Fix clog, clog10 inaccuracy (bug 19016).

For arguments with X^2 + Y^2 close to 1, clog and clog10 avoid large
errors from log(hypot) by computing X^2 + Y^2 - 1 in a way that avoids
cancellation error and then using log1p.

However, the thresholds for using that approach still result in log
being used on argument as large as sqrt(13/16) > 0.9, leading to
significant errors, in some cases above the 9ulp maximum allowed in
glibc libm.  This patch arranges for the approach using log1p to be
used in any cases where |X|, |Y| < 1 and X^2 + Y^2 >= 0.5 (with the
existing allowance for cases where one of X and Y is very small),
adjusting the __x2y2m1 functions to work with the wider range of
inputs.  This way, log only gets used on arguments below sqrt(1/2) (or
substantially above 1), where the error involved is much less.

Tested for x86_64, x86, mips64 and powerpc.  For the ulps regeneration
I removed the existing clog and clog10 ulps before regenerating to
allow any reduced ulps to appear.  Tests added include those found by
random test generation to produce large ulps either before or after
the patch, and some found by trying inputs close to the (0.75, 0.5)
threshold where the potential errors from using log are largest.

	[BZ #19016]
	* sysdeps/generic/math_private.h (__x2y2m1f): Update comment to
	allow more cases with X^2 + Y^2 >= 0.5.
	* sysdeps/ieee754/dbl-64/x2y2m1.c (__x2y2m1): Likewise.  Add -1 as
	normal element in sum instead of special-casing based on values of
	arguments.
	* sysdeps/ieee754/dbl-64/x2y2m1f.c (__x2y2m1f): Update comment.
	* sysdeps/ieee754/ldbl-128/x2y2m1l.c (__x2y2m1l): Likewise.  Add
	-1 as normal element in sum instead of special-casing based on
	values of arguments.
	* sysdeps/ieee754/ldbl-128ibm/x2y2m1l.c (__x2y2m1l): Likewise.
	* sysdeps/ieee754/ldbl-96/x2y2m1.c [FLT_EVAL_METHOD != 0]
	(__x2y2m1): Update comment.
	* sysdeps/ieee754/ldbl-96/x2y2m1l.c (__x2y2m1l): Likewise.  Add -1
	as normal element in sum instead of special-casing based on values
	of arguments.
	* math/s_clog.c (__clog): Handle more cases using log1p without
	hypot.
	* math/s_clog10.c (__clog10): Likewise.
	* math/s_clog10f.c (__clog10f): Likewise.
	* math/s_clog10l.c (__clog10l): Likewise.
	* math/s_clogf.c (__clogf): Likewise.
	* math/s_clogl.c (__clogl): Likewise.
	* math/auto-libm-test-in: Add more tests of clog and clog10.
	* math/auto-libm-test-out: Regenerated.
	* sysdeps/i386/fpu/libm-test-ulps: Update.
	* sysdeps/x86_64/fpu/libm-test-ulps: Likewise.

9 files changed, 119 insertions, 143 deletions

diff --git a/sysdeps/generic/math_private.h b/sysdeps/generic/math_private.h
index a8f1a8e4e5..cf1865dac8 100644
--- a/sysdeps/generic/math_private.h
+++ b/sysdeps/generic/math_private.h
@@ -365,8 +365,8 @@ extern double __slowpow (double __x, double __y, double __z);
 extern void __docos (double __x, double __dx, double __v[]);
 
 /* Return X^2 + Y^2 - 1, computed without large cancellation error.
-   It is given that 1 > X >= Y >= epsilon / 2, and that either X >=
-   0.75 or Y >= 0.5.  */
+   It is given that 1 > X >= Y >= epsilon / 2, and that X^2 + Y^2 >=
+   0.5.  */
 extern float __x2y2m1f (float x, float y);
 extern double __x2y2m1 (double x, double y);
 extern long double __x2y2m1l (long double x, long double y);
diff --git a/sysdeps/i386/fpu/libm-test-ulps b/sysdeps/i386/fpu/libm-test-ulps
index 32f24d07a2..438a390338 100644
--- a/sysdeps/i386/fpu/libm-test-ulps
+++ b/sysdeps/i386/fpu/libm-test-ulps
@@ -836,12 +836,12 @@ ildouble: 3
 ldouble: 3
 
