| How Computers Represent Floats [message #22640] |
Wed, 29 November 2000 00:00  |
davidf
Messages: 2866
Registered: September 1996
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Senior Member |
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Oh, dear. :-(
I have occasion to recall a discussion posted in this
forum some time ago about how computers represent
floating point numbers. How they appear inaccurate, etc.
I *know* I saved it, but I've searched on just about
every keyword I can think of on my local machines and
in Dejanews and I can't find what I am looking for.
(It's possible I dreamed the whole exchange. Stranger
things have happened.)
If anyone saved the discussion (or even recalls it
enough to supply me with some likely keywords to search
on), I would be grateful.
Cheers,
David
P.S. In my dream (apparently) someone who is not a
frequent poster to this newsgroup published a fabulously
informative article on the subject.
--
David Fanning, Ph.D.
Fanning Software Consulting
Phone: 970-221-0438 E-Mail: davidf@dfanning.com
Coyote's Guide to IDL Programming: http://www.dfanning.com/
Toll-Free IDL Book Orders: 1-888-461-0155
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| Re: How Computers Represent Floats [message #22751 is a reply to message #22640] |
Fri, 01 December 2000 00:00  |
Karl Schultz
Messages: 341
Registered: October 1999
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Senior Member |
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Here's another useful table if anyone is interested in pondering the
limits of floating point representations:
32-bit(single) 64-bit(double)
Mantissa 24 bits 53 bits
Exponent 8 bits 11 bits
EPS 1.19209e-007 2.22045e-016
Min 1.17549e-038 2.22507e-308
Max 3.40282e+038 1.79769e+308
Decimal Places 6 15
80-bit 128-bit(quad)
Mantissa 65 bits 113 bits
Exponent 15 bits 15 bits
EPS 1.08420e-019 1.92593e-034
Min 3.36210e-4932 3.36210e-4932
Max 1.18973e+4932 1.18973e+4932
Decimal Places 18 33
The mantissa bit counts include the sign bit.
I find it a little interesting that the number of bits in the exponent
remains the same between 80-bit and 128-bit.
I also think that the EPS (machine epsilon or machine precision) is
probably one of the most important values. It is the smallest value
that you can add to 1.0 and still have the result be something other
than 1.0. This can give you an idea of how closely you can resolve
(differentiate) floating point values at a given magnitude.
For example, see what this does in IDL 5.3:
PLOT,FINDGEN(100),FINDGEN(100)+2d8,YSTYLE=3
IDL 5.4 gives different results because PLOT works in double
precision. You can "simulate" the old 5.3 behavior with:
PLOT,FINDGEN(100),FLOAT(FINDGEN(100)+2d8), YSTYLE=3
Karl
Sent via Deja.com http://www.deja.com/
Before you buy.
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| Re: How Computers Represent Floats [message #22763 is a reply to message #22640] |
Fri, 01 December 2000 00:00 |
colinr
Messages: 30
Registered: July 1999
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Member |
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On Thu, 30 Nov 2000 13:23:41 -0700,
William B. Clodius <wclodius@lanl.gov> wrote:
>
>
> "William B. Clodius" wrote:
>> <snip> IEEE 754 requires that all intermediate calculations
>> be performed a higher precision so
> Ignore the above incomplete sentence. What I originally attempted to
> write was covered later.
>>
>> <snip>
>
> Some other surprises.
>
> The definition of the IEEE 754 mantisa, an integer with values from
> 2^n_mant to 2*2^n_mant-1, where n_mant is the number of bits available
> for the mantisa, is termed a normalized number. This is error prone for
> very small numbers. IEEE 754 mandates that there be available for such
> small numbers what are termed denorms where the mantissa is interpreted
> as an integer from 0 to 2^n_mant, so that accuracy degrades gradually
> for such values. However, this complicates the implementation of the
> floating point, so some processors, e.g., the DEC Alpha make this
> available only in software at a greatly reduced performance.
This sounds like it relates to my most recent problem - generating
real input files with IDL for a DEC Alpha fortran program. The
fortran program had big problems manipulating small numbers
generated by the IDL and I had to pepper the IDL code with WHERE
statements to set all very small numbers to zero. Has anyone else
seen stuff like this?
--
Colin Rosenthal
Astrophysics Institute
University of Oslo
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