Fix erfcf spurious underflows (bug 18217).

The flt-32 implementation of erfcf produces spurious underflow
exceptions for some arguments close to 0, because of calculations
squaring the argument and then multiplying by small constants.  This
patch fixes this by adjusting the threshold for arguments for which
the result is so close to 1 that 1 - x will give the right result from
2**-56 to 2**-26.  (If 1 - x * 2/sqrt(pi) were used, the errors would be
on the order of x^3 and a much larger threshold could be used.)

Tested for x86_64 and x86.

	[BZ #18217]
	* sysdeps/ieee754/flt-32/s_erff.c (__erfcf): Use 2**-26 not 2**-56
	as threshold for returning 1 - x.
	* math/auto-libm-test-in: Add more tests of erfc.
	* math/auto-libm-test-out: Regenerated.

5 files changed, 63 insertions, 3 deletions

diff --git a/ChangeLog b/ChangeLog
index 1179142c92..454a8d26e1 100644
--- a/ChangeLog
+++ b/ChangeLog
@@ -1,3 +1,11 @@
+2015-05-15  Joseph Myers  <joseph@codesourcery.com>
+
+	[BZ #18217]
+	* sysdeps/ieee754/flt-32/s_erff.c (__erfcf): Use 2**-26 not 2**-56
+	as threshold for returning 1 - x.
+	* math/auto-libm-test-in: Add more tests of erfc.
+	* math/auto-libm-test-out: Regenerated.
+
 2015-05-14  Joseph Myers  <joseph@codesourcery.com>
 
 	[BZ #18196]
diff --git a/NEWS b/NEWS
index c46ee35103..9251333195 100644
--- a/NEWS
+++ b/NEWS
@@ -16,8 +16,8 @@ Version 2.22
   17964, 17965, 17967, 17969, 17978, 17987, 17991, 17996, 17998, 17999,
   18007, 18019, 18020, 18029, 18030, 18032, 18036, 18038, 18039, 18042,
   18043, 18046, 18047, 18068, 18080, 18093, 18100, 18104, 18110, 18111,
-  18125, 18128, 18138, 18185, 18196, 18197, 18206, 18210, 18211, 18247,
-  18287, 18319, 18333, 18346, 18397, 18409.
+  18125, 18128, 18138, 18185, 18196, 18197, 18206, 18210, 18211, 18217,
+  18247, 18287, 18319, 18333, 18346, 18397, 18409.
 
 * Cache information can be queried via sysconf() function on s390 e.g. with
   _SC_LEVEL1_ICACHE_SIZE as argument.
diff --git a/math/auto-libm-test-in b/math/auto-libm-test-in
index 1deb99ad0d..c4bfe7439c 100644
--- a/math/auto-libm-test-in
+++ b/math/auto-libm-test-in
@@ -1129,6 +1129,8 @@ erf -0x1.3a0d48p+0
 
