Xorshift. Xorshift random number generators, also called shift-register generators, are a class of pseudorandom number generators that were invented by George Marsaglia. They are a subset of linear-feedback shift registers (LFSRs) which allow a particularly efficient implementation in software without the excessive use of sparse polynomials. They generate the next number in their sequence by repeatedly taking the exclusive or of a number with a bit-shifted version of itself. This makes execution extremely efficient on modern computer architectures, but it does not benefit efficiency in a hardware implementation. Like all LFSRs, the parameters have to be chosen very carefully in order to achieve a long period.

For execution in software, xorshift generators are among the fastest PRNGs, requiring very small code and state. However, they do not pass every statistical test without further refinement. This weakness is amended by combining them with a non-linear function, as described in the original paper. Because plain xorshift generators (without a non-linear step) fail some statistical tests, they have been accused of being unreliable.

Example implementation
A C version of three xorshift algorithms is given here. The first has one 32-bit word of state, and period 232−1. The second has one 64-bit word of state and period 264−1. The last one has four 32-bit words of state, and period 2128−1. The 128-bit algorithm passes the diehard tests. However, it fails the MatrixRank and LinearComp tests of the BigCrush test suite from the TestU01 framework.

All use three shifts and three or four exclusive-or operations:

#include <stdint.h>

struct xorshift32_state {
    uint32_t a;
};

/* The state must be initialized to non-zero */
uint32_t xorshift32(struct xorshift32_state *state)
{
	/* Algorithm "xor" from p. 4 of Marsaglia, "Xorshift RNGs" */
	uint32_t x = state->a;
	x ^= x << 13;
	x ^= x >> 17;
	x ^= x << 5;
	return state->a = x;
}

struct xorshift64_state {
    uint64_t a;
};

uint64_t xorshift64(struct xorshift64_state *state)
{
	uint64_t x = state->a;
	x ^= x << 13;
	x ^= x >> 7;
	x ^= x << 17;
	return state->a = x;
}

/* struct xorshift128_state can alternatively be defined as a pair
   of uint64_t or a uint128_t where supported */
struct xorshift128_state {
    uint32_t x[4];
};

/* The state must be initialized to non-zero */
uint32_t xorshift128(struct xorshift128_state *state)
{
	/* Algorithm "xor128" from p. 5 of Marsaglia, "Xorshift RNGs" */
	uint32_t t  = state->x[3];
    
    uint32_t s  = state->x[0];  /* Perform a contrived 32-bit shift. */
	state->x[3] = state->x[2];
	state->x[2] = state->x[1];
	state->x[1] = s;

	t ^= t << 11;
	t ^= t >> 8;
	return state->x[0] = t ^ s ^ (s >> 19);
}

In case of one 64-bit word of state, there exist parameters which hold period 264−1 with two pair of exclusive-or and shift.
#include <stdint.h>

struct xorshift64_state {
  uint64_t a;
};

uint64_t xorshift64(struct xorshift64_state *state)
{
	uint64_t x = state->a;
	x ^= x << 7;
	x ^= x >> 9;
	return state->a = x;
}

Non-linear variations
All xorshift generators fail some tests in the BigCrush test suite. This is true for all generators based on linear recurrences, such as the Mersenne Twister or WELL. However, it is easy to scramble the output of such generators to improve their quality.

The scramblers known as  and  still leave weakness in the low bits, so they are intended for floating point use, where the lowest bits of floating-point numbers have a smaller impact on the interpreted value. For general purpose, the scrambler  (pronounced starstar) makes the LFSR generators pass in all bits.

xorwow
Marsaglia suggested scrambling the output by combining it with a simple additive counter modulo 232 (which he calls a "Weyl sequence" after Weyl's equidistribution theorem). This also increases the period by a factor of 232, to 2192−232:
#include <stdint.h>

struct xorwow_state {
    uint32_t x[5];
    uint32_t counter;
};

/* The state array must be initialized to not be all zero in the first four words */
uint32_t xorwow(struct xorwow_state *state)
{
    /* Algorithm "xorwow" from p. 5 of Marsaglia, "Xorshift RNGs" */
    uint32_t t  = state->x[4];
 
    uint32_t s  = state->x[0];  /* Perform a contrived 32-bit rotate. */
    state->x[4] = state->x[3];
    state->x[3] = state->x[2];
    state->x[2] = state->x[1];
    state->x[1] = s;
 
    t ^= t >> 2;
    t ^= t << 1;
    t ^= s ^ (s << 4);
    state->x[0] = t;
    state->counter += 362437;
    return t + state->counter;
}
This performs well, but fails a few tests in BigCrush.  This generator is the default in Nvidia's CUDA toolkit.

xorshift*
An  generator applies an invertible multiplication (modulo the word size) as a non-linear transformation to the output of an  generator, as suggested by Marsaglia. All  generators emit a sequence of values that is equidistributed in the maximum possible dimension (except that they will never output zero for 16 calls, i.e. 128 bytes, in a row).

