Fix acosf underflow (bug 14153)

[Andreas Jaeger]

On 05/24/2012 11:52 PM, Joseph S. Myers wrote:
> Bug 14153 is a spurious underflow exception from acosf (2e-17); the
> cause is the acosf implementation calculating a value on the order of
> the cube of its argument, which underflows.  There is a test for very
> small arguments, maybe to avoid underflow, and if such an argument is
> encountered it returns pi/2 (computed by adding high and low parts, I
> suppose to get correct results for the current rounding mode and to
> raise the inexact exception).  However, this tests for values no more
> than 2**-57, as if the test was unchanged from a "double" version of
> the function; it suffices to test for values no more than 2**-26,
> which avoids the problem with internal underflows.  (For small x, the
> correct value is essentially pi/2 - x.  Compute pi/2 rounded to 25
> bits - one more bit than float - which is the nearest value that is
> representable as float or half way between two representable values.
> Compute the difference of pi/2 and that rounded values; if |x| is
> smaller than that difference then the result of rounding pi/2 - x will
> be the same in any rounding mode as the result of rounding pi/2.  Now
> that difference is more than 2**-26.)
>
> I propose this patch fixing the bug that way.  Tested x86_64 and x86.

thanks, this is fine,

Andreas
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