Article View: comp.lang.fortran
Article #6540Re: True probabilistic rounding [was Re: Precision ...]
From: Glen Herrmannsfeldt
Date: Wed, 01 Oct 2003 09:08
Date: Wed, 01 Oct 2003 09:08
41 lines
3556 bytes
3556 bytes
"Russell Wallace" <wallacethinmintr@eircom.net> wrote in message news:3f7a622d.113216013@news.eircom.net... > On 30 Sep 2003 10:42:58 GMT, nmm1@cus.cam.ac.uk (Nick Maclaren) wrote: > > [probabilistic floating point rounding] > > >The reason I say the last is that there is no need to calculate a > >multiplication to full accuracy - all you need is to generate > >enough bits for a decision. Of course, this means that operations > >can take variable times, which is Not Approved Of. But it might > >help, especially for extended precision multiplication. > > I'm no expert on numerical analysis, so tell me if I'm missing > something obvious, but: > > Let's suppose you have N bits of precision (N ~= 53 for IEEE double > precision IIRC). > > In the base case where rounding truncates; you have N bits of > precision. > > If rounding is to nearest, .5 rounding away from zero (the simplest > type of rounding, it requires the consideration of only one extra > bit), you effectively have something like N+0.5 bits of precision in > calculations. I am not sure that I understand it so well, but I saw an explanation that to get proper rounding for division with an N bit quotient, it is necessary to calculate 2N quotient bits. I believe that this is related to the statement about double rounding, where rounding to >=2N bits, and then subsequently to N bits gets the same result as rounding directly to N bits. Intermediate rounding to <2N bits before rounding to N bits can give the wrong answer. The statement about division is in the IBM Systems Journal article on extended precision floating point, and the DXR simulation routine. -- glen -- glen
Technical Details
Message-ID:
<HQweb.382949$2x.123651@rwcrnsc52.ops.asp.att.net>Path:
arc1-ca.pugleaf.net!archiver1.google.com!news2.google.com!newsfeed.stanford.edu!bloom-beacon.mit.edu!nycmny1-snh1.gtei.net!news.gtei.net!newsfeed.mathworks.com!wn13feed!wn11feed!worldnet.att.net!204.127.198.203!attbi_feed3!attbi_feed4!attbi.com!rwcrnsc52.ops.asp.att.net.POSTED!not-for-mailReferences:
<dG4db.598668$uu5.97984@sccrnsc04> <xrXdb.631209$Ho3.121609@sccrnsc03> <bl9ild$abv$1@pegasus.csx.cam.ac.uk> <Rh0eb.632709$Ho3.121617@sccrnsc03> <bla4k2$nas$1@pegasus.csx.cam.ac.uk> <FR0eb.637973$uu5.101567@sccrnsc04> <r%0eb.163529$3o3.11771639@bgtnsc05-news.ops.worldnet.att.net> <bY1eb.633341$Ho3.121512@sccrnsc03> <Zs2eb.157980$0v4.11847973@bgtnsc04-news.ops.worldnet.att.net> <blbcjf$lrn$1@pegasus.csx.cam.ac.uk> <QEbeb.377802$2x.116648@rwcrnsc52.ops.asp.att.net> <blbmni$g2$1@pegasus.csx.cam.ac.uk> <3f7a622d.113216013@news.eircom.net>Show Full Headers
Complete Article Headers
X-Google-Language: ENGLISH,ASCII-7-bit X-Google-Thread: fd4af,7830916812112af0 X-Google-Attributes: gidfd4af,public X-Google-Thread: 102b75,7830916812112af0 X-Google-Attributes: gid102b75,public X-Google-Thread: 1094ba,7830916812112af0 X-Google-Attributes: gid1094ba,public X-Google-ArrivalTime: 2003-10-01 02:08:56 PST Path: arc1-ca.pugleaf.net!archiver1.google.com!news2.google.com!newsfeed.stanford.edu!bloom-beacon.mit.edu!nycmny1-snh1.gtei.net!news.gtei.net!newsfeed.mathworks.com!wn13feed!wn11feed!worldnet.att.net!204.127.198.203!attbi_feed3!attbi_feed4!attbi.com!rwcrnsc52.ops.asp.att.net.POSTED!not-for-mail From: "Glen Herrmannsfeldt" <gah@ugcs.caltech.edu> Newsgroups: comp.lang.fortran,comp.arch,bit.listserv.ibm-main References: <dG4db.598668$uu5.97984@sccrnsc04> <xrXdb.631209$Ho3.121609@sccrnsc03> <bl9ild$abv$1@pegasus.csx.cam.ac.uk> <Rh0eb.632709$Ho3.121617@sccrnsc03> <bla4k2$nas$1@pegasus.csx.cam.ac.uk> <FR0eb.637973$uu5.101567@sccrnsc04> <r%0eb.163529$3o3.11771639@bgtnsc05-news.ops.worldnet.att.net> <bY1eb.633341$Ho3.121512@sccrnsc03> <Zs2eb.157980$0v4.11847973@bgtnsc04-news.ops.worldnet.att.net> <blbcjf$lrn$1@pegasus.csx.cam.ac.uk> <QEbeb.377802$2x.116648@rwcrnsc52.ops.asp.att.net> <blbmni$g2$1@pegasus.csx.cam.ac.uk> <3f7a622d.113216013@news.eircom.net> Subject: Re: True probabilistic rounding [was Re: Precision ...] Lines: 43 X-Priority: 3 X-MSMail-Priority: Normal X-Newsreader: Microsoft Outlook Express 5.50.4927.1200 X-MimeOLE: Produced By Microsoft MimeOLE V5.50.4927.1200 Message-ID: <HQweb.382949$2x.123651@rwcrnsc52.ops.asp.att.net> NNTP-Posting-Host: 12.207.204.17 X-Complaints-To: abuse@comcast.net X-Trace: rwcrnsc52.ops.asp.att.net 1064999335 12.207.204.17 (Wed, 01 Oct 2003 09:08:55 GMT) NNTP-Posting-Date: Wed, 01 Oct 2003 09:08:55 GMT Organization: Comcast Online Date: Wed, 01 Oct 2003 09:08:55 GMT