 Function: Real part of "clog":
-double: 3
-float: 2
-idouble: 3
-ifloat: 2
-ildouble: 4
-ldouble: 4
+double: 2
+float: 1
+idouble: 2
+ifloat: 1
+ildouble: 3
+ldouble: 3
 
 Function: Imaginary part of "clog":
 double: 1
@@ -864,10 +864,10 @@ ildouble: 2
 ldouble: 2
 
 Function: Real part of "clog10_downward":
-double: 5
-float: 4
-idouble: 5
-ifloat: 4
+double: 3
+float: 3
+idouble: 3
+ifloat: 3
 ildouble: 8
 ldouble: 8
 
@@ -876,14 +876,14 @@ double: 1
 float: 1
 idouble: 1
 ifloat: 1
-ildouble: 2
-ldouble: 2
+ildouble: 3
+ldouble: 3
 
 Function: Real part of "clog10_towardzero":
-double: 5
-float: 4
-idouble: 5
-ifloat: 4
+double: 3
+float: 3
+idouble: 3
+ifloat: 3
 ildouble: 8
 ldouble: 8
 
@@ -896,12 +896,12 @@ ildouble: 3
 ldouble: 3
 
 Function: Real part of "clog10_upward":
-double: 5
-float: 5
-idouble: 5
-ifloat: 5
-ildouble: 6
-ldouble: 6
+double: 3
+float: 3
+idouble: 3
+ifloat: 3
+ildouble: 7
+ldouble: 7
 
 Function: Imaginary part of "clog10_upward":
 double: 1
@@ -912,12 +912,12 @@ ildouble: 3
 ldouble: 3
 
 Function: Real part of "clog_downward":
-double: 5
-float: 5
-idouble: 5
-ifloat: 5
-ildouble: 7
-ldouble: 7
+double: 3
+float: 3
+idouble: 3
+ifloat: 3
+ildouble: 5
+ldouble: 5
 
 Function: Imaginary part of "clog_downward":
 double: 1
@@ -928,12 +928,12 @@ ildouble: 1
 ldouble: 1
 
 Function: Real part of "clog_towardzero":
-double: 5
-float: 5
-idouble: 5
-ifloat: 5
-ildouble: 8
-ldouble: 8
+double: 3
+float: 3
+idouble: 3
+ifloat: 3
+ildouble: 5
+ldouble: 5
 
 Function: Imaginary part of "clog_towardzero":
 double: 1
@@ -944,12 +944,12 @@ ildouble: 1
 ldouble: 1
 
 Function: Real part of "clog_upward":
-double: 5
-float: 5
-idouble: 5
-ifloat: 5
-ildouble: 6
-ldouble: 6
+double: 2
+float: 3
+idouble: 2
+ifloat: 3
+ildouble: 4
+ldouble: 4
 
 Function: Imaginary part of "clog_upward":
 double: 1
diff --git a/sysdeps/ieee754/dbl-64/x2y2m1.c b/sysdeps/ieee754/dbl-64/x2y2m1.c
index c96dae59e1..b040097d18 100644
--- a/sysdeps/ieee754/dbl-64/x2y2m1.c
+++ b/sysdeps/ieee754/dbl-64/x2y2m1.c
@@ -80,32 +80,26 @@ compare (const void *p, const void *q)
 }
 
