[J3] Draft interp on mathematical equivence

[Steven G. Kargl]

On Sat, Aug 10, 2024 at 11:29:35PM +0200, Toon Moene via J3 wrote:
> On 8/10/24 23:01, Robert Corbett via J3 wrote:
> 
> > The question assumes that the rules of mathematical equivalence need to accommodate NaNs and infinites.  That was not the intent of the committee when the concept of mathematical equivalence was added to the standard.  The rules of mathematical equivalence assume that expressions can be modified as if integer, real, and complex expressions are mathematical integer, real, and complex expressions, subject to a few additional restrictions stated in the standard, such as the requirement that the integrity of parentheses be honored.
> > 
> > Bob Corbett
> 
> From the Fortran 77 Standard (6.6 Evaluation of Expressions):
> 
> "Any arithmetic operation whose result is not mathematically defined is
> prohibited in the execution of an executable program. Examples are
> dividing by zero and raising a zero-valued primary to a zero-valued or
> negative-valued power. Raising a negative-valued primary to a real or
> double precision power is also prohibited."
> 
> It is my understanding that the concept of NaN or Infinity is only catered
> for when USEing the IEEE modules:
> 
> "17.1 Overview of IEEE arithmetic support
> 
> The intrinsic modules IEEE_EXCEPTIONS, IEEE_ARITHMETIC, and
> IEEE_FEATURES provide support3 for the facilities defined by ISO/IEC
> 60559:2020."
> 
> (From the latest version - 2023 - of the Standard).
> 

  10.1.5.2.4
  The execution of any numeric operation whose result is not
  defined by the arithmetic used by the processor is prohibited.

Let's call +-0, +-inf, and nan exceptional values.  Then, if the
exceptional values are not defined by the arithmetic used by
the processor, a programmer's code that could run into one of
these is nonconforming.  That is, in Z*X, none of REAL(Z), AIMAG(Z),
nor X can be exceptional and the results of either the short-circuited
evaluation or the standard conforming full evaluation of '*' cannot
have an exceptional value (if it does, the processor can do anything).

So, now you're down to the exceptional values are defined by the
arithmetic.  In that case, short-circuiting cannot occur under
mathematically equivalence, because

  10.1.5.2.4
  Two expressions of a numeric type are mathematically equivalent if,
  for all possible values of their primaries, their mathematical values
  are equal.

If one uses Z = (A,B), then Z*X = (A*X-B*0,A*0+B*X) for the evaluation
after type conversion and Z*X = (A*X,B*X) without type conversion.
A*X may not be equivalent to A*X-B*0 if any of A, X, or B is exceptional.
B*X may not be equivalent to A*0-B*X if any of A, X, or B is exceptional.

-- 
Steve
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