Re: floating point comparison

[jac@ibms48.scri.fsu.edu (Jim Carr)]

dewar@merv.cs.nyu.edu (Robert Dewar) writes:
} There are no *errors*, the discrepancies between IEEE arithmetic and
} real arithmetic are not errors, they are simply differences that come from
} two different arithmetic models.

christian.bau@isltd.insignia.comm (Christian Bau) writes:
>
>As an example: Given a floating point number with an absolute value less
>than 10^15, find an integer number that is close to it. 

 That is an interesting choice since you are not guaranteed to be 
 able to represent 15 decimal digits in a float data type in C.  
 Casting up to double to use your routine below will not change 
 that.  And don't tell me I pick nits; plenty of C programmers 
 will cast a float to a double to call the intrinsic double 
 routine to get a square root without narry a thought about it. 

>Solution in C with IEEE arithmetic: 
>
>double nearby_number (double x)
>{
>   x += 6.0 * 1024.0 * 1024.0 * 1024.0 * 1024.0 * 1024.0;
>   x -= 6.0 * 1024.0 * 1024.0 * 1024.0 * 1024.0 * 1024.0;
>   return x;
>}
>
>Adding 1.5 * 2^52 will produce a result that is rounded to an integer
>number. I cant call this a rounding error (or "uncertainty") since it is
>exactly the effect that I wanted to achieve. 

 The uncertainty is in the representation of the original real value, 
 say 1/10 or pi (or pi*10^14 in your case), as a float (or a double). 
 
jac@ibms48.scri.fsu.edu (Jim Carr) wrote:
|
| It could be useful to adopt the terminology of experimental science 
| here, and use "uncertainty" to denote the variance between the 
| results of real arithmetic and floating point arithmetic done 
| 'correctly' in a particular model.  The two have much in common, 
| since one propagates those uncertainties in a similar way, and 
| since the confusion associated with calling them "errors" is 
| equally strong.  And both are poorly known.  ;-) 

>That term "uncertainty" makes things much much worse. The difference
>between well designed floating point arithmetic and real arithmetic is in
>no way uncertain. It is absolutely certain and predictable. 

 So is the process of measurement with a device with less precision 
 than necessary to make the measurement.  For example, in your case 
 there is no uncertainty in the integer part of a 14 digit real in 
 a double precision IEEE representation that has been properly 
 rounded, but there is an uncertainty if it was stored as a float. 

>For example, with IEEE arithmetic it is relatively easy to compute the
>difference between a+b (using real arithmetic) and a+b (using IEEE
>arithmetic) _exactly_. But you cant call something "uncertainty" that can
>be computed exactly. 

 The uncertainty is that more than one value of a and b will give 
 the same a+b result on the IEEE side. 

-- 
 James A. Carr   <jac@scri.fsu.edu>     | Commercial e-mail is _NOT_ 
    http://www.scri.fsu.edu/~jac/       | desired to this or any address 
 Supercomputer Computations Res. Inst.  | that resolves to my account 
 Florida State, Tallahassee FL 32306    | for any reason at any time.