Re: Can someone please explain - real numbers

"John Dough" <nobody@xxxxxxxxxxxxxxx> wrote in message
news:8855l2lt9g8r5b68gt8i77l3nktgg9prrl@xxxxxxxxxx
On Wed, 08 Nov 2006 20:51:34 GMT, Chris Cheney
<XXChris.CheneyXX@xxxxxxxxx> wrote:

Note that, for example, the simple decimal fraction 0.1 is the
recurring binary fraction
0.0001100110011001100110011001100110 (0110 recurring) and cannot
therefore be stored precisely as a single/double/extended/real.

The FPU knows how to handle repeating fractions, regardless of their
base. The repeating portion is truncated (and rounded, if necessary)
at the last available digit of precision for any given float type. In
the case of "extended", the decimal portion terminates at the 18th
position.

It's rather generous to see 'truncating' and say 'handling'.


I'm not sure what leads you to believe that 0.1 cannot be stored
precisely.

The simple fact that not _all_ the bits in it are preserved.


Did you try it with code? I did...and it works fine.
I can store 0.1, and I get back 0.1
If you get back what you store, then by all accounts, it's precise.

It took some work, but here's a program that demonstrates the problem:

var X: Real;
begin
X:=0.2; { To start the recurring fraction immediately }
while True do begin
X:=16*X;
X:=X-3;
end;
end.

Sixteen times 0.2 equals 3.2, subtracting 3 should leave us with the
original 0.2. Ad infinitum. Right?

Not in finite precision. Looking at the finite bitstring above (minus
the first zero after the point to make it more regular), I'm shifting
it left four bits each time, then chopping off the bits that have
travelled to before the binary point. After a few iterations, the
roundoff error becomes visible. After 16 iterations, X had become 0.25
on my computer.

I think I'll try to convince Intel to recall all Pentium processors
again. There seems to be an FMUL bug in them, too.

Groetjes,
Maarten Wiltink


.

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