The following 64-bit generator has a maximal period of 264−1.
#include <stdint.h>

/* xorshift64s, variant A_1(12,25,27) with multiplier M_32 from line 3 of table 5 */
uint64_t xorshift64star(void) {
    /* initial seed must be nonzero, don't use a static variable for the state if multithreaded */
    static uint64_t x = 1;
    x ^= x >> 12;
    x ^= x << 25;
    x ^= x >> 27;
    return x * 0x2545F4914F6CDD1DULL;
}

The generator fails only the MatrixRank test of BigCrush, however if the generator is modified to return only the high 32 bits, then it passes BigCrush with zero failures. In fact, a reduced version with only 40 bits of internal state passes the suite, suggesting a large safety margin. A similar generator suggested in Numerical Recipes as RanQ1 also fails the BirthdaySpacings test.

Vigna suggests the following  generator with 1024 bits of state and a maximal period of 21024−1; however, it does not always pass BigCrush. xoshiro256** is therefore a much better option.
#include <stdint.h>

/* The state must be seeded so that there is at least one non-zero element in array */
struct xorshift1024s_state {
	uint64_t x[16];
	int index;
};

uint64_t xorshift1024s(struct xorshift1024s_state *state)
{
	int index = state->index;
	uint64_t const s = state->x[index++];
	uint64_t t = state->x[index &= 15];
	t ^= t << 31;		// a
	t ^= t >> 11;		// b  -- Again, the shifts and the multipliers are tunable
	t ^= s ^ (s >> 30);	// c
	state->x[index] = t;
	state->index = index;
	return t * 1181783497276652981ULL;
}

xorshift+
An  generator can achieve an order of magnitude fewer failures than Mersenne Twister or WELL. A native C implementation of an xorshift+ generator that passes all tests from the BigCrush suite can typically generate a random number in fewer than 10 clock cycles on x86, thanks to instruction pipelining.

Rather than using multiplication, it is possible to use addition as a faster non-linear transformation. The idea was first proposed by Saito and Matsumoto (also responsible for the Mersenne Twister) in the  generator, which adds two consecutive outputs of an underlying  generator based on 32-bit shifts. However, one disadvantage of adding consecutive outputs is that, while the underlying  generator is 2-dimensionally equidistributed, the  generator is only 1-dimensionally equidistributed.

 has some weakness in the low-order bits of its output; it fails several BigCrush tests when the output words are bit-reversed.  To correct this problem, Vigna introduced the  family, based on 64-bit shifts.  generators, even as large as , exhibit some detectable linearity in the low-order bits of their output; it passes BigCrush, but doesn't when the 32 lowest-order bits are used in reverse order from each 64-bit word. This generator is one of the fastest generators passing BigCrush.

The following  generator uses 128 bits of state and has a maximal period of 2128−1.
#include <stdint.h>

struct xorshift128p_state {
    uint64_t x[2];
};

/* The state must be seeded so that it is not all zero */
uint64_t xorshift128p(struct xorshift128p_state *state)
{   
	uint64_t t = state->x[0];
	uint64_t const s = state->x[1];
	state->x[0] = s;
	t ^= t << 23;		// a
	t ^= t >> 18;		// b -- Again, the shifts and the multipliers are tunable
	t ^= s ^ (s >> 5);	// c
	state->x[1] = t;
	return t + s;
}

xorshiftr+
 (r stands for reduced; reads "xorshifter plus") generator was mainly based on xorshift+ yet incorporates modifications making it significantly faster (especially on lightweight devices) and more successful in randomness tests (including TestU01 BigCrush suite) compared to its predecessors. It is one of the fastest generators passing all tests in TestU01's BigCrush suite. Like xorshift+, a native C implementation of an xorshiftr+ generator that passes all tests from the BigCrush suite can typically generate a random number in fewer than 10 clock cycles on x86, thanks to instruction pipelining.

Unlike ,  does not return the sum of two variables derived from the state using xorshift-style steps, rather it returns a single variable with the very last operation in its cycle; however, it features an addition just before returning a value, namely in the phase of adjusting the seed for the next cycle; hence the "+" in the name of the algorithm. The variable sizes, including the state, can be increased with no compromise to the randomness scores, but performance drops may be observed on lightweight devices. 

