Report a bug
If you spot a problem with this page, click here to create a Github issue.
Improve this page
Quickly fork, edit online, and submit a pull request for this page. Requires a signed-in GitHub account. This works well for small changes. If you'd like to make larger changes you may want to consider using a local clone.

mir.random.engine.xorshift

Generators

Generator name Description
Xoshiro256StarStar xoshiro256**: all-purpose, rock-solid generator
Xoshiro128StarStar_32 xoshiro128** (32-bit): 32-bit-oriented parameterization of xoshiro**
Xoroshiro128Plus xoroshiro128+: fast, small, and high-quality
Xorshift1024StarPhi xorshift1024*φ: when something larger than xoroshiro128+ is needed
Xorshift64Star32 xorshift64*/32: internal state of 64 bits and output of 32 bits
Xorshift32 .. Xorshift160 Basic xorshift generator with n bits of state (32, 64, 96, 128, 160)
Xorshift192 Generator from Marsaglia's paper combining 160-bit xorshift with a counter
Xorshift An alias to one of the generators in this package

Generic Templates

Template name Description
XoshiroEngine xoshiro** generator.
XorshiftStarEngine xorshift* generator with any word size and any number of bits of state.
XorshiftEngine xorshift generator with any word size and any number of bits of state.
Authors:
Masahiro Nakagawa, Ilya Yaroshenko (rework), Nathan Sashihara
struct XorshiftEngine(UIntType, uint bits, int sa, int sb, int sc) if (isUnsigned!UIntType);
Xorshift generator. Implemented according to Xorshift RNGs (Marsaglia, 2003) with Sebastino Vigna's optimization for large arrays.
Period is 2 ^^ bits - 1 except for a legacy 192-bit uint version (see note below).
Parameters:
UIntType Word size of this xorshift generator and the return type of opCall.
bits The number of bits of state of this generator. This must be a positive multiple of the size in bits of UIntType. If bits is large this struct may occupy slightly more memory than this so it can use a circular counter instead of shifting the entire array.
sa The direction and magnitude of the 1st shift. Positive means left, negative means right.
sb The direction and magnitude of the 2nd shift. Positive means left, negative means right.
sc The direction and magnitude of the 3rd shift. Positive means left, negative means right.

Note: For historical compatibility when bits == 192 and UIntType is uint a legacy hybrid PRNG is used consisting of a 160-bit xorshift combined with a 32-bit counter. This combined generator has period equal to the least common multiple of 2 ^^ 160 - 1 and 2 ^^ 32.

enum auto isRandomEngine;
Marker for mir.random.isRandomEngine
enum UIntType max;
Largest generated value.
pure nothrow @nogc @safe this()(UIntType x0)
if (UIntType.sizeof > (uint).sizeof);

pure nothrow @nogc @safe this()(uint x0)
if (UIntType.sizeof <= (uint).sizeof);
Constructs a XorshiftEngine generator seeded with x0.
pure nothrow @nogc @safe UIntType opCall();
Advances the random sequence.
alias Xorshift32 = XorshiftEngine!(uint, 32u, 13, -17, 15).XorshiftEngine;

alias Xorshift64 = XorshiftEngine!(uint, 64u, 10, -13, -10).XorshiftEngine;

alias Xorshift96 = XorshiftEngine!(uint, 96u, 10, -5, -26).XorshiftEngine;

alias Xorshift128 = XorshiftEngine!(uint, 128u, 11, -8, -19).XorshiftEngine;

alias Xorshift160 = XorshiftEngine!(uint, 160u, 2, -1, -4).XorshiftEngine;

alias Xorshift192 = XorshiftEngine!(uint, 192u, -2, 1, 4).XorshiftEngine;

alias Xorshift = XorshiftEngine!(uint, 128u, 11, -8, -19).XorshiftEngine;
Define XorshiftEngine generators with well-chosen parameters for 32-bit architectures. Xorshift is an alias of one of the generators in this module.
Examples:
import mir.random.engine;
auto rnd = Xorshift(cast(uint)unpredictableSeed);
auto num = rnd();

import std.traits;
static assert(is(ReturnType!rnd == uint));
static assert(isSaturatedRandomEngine!Xorshift);
struct XorshiftStarEngine(StateUInt, uint nbits, int sa, int sb, int sc, StateUInt multiplier, OutputUInt = StateUInt) if (isUnsigned!StateUInt && isUnsigned!OutputUInt && OutputUInt.sizeof <= StateUInt.sizeof && !(sa > 0 && sb > 0 && sc > 0));

