Fix cutoffs. This is just a cleanup and an optimization for unusual

cases which are used mainly by regression tests.

As usual, the cutoff for tiny args was not correctly translated to
float precision.  It was 2**-54 but 2**-24 works.  It must be about
2**-precision, since the error from approximating log(1+x) by x is
about the same as |x|.  Exhaustive testing shows that 2**-24 gives
perfect rounding in round-to-nearest mode.

Similarly for the cutoff for being small, except this is not used by
so many other functions.  It was 2**-29 but 2**-15 works.  It must be
a bit smaller than sqrt(2**-precision), since the error from
approximating log(1+x) by x-x*x/2 is about the same as x*x.  Exhaustive
testing shows that 2**-15 gives a maximum error of 0.5052 ulps in
round-to-nearest-mode.  The algorithm for the general case is only good
for 0.8388 ulps, so this is sufficient (but it loses slightly on i386 --
then extra precision gives 0.5032 ulps for the general case).

While investigating this, I noticed that optimizing the usual case by
falling into a middle case involving a simple polynomial evaluation
(return x-x*x/2 instead of x here) is not such a good idea since it
gives an enormous pessimization of tinier args on machines for which
denormals are slow.  Float x*x/2 is denormal when |x| ~< 2**-64 and
x*x/2 is evaluated in float precision, so it can easily be denormal
for normal x.  This is even more interesting for general polynomial
evaluations.  Multiplying out large powers of x is normally a good
optimization since it reduces dependencies, but it creates denormals
starting with quite large x.

1 files changed, 2 insertions, 2 deletions

diff --git a/lib/msun/src/s_log1pf.c b/lib/msun/src/s_log1pf.c
index 6b2fbc7..de5341d 100644
--- a/lib/msun/src/s_log1pf.c
+++ b/lib/msun/src/s_log1pf.c
@@ -50,9 +50,9 @@ log1pf(float x)
 		if(x==(float)-1.0) return -two25/zero; /* log1p(-1)=+inf */
 		else return (x-x)/(x-x);	/* log1p(x<-1)=NaN */
 	    }
-	    if(ax<0x31000000) {			/* |x| < 2**-29 */
+	    if(ax<0x38000000) {			/* |x| < 2**-15 */
 		if(two25+x>zero			/* raise inexact */
-	            &&ax<0x24800000) 		/* |x| < 2**-54 */
+	            &&ax<0x33800000) 		/* |x| < 2**-24 */
 		    return x;
 		else
 		    return x - x*x*(float)0.5;