The following  generator uses 128 bits of state (with two variables) and has a maximal period of 2128−1.
#include <stdint.h>

struct xorshiftr128plus_state {
    uint64_t s[2]; // seeds
};

/* The state must be seeded so that it is not all zero */
uint64_t xorshiftr128plus(struct xorshiftr128plus_state *state)
{ 
	uint64_t x = state->s[0];
	uint64_t const y = state->s[1];
	state->s[0] = y;
	x ^= x << 23; // shift & xor
	x ^= x >> 17; // shift & xor
	x ^= y; // xor
	state->s[1] = x + y;
	return x;
}

xoshiro
xoshiro (short for "xor, shift, rotate") and xoroshiro (short for "xor, rotate, shift, rotate") use rotations in addition to shifts. According to Vigna, they are faster and produce better quality output than xorshift.

This class of generator has variants for 32-bit and 64-bit integer and floating point output; for floating point, one takes the upper 53 bits (for binary64) or the upper 23 bits (for binary32), since the upper bits are of better quality than the lower bits in the floating point generators. The algorithms also include a jump function, which sets the state forward by some number of steps – usually a power of two that allows many threads of execution to start at distinct initial states.

For 32-bit output, xoshiro128** and xoshiro128+ are exactly equivalent to xoshiro256** and xoshiro256+, with  in place of , and with different shift/rotate constants.

More recently, the  generators have been made as an alternative to the  generators. They are used in some implementations of Fortran compilers such as GNU Fortran, Java, and Julia.

xoshiro256++ 
xoshiro256++ is the family's general-purpose random 64-bit number generator./* Adapted from the code included on Sebastiano Vigna's website */

#include <stdint.h>

uint64_t rol64(uint64_t x, int k) {
	return (x << k) | (x >> (64 - k));
}

struct xoshiro256pp_state {
	uint64_t s[4];
};

uint64_t xoshiro256pp(struct xoshiro256pp_state *state) {
	uint64_t *s = state->s;
	uint64_t const result = rol64(s[0] + s[3], 23) + s[0];
	uint64_t const t = s[1] << 17;

	s[2] ^= s[0];
	s[3] ^= s[1];
	s[1] ^= s[2];
	s[0] ^= s[3];

	s[2] ^= t;
	s[3] = rol64(s[3], 45);

	return result;
}

xoshiro256**
xoshiro256** uses multiplication rather than addition in its output function. It is worth noting, however, that the output function is invertible, allowing the underlying state to be trivially uncovered. It is used in GNU Fortran compiler, Lua (as of Lua 5.4), and the .NET framework (as of .NET 6.0).

/* Adapted from the code included on Sebastiano Vigna's website */

#include <stdint.h>

uint64_t rol64(uint64_t x, int k) {
	return (x << k) | (x >> (64 - k));
}

struct xoshiro256ss_state {
	uint64_t s[4];
};

uint64_t xoshiro256ss(struct xoshiro256ss_state *state) {
	uint64_t *s = state->s;
	uint64_t const result = rol64(s[1] * 5, 7) * 9;
	uint64_t const t = s[1] << 17;

	s[2] ^= s[0];
	s[3] ^= s[1];
	s[1] ^= s[2];
	s[0] ^= s[3];

	s[2] ^= t;
	s[3] = rol64(s[3], 45);

	return result;
}

xoshiro256+
xoshiro256+ is approximately 15% faster than xoshiro256**, but the lowest three bits have low linear complexity; therefore, it should be used only for floating point results by extracting the upper 53 bits.

#include <stdint.h>

uint64_t rol64(uint64_t x, int k) {
	return (x << k) | (x >> (64 - k));
}

struct xoshiro256p_state {
	uint64_t s[4];
};

uint64_t xoshiro256p(struct xoshiro256p_state *state) {
	uint64_t* s = state->s;
	uint64_t const result = s[0] + s[3];
	uint64_t const t = s[1] << 17;

	s[2] ^= s[0];
	s[3] ^= s[1];
	s[1] ^= s[2];
	s[0] ^= s[3];

	s[2] ^= t;
	s[3] = rol64(s[3], 45);

	return result;
}

xoroshiro
If space is at a premium, xoroshiro128** and xoroshiro128+ are equivalent to xoshiro256** and xoshiro256+. These have smaller state spaces, and thus are less useful for massively parallel programs. xoroshiro128+ also exhibits a mild dependency in the population count, generating a failure after  of output. The authors do not believe that this can be detected in real world programs. Instead of perpetuating Marsaglia's tradition of xorshift as a basic operation, xoroshiro128+ uses a shift/rotate-based linear transformation designed by Sebastiano Vigna in collaboration with David Blackman. The result is a significant improvement in speed and statistical quality.

xoroshiro64** and xoroshiro64* are equivalent to xoroshiro128** and xoroshiro128+. Unlike the xoshiro generators, they are not straightforward ports of their higher-precision counterparts.