template XorshiftStarEngine(StateUInt, uint nbits, int sa, int sb, int sc, StateUInt multiplier, OutputUInt = StateUInt) if (isUnsigned!StateUInt && isUnsigned!OutputUInt && OutputUInt.sizeof <= StateUInt.sizeof && (sa > 0 && sb > 0 && sc > 0))
Template for the xorshift* family of generators (Vigna, 2016; draft 2014).
xorshift* generators are very fast, high-quality PRNGs (pseudorandom number generators) obtained by scrambling the output of a Marsaglia xorshift generator with a 64-bit invertible multiplier (as suggested by Marsaglia in his paper). They are an excellent choice for all non-cryptographic applications, as they are incredibly fast, have long periods and their output passes strong statistical test suites.
Parameters:
StateUInt Word size of this xorshift generator.
nbits The number of bits of state of this generator. This must be a positive multiple of the size in bits of UIntType. If nbits is large this struct may occupy slightly more memory than this so it can use a circular counter instead of shifting the entire array.
sa The direction and magnitude of the 1st shift. Positive means left, negative means right.
sb The direction and magnitude of the 2nd shift. Positive means left, negative means right.
sc The direction and magnitude of the 3rd shift. Positive means left, negative means right.
multiplier Output of the internal xorshift engine is multiplied by a constant to eliminate linear artifacts except in the low-order bits. This constant must be an odd number other than 1.
OutputUInt Return type of opCall. By default same as StateUInt but can be set to a narrower unsigned type in which case the high bits of the multiplication result are returned.

Note: If sa, sb, and sc are all positive (which if interpreted as same-direction shifts could not result in a full-period xorshift generator) the shift directions are instead implicitly right-left-right when bits == UIntType.sizeof * 8 and in all other cases left-right-right. This maintains full compatibility with older versions of XorshiftStarEngine that took all shifts as unsigned magnitudes.

enum auto isRandomEngine;
Marker for mir.random.isRandomEngine
enum OutputUInt max;
Largest generated value.
enum bool preferHighBits;
Note that when StateUInt is the same size as OutputUInt the two lowest bits of this generator are LFSRs, and thus will fail binary rank tests. To provide some context, every bit of a Mersenne Twister generator (either the 32-bit or 64-bit variant) is an LFSR.
The rand!T functions in mir.random automatically will discard the low bits when generating output smaller than OutputUInt due to this generator having preferHighBits defined true.
pure nothrow @nogc @safe this()(StateUInt x0);
Constructs a XorshiftStarEngine generator seeded with x0.
pure nothrow @nogc @safe OutputUInt opCall()();
Advances the random sequence.
pure nothrow @nogc @safe void jump()()
if (nbits == 1024 && N == 16 && sa == 31 && sb == -11 && sc == -30);
This is the jump function for the standard 1024-bit generator. It is equivalent to 2 ^^ 512 invocations of opCall(); it can be used to generate 2 ^^ 512 non-overlapping subsequences for parallel computations. This function will only be defined if the shifts are the same as for Xorshift1024StarPhi.
alias Xorshift1024StarPhi = XorshiftStarEngine!(ulong, 1024u, 31, -11, -30, 11400714819323198483LU, ulong).XorshiftStarEngine;
Define XorshiftStarEngine with well-chosen parameters for large simulations on 64-bit machines.
Period of (2 ^^ 1024) - 1, 16-dimensionally equidistributed, and faster and statistically superior to Mt19937_64 while occupying significantly less memory. This generator is recommended for random number generation on 64-bit systems except when 1024 + 32 bits of state are excessive.
As described by Vigna in the 2014 draft of his paper published in 2016 detailing the xorshift* family, except with a better multiplier recommended by the author as of 2017-10-08.
Public domain reference implementation: xoroshiro.di.unimi.it/xorshift1024star.c.
Examples:
import mir.random.engine : EngineReturnType, isSaturatedRandomEngine;
auto rnd = Xorshift1024StarPhi(12434UL);
auto num = rnd();
assert(num != rnd());

static assert(is(EngineReturnType!Xorshift1024StarPhi == ulong));
static assert(isSaturatedRandomEngine!Xorshift1024StarPhi);

