To: J3                                                     J3/26-126
From: Malcolm Cohen
Subject: Integrity of parentheses
Date: 2026-February-25
Reference: 26-115r2, 24-007


Paper 26-115r2 asks questions about what the rules mean for parentheses,
and seems to deduce that being treated as a "data entity" means that the
result of evaluating a parenthesised expression means that it will be
rounded to the "storage format" of variables of that type and kind.

This is not what being "treated as a data entity" means. It means that the
value is computed before being used as part of a larger expression. This is
explained by the NOTE in 10.1.8.

Fortran has always permitted an expression to be evaluated entirely in a
higher precision than required by the type and kind. This has advantages
not only in obtaining a more accurate result, but also in performance.

For example, the popular x86 processor with its x87 co-processor only had
floating-point registers that were 80 bits wide. Rounding the result of an
operation to single precision or double precision necessarily involved a
round trip to memory along with the cost of the conversion itself (which
was non-trivial, especially for subnormal values). Requiring rounding of
intermediate results in an expression would thus have been prohibitively
expensive.

Nowadays, most processors have both single and double precision sized
registers, so there is little cost for those precisions on those processors,
but many do not have half precision sized registers. That means that rounding
intermediate values to half precision is very expensive, and possibly without
hardware support, which makes it even slower.
So the performance implications have not gone away.

A similar question was asked previously, about whether the REAL intrinsic
function was required to round intermediate results to main memory format,
and the answer was that because REAL of a floating-point expression is
essentially an identity operation, it was not required to round anything.
See interp F03/0121, paper 16-118r1.

Fortran has always been about providing high performance for mathematical
formulae evaluation. Thus the frequent mentions of "mathematically
equivalent expressions" and "processor-dependent approximations".
It would be astonishing if parentheses inserted rounding operations into the
middle of an expression, especially considering our answer for REAL.

However, sometimes people do want to round an intermediate result, and that is
why the IEEE_REAL function was added to Fortran 2018. This is not an identity
operation but a conversion that rounds an intermediate value to the specified
IEEE format.

Thus,
        IEEE_REAL(B*C,KIND(B*C)) + A
is guaranteed to round B*C to the specified precision (IEEE format), and so
(for example) the IEEE FMA operation cannot be used to evaluate that
expression, nor can it be evaluated entirely in a higher precision.

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