 /* Return X^2 + Y^2 - 1, computed without large cancellation error.
-   It is given that 1 > X >= Y >= epsilon / 2, and that either X >=
-   0.75 or Y >= 0.5.  */
+   It is given that 1 > X >= Y >= epsilon / 2, and that X^2 + Y^2 >=
+   0.5.  */
 
 double
 __x2y2m1 (double x, double y)
 {
-  double vals[4];
+  double vals[5];
   SET_RESTORE_ROUND (FE_TONEAREST);
   mul_split (&vals[1], &vals[0], x, x);
   mul_split (&vals[3], &vals[2], y, y);
-  if (x >= 0.75)
-    vals[1] -= 1.0;
-  else
-    {
-      vals[1] -= 0.5;
-      vals[3] -= 0.5;
-    }
-  qsort (vals, 4, sizeof (double), compare);
+  vals[4] = -1.0;
+  qsort (vals, 5, sizeof (double), compare);
   /* Add up the values so that each element of VALS has absolute value
      at most equal to the last set bit of the next nonzero
      element.  */
-  for (size_t i = 0; i <= 2; i++)
+  for (size_t i = 0; i <= 3; i++)
     {
       add_split (&vals[i + 1], &vals[i], vals[i + 1], vals[i]);
-      qsort (vals + i + 1, 3 - i, sizeof (double), compare);
+      qsort (vals + i + 1, 4 - i, sizeof (double), compare);
     }
   /* Now any error from this addition will be small.  */
-  return vals[3] + vals[2] + vals[1] + vals[0];
+  return vals[4] + vals[3] + vals[2] + vals[1] + vals[0];
 }
diff --git a/sysdeps/ieee754/dbl-64/x2y2m1f.c b/sysdeps/ieee754/dbl-64/x2y2m1f.c
index 43a8acf8a8..835f6a0e45 100644
--- a/sysdeps/ieee754/dbl-64/x2y2m1f.c
+++ b/sysdeps/ieee754/dbl-64/x2y2m1f.c
@@ -21,8 +21,8 @@
 #include <float.h>
 
 /* Return X^2 + Y^2 - 1, computed without large cancellation error.
-   It is given that 1 > X >= Y >= epsilon / 2, and that either X >=
-   0.75 or Y >= 0.5.  */
+   It is given that 1 > X >= Y >= epsilon / 2, and that X^2 + Y^2 >=
+   0.5.  */
 
 float
 __x2y2m1f (float x, float y)
diff --git a/sysdeps/ieee754/ldbl-128/x2y2m1l.c b/sysdeps/ieee754/ldbl-128/x2y2m1l.c
index 11757c6af8..a0498c30bb 100644
--- a/sysdeps/ieee754/ldbl-128/x2y2m1l.c
+++ b/sysdeps/ieee754/ldbl-128/x2y2m1l.c
@@ -80,32 +80,26 @@ compare (const void *p, const void *q)
 }
 
 /* Return X^2 + Y^2 - 1, computed without large cancellation error.
-   It is given that 1 > X >= Y >= epsilon / 2, and that either X >=
-   0.75 or Y >= 0.5.  */
+   It is given that 1 > X >= Y >= epsilon / 2, and that X^2 + Y^2 >=
+   0.5.  */
 
 long double
 __x2y2m1l (long double x, long double y)
 {
-  long double vals[4];
+  long double vals[5];
   SET_RESTORE_ROUNDL (FE_TONEAREST);
   mul_split (&vals[1], &vals[0], x, x);
   mul_split (&vals[3], &vals[2], y, y);
-  if (x >= 0.75L)
-    vals[1] -= 1.0L;
-  else
-    {
-      vals[1] -= 0.5L;
-      vals[3] -= 0.5L;
-    }
-  qsort (vals, 4, sizeof (long double), compare);
+  vals[4] = -1.0L;
+  qsort (vals, 5, sizeof (long double), compare);
   /* Add up the values so that each element of VALS has absolute value
      at most equal to the last set bit of the next nonzero
      element.  */
-  for (size_t i = 0; i <= 2; i++)
+  for (size_t i = 0; i <= 3; i++)
     {
       add_split (&vals[i + 1], &vals[i], vals[i + 1], vals[i]);
-      qsort (vals + i + 1, 3 - i, sizeof (long double), compare);
+      qsort (vals + i + 1, 4 - i, sizeof (long double), compare);
     }
   /* Now any error from this addition will be small.  */
-  return vals[3] + vals[2] + vals[1] + vals[0];
+  return vals[4] + vals[3] + vals[2] + vals[1] + vals[0];
 }
diff --git a/sysdeps/ieee754/ldbl-128ibm/x2y2m1l.c b/sysdeps/ieee754/ldbl-128ibm/x2y2m1l.c
index a001b58ca5..081fb98255 100644
--- a/sysdeps/ieee754/ldbl-128ibm/x2y2m1l.c
+++ b/sysdeps/ieee754/ldbl-128ibm/x2y2m1l.c
@@ -80,13 +80,13 @@ compare (const void *p, const void *q)
 }
 
