649789631480328192L, -1488330106139747584L, -1318513295725618176L,
/* [216] */ -1139685236927327232L, -951121376596854784L, -752016768184775936L, -541474585642866432L,
/* [220] */ -318492605725778432L, -81947227249193216L, 169425512612864512L, 437052607232193536L,
/* [224] */ 722551297568809984L, 1027761939299714304L, 1354787941622770432L, 1706044619203941632L,
/* [228] */ 2084319374409574144L, 2492846399593711360L, 2935400169348532480L, 3416413484613111552L,
/* [232] */ 3941127949860576256L, 4515787798793437952L, 5147892401439714304L, 5846529325380406016L,
/* [236] */ 6622819682216655360L, 7490522659874166016L, 8466869998277892096L, 8216968526387345408L,
/* [240] */ 4550693915488934656L, 7628019504138977280L, 6605080500908005888L, 7121156327650272512L,
/* [244] */ 2484871780331574272L, 7179104797032803328L, 7066086283830045440L, 1516500120817362944L,
/* [248] */ 216305945438803456L, 6295963418525324544L, 2889316805630113280L, -2712587580533804032L,
/* [252] */ 6562498853538167040L, 7975754821147501312L, -9223372036854775808L, -9223372036854775808L,
};
/**
* The precomputed ziggurat lengths, denoted X_i in the main text. X_i = length of
* ziggurat layer i. Values have been scaled by 2^-63.
*/
protected static final double[] X = {
/* [ 0] */ 3.9421662825398133e-19, 3.7204945004119012e-19, 3.5827024480628678e-19, 3.4807476236540249e-19,
/* [ 4] */ 3.3990177171882136e-19, 3.3303778360340139e-19, 3.270943881761755e-19, 3.21835771324951e-19,
/* [ 8] */ 3.1710758541840432e-19, 3.1280307407034065e-19, 3.0884520655804019e-19, 3.0517650624107352e-19,
/* [ 12] */ 3.01752902925846e-19, 2.985398344070532e-19, 2.9550967462801797e-19, 2.9263997988491663e-19,
/* [ 16] */ 2.8991225869977476e-19, 2.8731108780226291e-19, 2.8482346327101335e-19, 2.8243831535194389e-19,
/* [ 20] */ 2.8014613964727031e-19, 2.7793871261807797e-19, 2.7580886921411212e-19, 2.7375032698308758e-19,
/* [ 24] */ 2.7175754543391047e-19, 2.6982561247538484e-19, 2.6795015188771505e-19, 2.6612724730440033e-19,
/* [ 28] */ 2.6435337927976633e-19, 2.6262537282028438e-19, 2.6094035335224142e-19, 2.5929570954331002e-19,
/* [ 32] */ 2.5768906173214726e-19, 2.5611823497719608e-19, 2.5458123593393361e-19, 2.5307623292372459e-19,
/* [ 36] */ 2.51601538677984e-19, 2.5015559533646191e-19, 2.4873696135403158e-19, 2.4734430003079206e-19,
/* [ 40] */ 2.4597636942892726e-19, 2.446320134791245e-19, 2.4331015411139206e-19, 2.4200978427132955e-19,
/* [ 44] */ 2.4072996170445879e-19, 2.3946980340903347e-19, 2.3822848067252674e-19, 2.3700521461931801e-19,
/* [ 48] */ 2.357992722074133e-19, 2.3460996262069972e-19, 2.3343663401054455e-19, 2.322786705467384e-19,
/* [ 52] */ 2.3113548974303765e-19, 2.3000654002704238e-19, 2.2889129852797606e-19, 2.2778926905921897e-19,
/* [ 56] */ 2.2669998027527321e-19, 2.2562298398527416e-19, 2.245578536072726e-19, 2.2350418274933911e-19,
/* [ 60] */ 2.2246158390513294e-19, 2.2142968725296249e-19, 2.2040813954857555e-19, 2.1939660310297601e-19,
/* [ 64] */ 2.1839475483749618e-19, 2.1740228540916853e-19, 2.1641889840016519e-19, 2.1544430956570613e-19,
/* [ 68] */ 2.1447824613540345e-19, 2.1352044616350571e-19, 2.1257065792395107e-19, 2.1162863934653125e-19,
/* [ 72] */ 2.1069415749082026e-19, 2.0976698805483467e-19, 2.0884691491567363e-19, 2.0793372969963634e-19,
/* [ 76] */ 2.0702723137954107e-19, 2.0612722589717129e-19, 2.0523352580895635e-19, 2.0434594995315797e-19,
/* [ 80] */ 2.0346432313698148e-19, 2.0258847584216418e-19, 2.0171824394771313e-19, 2.0085346846857531e-19,
/* [ 84] */ 1.9999399530912015e-19, 1.9913967503040585e-19, 1.9829036263028144e-19, 1.9744591733545175e-19,
/* [ 88] */ 1.9660620240469857e-19, 1.9577108494251485e-19, 1.9494043572246307e-19, 1.9411412901962161e-19,
/* [ 92] */ 1.9329204245152935e-19, 1.9247405682708168e-19, 1.9166005600287074e-19, 1.9084992674649826e-19,
/* [ 96] */ 1.900435586064234e-19, 1.8924084378793725e-19, 1.8844167703488436e-19, 1.8764595551677749e-19,
/* [100] */ 1.868535787209745e-19, 1.8606444834960934e-19, 1.8527846822098793e-19, 1.8449554417517928e-19,
/* [104] */ 1.8371558398354868e-19, 1.8293849726199566e-19, 1.8216419538767393e-19, 1.8139259141898448e-19,
/* [108] */ 1.8062360001864453e-19, 1.7985713737964743e-19, 1.7909312115393845e-19, 1.78331470383642e-19,
/* [112] */ 1.7757210543468428e-19, 1.7681494793266395e-19, 1.760599207008314e-19, 1.7530694770004409e-19,
/* [116] */ 1.7455595397057217e-19, 1.7380686557563475e-19, 1.7305960954655264e-19, 1.7231411382940904e-19,
/* [120] */ 1.7157030723311378e-19, 1.7082811937877138e-19, 1.7008748065025788e-19, 1.6934832214591352e-19,
/* [124] */ 1.6861057563126349e-19, 1.6787417349268046e-19, 1.6713904869190636e-19, 1.6640513472135291e-19,
/* [128] */ 1.6567236556010242e-19, 1.6494067563053266e-19, 1.6420999975549115e-19, 1.6348027311594532e-19,
/* [132] */ 1.6275143120903661e-19, 1.6202340980646725e-19, 1.6129614491314931e-19, 1.6056957272604589e-19,
/* [136] */ 1.5984362959313479e-19, 1.5911825197242491e-19, 1.5839337639095554e-19, 1.57668939403708e-19,
/* [140] */ 1.5694487755235889e-19, 1.5622112732380261e-19, 1.554976251083707e-19, 1.5477430715767271e-19,
/* [144] */ 1.540511095419833e-19, 1.5332796810709688e-19, 1.5260481843056974e-19, 1.5188159577726683e-19,
/* [148] */ 1.5115823505412761e-19, 1.5043467076406199e-19, 1.4971083695888395e-19, 1.4898666719118714e-19,
/* [152] */ 1.4826209446506113e-19, 1.4753705118554365e-19, 1.468114691066983e-19, 1.4608527927820112e-19,
/* [156] */ 1.4535841199031451e-19, 1.4463079671711862e-19, 1.4390236205786415e-19, 1.4317303567630177e-19,
/* [160] */ 1.4244274423783481e-19, 1.4171141334433217e-19, 1.4097896746642792e-19, 1.4024532987312287e-19,
/* [164] */ 1.3951042255849034e-19, 1.3877416616527576e-19, 1.3803647990516385e-19, 1.3729728147547174e-19,
/* [168] */ 1.3655648697200824e-19, 1.3581401079782068e-19, 1.3506976556752901e-19, 1.3432366200692418e-19,
/* [172] */ 1.3357560884748263e-19, 1.3282551271542047e-19, 1.3207327801488087e-19, 1.3131880680481524e-19,
/* [176] */ 1.3056199866908076e-19, 1.2980275057923788e-19, 1.2904095674948608e-19, 1.2827650848312727e-19,
/* [180] */ 1.2750929400989213e-19, 1.2673919831340482e-19, 1.2596610294799512e-19, 1.2518988584399374e-19,
/* [184] */ 1.2441042110056523e-19, 1.2362757876504165e-19, 1.2284122459762072e-19, 1.2205121982017852e-19,
/* [188] */ 1.2125742084782245e-19, 1.2045967900166973e-19, 1.196578402011802e-19, 1.1885174463419555e-19,
/* [192] */ 1.1804122640264091e-19, 1.1722611314162064e-19, 1.1640622560939109e-19, 1.1558137724540874e-19,
/* [196] */ 1.1475137369333185e-19, 1.1391601228549047e-19, 1.1307508148492592e-19, 1.1222836028063025e-19,
/* [200] */ 1.1137561753107903e-19, 1.1051661125053526e-19, 1.0965108783189755e-19, 1.0877878119905372e-19,
/* [204] */ 1.0789941188076655e-19, 1.070126859970364e-19, 1.0611829414763286e-19, 1.0521591019102928e-19,
/* [208] */ 1.0430518990027552e-19, 1.0338576948035472e-19, 1.0245726392923699e-19, 1.015192652220931e-19,
/* [212] */ 1.0057134029488235e-19, 9.9613028799672809e-20, 9.8643840599459914e-20, 9.7663252964755816e-20,
/* [216] */ 9.6670707427623454e-20, 9.566560624086667e-20, 9.4647308380433213e-20, 9.3615125017323508e-20,
/* [220] */ 9.2568314370887282e-20, 9.1506075837638774e-20, 9.0427543267725716e-20, 8.933177723376368e-20,
/* [224] */ 8.8217756102327883e-20, 8.7084365674892319e-20, 8.5930387109612162e-20, 8.4754482764244349e-20,
/* [228] */ 8.3555179508462343e-20, 8.2330848933585364e-20, 8.1079683729129853e-20, 7.9799669284133864e-20,
/* [232] */ 7.8488549286072745e-20, 7.7143783700934692e-20, 7.5762496979467566e-20, 7.4341413578485329e-20,
/* [236] */ 7.2876776807378431e-20, 7.1364245443525374e-20, 6.9798760240761066e-20, 6.8174368944799054e-20,
/* [240] */ 6.6483992986198539e-20, 6.4719110345162767e-20, 6.2869314813103699e-20, 6.0921687548281263e-20,
/* [244] */ 5.8859873575576818e-20, 5.6662675116090981e-20, 5.4301813630894571e-20, 5.173817174449422e-20,
/* [248] */ 4.8915031722398545e-20, 4.5744741890755301e-20, 4.2078802568583416e-20, 3.7625986722404761e-20,
/* [252] */ 3.1628589805881879e-20, 0,
};
/** The precomputed ziggurat heights, denoted Y_i in the main text. Y_i = f(X_i).
* Values have been scaled by 2^-63. */
protected static final double[] Y = {
/* [ 0] */ 1.4598410796619063e-22, 3.0066613427942797e-22, 4.6129728815103466e-22, 6.2663350049234362e-22,
/* [ 4] */ 7.9594524761881544e-22, 9.6874655021705039e-22, 1.1446877002379439e-21, 1.3235036304379167e-21,
/* [ 8] */ 1.5049857692053131e-21, 1.6889653000719298e-21, 1.8753025382711626e-21, 2.0638798423695191e-21,
/* [ 12] */ 2.2545966913644708e-21, 2.4473661518801799e-21, 2.6421122727763533e-21, 2.8387681187879908e-21,
/* [ 16] */ 3.0372742567457284e-21, 3.2375775699986589e-21, 3.439630315794878e-21, 3.6433893657997798e-21,
/* [ 20] */ 3.8488155868912312e-21, 4.0558733309492775e-21, 4.264530010428359e-21, 4.4747557422305067e-21,
/* [ 24] */ 4.6865230465355582e-21, 4.8998065902775257e-21, 5.1145829672105489e-21, 5.3308305082046173e-21,
/* [ 28] */ 5.5485291167031758e-21, 5.7676601252690476e-21, 5.9882061699178461e-21, 6.2101510795442221e-21,
/* [ 32] */ 6.4334797782257209e-21, 6.6581781985713897e-21, 6.8842332045893181e-21, 7.1116325227957095e-21,
/* [ 36] */ 7.3403646804903092e-21, 7.5704189502886418e-21, 7.8017853001379744e-21, 8.0344543481570017e-21,
/* [ 40] */ 8.2684173217333118e-21, 8.5036660203915022e-21, 8.7401927820109521e-21, 8.9779904520281901e-21,
/* [ 44] */ 9.2170523553061439e-21, 9.457372270392882e-21, 9.698944405926943e-21, 9.9417633789758424e-21,
/* [ 48] */ 1.0185824195119818e-20, 1.043112223011477e-20, 1.0677653212987396e-20, 1.0925413210432004e-20,
/* [ 52] */ 1.1174398612392891e-20, 1.1424606118728715e-20, 1.1676032726866302e-20, 1.1928675720361027e-20,
/* [ 56] */ 1.2182532658289373e-20, 1.2437601365406785e-20, 1.2693879923010674e-20, 1.2951366660454145e-20,
/* [ 60] */ 1.3210060147261461e-20, 1.3469959185800733e-20, 1.3731062804473644e-20, 1.3993370251385596e-20,
/* [ 64] */ 1.4256880988463136e-20, 1.4521594685988369e-20, 1.4787511217522902e-20, 1.505463065519617e-20,
/* [ 68] */ 1.5322953265335218e-20, 1.5592479504415048e-20, 1.5863210015310328e-20, 1.6135145623830982e-20,
/* [ 72] */ 1.6408287335525592e-20, 1.6682636332737932e-20, 1.6958193971903124e-20, 1.7234961781071113e-20,
/* [ 76] */ 1.7512941457646084e-20, 1.7792134866331487e-20, 1.807254403727107e-20, 1.8354171164377277e-20,
/* [ 80] */ 1.8637018603838945e-20, 1.8921088872801004e-20, 1.9206384648209468e-20, 1.9492908765815636e-20,
/* [ 84] */ 1.9780664219333857e-20, 2.0069654159747839e-20, 2.0359881894760859e-20, 2.0651350888385696e-20,
/* [ 88] */ 2.0944064760670539e-20, 2.1238027287557466e-20, 2.1533242400870487e-20, 2.1829714188430474e-20,
/* [ 92] */ 2.2127446894294597e-20, 2.242644491911827e-20, 2.2726712820637798e-20, 2.3028255314272276e-20,
/* [ 96] */ 2.3331077273843558e-20, 2.3635183732413286e-20, 2.3940579883236352e-20, 2.4247271080830277e-20,
/* [100] */ 2.455526284216033e-20, 2.4864560847940368e-20, 2.5175170944049622e-20, 2.5487099143065929e-20,
/* [104] */ 2.5800351625915997e-20, 2.6114934743643687e-20, 2.6430855019297323e-20, 2.6748119149937411e-20,
/* [108] */ 2.7066734008766247e-20, 2.7386706647381193e-20, 2.7708044298153558e-20, 2.8030754376735269e-20,
/* [112] */ 2.8354844484695747e-20, 2.8680322412291631e-20, 2.9007196141372126e-20, 2.9335473848423219e-20,
/* [116] */ 2.9665163907753988e-20, 2.9996274894828624e-20, 3.0328815589748056e-20, 3.0662794980885287e-20,
/* [120] */ 3.099822226867876e-20, 3.1335106869588609e-20, 3.1673458420220558e-20, 3.2013286781622988e-20,
/* [124] */ 3.2354602043762612e-20, 3.2697414530184806e-20, 3.304173480286495e-20, 3.3387573667257349e-20,
/* [128] */ 3.3734942177548938e-20, 3.4083851642125208e-20, 3.4434313629256243e-20, 3.4786339973011376e-20,
/* [132] */ 3.5139942779411164e-20, 3.5495134432826171e-20, 3.585192760263246e-20, 3.6210335250134172e-20,
/* [136] */ 3.6570370635764384e-20, 3.6932047326575882e-20, 3.7295379204034252e-20, 3.7660380472126401e-20,
/* [140] */ 3.8027065665798284e-20, 3.8395449659736649e-20, 3.8765547677510167e-20, 3.9137375301086406e-20,
/* [144] */ 3.9510948480742172e-20, 3.988628354538543e-20, 4.0263397213308566e-20, 4.0642306603393541e-20,
/* [148] */ 4.1023029246790967e-20, 4.1405583099096438e-20, 4.1789986553048817e-20, 4.2176258451776819e-20,
/* [152] */ 4.2564418102621759e-20, 4.2954485291566197e-20, 4.3346480298300118e-20, 4.3740423911958146e-20,
/* [156] */ 4.4136337447563716e-20, 4.4534242763218286e-20, 4.4934162278076256e-20, 4.5336118991149025e-20,
/* [160] */ 4.5740136500984466e-20, 4.6146239026271279e-20, 4.6554451427421133e-20, 4.6964799229185088e-20,
/* [164] */ 4.7377308644364938e-20, 4.7792006598684169e-20, 4.8208920756888113e-20, 4.8628079550147814e-20,
/* [168] */ 4.9049512204847653e-20, 4.9473248772842596e-20, 4.9899320163277674e-20, 5.0327758176068971e-20,
/* [172] */ 5.0758595537153414e-20, 5.1191865935622696e-20, 5.1627604062866059e-20, 5.2065845653856416e-20,
/* [176] */ 5.2506627530725194e-20, 5.2949987648783448e-20, 5.3395965145159426e-20, 5.3844600390237576e-20,
/* [180] */ 5.4295935042099358e-20, 5.4750012104183868e-20, 5.5206875986405073e-20, 5.5666572569983821e-20,
/* [184] */ 5.6129149276275792e-20, 5.6594655139902476e-20, 5.7063140886520563e-20, 5.7534659015596918e-20,
/* [188] */ 5.8009263888591218e-20, 5.8487011822987583e-20, 5.8967961192659803e-20, 5.9452172535103471e-20,
/* [192] */ 5.9939708666122605e-20, 6.0430634802618929e-20, 6.0925018694200531e-20, 6.142293076440286e-20,
/* [196] */ 6.1924444262401531e-20, 6.2429635426193939e-20, 6.2938583658336214e-20, 6.3451371715447563e-20,
/* [200] */ 6.3968085912834963e-20, 6.4488816345752736e-20, 6.5013657128995346e-20, 6.5542706656731714e-20,
/* [204] */ 6.6076067884730717e-20, 6.6613848637404196e-20, 6.715616194241298e-20, 6.770312639595058e-20,
/* [208] */ 6.8254866562246408e-20, 6.8811513411327825e-20, 6.9373204799659681e-20, 6.9940085998959109e-20,
/* [212] */ 7.0512310279279503e-20, 7.1090039553397167e-20, 7.1673445090644796e-20, 7.2262708309655784e-20,
/* [216] */ 7.2858021661057338e-20, 7.34595896130358e-20, 7.4067629754967553e-20, 7.4682374037052817e-20,
/* [220] */ 7.5304070167226666e-20, 7.5932983190698547e-20, 7.6569397282483754e-20, 7.7213617789487678e-20,
/* [224] */ 7.7865973566417016e-20, 7.8526819659456755e-20, 7.919654040385056e-20, 7.9875553017037968e-20,
/* [228] */ 8.056431178890163e-20, 8.1263312996426176e-20, 8.1973100703706304e-20, 8.2694273652634034e-20,
/* [232] */ 8.3427493508836792e-20, 8.4173494807453416e-20, 8.4933097052832066e-20, 8.5707219578230905e-20,
/* [236] */ 8.6496899985930695e-20, 8.7303317295655327e-20, 8.8127821378859504e-20, 8.8971970928196666e-20,
/* [240] */ 8.9837583239314064e-20, 9.0726800697869543e-20, 9.1642181484063544e-20, 9.2586826406702765e-20,
/* [244] */ 9.3564561480278864e-20, 9.4580210012636175e-20, 9.5640015550850358e-20, 9.675233477050313e-20,
/* [248] */ 9.7928851697808831e-20, 9.9186905857531331e-20, 1.0055456271343397e-19, 1.0208407377305566e-19,
/* [252] */ 1.0390360993240711e-19, 1.0842021724855044e-19,
};
/** Underlying source of randomness. */
protected final UniformRandomProvider rng;
/** Exponential sampler used for the long tail. */
protected final ContinuousSampler exponential;
/**
* @param rng Generator of uniformly distributed random numbers.
*/
ModifiedZigguratNormalizedGaussianSampler(UniformRandomProvider rng) {
this.rng = rng;
exponential = new ModifiedZigguratExponentialSampler(rng);
}
/**
* Provide access to the precomputed ziggurat lengths.
*
* <p>This is package-private to allow usage in the interpolation test.
*
* @return x
*/
static double[] getX() {
return X;
}
/** {@inheritDoc} */
@Override
public double sample() {
final long xx = nextLong();
// Float multiplication squashes these last 8 bits, so they can be used to sample i
final int i = ((int) xx) & 0xff;
if (i < I_MAX) {
// Early exit.
return X[i] * xx;
}
// Recycle bits then advance RNG:
long u1 = xx & MAX_INT64;
// Another squashed, recyclable bit
// double sign_bit = u1 & 0x100 ? 1. : -1.
// Use 2 - 1 or 0 - 1
final double signBit = ((u1 >>> 7) & 0x2) - 1.0;
final int j = selectRegion();
// Four kinds of overhangs:
// j = 0 : Sample from tail
// 0 < j < J_INFLECTION : Overhang is concave; only sample from Lower-Left triangle
// j = J_INFLECTION : Must sample from entire overhang rectangle
// j > J_INFLECTION : Overhangs are convex; implicitly accept point in Lower-Left triangle
//
// Conditional statements are arranged such that the more likely outcomes are first.
double x;
if (j > J_INFLECTION) {
// Convex overhang
for (;;) {
x = sampleX(X, j, u1);
final long uDistance = randomInt63() - u1;
if (uDistance >= 0) {
// Lower-left triangle
break;
}
if (uDistance >= CONVEX_E_MAX &&
// Within maximum distance of f(x) from the triangle hypotenuse.
sampleY(Y, j, u1 + uDistance) < Math.exp(-0.5 * x * x)) {
break;
}
// uDistance < E_MAX (upper-right triangle) or rejected as above the curve
u1 = randomInt63();
}
} else if (j < J_INFLECTION) {
if (j == 0) {
// Tail
// Note: Although less frequent than the next branch, j == 0 is a subset of
// j < J_INFLECTION and must be first.
do {
x = ONE_OVER_X_0 * exponential.sample();
} while (exponential.sample() < 0.5 * x * x);
x += X_0;
} else {
// Concave overhang
for (;;) {
// U_x <- min(U_1, U_2)
// distance <- | U_1 - U_2 |
// U_y <- 1 - (U_x + distance)
long uDistance = randomInt63() - u1;
if (uDistance < 0) {
// Upper-right triangle. Reflect in hypotenuse.
uDistance = -uDistance;
u1 -= uDistance;
}
x = sampleX(X, j, u1);
if (uDistance > CONCAVE_E_MAX ||
sampleY(Y, j, u1 + uDistance) < Math.exp(-0.5 * x * x)) {
break;
}
u1 = randomInt63();
}
}
} else {
// Inflection point
for (;;) {
x = sampleX(X, j, u1);
if (sampleY(Y, j, randomInt63()) < Math.exp(-0.5 * x * x)) {
break;
}
u1 = randomInt63();
}
}
return signBit * x;
}
/**
* Select the overhang region or the tail using alias sampling.
*
* @return the region
*/
protected int selectRegion() {
final long x = nextLong();
// j in [0, 256)
final int j = ((int) x) & 0xff;
// map to j in [0, N] with N the number of layers of the ziggurat
return x >= IPMF[j] ? MAP[j] & 0xff : j;
}
/**
* Generates a {@code long}.
*
* @return the long
*/
protected long nextLong() {
return rng.nextLong();
}
/**
* Return a positive long in {@code [0, 2^63)}.
*
* @return the long
*/
protected long randomInt63() {
return rng.nextLong() & MAX_INT64;
}
}
/**
* Modified Ziggurat method for sampling from a Gaussian distribution with mean 0 and standard deviation 1.
*
* <p>Uses the algorithm from McFarland, C.D. (2016).
*
* <p>This implementation uses simple overhangs and does not exploit the precomputed
* distances of the concave and convex overhangs. The implementation matches the c-reference
* compiled using -DSIMPLE_OVERHANGS for the non-tail overhangs.
*
* <p>Note: The tail exponential sampler does not use simple overhangs. This facilitates
* performance comparison to the fast overhang method by keeping tail sampling the same.
*/
static class ModifiedZigguratNormalizedGaussianSamplerSimpleOverhangs
extends ModifiedZigguratNormalizedGaussianSampler {
/**
* @param rng Generator of uniformly distributed random numbers.
*/
ModifiedZigguratNormalizedGaussianSamplerSimpleOverhangs(UniformRandomProvider rng) {
super(rng);
}
/** {@inheritDoc} */
@Override
public double sample() {
final long xx = nextLong();
// Float multiplication squashes these last 8 bits, so they can be used to sample i
final int i = ((int) xx) & 0xff;
if (i < I_MAX) {
// Early exit.
return X[i] * xx;
}
// Another squashed, recyclable bit
// double sign_bit = u1 & 0x100 ? 1. : -1.
// Use 2 - 1 or 0 - 1
final double signBit = ((xx >>> 7) & 0x2) - 1.0;
final int j = selectRegion();
// Simple overhangs
double x;
if (j == 0) {
// Tail
do {
x = ONE_OVER_X_0 * exponential.sample();
} while (exponential.sample() < 0.5 * x * x);
x += X_0;
} else {
// Rejection sampling
// Recycle bits then advance RNG:
long u1 = xx & MAX_INT64;
for (;;) {
x = sampleX(X, j, u1);
if (sampleY(Y, j, randomInt63()) < Math.exp(-0.5 * x * x)) {
break;
}
u1 = randomInt63();
}
}
return signBit * x;
}
}
/**
* Modified Ziggurat method for sampling from a Gaussian distribution with mean 0 and standard deviation 1.
*
* <p>Uses the algorithm from McFarland, C.D. (2016).
*
* <p>This implementation separates sampling of the main ziggurat and sampling from the edge
* into different methods. This allows inlining of the main sample method.
*/
static class ModifiedZigguratNormalizedGaussianSamplerInlining
extends ModifiedZigguratNormalizedGaussianSampler {
/**
* @param rng Generator of uniformly distributed random numbers.
*/
ModifiedZigguratNormalizedGaussianSamplerInlining(UniformRandomProvider rng) {
super(rng);
}
/** {@inheritDoc} */
@Override
public double sample() {
// Ideally this method byte code size should be below -XX:MaxInlineSize
// (which defaults to 35 bytes). This compiles to 33 bytes.
final long xx = nextLong();
// Float multiplication squashes these last 8 bits, so they can be used to sample i
final int i = ((int) xx) & 0xff;
if (i < I_MAX) {
// Early exit.
return X[i] * xx;
}
return edgeSample(xx);
}
/**
* Create the sample from the edge of the ziggurat.
*
* <p>This method has been extracted to fit the main sample method within 35 bytes (the
* default size for a JVM to inline a method).
*
* @param xx Initial random deviate
* @return a sample
*/
private double edgeSample(long xx) {
// Recycle bits then advance RNG:
long u1 = xx & MAX_INT64;
// Another squashed, recyclable bit
// double sign_bit = u1 & 0x100 ? 1. : -1.
// Use 2 - 1 or 0 - 1
final double signBit = ((u1 >>> 7) & 0x2) - 1.0;
final int j = selectRegion();
// Four kinds of overhangs:
// j = 0 : Sample from tail
// 0 < j < J_INFLECTION : Overhang is concave; only sample from Lower-Left triangle
// j = J_INFLECTION : Must sample from entire overhang rectangle
// j > J_INFLECTION : Overhangs are convex; implicitly accept point in Lower-Left triangle
//
// Conditional statements are arranged such that the more likely outcomes are first.
double x;
if (j > J_INFLECTION) {
// Convex overhang
for (;;) {
x = sampleX(X, j, u1);
final long uDistance = randomInt63() - u1;
if (uDistance >= 0) {
// Lower-left triangle
break;
}
if (uDistance >= CONVEX_E_MAX &&
// Within maximum distance of f(x) from the triangle hypotenuse.
sampleY(Y, j, u1 + uDistance) < Math.exp(-0.5 * x * x)) {
break;
}
// uDistance < CONVEX_E_MAX (upper-right triangle) or rejected as above the curve
u1 = randomInt63();
}
} else if (j < J_INFLECTION) {
if (j == 0) {
// Tail
// Note: Although less frequent than the next branch, j == 0 is a subset of
// j < J_INFLECTION and must be first.
do {
x = ONE_OVER_X_0 * exponential.sample();
} while (exponential.sample() < 0.5 * x * x);
x += X_0;
} else {
// Concave overhang
for (;;) {
// U_x <- min(U_1, U_2)
// distance <- | U_1 - U_2 |
// U_y <- 1 - (U_x + distance)
long uDistance = randomInt63() - u1;
if (uDistance < 0) {
uDistance = -uDistance;
u1 -= uDistance;
}
x = sampleX(X, j, u1);
if (uDistance > CONCAVE_E_MAX ||
sampleY(Y, j, u1 + uDistance) < Math.exp(-0.5 * x * x)) {
break;
}
u1 = randomInt63();
}
}
} else {
// Inflection point
for (;;) {
x = sampleX(X, j, u1);
if (sampleY(Y, j, randomInt63()) < Math.exp(-0.5 * x * x)) {
break;
}
u1 = randomInt63();
}
}
return signBit * x;
}
}
/**
* Modified Ziggurat method for sampling from a Gaussian distribution with mean 0 and standard deviation 1.
*
* <p>Uses the algorithm from McFarland, C.D. (2016).
*
* <p>This implementation separates sampling of the main ziggurat and sampling from the edge
* into different methods. This allows inlining of the main sample method.
*
* <p>Positive longs are created using bit shifts (not masking). The y coordinate is
* generated with u2 not (1-u2) which avoids a subtraction.
*/
static class ModifiedZigguratNormalizedGaussianSamplerInliningShift
extends ModifiedZigguratNormalizedGaussianSampler {
/**
* @param rng Generator of uniformly distributed random numbers.
*/
ModifiedZigguratNormalizedGaussianSamplerInliningShift(UniformRandomProvider rng) {
super(rng);
}
/** {@inheritDoc} */
@Override
public double sample() {
// Ideally this method byte code size should be below -XX:MaxInlineSize
// (which defaults to 35 bytes). This compiles to 33 bytes.
final long xx = nextLong();
// Float multiplication squashes these last 8 bits, so they can be used to sample i
final int i = ((int) xx) & 0xff;
if (i < I_MAX) {
// Early exit.
return X[i] * xx;
}
return edgeSample(xx);
}
/**
* Create the sample from the edge of the ziggurat.
