## On intermediate precision required for correctly-rounding decimal-to-binary floating-point conversion

Venue: | In Proceedings of 6th Conference Real Numbers and Computers (RNC’6). Schloss Dagstuhl |

Citations: | 2 - 0 self |

### BibTeX

@INPROCEEDINGS{Hack_onintermediate,

author = {Michel Hack},

title = {On intermediate precision required for correctly-rounding decimal-to-binary floating-point conversion},

booktitle = {In Proceedings of 6th Conference Real Numbers and Computers (RNC’6). Schloss Dagstuhl},

year = {}

}

### OpenURL

### Abstract

The algorithms developed ten years ago in preparation for IBM’s support of IEEE Floating-Point on its mainframe S/390 processors use an overly conservative intermediate precision to guarantee correctly-rounded results across the entire exponent range. Here we study the minimal requirement for both bounded and unbounded precision on the decimal side (converting to machine precision on the binary side). An interesting new theorem on Continued Fraction expansions is offered, as well as an open problem on the growth of partial quotients for ratios of powers of two and five. Key words: Floating-Point conversion, Continued Fractions 1

### Citations

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Citation Context ...rational approximation with a smaller denominator necessarily leads to a larger approximation error. Continued Fraction theory is covered in many books on Number Theory, e.g. the classic Hardy&Wright =-=[5]-=-, but one of the most readable and complete expositions is Khinchin [6]. The basic idea is to get a sequence of successfully better rational approximations to a given real number (say positive, for th... |

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Citation Context ...ch includes a section on Floating-Point conversion, which gives the historical background. Briefly, there were several unpublished efforts dating back to 1977, as well as David Matula’s seminal paper =-=[8]-=- on the precision required for reversible conversions in 1968. Jerry Coonen’s Thesis [3] led to the IEEE 754 requirements for extended formats, so correctly-rounding conversion could be performed econ... |

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Citation Context ... but track the conversion error, and redo the conversion using a different, full-precision, method when the rounding threshold is dangerously nearby. Also from that period is Gordon Slishman’s method =-=[9]-=- which exploits the limited exponent range of IBM Hexadecimal Floating-Point for a table-driven approach, which picks carefully chosen multipliers so as to avoid dangerous rounding thresholds. Much la... |

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Citation Context ...se exploration of the search space. We conclude with a failed attempt to provide a bounding formula that could have avoided the need for a search altogether. 2 Prior Work This paper is a follow-up to =-=[1]-=- which includes a section on Floating-Point conversion, which gives the historical background. Briefly, there were several unpublished efforts dating back to 1977, as well as David Matula’s seminal pa... |

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Citation Context ... limited exponent range of IBM Hexadecimal Floating-Point for a table-driven approach, which picks carefully chosen multipliers so as to avoid dangerous rounding thresholds. Much later, Kenton Hanson =-=[4]-=- described a method for finding the most difficult numbers for each format, under the assumption of compatible decimal precision, by a suitably limited search. The method resembles Continued Fraction ... |

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Citation Context ...a larger approximation error. Continued Fraction theory is covered in many books on Number Theory, e.g. the classic Hardy&Wright [5], but one of the most readable and complete expositions is Khinchin =-=[6]-=-. The basic idea is to get a sequence of successfully better rational approximations to a given real number (say positive, for the sake of easier exposition). The crudest approximation is the integral... |