Refine documentation of libm exceptions goals.
This patch refines the math.texi documentation of the goals for when
libm function raise the inexact and underflow exceptions. The
previous text was problematic in some cases around the underflow
threshold.
* Strictly, it would have meant that if the mathematical result of pow
was very slightly below DBL_MIN, for example, it was required to
raise the underflow exception; although normally a few ulps error
would be OK, if that error meant the computed value was slightly
above DBL_MIN it would fail the previously described underflow
exception goal.
* Similarly, strict IEEE semantics would imply that sin (DBL_MIN), in
round-to-nearest mode, underflows on before-rounding but not
after-rounding architectures, while returning DBL_MIN; the previous
wording would have required an underflow exception, so preventing
checks for a result with absolute value below DBL_MIN from being
sufficient checks to determine whether the exception is required.
(Under the previous wording, checks for a result with absolute value
<= DBL_MIN wouldn't have been sufficient either, because in
FE_TOWARDZERO mode a result of DBL_MIN definitely does not result
from an underflowing infinite-precision result.)
* The previous wording about rounding infinite-precision values could
be taken to mean all exceptions including "inexact" must be
consistent with some such value. That would mean that a result of
DBL_MIN in FE_UPWARD mode with "inexact" raised must also have
"underflow" raised on before-rounding architectures. Again, that
would cause problems for computing a result (possibly with spurious
"inexact" exceptions) and then using a rounding-mode-independent
test for results with absolute value below DBL_MIN to determine
whether an underflow exception must be forced in case the underflows
from intermediate computations happened to be exact.
By refining the documentation, this patch avoids stating goals for
accuracy close to the underflow threshold that were stricter than
applied anywhere else, and allows the implementation strategy of:
compute a result within a few ulps, taking care to avoid underflows in
intermediate computations, then force an underflow exception if that
result was subnormal. Only fully-defined functions such as fma need
to take greater care about the exact underflow threshold (including
its dependence on whether the architecture is before-rounding or
after-rounding, and on the rounding mode on after-rounding
architectures).
(If the rounding mode is changed as part of the computation, it's
still necessary to ensure that not just intermediate computations, but
the final computation of the result to be returned, do not raise
underflow if that result is the least normal value and underflow would
be inconsistent with the original rounding mode. Since such code can
readily discard exceptions as part of saving and restoring the
rounding mode - SET_RESTORE_ROUND_NOEX etc. - I don't think that
should be a problem in practice.)
* manual/math.texi (Errors in Math Functions): Clarify goals
regarding inexact and underflow exceptions.