*
* <p>This method has been extracted to fit the main sample method within 35 bytes (the
* default size for a JVM to inline a method).
*
* @param xx Initial random deviate
* @return a sample
*/
private double edgeSample(long xx) {
// Recycle bits.
// Remove sign bit to create u1.
long u1 = (xx << 1) >>> 1;
// Extract the sign bit:
// Use 2 - 1 or 0 - 1
final double signBit = ((xx >>> 62) & 0x2) - 1.0;
final int j = selectRegion();
// Four kinds of overhangs:
// j = 0 : Sample from tail
// 0 < j < J_INFLECTION : Overhang is concave; only sample from Lower-Left triangle
// j = J_INFLECTION : Must sample from entire overhang rectangle
// j > J_INFLECTION : Overhangs are convex; implicitly accept point in Lower-Left triangle
//
// Conditional statements are arranged such that the more likely outcomes are first.
double x;
if (j > J_INFLECTION) {
// Convex overhang
for (;;) {
x = interpolateSample(X, j, u1);
final long uDistance = (nextLong() >>> 1) - u1;
if (uDistance >= 0) {
// Lower-left triangle
break;
}
if (uDistance >= CONVEX_E_MAX &&
// Within maximum distance of f(x) from the triangle hypotenuse.
// u2 = (u1 + uDistance)
interpolateSample(Y, j, u1 + uDistance) < Math.exp(-0.5 * x * x)) {
break;
}
// uDistance < CONVEX_E_MAX (upper-right triangle) or rejected as above the curve
u1 = nextLong() >>> 1;
}
} else if (j < J_INFLECTION) {
if (j == 0) {
// Tail
// Note: Although less frequent than the next branch, j == 0 is a subset of
// j < J_INFLECTION and must be first.
do {
x = ONE_OVER_X_0 * exponential.sample();
} while (exponential.sample() < 0.5 * x * x);
x += X_0;
} else {
// Concave overhang
for (;;) {
// U_x <- min(U_1, U_2)
// distance <- | U_1 - U_2 |
// U_y <- 1 - (U_x + distance)
long uDistance = (nextLong() >>> 1) - u1;
if (uDistance < 0) {
uDistance = -uDistance;
u1 -= uDistance;
}
x = interpolateSample(X, j, u1);
if (uDistance > CONCAVE_E_MAX ||
interpolateSample(Y, j, u1 + uDistance) < Math.exp(-0.5 * x * x)) {
break;
}
u1 = nextLong() >>> 1;
}
}
} else {
// Inflection point
for (;;) {
x = interpolateSample(X, j, u1);
if (interpolateSample(Y, j, nextLong() >>> 1) < Math.exp(-0.5 * x * x)) {
break;
}
u1 = nextLong() >>> 1;
}
}
return signBit * x;
}
/**
* Interpolate x from the uniform deviate.
* <pre>
* value = x[j] + u * (x[j-1] - x[j])
* </pre>
* <p>If x is the precomputed lengths of the ziggurat (X) then x[j-1] is larger and this adds
* a delta to x[j].
* <p>If x is the precomputed pdf for the lengths of the ziggurat (Y) then x[j-1] is smaller
* larger and this subtracts a delta from x[j].
*
* @param x x
* @param j j
* @param u uniform deviate
* @return the sample
*/
private static double interpolateSample(double[] x, int j, long u) {
return x[j] * TWO_POW_63 + (x[j - 1] - x[j]) * u;
}
}
/**
* Modified Ziggurat method for sampling from a Gaussian distribution with mean 0 and standard deviation 1.
*
* <p>Uses the algorithm from McFarland, C.D. (2016).
*
* <p>This implementation extracts the separates the sample method for the main ziggurat
* from the edge sampling. This allows inlining of the main sample method.
*
* <p>This implementation uses simple overhangs and does not exploit the precomputed
* distances of the concave and convex overhangs. The implementation matches the c-reference
* compiled using -DSIMPLE_OVERHANGS for the non-tail overhangs.
*
* <p>Note: The tail exponential sampler does not use simple overhangs. This facilitates
* performance comparison to the fast overhang method by keeping tail sampling the same.
*/
static class ModifiedZigguratNormalizedGaussianSamplerInliningSimpleOverhangs
extends ModifiedZigguratNormalizedGaussianSampler {
/**
* @param rng Generator of uniformly distributed random numbers.
*/
ModifiedZigguratNormalizedGaussianSamplerInliningSimpleOverhangs(UniformRandomProvider rng) {
super(rng);
}
/** {@inheritDoc} */
@Override
public double sample() {
// Ideally this method byte code size should be below -XX:MaxInlineSize
// (which defaults to 35 bytes). This compiles to 33 bytes.
final long xx = nextLong();
// Float multiplication squashes these last 8 bits, so they can be used to sample i
final int i = ((int) xx) & 0xff;
if (i < I_MAX) {
// Early exit.
return X[i] * xx;
}
return edgeSample(xx);
}
/**
* Create the sample from the edge of the ziggurat.
*
* <p>This method has been extracted to fit the main sample method within 35 bytes (the
* default size for a JVM to inline a method).
*
* @param xx Initial random deviate
* @return a sample
*/
private double edgeSample(long xx) {
// Another squashed, recyclable bit
// double sign_bit = u1 & 0x100 ? 1. : -1.
// Use 2 - 1 or 0 - 1
final double signBit = ((xx >>> 7) & 0x2) - 1.0;
final int j = selectRegion();
// Simple overhangs
double x;
if (j == 0) {
// Tail
do {
x = ONE_OVER_X_0 * exponential.sample();
} while (exponential.sample() < 0.5 * x * x);
x += X_0;
} else {
// Rejection sampling
// Recycle bits then advance RNG:
long u1 = xx & MAX_INT64;
for (;;) {
x = sampleX(X, j, u1);
if (sampleY(Y, j, randomInt63()) < Math.exp(-0.5 * x * x)) {
break;
}
u1 = randomInt63();
}
}
return signBit * x;
}
}
/**
* Modified Ziggurat method for sampling from a Gaussian distribution with mean 0 and standard deviation 1.
*
* <p>Uses the algorithm from McFarland, C.D. (2016).
*
* <p>This is a copy of {@link ModifiedZigguratNormalizedGaussianSampler} using
* an integer map in-place of a byte map look-up table. Note the underlying exponential
* sampler does not use an integer map. This sampler is used for the tail with a frequency
* of approximately 0.000276321 and should not impact performance comparisons.
*/
static class ModifiedZigguratNormalizedGaussianSamplerIntMap
extends ModifiedZigguratNormalizedGaussianSampler {
/** The alias map. An integer in [0, 255] stored as a byte to save space. */
private static final int[] INT_MAP = {
/* [ 0] */ 0, 0, 239, 2, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 253,
/* [ 16] */ 253, 253, 253, 253, 253, 253, 253, 253, 253, 253, 253, 253, 253, 253, 253, 253,
/* [ 32] */ 253, 253, 253, 253, 253, 253, 253, 253, 253, 253, 253, 253, 253, 253, 253, 253,
/* [ 48] */ 253, 253, 253, 253, 253, 253, 253, 253, 253, 253, 253, 253, 253, 253, 253, 253,
/* [ 64] */ 253, 253, 253, 253, 253, 253, 253, 253, 253, 253, 253, 253, 253, 253, 253, 253,
/* [ 80] */ 253, 253, 253, 253, 253, 253, 253, 253, 253, 253, 253, 253, 253, 253, 253, 253,
/* [ 96] */ 253, 253, 253, 253, 253, 253, 253, 253, 253, 253, 253, 253, 253, 253, 253, 253,
/* [112] */ 253, 253, 253, 253, 253, 253, 253, 253, 253, 253, 253, 253, 253, 253, 253, 253,
/* [128] */ 253, 253, 253, 253, 253, 253, 253, 253, 253, 253, 253, 253, 253, 253, 253, 253,
/* [144] */ 253, 253, 253, 253, 253, 253, 253, 253, 253, 253, 253, 253, 253, 253, 253, 253,
/* [160] */ 253, 253, 253, 253, 253, 253, 253, 253, 253, 253, 253, 253, 253, 253, 253, 253,
/* [176] */ 253, 253, 253, 253, 253, 253, 253, 253, 253, 253, 253, 253, 253, 253, 253, 253,
/* [192] */ 253, 253, 252, 252, 252, 252, 252, 252, 252, 252, 252, 252, 251, 251, 251, 251,
/* [208] */ 251, 251, 251, 250, 250, 250, 250, 250, 249, 249, 249, 248, 248, 248, 247, 247,
/* [224] */ 247, 246, 246, 245, 244, 244, 243, 242, 240, 2, 2, 3, 3, 0, 0, 240,
/* [240] */ 241, 242, 243, 244, 245, 246, 247, 248, 249, 250, 251, 252, 253, 1, 0, 0,
};
/**
* @param rng Generator of uniformly distributed random numbers.
*/
ModifiedZigguratNormalizedGaussianSamplerIntMap(UniformRandomProvider rng) {
super(rng);
}
/** {@inheritDoc} */
@Override
protected int selectRegion() {
final long x = nextLong();
// j in [0, 256)
final int j = ((int) x) & 0xff;
// map to j in [0, N] with N the number of layers of the ziggurat
return x >= IPMF[j] ? INT_MAP[j] : j;
}
}
/**
* Modified Ziggurat method for sampling from a Gaussian distribution with mean 0 and standard deviation 1.
*
* <p>Uses the algorithm from McFarland, C.D. (2016).
*
* <p>This is a copy of {@link ModifiedZigguratNormalizedGaussianSampler} using
* a table to store the maximum epsilon value for each overhang.
*/
static class ModifiedZigguratNormalizedGaussianSamplerEMaxTable
extends ModifiedZigguratNormalizedGaussianSampler {
/** The deviation of concave pdf(x) below the hypotenuse value for early exit scaled by 2^63.
* 253 entries with overhang {@code j} corresponding to entry {@code j-1}.
*
* <p>Stored as absolute distances. These are negated for the convex overhangs on initialization. */
private static final long[] E_MAX_TABLE = {
/* [ 0] */ 760463704284035184L, 448775940668074322L, 319432059030582634L, 248136886892509167L,
/* [ 4] */ 202879887821507213L, 171571664564813181L, 148614265661634224L, 131055242060438178L,
/* [ 8] */ 117188580518411866L, 105959220286966343L, 96679401617437000L, 88881528890695590L,
/* [ 12] */ 82236552845322455L, 76506171300171850L, 71513549901780258L, 67124696158442860L,
/* [ 16] */ 63236221616146940L, 59767073533081827L, 56652810532069337L, 53841553364303819L,
/* [ 20] */ 51291065360753073L, 48966611131444450L, 46839361730231995L, 44885190189779505L,
/* [ 24] */ 43083750299904955L, 41417763838249446L, 39872463215195657L, 38435151378929884L,
/* [ 28] */ 37094851170183015L, 35842023608135061L, 34668339797973811L, 33566494917490850L,
/* [ 32] */ 32530055495300595L, 31553333229961898L, 30631280119818406L, 29759400819129686L,
/* [ 36] */ 28933679006953617L, 28150515222628953L, 27406674137103216L, 26699239630262959L,
/* [ 40] */ 26025576358437882L, 25383296743791121L, 24770232513666098L, 24184410074620943L,
/* [ 44] */ 23624029131570806L, 23087444063833159L, 22573147652048958L, 22079756816885946L,
/* [ 48] */ 21606000085203013L, 21150706544370418L, 20712796082591192L, 20291270743857678L,
/* [ 52] */ 19885207051787867L, 19493749177966922L, 19116102848340569L, 18751529896268709L,
/* [ 56] */ 18399343383557859L, 18058903221544231L, 17729612233426885L, 17410912606822884L,
/* [ 60] */ 17102282692149351L, 16803234108116672L, 16513309120493202L, 16232078264496960L,
/* [ 64] */ 15959138184784472L, 15694109670143667L, 15436635862703561L, 15186380623831584L,
/* [ 68] */ 14943027040942155L, 14706276061226881L, 14475845239883253L, 14251467591786361L,
/* [ 72] */ 14032890536754294L, 13819874929609696L, 13612194167177793L, 13409633365176500L,
/* [ 76] */ 13211988598685834L, 13019066200522936L, 12830682112422215L, 12646661284424425L,
/* [ 80] */ 12466837118331739L, 12291050951483677L, 12119151577470956L, 11950994800721809L,
/* [ 84] */ 11786443022181594L, 11625364853565635L, 11467634757892067L, 11313132714210236L,
/* [ 88] */ 11161743904626184L, 11013358421893618L, 10867870995988906L, 10725180738227453L,
/* [ 92] */ 10585190901599062L, 10447808656112693L, 10312944878042918L, 10180513952058881L,
/* [ 96] */ 10050433585302955L, 9922624632558803L, 9797010931719358L, 9673519148825347L,
/* [100] */ 9552078632004422L, 9432621273689757L, 9315081380547200L, 9199395550580725L,
/* [104] */ 9085502556926974L, 8973343237884485L, 8862860392758412L, 8753998683127701L,
/* [108] */ 8646704539174329L, 8540926070735305L, 8436612982764049L, 8333716494908564L,
/* [112] */ 8232189264931586L, 8131985315719003L, 8033059965636513L, 7935369762012110L,
/* [116] */ 7838872417532931L, 7743526749360846L, 7649292620780855L, 7556130885207418L,
/* [120] */ 7464003332385685L, 7372872636629289L, 7282702306949403L, 7193456638934045L,
/* [124] */ 7105100668244386L, 7017600125602463L, 6930921393146734L, 6845031462041107L,
/* [128] */ 6759897891224838L, 6675488767195822L, 6591772664721566L, 6508718608380146L,
/* [132] */ 6426296034828168L, 6344474755702722L, 6263224921061057L, 6182516983265422L,
/* [136] */ 6102321661219813L, 6022609904868391L, 5943352859861905L, 5864521832301635L,
/* [140] */ 5786088253467533L, 5708023644436456L, 5630299580496017L, 5552887655255304L,
/* [144] */ 5475759444352708L, 5398886468659678L, 5322240156870143L, 5245791807368878L,
/* [148] */ 5169512549259789L, 5093373302437038L, 5017344736568650L, 4941397228861186L,
/* [152] */ 4865500820464432L, 4789625171364815L, 4713739513611018L, 4637812602700734L,
/* [156] */ 4561812666947717L, 4485707354637964L, 4409463678763447L, 4333047959114789L,
/* [160] */ 4256425761488008L, 4179561833749943L, 4102420038479365L, 4024963281881178L,
/* [164] */ 3947153438645186L, 3868951272390456L, 3790316351307711L, 3711206958575720L,
/* [168] */ 3631579997089249L, 3551390887994563L, 3470593462479259L, 3389139846213138L,
/* [172] */ 3306980335775941L, 3224063266344542L, 3140334869838942L, 3055739122646026L,
/* [176] */ 2970217581950400L, 2883709209602502L, 2796150182338912L, 2707473687051021L,
/* [180] */ 2617609699653324L, 2526484745953020L, 2434021642745634L, 2340139217177559L,
/* [184] */ 2244752002202266L, 2147769905731505L, 2049097850839701L, 1948635384132493L,
/* [188] */ 1846276249145715L, 1741907921439271L, 1635411101964517L, 1526659165422377L,
/* [192] */ 1415517561001961L, 1301843164664438L, 1185483586365029L, 1066276445521246L,
/* [196] */ 944048652015702L, 818615792062560L, 689781896027189L, 557340453963802L,
/* [200] */ 421079947136144L, 280810745547724L, 136572537955918L, 20305634136204L,
/* [204] */ 169472579551785L, 328795810543866L, 494085614296539L, 665530089254391L,
/* [208] */ 843517138081672L, 1028504363150172L, 1221002487033413L, 1421575123173034L,
/* [212] */ 1630842969010940L, 1849490036801539L, 2078271380701587L, 2318022280314205L,
/* [216] */ 2569669030578262L, 2834241595132002L, 3112888469933608L, 3406894199714926L,
/* [220] */ 3717700104287726L, 4046928914960108L, 4396414204558612L, 4768235732059441L,
/* [224] */ 5164762133803925L, 5588702804307930L, 6043171358058752L, 6531763802579722L,
/* [228] */ 7058655558989014L, 7628722851132691L, 8247695913715760L, 8922354192593482L,
/* [232] */ 9660777606582816L, 10472673600425443L, 11369808078043702L, 12366580875504986L,
/* [236] */ 13480805713009702L, 14734784789701036L, 16156816731707853L, 17783356724741365L,
/* [240] */ 19662183995497761L, 21857171972777613L, 24455696683167282L, 27580563854327149L,
/* [244] */ 31410046818804399L, 36213325371048583L, 42417257237736629L, 50742685211596715L,
/* [248] */ 62513643454971809L, 80469460944360062L, 111428640794724271L, 179355568638601057L,
/* [252] */ 2269182951627976004L,
};
static {
// Negate the E_MAX table for the convex overhangs
for (int j = J_INFLECTION; j < E_MAX_TABLE.length; j++) {
E_MAX_TABLE[j] = -E_MAX_TABLE[j];
}
}
/**
* @param rng Generator of uniformly distributed random numbers.
*/
ModifiedZigguratNormalizedGaussianSamplerEMaxTable(UniformRandomProvider rng) {
super(rng);
}
/** {@inheritDoc} */
@Override
public double sample() {
final long xx = nextLong();
// Float multiplication squashes these last 8 bits, so they can be used to sample i
final int i = ((int) xx) & 0xff;
if (i < I_MAX) {
// Early exit.
return X[i] * xx;
}
// Recycle bits then advance RNG:
long u1 = xx & MAX_INT64;
// Another squashed, recyclable bit
// double sign_bit = u1 & 0x100 ? 1. : -1.
// Use 2 - 1 or 0 - 1
final double signBit = ((u1 >>> 7) & 0x2) - 1.0;
final int j = selectRegion();
// Four kinds of overhangs:
// j = 0 : Sample from tail
// 0 < j < J_INFLECTION : Overhang is concave; only sample from Lower-Left triangle
// j = J_INFLECTION : Must sample from entire overhang rectangle
// j > J_INFLECTION : Overhangs are convex; implicitly accept point in Lower-Left triangle
//
// Conditional statements are arranged such that the more likely outcomes are first.
double x;
if (j > J_INFLECTION) {
// Convex overhang
final long eMax = E_MAX_TABLE[j - 1];
for (;;) {
x = sampleX(X, j, u1);
final long uDistance = randomInt63() - u1;
if (uDistance >= 0) {
// Lower-left triangle
break;
}
if (uDistance >= eMax &&
// Within maximum distance of f(x) from the triangle hypotenuse.
sampleY(Y, j, u1 + uDistance) < Math.exp(-0.5 * x * x)) {
break;
}
// uDistance < E_MAX (upper-right triangle) or rejected as above the curve
u1 = randomInt63();
}
} else if (j < J_INFLECTION) {
if (j == 0) {
// Tail
// Note: Although less frequent than the next branch, j == 0 is a subset of
// j < J_INFLECTION and must be first.
do {
x = ONE_OVER_X_0 * exponential.sample();
} while (exponential.sample() < 0.5 * x * x);
x += X_0;
} else {
// Concave overhang
final long eMax = E_MAX_TABLE[j - 1];
for (;;) {
// U_x <- min(U_1, U_2)
// distance <- | U_1 - U_2 |
// U_y <- 1 - (U_x + distance)
long uDistance = randomInt63() - u1;
if (uDistance < 0) {
// Upper-right triangle. Reflect in hypotenuse.
uDistance = -uDistance;
u1 -= uDistance;
}
x = sampleX(X, j, u1);
if (uDistance > eMax ||
sampleY(Y, j, u1 + uDistance) < Math.exp(-0.5 * x * x)) {
break;
}
u1 = randomInt63();
}
}
} else {
// Inflection point
for (;;) {
x = sampleX(X, j, u1);
if (sampleY(Y, j, randomInt63()) < Math.exp(-0.5 * x * x)) {
break;
}
u1 = randomInt63();
}
}
return signBit * x;
}
}
/**
* Modified Ziggurat method for sampling from a Gaussian distribution with mean 0 and standard deviation 1.
*
* <p>Uses the algorithm from McFarland, C.D. (2016).
*
* <p>This is a copy of {@link ModifiedZigguratNormalizedGaussianSampler} using
* two thresholds for the maximum epsilon value for convex and concave overhangs.
*
* <p>Note: The normal curve is very well approximated by the straight line of
* the triangle hypotenuse in the majority of cases. As the convex overhang approaches x=0
* the curve is significantly different. For the concave overhangs the curve is increasingly
* different approaching the tail. This can be exploited using two maximum deviation thresholds.
*/
static class ModifiedZigguratNormalizedGaussianSamplerEMax2
extends ModifiedZigguratNormalizedGaussianSampler {
// Ziggurat volumes:
// Inside the layers = 98.8281% (253/256)
// Fraction outside the layers:
// concave overhangs = 76.1941%
// inflection overhang = 0.1358%
// convex overhangs = 21.3072%
// tail = 2.3629%
// Separation of convex overhangs:
// (Cut made to separate large final overhang with a very different E max.)
// E_MAX
// j = 253 = 72.0882% 0.246
// 204 < j < 253 = 27.9118% 0.0194
// Separation of concave overhangs:
// (Cut made between overhangs requiring above or below 0.02 for E max.
// This choice is somewhat arbitrary. Increasing j will reduce the second E max but makes
// the branch less predictable as the major path is used less.)
// 1 <= j <= 5 = 12.5257% 0.0824
// 5 < j < 204 = 87.4743% 0.0186
/** The convex overhang region j below which the second maximum deviation constant is valid. */
private static final int CONVEX_J2 = 253;
/** The concave overhang region j above which the second maximum deviation constant is valid. */
private static final int CONCAVE_J2 = 5;
/** Maximum epsilon distance of convex pdf(x) above the hypotenuse value for early rejection
* for overhangs {@code 204 < j < 253}.
* Equal to approximately 0.0194 scaled by 2^63. This is negated on purpose.
*
* <p>This threshold increases the area of the early reject triangle by:
* (1-0.0194)^2 / (1-0.246)^2 = 69.1%. */
private static final long CONVEX_E_MAX_2 = -179355568638601057L;
/** Maximum deviation of concave pdf(x) below the hypotenuse value for early exit
* for overhangs {@code 5 < j < 204}.
* Equal to approximately 0.0186 scaled by 2^63.
*
* <p>This threshold increases the area of the early exit triangle by:
* (1-0.0186)^2 / (1-0.0824)^2 = 14.4%. */
private static final long CONCAVE_E_MAX_2 = 171571664564813181L;
/**
* @param rng Generator of uniformly distributed random numbers.
*/
ModifiedZigguratNormalizedGaussianSamplerEMax2(UniformRandomProvider rng) {
super(rng);
}
/** {@inheritDoc} */
@Override
public double sample() {
final long xx = nextLong();
// Float multiplication squashes these last 8 bits, so they can be used to sample i
final int i = ((int) xx) & 0xff;
if (i < I_MAX) {
// Early exit.
return X[i] * xx;
}
// Recycle bits then advance RNG:
long u1 = xx & MAX_INT64;
// Another squashed, recyclable bit
// double sign_bit = u1 & 0x100 ? 1. : -1.
// Use 2 - 1 or 0 - 1
final double signBit = ((u1 >>> 7) & 0x2) - 1.0;
final int j = selectRegion();
// Four kinds of overhangs:
// j = 0 : Sample from tail
// 0 < j < J_INFLECTION : Overhang is concave; only sample from Lower-Left triangle
// j = J_INFLECTION : Must sample from entire overhang rectangle
// j > J_INFLECTION : Overhangs are convex; implicitly accept point in Lower-Left triangle
//
// Conditional statements are arranged such that the more likely outcomes are first.
double x;
if (j > J_INFLECTION) {
// Convex overhang:
// j < J2 frequency: 0.279118
final long eMax = j < CONVEX_J2 ? CONVEX_E_MAX_2 : CONVEX_E_MAX;
for (;;) {
x = sampleX(X, j, u1);
final long uDistance = randomInt63() - u1;
if (uDistance >= 0) {
// Lower-left triangle
break;
}
if (uDistance >= eMax &&
// Within maximum distance of f(x) from the triangle hypotenuse.
sampleY(Y, j, u1 + uDistance) < Math.exp(-0.5 * x * x)) {
break;
}
// uDistance < E_MAX (upper-right triangle) or rejected as above the curve
u1 = randomInt63();
}
} else if (j < J_INFLECTION) {
if (j == 0) {
// Tail
// Note: Although less frequent than the next branch, j == 0 is a subset of
// j < J_INFLECTION and must be first.
do {
x = ONE_OVER_X_0 * exponential.sample();
} while (exponential.sample() < 0.5 * x * x);
x += X_0;
} else {
// Concave overhang
// j > J2 frequency: 0.874743
final long eMax = j > CONCAVE_J2 ? CONCAVE_E_MAX_2 : CONCAVE_E_MAX;
for (;;) {
// U_x <- min(U_1, U_2)
// distance <- | U_1 - U_2 |
// U_y <- 1 - (U_x + distance)
long uDistance = randomInt63() - u1;
if (uDistance < 0) {
// Upper-right triangle. Reflect in hypotenuse.
uDistance = -uDistance;
u1 -= uDistance;
}
x = sampleX(X, j, u1);
if (uDistance > eMax ||
sampleY(Y, j, u1 + uDistance) < Math.exp(-0.5 * x * x)) {
break;
}
u1 = randomInt63();
}
}
} else {
// Inflection point
for (;;) {
x = sampleX(X, j, u1);
if (sampleY(Y, j, randomInt63()) < Math.exp(-0.5 * x * x)) {
break;
}
u1 = randomInt63();
}
}
return signBit * x;
}
}
/**
* Modified Ziggurat method for sampling from a Gaussian distribution with mean 0 and standard deviation 1.
*
* <p>Uses the algorithm from McFarland, C.D. (2016).
*
* <p>This is a copy of {@link ModifiedZigguratNormalizedGaussianSampler} using
* a table size of 512.
*/
static class ModifiedZigguratNormalizedGaussianSampler512 implements ContinuousSampler {
// Ziggurat volumes:
// Inside the layers = 99.4141% (509/512)
// Fraction outside the layers:
// concave overhangs = 75.5775%
// inflection overhang = 0.0675%
// convex overhangs = 22.2196%
// tail = 2.1354%
/** The number of layers in the ziggurat. Maximum i value for early exit. */
protected static final int I_MAX = 509;
/** The point where the Gaussian switches from convex to concave.
* This is the largest value of X[j] below 1. */
protected static final int J_INFLECTION = 409;
/** Maximum epsilon distance of convex pdf(x) above the hypotenuse value for early rejection.
* Equal to approximately 0.2477 scaled by 2^63. This is negated on purpose as the
* distance for a point (x,y) above the hypotenuse is negative:
* {@code (|d| < max) == (d >= -max)}. */
protected static final long CONVEX_E_MAX = -2284356979160975476L;
/** Maximum distance of concave pdf(x) below the hypotenuse value for early exit.
