Division-Free Binary-to-Decimal Conversion

Abstract

This article presents algorithms that convert multiple precision integer or floating-point numbers from radix <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math>$2$</tex-math></inline-formula> to radix <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math>$10$</tex-math> </inline-formula> (or to any radix <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math>$b &gt; 2$</tex-math></inline-formula> ). Those algorithms, based on the “scaled remainder tree” technique, use multiplications instead of divisions in their critical part. Both quadratic and subquadratic algorithms are detailed, with proofs of correctness. Experimental results show that our implementation of those algorithms outperforms the GMP library by up to 50 percent (using the same low-level routines).