 /* Return X^2 + Y^2 - 1, computed without large cancellation error.
-   It is given that 1 > X >= Y >= epsilon / 2, and that either X >=
-   0.75 or Y >= 0.5.  */
+   It is given that 1 > X >= Y >= epsilon / 2, and that X^2 + Y^2 >=
+   0.5.  */
 
 long double
 __x2y2m1l (long double x, long double y)
 {
-  double vals[12];
+  double vals[13];
   SET_RESTORE_ROUND (FE_TONEAREST);
   union ibm_extended_long_double xu, yu;
   xu.ld = x;
@@ -105,25 +105,19 @@ __x2y2m1l (long double x, long double y)
   vals[8] *= 2.0;
   vals[9] *= 2.0;
   mul_split (&vals[11], &vals[10], yu.d[1].d, yu.d[1].d);
-  if (xu.d[0].d >= 0.75)
-    vals[1] -= 1.0;
-  else
-    {
-      vals[1] -= 0.5;
-      vals[7] -= 0.5;
-    }
-  qsort (vals, 12, sizeof (double), compare);
+  vals[12] = -1.0;
+  qsort (vals, 13, sizeof (double), compare);
   /* Add up the values so that each element of VALS has absolute value
      at most equal to the last set bit of the next nonzero
      element.  */
-  for (size_t i = 0; i <= 10; i++)
+  for (size_t i = 0; i <= 11; i++)
     {
       add_split (&vals[i + 1], &vals[i], vals[i + 1], vals[i]);
-      qsort (vals + i + 1, 11 - i, sizeof (double), compare);
+      qsort (vals + i + 1, 12 - i, sizeof (double), compare);
     }
   /* Now any error from this addition will be small.  */
-  long double retval = (long double) vals[11];
-  for (size_t i = 10; i != (size_t) -1; i--)
+  long double retval = (long double) vals[12];
+  for (size_t i = 11; i != (size_t) -1; i--)
     retval += (long double) vals[i];
   return retval;
 }
diff --git a/sysdeps/ieee754/ldbl-96/x2y2m1.c b/sysdeps/ieee754/ldbl-96/x2y2m1.c
index a6cc82cb29..2f6b0be8f9 100644
--- a/sysdeps/ieee754/ldbl-96/x2y2m1.c
+++ b/sysdeps/ieee754/ldbl-96/x2y2m1.c
@@ -27,8 +27,8 @@
 #else
 
 /* Return X^2 + Y^2 - 1, computed without large cancellation error.
-   It is given that 1 > X >= Y >= epsilon / 2, and that either X >=
-   0.75 or Y >= 0.5.  */
+   It is given that 1 > X >= Y >= epsilon / 2, and that X^2 + Y^2 >=
+   0.5.  */
 
 double
 __x2y2m1 (double x, double y)
diff --git a/sysdeps/ieee754/ldbl-96/x2y2m1l.c b/sysdeps/ieee754/ldbl-96/x2y2m1l.c
index 11757c6af8..a0498c30bb 100644
--- a/sysdeps/ieee754/ldbl-96/x2y2m1l.c
+++ b/sysdeps/ieee754/ldbl-96/x2y2m1l.c
@@ -80,32 +80,26 @@ compare (const void *p, const void *q)
 }
 