* Equal to approximately 0.08284 scaled by 2^63. */
protected static final long CONCAVE_E_MAX = 764138791244619676L;
/** Beginning of tail. Equal to X[0] * 2^63. */
protected static final double X_0 = 3.8358644648571882;
/** 1/X_0. Used for tail sampling. */
protected static final double ONE_OVER_X_0 = 1.0 / X_0;
/** The alias map. */
private static final int[] MAP = {
/* [ 0] */ 0, 0, 480, 2, 3, 4, 5, 6, 7, 0, 0, 0, 0, 0, 0, 0,
/* [ 16] */ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1,
/* [ 32] */ 1, 509, 509, 509, 509, 509, 509, 509, 509, 509, 509, 509, 509, 509, 509, 509,
/* [ 48] */ 509, 509, 509, 509, 509, 509, 509, 509, 509, 509, 509, 509, 509, 509, 509, 509,
/* [ 64] */ 509, 509, 509, 509, 509, 509, 509, 509, 509, 509, 509, 509, 509, 509, 509, 509,
/* [ 80] */ 509, 509, 509, 509, 509, 509, 509, 509, 509, 509, 509, 509, 509, 509, 509, 509,
/* [ 96] */ 509, 509, 509, 509, 509, 509, 509, 509, 509, 509, 509, 509, 509, 509, 509, 509,
/* [112] */ 509, 509, 509, 509, 509, 509, 509, 509, 509, 509, 509, 509, 509, 509, 509, 509,
/* [128] */ 509, 509, 509, 509, 509, 509, 509, 509, 509, 509, 509, 509, 509, 509, 509, 509,
/* [144] */ 509, 509, 509, 509, 509, 509, 509, 509, 509, 509, 509, 509, 509, 509, 509, 509,
/* [160] */ 509, 509, 509, 509, 509, 509, 509, 509, 509, 509, 509, 509, 509, 509, 509, 509,
/* [176] */ 509, 509, 509, 509, 509, 509, 509, 509, 509, 509, 509, 509, 509, 509, 509, 509,
/* [192] */ 509, 509, 509, 509, 509, 509, 509, 509, 509, 509, 509, 509, 509, 509, 509, 509,
/* [208] */ 509, 509, 509, 509, 509, 509, 509, 509, 509, 509, 509, 509, 509, 509, 509, 509,
/* [224] */ 509, 509, 509, 509, 509, 509, 509, 509, 509, 509, 509, 509, 509, 509, 509, 509,
/* [240] */ 509, 509, 509, 509, 509, 509, 509, 509, 509, 509, 509, 509, 509, 509, 509, 509,
/* [256] */ 509, 509, 509, 509, 509, 509, 509, 509, 509, 509, 509, 509, 509, 509, 509, 509,
/* [272] */ 509, 509, 509, 509, 509, 509, 509, 509, 509, 509, 509, 509, 509, 509, 509, 509,
/* [288] */ 509, 509, 509, 509, 509, 509, 509, 509, 509, 509, 509, 509, 509, 509, 509, 509,
/* [304] */ 509, 509, 509, 509, 509, 509, 509, 509, 509, 509, 509, 509, 509, 509, 509, 509,
/* [320] */ 509, 509, 509, 509, 509, 509, 509, 509, 509, 509, 509, 509, 509, 509, 509, 509,
/* [336] */ 509, 509, 509, 509, 509, 509, 509, 509, 509, 509, 509, 509, 509, 509, 509, 509,
/* [352] */ 509, 509, 509, 508, 508, 508, 508, 508, 508, 508, 508, 508, 508, 508, 508, 508,
/* [368] */ 508, 508, 508, 508, 508, 508, 508, 508, 508, 508, 507, 507, 507, 507, 507, 507,
/* [384] */ 507, 507, 507, 507, 507, 507, 507, 507, 506, 506, 506, 506, 506, 506, 506, 506,
/* [400] */ 506, 506, 505, 505, 505, 505, 505, 505, 505, 505, 504, 504, 504, 504, 504, 504,
/* [416] */ 503, 503, 503, 503, 503, 502, 502, 502, 502, 501, 501, 501, 501, 500, 500, 500,
/* [432] */ 500, 499, 499, 499, 498, 498, 498, 497, 497, 496, 496, 495, 495, 494, 494, 493,
/* [448] */ 493, 492, 491, 490, 489, 488, 487, 486, 484, 2, 2, 2, 2, 2, 3, 3,
/* [464] */ 3, 4, 4, 4, 5, 5, 6, 7, 8, 0, 0, 0, 0, 0, 0, 0,
/* [480] */ 481, 482, 483, 484, 485, 486, 487, 488, 489, 490, 491, 492, 493, 494, 495, 496,
/* [496] */ 497, 498, 499, 500, 501, 502, 503, 504, 505, 506, 507, 508, 509, 1, 0, 0,
};
/** The alias inverse PMF. */
private static final long[] IPMF = {
/* [ 0] */ 9223372036854775296L, 3232176979959565312L, 2248424156373566976L, 5572490326155894272L,
/* [ 4] */ 3878549168685540352L, 6596558088755235840L, 7126388452799016960L, 5909258326277172224L,
/* [ 8] */ 6789036995587703808L, 8550801776550481920L, 7272880214327450112L, 6197723851568405504L,
/* [ 12] */ 5279475843641833984L, 4485290838628302848L, 3790998004182467072L, 3178396845257682944L,
/* [ 16] */ 2633509168847815680L, 2145413694861947904L, 1705448049989382144L, 1306649397437548544L,
/* [ 20] */ 943354160611131392L, 610906287013685760L, 305441095029422592L, 23722736478678016L,
/* [ 24] */ -236979677660888064L, -478987634076675072L, -704287768537463808L, -914590611781799424L,
/* [ 28] */ -1111377331889550848L, -1295937239757415424L, -1469398133880157184L, -1632751030331424768L,
/* [ 32] */ -1786870462753421312L, -1932531249141885952L, -2070422432472262144L, -2201158928905888256L,
/* [ 36] */ -2325291321647773696L, -2443314131355272192L, -2555672836734271488L, -2662769858897092096L,
/* [ 40] */ -2764969686945558528L, -2862603283332330496L, -2955971885350600192L, -3045350299719277056L,
/* [ 44] */ -3130989762881722368L, -3213120439420999168L, -3291953604062304256L, -3367683558060156416L,
/* [ 48] */ -3440489316130714624L, -3510536092827323904L, -3577976620246562816L, -3642952314681374720L,
/* [ 52] */ -3705594315191049216L, -3766024408614111744L, -3824355856527138304L, -3880694134271785472L,
/* [ 56] */ -3935137594960746496L, -3987778065544580608L, -4038701385199136256L, -4087987887601244160L,
/* [ 60] */ -4135712841168144384L, -4181946844655158784L, -4226756186527602176L, -4270203172038951424L,
/* [ 64] */ -4312346420932761088L, -4353241136943226880L, -4392939356053354496L, -4431490172784719360L,
/* [ 68] */ -4468939945001798656L, -4505332485123679232L, -4540709232733563392L, -4575109415088551424L,
/* [ 72] */ -4608570193416988672L, -4641126797904314880L, -4672812652574308352L, -4703659490673809920L,
/* [ 76] */ -4733697460624237568L, -4762955224068688384L, -4791460047779233792L, -4819237887169053184L,
/* [ 80] */ -4846313464821579264L, -4872710343112062464L, -4898450992673160192L, -4923556853979476992L,
/* [ 84] */ -4948048396148470272L, -4971945172796024320L, -4995265870892981248L, -5018028360085760000L,
/* [ 88] */ -5040249735872740864L, -5061946361747427840L, -5083133907806415360L, -5103827387715595776L,
/* [ 92] */ -5124041191022579712L, -5143789117574014464L, -5163084406061926400L, -5181939762086439936L,
/* [ 96] */ -5200367384806330368L, -5218378992989818368L, -5235985846759801344L, -5253198770233210880L,
/* [100] */ -5270028172706144256L, -5286484068673344512L, -5302576093914087424L, -5318313526819882496L,
/* [104] */ -5333705300501412864L, -5348760022140801536L, -5363485985612222976L, -5377891185738658816L,
/* [108] */ -5391983331261941760L, -5405769858949649408L, -5419257942659722240L, -5432454506730834944L,
/* [112] */ -5445366236058309120L, -5457999585571643392L, -5470360790218524160L, -5482455874386169344L,
/* [116] */ -5494290660050575872L, -5505870774716076032L, -5517201660378635264L, -5528288579187974144L,
/* [120] */ -5539136622578390528L, -5549750716118886400L, -5560135626660535808L, -5570295969034180608L,
/* [124] */ -5580236210799400960L, -5589960678379259904L, -5599473562706778112L, -5608778923054202880L,
/* [128] */ -5617880692891545088L, -5626782684460370944L, -5635488592684399104L, -5644001999389030912L,
/* [132] */ -5652326378260390400L, -5660465096501542912L, -5668421421481986048L, -5676198522069054464L,
/* [136] */ -5683799472034193408L, -5691227254524439552L, -5698484764964738048L, -5705574811911234048L,
/* [140] */ -5712500123062644224L, -5719263345561367040L, -5725867049400364032L, -5732313730758826496L,
/* [144] */ -5738605812086640640L, -5744745647569117184L, -5750735522378352640L, -5756577656307920896L,
/* [148] */ -5762274206097785856L, -5767827266307192832L, -5773238871631092224L, -5778510999439789056L,
/* [152] */ -5783645569528274432L, -5788644448981466624L, -5793509449834629120L, -5798242334409694208L,
/* [156] */ -5802844813858308096L, -5807318551213207040L, -5811665162349513216L, -5815886217028536832L,
/* [160] */ -5819983240762728448L, -5823957715198579712L, -5827811080259974144L, -5831544734136989696L,
/* [164] */ -5835160035614893056L, -5838658304467378688L, -5842040821904171520L, -5845308832880873984L,
/* [168] */ -5848463546172629504L, -5851506134738812416L, -5854437738541255168L, -5857259462674584064L,
/* [172] */ -5859972381032173568L, -5862577534420617216L, -5865075933349880320L, -5867468558168720896L,
/* [176] */ -5869756358001916928L, -5871940255655389696L, -5874021142765756416L, -5875999885243720704L,
/* [180] */ -5877877321138562048L, -5879654262030006784L, -5881331493491406336L, -5882909775710032384L,
/* [184] */ -5884389844282934784L, -5885772409590032896L, -5887058159412298240L, -5888247756760180736L,
/* [188] */ -5889341841868359680L, -5890341033726286848L, -5891245927036046336L, -5892057096967517184L,
/* [192] */ -5892775095254208000L, -5893400454293159424L, -5893933684945675776L, -5894375277894875136L,
/* [196] */ -5894725704281316864L, -5894985415097101824L, -5895154842211865088L, -5895234398367037440L,
/* [200] */ -5895224478119281152L, -5895125456527683584L, -5894937691150572032L, -5894661521100592128L,
/* [204] */ -5894297268546692608L, -5893845237129657344L, -5893305713928932352L, -5892678968199164928L,
/* [208] */ -5891965253512273920L, -5891164805272775680L, -5890277843462263808L, -5889304570970719232L,
/* [212] */ -5888245175227536896L, -5887099827242714112L, -5885868681480997376L, -5884551878016265728L,
/* [216] */ -5883149540187518976L, -5881661776186659840L, -5880088679051159040L, -5878430325617481216L,
/* [220] */ -5876686779113256448L, -5874858086097725440L, -5872944278695313920L, -5870945374625903616L,
/* [224] */ -5868861375946597376L, -5866692270114584576L, -5864438029808936448L, -5862098613329829376L,
/* [228] */ -5859673963988850688L, -5857164010304659456L, -5854568667030923264L, -5851887832571710976L,
/* [232] */ -5849121393007995904L, -5846269217914838016L, -5843331163565322752L, -5840307071023708160L,
/* [236] */ -5837196767034538496L, -5834000063619345920L, -5830716757850101760L, -5827346633080182272L,
/* [240] */ -5823889457015247872L, -5820344982925702656L, -5816712949734398464L, -5812993080309509120L,
/* [244] */ -5809185084612210688L, -5805288655397134848L, -5801303472071718912L, -5797229197542626816L,
/* [248] */ -5793065480979245568L, -5788811954902762496L, -5784468237204753408L, -5780033929729368064L,
/* [252] */ -5775508619016246784L, -5770891876025206272L, -5766183254901072896L, -5761382294937170432L,
/* [256] */ -5756488518102026240L, -5751501431126126080L, -5746420523619011584L, -5741245267817476608L,
/* [260] */ -5735975121281433088L, -5730609522034737152L, -5725147892326605312L, -5719589637112302080L,
/* [264] */ -5713934142693986816L, -5708180778990178816L, -5702328896016054784L, -5696377827537145856L,
/* [268] */ -5690326887295941632L, -5684175370772383744L, -5677922555045085184L, -5671567696296352256L,
/* [272] */ -5665110033102450176L, -5658548782530205696L, -5651883142913629696L, -5645112290875286016L,
/* [276] */ -5638235383069228544L, -5631251554336663040L, -5624159919711433728L, -5616959570005672448L,
/* [280] */ -5609649575759112704L, -5602228984633701376L, -5594696820961281536L, -5587052085776743936L,
/* [284] */ -5579293757104064512L, -5571420787942931456L, -5563432107812764160L, -5555326619765914112L,
/* [288] */ -5547103202855799296L, -5538760709287606272L, -5530297965182673920L, -5521713769559916032L,
/* [292] */ -5513006894391551488L, -5504176082674265088L, -5495220049712259584L, -5486137481881783808L,
/* [296] */ -5476927034862822400L, -5467587334392330752L, -5458116976167262720L, -5448514522422561280L,
/* [300] */ -5438778504885873664L, -5428907421454137344L, -5418899736307948032L, -5408753879414383104L,
/* [304] */ -5398468245670757888L, -5388041193950201856L, -5377471046142717440L, -5366756087011503104L,
/* [308] */ -5355894562291467776L, -5344884679094671360L, -5333724604104164352L, -5322412462390110720L,
/* [312] */ -5310946337190221824L, -5299324269391752192L, -5287544254227943936L, -5275604243076108288L,
/* [316] */ -5263502140384954880L, -5251235803114780672L, -5238803040260463616L, -5226201609860500992L,
/* [320] */ -5213429220396639232L, -5200483526689941504L, -5187362130814278656L, -5174062579249043456L,
/* [324] */ -5160582362536272384L, -5146918913003260928L, -5133069603645442048L, -5119031746455840768L,
/* [328] */ -5104802590846193664L, -5090379322288435200L, -5075759059905230848L, -5060938855168388608L,
/* [332] */ -5045915690136155648L, -5030686475123909120L, -5015248047351192576L, -4999597168201581568L,
/* [336] */ -4983730521429875200L, -4967644711103290880L, -4951336259902648320L, -4934801604331100672L,
/* [340] */ -4918037096313518080L, -4901038996556133376L, -4883803475152384512L, -4866326606521733120L,
/* [344] */ -4848604368549652480L, -4830632638500906496L, -4812407189788950016L, -4793923690356844544L,
/* [348] */ -4775177697581510656L, -4756164656502554112L, -4736879895545858048L, -4717318622225770496L,
/* [352] */ -4697475921931613696L, -4677346750468523008L, -4656925933129603072L, -4636208158837948416L,
/* [356] */ -4615187976257695744L, -4593859789223797248L, -4572217851458716672L, -4550256263721420288L,
/* [360] */ -4527968965738787328L, -4505349732602470400L, -4482392170596992000L, -4459089707595035648L,
/* [364] */ -4435435591206243840L, -4411422880135513088L, -4387044438367173632L, -4362292928394443264L,
/* [368] */ -4337160804216032256L, -4311640303794398208L, -4285723442340192768L, -4259402002490114048L,
/* [372] */ -4232667527557880832L, -4205511312870738944L, -4177924395156510720L, -4149897544928282624L,
/* [376] */ -4121421255517056000L, -4092485732978126848L, -4063080884857030656L, -4033196309983527424L,
/* [380] */ -4002821285452889600L, -3971944754955299840L, -3940555315053852672L, -3908641203541062144L,
/* [384] */ -3876190282046383616L, -3843190024121122816L, -3809627498134461952L, -3775489350771450880L,
/* [388] */ -3740761791237469184L, -3705430570995169280L, -3669480966726291456L, -3632897760460073472L,
/* [392] */ -3595665216111472128L, -3557767061510936064L, -3519186461826909184L, -3479905997291117056L,
/* [396] */ -3439907636209809920L, -3399172709277578752L, -3357681881315035136L, -3315415118925273600L,
/* [400] */ -3272351662607639040L, -3228469989824430592L, -3183747783206633472L, -3138161891189001728L,
/* [404] */ -3091688288710896128L, -3044302038074020352L, -2995977242912180224L, -2946687001898157056L,
/* [408] */ -2896403361643595776L, -2845097261899269632L, -2792738483041489920L, -2739295585832944128L,
/* [412] */ -2684735849802800128L, -2629025208318833152L, -2572128177407577088L, -2514007780971290112L,
/* [416] */ -2454625474210264064L, -2393941056301004288L, -2331912582463081472L, -2268496267820575232L,
/* [420] */ -2203646385063500288L, -2137315156805725696L, -2069452637212809728L, -2000006589650226176L,
/* [424] */ -1928922352643083264L, -1856142697389204480L, -1781607676495585792L, -1705254458055283200L,
/* [428] */ -1627017152968592896L, -1546826624346926592L, -1464610284716681216L, -1380291880647352832L,
/* [432] */ -1293791253775522304L, -1205024091744495104L, -1113901652592291328L, -1020330470192921600L,
/* [436] */ -924212036524504576L, -825442455107712512L, -723912067910252544L, -619505050467681792L,
/* [440] */ -512098970851033088L, -401564310549775872L, -287763946399309312L, -170552579279989760L,
/* [444] */ -49776119384483328L, 74728993660240384L, 203136528144344576L, 335631271254235136L,
/* [448] */ 472409917558910464L, 613682045009508352L, 759671188745830912L, 910616024566752768L,
/* [452] */ 1066771674457471488L, 1228411150191648768L, 1395826950915523584L, 1569332837797146112L,
/* [456] */ 1749265802232386560L, 1935988261896280064L, 2129890506883624960L, 2331393435927328256L,
/* [460] */ 2540951622664266752L, 2759056758111070720L, 2986241520688130560L, 3223083948864353280L,
/* [464] */ 3470212380253329408L, 3728311054271952896L, 3998126479196178432L, 4280474694750398976L,
/* [468] */ 4576249573583467520L, 4886432342140325376L, 5212102538692988928L, 5554450665570580992L,
/* [472] */ 5914792845058133504L, 6294587870200374272L, 6695457110790894080L, 7119207852916254720L,
/* [476] */ 7567860785399318528L, 8043682516018548736L, 8549224231795344384L, 9087367898325182976L,
/* [480] */ 2686433895585048064L, 3738049383351049728L, 5447119168340997632L, 7862383433967084544L,
/* [484] */ 3564125970404768256L, 7561510489765503488L, 4795007330430237184L, 2822776087249664512L,
/* [488] */ 1739199416312530432L, 1651067950292976640L, 2680178586723521024L, 4966620264299254784L,
/* [492] */ 8672981687287425024L, 5102073364374541824L, 2973953244697943040L, 2586847729538911744L,
/* [496] */ 4294392489026305536L, 8524338069805147648L, 5657549542711693824L, 6039225349920109568L,
/* [500] */ -188437589949716992L, -1600155058499635712L, 3512467924097231872L, 5888549710424099840L,
/* [504] */ 8328892371572499456L, 3607211374099356160L, 952847323024169472L, -3124063727137777664L,
/* [508] */ -844200026382571008L, 1173948101715934208L, -9223372036854775808L, -9223372036854775808L,
};
/**
* The precomputed ziggurat lengths, denoted X_i in the main text. X_i = length of
* ziggurat layer i. Values have been scaled by 2^-63.
*/
private static final double[] X = {
/* [ 0] */ 4.1588525861581104e-19, 3.9503459916661627e-19, 3.821680975424891e-19, 3.7270045721185492e-19,
/* [ 4] */ 3.6514605982514084e-19, 3.5882746800626676e-19, 3.533765597048754e-19, 3.4857018027109719e-19,
/* [ 8] */ 3.4426246437790343e-19, 3.4035264389553312e-19, 3.3676808841410688e-19, 3.3345465900442888e-19,
/* [ 12] */ 3.3037088331351658e-19, 3.2748426481115448e-19, 3.2476884984205335e-19, 3.2220356973333147e-19,
/* [ 16] */ 3.1977107871867837e-19, 3.1745691935586742e-19, 3.1524891032273018e-19, 3.1313668890638164e-19,
/* [ 20] */ 3.1111136341647712e-19, 3.0916524520018658e-19, 3.0729163928347323e-19, 3.0548467885197293e-19,
/* [ 24] */ 3.0373919296832198e-19, 3.0205059980435001e-19, 3.0041481968537306e-19, 2.9882820368030914e-19,
/* [ 28] */ 2.9728747450808126e-19, 2.9578967728884489e-19, 2.9433213822959269e-19, 2.9291242975352861e-19,
/* [ 32] */ 2.9152834090005319e-19, 2.9017785206455541e-19, 2.8885911333390034e-19, 2.8757042581852451e-19,
/* [ 36] */ 2.8631022549560207e-19, 2.8507706916730813e-19, 2.8386962220934334e-19, 2.8268664784176013e-19,
/* [ 40] */ 2.815269976998848e-19, 2.8038960352015381e-19, 2.7927346978580998e-19, 2.7817766720204774e-19,
/* [ 44] */ 2.7710132689045585e-19, 2.7604363520934134e-19, 2.7500382912040458e-19, 2.739811920338075e-19,
/* [ 48] */ 2.7297505007336127e-19, 2.7198476871169526e-19, 2.710097497321303e-19, 2.7004942847978509e-19,
/* [ 52] */ 2.6910327136937779e-19, 2.6817077362138452e-19, 2.6725145720181144e-19, 2.6634486894391515e-19,
/* [ 56] */ 2.6545057883285584e-19, 2.6456817843655206e-19, 2.636972794679811e-19, 2.6283751246588273e-19,
/* [ 60] */ 2.6198852558231173e-19, 2.6114998346678357e-19, 2.6032156623788965e-19, 2.5950296853425164e-19,
/* [ 64] */ 2.5869389863755409e-19, 2.5789407766116098e-19, 2.5710323879849453e-19, 2.5632112662595285e-19,
/* [ 68] */ 2.555474964556656e-19, 2.5478211373385828e-19, 2.5402475348100586e-19, 2.532751997703282e-19,
/* [ 72] */ 2.5253324524150594e-19, 2.5179869064679051e-19, 2.5107134442694197e-19, 2.503510223146645e-19,
/* [ 76] */ 2.4963754696341833e-19, 2.4893074759967754e-19, 2.4823045969687053e-19, 2.4753652466939529e-19,
/* [ 80] */ 2.4684878958523816e-19, 2.4616710689584975e-19, 2.4549133418204393e-19, 2.4482133391478862e-19,
/* [ 84] */ 2.4415697322984843e-19, 2.4349812371532436e-19, 2.4284466121121044e-19, 2.42196465620158e-19,
/* [ 88] */ 2.4155342072870043e-19, 2.4091541403824951e-19, 2.402823366052261e-19, 2.3965408288973736e-19,
/* [ 92] */ 2.3903055061225481e-19, 2.3841164061778912e-19, 2.3779725674709409e-19, 2.3718730571446517e-19,
/* [ 96] */ 2.3658169699173055e-19, 2.3598034269805933e-19, 2.3538315749523948e-19, 2.3479005848810095e-19,
/* [100] */ 2.3420096512978219e-19, 2.3361579913155922e-19, 2.3303448437697401e-19, 2.3245694684001817e-19,
/* [104] */ 2.3188311450714236e-19, 2.3131291730287861e-19, 2.3074628701887441e-19, 2.3018315724615257e-19,
/* [108] */ 2.2962346331042115e-19, 2.290671422102686e-19, 2.285141325580919e-19, 2.2796437452361119e-19,
/* [112] */ 2.2741780977983687e-19, 2.2687438145136136e-19, 2.2633403406485519e-19, 2.2579671350165625e-19,
/* [116] */ 2.2526236695234495e-19, 2.2473094287320652e-19, 2.2420239094448635e-19, 2.2367666203034958e-19,
/* [120] */ 2.231537081404625e-19, 2.2263348239311565e-19, 2.2211593897981584e-19, 2.2160103313127622e-19,
/* [124] */ 2.2108872108473813e-19, 2.2057896005256279e-19, 2.2007170819203281e-19, 2.1956692457630854e-19,
/* [128] */ 2.1906456916648504e-19, 2.1856460278470111e-19, 2.1806698708825153e-19, 2.1757168454465786e-19,
/* [132] */ 2.1707865840765572e-19, 2.1658787269405759e-19, 2.1609929216145222e-19, 2.1561288228670561e-19,
/* [136] */ 2.1512860924522763e-19, 2.1464643989097195e-19, 2.1416634173713822e-19, 2.1368828293754638e-19,
/* [140] */ 2.1321223226865536e-19, 2.1273815911219872e-19, 2.122660334384123e-19, 2.1179582578982903e-19,
/* [144] */ 2.1132750726561794e-19, 2.1086104950644545e-19, 2.103964246798376e-19, 2.0993360546602319e-19,
/* [148] */ 2.09472565044239e-19, 2.0901327707947851e-19, 2.0855571570966692e-19, 2.0809985553324565e-19,
/* [152] */ 2.0764567159715066e-19, 2.0719313938516928e-19, 2.0674223480666103e-19, 2.0629293418562883e-19,
/* [156] */ 2.0584521425012706e-19, 2.05399052121994e-19, 2.0495442530689639e-19, 2.0451131168467476e-19,
/* [160] */ 2.0406968949997815e-19, 2.0362953735317784e-19, 2.0319083419154967e-19, 2.027535593007156e-19,
/* [164] */ 2.0231769229633464e-19, 2.0188321311603464e-19, 2.0145010201157619e-19, 2.0101833954124038e-19,
/* [168] */ 2.0058790656243254e-19, 2.0015878422449446e-19, 1.9973095396171768e-19, 1.9930439748655098e-19,
/* [172] */ 1.9887909678299541e-19, 1.984550341001801e-19, 1.9803219194611336e-19, 1.9761055308160207e-19,
/* [176] */ 1.9719010051433485e-19, 1.967708174931225e-19, 1.9635268750229097e-19, 1.9593569425622174e-19,
/* [180] */ 1.955198216940345e-19, 1.9510505397440756e-19, 1.9469137547053159e-19, 1.9427877076519196e-19,
/* [184] */ 1.9386722464597592e-19, 1.9345672210060023e-19, 1.9304724831235547e-19, 1.9263878865566334e-19,
/* [188] */ 1.9223132869174325e-19, 1.9182485416438451e-19, 1.9141935099582133e-19, 1.9101480528270662e-19,
/* [192] */ 1.9061120329218205e-19, 1.9020853145804105e-19, 1.8980677637698202e-19, 1.8940592480494855e-19,
/* [196] */ 1.8900596365355448e-19, 1.8860687998659074e-19, 1.8820866101661152e-19, 1.8781129410159744e-19,
/* [200] */ 1.8741476674169321e-19, 1.8701906657601756e-19, 1.8662418137954315e-19, 1.8623009906004439e-19,
/* [204] */ 1.85836807655111e-19, 1.8544429532922543e-19, 1.8505255037090195e-19, 1.8466156118988594e-19,
/* [208] */ 1.8427131631441095e-19, 1.8388180438851245e-19, 1.8349301416939593e-19, 1.8310493452485817e-19,
/* [212] */ 1.8271755443075968e-19, 1.823308629685471e-19, 1.8194484932282376e-19, 1.8155950277896707e-19,
/* [216] */ 1.8117481272079125e-19, 1.8079076862825417e-19, 1.804073600752067e-19, 1.8002457672718337e-19,
/* [220] */ 1.7964240833923322e-19, 1.7926084475378945e-19, 1.7887987589857656e-19, 1.7849949178455405e-19,
/* [224] */ 1.7811968250389552e-19, 1.7774043822800183e-19, 1.773617492055474e-19, 1.7698360576055882e-19,
/* [228] */ 1.7660599829052424e-19, 1.7622891726453298e-19, 1.7585235322144435e-19, 1.754762967680843e-19,
/* [232] */ 1.751007385774697e-19, 1.7472566938705867e-19, 1.743510799970264e-19, 1.7397696126856576e-19,
/* [236] */ 1.7360330412221144e-19, 1.7323009953618705e-19, 1.7285733854477456e-19, 1.7248501223670475e-19,
/* [240] */ 1.7211311175356858e-19, 1.7174162828824813e-19, 1.7137055308336676e-19, 1.7099987742975751e-19,
/* [244] */ 1.7062959266494937e-19, 1.7025969017167022e-19, 1.6989016137636627e-19, 1.6952099774773704e-19,
/* [248] */ 1.6915219079528517e-19, 1.6878373206788065e-19, 1.6841561315233867e-19, 1.6804782567201046e-19,
/* [252] */ 1.6768036128538652e-19, 1.6731321168471171e-19, 1.6694636859461146e-19, 1.6657982377072854e-19,
/* [256] */ 1.6621356899836991e-19, 1.6584759609116298e-19, 1.6548189688972057e-19, 1.6511646326031438e-19,
/* [260] */ 1.6475128709355596e-19, 1.6438636030308492e-19, 1.6402167482426369e-19, 1.6365722261287829e-19,
/* [264] */ 1.632929956438448e-19, 1.6292898590992037e-19, 1.625651854204191e-19, 1.6220158619993133e-19,
/* [268] */ 1.618381802870466e-19, 1.6147495973307924e-19, 1.6111191660079618e-19, 1.6074904296314645e-19,
/* [272] */ 1.6038633090199211e-19, 1.600237725068394e-19, 1.5966135987357025e-19, 1.5929908510317342e-19,
/* [276] */ 1.5893694030047434e-19, 1.5857491757286376e-19, 1.5821300902902412e-19, 1.5785120677765333e-19,
/* [280] */ 1.5748950292618545e-19, 1.5712788957950753e-19, 1.5676635883867226e-19, 1.5640490279960565e-19,
/* [284] */ 1.560435135518094e-19, 1.5568218317705714e-19, 1.5532090374808409e-19, 1.5495966732726971e-19,
/* [288] */ 1.545984659653122e-19, 1.5423729169989499e-19, 1.5387613655434405e-19, 1.5351499253627542e-19,
/* [292] */ 1.531538516362328e-19, 1.5279270582631381e-19, 1.5243154705878517e-19, 1.5207036726468514e-19,
/* [296] */ 1.5170915835241339e-19, 1.5134791220630703e-19, 1.5098662068520251e-19, 1.5062527562098241e-19,
/* [300] */ 1.5026386881710618e-19, 1.499023920471249e-19, 1.4954083705317816e-19, 1.4917919554447328e-19,
/* [304] */ 1.4881745919574531e-19, 1.4845561964569755e-19, 1.4809366849542141e-19, 1.4773159730679478e-19,
/* [308] */ 1.4736939760085815e-19, 1.470070608561676e-19, 1.4664457850712328e-19, 1.4628194194227331e-19,
/* [312] */ 1.4591914250259098e-19, 1.4555617147972526e-19, 1.4519302011422298e-19, 1.4482967959372176e-19,
/* [316] */ 1.4446614105111258e-19, 1.4410239556267103e-19, 1.4373843414615571e-19, 1.4337424775887289e-19,
/* [320] */ 1.4300982729570602e-19, 1.4264516358710902e-19, 1.4228024739706157e-19, 1.4191506942098562e-19,
/* [324] */ 1.4154962028362133e-19, 1.4118389053686103e-19, 1.4081787065753974e-19, 1.4045155104518092e-19,
/* [328] */ 1.4008492201969533e-19, 1.3971797381903188e-19, 1.3935069659677833e-19, 1.3898308041971047e-19,
/* [332] */ 1.3861511526528749e-19, 1.3824679101909196e-19, 1.3787809747221246e-19, 1.3750902431856644e-19,
/* [336] */ 1.3713956115216185e-19, 1.3676969746429448e-19, 1.3639942264067964e-19, 1.3602872595851507e-19,
/* [340] */ 1.3565759658347313e-19, 1.3528602356661948e-19, 1.3491399584125558e-19, 1.3454150221968241e-19,
/* [344] */ 1.3416853138988266e-19, 1.3379507191211783e-19, 1.3342111221543818e-19, 1.3304664059410112e-19,
/* [348] */ 1.3267164520389587e-19, 1.3229611405836999e-19, 1.3192003502495473e-19, 1.3154339582098531e-19,
/* [352] */ 1.3116618400961209e-19, 1.3078838699559864e-19, 1.3040999202100255e-19, 1.3003098616073445e-19,
/* [356] */ 1.2965135631799062e-19, 1.2927108921955436e-19, 1.2889017141096131e-19, 1.2850858925152309e-19,
/* [360] */ 1.2812632890920429e-19, 1.2774337635534646e-19, 1.2735971735923388e-19, 1.2697533748249419e-19,
/* [364] */ 1.2659022207332785e-19, 1.262043562605593e-19, 1.2581772494750294e-19, 1.2543031280563616e-19,
/* [368] */ 1.250421042680721e-19, 1.2465308352282341e-19, 1.2426323450584884e-19, 1.238725408938736e-19,
/* [372] */ 1.2348098609697396e-19, 1.2308855325091651e-19, 1.2269522520924136e-19, 1.22300984535079e-19,
/* [376] */ 1.2190581349268875e-19, 1.2150969403870742e-19, 1.2111260781309555e-19, 1.2071453612976755e-19,
/* [380] */ 1.2031545996689279e-19, 1.1991535995685211e-19, 1.1951421637583513e-19, 1.191120091330619e-19,
/* [384] */ 1.1870871775961209e-19, 1.1830432139684355e-19, 1.1789879878438187e-19, 1.1749212824766058e-19,
/* [388] */ 1.1708428768499157e-19, 1.1667525455414306e-19, 1.1626500585840243e-19, 1.1585351813209874e-19,
/* [392] */ 1.1544076742555945e-19, 1.150267292894733e-19, 1.1461137875863088e-19, 1.1419469033501178e-19,
/* [396] */ 1.1377663797018575e-19, 1.1335719504699387e-19, 1.1293633436047259e-19, 1.1251402809798253e-19,
/* [400] */ 1.1209024781850075e-19, 1.1166496443103302e-19, 1.1123814817209981e-19, 1.1080976858224708e-19,
/* [404] */ 1.1037979448152952e-19, 1.0994819394391104e-19, 1.0951493427052315e-19, 1.0907998196171864e-19,
/* [408] */ 1.0864330268785362e-19, 1.0820486125872634e-19, 1.0776462159159718e-19, 1.0732254667770796e-19,
/* [412] */ 1.0687859854721439e-19, 1.0643273823243869e-19, 1.059849257293434e-19, 1.0553511995712021e-19,
/* [416] */ 1.0508327871578037e-19, 1.0462935864162494e-19, 1.0417331516046436e-19, 1.0371510243844727e-19,
/* [420] */ 1.0325467333034829e-19, 1.0279197932515293e-19, 1.0232697048876598e-19, 1.0185959540365581e-19,
/* [424] */ 1.013898011052333e-19, 1.009175330147477e-19, 1.0044273486846454e-19, 9.9965348642872442e-20,
/* [428] */ 9.9485314475644288e-20, 9.900257058205621e-20, 9.8517053166543133e-20, 9.8028696329042194e-20,
/* [432] */ 9.7537431965745965e-20, 9.7043189663854526e-20, 9.654589658987947e-20, 9.6045477371013195e-20,
/* [436] */ 9.5541853969033159e-20, 9.5034945546162292e-20, 9.4524668322253246e-20, 9.4010935422604966e-20,
/* [440] */ 9.3493656715654088e-20, 9.2972738639710746e-20, 9.2448084017826916e-20, 9.1919591859794958e-20,
/* [444] */ 9.1387157150172602e-20, 9.0850670621117885e-20, 9.0310018508690594e-20, 8.9765082291135159e-20,
/* [448] */ 8.9215738407499983e-20, 8.8661857954768974e-20, 8.8103306361478308e-20, 8.7539943035562694e-20,
/* [452] */ 8.6971620983916387e-20, 8.6398186400860379e-20, 8.5819478222373023e-20, 8.5235327642560774e-20,
/* [456] */ 8.4645557588411128e-20, 8.4049982148372247e-20, 8.3448405949733243e-20, 8.2840623479121996e-20,
/* [460] */ 8.2226418339680758e-20, 8.1605562437603462e-20, 8.0977815089703785e-20, 8.0342922042501121e-20,
/* [464] */ 7.9700614391933997e-20, 7.9050607391197453e-20, 7.8392599132306843e-20, 7.772626908475868e-20,
/* [468] */ 7.7051276472020123e-20, 7.6367258463445082e-20, 7.5673828155481347e-20, 7.497057231156425e-20,
/* [472] */ 7.4257048824721583e-20, 7.353278386042921e-20, 7.2797268629389381e-20, 7.2049955730309961e-20,
/* [476] */ 7.1290254991002794e-20, 7.0517528721622442e-20, 6.9731086275889195e-20, 6.8930177793707327e-20,
/* [480] */ 6.8113986970409562e-20, 6.7281622662207744e-20, 6.6432109091987626e-20, 6.5564374361194866e-20,
/* [484] */ 6.4677236897885373e-20, 6.3769389372032666e-20, 6.2839379478434802e-20, 6.1885586812998988e-20,
/* [488] */ 6.0906194832423162e-20, 5.9899156564895461e-20, 5.8862152292532516e-20, 5.7792536797552712e-20,
/* [492] */ 5.6687272865166578e-20, 5.5542846427447531e-20, 5.4355156789160883e-20, 5.3119372426547794e-20,
/* [496] */ 5.1829738259578428e-20, 5.047931295220812e-20, 4.9059602658819333e-20, 4.7560036835100967e-20,
/* [500] */ 4.5967194527575314e-20, 4.4263619567465964e-20, 4.2425923207137379e-20, 4.0421571557166738e-20,
/* [504] */ 3.820304293025196e-20, 3.5696135223547374e-20, 3.2773159946159168e-20, 2.9176892376585344e-20,
/* [508] */ 2.4195231151204545e-20, 0,
};
/** The precomputed ziggurat heights, denoted Y_i in the main text. Y_i = f(X_i).
