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Re: Refine documentation of libm exceptions goals [committed]

* Joseph Myers <> [2015-02-17 23:42:32 +0000]:
> 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).

this is an interesting issue because iso c annex f forbids the
omission of underflow right now, which is hard to achive unless
libm checks <= DBL_MIN instead of the sensible < DBL_MIN and
raises spurious underflow for a lot of exact (or non-tiny) DBL_MIN
results (which is allowed by the standard)

(FLT_EVAL_METHOD!=0 platforms add further complication to this
issue on the implementation side, and there builtin operations
like 'a=b*c;' can omit underflow because of double rounding,
but fma in libm is not allowed to..)

i think the requirement in the standard is suboptimal: reasonable
implementations are hard to do, but the slow and useless 'always
raise underflow' is correct.

(this may be worth a wg14 defect report, my guess is that there
does not exist an implementation that has conforming and
reasonable underflow behaviour)

> (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.)
> Committed.
> 2015-02-17  Joseph Myers  <>
> 	* manual/math.texi (Errors in Math Functions): Clarify goals
> 	regarding inexact and underflow exceptions.
> diff --git a/manual/math.texi b/manual/math.texi
> index 206021c..72f3fda 100644
> --- a/manual/math.texi
> +++ b/manual/math.texi
> @@ -1313,7 +1313,9 @@ exact value passed as the input.  Exceptions are raised appropriately
>  for this value and in accordance with IEEE 754 / ISO C / POSIX
>  semantics, and it is then rounded according to the current rounding
>  direction to the result that is returned to the user.  @code{errno}
> -may also be set (@pxref{Math Error Reporting}).
> +may also be set (@pxref{Math Error Reporting}).  (The ``inexact''
> +exception may be raised, or not raised, even if this is inconsistent
> +with the infinite-precision value.)
>  @item
>  For the IBM @code{long double} format, as used on PowerPC GNU/Linux,
> @@ -1344,11 +1346,16 @@ subnormal (in each case, with the correct sign), according to the
>  current rounding direction and with the underflow exception raised.
>  @item
> -Where the mathematical result underflows and is not exactly
> -representable as a floating-point value, the underflow exception is
> -raised (so there may be spurious underflow exceptions in cases where
> -the underflowing result is exact, but not missing underflow exceptions
> -in cases where it is inexact).
> +Where the mathematical result underflows (before rounding) and is not
> +exactly representable as a floating-point value, the function does not
> +behave as if the computed infinite-precision result is an exact value
> +in the subnormal range.  This means that the underflow exception is
> +raised other than possibly for cases where the mathematical result is
> +very close to the underflow threshold and the function behaves as if
> +it computes an infinite-precision result that does not underflow.  (So
> +there may be spurious underflow exceptions in cases where the
> +underflowing result is exact, but not missing underflow exceptions in
> +cases where it is inexact.)
>  @item
>  @Theglibc{} does not aim for functions to satisfy other properties of
> -- 
> Joseph S. Myers

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