src/java.base/share/classes/java/util/random/package-info.java - master - githubmirror/jdk17u-dev

src/java.base/share/classes/java/util/random/package-info.java
633 строки35 KB
Goetz Lindenmaier
8272866: java.util.random package summary contains incorrect mixing function in table
28 фев 2022, 14:03
28 фев 2022, 14:0390d83a2
/*
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 * DO NOT ALTER OR REMOVE COPYRIGHT NOTICES OR THIS FILE HEADER.
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 *
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 *
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/**
 * This package contains classes and interfaces that support a generic API
 * for random number generation.
 *
 * <p>These classes and interfaces support the definition and use of "random
 * generators", a term covering what have traditionally been called "random
 * number generators" as well as generators of other sorts of randomly chosen
 * values (eg. booleans). These classes and interfaces cover not only
 * deterministic (pseudorandom) algorithms but also generators of values that
 * use some "truly random" physical source (stochastic algorithms perhaps making
 * use of thermal noise, for example, or quantum-mechanical effects).
 *
 * <p> The principal interface is {@link RandomGenerator}, which provides
 * methods for requesting individual values of type {@code int}, {@code long},
 * {@code float}, {@code double}, or {@code boolean} chosen pseudorandomly
 * from a uniform distribution; methods for requesting values of type
 * {@code double} chosen pseudorandomly from a normal distribution or from an
 * exponential distribution; and methods for creating streams of values of type
 * {@code int}, {@code long}, or {@code double} chosen pseudorandomly from a
 * uniform distribution (such streams are spliterator-based, allowing for
 * parallel processing of their elements). There are also static factory methods
 * for creating an instance of a specific random number generator algorithm
 * given its name.
 *
 * <p> The principal supporting class is {@link RandomGeneratorFactory}. This
 * can be used to generate multiple random number generators for a specific
 * algorithm. {@link RandomGeneratorFactory} also provides methods for
 * selecting random number generator algorithms. RandomGeneratorFactory
 * registers implementations of {@link RandomGenerator} interface using the
 * service provider API.
 *
 * <p> An important subsidiary interface is
 * {@link RandomGenerator.StreamableGenerator}, which provides methods for
 * creating spliterator-based streams of {@link RandomGenerator} objects,
 * allowing for parallel processing of these objects using multiple threads.
 * Unlike {@link java.util.Random}, most implementations of
 * {@link RandomGenerator} are <i>not</i> thread-safe. The intent is that
 * instances should not be shared among threads; rather, each thread should have
 * its own random generator(s) to use. The various pseudorandom algorithms
 * provided by this package are designed so that multiple instances will (with
 * very high probability) behave as if statistically independent.
 *
 * <p> For many purposes, these are the only two interfaces that a consumer of
 * pseudorandom values will need. There are also some more specialized
 * interfaces that describe more specialized categories of random number
 * generators {@link RandomGenerator.SplittableGenerator SplittableGenerator},
 * {@link RandomGenerator.JumpableGenerator JumpableGenerator},
 * {@link RandomGenerator.LeapableGenerator LeapableGenerator}, and
 * {@link RandomGenerator.ArbitrarilyJumpableGenerator ArbitrarilyJumpableGenerator}
 * that have specific strategies for creating statistically independent instances.
 *
 * <h2>Using the Random Number Generator Interfaces</h2>
 *
 * To get started, an application should first create one instance of a
 * generator class. Assume that the contents of the package
 * {@link java.util.random} has been imported:
 *
 * <blockquote>{@code import java.util.random.*;}</blockquote>
 *
 * Then one can choose a specific implementation by giving the name of a generator
 * algorithm to the static method {@link RandomGenerator#of}, in which case the
 * no-arguments constructor for that implementation is used:
 *
 * <blockquote>{@code RandomGenerator g = RandomGenerator.of("L64X128MixRandom");}</blockquote>
 *
 * For a single-threaded application, this is all that is needed. One can then
 * invoke methods of {@code g} such as
 * {@link RandomGenerator#nextLong nextLong()},
 * {@link RandomGenerator#nextInt nextInt()},
 * {@link RandomGenerator#nextFloat nextFloat()},
 * {@link RandomGenerator#nextDouble nextDouble()} and
 * {@link RandomGenerator#nextBoolean nextBoolean()} to generate individual
 * randomly chosen values. One can also use the methods
 * {@link RandomGenerator#ints ints()}, {@link RandomGenerator#longs longs()}
 * and {@link RandomGenerator#doubles doubles()} to create streams of randomly
 * chosen values. The methods
 * {@link RandomGenerator#nextGaussian nextGaussian()} and
 * {@link RandomGenerator#nextExponential nextExponential()} draw floating-point
 * values from nonuniform distributions.