* Values have been scaled by 2^-63. */
private static final double[] Y = {
/* [ 0] */ 6.9188990988329477e-23, 1.4202990535697683e-22, 2.1732316399259404e-22, 2.9452908326033447e-22,
/* [ 4] */ 3.7333229265092159e-22, 4.5352314752172509e-22, 5.3495096331850513e-22, 6.175015744699501e-22,
/* [ 8] */ 7.0108513174955496e-22, 7.8562885992867391e-22, 8.7107247053338667e-22, 9.5736510622922269e-22,
/* [ 12] */ 1.0444632219043918e-21, 1.1323290661676009e-21, 1.2209295628733119e-21, 1.3102354679393653e-21,
/* [ 16] */ 1.4002207209079843e-21, 1.4908619375774252e-21, 1.5821380069573031e-21, 1.6740297667860313e-21,
/* [ 20] */ 1.7665197391694204e-21, 1.8595919128929468e-21, 1.9532315624377367e-21, 2.0474250961975478e-21,
/* [ 24] */ 2.1421599281741028e-21, 2.2374243687320324e-21, 2.3332075309631245e-21, 2.4294992499379909e-21,
/* [ 28] */ 2.5262900126775297e-21, 2.6235708971028823e-21, 2.7213335185536967e-21, 2.8195699827241005e-21,
/* [ 32] */ 2.9182728440709958e-21, 3.0174350689128386e-21, 3.1170500025683573e-21, 3.2171113399908191e-21,
/* [ 36] */ 3.3176130994398206e-21, 3.4185495988032995e-21, 3.5199154342406966e-21, 3.6217054608664173e-21,
/* [ 40] */ 3.7239147752328608e-21, 3.8265386994058555e-21, 3.9295727664535323e-21, 4.0330127071934498e-21,
/* [ 44] */ 4.1368544380629805e-21, 4.2410940499950823e-21, 4.345727798196275e-21, 4.4507520927361638e-21,
/* [ 48] */ 4.5561634898686684e-21, 4.6619586840144468e-21, 4.7681345003420505e-21, 4.8746878878923688e-21,
/* [ 52] */ 4.9816159131970167e-21, 5.0889157543466486e-21, 5.1965846954698379e-21, 5.3046201215872733e-21,
/* [ 56] */ 5.413019513809608e-21, 5.5217804448504855e-21, 5.6309005748290817e-21, 5.7403776473389792e-21,
/* [ 60] */ 5.8502094857624066e-21, 5.9603939898108565e-21, 6.0709291322748242e-21, 6.1818129559670009e-21,
/* [ 64] */ 6.2930435708446384e-21, 6.4046191512980779e-21, 6.5165379335935457e-21, 6.6287982134593532e-21,
/* [ 68] */ 6.7413983438055354e-21, 6.8543367325678093e-21, 6.9676118406674612e-21, 7.0812221800794591e-21,
/* [ 72] */ 7.1951663120016937e-21, 7.3094428451188258e-21, 7.4240504339546813e-21, 7.5389877773076551e-21,
/* [ 76] */ 7.6542536167639494e-21, 7.7698467352838953e-21, 7.8857659558569342e-21, 8.0020101402211584e-21,
/* [ 80] */ 8.1185781876436157e-21, 8.235469033757848e-21, 8.3526816494553559e-21, 8.4702150398279595e-21,
/* [ 84] */ 8.5880682431581806e-21, 8.706240329954993e-21, 8.8247304020324768e-21, 8.9435375916290263e-21,
/* [ 88] */ 9.0626610605649797e-21, 9.1820999994366113e-21, 9.3018536268446223e-21, 9.4219211886553146e-21,
/* [ 92] */ 9.5423019572928162e-21, 9.6629952310607645e-21, 9.7840003334919972e-21, 9.905316612724868e-21,
/* [ 96] */ 1.0026943440904879e-20, 1.0148880213610417e-20, 1.0271126349301457e-20, 1.0393681288790118e-20,
/* [100] */ 1.0516544494732088e-20, 1.0639715451137926e-20, 1.076319366290336e-20, 1.088697865535769e-20,
/* [104] */ 1.1011069973829531e-20, 1.1135467183229089e-20, 1.1260169867646259e-20, 1.1385177629963887e-20,
/* [108] */ 1.1510490091485507e-20, 1.1636106891576963e-20, 1.176202768732133e-20, 1.1888252153186578e-20,
/* [112] */ 1.2014779980705474e-20, 1.2141610878167194e-20, 1.2268744570320195e-20, 1.2396180798085917e-20,
/* [116] */ 1.2523919318282839e-20, 1.2651959903360538e-20, 1.2780302341143342e-20, 1.2908946434583201e-20,
/* [120] */ 1.3037892001521468e-20, 1.3167138874459203e-20, 1.3296686900335739e-20, 1.342653594031518e-20,
/* [124] */ 1.3556685869580544e-20, 1.3687136577135297e-20, 1.3817887965612001e-20, 1.3948939951087837e-20,
/* [128] */ 1.4080292462906762e-20, 1.421194544350808e-20, 1.4343898848261192e-20, 1.447615264530638e-20,
/* [132] */ 1.4608706815401313e-20, 1.4741561351773229e-20, 1.4874716259976492e-20, 1.5008171557755435e-20,
/* [136] */ 1.5141927274912272e-20, 1.527598345317996e-20, 1.5410340146099826e-20, 1.5544997418903869e-20,
/* [140] */ 1.5679955348401513e-20, 1.5815214022870783e-20, 1.5950773541953694e-20, 1.6086634016555787e-20,
/* [144] */ 1.6222795568749659e-20, 1.6359258331682407e-20, 1.6496022449486859e-20, 1.6633088077196497e-20,
/* [148] */ 1.6770455380664001e-20, 1.690812453648326e-20, 1.7046095731914825e-20, 1.7184369164814694e-20,
/* [152] */ 1.7322945043566328e-20, 1.7461823587015858e-20, 1.7601005024410389e-20, 1.7740489595339301e-20,
/* [156] */ 1.7880277549678543e-20, 1.8020369147537807e-20, 1.8160764659210518e-20, 1.8301464365126615e-20,
/* [160] */ 1.8442468555808009e-20, 1.8583777531826755e-20, 1.8725391603765759e-20, 1.8867311092182093e-20,
/* [164] */ 1.9009536327572789e-20, 1.9152067650343099e-20, 1.9294905410777168e-20, 1.9438049969011095e-20,
/* [168] */ 1.9581501695008306e-20, 1.9725260968537243e-20, 1.9869328179151295e-20, 2.0013703726170975e-20,
/* [172] */ 2.0158388018668269e-20, 2.030338147545316e-20, 2.0448684525062275e-20, 2.0594297605749654e-20,
/* [176] */ 2.0740221165479551e-20, 2.0886455661921353e-20, 2.1033001562446463e-20, 2.1179859344127223e-20,
/* [180] */ 2.132702949373782e-20, 2.1474512507757144e-20, 2.1622308892373594e-20, 2.177041916349182e-20,
/* [184] */ 2.1918843846741371e-20, 2.2067583477487234e-20, 2.221663860084227e-20, 2.2366009771681506e-20,
/* [188] */ 2.2515697554658291e-20, 2.2665702524222297e-20, 2.2816025264639365e-20, 2.2966666370013165e-20,
/* [192] */ 2.3117626444308697e-20, 2.3268906101377579e-20, 2.3420505964985179e-20, 2.3572426668839511e-20,
/* [196] */ 2.3724668856621966e-20, 2.3877233182019821e-20, 2.4030120308760546e-20, 2.4183330910647897e-20,
/* [200] */ 2.4336865671599831e-20, 2.4490725285688176e-20, 2.4644910457180119e-20, 2.4799421900581494e-20,
/* [204] */ 2.4954260340681856e-20, 2.5109426512601375e-20, 2.5264921161839525e-20, 2.5420745044325602e-20,
/* [208] */ 2.5576898926471056e-20, 2.5733383585223655e-20, 2.5890199808123478e-20, 2.6047348393360769e-20,
/* [212] */ 2.620483014983564e-20, 2.6362645897219624e-20, 2.6520796466019148e-20, 2.6679282697640851e-20,
/* [216] */ 2.6838105444458826e-20, 2.6997265569883789e-20, 2.7156763948434162e-20, 2.7316601465809108e-20,
/* [220] */ 2.7476779018963542e-20, 2.7637297516185117e-20, 2.7798157877173203e-20, 2.7959361033119882e-20,
/* [224] */ 2.8120907926793008e-20, 2.8282799512621313e-20, 2.8445036756781554e-20, 2.8607620637287851e-20,
/* [228] */ 2.8770552144083069e-20, 2.8933832279132388e-20, 2.909746205651907e-20, 2.9261442502542373e-20,
/* [232] */ 2.9425774655817774e-20, 2.9590459567379366e-20, 2.975549830078463e-20, 2.9920891932221431e-20,
/* [236] */ 3.0086641550617475e-20, 3.0252748257752036e-20, 3.0419213168370193e-20, 3.0586037410299459e-20,
/* [240] */ 3.0753222124568922e-20, 3.0920768465530904e-20, 3.1088677600985173e-20, 3.1256950712305787e-20,
/* [244] */ 3.1425588994570519e-20, 3.1594593656693039e-20, 3.1763965921557762e-20, 3.1933707026157491e-20,
/* [248] */ 3.210381822173386e-20, 3.2274300773920679e-20, 3.2445155962890127e-20, 3.261638508350196e-20,
/* [252] */ 3.2787989445455682e-20, 3.2959970373445826e-20, 3.3132329207320317e-20, 3.3305067302242021e-20,
/* [256] */ 3.3478186028853508e-20, 3.3651686773445136e-20, 3.3825570938126452e-20, 3.3999839941001009e-20,
/* [260] */ 3.4174495216344673e-20, 3.434953821478746e-20, 3.4524970403498974e-20, 3.4700793266377521e-20,
/* [264] */ 3.4877008304243007e-20, 3.5053617035033593e-20, 3.5230620994006313e-20, 3.5408021733941619e-20,
/* [268] */ 3.5585820825352012e-20, 3.5764019856694792e-20, 3.5942620434589016e-20, 3.6121624184036813e-20,
/* [272] */ 3.6301032748649041e-20, 3.6480847790875463e-20, 3.6661070992239485e-20, 3.6841704053577609e-20,
/* [276] */ 3.7022748695283621e-20, 3.7204206657557678e-20, 3.7386079700660407e-20, 3.7568369605172066e-20,
/* [280] */ 3.7751078172256915e-20, 3.7934207223932926e-20, 3.8117758603346923e-20, 3.8301734175055289e-20,
/* [284] */ 3.8486135825310328e-20, 3.8670965462352503e-20, 3.8856225016708545e-20, 3.9041916441495727e-20,
/* [288] */ 3.9228041712732281e-20, 3.9414602829654225e-20, 3.9601601815038723e-20, 3.9789040715534053e-20,
/* [292] */ 3.9976921601996467e-20, 4.0165246569833989e-20, 4.0354017739357392e-20, 4.0543237256138495e-20,
/* [296] */ 4.0732907291375956e-20, 4.092303004226878e-20, 4.1113607732397641e-20, 4.1304642612114334e-20,
/* [300] */ 4.1496136958939466e-20, 4.1688093077968586e-20, 4.1880513302287083e-20, 4.2073399993393892e-20,
/* [304] */ 4.226675554163437e-20, 4.2460582366642555e-20, 4.2654882917792981e-20, 4.2849659674662346e-20,
/* [308] */ 4.3044915147501306e-20, 4.3240651877716606e-20, 4.3436872438363862e-20, 4.3633579434651235e-20,
/* [312] */ 4.3830775504454331e-20, 4.402846331884258e-20, 4.4226645582617465e-20, 4.44253250348628e-20,
/* [316] */ 4.4624504449507564e-20, 4.4824186635901433e-20, 4.502437443940352e-20, 4.5225070741984591e-20,
/* [320] */ 4.5426278462843173e-20, 4.5628000559035919e-20, 4.5830240026122638e-20, 4.6032999898826428e-20,
/* [324] */ 4.6236283251709275e-20, 4.6440093199863635e-20, 4.66444328996204e-20, 4.6849305549273747e-20,
/* [328] */ 4.7054714389823359e-20, 4.7260662705734508e-20, 4.7467153825716568e-20, 4.7674191123520394e-20,
/* [332] */ 4.7881778018755285e-20, 4.8089917977726014e-20, 4.8298614514290481e-20, 4.8507871190738755e-20,
/* [336] */ 4.8717691618694051e-20, 4.8928079460036324e-20, 4.9139038427849191e-20, 4.9350572287390917e-20,
/* [340] */ 4.9562684857090195e-20, 4.9775380009567483e-20, 4.9988661672682747e-20, 5.0202533830610375e-20,
/* [344] */ 5.0417000524942233e-20, 5.0632065855819656e-20, 5.0847733983095388e-20, 5.1064009127526389e-20,
/* [348] */ 5.1280895571998574e-20, 5.1498397662784496e-20, 5.1716519810835054e-20, 5.1935266493106415e-20,
/* [352] */ 5.2154642253923237e-20, 5.2374651706379573e-20, 5.2595299533778521e-20, 5.281659049111218e-20,
/* [356] */ 5.3038529406583093e-20, 5.3261121183168789e-20, 5.3484370800230758e-20, 5.3708283315169579e-20,
/* [360] */ 5.3932863865127733e-20, 5.4158117668741825e-20, 5.4384050027946003e-20, 5.4610666329828364e-20,
/* [364] */ 5.4837972048542409e-20, 5.5065972747275354e-20, 5.529467408027557e-20, 5.5524081794941295e-20,
/* [368] */ 5.5754201733972871e-20, 5.5985039837590894e-20, 5.6216602145822901e-20, 5.6448894800860957e-20,
/* [372] */ 5.6681924049493214e-20, 5.6915696245611979e-20, 5.7150217852801476e-20, 5.7385495447008377e-20,
/* [376] */ 5.7621535719298375e-20, 5.7858345478702286e-20, 5.8095931655155148e-20, 5.8334301302532288e-20,
/* [380] */ 5.8573461601786159e-20, 5.8813419864188086e-20, 5.9054183534679474e-20, 5.9295760195336789e-20,
/* [384] */ 5.9538157568955278e-20, 5.9781383522756458e-20, 6.00254460722246e-20, 6.0270353385077896e-20,
/* [388] */ 6.0516113785379988e-20, 6.0762735757798273e-20, 6.1010227952015272e-20, 6.1258599187299976e-20,
/* [392] */ 6.1507858457246388e-20, 6.1758014934686814e-20, 6.2009077976787964e-20, 6.2261057130338207e-20,
/* [396] */ 6.2513962137235034e-20, 6.2767802940182038e-20, 6.3022589688605306e-20, 6.3278332744799953e-20,
/* [400] */ 6.3535042690317614e-20, 6.3792730332606894e-20, 6.4051406711919054e-20, 6.4311083108492257e-20,
/* [404] */ 6.4571771050028178e-20, 6.4833482319475923e-20, 6.5096228963138882e-20, 6.5360023299121136e-20,
/* [408] */ 6.562487792613134e-20, 6.5890805732662563e-20, 6.6157819906568342e-20, 6.6425933945056075e-20,
/* [412] */ 6.669516166512051e-20, 6.6965517214441315e-20, 6.7237015082770543e-20, 6.7509670113837357e-20,
/* [416] */ 6.778349751779927e-20, 6.8058512884271187e-20, 6.8334732195965602e-20, 6.861217184297964e-20,
/* [420] */ 6.8890848637767307e-20, 6.9170779830837617e-20, 6.9451983127222779e-20, 6.973447670376329e-20,
/* [424] */ 7.0018279227260534e-20, 7.0303409873551134e-20, 7.0589888347561524e-20, 7.0877734904405408e-20,
/* [428] */ 7.1166970371591799e-20, 7.1457616172416729e-20, 7.1749694350617054e-20, 7.2043227596371579e-20,
/* [432] */ 7.2338239273741285e-20, 7.2634753449648124e-20, 7.2932794924500039e-20, 7.323238926457911e-20,
/* [436] */ 7.3533562836319459e-20, 7.3836342842612738e-20, 7.4140757361291078e-20, 7.4446835385950638e-20,
/* [440] */ 7.4754606869293753e-20, 7.5064102769183896e-20, 7.5375355097625875e-20, 7.5688396972903551e-20,
/* [444] */ 7.6003262675129782e-20, 7.6319987705488118e-20, 7.663860884947315e-20, 7.6959164244467646e-20,
/* [448] */ 7.7281693452028738e-20, 7.7606237535294258e-20, 7.7932839141963691e-20, 7.8261542593356837e-20,
/* [452] */ 7.8592393980108262e-20, 7.8925441265117724e-20, 7.9260734394446575e-20, 7.9598325416929843e-20,
/* [456] */ 7.9938268613363919e-20, 8.0280620636232317e-20, 8.0625440661049589e-20, 8.0972790550537079e-20,
/* [460] */ 8.132273503299865e-20, 8.1675341896440717e-20, 8.20306822001853e-20, 8.2388830505960473e-20,
/* [464] */ 8.2749865130725681e-20, 8.3113868423807933e-20, 8.3480927071295611e-20, 8.3851132431071348e-20,
/* [468] */ 8.4224580902376006e-20, 8.4601374334397956e-20, 8.498162047909447e-20, 8.5365433494299503e-20,
/* [472] */ 8.5752934504183006e-20, 8.6144252225339393e-20, 8.6539523668242568e-20, 8.6938894925572055e-20,
/* [476] */ 8.7342522061064557e-20, 8.7750572115174375e-20, 8.816322424706147e-20, 8.8580671036429462e-20,
/* [480] */ 8.9003119973724073e-20, 8.943079517345898e-20, 8.9863939353342e-20, 9.030281613194174e-20,
/* [484] */ 9.0747712710563048e-20, 9.1198943021750057e-20, 9.1656851448748822e-20, 9.2121817249226692e-20,
/* [488] */ 9.2594259855265681e-20, 9.3074645274038203e-20, 9.3563493885409803e-20, 9.4061390032646715e-20,
/* [492] */ 9.4568993943651402e-20, 9.5087056723313496e-20, 9.56164394555198e-20, 9.6158137899911961e-20,
/* [496] */ 9.6713314954182168e-20, 9.7283344135003048e-20, 9.7869869093443078e-20, 9.8474887157460046e-20,
/* [500] */ 9.9100870137415168e-20, 9.9750945338940223e-20, 1.004291788159128e-19, 1.0114104330564085e-19,
/* [504] */ 1.0189424721829993e-19, 1.0270034796927329e-19, 1.0357834330014222e-19, 1.0456455804293095e-19,
/* [508] */ 1.0575382882239675e-19, 1.0842021724855044e-19,
};
/** Underlying source of randomness. */
protected final UniformRandomProvider rng;
/** Exponential sampler used for the long tail. */
protected final ContinuousSampler exponential;
/**
* @param rng Generator of uniformly distributed random numbers.
*/
ModifiedZigguratNormalizedGaussianSampler512(UniformRandomProvider rng) {
this.rng = rng;
exponential = new ModifiedZigguratExponentialSampler512(rng);
}
/** {@inheritDoc} */
@Override
public double sample() {
final long xx = nextLong();
// Float multiplication squashes these last 9 bits, so they can be used to sample i
final int i = ((int) xx) & 0x1ff;
if (i < I_MAX) {
// Early exit.
return X[i] * xx;
}
// Recycle bits then advance RNG:
long u1 = xx & MAX_INT64;
// Another squashed, recyclable bit
// Use 2 - 1 or 0 - 1
final double signBit = ((u1 >>> 8) & 0x2) - 1.0;
final int j = selectRegion();
// Four kinds of overhangs:
// j = 0 : Sample from tail
// 0 < j < J_INFLECTION : Overhang is concave; only sample from Lower-Left triangle
// j = J_INFLECTION : Must sample from entire overhang rectangle
// j > J_INFLECTION : Overhangs are convex; implicitly accept point in Lower-Left triangle
//
// Conditional statements are arranged such that the more likely outcomes are first.
double x;
if (j > J_INFLECTION) {
// Convex overhang
for (;;) {
x = sampleX(X, j, u1);
final long uDistance = randomInt63() - u1;
if (uDistance >= 0) {
// Lower-left triangle
break;
}
if (uDistance >= CONVEX_E_MAX &&
// Within maximum distance of f(x) from the triangle hypotenuse.
sampleY(Y, j, u1 + uDistance) < Math.exp(-0.5 * x * x)) {
break;
}
// uDistance < E_MAX (upper-right triangle) or rejected as above the curve
u1 = randomInt63();
}
} else if (j < J_INFLECTION) {
if (j == 0) {
// Tail
// Note: Although less frequent than the next branch, j == 0 is a subset of
// j < J_INFLECTION and must be first.
do {
x = ONE_OVER_X_0 * exponential.sample();
} while (exponential.sample() < 0.5 * x * x);
x += X_0;
} else {
// Concave overhang
for (;;) {
// U_x <- min(U_1, U_2)
// distance <- | U_1 - U_2 |
// U_y <- 1 - (U_x + distance)
long uDistance = randomInt63() - u1;
if (uDistance < 0) {
uDistance = -uDistance;
u1 -= uDistance;
}
x = sampleX(X, j, u1);
if (uDistance > CONCAVE_E_MAX ||
sampleY(Y, j, u1 + uDistance) < Math.exp(-0.5 * x * x)) {
break;
}
u1 = randomInt63();
}
}
} else {
// Inflection point
for (;;) {
x = sampleX(X, j, u1);
if (sampleY(Y, j, randomInt63()) < Math.exp(-0.5 * x * x)) {
break;
}
u1 = randomInt63();
}
}
return signBit * x;
}
/**
* Select the overhang region or the tail using alias sampling.
*
* @return the region
*/
protected int selectRegion() {
final long x = nextLong();
// j in [0, 512)
final int j = ((int) x) & 0x1ff;
// map to j in [0, N] with N the number of layers of the ziggurat
return x >= IPMF[j] ? MAP[j] : j;
}
/**
* Generates a {@code long}.
*
* @return the long
*/
protected long nextLong() {
return rng.nextLong();
}
/**
* Return a positive long in {@code [0, 2^63)}.
*
* @return the long
*/
protected long randomInt63() {
return rng.nextLong() & MAX_INT64;
}
}
/**
* Modified Ziggurat method for sampling from an exponential distribution.
*
* <p>Uses the algorithm from:
*
* <blockquote>
* McFarland, C.D. (2016)<br>
* "A modified ziggurat algorithm for generating exponentially and normally distributed pseudorandom numbers".<br>
* <i>Journal of Statistical Computation and Simulation</i> <b>86</b>, 1281-1294.
* </blockquote>
*
* <p>This class uses the same tables as the production version
* {@link org.apache.commons.rng.sampling.distribution.ZigguratSampler.Exponential}
* with the overhang sampling matching the reference c implementation. Methods and members
* are protected to allow the implementation to be modified in sub-classes.
*
* @see <a href="https://www.tandfonline.com/doi/abs/10.1080/00949655.2015.1060234">
* McFarland (2016) JSCS 86, 1281-1294</a>
*/
static class ModifiedZigguratExponentialSampler implements ContinuousSampler {
// Ziggurat volumes:
// Inside the layers = 98.4375% (252/256)
// Fraction outside the layers:
// concave overhangs = 96.6972%
// tail = 3.3028%
/** The number of layers in the ziggurat. Maximum i value for early exit. */
protected static final int I_MAX = 252;
/** Maximum deviation of concave pdf(x) below the hypotenuse value for early exit.
