The binary expansions of irrational algebraic numbers can serve as attractive candidates for pseudorandom binary sequences. This study presents an efficient method for computing the exact binary expansions of real quadratic algebraic integers using Newton’s method. To this end, we clarify conditions under which the first N bits of the binary expansion of an irrational number match those of its upper rational approximation. Furthermore, we establish that the worst-case time complexity of generating a sequence of length N with the proposed method is equivalent to the complexity of multiplying two N-bit integers, showing its efficiency compared to a previously proposed true orbit generator. We report the results of numerical experiments on computation time and memory usage, highlighting in particular that the proposed method successfully accelerates true orbit pseudorandom bit generation. We also confirm that a generated pseudorandom sequence successfully passes all the statistical tests included in Rabbit of TestU01.