 *
 * <p> For a multi-threaded application, one can repeat the preceding steps
 * to create additional {@linkplain RandomGenerator RandomGenerators}, but
 * often it is preferable to use methods of the one single initially
 * created generator to create others like it. (One reason is that some
 * generator algorithms, if asked to create a new set of generators all at
 * once, can make a special effort to ensure that the new generators are
 * statistically independent.) If the initial generator implements the
 * interface {@link RandomGenerator.StreamableGenerator}, then the method
 * {@link RandomGenerator.StreamableGenerator#rngs rngs()} can be used to
 * create a stream of generators. If this is a parallel stream, then it is
 * easy to get parallel execution by using the
 * {@link java.util.stream.Stream#map map()} method on the stream.
 * <p> For a multi-threaded application that forks new threads dynamically,
 * another approach is to use an initial generator that implements the interface
 * {@link RandomGenerator.SplittableGenerator}, which is then considered to
 * "belong" to the initial thread for its exclusive use; then whenever any
 * thread needs to fork a new thread, it first uses the
 * {@link RandomGenerator.SplittableGenerator#split split()} method of its own
 * generator to create a new generator, which is then passed to the newly
 * created thread for exclusive use by that new thread.
 *
 *
 * <h2>Choosing a Random Number Generator Algorithm</h2>
 *
 * <p> There are three groups of random number generator algorithm provided
 * in Java: the Legacy group, the LXM group, and the Xoroshiro/Xoshiro group.
 *
 * <p> The legacy group includes random number generators that existed
 * before JDK 17: Random, ThreadLocalRandom, SplittableRandom, and
 * SecureRandom. Random (LCG) is the weakest of the available algorithms, and it
 * is recommended that users migrate to newer algorithms. If an application
 * requires a random number generator algorithm that is cryptographically
 * secure, then it should continue to use an instance of the class {@link
 * java.security.SecureRandom}.
 *
 * <p> The algorithms in the LXM group are similar to each other. The parameters
 * of each algorithm can be found in the algorithm name. The number after "L" indicates the
 * number of state bits for the LCG subgenerator, and the number after "X" indicates the
 * number of state bits for the XBG subgenerator. "Mix" indicates that
 * the algorithm uses an 8-operation bit-mixing function; "StarStar" indicates use
 * of a 3-operation bit-scrambler.
 *
 * <p> The algorithms in the Xoroshiro/Xoshiro group are more traditional algorithms
 * (see David Blackman and Sebastiano Vigna, "Scrambled Linear Pseudorandom
 * Number Generators," ACM Transactions on Mathematical Software, 2021);
 * the number in the name indicates the number of state bits.
 *
 * <p> For applications (such as physical simulation, machine learning, and
 * games) that do not require a cryptographically secure algorithm, this package
 * provides multiple implementations of interface {@link RandomGenerator} that
 * provide trade-offs among speed, space, period, accidental correlation, and
 * equidistribution properties.
 *
 * <p> For applications with no special requirements,
 * {@code L64X128MixRandom} has a good balance among speed, space,
 * and period, and is suitable for both single-threaded and multi-threaded
 * applications when used properly (a separate instance for each thread).
 *
 * <p> If the application uses only a single thread, then
 * {@code Xoroshiro128PlusPlus} is even smaller and faster, and
 * certainly has a sufficiently long period.
 *
 * <p> For an application running in a 32-bit hardware environment and using
 * only one thread or a small number of threads, {@code L32X64MixRandom} may be a good
 * choice.
 *
 * <p> For an application that uses many threads that are allocated in one batch
 * at the start of the computation, either a "jumpable" generator such as
 * {@code Xoroshiro128PlusPlus} or
 * {@code Xoshiro256PlusPlus} may be used, or a "splittable"
 * generator such as {@code L64X128MixRandom} or
 * {@code L64X256MixRandom} may be used.
 *
 * <p> For an application that creates many threads dynamically, perhaps through
 * the use of spliterators, a "splittable" generator such as
 * {@code L64X128MixRandom} or {@code L64X256MixRandom} is
 * recommended. If the number of generators created dynamically may
 * be very large (millions or more), then using generators such as
 * {@code L128X128MixRandom} or {@code L128X256MixRandom},
 * which use a 128-bit parameter rather than a 64-bit parameter for their LCG
 * subgenerator, will make it much less likely that two instances use the same
 * state cycle.