* Equal to approximately 0.0926 scaled by 2^63. */
protected static final long E_MAX = 853965788476313646L;
/** Beginning of tail. Equal to X[0] * 2^63. */
protected static final double X_0 = 7.569274694148063;
/** The alias map. An integer in [0, 255] stored as a byte to save space. */
protected static final byte[] MAP = {
/* [ 0] */ (byte) 0, (byte) 0, (byte) 1, (byte) 235, (byte) 3, (byte) 4, (byte) 5, (byte) 0,
/* [ 8] */ (byte) 0, (byte) 0, (byte) 0, (byte) 0, (byte) 0, (byte) 0, (byte) 0, (byte) 0,
/* [ 16] */ (byte) 0, (byte) 0, (byte) 1, (byte) 1, (byte) 1, (byte) 1, (byte) 2, (byte) 2,
/* [ 24] */ (byte) 252, (byte) 252, (byte) 252, (byte) 252, (byte) 252, (byte) 252, (byte) 252, (byte) 252,
/* [ 32] */ (byte) 252, (byte) 252, (byte) 252, (byte) 252, (byte) 252, (byte) 252, (byte) 252, (byte) 252,
/* [ 40] */ (byte) 252, (byte) 252, (byte) 252, (byte) 252, (byte) 252, (byte) 252, (byte) 252, (byte) 252,
/* [ 48] */ (byte) 252, (byte) 252, (byte) 252, (byte) 252, (byte) 252, (byte) 252, (byte) 252, (byte) 252,
/* [ 56] */ (byte) 252, (byte) 252, (byte) 252, (byte) 252, (byte) 252, (byte) 252, (byte) 252, (byte) 252,
/* [ 64] */ (byte) 252, (byte) 252, (byte) 252, (byte) 252, (byte) 252, (byte) 252, (byte) 252, (byte) 252,
/* [ 72] */ (byte) 252, (byte) 252, (byte) 252, (byte) 252, (byte) 252, (byte) 252, (byte) 252, (byte) 252,
/* [ 80] */ (byte) 252, (byte) 252, (byte) 252, (byte) 252, (byte) 252, (byte) 252, (byte) 252, (byte) 252,
/* [ 88] */ (byte) 252, (byte) 252, (byte) 252, (byte) 252, (byte) 252, (byte) 252, (byte) 252, (byte) 252,
/* [ 96] */ (byte) 252, (byte) 252, (byte) 252, (byte) 252, (byte) 252, (byte) 252, (byte) 252, (byte) 252,
/* [104] */ (byte) 252, (byte) 252, (byte) 252, (byte) 252, (byte) 252, (byte) 252, (byte) 252, (byte) 252,
/* [112] */ (byte) 252, (byte) 252, (byte) 252, (byte) 252, (byte) 252, (byte) 252, (byte) 252, (byte) 252,
/* [120] */ (byte) 252, (byte) 252, (byte) 252, (byte) 252, (byte) 252, (byte) 252, (byte) 252, (byte) 252,
/* [128] */ (byte) 252, (byte) 252, (byte) 252, (byte) 252, (byte) 252, (byte) 252, (byte) 252, (byte) 252,
/* [136] */ (byte) 252, (byte) 252, (byte) 252, (byte) 252, (byte) 252, (byte) 252, (byte) 252, (byte) 252,
/* [144] */ (byte) 252, (byte) 252, (byte) 252, (byte) 252, (byte) 252, (byte) 252, (byte) 252, (byte) 252,
/* [152] */ (byte) 252, (byte) 252, (byte) 252, (byte) 252, (byte) 252, (byte) 252, (byte) 252, (byte) 252,
/* [160] */ (byte) 252, (byte) 252, (byte) 252, (byte) 252, (byte) 252, (byte) 252, (byte) 252, (byte) 252,
/* [168] */ (byte) 252, (byte) 252, (byte) 252, (byte) 252, (byte) 252, (byte) 252, (byte) 252, (byte) 252,
/* [176] */ (byte) 252, (byte) 251, (byte) 251, (byte) 251, (byte) 251, (byte) 251, (byte) 251, (byte) 251,
/* [184] */ (byte) 251, (byte) 251, (byte) 251, (byte) 251, (byte) 251, (byte) 251, (byte) 250, (byte) 250,
/* [192] */ (byte) 250, (byte) 250, (byte) 250, (byte) 250, (byte) 250, (byte) 249, (byte) 249, (byte) 249,
/* [200] */ (byte) 249, (byte) 249, (byte) 249, (byte) 248, (byte) 248, (byte) 248, (byte) 248, (byte) 247,
/* [208] */ (byte) 247, (byte) 247, (byte) 247, (byte) 246, (byte) 246, (byte) 246, (byte) 245, (byte) 245,
/* [216] */ (byte) 244, (byte) 244, (byte) 243, (byte) 243, (byte) 242, (byte) 241, (byte) 241, (byte) 240,
/* [224] */ (byte) 239, (byte) 237, (byte) 3, (byte) 3, (byte) 4, (byte) 4, (byte) 6, (byte) 0,
/* [232] */ (byte) 0, (byte) 0, (byte) 0, (byte) 236, (byte) 237, (byte) 238, (byte) 239, (byte) 240,
/* [240] */ (byte) 241, (byte) 242, (byte) 243, (byte) 244, (byte) 245, (byte) 246, (byte) 247, (byte) 248,
/* [248] */ (byte) 249, (byte) 250, (byte) 251, (byte) 252, (byte) 2, (byte) 0, (byte) 0, (byte) 0,
};
/** The alias inverse PMF. */
protected static final long[] IPMF = {
/* [ 0] */ 9223372036854774016L, 1623796909450834944L, 2664290944894291200L, 7387971354164060928L,
/* [ 4] */ 6515064486552723200L, 8840508362680718848L, 6099647593382936320L, 7673130333659513856L,
/* [ 8] */ 6220332867583438080L, 5045979640552813824L, 4075305837223955456L, 3258413672162525440L,
/* [ 12] */ 2560664887087762432L, 1957224924672899584L, 1429800935350577408L, 964606309710808320L,
/* [ 16] */ 551043923599587072L, 180827629096890368L, -152619738120023552L, -454588624410291456L,
/* [ 20] */ -729385126147774976L, -980551509819447040L, -1211029700667463936L, -1423284293868548352L,
/* [ 24] */ -1619396356369050368L, -1801135830956211712L, -1970018048575618048L, -2127348289059705344L,
/* [ 28] */ -2274257249303686400L, -2411729520096655360L, -2540626634159181056L, -2661705860113406464L,
/* [ 32] */ -2775635634532450560L, -2883008316030465280L, -2984350790383654912L, -3080133339198116352L,
/* [ 36] */ -3170777096303091200L, -3256660348483819008L, -3338123885075136256L, -3415475560473299200L,
/* [ 40] */ -3488994201966428160L, -3558932970354473216L, -3625522261068041216L, -3688972217741989376L,
/* [ 44] */ -3749474917563782656L, -3807206277531056128L, -3862327722496843520L, -3914987649156779776L,
/* [ 48] */ -3965322714631865344L, -4013458973776895488L, -4059512885612783360L, -4103592206186241024L,
/* [ 52] */ -4145796782586128128L, -4186219260694347008L, -4224945717447275264L, -4262056226866285568L,
/* [ 56] */ -4297625367836519680L, -4331722680528537344L, -4364413077437472512L, -4395757214229401600L,
/* [ 60] */ -4425811824915135744L, -4454630025296932608L, -4482261588141290496L, -4508753193105288192L,
/* [ 64] */ -4534148654077808896L, -4558489126279958272L, -4581813295192216576L, -4604157549138257664L,
/* [ 68] */ -4625556137145255168L, -4646041313519104512L, -4665643470413305856L, -4684391259530326528L,
/* [ 72] */ -4702311703971761664L, -4719430301145103360L, -4735771117539946240L, -4751356876102087168L,
/* [ 76] */ -4766209036859133952L, -4780347871386013440L, -4793792531638892032L, -4806561113635132672L,
/* [ 80] */ -4818670716409306624L, -4830137496634465536L, -4840976719260837888L, -4851202804490348800L,
/* [ 84] */ -4860829371376460032L, -4869869278311657472L, -4878334660640771072L, -4886236965617427200L,
/* [ 88] */ -4893586984900802560L, -4900394884772702720L, -4906670234238885376L, -4912422031164496896L,
/* [ 92] */ -4917658726580119808L, -4922388247283532288L, -4926618016851066624L, -4930354975163335168L,
/* [ 96] */ -4933605596540651264L, -4936375906575303936L, -4938671497741366016L, -4940497543854575616L,
/* [100] */ -4941858813449629440L, -4942759682136114944L, -4943204143989086720L, -4943195822025528064L,
/* [104] */ -4942737977813206528L, -4941833520255033344L, -4940485013586738944L, -4938694684624359424L,
/* [108] */ -4936464429291795968L, -4933795818458825728L, -4930690103114057984L, -4927148218896864000L,
/* [112] */ -4923170790008275968L, -4918758132519213568L, -4913910257091645696L, -4908626871126539264L,
/* [116] */ -4902907380349533952L, -4896750889844272896L, -4890156204540531200L, -4883121829162554368L,
/* [120] */ -4875645967641781248L, -4867726521994927104L, -4859361090668103424L, -4850546966345113600L,
/* [124] */ -4841281133215539200L, -4831560263698491904L, -4821380714613447424L, -4810738522790066176L,
/* [128] */ -4799629400105481984L, -4788048727936307200L, -4775991551010514944L, -4763452570642114304L,
/* [132] */ -4750426137329494528L, -4736906242696389120L, -4722886510751377664L, -4708360188440089088L,
/* [136] */ -4693320135461421056L, -4677758813316108032L, -4661668273553489152L, -4645040145179241472L,
/* [140] */ -4627865621182772224L, -4610135444140930048L, -4591839890849345536L, -4572968755929961472L,
/* [144] */ -4553511334358205696L, -4533456402849101568L, -4512792200036279040L, -4491506405372580864L,
/* [148] */ -4469586116675402496L, -4447017826233107968L, -4423787395382284800L, -4399880027458416384L,
/* [152] */ -4375280239014115072L, -4349971829190472192L, -4323937847117721856L, -4297160557210933504L,
/* [156] */ -4269621402214949888L, -4241300963840749312L, -4212178920821861632L, -4182234004204451584L,
/* [160] */ -4151443949668877312L, -4119785446662287616L, -4087234084103201536L, -4053764292396156928L,
/* [164] */ -4019349281473081856L, -3983960974549692672L, -3947569937258423296L, -3910145301787345664L,
/* [168] */ -3871654685619032064L, -3832064104425388800L, -3791337878631544832L, -3749438533114327552L,
/* [172] */ -3706326689447984384L, -3661960950051848192L, -3616297773528534784L, -3569291340409189376L,
/* [176] */ -3520893408440946176L, -3471053156460654336L, -3419717015797782528L, -3366828488034805504L,
/* [180] */ -3312327947826460416L, -3256152429334010368L, -3198235394669719040L, -3138506482563172864L,
/* [184] */ -3076891235255162880L, -3013310801389730816L, -2947681612411374848L, -2879915029671670784L,
/* [188] */ -2809916959107513856L, -2737587429961866240L, -2662820133571325696L, -2585501917733380096L,
/* [192] */ -2505512231579385344L, -2422722515205211648L, -2336995527534088448L, -2248184604988727552L,
/* [196] */ -2156132842510765056L, -2060672187261025536L, -1961622433929371904L, -1858790108950105600L,
/* [200] */ -1751967229002895616L, -1640929916937142784L, -1525436855617582592L, -1405227557075253248L,
/* [204] */ -1280020420662650112L, -1149510549536596224L, -1013367289578704896L, -871231448632104192L,
/* [208] */ -722712146453667840L, -567383236774436096L, -404779231966938368L, -234390647591545856L,
/* [212] */ -55658667960119296L, 132030985907841280L, 329355128892811776L, 537061298001085184L,
/* [216] */ 755977262693564160L, 987022116608033280L, 1231219266829431296L, 1489711711346518528L,
/* [220] */ 1763780090187553792L, 2054864117341795072L, 2364588157623768832L, 2694791916990503168L,
/* [224] */ 3047567482883476224L, 3425304305830816256L, 3830744187097297920L, 4267048975685830400L,
/* [228] */ 4737884547990017280L, 5247525842198998272L, 5800989391535355392L, 6404202162993295360L,
/* [232] */ 7064218894258540544L, 7789505049452331520L, 8590309807749444864L, 7643763810684489984L,
/* [236] */ 8891950541491446016L, 5457384281016206080L, 9083704440929284096L, 7976211653914433280L,
/* [240] */ 8178631350487117568L, 2821287825726744832L, 6322989683301709568L, 4309503753387611392L,
/* [244] */ 4685170734960170496L, 8404845967535199744L, 7330522972447554048L, 1960945799076992000L,
/* [248] */ 4742910674644899072L, -751799822533509888L, 7023456603741959936L, 3843116882594676224L,
/* [252] */ 3927231442413903104L, -9223372036854775808L, -9223372036854775808L, -9223372036854775808L,
};
/**
* The precomputed ziggurat lengths, denoted X_i in the main text. X_i = length of
* ziggurat layer i. Values have been scaled by 2^-63.
*/
protected static final double[] X = {
/* [ 0] */ 8.2066240675348816e-19, 7.3973732351607284e-19, 6.9133313377915293e-19, 6.5647358820964533e-19,
/* [ 4] */ 6.2912539959818508e-19, 6.0657224129604964e-19, 5.8735276103737269e-19, 5.7058850528536941e-19,
/* [ 8] */ 5.557094569162239e-19, 5.4232438903743953e-19, 5.3015297696508776e-19, 5.1898739257708062e-19,
/* [ 12] */ 5.086692261799833e-19, 4.9907492938796469e-19, 4.9010625894449536e-19, 4.8168379010649187e-19,
/* [ 16] */ 4.7374238653644714e-19, 4.6622795807196824e-19, 4.5909509017784048e-19, 4.5230527790658154e-19,
/* [ 20] */ 4.458255881635396e-19, 4.3962763126368381e-19, 4.336867596710647e-19, 4.2798143618469714e-19,
/* [ 24] */ 4.2249273027064889e-19, 4.172039125346411e-19, 4.1210012522465616e-19, 4.0716811225869233e-19,
/* [ 28] */ 4.0239599631006903e-19, 3.9777309342877357e-19, 3.9328975785334499e-19, 3.8893725129310323e-19,
/* [ 32] */ 3.8470763218720385e-19, 3.8059366138180143e-19, 3.765887213854473e-19, 3.7268674692030177e-19,
/* [ 36] */ 3.6888216492248162e-19, 3.6516984248800068e-19, 3.6154504153287473e-19, 3.5800337915318032e-19,
/* [ 40] */ 3.5454079284533432e-19, 3.5115350988784242e-19, 3.4783802030030962e-19, 3.4459105288907336e-19,
/* [ 44] */ 3.4140955396563316e-19, 3.3829066838741162e-19, 3.3523172262289001e-19, 3.3223020958685874e-19,
/* [ 48] */ 3.2928377502804472e-19, 3.2639020528202049e-19, 3.2354741622810815e-19, 3.2075344331080789e-19,
/* [ 52] */ 3.1800643250478609e-19, 3.1530463211820845e-19, 3.1264638534265134e-19, 3.1003012346934211e-19,
/* [ 56] */ 3.0745435970137301e-19, 3.0491768350005559e-19, 3.0241875541094565e-19, 2.999563023214455e-19,
/* [ 60] */ 2.9752911310742592e-19, 2.9513603463113224e-19, 2.9277596805684267e-19, 2.9044786545442563e-19,
/* [ 64] */ 2.8815072666416712e-19, 2.8588359639906928e-19, 2.8364556156331615e-19, 2.8143574876779799e-19,
/* [ 68] */ 2.7925332202553125e-19, 2.7709748061152879e-19, 2.7496745707320232e-19, 2.7286251537873397e-19,
/* [ 72] */ 2.7078194919206054e-19, 2.687250802641905e-19, 2.6669125693153442e-19, 2.6467985271278891e-19,
/* [ 76] */ 2.6269026499668434e-19, 2.6072191381359757e-19, 2.5877424068465143e-19, 2.5684670754248168e-19,
/* [ 80] */ 2.5493879571835479e-19, 2.5305000499077481e-19, 2.511798526911271e-19, 2.4932787286227806e-19,
/* [ 84] */ 2.474936154663866e-19, 2.4567664563848669e-19, 2.4387654298267842e-19, 2.4209290090801527e-19,
/* [ 88] */ 2.4032532600140538e-19, 2.3857343743505147e-19, 2.3683686640614648e-19, 2.3511525560671253e-19,
/* [ 92] */ 2.3340825872163284e-19, 2.3171553995306794e-19, 2.3003677356958333e-19, 2.2837164347843482e-19,
/* [ 96] */ 2.2671984281957174e-19, 2.2508107358001938e-19, 2.2345504622739592e-19, 2.2184147936140775e-19,
/* [100] */ 2.2024009938224424e-19, 2.1865064017486842e-19, 2.1707284280826716e-19, 2.1550645524878675e-19,
/* [104] */ 2.1395123208673778e-19, 2.124069342755064e-19, 2.1087332888245875e-19, 2.0935018885097035e-19,
/* [108] */ 2.0783729277295508e-19, 2.0633442467130712e-19, 2.0484137379170616e-19, 2.0335793440326865e-19,
/* [112] */ 2.018839056075609e-19, 2.0041909115551697e-19, 1.9896329927183254e-19, 1.975163424864309e-19,
/* [116] */ 1.9607803747261946e-19, 1.9464820489157862e-19, 1.9322666924284314e-19, 1.9181325872045647e-19,
/* [120] */ 1.9040780507449479e-19, 1.8901014347767504e-19, 1.8762011239677479e-19, 1.8623755346860768e-19,
/* [124] */ 1.8486231138030984e-19, 1.8349423375370566e-19, 1.8213317103353295e-19, 1.8077897637931708e-19,
/* [128] */ 1.7943150556069476e-19, 1.7809061685599652e-19, 1.7675617095390567e-19, 1.7542803085801941e-19,
/* [132] */ 1.7410606179414531e-19, 1.727901311201724e-19, 1.7148010823836362e-19, 1.7017586450992059e-19,
/* [136] */ 1.6887727317167824e-19, 1.6758420925479093e-19, 1.6629654950527621e-19, 1.6501417230628659e-19,
/* [140] */ 1.6373695760198277e-19, 1.624647868228856e-19, 1.6119754281258616e-19, 1.5993510975569615e-19,
/* [144] */ 1.5867737310692309e-19, 1.5742421952115544e-19, 1.5617553678444595e-19, 1.5493121374578016e-19,
/* [148] */ 1.5369114024951992e-19, 1.5245520706841019e-19, 1.5122330583703858e-19, 1.4999532898563561e-19,
/* [152] */ 1.4877116967410352e-19, 1.4755072172615974e-19, 1.4633387956347966e-19, 1.4512053813972103e-19,
/* [156] */ 1.4391059287430991e-19, 1.4270393958586506e-19, 1.4150047442513381e-19, 1.4030009380730888e-19,
/* [160] */ 1.3910269434359025e-19, 1.3790817277185197e-19, 1.3671642588626657e-19, 1.3552735046573446e-19,
/* [164] */ 1.3434084320095729e-19, 1.3315680061998685e-19, 1.3197511901207148e-19, 1.3079569434961214e-19,
/* [168] */ 1.2961842220802957e-19, 1.2844319768333099e-19, 1.2726991530715219e-19, 1.2609846895903523e-19,
/* [172] */ 1.2492875177568625e-19, 1.237606560569394e-19, 1.2259407316813331e-19, 1.2142889343858445e-19,
/* [176] */ 1.2026500605581765e-19, 1.1910229895518744e-19, 1.1794065870449425e-19, 1.1677997038316715e-19,
/* [180] */ 1.1562011745554883e-19, 1.1446098163777869e-19, 1.1330244275772562e-19, 1.1214437860737343e-19,
/* [184] */ 1.109866647870073e-19, 1.0982917454048923e-19, 1.0867177858084351e-19, 1.0751434490529747e-19,
/* [188] */ 1.0635673859884002e-19, 1.0519882162526621e-19, 1.0404045260457141e-19, 1.0288148657544097e-19,
/* [192] */ 1.0172177474144965e-19, 1.0056116419943559e-19, 9.9399497648346677e-20, 9.8236613076667446e-20,
/* [196] */ 9.7072343426320094e-20, 9.5906516230690634e-20, 9.4738953224154196e-20, 9.3569469920159036e-20,
/* [200] */ 9.2397875154569468e-20, 9.1223970590556472e-20, 9.0047550180852874e-20, 8.8868399582647627e-20,
/* [204] */ 8.768629551976745e-20, 8.6501005086071005e-20, 8.5312284983141187e-20, 8.4119880684385214e-20,
/* [208] */ 8.292352551651342e-20, 8.1722939648034506e-20, 8.0517828972839211e-20, 7.9307883875099226e-20,
/* [212] */ 7.8092777859524425e-20, 7.6872166028429042e-20, 7.5645683383965122e-20, 7.4412942930179128e-20,
/* [216] */ 7.3173533545093332e-20, 7.1927017587631075e-20, 7.0672928197666785e-20, 6.9410766239500362e-20,
/* [220] */ 6.8139996829256425e-20, 6.6860045374610234e-20, 6.5570293040210081e-20, 6.4270071533368528e-20,
/* [224] */ 6.2958657080923559e-20, 6.1635263438143136e-20, 6.02990337321517e-20, 5.8949030892850181e-20,
/* [228] */ 5.758422635988593e-20, 5.6203486669597397e-20, 5.4805557413499315e-20, 5.3389043909003295e-20,
/* [232] */ 5.1952387717989917e-20, 5.0493837866338355e-20, 4.9011415222629489e-20, 4.7502867933366117e-20,
/* [236] */ 4.5965615001265455e-20, 4.4396673897997565e-20, 4.2792566302148588e-20, 4.1149193273430015e-20,
/* [240] */ 3.9461666762606287e-20, 3.7724077131401685e-20, 3.592916408620436e-20, 3.4067836691100565e-20,
/* [244] */ 3.2128447641564046e-20, 3.0095646916399994e-20, 2.7948469455598328e-20, 2.5656913048718645e-20,
/* [248] */ 2.3175209756803909e-20, 2.0426695228251291e-20, 1.7261770330213488e-20, 1.3281889259442579e-20,
/* [252] */ 0,
};
/** The precomputed ziggurat heights, denoted Y_i in the main text. Y_i = f(X_i).
* Values have been scaled by 2^-63. */
protected static final double[] Y = {
/* [ 0] */ 5.595205495112736e-23, 1.1802509982703313e-22, 1.8444423386735829e-22, 2.5439030466698309e-22,
/* [ 4] */ 3.2737694311509334e-22, 4.0307732132706715e-22, 4.8125478319495115e-22, 5.6172914896583308e-22,
/* [ 8] */ 6.4435820540443526e-22, 7.2902662343463681e-22, 8.1563888456321941e-22, 9.0411453683482223e-22,
/* [ 12] */ 9.9438488486399206e-22, 1.0863906045969114e-21, 1.1800799775461269e-21, 1.2754075534831208e-21,
/* [ 16] */ 1.372333117637729e-21, 1.4708208794375214e-21, 1.5708388257440445e-21, 1.6723581984374566e-21,
/* [ 20] */ 1.7753530675030514e-21, 1.8797999785104595e-21, 1.9856776587832504e-21, 2.0929667704053244e-21,
/* [ 24] */ 2.201649700995824e-21, 2.3117103852306179e-21, 2.4231341516125464e-21, 2.5359075901420891e-21,
/* [ 28] */ 2.6500184374170538e-21, 2.7654554763660391e-21, 2.8822084483468604e-21, 3.0002679757547711e-21,
/* [ 32] */ 3.1196254936130377e-21, 3.2402731888801749e-21, 3.3622039464187092e-21, 3.4854113007409036e-21,
/* [ 36] */ 3.6098893927859475e-21, 3.7356329310971768e-21, 3.8626371568620053e-21, 3.9908978123552837e-21,
/* [ 40] */ 4.1204111123918948e-21, 4.2511737184488913e-21, 4.3831827151633737e-21, 4.5164355889510656e-21,
/* [ 44] */ 4.6509302085234806e-21, 4.7866648071096003e-21, 4.9236379662119969e-21, 5.0618486007478993e-21,
/* [ 48] */ 5.2012959454434732e-21, 5.3419795423648946e-21, 5.4838992294830959e-21, 5.6270551301806347e-21,
/* [ 52] */ 5.7714476436191935e-21, 5.9170774358950678e-21, 6.0639454319177027e-21, 6.2120528079531677e-21,
/* [ 56] */ 6.3614009847804375e-21, 6.5119916214136427e-21, 6.6638266093481696e-21, 6.8169080672926277e-21,
/* [ 60] */ 6.9712383363524377e-21, 7.1268199756340822e-21, 7.2836557582420336e-21, 7.4417486676430174e-21,
/* [ 64] */ 7.6011018943746355e-21, 7.7617188330775411e-21, 7.9236030798322572e-21, 8.0867584297834842e-21,
/* [ 68] */ 8.2511888750363333e-21, 8.4168986028103258e-21, 8.5838919938383098e-21, 8.7521736209986459e-21,
/* [ 72] */ 8.9217482481700712e-21, 9.0926208292996504e-21, 9.2647965076751277e-21, 9.4382806153938292e-21,
/* [ 76] */ 9.6130786730210328e-21, 9.7891963894314161e-21, 9.966639661827884e-21, 1.0145414575932636e-20,
/* [ 80] */ 1.0325527406345955e-20, 1.0506984617068672e-20, 1.0689792862184811e-20, 1.0873958986701341e-20,
/* [ 84] */ 1.10594900275424e-20, 1.1246393214695825e-20, 1.1434675972510121e-20, 1.1624345921140471e-20,
/* [ 88] */ 1.1815410878142659e-20, 1.2007878860214202e-20, 1.2201758085082226e-20, 1.239705697353804e-20,
/* [ 92] */ 1.2593784151618565e-20, 1.2791948452935152e-20, 1.29915589211506e-20, 1.3192624812605428e-20,
/* [ 96] */ 1.3395155599094805e-20, 1.3599160970797774e-20, 1.3804650839360727e-20, 1.4011635341137284e-20,
/* [100] */ 1.4220124840587164e-20, 1.4430129933836705e-20, 1.4641661452404201e-20, 1.485473046709328e-20,
/* [104] */ 1.5069348292058084e-20, 1.5285526489044053e-20, 1.5503276871808626e-20, 1.5722611510726402e-20,
/* [108] */ 1.5943542737583543e-20, 1.6166083150566702e-20, 1.6390245619451956e-20, 1.6616043290999594e-20,
/* [112] */ 1.6843489594561079e-20, 1.7072598247904713e-20, 1.7303383263267072e-20, 1.7535858953637607e-20,
/* [116] */ 1.7770039939284241e-20, 1.8005941154528286e-20, 1.8243577854777398e-20, 1.8482965623825808e-20,
/* [120] */ 1.8724120381431627e-20, 1.8967058391181452e-20, 1.9211796268653192e-20, 1.9458350989888484e-20,
/* [124] */ 1.9706739900186868e-20, 1.9956980723234356e-20, 2.0209091570579904e-20, 2.0463090951473895e-20,
/* [128] */ 2.0718997783083593e-20, 2.097683140110135e-20, 2.123661157076213e-20, 2.1498358498287976e-20,
/* [132] */ 2.1762092842777868e-20, 2.2027835728562592e-20, 2.2295608758045219e-20, 2.2565434025049041e-20,
/* [136] */ 2.2837334128696004e-20, 2.311133218784001e-20, 2.3387451856080863e-20, 2.3665717337386111e-20,
/* [140] */ 2.394615340234961e-20, 2.422878540511741e-20, 2.4513639301013211e-20, 2.4800741664897764e-20,
/* [144] */ 2.5090119710298442e-20, 2.5381801309347597e-20, 2.56758150135705e-20, 2.5972190075566336e-20,
/* [148] */ 2.6270956471628253e-20, 2.6572144925351523e-20, 2.6875786932281841e-20, 2.7181914785659148e-20,
/* [152] */ 2.7490561603315974e-20, 2.7801761355793055e-20, 2.8115548895739172e-20, 2.8431959988666534e-20,
/* [156] */ 2.8751031345137833e-20, 2.9072800654466307e-20, 2.9397306620015486e-20, 2.9724588996191657e-20,
/* [160] */ 3.0054688627228112e-20, 3.0387647487867642e-20, 3.0723508726057078e-20, 3.1062316707775905e-20,
/* [164] */ 3.1404117064129991e-20, 3.1748956740850969e-20, 3.2096884050352357e-20, 3.2447948726504914e-20,
/* [168] */ 3.2802201982306013e-20, 3.3159696570631373e-20, 3.352048684827223e-20, 3.3884628843476888e-20,
/* [172] */ 3.4252180327233346e-20, 3.4623200888548644e-20, 3.4997752014001677e-20, 3.537589717186906e-20,
/* [176] */ 3.5757701901149035e-20, 3.6143233905835799e-20, 3.65325631548274e-20, 3.6925761987883572e-20,
/* [180] */ 3.7322905228086981e-20, 3.7724070301302117e-20, 3.8129337363171041e-20, 3.8538789434235234e-20,
/* [184] */ 3.8952512543827862e-20, 3.9370595883442399e-20, 3.9793131970351439e-20, 4.0220216822325769e-20,
/* [188] */ 4.0651950144388133e-20, 4.1088435528630944e-20, 4.1529780668232712e-20, 4.1976097586926582e-20,
/* [192] */ 4.2427502885307452e-20, 4.2884118005513604e-20, 4.3346069515987453e-20, 4.3813489418210257e-20,
/* [196] */ 4.4286515477520838e-20, 4.4765291580372353e-20, 4.5249968120658306e-20, 4.5740702418054417e-20,
/* [200] */ 4.6237659171683015e-20, 4.6741010952818368e-20, 4.7250938740823415e-20, 4.7767632507051219e-20,
/* [204] */ 4.8291291852069895e-20, 4.8822126702292804e-20, 4.9360358072933852e-20, 4.9906218905182021e-20,
/* [208] */ 5.0459954986625539e-20, 5.1021825965285324e-20, 5.1592106469178258e-20, 5.2171087345169234e-20,
/* [212] */ 5.2759077033045284e-20, 5.3356403093325858e-20, 5.3963413910399511e-20, 5.4580480596259246e-20,
/* [216] */ 5.5207999124535584e-20, 5.584639272987383e-20, 5.649611461419377e-20, 5.7157651009290713e-20,
/* [220] */ 5.7831524654956632e-20, 5.8518298763794323e-20, 5.9218581558791713e-20, 5.99330314883387e-20,
/* [224] */ 6.0662363246796887e-20, 6.1407354758435e-20, 6.2168855320499763e-20, 6.2947795150103727e-20,
/* [228] */ 6.3745196643214394e-20, 6.4562187737537985e-20, 6.5400017881889097e-20, 6.6260077263309343e-20,
/* [232] */ 6.714392014514662e-20, 6.8053293447301698e-20, 6.8990172088133e-20, 6.9956803158564498e-20,
/* [236] */ 7.095576179487843e-20, 7.199002278894508e-20, 7.3063053739105458e-20, 7.4178938266266881e-20,
/* [240] */ 7.5342542134173124e-20, 7.6559742171142969e-20, 7.783774986341285e-20, 7.9185582674029512e-20,
/* [244] */ 8.06147755373533e-20, 8.2140502769818073e-20, 8.3783445978280519e-20, 8.5573129249678161e-20,
/* [248] */ 8.75544596695901e-20, 8.9802388057706877e-20, 9.2462471421151086e-20, 9.5919641344951721e-20,
/* [252] */ 1.0842021724855044e-19,
};
/** Underlying source of randomness. */
protected final UniformRandomProvider rng;
/**
* @param rng Generator of uniformly distributed random numbers.
*/
ModifiedZigguratExponentialSampler(UniformRandomProvider rng) {
this.rng = rng;
}
/** {@inheritDoc} */
@Override
public double sample() {
return createSample();
}
/**
* Creates the exponential sample with {@code mean = 1}.
*
* <p>Note: This has been extracted to a separate method so that the recursive call
* when sampling tries again targets this function. This allows sub-classes
* to override the sample method to generate a sample with a different mean using:
* <pre>
* @Override
* public double sample() {
* return super.sample() * mean;
* }
* </pre>
* Otherwise the sub-class {@code sample()} method will recursively call
* the overloaded sample() method when trying again which creates a bad sample due
* to compound multiplication of the mean.
*
* @return the sample
*/
protected double createSample() {
final long x = nextLong();
// Float multiplication squashes these last 8 bits, so they can be used to sample i
final int i = ((int) x) & 0xff;
if (i < I_MAX) {
// Early exit.
return X[i] * (x & MAX_INT64);
}
final int j = selectRegion();
return j == 0 ? X_0 + createSample() : sampleOverhang(j);
}
/**
* Select the overhang region or the tail using alias sampling.
*
* @return the region
*/
protected int selectRegion() {
final long x = nextLong();
// j in [0, 256)
final int j = ((int) x) & 0xff;
// map to j in [0, N] with N the number of layers of the ziggurat
return x >= IPMF[j] ? MAP[j] & 0xff : j;
}
/**
* Sample from overhang region {@code j}.