 erfc 0.0
 erfc -0
+erfc 0x1p-55
+erfc -0x1p-55
 erfc 0.125
 erfc 0.75
 erfc 1.25
diff --git a/math/auto-libm-test-out b/math/auto-libm-test-out
index 9f6bbbbf6f..d717ea4431 100644
--- a/math/auto-libm-test-out
+++ b/math/auto-libm-test-out
@@ -101555,6 +101555,56 @@ erfc -0
 = erfc tonearest ldbl-128ibm -0x0p+0L : 0x1p+0L : inexact-ok
 = erfc towardzero ldbl-128ibm -0x0p+0L : 0x1p+0L : inexact-ok
 = erfc upward ldbl-128ibm -0x0p+0L : 0x1p+0L : inexact-ok
+erfc 0x1p-55
+= erfc downward flt-32 0x2p-56f : 0xf.fffffp-4f : inexact-ok
+= erfc tonearest flt-32 0x2p-56f : 0x1p+0f : inexact-ok
+= erfc towardzero flt-32 0x2p-56f : 0xf.fffffp-4f : inexact-ok
+= erfc upward flt-32 0x2p-56f : 0x1p+0f : inexact-ok
+= erfc downward dbl-64 0x2p-56 : 0xf.ffffffffffff8p-4 : inexact-ok
+= erfc tonearest dbl-64 0x2p-56 : 0x1p+0 : inexact-ok
+= erfc towardzero dbl-64 0x2p-56 : 0xf.ffffffffffff8p-4 : inexact-ok
+= erfc upward dbl-64 0x2p-56 : 0x1p+0 : inexact-ok
+= erfc downward ldbl-96-intel 0x2p-56L : 0xf.ffffffffffffdbep-4L : inexact-ok
+= erfc tonearest ldbl-96-intel 0x2p-56L : 0xf.ffffffffffffdbep-4L : inexact-ok
+= erfc towardzero ldbl-96-intel 0x2p-56L : 0xf.ffffffffffffdbep-4L : inexact-ok
+= erfc upward ldbl-96-intel 0x2p-56L : 0xf.ffffffffffffdbfp-4L : inexact-ok
+= erfc downward ldbl-96-m68k 0x2p-56L : 0xf.ffffffffffffdbep-4L : inexact-ok
+= erfc tonearest ldbl-96-m68k 0x2p-56L : 0xf.ffffffffffffdbep-4L : inexact-ok
+= erfc towardzero ldbl-96-m68k 0x2p-56L : 0xf.ffffffffffffdbep-4L : inexact-ok
+= erfc upward ldbl-96-m68k 0x2p-56L : 0xf.ffffffffffffdbfp-4L : inexact-ok
+= erfc downward ldbl-128 0x2p-56L : 0xf.ffffffffffffdbe4515f7ac925d8p-4L : inexact-ok
+= erfc tonearest ldbl-128 0x2p-56L : 0xf.ffffffffffffdbe4515f7ac925ep-4L : inexact-ok
+= erfc towardzero ldbl-128 0x2p-56L : 0xf.ffffffffffffdbe4515f7ac925d8p-4L : inexact-ok
+= erfc upward ldbl-128 0x2p-56L : 0xf.ffffffffffffdbe4515f7ac925ep-4L : inexact-ok
+= erfc downward ldbl-128ibm 0x2p-56L : 0xf.ffffffffffffdbe4515f7ac924p-4L : inexact-ok
+= erfc tonearest ldbl-128ibm 0x2p-56L : 0xf.ffffffffffffdbe4515f7ac924p-4L : inexact-ok
+= erfc towardzero ldbl-128ibm 0x2p-56L : 0xf.ffffffffffffdbe4515f7ac924p-4L : inexact-ok
+= erfc upward ldbl-128ibm 0x2p-56L : 0xf.ffffffffffffdbe4515f7ac928p-4L : inexact-ok
+erfc -0x1p-55
+= erfc downward flt-32 -0x2p-56f : 0x1p+0f : inexact-ok
+= erfc tonearest flt-32 -0x2p-56f : 0x1p+0f : inexact-ok
+= erfc towardzero flt-32 -0x2p-56f : 0x1p+0f : inexact-ok
+= erfc upward flt-32 -0x2p-56f : 0x1.000002p+0f : inexact-ok
+= erfc downward dbl-64 -0x2p-56 : 0x1p+0 : inexact-ok
+= erfc tonearest dbl-64 -0x2p-56 : 0x1p+0 : inexact-ok
+= erfc towardzero dbl-64 -0x2p-56 : 0x1p+0 : inexact-ok
+= erfc upward dbl-64 -0x2p-56 : 0x1.0000000000001p+0 : inexact-ok
+= erfc downward ldbl-96-intel -0x2p-56L : 0x1.000000000000024p+0L : inexact-ok
+= erfc tonearest ldbl-96-intel -0x2p-56L : 0x1.0000000000000242p+0L : inexact-ok
+= erfc towardzero ldbl-96-intel -0x2p-56L : 0x1.000000000000024p+0L : inexact-ok
+= erfc upward ldbl-96-intel -0x2p-56L : 0x1.0000000000000242p+0L : inexact-ok
+= erfc downward ldbl-96-m68k -0x2p-56L : 0x1.000000000000024p+0L : inexact-ok
+= erfc tonearest ldbl-96-m68k -0x2p-56L : 0x1.0000000000000242p+0L : inexact-ok
+= erfc towardzero ldbl-96-m68k -0x2p-56L : 0x1.000000000000024p+0L : inexact-ok
+= erfc upward ldbl-96-m68k -0x2p-56L : 0x1.0000000000000242p+0L : inexact-ok
+= erfc downward ldbl-128 -0x2p-56L : 0x1.0000000000000241baea08536da2p+0L : inexact-ok
+= erfc tonearest ldbl-128 -0x2p-56L : 0x1.0000000000000241baea08536da2p+0L : inexact-ok
+= erfc towardzero ldbl-128 -0x2p-56L : 0x1.0000000000000241baea08536da2p+0L : inexact-ok
+= erfc upward ldbl-128 -0x2p-56L : 0x1.0000000000000241baea08536da3p+0L : inexact-ok
+= erfc downward ldbl-128ibm -0x2p-56L : 0x1.0000000000000241baea08536d8p+0L : inexact-ok
+= erfc tonearest ldbl-128ibm -0x2p-56L : 0x1.0000000000000241baea08536d8p+0L : inexact-ok
+= erfc towardzero ldbl-128ibm -0x2p-56L : 0x1.0000000000000241baea08536d8p+0L : inexact-ok
+= erfc upward ldbl-128ibm -0x2p-56L : 0x1.0000000000000241baea08536ep+0L : inexact-ok
 erfc 0.125
 = erfc downward flt-32 0x2p-4f : 0xd.c143cp-4f : inexact-ok
 = erfc tonearest flt-32 0x2p-4f : 0xd.c143dp-4f : inexact-ok
diff --git a/sysdeps/ieee754/flt-32/s_erff.c b/sysdeps/ieee754/flt-32/s_erff.c
index d23cff9293..2be44cc40c 100644
--- a/sysdeps/ieee754/flt-32/s_erff.c
+++ b/sysdeps/ieee754/flt-32/s_erff.c
@@ -169,7 +169,7 @@ float __erfcf(float x)
 	}
 
 	if(ix < 0x3f580000) {		/* |x|<0.84375 */
-	    if(ix < 0x23800000)  	/* |x|<2**-56 */
+	    if(ix < 0x32800000)  	/* |x|<2**-26 */
 		return one-x;
 	    z = x*x;
 	    r = pp0+z*(pp1+z*(pp2+z*(pp3+z*pp4)));