Main article: Xorshift#xoroshiro128+
xoroshiro128+ (named after its operations: XOR, rotate, shift, rotate) is a pseudorandom number generator intended as a successor to xorshift+. 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 (well below a nanosecond per integer) and a significant improvement in statistical 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, b=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 weak low order bit, 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 the Mersenne Twister, WELL, etc., so the issue is mainly of academic concern.
As stated in the comments, the generator fails a Hamming-weight dependency test developed by Blackman and Vigna after 8 TB of data. As a comparison, for some choice of parameters the Mersenne Twister at 607 bits fails the same test after less than a gigabyte of data.
David Meister, who implemented it in Clojure, made some valuable statements:
Matt Gallagher, in his study on random number generators in Swift made the following conclusion:
It looks like Xoroshiro is the best general purpose algorithm currently available. Low memory (just 128 bits of storage), extremely high performance (1.2 nanoseconds per 64-bit number, after subtracting baseline overheads) and very well distributed (beating other algorithms on a range of automated tests). Mersenne Twister might still be a better choice for highly conservative projects unwilling to switch to such a new algorithm, but the current generation of statistically tested algorithms brings a baseline of assurance from the outset that previous generations lacked.
- xoroshiro128** prevents linear artifacts in the low bits
- xoshiro256+ has 256 bits of state allowing for more parallelism
- xoshiro256** — "our all-purpose, rock-solid generator"
The generators ending with + have weak low bits, so they are recommended for floating point number generation, using only the 53 most significant bits.
- Blackman, David; Vigna, Sebastiano. "Scrambled Linear Pseudorandom Generators". arXiv:1805.01407 [cs.DS].
- Blackman, David; Vigna, Sebastiano (2018). "Original C source code implementation of xoroshiro128+". Retrieved May 4, 2018.
- "Testing Hamming-weight dependencies". May 3, 2018. Retrieved May 3, 2018.
- Meister, David (August 1, 2016). "Clojure implementation of the xoroshiro128+ PRNG described at web site xoroshiro.di.unimi.it". github.com. Retrieved November 2, 2016.
- Gallagher, Matt (May 19, 2016). "Random number generators in Swift". www.cocoawithlove.com. Retrieved November 2, 2016.
- Vigna, Sebastiano (2018). "xoshiro / xoroshiro generators and the PRNG shootout". Retrieved 2018-05-04.