Mon, 17 Jun 1991 18:18:47 GMT

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Optimizing IEEE Floating-Point Operations daryl@hpclopt.cup.hp.com (1991-06-06) |

Re: Optimizing IEEE Floating-Point Operations bron@sgi.com (1991-06-11) |

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Optimizing IEEE Floating-Point Operations eggert@twinsun.com (1991-06-14) |

Optimizing IEEE Floating-Point Operations cfarnum@valhalla.cs.wright.edu (1991-06-17) |

Re: Optimizing IEEE Floating-Point Operations henry@zoo.toronto.edu (1991-06-17) |

Re: Optimizing IEEE Floating-Point Operations bill@hcx2.ssd.csd.harris.com (1991-06-18) |

Re: Optimizing IEEE Floating-Point Operations jbc@hpcupt3.cup.hp.com (1991-06-19) |

Newsgroups: | comp.compilers |

From: | henry@zoo.toronto.edu (Henry Spencer) |

Keywords: | arithmetic, Fortran |

Organization: | U of Toronto Zoology |

References: | 91-06-011 91-06-016 |

Date: | Mon, 17 Jun 1991 18:18:47 GMT |

In article 91-06-016 bill@hcx2.SSD.CSD.HARRIS.COM (Bill Leonard) writes:

*>As far as I know, there is only _one_ kind of mathematics...*

Sorry, that view has been obsolete for a century, ever since non-Euclidean

geometry started being taken seriously. You choose whichever mathematical

system is suited to the problems you want to tackle. It is not at all

difficult to find extended versions of the real numbers which feature things

like infinities as part of the number system. In fact, if you start looking

at the extended-real-number systems used in things like non-standard

analysis, you find "numbers" much stranger than anything in IEEE arithmetic.

*>... NaNs and INFs represent a failure of the machine model to*

*>adequately represent the _mathematical_ result (i.e., the result you would*

*>get with infinite precision)...*

Um, what *is* the "mathematical result" of, say, 1/0? Even in high-school

mathematics, that's illegal, i.e. NaN. In mathematical systems like the one

underlying IEEE arithmetic, it is +infinity. There is no approximation

involved; either one is an exact, mathematically correct result that would

not be affected in any way by use of infinite precision. Which is right, and

whether NaN is a representable value or simply results in an immediate

failure, depends on the number system in use.

It is important to realize that IEEE arithmetic is based on a slightly more

sophisticated view of the numerical world than that taught in high school,

and its implications cannot be understood in terms of high-school

mathematics.

It is also important to realize that you *cannot* reconcile the FORTRAN

standard with the IEEE arithmetic standard just by reading between the lines

intensively. Actual changes to FORTRAN would probably be needed to make it

consistent with IEEE arithmetic.

--

Henry Spencer @ U of Toronto Zoology, henry@zoo.toronto.edu utzoo!henry

--

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