Fix erf underflow handling near 0 (bug 16516).

Bug 16516 reports spurious underflows from erf (for all floating-point
types), when the result is close to underflowing but does not actually
underflow.

erf (x) is about (2/sqrt(pi))*x for x close to 0, so there are
subnormal arguments for which it does not underflow.  The various
implementations do (x + efx*x) (for efx = 2/sqrt(pi) - 1), for greater
accuracy than if just using a single multiplication by an
approximation to 2/sqrt(pi) (effectively, this way there are a few
more bits in the approximation to 2/sqrt(pi)).  This can introduce
underflows when efx*x underflows even though the final result does
not, so a scaled calculation with 8*efx is done in these cases - but 8
is not a big enough scale factor to avoid all such underflows.  16 is
(any underflows with a scale factor of 16 would only occur when the
final result underflows), so this patch changes the code to use that
factor.  Rather than recomputing all the values of the efx8 variable,
it is removed, leaving it to the compiler's constant folding to
compute 16*efx.  As such scaling can also lose underflows when the
final scaling down happens to be exact, appropriate checks are added
to ensure underflow exceptions occur when required in such cases.

Tested x86_64 and x86; no ulps updates needed.  Also spot-checked for
powerpc32 and mips64 to verify the changes to the ldbl-128ibm and
ldbl-128 implementations.

	[BZ #16516]
	* sysdeps/ieee754/dbl-64/s_erf.c (efx8): Remove variable.
	(__erf): Scale by 16 instead of 8 in potentially underflowing
	case.  Ensure exception if result actually underflows.
	* sysdeps/ieee754/flt-32/s_erff.c (efx8): Remove variable.
	(__erff): Scale by 16 instead of 8 in potentially underflowing
	case.  Ensure exception if result actually underflows.
	* sysdeps/ieee754/ldbl-128/s_erfl.c: Include <float.h>.
	(efx8): Remove variable.
	(__erfl): Scale by 16 instead of 8 in potentially underflowing
	case.  Ensure exception if result actually underflows.
	* sysdeps/ieee754/ldbl-128ibm/s_erfl.c: Include <float.h>.
	(efx8): Remove variable.
	(__erfl): Scale by 16 instead of 8 in potentially underflowing
	case.  Ensure exception if result actually underflows.
	* sysdeps/ieee754/ldbl-96/s_erfl.c: Include <float.h>.
	(efx8): Remove variable.
	(__erfl): Scale by 16 instead of 8 in potentially underflowing
	case.  Ensure exception if result actually underflows.
	* math/auto-libm-test-in: Add more tests of erf.
	* math/auto-libm-test-out: Regenerated.

5 files changed, 54 insertions, 17 deletions

diff --git a/sysdeps/ieee754/dbl-64/s_erf.c b/sysdeps/ieee754/dbl-64/s_erf.c
index 3f37397224..ea0a73424e 100644
--- a/sysdeps/ieee754/dbl-64/s_erf.c
+++ b/sysdeps/ieee754/dbl-64/s_erf.c
@@ -128,7 +128,6 @@ static const double
  * Coefficients for approximation to  erf on [0,0.84375]
  */
   efx = 1.28379167095512586316e-01, /* 0x3FC06EBA, 0x8214DB69 */
-  efx8 = 1.02703333676410069053e+00, /* 0x3FF06EBA, 0x8214DB69 */
   pp[] = { 1.28379167095512558561e-01, /* 0x3FC06EBA, 0x8214DB68 */
 	   -3.25042107247001499370e-01, /* 0xBFD4CD7D, 0x691CB913 */
 	   -2.84817495755985104766e-02, /* 0xBF9D2A51, 0xDBD7194F */
@@ -211,7 +210,16 @@ __erf (double x)
       if (ix < 0x3e300000)              /* |x|<2**-28 */
 	{
 	  if (ix < 0x00800000)
-	    return 0.125 * (8.0 * x + efx8 * x);  /*avoid underflow */
+	    {
+	      /* Avoid spurious underflow.  */
+	      double ret = 0.0625 * (16.0 * x + (16.0 * efx) * x);
+	      if (fabs (ret) < DBL_MIN)
+		{
+		  double force_underflow = ret * ret;
+		  math_force_eval (force_underflow);
+		}
+	      return ret;
+	    }
 	  return x + efx * x;
 	}
       z = x * x;
diff --git a/sysdeps/ieee754/flt-32/s_erff.c b/sysdeps/ieee754/flt-32/s_erff.c
index 7c09589648..d23cff9293 100644
--- a/sysdeps/ieee754/flt-32/s_erff.c
+++ b/sysdeps/ieee754/flt-32/s_erff.c
@@ -33,7 +33,6 @@ erx =  8.4506291151e-01, /* 0x3f58560b */
  * Coefficients for approximation to  erf on [0,0.84375]
  */
 efx =  1.2837916613e-01, /* 0x3e0375d4 */
-efx8=  1.0270333290e+00, /* 0x3f8375d4 */
 pp0  =  1.2837916613e-01, /* 0x3e0375d4 */
 pp1  = -3.2504209876e-01, /* 0xbea66beb */
 pp2  = -2.8481749818e-02, /* 0xbce9528f */
@@ -111,8 +110,16 @@ float __erff(float x)
 	if(ix < 0x3f580000) {		/* |x|<0.84375 */
 	    if(ix < 0x31800000) { 	/* |x|<2**-28 */
 	        if (ix < 0x04000000)
-		    /*avoid underflow */
-		    return (float)0.125*((float)8.0*x+efx8*x);
+		  {
+		    /* Avoid spurious underflow.  */
+		    float ret = 0.0625f * (16.0f * x + (16.0f * efx) * x);
+		    if (fabsf (ret) < FLT_MIN)
+		      {
+			float force_underflow = ret * ret;
+			math_force_eval (force_underflow);
+		      }
+		    return ret;
+		  }
 		return x + efx*x;
 	    }
 	    z = x*x;
diff --git a/sysdeps/ieee754/ldbl-128/s_erfl.c b/sysdeps/ieee754/ldbl-128/s_erfl.c
index ef65ed8922..35ca8c1360 100644
--- a/sysdeps/ieee754/ldbl-128/s_erfl.c
+++ b/sysdeps/ieee754/ldbl-128/s_erfl.c
@@ -97,6 +97,7 @@
  */
 
