numpy.allclose()

[Steven D'Aprano]

Ian Kelly wrote:

> On Sat, Jan 17, 2015 at 7:26 AM, Steven D'Aprano
> <steve+comp.lang.python at pearwood.info> wrote:
>> I don't understand why they add the error tolerances together. I can
>> understand taking the minimum, or the maximum:
> 
> The usual idea is that the tolerance is calculated as a relative
> value, but an absolute tolerance is used instead when the values get
> close to zero, so under normal circumstances the relative tolerance
> will dominate and the absolute tolerance will be very small in
> comparison. 

That suggests that approx_equal() should return True if the actual error is
smaller than *either* the absolute or relative error, that is, using the
maximum.


> So why add them instead of taking the maximum? I'm not 
> sure, but I would guess it's probably for efficiency. Determining the
> maximum requires the CPU to perform a conditional branch that a simple
> addition does not.

That's... horrible. 

"Tell me whether these two numbers differ by at most X or Y%."

"Oh, that would take a nanosecond longer than I think is reasonable, so I'm
instead going to tell you whether they differ by some arbitrary Z instead."

I'm guessing that can only have come from the mindset of C/C++ programmers,
where this sort of thing is considered acceptable:

http://blogs.msdn.com/b/oldnewthing/archive/2014/06/27/10537746.aspx

:-)


> In any case, this is not unique to numpy.  See 
> e.g.:
> 
> https://books.google.com/books?id=CbH0AwAAQBAJ&pg=PA164
> (equation 8.26)

Ah, well as soon as you start using GPUs for numerical computation, you're
already living in a state of sin. GPU's attitude to numerical correctness
is best described as "meh, close enough".

Seriously, anyone familiar with the horrible state of numeric computing
prior to IEEE-756 surely cannot look at GPU numerics without having
flashbacks to the bad old days where:

    if x != 0:
        y = 1/x

could crash with a divide by zero!


[...]
>> * taking the minimum is equivalent to the rule "numbers are close if they
>>   differ by no more than BOTH the absolute tolerance AND the relative
>>   tolerance".
> 
> This is generally not the desired semantic, and it's not unheard of to
> pass 0 for one of the tolerances, in which case the minimum would
> clearly be incorrect.

True. In that case, you would have to treat 0 as a special case, which
starts getting messy.




-- 
Steven