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28 <h1>Re: round32 ( round64 ( X ) ) ?= round32 ( X )</h1>
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34 <li><em>Subject</em>: Re: round32 ( round64 ( X ) ) ?= round32 ( X )</li>
35 <li><em>From</em>: "Nelson H. F. Beebe" <<a href="mailto:
[email protected]">beebe@xxxxxxxxxxxxx</a>></li>
36 <li><em>Date</em>: Thu, 31 Mar 2011 17:41:22 -0600 (MDT)</li>
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49 <pre style="margin: 0em;">Peter Lawrence asks about the infamous problem of double rounding on
50 systems with long internal registers (Honeywell mainframes of 1970s,
51 Motorola 68K, and current Intel x86 and x86_64 families).
53 Double rounding is indeed a nuisance, and there is a surprising recent
54 discovery that it could have been prevented if there were an unusual
55 rounding mode, round-to-odd (RO(x)). The authors of the paper below
56 show how to implement that rounding in software, and discuss how it
57 can be used to fix the double-rounding problem.
59 It is too late now to repair the mistakes of the past that are present
60 in millions of installed systems, but it is good to know that careful
61 research before designing hardware can be helpful.
63 @String{j-IEEE-TRANS-COMPUT = "IEEE Transactions on Computers"}
65 @Article{Boldo:2008:EFC,
66 author = "Sylvie Boldo and Guillaume Melquiond",
67 title = "Emulation of a {FMA} and Correctly Rounded Sums:
68 Proved Algorithms Using Rounding to Odd",
69 journal = j-IEEE-TRANS-COMPUT,
76 DOI = "<a rel="nofollow" href="http://dx.doi.org/10.1109/TC.2007.70819%22">http://dx.doi.org/10.1109/TC.2007.70819"</a>;,
78 bibdate = "Sat Feb 19 18:44:18 2011",
79 abstract = "Rounding to odd is a nonstandard rounding on
80 floating-point numbers. By using it for some
81 intermediate values instead of rounding to nearest,
82 correctly rounded results can be obtained at the end of
83 computations. We present an algorithm for emulating the
84 fused multiply-and-add operator. We also present an
85 iterative algorithm for computing the correctly rounded
86 sum of a set of floating-point numbers under mild
87 assumptions. A variation on both previous algorithms is
88 the correctly rounded sum of any three floating-point
89 numbers. This leads to efficient implementations, even
90 when this rounding is not available. In order to
91 guarantee the correctness of these properties and
92 algorithms, we formally proved them by using the Coq
94 acknowledgement = ack-nhfb,
95 fjournal = "IEEE Transactions on Computers",
96 keyword = "round-to-odd (RO(x))",
99 See also discussions of the double-rounding problem in this recent
102 @String{pub-BIRKHAUSER-BOSTON = "Birkh{\"a}user Boston Inc."}
103 @String{pub-BIRKHAUSER-BOSTON:adr = "Cambridge, MA, USA"}
105 @Book{Muller:2010:HFP,
106 author = "Jean-Michel Muller and Nicolas Brisebarre and Florent
107 de Dinechin and Claude-Pierre Jeannerod and Vincent
108 Lef{\`e}vre and Guillaume Melquiond and Nathalie Revol
109 and Damien Stehl{\'e} and Serge Torres",
110 title = "Handbook of Floating-Point Arithmetic",
111 publisher = pub-BIRKHAUSER-BOSTON,
112 address = pub-BIRKHAUSER-BOSTON:adr,
113 pages = "xxiii + 572",
115 DOI = "<a rel="nofollow" href="http://dx.doi.org/10.1007/978-0-8176-4704-9%22">http://dx.doi.org/10.1007/978-0-8176-4704-9"</a>;,
116 ISBN = "0-8176-4704-X",
117 ISBN-13 = "978-0-8176-4704-9",
118 LCCN = "QA76.9.C62 H36 2010",
119 bibdate = "Thu Jan 27 16:18:58 2011",
120 price = "US\$90 (est.)",
121 acknowledgement = ack-nhfb,
124 -------------------------------------------------------------------------------
125 - Nelson H. F. Beebe Tel: +1 801 581 5254 -
126 - University of Utah FAX: +1 801 581 4148 -
127 - Department of Mathematics, 110 LCB Internet e-mail: beebe@xxxxxxxxxxxxx -
128 - 155 S 1400 E RM 233 beebe@xxxxxxx beebe@xxxxxxxxxxxx -
129 - Salt Lake City, UT 84112-0090, USA URL: <a rel="nofollow" href="http://www.math.utah.edu/%7Ebeebe/">http://www.math.utah.edu/~beebe/</a> -
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143 <strong><a href="http://grouper.ieee.org/groups/754/email/msg04165.html">RE: round32 ( round64 ( X ) ) ?= round32 ( X )</a></strong>
146 <strong><a href="http://grouper.ieee.org/groups/754/email/msg04164.html">Re: round32 ( round64 ( X ) ) ?= round32 ( X )</a></strong>
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149 <strong><a href="http://grouper.ieee.org/groups/754/email/msg04171.html">round32 ( round64 ( X ) ) ?= round32 ( X )</a></strong>
152 <strong><a href="http://grouper.ieee.org/groups/754/email/msg04167.html">FW: ISO/IEC/IEEE 60559 (IEEE Std 754-2008)</a></strong>
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