 *
 * <p> For an application that uses tuples of consecutively generated values, it
 * may be desirable to use a generator that is <i>k</i>-equidistributed such
 * that <i>k</i> is at least as large as the length of the tuples being
 * generated. The generator {@code L64X256MixRandom} is provably
 * 4-equidistributed, and {@code L64X1024MixRandom} is provably
 * 16-equidistributed.
 *
 * <p> For applications that generate large permutations, it may be best to use
 * a generator whose period is much larger than the total number of possible
 * permutations; otherwise it will be impossible to generate some of the
 * intended permutations. For example, if the goal is to shuffle a deck of 52
 * cards, the number of possible permutations is 52! (52 factorial), which is
 * larger than 2<sup>225</sup> (but smaller than 2<sup>226</sup>), so it may be
 * best to use a generator whose period at least 2<sup>256</sup>, such as
 * {@code L64X256MixRandom} or {@code L64X1024MixRandom}
 * or {@code L128X256MixRandom} or
 * {@code L128X1024MixRandom}. (It is of course also necessary to
 * provide sufficiently many seed bits when the generator is initialized, or
 * else it will still be impossible to generate some of the intended
 * permutations.)
 *
 *
 * <h2><a id="algorithms">Random Number Generator Algorithms Available</a></h2>
 *
 * These algorithms [in the table below] must be found with the current version
 * of Java SE. A particular JDK implementation may recognize additional
 * algorithms; check the JDK's documentation for details. The set of algorithms
 * required by Java SE may be updated by changes to the Java SE specification.
 * Over time, new algorithms may be added and old algorithms may be removed.
 * <p>In addition, as another life-cycle phase, an algorithm may be {@linkplain
 * RandomGeneratorFactory#isDeprecated() deprecated}. A deprecated algorithm is
 * not recommended for use. If a required algorithm is deprecated, it may be
 * removed in a future release. Due to advances in random number generator
 * algorithm development and analysis, an algorithm may be deprecated during the
 * lifetime of a particular Java SE release. Changing the deprecation status of
 * an algorithm is <em>not</em> a specification change.
 *
 * <table style="padding:0px 20px 0px 0px">
 *  <caption>Available Algorithms</caption>
 *  <thead>
 *  <tr>
 *      <th style="text-align:left">Algorithm</th>
 *      <th style="text-align:left">Group</th>
 *      <th style="text-align:left">Period</th>
 *      <th style="text-align:right">StateBits</th>
 *      <th style="text-align:right">Equidistribution</th>
 *  </tr>
 *  </thead>
 *  <tbody>
 *  <tr>
 *      <td style="text-align:left">L128X1024MixRandom</td>
 *      <td style="text-align:left">LXM</td>
 *      <td style="text-align:left">BigInteger.ONE.shiftLeft(1024).subtract(BigInteger.ONE).shiftLeft(128)</td>
 *      <td style="text-align:right">1152</td>
 *      <td style="text-align:right">1</td>
 *  </tr>
 *  <tr>
 *      <td style="text-align:left">L128X128MixRandom</td>
 *      <td style="text-align:left">LXM</td>
 *      <td style="text-align:left">BigInteger.ONE.shiftLeft(128).subtract(BigInteger.ONE).shiftLeft(128)</td>
 *      <td style="text-align:right">256</td>
 *      <td style="text-align:right">1</td>
 *  </tr>
 *  <tr>
 *      <td style="text-align:left">L128X256MixRandom</td>
 *      <td style="text-align:left">LXM</td>
 *      <td style="text-align:left">BigInteger.ONE.shiftLeft(256).subtract(BigInteger.ONE).shiftLeft(128)</td>
 *      <td style="text-align:right">384</td>
 *      <td style="text-align:right">1</td>
 *  </tr>
 *  <tr>
 *      <td style="text-align:left">L32X64MixRandom</td>
 *      <td style="text-align:left">LXM</td>
 *      <td style="text-align:left">BigInteger.ONE.shiftLeft(64).subtract(BigInteger.ONE).shiftLeft(32)</td>
 *      <td style="text-align:right">96</td>
 *      <td style="text-align:right">1</td>
 *  </tr>
 *  <tr>
 *      <td style="text-align:left">L64X1024MixRandom</td>
 *      <td style="text-align:left">LXM</td>
 *      <td style="text-align:left">BigInteger.ONE.shiftLeft(1024).subtract(BigInteger.ONE).shiftLeft(64)</td>