*
* @param j Index j (must be {@code > 0})
* @return the sample
*/
protected double sampleOverhang(int j) {
// Sample from the triangle:
// X[j],Y[j]
// |\-->u1
// | \ |
// | \ |
// | \| Overhang j (with hypotenuse not pdf(x))
// | \
// | |\
// | | \
// | u2 \
// +-------- X[j-1],Y[j-1]
// u2 = u1 + (u2 - u1) = u1 + uDistance
// If u2 < u1 then reflect in the hypotenuse by swapping u1 and u2.
long u1 = randomInt63();
long uDistance = randomInt63() - u1;
if (uDistance < 0) {
uDistance = -uDistance;
u1 -= uDistance;
}
final double x = sampleX(X, j, u1);
if (uDistance >= E_MAX) {
// Early Exit: x < y - epsilon
return x;
}
return sampleY(Y, j, u1 + uDistance) <= Math.exp(-x) ? x : sampleOverhang(j);
}
/**
* Generates a {@code long}.
*
* @return the long
*/
protected long nextLong() {
return rng.nextLong();
}
/**
* Return a positive long in {@code [0, 2^63)}.
*
* @return the long
*/
protected long randomInt63() {
return rng.nextLong() & MAX_INT64;
}
}
/**
* Modified Ziggurat method for sampling from an exponential distribution.
*
* <p>Uses the algorithm from McFarland, C.D. (2016).
*
* <p>This implementation uses simple overhangs and does not exploit the precomputed
* distances of the convex overhang.
*
* <p>Note: It is not expected that this method is faster as the simple overhangs do
* not exploit the property that the exponential PDF is always concave. Thus any upper-right
* triangle sample at the edge of the ziggurat can be reflected and tested against the
* lower-left triangle limit.
*/
static class ModifiedZigguratExponentialSamplerSimpleOverhangs
extends ModifiedZigguratExponentialSampler {
/**
* @param rng Generator of uniformly distributed random numbers.
*/
ModifiedZigguratExponentialSamplerSimpleOverhangs(UniformRandomProvider rng) {
super(rng);
}
/** {@inheritDoc} */
@Override
protected double sampleOverhang(int j) {
final double x = sampleX(X, j, randomInt63());
return sampleY(Y, j, randomInt63()) <= Math.exp(-x) ? x : sampleOverhang(j);
}
}
/**
* Modified Ziggurat method for sampling from an exponential distribution.
*
* <p>Uses the algorithm from McFarland, C.D. (2016).
*
* <p>This implementation separates sampling of the main ziggurat and sampling from the edge
* into different methods. This allows inlining of the main sample method.
*
* <p>The sampler will output different values due to the use of a bit shift to generate
* unsigned integers. This removes the requirement to load the mask MAX_INT64
* and ensures the method is under 35 bytes.
*/
static class ModifiedZigguratExponentialSamplerInlining
extends ModifiedZigguratExponentialSampler {
/**
* @param rng Generator of uniformly distributed random numbers.
*/
ModifiedZigguratExponentialSamplerInlining(UniformRandomProvider rng) {
super(rng);
}
/** {@inheritDoc} */
@Override
public double sample() {
// Ideally this method byte code size should be below -XX:MaxInlineSize
// (which defaults to 35 bytes). This compiles to 34 bytes.
final long x = nextLong();
// Float multiplication squashes these last 8 bits, so they can be used to sample i
final int i = ((int) x) & 0xff;
if (i < I_MAX) {
// Early exit.
return X[i] * (x >>> 1);
}
return edgeSample();
}
/**
* Create the sample from the edge of the ziggurat.
*
* <p>This method has been extracted to fit the main sample method within 35 bytes (the
* default size for a JVM to inline a method).
*
* @return a sample
*/
private double edgeSample() {
final int j = selectRegion();
return j == 0 ? sampleAdd(X_0) : sampleOverhang(j);
}
/**
* Creates a sample and adds it to the current value. This exploits the memoryless
* exponential distribution by generating a sample and adding it to the existing end
* of the previous ziggurat.
*
* <p>The method will use recursion in the event of a new tail. Passing the
* existing value allows the potential to optimise the memory stack.
*
* @param x0 Current value
* @return the sample
*/
private double sampleAdd(double x0) {
final long x = nextLong();
final int i = ((int) x) & 0xff;
if (i < I_MAX) {
// Early exit.
return x0 + X[i] * (x >>> 1);
}
// Edge of the ziggurat
final int j = selectRegion();
return j == 0 ? sampleAdd(x0 + X_0) : x0 + sampleOverhang(j);
}
}
/**
* Modified Ziggurat method for sampling from an exponential distribution.
*
* <p>Uses the algorithm from McFarland, C.D. (2016).
*
* <p>This implementation separates sampling of the main ziggurat and sampling from the edge
* into different methods. This allows inlining of the main sample method.
*
* <p>The sampler will output different values due to the use of a bit shift to generate
* unsigned integers. This removes the requirement to load the mask MAX_INT64
* and ensures the method is under 35 bytes.
*
* <p>Tail sampling outside of the main sample method is performed in a loop.
*/
static class ModifiedZigguratExponentialSamplerLoop
extends ModifiedZigguratExponentialSampler {
/**
* @param rng Generator of uniformly distributed random numbers.
*/
ModifiedZigguratExponentialSamplerLoop(UniformRandomProvider rng) {
super(rng);
}
/** {@inheritDoc} */
@Override
public double sample() {
// Ideally this method byte code size should be below -XX:MaxInlineSize
// (which defaults to 35 bytes). This compiles to 34 bytes.
final long x = nextLong();
// Float multiplication squashes these last 8 bits, so they can be used to sample i
final int i = ((int) x) & 0xff;
if (i < I_MAX) {
// Early exit.
return X[i] * (x >>> 1);
}
return edgeSample();
}
/**
* Create the sample from the edge of the ziggurat.
*
* <p>This method has been extracted to fit the main sample method within 35 bytes (the
* default size for a JVM to inline a method).
*
* @return a sample
*/
private double edgeSample() {
int j = selectRegion();
if (j != 0) {
return sampleOverhang(j);
}
// Perform a new sample and add it to the start of the tail.
double x0 = X_0;
for (;;) {
final long x = nextLong();
final int i = ((int) x) & 0xff;
if (i < I_MAX) {
// Early exit.
return x0 + X[i] * (x >>> 1);
}
// Edge of the ziggurat
j = selectRegion();
if (j != 0) {
return x0 + sampleOverhang(j);
}
// Another tail sample
x0 += X_0;
}
}
}
/**
* Modified Ziggurat method for sampling from an exponential distribution.
*
* <p>Uses the algorithm from McFarland, C.D. (2016).
*
* <p>This implementation separates sampling of the main ziggurat and sampling from the edge
* into different methods. This allows inlining of the main sample method.
*
* <p>The sampler will output different values due to the use of a bit shift to generate
* unsigned integers. This removes the requirement to load the mask MAX_INT64
* and ensures the method is under 35 bytes.
*
* <p>Tail sampling outside of the main sample method is performed in a loop. No recursion
* is used. The first random deviate is recycled if the sample if from the edge.
*/
static class ModifiedZigguratExponentialSamplerLoop2
extends ModifiedZigguratExponentialSampler {
/**
* @param rng Generator of uniformly distributed random numbers.
*/
ModifiedZigguratExponentialSamplerLoop2(UniformRandomProvider rng) {
super(rng);
}
/** {@inheritDoc} */
@Override
public double sample() {
// Ideally this method byte code size should be below -XX:MaxInlineSize
// (which defaults to 35 bytes). This compiles to 35 bytes.
final long x = nextLong();
// Float multiplication squashes these last 8 bits, so they can be used to sample i
final int i = ((int) x) & 0xff;
if (i < I_MAX) {
// Early exit.
return X[i] * (x >>> 1);
}
// Recycle x as the upper 56 bits have not been used.
return edgeSample(x);
}
/**
* Create the sample from the edge of the ziggurat.
*
* <p>This method has been extracted to fit the main sample method within 35 bytes (the
* default size for a JVM to inline a method).
*
* @param xx Initial random deviate
* @return a sample
*/
private double edgeSample(long xx) {
int j = selectRegion();
if (j != 0) {
return sampleOverhang(j, xx);
}
// Perform a new sample and add it to the start of the tail.
double x0 = X_0;
for (;;) {
final long x = nextLong();
final int i = ((int) x) & 0xff;
if (i < I_MAX) {
// Early exit.
return x0 + X[i] * (x >>> 1);
}
// Edge of the ziggurat
j = selectRegion();
if (j != 0) {
return x0 + sampleOverhang(j, x);
}
// Another tail sample
x0 += X_0;
}
}
/**
* Sample from overhang region {@code j}.
*
* <p>This does not use recursion.
*
* @param j Index j (must be {@code > 0})
* @param xx Initial random deviate
* @return the sample
*/
protected double sampleOverhang(int j, long xx) {
// Recycle the initial random deviate.
// Shift right to make an unsigned long.
for (long u1 = xx >>> 1;; u1 = nextLong() >>> 1) {
// Sample from the triangle:
// X[j],Y[j]
// |\-->u1
// | \ |
// | \ |
// | \| Overhang j (with hypotenuse not pdf(x))
// | \
// | |\
// | | \
// | u2 \
// +-------- X[j-1],Y[j-1]
// u2 = u1 + (u2 - u1) = u1 + uDistance
// If u2 < u1 then reflect in the hypotenuse by swapping u1 and u2.
long uDistance = (nextLong() >>> 1) - u1;
if (uDistance < 0) {
uDistance = -uDistance;
u1 -= uDistance;
}
final double x = sampleX(X, j, u1);
if (uDistance >= E_MAX) {
// Early Exit: x < y - epsilon
return x;
}
if (sampleY(Y, j, u1 + uDistance) <= Math.exp(-x)) {
return x;
}
}
}
}
/**
* Modified Ziggurat method for sampling from an exponential distribution.
*
* <p>Uses the algorithm from McFarland, C.D. (2016).
*
* <p>This implementation separates sampling of the main ziggurat and the recursive
* sampling from the edge into different methods.
*/
static class ModifiedZigguratExponentialSamplerRecursion
extends ModifiedZigguratExponentialSampler {
/**
* @param rng Generator of uniformly distributed random numbers.
*/
ModifiedZigguratExponentialSamplerRecursion(UniformRandomProvider rng) {
super(rng);
}
/** {@inheritDoc} */
@Override
public double sample() {
final long x = nextLong();
// Float multiplication squashes these last 8 bits, so they can be used to sample i
final int i = ((int) x) & 0xff;
if (i < I_MAX) {
// Early exit.
return X[i] * (x & MAX_INT64);
}
final int j = selectRegion();
return j == 0 ? sampleAdd(X_0) : sampleOverhang(j);
}
/**
* Creates a sample and adds it to the current value. This exploits the memoryless
* exponential distribution by generating a sample and adding it to the existing end
* of the previous ziggurat.
*
* <p>The method will use recursion in the event of a new tail. Passing the
* existing value allows the potential to optimise the memory stack.
*
* @param x0 Current value
* @return the sample
*/
private double sampleAdd(double x0) {
final long x = nextLong();
final int i = ((int) x) & 0xff;
if (i < I_MAX) {
// Early exit.
return x0 + X[i] * (x & MAX_INT64);
}
// Edge of the ziggurat
final int j = selectRegion();
return j == 0 ? sampleAdd(x0 + X_0) : x0 + sampleOverhang(j);
}
}
/**
* Modified Ziggurat method for sampling from an exponential distribution.
*
* <p>Uses the algorithm from McFarland, C.D. (2016).
*
* <p>This is a copy of {@link ModifiedZigguratExponentialSampler} using
* an integer map in-place of a byte map look-up table.
*/
static class ModifiedZigguratExponentialSamplerIntMap
extends ModifiedZigguratExponentialSampler {
/** The alias map. */
private static final int[] INT_MAP = {
/* [ 0] */ 0, 0, 1, 235, 3, 4, 5, 0, 0, 0, 0, 0, 0, 0, 0, 0,
/* [ 16] */ 0, 0, 1, 1, 1, 1, 2, 2, 252, 252, 252, 252, 252, 252, 252, 252,
/* [ 32] */ 252, 252, 252, 252, 252, 252, 252, 252, 252, 252, 252, 252, 252, 252, 252, 252,
/* [ 48] */ 252, 252, 252, 252, 252, 252, 252, 252, 252, 252, 252, 252, 252, 252, 252, 252,
/* [ 64] */ 252, 252, 252, 252, 252, 252, 252, 252, 252, 252, 252, 252, 252, 252, 252, 252,
/* [ 80] */ 252, 252, 252, 252, 252, 252, 252, 252, 252, 252, 252, 252, 252, 252, 252, 252,
/* [ 96] */ 252, 252, 252, 252, 252, 252, 252, 252, 252, 252, 252, 252, 252, 252, 252, 252,
/* [112] */ 252, 252, 252, 252, 252, 252, 252, 252, 252, 252, 252, 252, 252, 252, 252, 252,
/* [128] */ 252, 252, 252, 252, 252, 252, 252, 252, 252, 252, 252, 252, 252, 252, 252, 252,
/* [144] */ 252, 252, 252, 252, 252, 252, 252, 252, 252, 252, 252, 252, 252, 252, 252, 252,
/* [160] */ 252, 252, 252, 252, 252, 252, 252, 252, 252, 252, 252, 252, 252, 252, 252, 252,
/* [176] */ 252, 251, 251, 251, 251, 251, 251, 251, 251, 251, 251, 251, 251, 251, 250, 250,
/* [192] */ 250, 250, 250, 250, 250, 249, 249, 249, 249, 249, 249, 248, 248, 248, 248, 247,
/* [208] */ 247, 247, 247, 246, 246, 246, 245, 245, 244, 244, 243, 243, 242, 241, 241, 240,
/* [224] */ 239, 237, 3, 3, 4, 4, 6, 0, 0, 0, 0, 236, 237, 238, 239, 240,
/* [240] */ 241, 242, 243, 244, 245, 246, 247, 248, 249, 250, 251, 252, 2, 0, 0, 0,
};
/**
* @param rng Generator of uniformly distributed random numbers.
*/
ModifiedZigguratExponentialSamplerIntMap(UniformRandomProvider rng) {
super(rng);
}
/** {@inheritDoc} */
@Override
protected int selectRegion() {
final long x = nextLong();
// j in [0, 256)
final int j = ((int) x) & 0xff;
// map to j in [0, N] with N the number of layers of the ziggurat
return x >= IPMF[j] ? INT_MAP[j] : j;
}
}
/**
* Modified Ziggurat method for sampling from an exponential distribution.
*
* <p>Uses the algorithm from McFarland, C.D. (2016).
*
* <p>This is a copy of {@link ModifiedZigguratExponentialSampler} using
* a table to store the maximum epsilon value for each overhang.
*/
static class ModifiedZigguratExponentialSamplerEMaxTable
extends ModifiedZigguratExponentialSampler {
/** The deviation of concave pdf(x) below the hypotenuse value for early exit scaled by 2^63.
* 252 entries with overhang {@code j} corresponding to entry {@code j-1}. */
private static final long[] E_MAX_TABLE = {
/* [ 0] */ 853965788476313647L, 513303011048449571L, 370159258027176883L, 290559192811411702L,
/* [ 4] */ 239682322342140259L, 204287432757591023L, 178208980447816578L, 158179811652601808L,
/* [ 8] */ 142304335868356315L, 129406004988661986L, 118715373346558305L, 109707765835789898L,
/* [ 12] */ 102012968856398207L, 95362200070363889L, 89555545194518719L, 84441195786718132L,
/* [ 16] */ 79901778875942314L, 75845102376072521L, 72197735928174183L, 68900462142397564L,
/* [ 20] */ 65904991388597543L, 63171548466382494L, 60667072432484929L, 58363855085733832L,
/* [ 24] */ 56238498180198598L, 54271105522401119L, 52444650416313533L, 50744475573288311L,
/* [ 28] */ 49157894191680121L, 47673869089531616L, 46282752622782375L, 44976074355904424L,
/* [ 32] */ 43746366552358124L, 42587019846569729L, 41492163173714974L, 40456563326820177L,
/* [ 36] */ 39475540494600872L, 38544896888152443L, 37660856147942019L, 36820011676708837L,
/* [ 40] */ 36019282399893713L, 35255874736108745L, 34527249783140581L, 33831094903027707L,
/* [ 44] */ 33165299032708110L, 32527931162123068L, 31917221515265728L, 31331545045961871L,
/* [ 48] */ 30769406922648640L, 30229429727799711L, 29710342140081525L, 29210968902516770L,
/* [ 52] */ 28730221909221458L, 28267092267755223L, 27820643214642802L, 27390003778888310L,
/* [ 56] */ 26974363102873684L, 26572965342371777L, 26185105077881714L, 25810123178421168L,
/* [ 60] */ 25447403066532673L, 25096367339794542L, 24756474709733050L, 24427217223862121L,
/* [ 64] */ 24108117740746397L, 23798727631587496L, 23498624684961198L, 23207411194051023L,
/* [ 68] */ 22924712208092083L, 22650173931802985L, 22383462258391643L, 22124261423306770L,
/* [ 72] */ 21872272767292587L, 21627213598531964L, 21388816144739113L, 21156826587012934L,
/* [ 76] */ 20931004168108672L, 20711120368524119L, 20496958144463729L, 20288311222329444L,
/* [ 80] */ 20084983444908931L, 19886788164900174L, 19693547681827322L, 19505092718772996L,
/* [ 84] */ 19321261935686425L, 19141901476327133L, 18966864546167569L, 18796011018823163L,
/* [ 88] */ 18629207068791541L, 18466324828481792L, 18307242067684946L, 18151841893803319L,
/* [ 92] */ 18000012471293326L, 17851646758910343L, 17706642263461846L, 17564900808880742L,
/* [ 96] */ 17426328319528561L, 17290834616728420L, 17158333227603225L, 17028741205374556L,
/* [100] */ 16901978960337609L, 16777970100794092L, 16656641283277552L, 16537922071455934L,
/* [104] */ 16421744803147429L, 16308044464921141L, 16196758573800601L, 16087827065618528L,
/* [108] */ 15981192189607877L, 15876798408841619L, 15774592306164744L, 15674522495285722L,
/* [112] */ 15576539536717537L, 15480595858282956L, 15386645679917351L, 15294644942520693L,
/* [116] */ 15204551240628355L, 15116323758686143L, 15029923210730485L, 14945311783286871L,
/* [120] */ 14862453081314005L, 14781312077032665L, 14701855061488253L, 14624049598708474L,
/* [124] */ 14547864482325100L, 14473269694538314L, 14400236367311981L, 14328736745694203L,
/* [128] */ 14258744153165819L, 14190232958926857L, 14123178547035436L, 14057557287324133L,
/* [132] */ 13993346508018831L, 13930524469996132L, 13869070342616606L, 13808964181079506L,
/* [136] */ 13750186905245534L, 13692720279883516L, 13636546896296650L, 13581650155291926L,
/* [140] */ 13528014251457894L, 13475624158722351L, 13424465617162582L, 13374525121048117L,
/* [144] */ 13325789908096374L, 13278247949928556L, 13231887943714024L, 13186699304997313L,
/* [148] */ 13142672161706220L, 13099797349339341L, 13058066407342105L, 13017471576677033L,
/* [152] */ 12978005798604683L, 12939662714691649L, 12902436668069432L, 12866322705969175L,
/* [156] */ 12831316583568516L, 12797414769185138L, 12764614450860914L, 12732913544387857L,
/* [160] */ 12702310702829356L, 12672805327602182L, 12644397581185285L, 12617088401539963L,
/* [164] */ 12590879518319943L, 12565773470976553L, 12541773628858256L, 12518884213426018L,
/* [168] */ 12497110322714335L, 12476457958178394L, 12456934054089150L, 12438546509646839L,
/* [172] */ 12421304224007330L, 12405217134429885L, 12390296257782254L, 12376553735658483L,
/* [176] */ 12364002883391013L, 12352658243275075L, 12342535642345809L, 12333652255094142L,
/* [180] */ 12326026671543609L, 12319678971157050L, 12314630803093705L, 12310905473395866L,
/* [184] */ 12308528039744429L, 12307525414502757L, 12307926476843287L, 12309762194847939L,
/* [188] */ 12313065758579233L, 12317872725234489L, 12324221177635088L, 12332151897455469L,
/* [192] */ 12341708554772055L, 12352937915715590L, 12365890070240929L, 12380618682293187L,
/* [196] */ 12397181264956129L, 12415639483521735L, 12436059489828363L, 12458512291688487L,
/* [200] */ 12483074161780576L, 12509827091021440L, 12538859292191186L, 12570265760462982L,
/* [204] */ 12604148898533980L, 12640619215280889L, 12679796108318231L, 12721808742570859L,
/* [208] */ 12766797039034927L, 12814912790376408L, 12866320922993068L, 12921200928753560L,
/* [212] */ 12979748493987873L, 13042177358607040L, 13108721444725108L, 13179637302147239L,
/* [216] */ 13255206927961732L, 13335741029756972L, 13421582817340701L, 13513112427153506L,
/* [220] */ 13610752108037106L, 13714972328197193L, 13826299003251241L, 13945322097066639L,
/* [224] */ 14072705914699465L, 14209201495705644L, 14355661634264763L, 14513059211087015L,
/* [228] */ 14682509737034964L, 14865299303251008L, 15062919542085557L, 15277111779554570L,
/* [232] */ 15509923383418374L, 15763780505978643L, 16041583185668067L, 16346831428994830L,
/* [236] */ 16683794981864260L, 17057745936956694L, 17475283735768697L, 17944799475531562L,
/* [240] */ 18477156350176671L, 19086716702655372L, 19792946841346963L, 20623030105811966L,
/* [244] */ 21616339529831424L, 22832582766564173L, 24367856249313960L, 26389803907514902L,
/* [248] */ 29226958795272911L, 33654855924781132L, 42320562697765039L, 141207843617418268L,
};
/**
* @param rng Generator of uniformly distributed random numbers.
*/
ModifiedZigguratExponentialSamplerEMaxTable(UniformRandomProvider rng) {
super(rng);
}
@Override
protected double sampleOverhang(int j) {
// Look-up E_MAX
return sampleOverhang(j, E_MAX_TABLE[j - 1]);
}
/**
* Sample from overhang region {@code j}.
*
* @param j Index j (must be {@code > 0})
* @param eMax Maximum deviation of concave pdf(x) below the hypotenuse value for early exit
* @return the sample
*/
private double sampleOverhang(int j, long eMax) {
long u1 = randomInt63();
long uDistance = randomInt63() - u1;
if (uDistance < 0) {
uDistance = -uDistance;
u1 -= uDistance;
}
final double x = sampleX(X, j, u1);
if (uDistance >= eMax) {
// Early Exit: x < y - epsilon
return x;
}
return sampleY(Y, j, u1 + uDistance) <= Math.exp(-x) ? x : sampleOverhang(j, eMax);
}
}
/**
* Modified Ziggurat method for sampling from an exponential distribution.
*
* <p>Uses the algorithm from McFarland, C.D. (2016).
*
* <p>This is a copy of {@link ModifiedZigguratExponentialSampler} using
* two values for the maximum epsilon value for the overhangs.
*
* <p>Note: The exponential curve is very well approximated by the straight line of
* the triangle hypotenuse in the majority of cases. Only as the overhang approaches the
* tail or close to x=0 does the curve differ more than 0.01 (expressed as a fraction of
* the triangle height). This can be exploited using two maximum deviation thresholds.
*/
static class ModifiedZigguratExponentialSamplerEMax2
extends ModifiedZigguratExponentialSampler {
// Ziggurat volumes:
// Inside the layers = 98.4375% (252/256)
// Fraction outside the layers:
// concave overhangs = 96.6972%
// tail (j=0) = 3.3028%
//
// Separation of tail with large maximum deviation from hypotenuse
// 0 <= j <= 8 = 7.9913%
// 1 <= j <= 8 = 4.6885%
// 9 <= j = 92.0087%
/** The overhang region j where the second maximum deviation constant is valid. */
private static final int J2 = 9;
/** Maximum deviation of concave pdf(x) below the hypotenuse value for early exit
* for overhangs {@code 9 <= j <= 253}.
* Equal to approximately 0.0154 scaled by 2^63.
*
* <p>This threshold increases the area of the early exit triangle by:
* (1-0.0154)^2 / (1-0.0926)^2 = 17.74%. */
private static final long E_MAX_2 = 142304335868356315L;
/**
* @param rng Generator of uniformly distributed random numbers.
*/
ModifiedZigguratExponentialSamplerEMax2(UniformRandomProvider rng) {
super(rng);
}
@Override
protected double createSample() {
final long x = nextLong();
// Float multiplication squashes these last 8 bits, so they can be used to sample i
final int i = ((int) x) & 0xff;
if (i < I_MAX) {
// Early exit.
return X[i] * (x & MAX_INT64);
}
final int j = selectRegion();
if (j < J2) {
// Long tail frequency: 0.0799
// j==0 (tail) frequency: 0.033028
return j == 0 ? X_0 + createSample() : sampleOverhang(j, E_MAX);
}
// Majority of curve can use a smaller threshold.
// Frequency: 0.920087
return sampleOverhang(j, E_MAX_2);
}
/**
* Sample from overhang region {@code j}.
*
* @param j Index j (must be {@code > 0})
* @param eMax Maximum deviation of concave pdf(x) below the hypotenuse value for early exit
* @return the sample
*/
private double sampleOverhang(int j, long eMax) {
long u1 = randomInt63();
long uDistance = randomInt63() - u1;
if (uDistance < 0) {
uDistance = -uDistance;
u1 -= uDistance;
}
final double x = sampleX(X, j, u1);
if (uDistance >= eMax) {
// Early Exit: x < y - epsilon
return x;
}
return sampleY(Y, j, u1 + uDistance) <= Math.exp(-x) ? x : sampleOverhang(j, eMax);
}
}
/**
* Modified Ziggurat method for sampling from an exponential distribution.
*
* <p>Uses the algorithm from McFarland, C.D. (2016).
*
* <p>This is a copy of {@link ModifiedZigguratExponentialSampler} using
* a table size of 512.
*/
static class ModifiedZigguratExponentialSampler512 implements ContinuousSampler {
// Ziggurat volumes:
// Inside the layers = 99.2188% (508/512)
// Fraction outside the layers:
// concave overhangs = 97.0103%
// tail = 2.9897%
/** The number of layers in the ziggurat. Maximum i value for early exit. */
protected static final int I_MAX = 508;
/** Maximum deviation of concave pdf(x) below the hypotenuse value for early exit.
