LXM: better splittable pseudorandom number generators (and almost as fast)

In 2014, Steele, Lea, and Flood presented SplitMix, an object-oriented pseudorandom number generator (prng) that is quite fast (9 64-bit arithmetic/logical operations per 64 bits generated) and also splittable. A conventional prng object provides a generate method that returns one pseudorandom value and updates the state of the prng; a splittable prng object also has a second operation, split, that replaces the original prng object with two (seemingly) independent prng objects, by creating and returning a new such object and updating the state of the original object. Splittable prng objects make it easy to organize the use of pseudorandom numbers in multithreaded programs structured using fork-join parallelism. This overall strategy still appears to be sound, but the specific arithmetic calculation used for generate in the SplitMix algorithm has some detectable weaknesses, and the period of any one generator is limited to 2⁶⁴.

Here we present the LXM family of prng algorithms. The idea is an old one: combine the outputs of two independent prng algorithms, then (optionally) feed the result to a mixing function. An LXM algorithm uses a linear congruential subgenerator and an F₂-linear subgenerator; the examples studied in this paper use a linear congruential generator (LCG) of period 2¹⁶, 2³², 2⁶⁴, or 2¹²⁸ with one of the multipliers recommended by L’Ecuyer or by Steele and Vigna, and an F₂-linear xor-based generator (XBG) of the xoshiro family or xoroshiro family as described by Blackman and Vigna. For mixing functions we study the MurmurHash3 finalizer function; variants by David Stafford, Doug Lea, and degski; and the null (identity) mixing function.

Like SplitMix, LXM provides both a generate operation and a split operation. Also like SplitMix, LXM requires no locking or other synchronization (other than the usual memory fence after instance initialization), and is suitable for use with simd instruction sets because it has no branches or loops.

We analyze the period and equidistribution properties of LXM generators, and present the results of thorough testing of specific members of this family, using the TestU01 and PractRand test suites, not only on single instances of the algorithm but also for collections of instances, used in parallel, ranging in size from 2 to 2²⁴. Single instances of LXM that include a strong mixing function appear to have no major weaknesses, and LXM is significantly more robust than SplitMix against accidental correlation in a multithreaded setting. We believe that LXM, like SplitMix, is suitable for “everyday” scientific and machine-learning applications (but not cryptographic applications), especially when concurrent threads or distributed processes are involved.