 *      <td style="text-align:right">1088</td>
 *      <td style="text-align:right">16</td>
 *  </tr>
 *  <tr>
 *      <td style="text-align:left">L64X128MixRandom</td>
 *      <td style="text-align:left">LXM</td>
 *      <td style="text-align:left">BigInteger.ONE.shiftLeft(128).subtract(BigInteger.ONE).shiftLeft(64)</td>
 *      <td style="text-align:right">192</td>
 *      <td style="text-align:right">2</td>
 *  </tr>
 *  <tr>
 *      <td style="text-align:left">L64X128StarStarRandom</td>
 *      <td style="text-align:left">LXM</td>
 *      <td style="text-align:left">BigInteger.ONE.shiftLeft(128).subtract(BigInteger.ONE).shiftLeft(64)</td>
 *      <td style="text-align:right">192</td>
 *      <td style="text-align:right">2</td>
 *  </tr>
 *  <tr>
 *      <td style="text-align:left">L64X256MixRandom</td>
 *      <td style="text-align:left">LXM</td>
 *      <td style="text-align:left">BigInteger.ONE.shiftLeft(256).subtract(BigInteger.ONE).shiftLeft(64)</td>
 *      <td style="text-align:right">320</td>
 *      <td style="text-align:right">4</td>
 *  </tr>
 *  <tr>
 *      <td style="text-align:left">Random</td>
 *      <td style="text-align:left">Legacy</td>
 *      <td style="text-align:left">BigInteger.ONE.shiftLeft(48)</td>
 *      <td style="text-align:right">48</td>
 *      <td style="text-align:right">0</td>
 *  </tr>
 *  <tr>
 *      <td style="text-align:left">SplittableRandom</td>
 *      <td style="text-align:left">Legacy</td>
 *      <td style="text-align:left">BigInteger.ONE.shiftLeft(64)</td>
 *      <td style="text-align:right">64</td>
 *      <td style="text-align:right">1</td>
 *  </tr>
 *  <tr>
 *      <td style="text-align:left">ThreadLocalRandom <sup>*</sup></td>
 *      <td style="text-align:left">Legacy</td>
 *      <td style="text-align:left">BigInteger.ONE.shiftLeft(64)</td>
 *      <td style="text-align:right">64</td>
 *      <td style="text-align:right">1</td>
 *  </tr>
 *  <tr>
 *      <td style="text-align:left">Xoroshiro128PlusPlus</td>
 *      <td style="text-align:left">Xoroshiro</td>
 *      <td style="text-align:left">BigInteger.ONE.shiftLeft(128).subtract(BigInteger.ONE)</td>
 *      <td style="text-align:right">128</td>
 *      <td style="text-align:right">1</td>
 *  </tr>
 *  <tr>
 *      <td style="text-align:left">Xoshiro256PlusPlus</td>
 *      <td style="text-align:left">Xoshiro</td>
 *      <td style="text-align:left">BigInteger.ONE.shiftLeft(256).subtract(BigInteger.ONE)</td>
 *      <td style="text-align:right">256</td>
 *      <td style="text-align:right">3</td>
 *  </tr>
 *  </tbody>
 * </table>
 *
 * <p><sup>*</sup> ThreadLocalRandom can only be accessed via
 * {@link java.util.concurrent.ThreadLocalRandom#current()}.
 *
 * <h2>Categories of Random Number Generator Algorithms</h2>
 *
 * Historically, most pseudorandom generator algorithms have been based on some
 * sort of finite-state machine with a single, large cycle of states; when it is
 * necessary to have multiple threads use the same algorithm simultaneously, the
 * usual technique is to arrange for each thread to traverse a different region
 * of the state cycle. These regions may be doled out to threads by starting
 * with a single initial state and then using a "jump function" that travels a
 * long distance around the cycle (perhaps 2<sup>64</sup> steps or more); the
 * jump function is applied repeatedly and sequentially, to identify widely
 * spaced states that are then doled out, one to each thread, to serve as the
 * initial state for the generator to be used by that thread. This strategy is
 * supported by the interface {@link RandomGenerator.JumpableGenerator}.
 * Sometimes it is desirable to support two levels of jumping (by long distances
 * and by <i>really</i> long distances); this strategy is supported by the
 * interface {@link RandomGenerator.LeapableGenerator}. There is also an interface
 * {@link RandomGenerator.ArbitrarilyJumpableGenerator} for algorithms that allow
 * jumping along the state cycle by any user-specified distance. In this package,
 * implementations of these interfaces include
 * "Xoroshiro128PlusPlus", and
 * "Xoshiro256PlusPlus".
