Re: _Type vs no _Type

[Shark8 <onewingedshark@gmail.com>]

On Nov 6, 3:58 pm, Yannick Duchêne (Hibou57)
<yannick_duch...@yahoo.fr> wrote:
> > Excepting floating point numbers; addition is commutative, isn't it?
>
> I would welcome an example here. Does not mean I do not believe that (I  
> know floating points easily leads to trouble), just that I would like to  
> understand, as I always though even with floating points, the addition is  
> commutative.

The general way to show the non-commutativity of floating-point would
be
something like the initialization of the number to something large,
say
a hundred-million, and some repetitive additions of some small number
say a hundred-thousand hundred-thousandths, so that what you would
expect
mathematically would be a hundred-million and one would actually have
the
value one hundred million.

The reason for this is that the "largeness" of the floating point
number
at the start causes all the small values to be dropped so that after
the
addition you are left with the large number itself.

Whereas if you were to reverse the order of the additions; initialize
to
zero and do the repeated additions of the smaller number followed by
the
large number you are likely to get the correct result.

I've heard someone tell a story where they had to disable optimization
on
their compiler because it was changing the order of operations for a
floating-point number and giving results as described.

>
> What kind of addition algorithm can end into a none-commutative addition ?
>

http://en.wikipedia.org/wiki/Quaternion
These are non-communative on multipclation and are, basically, to the
complex-numbers what the complex-numbers are to real numbers.