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Re: Binary128 - PI and E



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


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