//Xorshift1024StarPhi has a jump function that is equivalent
//to 2 ^^ 512 invocations of opCall.
rnd.jump();
num = rnd();
assert(num != rnd());
alias Xorshift64Star32 = XorshiftStarEngine!(ulong, 64u, -12, 25, -27, 2685821657736338717LU, uint).XorshiftStarEngine;
Generates 32 bits of output from 64 bits of state. A fast generator with excellent statistical properties for memory-constrained situations where more than 64 bits of state would be too much and generating only 32 bits with each opCall will not cause a slowdown. If you need a generator with 64 bits of state that produces output 64 bits at a time SplitMix64 is an option.
Note that xorshift64*/32 is slower than xorshift1024* even when only 32 bits of output are needed at a time. Per Vigna:
The three xor/shifts of a xorshift64* generator must be executed sequentially, as each one is dependent on the result of the previous one. In a xorshift1024* generator two of the xor/shifts are completely independent and can be parallelized internally by the CPU.
Public domain xorshift64* reference implementation (Internet Archive).
Examples:
import mir.random.engine : isSaturatedRandomEngine;
static assert(isSaturatedRandomEngine!Xorshift64Star32);
Xorshift64Star32 rnd = Xorshift64Star32(123456789);
uint x = rnd();
assert(x == 3988833114);
struct Xoroshiro128Plus;
xoroshiro128+ (XOR/rotate/shift/rotate) generator. 64 bit output. 128 bits of state. Period of (2 ^^ 128) - 1.
Created in 2016 by David Blackman and Sebastiano Vigna as the successor to Vigna's extremely popular xorshift128+ generator used in the JavaScript engines of Google Chrome, Mozilla Firefox, Safari, and Microsoft Edge. From the authors:
This is the successor to xorshift128+. It is the fastest full-period generator passing BigCrush without systematic failures, but due to the relatively short period it is acceptable only for applications with a mild amount of parallelism; otherwise, use a xorshift1024* generator.
Beside passing BigCrush, this generator passes the PractRand test suite up to (and included) 16TB, with the exception of binary rank tests, as the lowest bit of this generator is an LFSR. The next bit is not an LFSR, but in the long run it will fail binary rank tests, too. The other bits have no LFSR artifacts.
We suggest to use a sign test to extract a random Boolean value, and right shifts to extract subsets of bits.
Public domain reference implementation: xoroshiro.di.unimi.it/xoroshiro128plus.c.
Examples:
import mir.random.engine : isSaturatedRandomEngine;
static assert(isSaturatedRandomEngine!Xoroshiro128Plus);
auto gen = Xoroshiro128Plus(1234u);//Seed with constant.
assert(gen() == 5968561782418604543);//Generate number.
foreach (i; 0 .. 8)
    gen();
assert(gen() == 8335647863237943914uL);
//Xoroshiro128Plus has a jump function that is equivalent
//to 2 ^^ 64 invocations of opCall.
gen.jump();
auto n = gen();
enum bool isRandomEngine;
enum ulong max;
Largest generated value.
ulong[2] s;
State must not be entirely zero. The constructor ensures this condition is met.
enum bool preferHighBits;
The lowest bit of this generator is an LFSR. The next bit is not an LFSR, but in the long run it will fail binary rank tests, too. The other bits have no LFSR artifacts. To provide some context, every bit of a Mersenne Twister generator (either the 32-bit or 64-bit variant) is an LFSR.
The rand!T functions in mir.random automatically will discard the low bits when generating output smaller than ulong due to this generator having preferHighBits defined true.
pure nothrow @nogc @safe this()(ulong x0);
Constructs an Xoroshiro128Plus generator seeded with x0.
ulong opCall()();
Advances the random sequence.
pure nothrow @nogc @safe void jump()();
This is the jump function for the generator. It is equivalent to 2^^64 calls to opCall(); it can be used to generate 2^^64 non-overlapping subsequences for parallel computations.
enum bool isUniformRandom;

enum typeof(this.max) min;

enum bool empty;

const @property ulong front()();

void popFront()();

void seed()(ulong x0);
Compatibility with Phobos library methods. Presents this RNG as an InputRange.
This struct disables its default copy constructor and so will only work with Phobos functions that "do the right thing" and take RNGs by reference and do not accidentally make implicit copies.
alias Xoshiro256StarStar = XoshiroEngine!(ulong, 256u, "**", 17u, 45u, 1u, 7u, 5LU, 9LU).XoshiroEngine;
xoshiro256** (XOR/shift/rotate) as described in Scrambled linear pseudorandom number generators (Blackman and Vigna, 2018). 64 bit output. 256 bits of state. Period of 2^^256-1. 4-dimensionally equidistributed. It is 15% slower than xoroshiro128+ but none of its bits fail binary rank tests and it passes tests for Hamming-weight dependencies introduced in the linked paper. From the authors:
This is xoshiro256** 1.0, our all-purpose, rock-solid generator. It has excellent (sub-ns) speed, a state (256 bits) that is large enough for any parallel application, and it passes all tests we are aware of.
A jump() function is included that skips ahead by 2^^128 calls, to generate non-overlapping subsequences for parallel computations.
Public domain reference implementation: xoshiro.di.unimi.it/xoshiro256starstar.c.
Examples:
import mir.random : isSaturatedRandomEngine, rand;
import mir.random.engine.xorshift : Xoshiro256StarStar;
import mir.math.common: fabs;

