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Bill, Thank you. Also, I wish to apologize for my former confusion with the terminology. I had earlier interpreted "correctly rounded" to mean to the greatest floating point number less than, the nearest floating point number, or the smallest floating point number greater than, depending on the rounding mode. Actually, IEEE 754-2008 uses the term to mean simply "less than the exact result" in the case of downward rounding, etc. 754-2008 only requires the "nearest" such floating point number for the basic operations, and for binary to decimal and decimal to binary conversions, but allows the language to define the "quantum" for rounding for the recommended elementary functions. I interpret this 754 requirement for the recommended functions to mean that the language shall explicitly state the quantum. You are saying that the quantum is 2ULP in the case of the Sun environment. I don't know how many environments are compliant with this aspect of 754 (I suspect many are not), but the requirement certainly would be useful in constructing a compliant interval system. Best regards, Baker
Baker,Yes. Sun's elementary function library returns correctly rounded results.In addition, I believe it is the case that in almost all cases the interval function library correctly returns 1 ulp width and at most 2 ulp width.Gregory Tarsy can confirm this. Cheers, Bill On 3/14/13 6:07 AM, Ralph Baker Kearfott wrote:P-1788: Bill's reference to the Sun interval libraries for Fortran is interesting. I do recall from the 1990's that the Sun elementary function library was of a higher quality than many in the sense that the values returned represented correctly rounded results, and this was actually stated. I found it frustrating that no such published statements (other than statistical studies on many randomly produced arguments) were made by other vendors, and that computer languages said nothing about accuracy of the elementary functions, or whether or not the underlying arithmetic was 754-compliant. Such facts were a major consideration in my producing INTLIB, a portable Fortran-77 package for elementary functions making minimal assumptions on the machine's +, -, x, /. INTLIB produces mathematically rigorous enclosures that are nonetheless very far from "tight." (Note: Simple precursors to INTLIB were developed prior to widespread implementation of 754 arithmetic.) The interval Fortran library probably made use of techniques in Sun's high-quality floating point library, but with some modifications. In particular, for some functions, computation of the interval result, on average, takes less than twice the amount of time as that required to produce a floating point result. It could be worthwhile to examine Bill's links, as background for implementation, etc. Best regards, Baker On 03/09/2013 10:30 PM, G. William (Bill) Walster wrote: . . .<http://www.lncc.br/sta/manuais/Fortran95_Prog_Ref.pdf> <http://docs.oracle.com/cd/E19422-01/819-3695/iapgFusing.html> and<http://dsc.sun.com/sunstudio/codesamples/s1scc7_examples/IAF95codesamples.html>code examples for all the examples in the Programmer's Reference Manual.
-- --------------------------------------------------------------- R. Baker Kearfott, rbk@xxxxxxxxxxxxx (337) 482-5346 (fax) (337) 482-5270 (work) (337) 993-1827 (home) URL: http://interval.louisiana.edu/kearfott.html Department of Mathematics, University of Louisiana at Lafayette (Room 217 Maxim D. Doucet Hall, 1403 Johnston Street) Box 4-1010, Lafayette, LA 70504-1010, USA ---------------------------------------------------------------