[CLN-list] Operations with different precisions

[Bruno Haible]

Sebastian Macke wrote:
[Please keep cln-list in CC. If you write to me privately, the probability that
 you get a response is less than 0.3.]

> I used your library with a template class.
>    template <class T>
> Here I define one and zero as
>  T(1.)
>  T(0.)

Please read again the essential, central chapter "Ordinary number types"
of the CLN documentation.

From it, it follows that
  - Numbers such as 1.0 or 0.0 are floating-points and thus are approximate.
  - If you want a "one" and "zero" in a ring, you should take exact numbers,
    (cl_I)1 or (cl_I)0.
  - You don't need templates, because the cl_F, cl_R, cl_N types already
    comprise 'float' and 'double' (and longer) precisions.

> This works without problem with float, double and long double
> But with your implementation I can lose precision.

In the previous mail I explained to you that when you combine two floating-
point numbers with different precision, the result will have an accuracy
that approximately equals the minimum of the two precisions. This is a
mathematical property of the definition of accuracy. (You can verify it
with interval arithmetic or with statistics, if you like.)

You then have the choice:
  - Return a result with a precision approximately equal to its accuracy.
    This is what CLN does. Or
  - Return a result whose precision is the *maximum* of the two precisions.
    If you do this, you are betraying the user of your program.

Bruno