approxEqual() has fooled me for a long time...

[Fawzi Mohamed]

On 20-ott-10, at 20:53, Don wrote:

> Andrei Alexandrescu wrote:
>> On 10/20/10 10:52 CDT, Don wrote:
>>> I don't think it's possible to have a sensible default for absolute
>>> tolerance, because you never know what scale is important. You can  
>>> do a
>>> default for relative tolerance, because floating point numbers  
>>> work that
>>> way (eg, you can say they're equal if they differ in only the last 4
>>> bits, or if half of the mantissa bits are equal).
>>>
>>> I would even think that the acceptable relative error is almost  
>>> always
>>> known at compile time, but the absolute error may not be.
>> I wonder if it could work to set either number, if zero, to the  
>> smallest normalized value. Then proceed with the feqrel algorithm.  
>> Would that work?
>> Andrei
>
> feqrel actually treats zero fairly. There are exactly as many  
> possible values almost equal to zero, as there are near any other  
> number.
> So in terms of the floating point number representation, the  
> behaviour is perfect.
>
> Thinking out loud here...
>
> I think that you use absolute error to deal with the difference  
> between the computer's representation, and the real world. You're  
> almost pretending that they are fixed point numbers.
> Pretty much any real-world data set has a characteristic magnitude,  
> and anything which is more than (say) 10^^50 times smaller than the  
> average is probably equivalent to zero.

The thing is two fold, from one thing, yes numbers 10^^50 smaller are  
not important, but the real problem is another, you will probably add  
and subtract numbers of magnitude x, on this operation the *absolute*  
error is x*epsilon.

Note that the error is relative to the magnitude of the operands, not  
of the result, it is really an absolute error.
Now the end result might have a relative error, but also an absolute  
error whose size depends on the magnitude of the operands.
If the result is close to 0 the absolute error is likely to dominate,  
and checking the relative error will fail.
This is the case for example for matrix multiplication.
In NArray I wanted to check the linar algebra routines with matrixes  
of random numbers, feqrel did fail too much for number close to 0.

Obviously the right thing as Walter said is to let the user choose the  
magnitude of its results.
In the code I posted I did choose simply 0.5**(mantissa_bits/4) which  
is smaller than 1 but not horribly so.
One can easily make that an input parameter (it is the shift parameter  
in my code)

Fawzi