 *
 * <p> A more recent category of "splittable" pseudorandom generator algorithms
 * uses a large family of state cycles and makes some attempt to ensure that
 * distinct instances use different state cycles; but even if two instances
 * "accidentally" use the same state cycle, they are highly likely to traverse
 * different regions parts of that shared state cycle. This strategy is
 * supported by the interface {@link RandomGenerator.SplittableGenerator}.
 * In this package, implementations of this interface include
 * "L32X64MixRandom",
 * "L64X128StarStarRandom",
 * "L64X128MixRandom",
 * "L64X256MixRandom",
 * "L64X1024MixRandom",
 * "L128X128MixRandom",
 * "L128X256MixRandom", and
 * "L128X1024MixRandom"; note that the class
 * {@link java.util.SplittableRandom} also implements this interface.
 *
 *
 * <h2>The LXM Family of Random Number Generator Algorithms</h2>
 *
 * The structure of the central nextLong (or nextInt) method of an LXM
 * algorithm follows a suggestion in December 2017 by Sebastiano Vigna
 * that using one Linear Congruential Generator (LCG) as a first subgenerator
 * and one Xor-Based Generator (XBG) as a second subgenerator (rather
 * than using two LCG subgenerators) would provide a longer period, superior
 * equidistribution, scalability, and better quality.  Each of the
 * specific implementations here combines one of the best currently known
 * XBG algorithms (xoroshiro128 or xoshiro256, described by Blackman and
 * Vigna in "Scrambled Linear Pseudorandom Number Generators", ACM Transactions
 * on Mathematical Software, 2021) with an LCG that uses one of the best
 * currently known multipliers (found by a search for better multipliers
 * in 2019 by Steele and Vigna), and then applies either a mixing function
 * identified by Doug Lea or a simple scrambler proposed by Blackman and Vigna.
 * Testing has confirmed that the LXM algorithm is far superior in quality to
 * the SplitMix algorithm (2014) used by {@code SplittableRandom}.
 *
 * Each class with a name of the form
 * {@code L}<i>p</i>{@code X}<i>q</i>{@code SomethingRandom}
 * uses some specific member of the LXM family of random number
 * algorithms; "LXM" is short for "LCG, XBG, Mixer". Every LXM
 * generator has two subgenerators; one is an LCG (Linear Congruential
 * Generator) and the other is an XBG (Xor-Based Generator). Each output of an LXM
 * generator is the result of combining state from the LCG with state from the
 * XBG using a Mixing function (and then the state of the LCG
 * and the state of the XBG are advanced).
 *
 * <p> The LCG subgenerator has an update step of the form {@code s = m*s + a},
 * where {@code s}, {@code m}, and {@code a} are all binary integers of the same
 * size, each having <i>p</i> bits; {@code s} is the mutable state, the
 * multiplier {@code m} is fixed (the same for all instances of a class) and the
 * addend {@code a} is a parameter (a final field of the instance). The
 * parameter {@code a} is required to be odd (this allows the LCG to have the
 * maximal period, namely 2<sup><i>p</i></sup>); therefore there are
 * 2<sup><i>p</i>&minus;1</sup> distinct choices of parameter. (When the size of
 * {@code s} is 128 bits, then we use the name "{@code sh}" below to refer to
 * the high half of {@code s}, that is, the high-order 64 bits of {@code s}.)
 *
 * <p> The XBG subgenerator can in principle be any one of a wide variety
 * of XBG algorithms; in this package it is always either
 * {@code xoroshiro128}, {@code xoshiro256}, or {@code xoroshiro1024}, in each
 * case without any final scrambler (such as "+" or "**") because LXM uses
 * a separate Mixer later in the process. The XBG state consists of
 * some fixed number of {@code int} or {@code long} fields, generally named
 * {@code x0}, {@code x1}, and so on, which can take on any values provided that
 * they are not all zero. The collective total size of these fields is <i>q</i>
 * bits; therefore the period of this subgenerator is
 * 2<sup><i>q</i></sup>&minus;1.
