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 mathbf Z/2mathbf Z-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 mathbf Z/232mathbf Z, leading to the XSadd generator. Starting from the observation that the lower bits of XSadd are very weak, as its reverse fails systematically several statistical tests, we explore xorshift+, a variant of XSadd using 64-bit operations, which leads, in small dimension, to extremely fast high-quality generators.