* Equal to approximately 0.0919 scaled by 2^63. */
protected static final long E_MAX = 847415790149374213L;
/** Beginning of tail. Equal to X[0] * 2^63. */
protected static final double X_0 = 8.362025281328359;
/** The alias map. */
private static final int[] MAP = {
/* [ 0] */ 0, 0, 1, 473, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 0, 0,
/* [ 16] */ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
/* [ 32] */ 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1,
/* [ 48] */ 1, 2, 2, 2, 2, 508, 508, 508, 508, 508, 508, 508, 508, 508, 508, 508,
/* [ 64] */ 508, 508, 508, 508, 508, 508, 508, 508, 508, 508, 508, 508, 508, 508, 508, 508,
/* [ 80] */ 508, 508, 508, 508, 508, 508, 508, 508, 508, 508, 508, 508, 508, 508, 508, 508,
/* [ 96] */ 508, 508, 508, 508, 508, 508, 508, 508, 508, 508, 508, 508, 508, 508, 508, 508,
/* [112] */ 508, 508, 508, 508, 508, 508, 508, 508, 508, 508, 508, 508, 508, 508, 508, 508,
/* [128] */ 508, 508, 508, 508, 508, 508, 508, 508, 508, 508, 508, 508, 508, 508, 508, 508,
/* [144] */ 508, 508, 508, 508, 508, 508, 508, 508, 508, 508, 508, 508, 508, 508, 508, 508,
/* [160] */ 508, 508, 508, 508, 508, 508, 508, 508, 508, 508, 508, 508, 508, 508, 508, 508,
/* [176] */ 508, 508, 508, 508, 508, 508, 508, 508, 508, 508, 508, 508, 508, 508, 508, 508,
/* [192] */ 508, 508, 508, 508, 508, 508, 508, 508, 508, 508, 508, 508, 508, 508, 508, 508,
/* [208] */ 508, 508, 508, 508, 508, 508, 508, 508, 508, 508, 508, 508, 508, 508, 508, 508,
/* [224] */ 508, 508, 508, 508, 508, 508, 508, 508, 508, 508, 508, 508, 508, 508, 508, 508,
/* [240] */ 508, 508, 508, 508, 508, 508, 508, 508, 508, 508, 508, 508, 508, 508, 508, 508,
/* [256] */ 508, 508, 508, 508, 508, 508, 508, 508, 508, 508, 508, 508, 508, 508, 508, 508,
/* [272] */ 508, 508, 508, 508, 508, 508, 508, 508, 508, 508, 508, 508, 508, 508, 508, 508,
/* [288] */ 508, 508, 508, 508, 508, 508, 508, 508, 508, 508, 508, 508, 508, 508, 508, 508,
/* [304] */ 508, 508, 508, 508, 508, 508, 508, 508, 507, 507, 507, 507, 507, 507, 507, 507,
/* [320] */ 507, 507, 507, 507, 507, 507, 507, 507, 507, 507, 507, 507, 507, 507, 507, 507,
/* [336] */ 507, 507, 507, 507, 507, 506, 506, 506, 506, 506, 506, 506, 506, 506, 506, 506,
/* [352] */ 506, 506, 506, 506, 506, 506, 505, 505, 505, 505, 505, 505, 505, 505, 505, 505,
/* [368] */ 505, 504, 504, 504, 504, 504, 504, 504, 504, 504, 503, 503, 503, 503, 503, 503,
/* [384] */ 503, 503, 502, 502, 502, 502, 502, 502, 501, 501, 501, 501, 501, 500, 500, 500,
/* [400] */ 500, 500, 499, 499, 499, 499, 498, 498, 498, 498, 497, 497, 497, 496, 496, 496,
/* [416] */ 495, 495, 495, 494, 494, 493, 493, 492, 492, 491, 491, 490, 490, 489, 489, 488,
/* [432] */ 487, 486, 485, 485, 484, 482, 481, 479, 477, 3, 3, 3, 3, 3, 4, 4,
/* [448] */ 4, 5, 5, 5, 6, 6, 6, 7, 7, 8, 9, 9, 10, 13, 0, 0,
/* [464] */ 0, 0, 0, 0, 0, 0, 0, 0, 0, 474, 475, 476, 477, 478, 479, 480,
/* [480] */ 481, 482, 483, 484, 485, 486, 487, 488, 489, 490, 491, 492, 493, 494, 495, 496,
/* [496] */ 497, 498, 499, 500, 501, 502, 503, 504, 505, 506, 507, 508, 2, 0, 0, 0,
};
/** The alias inverse PMF. */
private static final long[] IPMF = {
/* [ 0] */ 9223372036854775296L, 2636146522269100032L, 3561072052814771200L, 3481882573634022912L,
/* [ 4] */ 7116525960784167936L, 3356999026233157120L, 4185343482988790272L, 7712945794725800960L,
/* [ 8] */ 7827282421678200832L, 5031524219672503296L, 8479703117927504896L, 8798557826262976512L,
/* [ 12] */ 6461753601163497984L, 5374576640182409216L, 8285209933846173696L, 7451775918468278784L,
/* [ 16] */ 6710527725555065344L, 6046590166603477504L, 5448165515863960576L, 4905771306157685248L,
/* [ 20] */ 4411694943948817408L, 3959596233157434880L, 3544212893101320192L, 3161139128795137536L,
/* [ 24] */ 2806656884941820416L, 2477605671376185344L, 2171281018524273664L, 1885354452557434880L,
/* [ 28] */ 1617809833614047744L, 1366892269237658112L, 1131066787615028736L, 908984654997058560L,
/* [ 32] */ 699455731348318720L, 501425633583915520L, 313956755007992832L, 136212399366961152L,
/* [ 36] */ -32556553021349888L, -193022908001625600L, -345792716399554560L, -491413423379838464L,
/* [ 40] */ -630380845888988672L, -763145170929260544L, -890116131912690176L, -1011667492054083072L,
/* [ 44] */ -1128140941080192000L, -1239849493252469760L, -1347080459893031424L, -1450098057542626304L,
/* [ 48] */ -1549145703018891776L, -1644448038535791104L, -1736212723361025536L, -1824632022940825600L,
/* [ 52] */ -1909884221816344064L, -1992134882804800512L, -2071537971688286208L, -2148236863947458560L,
/* [ 56] */ -2222365247780633088L, -2294047935714820096L, -2363401595468253696L, -2430535409323403776L,
/* [ 60] */ -2495551670069566976L, -2558546320552583680L, -2619609442985608192L, -2678825703417424896L,
/* [ 64] */ -2736274756101145600L, -2792031611938902016L, -2846166974686572544L, -2898747548177031680L,
/* [ 68] */ -2949836317447342592L, -2999492806329770496L, -3047773313785532928L, -3094731131006235648L,
/* [ 72] */ -3140416741094473728L, -3184878002938374144L, -3228160320727380992L, -3270306800406949376L,
/* [ 76] */ -3311358394234071552L, -3351354034484032512L, -3390330757247591424L, -3428323817169865728L,
/* [ 80] */ -3465366793898720256L, -3501491690934902784L, -3536729027513080832L, -3571107924081618944L,
/* [ 84] */ -3604656181899062784L, -3637400357214628352L, -3669365830461451264L, -3700576870849548288L,
/* [ 88] */ -3731056696714091520L, -3760827531940795392L, -3789910658766305792L, -3818326467220366848L,
/* [ 92] */ -3846094501460680704L, -3873233503224223232L, -3899761452604410880L, -3925695606345008640L,
/* [ 96] */ -3951052533825223680L, -3975848150898364416L, -4000097751731727360L, -4023816038786208256L,
/* [100] */ -4047017151058499584L, -4069714690707166208L, -4091921748165281792L, -4113650925843118592L,
/* [104] */ -4134914360510158336L, -4155723744443228160L, -4176090345419594752L, -4196025025626374656L,
/* [108] */ -4215538259557954560L, -4234640150960326144L, -4253340448884000768L, -4271648562898405376L,
/* [112] */ -4289573577519592448L, -4307124265897231872L, -4324309102806436352L, -4341136276983992320L,
/* [116] */ -4357613702847894016L, -4373749031636836352L, -4389549662001094144L, -4405022750077366784L,
/* [120] */ -4420175219077067264L, -4435013768413905920L, -4449544882397231104L, -4463774838515782144L,
/* [124] */ -4477709715332459008L, -4491355400012626944L, -4504717595505328640L, -4517801827395838976L,
/* [128] */ -4530613450446389760L, -4543157654842793472L, -4555439472160934400L, -4567463781068195840L,
/* [132] */ -4579235312773997568L, -4590758656240654848L, -4602038263168300032L, -4613078452764409856L,
/* [136] */ -4623883416308989952L, -4634457221524290048L, -4644803816761235456L, -4654927035008246784L,
/* [140] */ -4664830597734270976L, -4674518118570737664L, -4683993106843208704L, -4693258970957511168L,
/* [144] */ -4702319021648195072L, -4711176475095642112L, -4719834455917983232L, -4728296000042890240L,
/* [148] */ -4736564057465815040L, -4744641494898972160L, -4752531098316131328L, -4760235575398025216L,
/* [152] */ -4767757557883156992L, -4775099603826595840L, -4782264199773566464L, -4789253762848160256L,
/* [156] */ -4796070642763833856L, -4802717123757088768L, -4809195426448159744L, -4815507709632380416L,
/* [160] */ -4821656072004040192L, -4827642553816744448L, -4833469138481454080L, -4839137754106220032L,
/* [164] */ -4844650274978936320L, -4850008522996425728L, -4855214269039884288L, -4860269234302464000L,
/* [168] */ -4865175091566906368L, -4869933466437730816L, -4874545938529117696L, -4879014042609741312L,
/* [172] */ -4883339269707099136L, -4887523068171446784L, -4891566844702813184L, -4895471965340231168L,
/* [176] */ -4899239756417071616L, -4902871505481196544L, -4906368462183164928L, -4909731839132712448L,
/* [180] */ -4912962812724644864L, -4916062523935916544L, -4919032079093866496L, -4921872550617327616L,
/* [184] */ -4924584977732045824L, -4927170367159303168L, -4929629693781258752L, -4931963901281752576L,
/* [188] */ -4934173902764700160L, -4936260581349647360L, -4938224790746291200L, -4940067355807971840L,
/* [192] */ -4941789073065357824L, -4943390711239933952L, -4944873011739753984L, -4946236689135308800L,
/* [196] */ -4947482431619080192L, -4948610901446864896L, -4949622735363246080L, -4950518545009749504L,
/* [200] */ -4951298917317924864L, -4951964414887600640L, -4952515576348487168L, -4952952916709260288L,
/* [204] */ -4953276927691216384L, -4953488078048885760L, -4953586813876376064L, -4953573558902138368L,
/* [208] */ -4953448714769485312L, -4953212661305179136L, -4952865756776645632L, -4952408338136501760L,
/* [212] */ -4951840721255068672L, -4951163201143173120L, -4950376052162316800L, -4949479528224847360L,
/* [216] */ -4948473862982736384L, -4947359270007034368L, -4946135942956237312L, -4944804055734229504L,
/* [220] */ -4943363762640024064L, -4941815198506020352L, -4940158478827333120L, -4938393699882143232L,
/* [224] */ -4936520938842464768L, -4934540253875459072L, -4932451684236201472L, -4930255250350992896L,
/* [228] */ -4927950953893274112L, -4925538777848453632L, -4923018686572188672L, -4920390625839525888L,
/* [232] */ -4917654522885157888L, -4914810286435288576L, -4911857806732278272L, -4908796955549192192L,
/* [236] */ -4905627586197715456L, -4902349533527011840L, -4898962613914057216L, -4895466625247106560L,
/* [240] */ -4891861346899963904L, -4888146539697684992L, -4884321945875628032L, -4880387289029549568L,
/* [244] */ -4876342274056310272L, -4872186587089345536L, -4867919895422610944L, -4863541847428798464L,
/* [248] */ -4859052072467801600L, -4854450180787865088L, -4849735763417182208L, -4844908392047856128L,
/* [252] */ -4839967618911950848L, -4834912976647099392L, -4829743978155096576L, -4824460116451147776L,
/* [256] */ -4819060864504044544L, -4813545675067768320L, -4807913980504056832L, -4802165192594749952L,
/* [260] */ -4796298702346892288L, -4790313879785735168L, -4784210073740736512L, -4777986611619728896L,
/* [264] */ -4771642799174403072L, -4765177920255393792L, -4758591236556836352L, -4751881987350666752L,
/* [268] */ -4745049389210733568L, -4738092635724898304L, -4731010897197732864L, -4723803320339894784L,
/* [272] */ -4716469027948174336L, -4709007118571786240L, -4701416666168034304L, -4693696719744952320L,
/* [276] */ -4685846302991571968L, -4677864413895422464L, -4669750024346546688L, -4661502079728433664L,
/* [280] */ -4653119498494611968L, -4644601171731585024L, -4635945962706888192L, -4627152706402486784L,
/* [284] */ -4618220209032402944L, -4609147247545243136L, -4599932569110216704L, -4590574890586744832L,
/* [288] */ -4581072897977050112L, -4571425245860782592L, -4561630556812469760L, -4551687420800102912L,
/* [292] */ -4541594394564215808L, -4531350000978089984L, -4520952728387175936L, -4510401029928645120L,
/* [296] */ -4499693322828163584L, -4488827987675754496L, -4477803367678713856L, -4466617767890585600L,
/* [300] */ -4455269454416637440L, -4443756653594040832L, -4432077551146284544L, -4420230291311382528L,
/* [304] */ -4408212975942271488L, -4396023663579625472L, -4383660368494419456L, -4371121059701586432L,
/* [308] */ -4358403659940780544L, -4345506044627643904L, -4332426040768207872L, -4319161425842345472L,
/* [312] */ -4305709926649207296L, -4292069218116598784L, -4278236922073029120L, -4264210605978586624L,
/* [316] */ -4249987781616803840L, -4235565903742823424L, -4220942368688843264L, -4206114512923164672L,
/* [320] */ -4191079611563289088L, -4175834876839676928L, -4160377456508627456L, -4144704432214183936L,
/* [324] */ -4128812817794444800L, -4112699557531435520L, -4096361524344448000L, -4079795517919899648L,
/* [328] */ -4062998262780994048L, -4045966406289958912L, -4028696516583542784L, -4011185080437596160L,
/* [332] */ -3993428501058901504L, -3975423095799736320L, -3957165093794900480L, -3938650633514732032L,
/* [336] */ -3919875760234452992L, -3900836423412657152L, -3881528473978691584L, -3861947661522247168L,
/* [340] */ -3842089631384604160L, -3821949921642690048L, -3801523959987316736L, -3780807060485497856L,
/* [344] */ -3759794420226106880L, -3738481115842412032L, -3716862099905139200L, -3694932197183221248L,
/* [348] */ -3672686100763535872L, -3650118368025705472L, -3627223416464823808L, -3603995519354882048L,
/* [352] */ -3580428801247209472L, -3556517233294936064L, -3532254628397208576L, -3507634636153634816L,
/* [356] */ -3482650737621700608L, -3457296239866780160L, -3431564270296417792L, -3405447770767880704L,
/* [360] */ -3378939491458745856L, -3352031984490172416L, -3324717597289475072L, -3296988465680861184L,
/* [364] */ -3268836506691269120L, -3240253411056924672L, -3211230635415641088L, -3181759394171538432L,
/* [368] */ -3151830651013152256L, -3121435110069826048L, -3090563206687621632L, -3059205097804578816L,
/* [372] */ -3027350651907195904L, -2994989438544094720L, -2962110717375283712L, -2928703426733917184L,
/* [376] */ -2894756171672258048L, -2860257211467748352L, -2825194446557172224L, -2789555404872457216L,
/* [380] */ -2753327227540304896L, -2716496653917054464L, -2679050005916396544L, -2640973171596317184L,
/* [384] */ -2602251587958849024L, -2562870222922691584L, -2522813556418801152L, -2482065560560435712L,
/* [388] */ -2440609678833421312L, -2398428804250509824L, -2355505256407986688L, -2311820757381316608L,
/* [392] */ -2267356406388873216L, -2222092653150109696L, -2176009269860336640L, -2129085321695611904L,
/* [396] */ -2081299135757883392L, -2032628268363105280L, -1983049470568565760L, -1932538651826487296L,
/* [400] */ -1881070841645808640L, -1828620149131529728L, -1775159720264906752L, -1720661692774047744L,
/* [404] */ -1665097148437127680L, -1608436062642298880L, -1550647251022831104L, -1491698312963664384L,
/* [408] */ -1431555571764694528L, -1370184011228092416L, -1307547208415545344L, -1243607262304812544L,
/* [412] */ -1178324718048254976L, -1111658486516102144L, -1043565758774211584L, -974001915124345856L,
/* [416] */ -902920428295306752L, -830272760342546432L, -756008252773183488L, -680074009369660928L,
/* [420] */ -602414771142033408L, -522972782779506176L, -441687649923274752L, -358496186511785984L,
/* [424] */ -273332251384862720L, -186126573250838016L, -96806563039876096L, -5296112566262784L,
/* [428] */ 88484621680143872L, 184619450912608768L, 283196404027497984L, 384307996761360384L,
/* [432] */ 488051521767595008L, 594529361535797760L, 703849326283634688L, 816125019179379200L,
/* [436] */ 931476231515671552L, 1050029370739642368L, 1171917924579436544L, 1297282964870890496L,
/* [440] */ 1426273695110094848L, 1559048046230309888L, 1695773325638197760L, 1836626925155011584L,
/* [444] */ 1981797094206053888L, 2131483785391109632L, 2285899580478475776L, 2445270705902480384L,
/* [448] */ 2609838148036959744L, 2779858879886981120L, 2955607212425994240L, 3137376285625809920L,
/* [452] */ 3325479716351016448L, 3520253422737882112L, 3722057647546029568L, 3931279206301515776L,
/* [456] */ 4148333989964127232L, 4373669756431084544L, 4607769250590803968L, 4851153699010013696L,
/* [460] */ 5104386732888089088L, 5368078801878179840L, 5642892152071508992L, 5929546454235676672L,
/* [464] */ 6228825183767088640L, 6541582872345928704L, 6868753373709051392L, 7211359313221728768L,
/* [468] */ 7570522924211155456L, 7947478514849258496L, 8343586859683210240L, 8760351872223335936L,
/* [472] */ 9199439992591985152L, 3921213358161086464L, 4850044974529517568L, 6296921656268977664L,
/* [476] */ 8292973759562619904L, 3075143244722959872L, 6274953360832545280L, 2210268915646535680L,
/* [480] */ 6778561557392493568L, 4052431235721115136L, 2017158148825034240L, 9030121550070927872L,
/* [484] */ 8695700292397356032L, 749256839723897344L, 2212185722591297024L, 4780810392134832640L,
/* [488] */ 8573064486620626432L, 4685427019526152192L, 2026848442171050496L, 795427058175291392L,
/* [492] */ 1220057396927800832L, 3568726050706383360L, 8159594562977490432L, 5249819115171273728L,
/* [496] */ 5086123243830243840L, 8282283418103647744L, 4836310890809390080L, 6126145310249794560L,
/* [500] */ 2267629071146367488L, 5974727006028887552L, 8264939763687273472L, 608340989510339584L,
/* [504] */ 828635949469412864L, 8856898102852928512L, 1720475810403128832L, 3683415679219046400L,
/* [508] */ -1284830592014387712L, -9223372036854775808L, -9223372036854775808L, -9223372036854775808L,
};
/**
* The precomputed ziggurat lengths, denoted X_i in the main text. X_i = length of
* ziggurat layer i. Values have been scaled by 2^-63.
*/
private static final double[] X = {
/* [ 0] */ 9.0661259763949181e-19, 8.263176964596684e-19, 7.7842821042069184e-19, 7.4401381372447472e-19,
/* [ 4] */ 7.1706345512475669e-19, 6.9487349626981025e-19, 6.7599059941438532e-19, 6.5954173866761093e-19,
/* [ 8] */ 6.4496076486975596e-19, 6.3185931723986094e-19, 6.1995926140436088e-19, 6.0905448556085982e-19,
/* [ 12] */ 5.9898795313195317e-19, 5.8963723366494962e-19, 5.8090500779440151e-19, 5.727126242519077e-19,
/* [ 16] */ 5.6499560161437446e-19, 5.5770040980809757e-19, 5.5078211755959678e-19, 5.4420264022055456e-19,
/* [ 20] */ 5.3792941286269432e-19, 5.319343704005985e-19, 5.2619315318369782e-19, 5.2068448072048437e-19,
/* [ 24] */ 5.1538965252883907e-19, 5.1029214632636893e-19, 5.0537729161623905e-19, 5.0063200229055628e-19,
/* [ 28] */ 4.9604455588176776e-19, 4.9160441001706848e-19, 4.8730204879053506e-19, 4.8312885338060169e-19,
/* [ 32] */ 4.7907699245760014e-19, 4.7513932885346785e-19, 4.7130933967875306e-19, 4.6758104762491562e-19,
/* [ 36] */ 4.6394896162201668e-19, 4.6040802536211077e-19, 4.5695357246842147e-19, 4.5358128730569534e-19,
/* [ 40] */ 4.5028717060006086e-19, 4.4706750917642919e-19, 4.4391884923497501e-19, 4.4083797268094136e-19,
/* [ 44] */ 4.3782187609810469e-19, 4.3486775201900267e-19, 4.3197297219702936e-19, 4.2913507262878116e-19,
/* [ 48] */ 4.263517401111986e-19, 4.2362080014839232e-19, 4.2094020604858992e-19, 4.1830802907323973e-19,
/* [ 52] */ 4.1572244951862374e-19, 4.1318174862592063e-19, 4.1068430122896932e-19, 4.0822856906037985e-19,
/* [ 56] */ 4.058130946464292e-19, 4.0343649572961074e-19, 4.0109746016499096e-19, 3.9879474124283382e-19,
/* [ 60] */ 3.9652715339543132e-19, 3.9429356825084459e-19, 3.920929110004198e-19, 3.8992415705058057e-19,
/* [ 64] */ 3.8778632893258511e-19, 3.8567849344673741e-19, 3.8359975902000546e-19, 3.8154927325817224e-19,
/* [ 68] */ 3.7952622067556671e-19, 3.7752982058712154e-19, 3.7555932514901226e-19, 3.7361401753547253e-19,
/* [ 72] */ 3.7169321024057316e-19, 3.6979624349481498e-19, 3.6792248378733595e-19, 3.6607132248538204e-19,
/* [ 76] */ 3.6424217454345153e-19, 3.6243447729520606e-19, 3.6064768932185395e-19, 3.5888128939126416e-19,
/* [ 80] */ 3.5713477546256509e-19, 3.5540766375143278e-19, 3.5369948785167605e-19, 3.5200979790909461e-19,
/* [ 84] */ 3.5033815984391743e-19, 3.4868415461842876e-19, 3.4704737754666438e-19, 3.4542743764330739e-19,
/* [ 88] */ 3.4382395700913942e-19, 3.4223657025060954e-19, 3.4066492393126986e-19, 3.3910867605299972e-19,
/* [ 92] */ 3.3756749556509512e-19, 3.3604106189944498e-19, 3.3452906453014644e-19, 3.3303120255603112e-19,
/* [ 96] */ 3.3154718430468554e-19, 3.300767269566494e-19, 3.2861955618856956e-19, 3.2717540583417136e-19,
/* [100] */ 3.2574401756198997e-19, 3.2432514056887605e-19, 3.2291853128835623e-19, 3.2152395311299247e-19,
/* [104] */ 3.2014117612994074e-19, 3.1876997686896117e-19, 3.1741013806218296e-19, 3.1606144841497054e-19,
/* [108] */ 3.1472370238728068e-19, 3.1339669998493811e-19, 3.1208024656029448e-19, 3.1077415262176789e-19,
/* [112] */ 3.0947823365179181e-19, 3.0819230993273112e-19, 3.0691620638035022e-19, 3.0564975238444241e-19,
/* [116] */ 3.0439278165625384e-19, 3.0314513208235683e-19, 3.0190664558464783e-19, 3.0067716798616362e-19,
/* [120] */ 2.9945654888242782e-19, 2.9824464151805648e-19, 2.97041302668365e-19, 2.9584639252573667e-19,
/* [124] */ 2.9465977459052242e-19, 2.9348131556625803e-19, 2.9231088525899381e-19, 2.9114835648054477e-19,
/* [128] */ 2.8999360495547921e-19, 2.888465092316728e-19, 2.8770695059426596e-19, 2.8657481298286891e-19,
/* [132] */ 2.8544998291186936e-19, 2.8433234939370269e-19, 2.8322180386495424e-19, 2.8211824011516853e-19,
/* [136] */ 2.8102155421824659e-19, 2.7993164446631984e-19, 2.788484113059931e-19, 2.7777175727685546e-19,
/* [140] */ 2.7670158695216292e-19, 2.7563780688160088e-19, 2.7458032553603932e-19, 2.7352905325419784e-19,
/* [144] */ 2.7248390219114179e-19, 2.7144478626853394e-19, 2.7041162112657045e-19, 2.6938432407753285e-19,
/* [148] */ 2.6836281406089111e-19, 2.6734701159989559e-19, 2.6633683875959909e-19, 2.6533221910625232e-19,
/* [152] */ 2.6433307766801955e-19, 2.6333934089696193e-19, 2.6235093663224118e-19, 2.6136779406449473e-19,
/* [156] */ 2.6038984370133962e-19, 2.5941701733396064e-19, 2.5844924800474283e-19, 2.5748646997590919e-19,
/* [160] */ 2.5652861869912541e-19, 2.5557563078603673e-19, 2.5462744397970219e-19, 2.5368399712689353e-19,
/* [164] */ 2.5274523015122715e-19, 2.5181108402709945e-19, 2.5088150075439651e-19, 2.4995642333395008e-19,
/* [168] */ 2.4903579574371396e-19, 2.4811956291563531e-19, 2.4720767071319582e-19, 2.4630006590960037e-19,
/* [172] */ 2.4539669616659003e-19, 2.4449751001385796e-19, 2.4360245682904773e-19, 2.4271148681831419e-19,
/* [176] */ 2.4182455099742769e-19, 2.4094160117340305e-19, 2.4006258992663645e-19, 2.3918747059353259e-19,
/* [180] */ 2.3831619724960594e-19, 2.3744872469304068e-19, 2.3658500842869427e-19, 2.3572500465252978e-19,
/* [184] */ 2.3486867023646363e-19, 2.3401596271361442e-19, 2.3316684026394113e-19, 2.3232126170025724e-19,
/* [188] */ 2.3147918645460906e-19, 2.3064057456500724e-19, 2.298053866624993e-19, 2.2897358395857378e-19,
/* [192] */ 2.2814512823288434e-19, 2.2731998182128476e-19, 2.2649810760416509e-19, 2.2567946899507905e-19,
/* [196] */ 2.2486402992965488e-19, 2.2405175485477971e-19, 2.2324260871805039e-19, 2.2243655695748146e-19,
/* [200] */ 2.2163356549146385e-19, 2.2083360070896563e-19, 2.2003662945996843e-19, 2.1924261904613178e-19,
/* [204] */ 2.1845153721167946e-19, 2.1766335213450081e-19, 2.1687803241746032e-19, 2.1609554707991062e-19,
/* [208] */ 2.153158655494016e-19, 2.145389576535809e-19, 2.1376479361227967e-19, 2.1299334402977885e-19,
/* [212] */ 2.1222457988725017e-19, 2.1145847253536737e-19, 2.1069499368708242e-19, 2.0993411541056253e-19,
/* [216] */ 2.0917581012228278e-19, 2.0842005058027066e-19, 2.076668098774977e-19, 2.0691606143541438e-19,
/* [220] */ 2.061677789976242e-19, 2.0542193662369327e-19, 2.0467850868309113e-19, 2.0393746984925989e-19,
/* [224] */ 2.0319879509380751e-19, 2.0246245968082214e-19, 2.0172843916130424e-19, 2.0099670936771298e-19,
/* [228] */ 2.0026724640862402e-19, 1.9954002666349563e-19, 1.9881502677754006e-19, 1.9809222365669745e-19,
/* [232] */ 1.9737159446270942e-19, 1.966531166082896e-19, 1.9593676775238861e-19, 1.9522252579555087e-19,
/* [236] */ 1.9451036887536058e-19, 1.9380027536197503e-19, 1.9309222385374197e-19, 1.9238619317289982e-19,
/* [240] */ 1.9168216236135756e-19, 1.9098011067655294e-19, 1.9028001758738615e-19, 1.8958186277022764e-19,
/* [244] */ 1.8888562610499751e-19, 1.8819128767131499e-19, 1.8749882774471561e-19, 1.8680822679293485e-19,
/* [248] */ 1.8611946547225595e-19, 1.8543252462392036e-19, 1.8474738527059914e-19, 1.8406402861292351e-19,
/* [252] */ 1.833824360260732e-19, 1.8270258905642057e-19, 1.8202446941822951e-19, 1.8134805899040715e-19,
/* [256] */ 1.8067333981330703e-19, 1.8000029408558255e-19, 1.7932890416108891e-19, 1.786591525458323e-19,
/* [260] */ 1.7799102189496529e-19, 1.7732449500982641e-19, 1.7665955483502355e-19, 1.7599618445555899e-19,
/* [264] */ 1.7533436709399558e-19, 1.7467408610766236e-19, 1.7401532498589862e-19, 1.7335806734733532e-19,
/* [268] */ 1.7270229693721261e-19, 1.7204799762473213e-19, 1.7139515340044364e-19, 1.7074374837366399e-19,
/* [272] */ 1.7009376676992816e-19, 1.694451929284709e-19, 1.6879801129973803e-19, 1.6815220644292632e-19,
/* [276] */ 1.6750776302355125e-19, 1.6686466581104123e-19, 1.6622289967635762e-19, 1.655824495896394e-19,
/* [280] */ 1.6494330061787188e-19, 1.64305437922578e-19, 1.6366884675753161e-19, 1.6303351246649206e-19,
/* [284] */ 1.6239942048095857e-19, 1.6176655631794399e-19, 1.6113490557776692e-19, 1.6050445394186123e-19,
/* [288] */ 1.598751871706023e-19, 1.5924709110114872e-19, 1.5862015164529922e-19, 1.5799435478736328e-19,
/* [292] */ 1.5736968658204502e-19, 1.5674613315233954e-19, 1.5612368068744041e-19, 1.55502315440658e-19,
/* [296] */ 1.5488202372734756e-19, 1.5426279192284608e-19, 1.536446064604174e-19, 1.5302745382920442e-19,
/* [300] */ 1.5241132057218782e-19, 1.5179619328415014e-19, 1.5118205860964477e-19, 1.5056890324096861e-19,
/* [304] */ 1.4995671391613777e-19, 1.4934547741686533e-19, 1.4873518056654034e-19, 1.4812581022820734e-19,
/* [308] */ 1.4751735330254499e-19, 1.4690979672584366e-19, 1.4630312746798053e-19, 1.456973325303913e-19,
/* [312] */ 1.4509239894403808e-19, 1.4448831376737181e-19, 1.438850640842889e-19, 1.4328263700208072e-19,
/* [316] */ 1.4268101964937506e-19, 1.4208019917406874e-19, 1.4148016274125018e-19, 1.4088089753111074e-19,
/* [320] */ 1.4028239073684423e-19, 1.396846295625332e-19, 1.3908760122102091e-19, 1.3849129293176794e-19,
/* [324] */ 1.3789569191869245e-19, 1.3730078540799268e-19, 1.3670656062595043e-19, 1.3611300479671491e-19,
/* [328] */ 1.3552010514006475e-19, 1.3492784886914775e-19, 1.3433622318819671e-19, 1.3374521529021971e-19,
/* [332] */ 1.3315481235466403e-19, 1.3256500154505191e-19, 1.3197577000658667e-19, 1.3138710486372786e-19,
/* [336] */ 1.3079899321773381e-19, 1.3021142214417001e-19, 1.2962437869038141e-19, 1.2903784987292747e-19,
/* [340] */ 1.2845182267497766e-19, 1.2786628404366604e-19, 1.2728122088740255e-19, 1.2669662007313963e-19,
/* [344] */ 1.2611246842359178e-19, 1.2552875271440607e-19, 1.2494545967128163e-19, 1.2436257596703557e-19,
/* [348] */ 1.2378008821861334e-19, 1.231979829840411e-19, 1.2261624675931726e-19, 1.2203486597524123e-19,
/* [352] */ 1.2145382699417624e-19, 1.2087311610674357e-19, 1.2029271952844551e-19, 1.1971262339621383e-19,
/* [356] */ 1.1913281376488073e-19, 1.1855327660356906e-19, 1.1797399779199848e-19, 1.1739496311670397e-19,
/* [360] */ 1.1681615826716313e-19, 1.1623756883182844e-19, 1.1565918029406041e-19, 1.1508097802795774e-19,
/* [364] */ 1.1450294729408e-19, 1.1392507323505832e-19, 1.1334734087108959e-19, 1.1276973509530898e-19,
/* [368] */ 1.1219224066903596e-19, 1.1161484221688824e-19, 1.1103752422175824e-19, 1.1046027101964606e-19,
/* [372] */ 1.0988306679434292e-19, 1.0930589557195864e-19, 1.0872874121528646e-19, 1.0815158741799823e-19,
/* [376] */ 1.0757441769866241e-19, 1.0699721539457744e-19, 1.0641996365541205e-19, 1.0584264543664434e-19,
/* [380] */ 1.0526524349279041e-19, 1.0468774037041343e-19, 1.0411011840090299e-19, 1.0353235969301472e-19,
/* [384] */ 1.0295444612515902e-19, 1.0237635933742761e-19, 1.0179808072334579e-19, 1.0121959142133761e-19,
/* [388] */ 1.0064087230589085e-19, 1.0006190397840746e-19, 9.9482666757724647e-20, 9.890314067029116e-20,
/* [392] */ 9.8323305439981917e-20, 9.7743140477533696e-20, 9.7162624869583486e-20, 9.6581737367289816e-20,
/* [396] */ 9.6000456374516673e-20, 9.5418759935558063e-20, 9.4836625722380091e-20, 9.4254031021356359e-20,
/* [400] */ 9.3670952719470503e-20, 9.3087367289958548e-20, 9.2503250777362024e-20, 9.1918578781960824e-20,
/* [404] */ 9.1333326443553021e-20, 9.074746842454686e-20, 9.0160978892327756e-20, 8.9573831500860881e-20,
/* [408] */ 8.8985999371487523e-20, 8.8397455072870296e-20, 8.7808170600039778e-20, 8.7218117352491731e-20,
/* [412] */ 8.6627266111280592e-20, 8.6035587015051616e-20, 8.5443049534949641e-20, 8.4849622448338465e-20,
/* [416] */ 8.4255273811260126e-20, 8.3659970929558324e-20, 8.3063680328584881e-20, 8.246636772140225e-20,
/* [420] */ 8.186799797538864e-20, 8.1268535077145628e-20, 8.0667942095600287e-20, 8.0066181143186217e-20,
/* [424] */ 7.9463213334978367e-20, 7.885899874564743e-20, 7.8253496364088652e-20, 7.7646664045568686e-20,
/* [428] */ 7.7038458461221292e-20, 7.642883504470902e-20, 7.5817747935853203e-20, 7.5205149921017434e-20,
/* [432] */ 7.4590992370012353e-20, 7.3975225169269108e-20, 7.335779665100703e-20, 7.2738653518097223e-20,
/* [436] */ 7.2117740764296587e-20, 7.1495001589498138e-20, 7.0870377309610182e-20, 7.0243807260641471e-20,
/* [440] */ 6.96152286965294e-20, 6.8984576680203705e-20, 6.8351783967329228e-20, 6.771678088211617e-20,
/* [444] */ 6.7079495184524855e-20, 6.6439851928123895e-20, 6.5797773307783692e-20, 6.5153178496301541e-20,
/* [448] */ 6.4505983468957815e-20, 6.3856100814894436e-20, 6.320343953408397e-20, 6.2547904818519893e-20,
/* [452] */ 6.1889397816101516e-20, 6.1227815375509665e-20, 6.056304977016756e-20, 5.9894988399150965e-20,
/* [456] */ 5.9223513462649234e-20, 5.8548501609278489e-20, 5.7869823552202679e-20, 5.7187343650622052e-20,
/* [460] */ 5.6500919452730109e-20, 5.5810401195711026e-20, 5.5115631257734864e-20, 5.4416443556192869e-20,
/* [464] */ 5.3712662885580816e-20, 5.3004104187460606e-20, 5.2290571743781936e-20, 5.1571858283490972e-20,
/* [468] */ 5.0847743990749481e-20, 5.0117995401181822e-20, 4.9382364170293304e-20, 4.8640585695477616e-20,
/* [472] */ 4.7892377569750159e-20, 4.7137437841375266e-20, 4.6375443048730777e-20, 4.5606045993857614e-20,
/* [476] */ 4.4828873210896671e-20, 4.4043522076659498e-20, 4.3249557499439262e-20, 4.2446508108220653e-20,
/* [480] */ 4.1633861846859842e-20, 4.0811060855462851e-20, 3.997749549257881e-20, 3.9132497314870153e-20,
/* [484] */ 3.8275330782752969e-20, 3.7405183397092119e-20, 3.6521153887674253e-20, 3.5622237960646127e-20,
/* [488] */ 3.4707310957388568e-20, 3.3775106563577917e-20, 3.2824190407548451e-20, 3.1852926960062177e-20,
/* [492] */ 3.0859437528023669e-20, 2.9841546217580957e-20, 2.8796709353961079e-20, 2.772192169113539e-20,
/* [496] */ 2.6613589304954131e-20, 2.5467353391178318e-20, 2.4277839478989958e-20, 2.3038289202859939e-20,
/* [500] */ 2.1739999061414015e-20, 2.037142499071807e-20, 1.8916669455626665e-20, 1.7352728004959384e-20,
/* [504] */ 1.5643946691546401e-20, 1.3729109753658277e-20, 1.1483304366567183e-20, 8.5324113192621202e-21,
/* [508] */ 0,
};
/** The precomputed ziggurat heights, denoted Y_i in the main text. Y_i = f(X_i).