 *
 * <p> Because the periods 2<sup><i>p</i></sup> and 2<sup><i>q</i></sup>&minus;1
 * of the two subgenerators are relatively prime, the <em>period</em> of any
 * single instance of an LXM algorithm (the length of the series of generated
 * values before it repeats) is the product of the periods of the subgenerators,
 * that is, 2<sup><i>p</i></sup>(2<sup><i>q</i></sup>&minus;1), which is just
 * slightly smaller than 2<sup>(<i>p</i>+<i>q</i>)</sup>. Moreover, if two
 * distinct instances of the same LXM algorithm have different {@code a}
 * parameters, then their cycles of produced values will be different.
 *
 * <p> Generally speaking, among the "{@code L}<i>p</i>{@code X}<i>q</i>"
 * generators, the memory required for an instance is 2<i>p</i>+<i>q</i> bits.
 * (If <i>q</i> is 1024 or larger, the XBG state is represented as an
 * array, so additional bits are needed for the array object header, and another
 * 32 bits are used for an array index.)
 *
 * <p> Larger values of <i>p</i> imply a lower probability that two distinct
 * instances will traverse the same state cycle, and larger values of <i>q</i>
 * imply that the generator is equidistributed in a larger number of dimensions
 * (this is provably true when <i>p</i> is 64, and conjectured to be
 * approximately true when <i>p</i> is 128). A class with "{@code Mix}" in its
 * name uses a fairly strong mixing function with excellent avalanche
 * characteristics; a class with "{@code StarStar}" in its name uses a weaker
 * but faster mixing function.
 *
 * <p> The specific LXM algorithms used in this package are all chosen so that
 * the 64-bit values produced by the {@link RandomGenerator#nextLong nextLong()}
 * method are exactly equidistributed (for example, for any specific instance of
 * "L64X128MixRandom", over the course of its cycle each of the
 * 2<sup>64</sup> possible {@code long} values will be produced
 * 2<sup>128</sup>&minus;1 times). The values produced by the
 * {@link RandomGenerator#nextInt nextInt()},
 * {@link RandomGenerator#nextFloat nextFloat()}, and
 * {@link RandomGenerator#nextDouble nextDouble()} methods are likewise exactly
 * equidistributed. Some algorithms provide a further guarantee of
 * <i>k</i>-equidistribution for some <i>k</i> greater than 1, meaning that successive
 * non-overlapping <i>k</i>-tuples of 64-bit values produced by the
 * {@link RandomGenerator#nextLong nextLong()} method are exactly
 * equidistributed (equally likely to occur).
 *
 * <p> The following table gives the period, state size (in bits), parameter
 * size (in bits, including the low-order bit that is required always to be a
 * 1-bit), and equidistribution property for each of the specific LXM algorithms
 * used in this package.
 *
 * <table style="padding:0px 20px 0px 0px">
 * <caption>Algorithm Properties</caption>
 * <thead>
 *   <tr><th style="text-align:left">Implementation</th>
 *       <th style="text-align:right">Period</th>
 *       <th style="text-align:right">State size</th>
 *       <th style="text-align:right">Parameter size</th>
 *       <th style="text-align:left">{@link RandomGenerator#nextLong nextLong()} values are</th></tr>
 * </thead>
 * <tbody>
 *   <tr><td style="text-align:left">"L32X64MixRandom"</td>
 *       <td style="text-align:right">2<sup>32</sup>(2<sup>64</sup>&minus;1)</td>
 *       <td style="text-align:right">96 bits</td>
 *       <td style="text-align:right">32 bits</td>
 *       <td style="text-align:left"></td></tr>
 *   <tr><td style="text-align:left">"L64X128StarStarRandom"</td>
 *       <td style="text-align:right">2<sup>64</sup>(2<sup>128</sup>&minus;1)</td>
 *       <td style="text-align:right">192 bits</td>
 *       <td style="text-align:right">64 bits</td>
 *       <td style="text-align:left">2-equidistributed and exactly equidistributed</td></tr>
 *   <tr><td style="text-align:left">"L64X128MixRandom"</td>
 *       <td style="text-align:right">2<sup>64</sup>(2<sup>128</sup>&minus;1)</td>
 *       <td style="text-align:right">192 bits</td>
 *       <td style="text-align:right">64 bits</td>
 *       <td style="text-align:left">2-equidistributed and exactly equidistributed</td></tr>
 *   <tr><td style="text-align:left">"L64X256MixRandom"</td>
 *       <td style="text-align:right">2<sup>64</sup>(2<sup>256</sup>&minus;1)</td>
