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*To*: IEEE 754 <stds-754@xxxxxxxxxxxxxxxxx>*Subject*: Re: Binary128 - PI and E*From*: Eric Postpischil <stds-754+ieee.org@xxxxxxx>*Date*: Thu, 12 May 2011 12:14:29 -0400*In-reply-to*: <7249AA8A-98FA-4AAB-9127-3DAF2E0970F0@edp.org>*List-help*: <http://listserv.ieee.org/cgi-bin/wa?LIST=STDS-754>, <mailto:LISTSERV@LISTSERV.IEEE.ORG?body=INFO%20STDS-754>*List-owner*: <mailto:STDS-754-request@LISTSERV.IEEE.ORG>*List-subscribe*: <mailto:STDS-754-subscribe-request@LISTSERV.IEEE.ORG>*List-unsubscribe*: <mailto:STDS-754-unsubscribe-request@LISTSERV.IEEE.ORG>*References*: <4DCAFA56.1020402@gmx.de> <COL112-W48A245E60FB737FB19F08183860@phx.gbl> <7249AA8A-98FA-4AAB-9127-3DAF2E0970F0@edp.org>*Sender*: stds-754@xxxxxxxx

On May 11, 2011, at 22:09, Eric Postpischil wrote:

Now, suppose x rounded to 35 decimal places is 2.000...00116. This tells you x is between 2.000...00115,5 and 2.000...00116,5. Some of the numbers in that interval are below 2.000...00115,55 and some are above it. So knowing x to 35 decimal places does not give you enough information to know which 128-bit floating-point number is closest to x. Some number theory can be used to find where the representable numbers most closely approach decimal values, and that would tell you how many decimal digits you need in the worst case. However, I have run out of time for tonight.

It was definitely time to sleep rather than post. Those points are the representable numbers. The actual decision points are midway between representable numbers, since we want to know whether x is closer to one representable number or another. However, the idea is the same: some of the decision points are within the interval that funnels into a particular 35-decimal-place numeral, so the numeral alone is inconclusive about which representable number is closest. — edp (Eric Postpischil) http://edp.org ... when life ceases to be a fraction and becomes an integer. — Harry Emerson Fosdick

**References**:**Fwd: Re: Binary128 - PI and E***From:*Thorsten Siebenborn

**RE: Binary128 - PI and E***From:*Charles Stevens

**Re: Binary128 - PI and E***From:*Eric Postpischil

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