* Values have been scaled by 2^-63. */
private static final double[] Y = {
/* [ 0] */ 2.5323797727138181e-23, 5.3108358239232209e-23, 8.26022457065042e-23, 1.1346037443474311e-22,
/* [ 4] */ 1.4547828564666417e-22, 1.7851865078182134e-22, 2.1248195450141036e-22, 2.4729229734018009e-22,
/* [ 8] */ 2.8288961626257085e-22, 3.1922503684288075e-22, 3.5625791217647975e-22, 3.9395384018258527e-22,
/* [ 12] */ 4.3228328223471963e-22, 4.7122056897421988e-22, 5.1074316514991609e-22, 5.5083111339357016e-22,
/* [ 16] */ 5.914666050219951e-22, 6.3263364315882896e-22, 6.7431777433800354e-22, 7.1650587182719069e-22,
/* [ 20] */ 7.5918595863879997e-22, 8.0234706143087472e-22, 8.4597908875883711e-22, 8.9007272874543527e-22,
/* [ 24] */ 9.3461936239794752e-22, 9.7961098965456997e-22, 1.0250401658767016e-21, 1.0708999469822909e-21,
/* [ 28] */ 1.117183841780181e-21, 1.1638857703464862e-21, 1.2110000275027631e-21, 1.2585212506274999e-21,
/* [ 32] */ 1.3064443911684845e-21, 1.3547646893321801e-21, 1.4034776515135506e-21, 1.4525790301004666e-21,
/* [ 36] */ 1.5020648053444311e-21, 1.5519311690365945e-21, 1.60217450976698e-21, 1.6527913995771288e-21,
/* [ 40] */ 1.7037785818432865e-21, 1.7551329602497868e-21, 1.8068515887312395e-21, 1.858931662278158e-21,
/* [ 44] */ 1.9113705085142316e-21, 1.9641655799650449e-21, 2.0173144469479217e-21, 2.0708147910210758e-21,
/* [ 48] */ 2.1246643989375575e-21, 2.1788611570557979e-21, 2.2334030461640281e-21, 2.2882881366806101e-21,
/* [ 52] */ 2.3435145841964528e-21, 2.3990806253293198e-21, 2.4549845738630073e-21, 2.5112248171471613e-21,
/* [ 56] */ 2.5677998127359586e-21, 2.6247080852460518e-21, 2.6819482234160959e-21, 2.7395188773518701e-21,
/* [ 60] */ 2.7974187559425357e-21, 2.8556466244349068e-21, 2.9142013021538138e-21, 2.9730816603577289e-21,
/* [ 64] */ 3.0322866202197593e-21, 3.0918151509250072e-21, 3.1516662678760491e-21, 3.2118390309989992e-21,
/* [ 68] */ 3.2723325431432467e-21, 3.3331459485685267e-21, 3.3942784315135004e-21, 3.4557292148404881e-21,
/* [ 72] */ 3.5174975587514197e-21, 3.579582759570448e-21, 3.6419841485890333e-21, 3.7047010909696115e-21,
/* [ 76] */ 3.7677329847042535e-21, 3.8310792596249961e-21, 3.8947393764627574e-21, 3.9587128259519763e-21,
/* [ 80] */ 4.0229991279783226e-21, 4.0875978307670022e-21, 4.152508510109364e-21, 4.2177307686256654e-21,
/* [ 84] */ 4.2832642350619977e-21, 4.3491085636195201e-21, 4.4152634333142514e-21, 4.4817285473658032e-21,
/* [ 88] */ 4.5485036326135354e-21, 4.6155884389587065e-21, 4.6829827388312992e-21, 4.7506863266802562e-21,
/* [ 92] */ 4.8186990184859766e-21, 4.8870206512939572e-21, 4.955651082768556e-21, 5.0245901907659065e-21,
/* [ 96] */ 5.0938378729250747e-21, 5.1633940462765908e-21, 5.2332586468675656e-21, 5.3034316294026107e-21,
/* [100] */ 5.3739129668998646e-21, 5.4447026503614276e-21, 5.5158006884575949e-21, 5.5872071072242535e-21,
/* [104] */ 5.6589219497729013e-21, 5.7309452760127363e-21, 5.8032771623843054e-21, 5.875917701604242e-21,
/* [108] */ 5.9488670024206328e-21, 6.0221251893785718e-21, 6.0956924025955166e-21, 6.1695687975460331e-21,
/* [112] */ 6.2437545448555828e-21, 6.3182498301029956e-21, 6.3930548536312928e-21, 6.4681698303665609e-21,
/* [116] */ 6.5435949896445654e-21, 6.61933057504483e-21, 6.6953768442319073e-21, 6.7717340688035926e-21,
/* [120] */ 6.8484025341458281e-21, 6.9253825392940775e-21, 7.0026743968009459e-21, 7.0802784326098346e-21,
/* [124] */ 7.1581949859344372e-21, 7.2364244091438833e-21, 7.3149670676533519e-21, 7.3938233398199853e-21,
/* [128] */ 7.472993616843931e-21, 7.5524783026743669e-21, 7.6322778139203621e-21, 7.7123925797664099e-21,
/* [132] */ 7.7928230418925412e-21, 7.8735696543988364e-21, 7.9546328837342629e-21, 8.0360132086296871e-21,
/* [136] */ 8.1177111200349716e-21, 8.1997271210600361e-21, 8.2820617269197917e-21, 8.3647154648828443e-21,
/* [140] */ 8.4476888742238858e-21, 8.5309825061796779e-21, 8.614596923908543e-21, 8.6985327024532865e-21,
/* [144] */ 8.7827904287074878e-21, 8.8673707013850592e-21, 8.9522741309930325e-21, 9.0375013398074916e-21,
/* [148] */ 9.1230529618525954e-21, 9.2089296428826335e-21, 9.2951320403670488e-21, 9.3816608234783977e-21,
/* [152] */ 9.4685166730831613e-21, 9.5557002817353942e-21, 9.6432123536731481e-21, 9.7310536048176277e-21,
/* [156] */ 9.8192247627750442e-21, 9.9077265668411266e-21, 9.9965597680082532e-21, 1.0085725128975164e-20,
/* [160] */ 1.0175223424159242e-20, 1.0265055439711297e-20, 1.0355221973532876e-20, 1.0445723835296011e-20,
/* [164] */ 1.0536561846465444e-20, 1.0627736840323247e-20, 1.0719249661995874e-20, 1.0811101168483563e-20,
/* [168] */ 1.0903292228692131e-20, 1.0995823723467109e-20, 1.1088696545630196e-20, 1.118191160001806e-20,
/* [172] */ 1.1275469803523424e-20, 1.1369372085138467e-20, 1.14636193860005e-20, 1.155821265943994e-20,
/* [176] */ 1.1653152871030541e-20, 1.1748440998641906e-20, 1.1844078032494261e-20, 1.1940064975215492e-20,
/* [180] */ 1.2036402841900448e-20, 1.2133092660172487e-20, 1.2230135470247318e-20, 1.2327532324999065e-20,
/* [184] */ 1.2425284290028643e-20, 1.2523392443734356e-20, 1.2621857877384821e-20, 1.2720681695194142e-20,
/* [188] */ 1.2819865014399386e-20, 1.2919408965340364e-20, 1.3019314691541709e-20, 1.311958334979729e-20,
/* [192] */ 1.3220216110256947e-20, 1.3321214156515581e-20, 1.342257868570459e-20, 1.35243109085857e-20,
/* [196] */ 1.3626412049647167e-20, 1.3728883347202408e-20, 1.3831726053491038e-20, 1.3934941434782369e-20,
/* [200] */ 1.403853077148138e-20, 1.4142495358237157e-20, 1.4246836504053859e-20, 1.4351555532404212e-20,
/* [204] */ 1.4456653781345581e-20, 1.4562132603638601e-20, 1.4667993366868439e-20, 1.4774237453568695e-20,
/* [208] */ 1.4880866261347982e-20, 1.4987881203019181e-20, 1.5095283706731464e-20, 1.5203075216105051e-20,
/* [212] */ 1.5311257190368801e-20, 1.5419831104500607e-20, 1.5528798449370678e-20, 1.5638160731887733e-20,
/* [216] */ 1.5747919475148132e-20, 1.5858076218588015e-20, 1.5968632518138439e-20, 1.607958994638363e-20,
/* [220] */ 1.6190950092722321e-20, 1.6302714563532247e-20, 1.6414884982337896e-20, 1.6527462989981453e-20,
/* [224] */ 1.6640450244797108e-20, 1.6753848422788681e-20, 1.6867659217810695e-20, 1.6981884341752875e-20,
/* [228] */ 1.7096525524728213e-20, 1.721158451526456e-20, 1.7327063080499912e-20, 1.7442963006381335e-20,
/* [232] */ 1.7559286097867713e-20, 1.7676034179136259e-20, 1.7793209093792969e-20, 1.7910812705086994e-20,
/* [236] */ 1.8028846896129066e-20, 1.8147313570114001e-20, 1.82662146505474e-20, 1.8385552081476575e-20,
/* [240] */ 1.8505327827725806e-20, 1.8625543875136006e-20, 1.8746202230808869e-20, 1.8867304923355594e-20,
/* [244] */ 1.8988854003150238e-20, 1.9110851542587859e-20, 1.9233299636347447e-20, 1.9356200401659812e-20,
/* [248] */ 1.9479555978580487e-20, 1.9603368530267746e-20, 1.9727640243265847e-20, 1.9852373327793618e-20,
/* [252] */ 1.9977570018038445e-20, 2.010323257245582e-20, 2.0229363274074546e-20, 2.0355964430807701e-20,
/* [256] */ 2.0483038375769486e-20, 2.0610587467598089e-20, 2.0738614090784689e-20, 2.0867120656008704e-20,
/* [260] */ 2.099610960047943e-20, 2.1125583388284243e-20, 2.1255544510743418e-20, 2.1385995486771794e-20,
/* [264] */ 2.1516938863247379e-20, 2.1648377215387072e-20, 2.1780313147129627e-20, 2.1912749291526057e-20,
/* [268] */ 2.2045688311137619e-20, 2.2179132898441537e-20, 2.2313085776244664e-20, 2.2447549698105236e-20,
/* [272] */ 2.2582527448762922e-20, 2.2718021844577323e-20, 2.2854035733975161e-20, 2.2990571997906318e-20,
/* [276] */ 2.3127633550308938e-20, 2.3265223338583805e-20, 2.3403344344078233e-20, 2.3541999582579633e-20,
/* [280] */ 2.3681192104819063e-20, 2.3820924996984908e-20, 2.396120138124703e-20, 2.4102024416291538e-20,
/* [284] */ 2.4243397297866505e-20, 2.4385323259338871e-20, 2.452780557226279e-20, 2.4670847546959731e-20,
/* [288] */ 2.4814452533110573e-20, 2.4958623920360058e-20, 2.5103365138933835e-20, 2.5248679660268454e-20,
/* [292] */ 2.539457099765463e-20, 2.5541042706894083e-20, 2.5688098386970325e-20, 2.583574168073375e-20,
/* [296] */ 2.5983976275601352e-20, 2.6132805904271501e-20, 2.6282234345454126e-20, 2.6432265424616709e-20,
/* [300] */ 2.6582903014746527e-20, 2.6734151037129546e-20, 2.6886013462146377e-20, 2.7038494310085842e-20,
/* [304] */ 2.7191597651976473e-20, 2.7345327610436566e-20, 2.7499688360543205e-20, 2.7654684130720776e-20,
/* [308] */ 2.7810319203649531e-20, 2.7966597917194705e-20, 2.8123524665356816e-20, 2.8281103899243675e-20,
/* [312] */ 2.8439340128064684e-20, 2.8598237920148148e-20, 2.8757801903982133e-20, 2.8918036769279628e-20,
/* [316] */ 2.9078947268068582e-20, 2.9240538215807684e-20, 2.9402814492528446e-20, 2.9565781044004507e-20,
/* [320] */ 2.9729442882948833e-20, 2.9893805090239645e-20, 3.005887281617601e-20, 3.0224651281763803e-20,
/* [324] */ 3.0391145780033081e-20, 3.0558361677387748e-20, 3.072630441498844e-20, 3.0894979510169736e-20,
/* [328] */ 3.1064392557892572e-20, 3.1234549232233119e-20, 3.1405455287909063e-20, 3.1577116561844568e-20,
/* [332] */ 3.1749538974775062e-20, 3.1922728532893043e-20, 3.2096691329536337e-20, 3.2271433546919926e-20,
/* [336] */ 3.2446961457912936e-20, 3.26232814278621e-20, 3.2800399916463208e-20, 3.2978323479682113e-20,
/* [340] */ 3.3157058771726903e-20, 3.3336612547072853e-20, 3.3516991662541965e-20, 3.3698203079438833e-20,
/* [344] */ 3.3880253865744772e-20, 3.4063151198372078e-20, 3.4246902365480509e-20, 3.4431514768858065e-20,
/* [348] */ 3.4616995926368224e-20, 3.4803353474466002e-20, 3.4990595170785116e-20, 3.517872889679876e-20,
/* [352] */ 3.5367762660556557e-20, 3.5557704599500393e-20, 3.5748562983361843e-20, 3.5940346217144218e-20,
/* [356] */ 3.6133062844192156e-20, 3.6326721549351945e-20, 3.6521331162225938e-20, 3.6716900660524377e-20,
/* [360] */ 3.6913439173518358e-20, 3.7110955985597548e-20, 3.7309460539936645e-20, 3.7508962442274668e-20,
/* [364] */ 3.7709471464811288e-20, 3.7910997550224699e-20, 3.8113550815815718e-20, 3.8317141557782911e-20,
/* [368] */ 3.8521780255633917e-20, 3.8727477576738257e-20, 3.8934244381027282e-20, 3.914209172584698e-20,
/* [372] */ 3.935103087096993e-20, 3.9561073283772659e-20, 3.977223064458528e-20, 3.9984514852220283e-20,
/* [376] */ 4.0197938029688092e-20, 4.0412512530106928e-20, 4.0628250942815264e-20, 4.0845166099695399e-20,
/* [380] */ 4.1063271081717031e-20, 4.1282579225710425e-20, 4.1503104131378944e-20, 4.1724859668561458e-20,
/* [384] */ 4.1947859984755547e-20, 4.2172119512913083e-20, 4.2397652979520307e-20, 4.2624475412975199e-20,
/* [388] */ 4.2852602152275669e-20, 4.3082048856032696e-20, 4.3312831511823486e-20, 4.35449664459004e-20,
/* [392] */ 4.3778470333272371e-20, 4.4013360208176433e-20, 4.4249653474957918e-20, 4.4487367919379097e-20,
/* [396] */ 4.4726521720376992e-20, 4.4967133462292346e-20, 4.5209222147593173e-20, 4.5452807210117408e-20,
/* [400] */ 4.5697908528860855e-20, 4.5944546442338236e-20, 4.6192741763546519e-20, 4.6442515795561947e-20,
/* [404] */ 4.6693890347803697e-20, 4.6946887752999492e-20, 4.7201530884890459e-20, 4.7457843176715081e-20,
/* [408] */ 4.7715848640514521e-20, 4.797557188730451e-20, 4.8237038148161644e-20, 4.8500273296275537e-20,
/* [412] */ 4.8765303870021277e-20, 4.903215709711068e-20, 4.930086091988455e-20, 4.957144402181263e-20,
/* [416] */ 4.9843935855272545e-20, 5.0118366670684009e-20, 5.0394767547080008e-20, 5.0673170424202713e-20,
/* [420] */ 5.0953608136218094e-20, 5.1236114447150163e-20, 5.1520724088143471e-20, 5.1807472796670477e-20,
/* [424] */ 5.2096397357809406e-20, 5.2387535647727967e-20, 5.2680926679518772e-20, 5.2976610651543945e-20,
/* [428] */ 5.3274628998459006e-20, 5.3575024445099889e-20, 5.387784106343208e-20, 5.4183124332777464e-20,
/* [432] */ 5.4490921203552541e-20, 5.4801280164771845e-20, 5.5114251315592008e-20, 5.5429886441196666e-20,
/* [436] */ 5.5748239093348294e-20, 5.6069364675963432e-20, 5.6393320536099437e-20, 5.672016606077763e-20,
/* [440] */ 5.7049962780107257e-20, 5.7382774477219359e-20, 5.7718667305568524e-20, 5.805770991421623e-20,
/* [444] */ 5.839997358176981e-20, 5.8745532359720356e-20, 5.9094463225999108e-20, 5.9446846249657746e-20,
/* [448] */ 5.9802764767674702e-20, 6.0162305574997887e-20, 6.0525559129056712e-20, 6.0892619770114342e-20,
/* [452] */ 6.1263585958987563e-20, 6.1638560533838738e-20, 6.2017650987946062e-20, 6.240096977058771e-20,
/* [456] */ 6.2788634613437286e-20, 6.3180768885168317e-20, 6.3577501977308952e-20, 6.3978969724784013e-20,
/* [460] */ 6.4385314865037979e-20, 6.4796687540159733e-20, 6.5213245847042516e-20, 6.5635156441324361e-20,
/* [464] */ 6.60625952016853e-20, 6.6495747962050781e-20, 6.6934811320393724e-20, 6.7379993534175763e-20,
/* [468] */ 6.7831515514062906e-20, 6.8289611929446599e-20, 6.8754532441561504e-20, 6.9226543082700484e-20,
/* [472] */ 6.9705927803286983e-20, 7.0192990212507397e-20, 7.0688055542996486e-20, 7.1191472875921819e-20,
/* [476] */ 7.1703617670003266e-20, 7.2224894646888467e-20, 7.2755741096353193e-20, 7.3296630678623713e-20,
/* [480] */ 7.3848077818549056e-20, 7.4410642808486694e-20, 7.4984937765101967e-20, 7.5571633621866871e-20,
/* [484] */ 7.6171468386713451e-20, 7.6785256957023811e-20, 7.741390286755859e-20, 7.8058412459147469e-20,
/* [488] */ 7.8719912108823151e-20, 7.9399669373134973e-20, 8.0099119192138521e-20, 8.081989672282774e-20,
/* [492] */ 8.1563878981695832e-20, 8.2333238379901939e-20, 8.3130512601664359e-20, 8.3958697396906725e-20,
/* [496] */ 8.4821372242318269e-20, 8.5722874400271225e-20, 8.6668546442438178e-20, 8.7665099347792271e-20,
/* [500] */ 8.8721165356564412e-20, 8.9848179006810222e-20, 9.1061863799122794e-20, 9.2384933810531728e-20,
/* [504] */ 9.3852522168647537e-20, 9.5524799137180048e-20, 9.7524125577254293e-20, 1.0021490936397442e-19,
/* [508] */ 1.0842021724855044e-19,
};
/** Underlying source of randomness. */
protected final UniformRandomProvider rng;
/**
* @param rng Generator of uniformly distributed random numbers.
*/
ModifiedZigguratExponentialSampler512(UniformRandomProvider rng) {
this.rng = rng;
}
/** {@inheritDoc} */
@Override
public double sample() {
return createSample();
}
/**
* Creates the exponential sample with {@code mean = 1}.
*
* <p>Note: This has been extracted to a separate method so that the recursive call
* when sampling tries again targets this function. This allows sub-classes
* to override the sample method to generate a sample with a different mean using:
* <pre>
* @Override
* public double sample() {
* return super.sample() * mean;
* }
* </pre>
* Otherwise the sub-class {@code sample()} method will recursively call
* the overloaded sample() method when trying again which creates a bad sample due
* to compound multiplication of the mean.
*
* @return the sample
*/
final double createSample() {
final long x = nextLong();
// Float multiplication squashes these last 9 bits, so they can be used to sample i
final int i = ((int) x) & 0x1ff;
if (i < I_MAX) {
// Early exit.
return X[i] * (x & MAX_INT64);
}
final int j = selectRegion();
return j == 0 ? X_0 + createSample() : sampleOverhang(j);
}
/**
* Select the overhang region or the tail using alias sampling.
*
* @return the region
*/
protected int selectRegion() {
final long x = nextLong();
// j in [0, 512)
final int j = ((int) x) & 0x1ff;
// map to j in [0, N] with N the number of layers of the ziggurat
return x >= IPMF[j] ? MAP[j] : j;
}
/**
* Sample from overhang region {@code j}.
*
* @param j Index j (must be {@code > 0})
* @return the sample
*/
protected double sampleOverhang(int j) {
// Sample from the triangle:
// X[j],Y[j]
// |\-->u1
// | \ |
// | \ |
// | \| Overhang j (with hypotenuse not pdf(x))
// | \
// | |\
// | | \
// | u2 \
// +-------- X[j-1],Y[j-1]
// u2 = u1 + (u2 - u1) = u1 + uDistance
// If u2 < u1 then reflect in the hypotenuse by swapping u1 and u2.
long u1 = randomInt63();
long uDistance = randomInt63() - u1;
if (uDistance < 0) {
uDistance = -uDistance;
u1 -= uDistance;
}
final double x = sampleX(X, j, u1);
if (uDistance >= E_MAX) {
// Early Exit: x < y - epsilon
return x;
}
return sampleY(Y, j, u1 + uDistance) <= Math.exp(-x) ? x : sampleOverhang(j);
}
/**
* Generates a {@code long}.
*
* @return the long
*/
protected long nextLong() {
return rng.nextLong();
}
/**
* Return a positive long in {@code [0, 2^63)}.
*
* @return the long
*/
protected long randomInt63() {
return rng.nextLong() & MAX_INT64;
}
}
/**
* Compute the x value of the point in the overhang region from the uniform deviate.
* <pre>{@code
* X[j],Y[j]
* |\ |
* | \|
* | \
* | |\ Ziggurat overhang j (with hypotenuse not pdf(x))
* | | \
* | u2 \
* | \
* |-->u1 \
* +-------- X[j-1],Y[j-1]
*
* x = X[j] + u1 * (X[j-1] - X[j])
* }</pre>
*
* @param x Ziggurat data table X. Values assumed to be scaled by 2^-63.
* @param j Index j. Value assumed to be above zero.
* @param u1 Uniform deviate. Value assumed to be in {@code [0, 2^63)}.
* @return y
*/
static double sampleX(double[] x, int j, long u1) {
return x[j] * TWO_POW_63 + u1 * (x[j - 1] - x[j]);
}
/**
* Compute the y value of the point in the overhang region from the uniform deviate.
* <pre>{@code
* X[j],Y[j]
* |\ |
* | \|
* | \
* | |\ Ziggurat overhang j (with hypotenuse not pdf(x))
* | | \
* | u2 \
* | \
* |-->u1 \
* +-------- X[j-1],Y[j-1]
*
* y = Y[j-1] + (1-u2) * (Y[j] - Y[j-1])
* }</pre>
*
* @param y Ziggurat data table Y. Values assumed to be scaled by 2^-63.
* @param j Index j. Value assumed to be above zero.
* @param u2 Uniform deviate. Value assumed to be in {@code [0, 2^63)}.
* @return y
*/
static double sampleY(double[] y, int j, long u2) {
// Note: u2 is in [0, 2^63)
// Long.MIN_VALUE is used as an unsigned int with value 2^63:
// 1 - u2 = Long.MIN_VALUE - u2
//
// The subtraction from 1 can be avoided with:
// y = Y[j] + u2 * (Y[j-1] - Y[j])
// This is effectively sampleX(y, j, u2)
// Tests show the alternative is 1 ULP different with approximately 3% frequency.
// It has not been measured more than 1 ULP different.
return y[j - 1] * TWO_POW_63 + (Long.MIN_VALUE - u2) * (y[j] - y[j - 1]);
}
/**
* Throw an illegal state exception for the unknown parameter.
*
* @param parameter Parameter name
*/
private static void throwIllegalStateException(String parameter) {
throw new IllegalStateException("Unknown: " + parameter);
}
/**
* Baseline for the JMH timing overhead for production of an {@code double} value.
*
* @return the {@code double} value
*/
@Benchmark
public double baseline() {
return value;
}
/**
* Benchmark methods for obtaining an index from the lower bits of a long.
*
* <p>Note: This is disabled as there is no measurable difference between methods.
*
* @param sources Source of randomness.
* @return the sample value
*/
//@Benchmark
public int getIndex(IndexSources sources) {
return sources.getSampler().sample();
}
/**
* Benchmark methods for obtaining an index from the lower bits of a long and
* comparing them to a limit then returning the index as an {@code int}.
*
* <p>Note: This is disabled as there is no measurable difference between methods.
*
* @param sources Source of randomness.
* @return the sample value
*/
//@Benchmark
public int compareIndex(IndexCompareSources sources) {
return sources.getSampler().sample();
}
/**
* Benchmark methods for obtaining an unsigned long.
*
* <p>Note: This is disabled. Either there is no measurable difference between methods
* or the bit shift method is marginally faster depending on JDK and platform.
*
* @param sources Source of randomness.
* @return the sample value
*/
//@Benchmark
public long getUnsignedLong(LongSources sources) {
return sources.getSampler().sample();
}
/**
* Benchmark methods for obtaining an interpolated value from an unsigned long.
*
* <p>Note: This is disabled. The speed is typically: U1, 1minusU2, U_1minusU.
*
* @param sources Source of randomness.
* @return the sample value
*/
//@Benchmark
public double interpolate(InterpolationSources sources) {
return sources.getSampler().sample();
}
/**
* Benchmark methods for obtaining a sign value from a long.
*
* <p>Note: This is disabled. The branchless versions using a subtraction of
* 2 - 1 or 0 - 1 are faster.
*
* @param sources Source of randomness.
* @return the sample value
*/
//@Benchmark
public double signBit(SignBitSources sources) {
return sources.getSampler().sample();
}
/**
* Benchmark methods for obtaining {@code exp(z)} when {@code -8 <= z <= 0}.
*
* <p>Note: This is disabled. On JDK 8 FastMath is faster. On JDK 11 Math.exp is
* a hotspot intrinsic and may be faster based on the platform architecture.
* Example results:
*
* <pre>
* JDK 8 JDK 11
* i7-6820HQ E5-1680 i7-6820HQ E5-1680 3960X
* noop 4.523 3.871 4.351 4.206 3.936
* Math.exp 61.350 48.519 22.552 19.195 12.568
* FastMath.exp 33.858 24.469 31.396 24.568 11.159
* </pre>
*
* <p>The ziggurat sampler avoids calls to Math.exp in the majority of cases
* when using McFarland's fast method for overhangs which exploit the known
* maximum difference between pdf(x) and the triangle hypotenuse. Example data
* of the frequency that Math.exp is called per sample from the base
* implementations (using n=2^31):
*
* <pre>
* Calls FastOverhangs SimpleOverhangs
* Exp exp(-x) 0.00271 0.0307
* Normal exp(-0.5*x*x) 0.00359 0.0197*
*
* * Increases to 0.0198 if the tail exponential sampler uses simple overhangs
* </pre>
*
* <p>Note that the maximum difference between pdf(x) and the triangle
* hypotenuse is smaller for the exponential distribution; thus the fast method
* can avoid using Math.exp more often. In the case of simple overhangs the
* normal distribution has more region covered by the ziggurat thus has a lower
* frequency of overhang sampling.
*
* <p>A significant speed-up of the exp function may improve run-time of the
* simple overhangs method. Any effect on the fast method is less likely to be
* noticed. This is observed in benchmark times which show an improvement of the
* simple overhangs method for the exponential relative to other methods on JDK
* 11 vs 8. However the simple overhangs is still slower as the exponential
* distribution is always concave and upper-right triangle samples are avoided
* by the fast overhang method.
*
* @param sources Source of randomness.
* @return the sample value
*/
//@Benchmark
public double exp(ExpSources sources) {
return sources.getSampler().sample();
}
/**
* Run the sampler.
*
* @param sources Source of randomness.
* @return the sample value
*/
@Benchmark
public double sample(SingleSources sources) {
return sources.getSampler().sample();
}
/**
* Run the sampler to generate a number of samples sequentially.
*
* @param sources Source of randomness.
* @return the sample value
*/
@Benchmark
public double sequentialSample(SequentialSources sources) {
return sources.getSampler().sample();
}
}