Statistical quality 

The lowest bits of the output generated by xoroshiro128+ have low quality. The authors of xoroshiro128+ acknowledge that it does not pass all statistical tests, stating

   This is xoroshiro128+ 1.0, our best and fastest small-state generator
   for floating-point numbers. We suggest to use its upper bits for
   floating-point generation, as it is slightly faster than
   xoroshiro128**. It passes all tests we are aware of except for the four
   lower bits, which might fail linearity tests (and just those), so if
   low linear complexity is not considered an issue (as it is usually the
   case) it can be used to generate 64-bit outputs, too; moreover, this
   generator has a very mild Hamming-weight dependency making our test
   (http://prng.di.unimi.it/hwd.php) fail after 5 TB of output; we believe
   this slight bias cannot affect any application. If you are concerned,
   use xoroshiro128** or xoshiro256+.

   We suggest to use a sign test to extract a random Boolean value, and
   right shifts to extract subsets of bits.

   The state must be seeded so that it is not everywhere zero. If you have
   a 64-bit seed, we suggest to seed a splitmix64 generator and use its
   output to fill s.

   NOTE: the parameters (a=24, b=16, c=37) of this version give slightly
 
  better results in our test than the 2016 version (a=55, b=14, c=36).

These claims about not passing tests can be confirmed by running PractRand on the input, resulting in output like that shown below:
RNG_test using PractRand version 0.93
RNG = RNG_stdin64, seed = 0xfac83126
test set = normal, folding = standard (64 bit)

rng=RNG_stdin64, seed=0xfac83126
length= 128 megabytes (2^27 bytes), time= 2.1 seconds
  Test Name                         Raw       Processed     Evaluation
  [Low1/64]BRank(12):256(2)         R= +3748  p~=  3e-1129    FAIL !!!!!!!!  
  [Low1/64]BRank(12):384(1)         R= +5405  p~=  3e-1628    FAIL !!!!!!!!  
  ...and 146 test result(s) without anomalies

Acknowledging the authors go on to say:
We suggest to use a sign test to extract a random Boolean value
Thus, programmers should prefer the highest bits (e.g., making a heads/tails by writing random_number < 0 rather than  random_number & 1). It must be noted, though, that the same test is failed by some instances of the Mersenne Twister and WELL.

The statistical problems extend far beyond the bottom few bits, because it fails the PractRand test even when truncated  and fails multiple tests in BigCrush even when the bits are reversed.

Initialization
In the xoshiro paper, it is recommended to initialize the state of the generators using a generator which is radically different from the initialized generators, as well as one which will never give the "all-zero" state; for shift-register generators, this state is impossible to escape from. The authors specifically recommend using the SplitMix64 generator, from a 64-bit seed, as follows:

#include <stdint.h>

struct splitmix64_state {
	uint64_t s;
};

uint64_t splitmix64(struct splitmix64_state *state) {
	uint64_t result = (state->s += 0x9E3779B97f4A7C15);
	result = (result ^ (result >> 30)) * 0xBF58476D1CE4E5B9;
	result = (result ^ (result >> 27)) * 0x94D049BB133111EB;
	return result ^ (result >> 31);
}

struct xorshift128_state {
    uint32_t x[4];
};

// one could do the same for any of the other generators
void xorshift128_init(struct xorshift128_state *state, uint64_t seed) {
	struct splitmix64_state smstate = {seed};

	uint64_t tmp = splitmix64(&smstate);
	state->x[0] = (uint32_t)tmp;
	state->x[1] = (uint32_t)(tmp >> 32);

	tmp = splitmix64(&smstate);
	state->x[2] = (uint32_t)tmp;
	state->x[3] = (uint32_t)(tmp >> 32);
}

See also 
 List of random number generators
 Pseudorandom number generator
 Mersenne Twister
 Well equidistributed long-period linear

Notes

References

Further reading
  Lists generators of various sizes with four shifts (two per feedback word).

External links

Articles with example C code
Pseudorandom number generators