 #include <errno.h>
+#include <float.h>
 #include <math.h>
 #include <math_private.h>
 
@@ -143,9 +144,7 @@ tiny = 1e-4931L,
   one = 1.0L,
   two = 2.0L,
   /* 2/sqrt(pi) - 1 */
-  efx = 1.2837916709551257389615890312154517168810E-1L,
-  /* 8 * (2/sqrt(pi) - 1) */
-  efx8 = 1.0270333367641005911692712249723613735048E0L;
+  efx = 1.2837916709551257389615890312154517168810E-1L;
 
 
 /* erf(x)  = x  + x R(x^2)
@@ -782,7 +781,16 @@ __erfl (long double x)
       if (ix < 0x3fc60000) /* |x|<2**-57 */
 	{
 	  if (ix < 0x00080000)
-	    return 0.125 * (8.0 * x + efx8 * x);	/*avoid underflow */
+	    {
+	      /* Avoid spurious underflow.  */
+	      long double ret =  0.0625 * (16.0 * x + (16.0 * efx) * x);
+	      if (fabsl (ret) < LDBL_MIN)
+		{
+		  long double force_underflow = ret * ret;
+		  math_force_eval (force_underflow);
+		}
+	      return ret;
+	    }
 	  return x + efx * x;
 	}
       y = a + a * neval (z, TN1, NTN1) / deval (z, TD1, NTD1);
diff --git a/sysdeps/ieee754/ldbl-128ibm/s_erfl.c b/sysdeps/ieee754/ldbl-128ibm/s_erfl.c
index 95dc415845..f55e8b7879 100644
--- a/sysdeps/ieee754/ldbl-128ibm/s_erfl.c
+++ b/sysdeps/ieee754/ldbl-128ibm/s_erfl.c
@@ -102,6 +102,7 @@
  */
 
 #include <errno.h>
+#include <float.h>
 #include <math.h>
 #include <math_private.h>
 #include <math_ldbl_opt.h>
@@ -149,9 +150,7 @@ tiny = 1e-300L,
   one = 1.0L,
   two = 2.0L,
   /* 2/sqrt(pi) - 1 */
-  efx = 1.2837916709551257389615890312154517168810E-1L,
-  /* 8 * (2/sqrt(pi) - 1) */
-  efx8 = 1.0270333367641005911692712249723613735048E0L;
+  efx = 1.2837916709551257389615890312154517168810E-1L;
 
 
 /* erf(x)  = x  + x R(x^2)
@@ -801,10 +800,17 @@ __erfl (long double x)
 	  if (ix < 0x00800000)
 	    {
 	      /* erf (-0) = -0.  Unfortunately, for IBM extended double
-		 0.125 * (8.0 * x + efx8 * x) for x = -0 evaluates to 0.  */
+		 0.0625 * (16.0 * x + (16.0 * efx) * x) for x = -0
+		 evaluates to 0.  */
 	      if (x == 0)
 		return x;
-	      return 0.125 * (8.0 * x + efx8 * x);	/*avoid underflow */
+	      long double ret = 0.0625 * (16.0 * x + (16.0 * efx) * x);
+	      if (fabsl (ret) < LDBL_MIN)
+		{
+		  long double force_underflow = ret * ret;
+		  math_force_eval (force_underflow);
+		}
+	      return ret;
 	    }
 	  return x + efx * x;
 	}
diff --git a/sysdeps/ieee754/ldbl-96/s_erfl.c b/sysdeps/ieee754/ldbl-96/s_erfl.c
index 47e4b9e909..c27de812ca 100644
--- a/sysdeps/ieee754/ldbl-96/s_erfl.c
+++ b/sysdeps/ieee754/ldbl-96/s_erfl.c
@@ -105,6 +105,7 @@
 
 
 #include <errno.h>
+#include <float.h>
 #include <math.h>
 #include <math_private.h>
 
@@ -120,8 +121,6 @@ tiny = 1e-4931L,
  */
   /* 2/sqrt(pi) - 1 */
   efx = 1.2837916709551257389615890312154517168810E-1L,
-  /* 8 * (2/sqrt(pi) - 1) */
-  efx8 = 1.0270333367641005911692712249723613735048E0L,
 
   pp[6] = {
     1.122751350964552113068262337278335028553E6L,
@@ -272,7 +271,16 @@ __erfl (long double x)
       if (ix < 0x3fde8000) /* |x|<2**-33 */
 	{
 	  if (ix < 0x00080000)
-	    return 0.125 * (8.0 * x + efx8 * x);	/*avoid underflow */
+	    {
+	      /* Avoid spurious underflow.  */
+	      long double ret = 0.0625 * (16.0 * x + (16.0 * efx) * x);
+	      if (fabsl (ret) < LDBL_MIN)
+		{
+		  long double force_underflow = ret * ret;
+		  math_force_eval (force_underflow);
+		}
+	      return ret;
+	    }
 	  return x + efx * x;
 	}
       z = x * x;