 *       <td style="text-align:right">320 bits</td>
 *       <td style="text-align:right">64 bits</td>
 *       <td style="text-align:left">4-equidistributed and exactly equidistributed</td></tr>
 *   <tr><td style="text-align:left">"L64X1024MixRandom"</td>
 *       <td style="text-align:right">2<sup>64</sup>(2<sup>1024</sup>&minus;1)</td>
 *       <td style="text-align:right">1088 bits</td>
 *       <td style="text-align:right">64 bits</td>
 *       <td style="text-align:left">16-equidistributed and exactly equidistributed</td></tr>
 *   <tr><td style="text-align:left">"L128X128MixRandom"</td>
 *       <td style="text-align:right">2<sup>128</sup>(2<sup>128</sup>&minus;1)</td>
 *       <td style="text-align:right">256 bits</td>
 *       <td style="text-align:right">128 bits</td>
 *       <td style="text-align:left">exactly equidistributed</td></tr>
 *   <tr><td style="text-align:left">"L128X256MixRandom"</td>
 *       <td style="text-align:right">2<sup>128</sup>(2<sup>256</sup>&minus;1)</td>
 *       <td style="text-align:right">384 bits</td>
 *       <td style="text-align:right">128 bits</td>
 *       <td style="text-align:left">exactly equidistributed</td></tr>
 *   <tr><td style="text-align:left">"L128X1024MixRandom"</td>
 *       <td style="text-align:right">2<sup>128</sup>(2<sup>1024</sup>&minus;1)</td>
 *       <td style="text-align:right">1152 bits</td>
 *       <td style="text-align:right">128 bits</td>
 *       <td style="text-align:left">exactly equidistributed</td></tr>
 * </tbody>
 * </table>
 *
 * For the algorithms listed above whose names begin with {@code L32}, the
 * 32-bit values produced by the {@link RandomGenerator#nextInt nextInt()}
 * method are exactly equidistributed, but the 64-bit values produced by the
 * {@link RandomGenerator#nextLong nextLong()} method are not exactly
 * equidistributed.
 *
 * <p> For the algorithms listed above whose names begin with {@code L64} or
 * {@code L128}, the 64-bit values produced by the
 * {@link RandomGenerator#nextLong nextLong()} method are <i>exactly
 * equidistributed</i>: every instance, over the course of its cycle, will
 * produce each of the 2<sup>64</sup> possible {@code long} values exactly the
 * same number of times. For example, any specific instance of
 * "L64X256MixRandom", over the course of its cycle each of the
 * 2<sup>64</sup> possible {@code long} values will be produced
 * 2<sup>256</sup>&minus;1 times. The values produced by the
 * {@link RandomGenerator#nextInt nextInt()},
 * {@link RandomGenerator#nextFloat nextFloat()}, and
 * {@link RandomGenerator#nextDouble nextDouble()} methods are likewise exactly
 * equidistributed.
 *
 * <p> In addition, for the algorithms listed above whose names begin with
 * {@code L64}, the 64-bit values produced by the
 * {@link RandomGenerator#nextLong nextLong()} method are
 * <i>k</i>-equidistributed (but not exactly <i>k</i>-equidistributed). To be
 * precise, and taking "L64X256MixRandom" as an example: for
 * any specific instance of "L64X256MixRandom", consider the
 * (overlapping) length-4 subsequences of the cycle of 64-bit values produced by
 * {@link RandomGenerator#nextLong nextLong()} (assuming no other methods are
 * called that would affect the state). There are
 * 2<sup>64</sup>(2<sup>256</sup>&minus;1) such subsequences, and each
 * subsequence, which consists of 4 64-bit values, can have one of
 * 2<sup>256</sup> values. Of those 2<sup>256</sup> subsequence values, nearly
 * all of them (2<sup>256</sup>&minus;2<sup>64</sup>) occur 2<sup>64</sup> times
 * over the course of the entire cycle, and the other 2<sup>64</sup> subsequence
 * values occur only 2<sup>64</sup>&minus;1 times. So the ratio of the
 * probability of getting any specific one of the less common subsequence values
 * and the probability of getting any specific one of the more common
 * subsequence values is 1&minus;2<sup>-64</sup>. (Note that the set of
 * 2<sup>64</sup> less-common subsequence values will differ from one instance
 * of "L64X256MixRandom" to another, as a function of the
 * additive parameter of the LCG.) The values produced by the
 * {@link RandomGenerator#nextInt nextInt()},
 * {@link RandomGenerator#nextFloat nextFloat()}, and
 * {@link RandomGenerator#nextDouble nextDouble()} methods are likewise
 * 4-equidistributed (but not exactly 4-equidistributed).
