Mathematica: a system for doing mathematics by computer (2nd ed.)
Mathematica: a system for doing mathematics by computer (2nd ed.)
Fast high-precision computation of complex square roots
ISSAC '96 Proceedings of the 1996 international symposium on Symbolic and algebraic computation
Elementary functions: algorithms and implementation
Elementary functions: algorithms and implementation
Complexity and real computation
Complexity and real computation
Fast Multiple-Precision Evaluation of Elementary Functions
Journal of the ACM (JACM)
Comparison of arithmetic functions with respect to boolean circuit depth
STOC '84 Proceedings of the sixteenth annual ACM symposium on Theory of computing
Logarithmic depth circuits for algebraic functions
SFCS '83 Proceedings of the 24th Annual Symposium on Foundations of Computer Science
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In this paper we present an algorithm which shows that the exponential function has algebraic complexity O(log2 n), i.e., can be evaluated with relative error O(2-n) using O(log2 n) infinite-precision additions, subtractions, multiplications and divisions. This solves a question of J. M. Borwein and P. B. Borwein [9]. The best known lower bound for the algebraic complexity of the exponential function is Ω(log n). The best known upper and lower bounds for the bit complexity of the exponential function are O(µ(n) log n) [10] and (Ω(v(n)) [4], respectively, where µ(n) denotes an upper bound and v(n) denotes a lower bound for the bit complexity of n-bit integer multiplication. The presented algorithm has bit complexity O(µ(n) log n).