static assert(isSaturatedRandomEngine!Xoshiro256StarStar);
auto gen = Xoshiro256StarStar(1234u);//Seed with constant.
assert(gen.rand!double.fabs == 0x1.b45d9a0e3ae53p-2);//Generate number from 0 inclusive to 1 exclusive.
assert(gen.rand!ulong == 15548185570577040190UL);
//Xoshiro256StarStar has a jump function that is equivalent
//to 2 ^^ 128 invocations of opCall.
gen.jump();
assert(gen.rand!ulong == 10759542936515257968UL);
alias Xoshiro128StarStar_32 = XoshiroEngine!(uint, 128u, "**", 9u, 11u, 0u, 7u, 5u, 9u).XoshiroEngine;
32-bit-oriented xoshiro** with 128 bits of state. In general Xoshiro256StarStar is preferable except if you are tight on space and know that the generator's output will be consumed 32 bits at a time. (If you need a generator with 128 bits of state that is geared towards producing 64 bits at a time, Xoroshiro128Plus is an option.) 32 bit output. 128 bits of state. Period of 2^^128-1. 4-dimensionally equidistributed. None of its bits fail binary rank tests and it passes tests for Hamming-weight dependencies introduced in the xoshiro paper. From the authors:
This is xoshiro128** 1.0, our 32-bit all-purpose, rock-solid generator. It has excellent (sub-ns) speed, a state size (128 bits) that is large enough for mild parallelism, and it passes all tests we are aware of.
A jump() function is included that skips ahead by 2^^64 calls, to generate non-overlapping subsequences for parallel computations.
Public domain reference implementation: xoshiro.di.unimi.it/xoshiro128starstar.c.
Examples:
import mir.random : isSaturatedRandomEngine, rand;
import mir.random.engine.xorshift : Xoshiro128StarStar_32;

static assert(isSaturatedRandomEngine!Xoshiro128StarStar_32);
auto gen = Xoshiro128StarStar_32(1234u);//Seed with constant.
assert(gen.rand!uint == 1751597702U);
//Xoshiro128StarStar_32 has a jump function that is equivalent
//to 2 ^^ 64 invocations of opCall.
gen.jump();
assert(gen.rand!uint == 1248004684U);
struct XoshiroEngine(UIntType, uint nbits, string scrambler, uint A, uint B, uint I, uint R, UIntType S, UIntType T) if ((is(UIntType == uint) || is(UIntType == ulong)) && "**" == scrambler && (UIntType.sizeof * 8 * 4 == nbits || UIntType.sizeof * 8 * 8 == nbits));
Template for the xoshiro family of generators. See the paper introducing xoshiro and xoroshiro.
Xoshiro256StarStar and Xoshiro128StarStar_32 are aliases for XoshiroEngine instantiated with recommended parameters for 64-bit and 32-bit architectures, respectively.
Parameters:
UIntType uint or ulong
nbits number of bits (128, 256, 512; must be 4x or 8x bit size of UIntType)
scrambler "**" (in the future "+" may be added)
A state xor-lshift
B state rotate left
I index of element used for output
R output scramble rotate left
S output scramble pre-rotate multiplier (must be odd)
T output scramble post-rotate multiplier (must be odd)
enum auto isRandomEngine;
enum UIntType max;
Largest generated value.
UIntType[nbits / (UIntType.sizeof * 8)] s;
State must not be entirely zero. The constructor ensures this condition is met.
pure nothrow @nogc @safe this()(UIntType x0);
Initializes the generator with a seed.
pure nothrow @nogc @safe UIntType opCall()();
Advances the random sequence.
Returns:
A uniformly-distributed integer in the closed range [0, UIntType.max].
pure nothrow @nogc @safe void jump()();
Jump functions are defined for certain UIntType, A, B

combinations:

UIntTypenbitsABNum. calls skipped
ulong25617452^^128
ulong51211212^^256
uint1289112^^64
These can be used to generate non-overlapping subsequences for parallel computations.

enum bool isUniformRandom;

enum typeof(this.max) min;

enum bool empty;

const @property UIntType front()();

void popFront()();

void seed()(UIntType x0);
Compatibility with Phobos library methods. Presents this RNG as an InputRange.
This struct disables its default copy constructor and so will only work with Phobos functions that "do the right thing" and take RNGs by reference and do not accidentally make implicit copies.