 *
 * <p> The next table gives the LCG multiplier value, the name of the specific
 * XBG algorithm used, the specific numeric parameters for that XBG
 * algorithm, and the mixing function for each of the specific LXM algorithms
 * used in this package. (Note that the multiplier used for the 128-bit LCG
 * cases is 65 bits wide, so the constant {@code 0x1d605bbb58c8abbfdL} shown in
 * the table cannot actually be used in code; instead, only the 64 low-order
 * bits {@code 0xd605bbb58c8abbfdL} are represented in the source code, and the
 * missing 1-bit is handled through special coding of the multiply-add algorithm
 * used in the LCG.)
 *
 * <table style="padding:0px 20px 0px 0px">
 * <caption>LXM Multipliers</caption>
 * <thead>
 *   <tr><th style="text-align:left">Implementation</th>
 *       <th style="text-align:right">LCG multiplier {@code m}</th>
 *       <th style="text-align:left">XBG algorithm</th>
 *       <th style="text-align:left">XBG parameters</th>
 *       <th style="text-align:left">Mixing function</th></tr>
 * </thead>
 * <tbody>
 *   <tr><td style="text-align:left">"L32X64MixRandom"</td>
 *       <td style="text-align:right">{@code 0xadb4a92d}</td>
 *       <td style="text-align:left">{@code xoroshiro64}, version 1.0</td>
 *       <td style="text-align:left">{@code (26, 9, 13)}</td>
 *       <td style="text-align:left">mixLea32{@code (s+x0)}</td></tr>
 *   <tr><td style="text-align:left">"L64X128StarStarRandom" </td>
 *       <td style="text-align:right">{@code 0xd1342543de82ef95L}</td>
 *       <td style="text-align:left">{@code xoroshiro128}, version 1.0</td>
 *       <td style="text-align:left">{@code (24, 16, 37)}</td>
 *       <td style="text-align:left">{@code Long.rotateLeft((s+x0)* 5, 7) * 9}</td></tr>
 *   <tr><td style="text-align:left">"L64X128MixRandom"</td>
 *       <td style="text-align:right">{@code 0xd1342543de82ef95L}</td>
 *       <td style="text-align:left">{@code xoroshiro128}, version 1.0</td>
 *       <td style="text-align:left">{@code (24, 16, 37)}</td>
 *       <td style="text-align:left">mixLea64{@code (s+x0)}</td></tr>
 *   <tr><td style="text-align:left">"L64X256MixRandom"</td>
 *       <td style="text-align:right">{@code 0xd1342543de82ef95L}</td>
 *       <td style="text-align:left">{@code xoshiro256}, version 1.0</td>
 *       <td style="text-align:left">{@code (17, 45)}</td>
 *       <td style="text-align:left">mixLea64{@code (s+x0)}</td></tr>
 *   <tr><td style="text-align:left">"L64X1024MixRandom"</td>
 *       <td style="text-align:right">{@code 0xd1342543de82ef95L}</td>
 *       <td style="text-align:left">{@code xoroshiro1024}, version 1.0</td>
 *       <td style="text-align:left">{@code (25, 27, 36)}</td>
 *       <td style="text-align:left">mixLea64{@code (s+x0)}</td></tr>
 *   <tr><td style="text-align:left">"L128X128MixRandom"</td>
 *       <td style="text-align:right">{@code 0x1d605bbb58c8abbfdL}</td>
 *       <td style="text-align:left">{@code xoroshiro128}, version 1.0</td>
 *       <td style="text-align:left">{@code (24, 16, 37)}</td>
 *       <td style="text-align:left">mixLea64{@code (sh+x0)}</td></tr>
 *   <tr><td style="text-align:left">"L128X256MixRandom"</td>
 *       <td style="text-align:right">{@code 0x1d605bbb58c8abbfdL}</td>
 *       <td style="text-align:left">{@code xoshiro256}, version 1.0</td>
 *       <td style="text-align:left">{@code (17, 45)}</td>
 *       <td style="text-align:left">mixLea64{@code (sh+x0)}</td></tr>
 *   <tr><td style="text-align:left">"L128X1024MixRandom"</td>
 *       <td style="text-align:right">{@code 0x1d605bbb58c8abbfdL}</td>
 *       <td style="text-align:left">{@code xoroshiro1024}, version 1.0</td>
 *       <td style="text-align:left">{@code (25, 27, 36)}</td>
 *       <td style="text-align:left">mixLea64{@code (sh+x0)}</td></tr>
 * </tbody>
 * </table>
 *
 * @since   17
 */
package java.util.random;