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

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

[Corinna Vinschen]

https://sourceware.org/git/gitweb.cgi?p=newlib-cygwin.git;h=a8a40ee5750cb8a4280379a38c1ed490262481c4

commit a8a40ee5750cb8a4280379a38c1ed490262481c4
Author: Fabian Schriever <fabian.schriever@gtd-gmbh.de>
Date:   Fri Mar 6 15:46:33 2020 +0100

    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).

Diff:
---
 newlib/libm/math/e_exp.c  | 2 +-
 newlib/libm/math/ef_exp.c | 2 +-
 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 */
 	}