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Date: Mon, 06 Mar 2006 10:29:31 +0100
From: Roberto Bagnara <bagnara AT cs DOT unipr DOT it>
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To: Jim Easton <jim AT cs DOT ualberta DOT ca>
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   "The Parma Polyhedra Library developers' list" <ppl-devel AT cs DOT unipr DOT it>
Subject: Re: Precision of doubles and stdio
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Jim Easton wrote:
> Dear Mr. Bagnara,
> 
> Roberto Bagnara wrote: 
> ...
>> does this on Linux/i686
>>
>> $ a.out
>> 70.9
>> 70.900000000000005684341886080801486968994140625
>>
>> and does the following under Cygwin on the same machine:
> ...
>> $ ./a.exe
>> 70.9
>> 70.90000000000000568434188608080148696899414
>>
>> Why?  Is there a way to reconcile the two behaviors?
>> Notice that I know about the x87 and its vaguaries:
>> nonetheless I wonder why such a scanf immediately
>> followed by a printf shows a difference between
>> Cygwin and Linux.
> 
> With all due respect, why would you want to?  With double you are
> guaranteed only 16 or so digits - the rest is noise.  Frankly I am
> amazed that it agrees as far as it does.

Dear Jim,

what is and what is not noise depends on the application.
In our applications we systematically use controlled rounding
on IEEE 754 floating point numbers.  In the end, what we obtain
(in memory) are definite (i.e., provably correct) lower or upper
bounds of some quantities.

Call `x' such a quantity, and suppose we have that our computed
upper bound for `x' is the IEEE 754 Double Precision number

     0x4051b9999999999a,

that is (exactly!),

     70.900000000000005684341886080801486968994140625.

If that number is (wrongly!) printed as

     70.90000000000000568434188608080148696899414

then we lose correctness, since
x <= 70.900000000000005684341886080801486968994140625
does not imply
x <= 70.90000000000000568434188608080148696899414.
So, the final "0625" is not "noise" in our applications:
it is what may make the difference between a correct statement
and an incorrect one.

Notice also that any IEEE 754 number can be (exactly!) printed
with a finite number of decimal digits.  Finally, notice that
writing an algorithm to print them correctly is not rocket science.
Hence my astonishment when Tim showed me that the C standard
decided, instead, to allow blatant violations of the principle
of least astonishment :-)
All the best,

    Roberto

-- 
Prof. Roberto Bagnara
Computer Science Group
Department of Mathematics, University of Parma, Italy
http://www.cs.unipr.it/~bagnara/
mailto:bagnara AT cs DOT unipr DOT it

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