Further scramblings of Marsaglia's xorshift generators

xorshift* generators are a variant of Marsaglia's xorshift generators that eliminate linear artifacts typical of generators based on Z/2Z-linear operations using multiplication by a suitable constant. Shortly after high-dimensional xorshift* generators were introduced, Saito and Matsumoto suggested a different way to eliminate linear artifacts based on addition in Z/232Z, leading to the XSadd generator. Starting from the observation that the lower bits of XSadd are very weak, as its reverse fails several statistical tests, we explore variants of XSadd using 64-bit operations, and describe in detail xorshift128+, an extremely fast generator that passes strong statistical tests using only three shifts, four xors and an addition.