Printing floating-point numbers quickly and accurately with integers

Abstract

We present algorithms for accurately converting floating-point numbers to decimal representation. They are fast (up to 4 times faster than commonly used algorithms that use high-precision integers) and correct: any printed number will evaluate to the same number, when read again. Our algorithms are fast, because they require only fixed-size integer arithmetic. The sole requirement for the integer type is that it has at least two more bits than the significand of the floating-point number. Hence, for IEEE 754 double-precision numbers (having a 53-bit significand) an integer type with 55 bits is sufficient. Moreover we show how to exploit additional bits to improve the generated output. We present three algorithms with different properties: the first algorithm is the most basic one, and does not take advantage of any extra bits. It simply shows how to perform the binary-to-decimal transformation with the minimal number of bits. Our second algorithm improves on the first one by using the additional bits to produce a shorter (often the shortest) result. Finally we propose a third version that can be used when the shortest output is a requirement. The last algorithm either produces optimal decimal representations (with respect to shortness and rounding) or rejects its input. For IEEE 754 double-precision numbers and 64-bit integers roughly 99.4% of all numbers can be processed efficiently. The remaining 0.6% are rejected and need to be printed by a slower complete algorithm.