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

[Fawzi Mohamed]

On 20-ott-10, at 23:28, Don wrote:

> Fawzi Mohamed wrote:
>> 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.
>
> You have just lost precision.
> BTW -- I haven't yet worked out if we are disagreeing with each  
> other, or not.

eheh I think we both know the problems, and it is just the matter of  
the kind of tests we do more often.
feqrel is a very important primitive, and it should be available, that  
is what should have been used by Lars.
I happen to test often things where the result is a postion, an energy  
difference,... and in those cases I have a magnitude, and so  
implicitly also an absolute error.
In any case all this discussion was useful, as it made me improve my  
code by making the magnitude an explicit argument.

Fawzi