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Re: round32 ( round64 ( X ) ) ?= round32 ( X )
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Peter Lawrence asks about the infamous problem of double rounding on
+systems with long internal registers (Honeywell mainframes of 1970s,
+Motorola 68K, and current Intel x86 and x86_64 families).
+
+Double rounding is indeed a nuisance, and there is a surprising recent
+discovery that it could have been prevented if there were an unusual
+rounding mode, round-to-odd (RO(x)). The authors of the paper below
+show how to implement that rounding in software, and discuss how it
+can be used to fix the double-rounding problem.
+
+It is too late now to repair the mistakes of the past that are present
+in millions of installed systems, but it is good to know that careful
+research before designing hardware can be helpful.
+
+@String{j-IEEE-TRANS-COMPUT = "IEEE Transactions on Computers"}
+
+@Article{Boldo:2008:EFC,
+ author = "Sylvie Boldo and Guillaume Melquiond",
+ title = "Emulation of a {FMA} and Correctly Rounded Sums:
+ Proved Algorithms Using Rounding to Odd",
+ journal = j-IEEE-TRANS-COMPUT,
+ volume = "54",
+ number = "4",
+ pages = "462--471",
+ month = apr,
+ year = "2008",
+ CODEN = "ITCOB4",
+ DOI = "http://dx.doi.org/10.1109/TC.2007.70819";,
+ ISSN = "0018-9340",
+ bibdate = "Sat Feb 19 18:44:18 2011",
+ abstract = "Rounding to odd is a nonstandard rounding on
+ floating-point numbers. By using it for some
+ intermediate values instead of rounding to nearest,
+ correctly rounded results can be obtained at the end of
+ computations. We present an algorithm for emulating the
+ fused multiply-and-add operator. We also present an
+ iterative algorithm for computing the correctly rounded
+ sum of a set of floating-point numbers under mild
+ assumptions. A variation on both previous algorithms is
+ the correctly rounded sum of any three floating-point
+ numbers. This leads to efficient implementations, even
+ when this rounding is not available. In order to
+ guarantee the correctness of these properties and
+ algorithms, we formally proved them by using the Coq
+ proof checker.",
+ acknowledgement = ack-nhfb,
+ fjournal = "IEEE Transactions on Computers",
+ keyword = "round-to-odd (RO(x))",
+}
+
+See also discussions of the double-rounding problem in this recent
+useful book:
+
+@String{pub-BIRKHAUSER-BOSTON = "Birkh{\"a}user Boston Inc."}
+@String{pub-BIRKHAUSER-BOSTON:adr = "Cambridge, MA, USA"}
+
+@Book{Muller:2010:HFP,
+ author = "Jean-Michel Muller and Nicolas Brisebarre and Florent
+ de Dinechin and Claude-Pierre Jeannerod and Vincent
+ Lef{\`e}vre and Guillaume Melquiond and Nathalie Revol
+ and Damien Stehl{\'e} and Serge Torres",
+ title = "Handbook of Floating-Point Arithmetic",
+ publisher = pub-BIRKHAUSER-BOSTON,
+ address = pub-BIRKHAUSER-BOSTON:adr,
+ pages = "xxiii + 572",
+ year = "2010",
+ DOI = "http://dx.doi.org/10.1007/978-0-8176-4704-9";,
+ ISBN = "0-8176-4704-X",
+ ISBN-13 = "978-0-8176-4704-9",
+ LCCN = "QA76.9.C62 H36 2010",
+ bibdate = "Thu Jan 27 16:18:58 2011",
+ price = "US\$90 (est.)",
+ acknowledgement = ack-nhfb,
+}
+
+-------------------------------------------------------------------------------
+- Nelson H. F. Beebe Tel: +1 801 581 5254 -
+- University of Utah FAX: +1 801 581 4148 -
+- Department of Mathematics, 110 LCB Internet e-mail: beebe@xxxxxxxxxxxxx -
+- 155 S 1400 E RM 233 beebe@xxxxxxx beebe@xxxxxxxxxxxx -
+- Salt Lake City, UT 84112-0090, USA URL: http://www.math.utah.edu/~beebe/ -
+-------------------------------------------------------------------------------
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