Fix error in exp in magnitude [2e-32,2e-28]

While testing the exp function we noticed some errors at the specified
magnitude. Within this range the exp function returns the input value +1
as an output. We chose to run a test of 1m exponentially spaced values
in the ranges [-2^-27,-2^-32] and [2^-32,2^-27] which showed 7603 and
3912 results with an error of >=0.5 ULP (compared with MPFR in 128 bit)
with the highest being 0.56 ULP and 0.53 ULP.

It's easy to fix by changing the magnitude at which the input value +1
is returned from <2^-28 to <2^-32 and using the polynomial instead. This
reduces the number of results with an error of >=0.5 ULP to 485 and 479
in above tests, all of which are exactly 0.5 ULP.

As we were already checking on exp we also took a look at expf. For expf
the magnitude where the input value +1 is returned can be increased from
<2^-28 to <2^-23 without accuracy loss for a slight performance
improvement. To ensure this was the correct value we tested all values
in the ranges [-2^-17,-2^-28] and [2^-28,2^-17] (~92.3m values each).

2 files changed, 2 insertions, 2 deletions

diff --git a/newlib/libm/math/e_exp.c b/newlib/libm/math/e_exp.c
index 81ea64d..d23b116 100644
--- a/newlib/libm/math/e_exp.c
+++ b/newlib/libm/math/e_exp.c
@@ -142,7 +142,7 @@ P5   =  4.13813679705723846039e-08; /* 0x3E663769, 0x72BEA4D0 */
 	    }
 	    x  = hi - lo;
 	} 
-	else if(hx < 0x3e300000)  {	/* when |x|<2**-28 */
+	else if(hx < 0x3df00000)  {	/* when |x|<2**-32 */
 	    if(huge+x>one) return one+x;/* trigger inexact */
 	}
 
diff --git a/newlib/libm/math/ef_exp.c b/newlib/libm/math/ef_exp.c
index e817370..fb3e2ff 100644
--- a/newlib/libm/math/ef_exp.c
+++ b/newlib/libm/math/ef_exp.c
@@ -77,7 +77,7 @@ P5   =  4.1381369442e-08; /* 0x3331bb4c */
 	    }
 	    x  = hi - lo;
 	} 
-	else if(hx < 0x31800000)  {	/* when |x|<2**-28 */
+	else if(hx < 0x34000000)  {	/* when |x|<2**-23 */
 	    if(huge+x>one) return one+x;/* trigger inexact */
 	}