xorshift* generators are 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 pass BigCrush from TestU01, but their lowest bits have mild linear artifacts (like the Mersenne Twister, WELL, etc.), and will fail linearity tests. This has no impact in practice, and has no impact at all if you generate floating-point numbers using the highest 53 bits. The exact linear complexity of the lower bits can be estimated using results from this paper.

xorshift+ generators are a 64-bit version of Saito and Matsumoto's XSadd generator. Instead of using a multiplication, we return the sum (in Z/264Z) of two consecutive output of a xorshift generator. As in the previous case, the generators pass BigCrush from TestU01, but again their lowest bits have mild (in fact, even milder) linear artifacts. Albeit superseded by our new generators, xorshift128+ is presently used in the JavaScript engines of Chrome, Node.js, Firefox, Safari and Microsoft Edge. xorshift128+ is a weaker generator than, say, xoroshiro128+, as it fails this test after 6 GB of output.