 /* Return X^2 + Y^2 - 1, computed without large cancellation error.
-   It is given that 1 > X >= Y >= epsilon / 2, and that either X >=
-   0.75 or Y >= 0.5.  */
+   It is given that 1 > X >= Y >= epsilon / 2, and that X^2 + Y^2 >=
+   0.5.  */
 
 long double
 __x2y2m1l (long double x, long double y)
 {
-  long double vals[4];
+  long double vals[5];
   SET_RESTORE_ROUNDL (FE_TONEAREST);
   mul_split (&vals[1], &vals[0], x, x);
   mul_split (&vals[3], &vals[2], y, y);
-  if (x >= 0.75L)
-    vals[1] -= 1.0L;
-  else
-    {
-      vals[1] -= 0.5L;
-      vals[3] -= 0.5L;
-    }
-  qsort (vals, 4, sizeof (long double), compare);
+  vals[4] = -1.0L;
+  qsort (vals, 5, sizeof (long double), compare);
   /* Add up the values so that each element of VALS has absolute value
      at most equal to the last set bit of the next nonzero
      element.  */
-  for (size_t i = 0; i <= 2; i++)
+  for (size_t i = 0; i <= 3; i++)
     {
       add_split (&vals[i + 1], &vals[i], vals[i + 1], vals[i]);
-      qsort (vals + i + 1, 3 - i, sizeof (long double), compare);
+      qsort (vals + i + 1, 4 - i, sizeof (long double), compare);
     }
   /* Now any error from this addition will be small.  */
-  return vals[3] + vals[2] + vals[1] + vals[0];
+  return vals[4] + vals[3] + vals[2] + vals[1] + vals[0];
 }
diff --git a/sysdeps/x86_64/fpu/libm-test-ulps b/sysdeps/x86_64/fpu/libm-test-ulps
index 12c3dd1e01..ef3ab70369 100644
--- a/sysdeps/x86_64/fpu/libm-test-ulps
+++ b/sysdeps/x86_64/fpu/libm-test-ulps
@@ -869,11 +869,11 @@ ldouble: 3
 
 Function: Real part of "clog":
 double: 3
-float: 2
+float: 3
 idouble: 3
-ifloat: 2
-ildouble: 4
-ldouble: 4
+ifloat: 3
+ildouble: 3
+ldouble: 3
 
 Function: Imaginary part of "clog":
 float: 1
@@ -883,9 +883,9 @@ ldouble: 1
 
 Function: Real part of "clog10":
 double: 3
-float: 3
+float: 4
 idouble: 3
-ifloat: 3
+ifloat: 4
 ildouble: 4
 ldouble: 4
 
@@ -898,10 +898,10 @@ ildouble: 2
 ldouble: 2
 
 Function: Real part of "clog10_downward":
-double: 6
-float: 6
-idouble: 6
-ifloat: 6
+double: 5
+float: 4
+idouble: 5
+ifloat: 4
 ildouble: 8
 ldouble: 8
 
@@ -910,14 +910,14 @@ double: 2
 float: 4
 idouble: 2
 ifloat: 4
-ildouble: 2
-ldouble: 2
+ildouble: 3
+ldouble: 3
 
 Function: Real part of "clog10_towardzero":
 double: 5
-float: 4
+float: 5
 idouble: 5
-ifloat: 4
+ifloat: 5
 ildouble: 8
 ldouble: 8
 
@@ -930,28 +930,28 @@ ildouble: 3
 ldouble: 3
 
 Function: Real part of "clog10_upward":
-double: 8
+double: 6
 float: 5
-idouble: 8
+idouble: 6
 ifloat: 5
-ildouble: 6
-ldouble: 6
+ildouble: 7
+ldouble: 7
 
 Function: Imaginary part of "clog10_upward":
 double: 2
-float: 3
+float: 4
 idouble: 2
-ifloat: 3
+ifloat: 4
 ildouble: 3
 ldouble: 3
 
 Function: Real part of "clog_downward":
-double: 7
-float: 5
-idouble: 7
-ifloat: 5
-ildouble: 7
-ldouble: 7
+double: 4
+float: 3
+idouble: 4
+ifloat: 3
+ildouble: 5
+ldouble: 5
 
 Function: Imaginary part of "clog_downward":
 double: 1
@@ -962,28 +962,28 @@ ildouble: 1
 ldouble: 1
 
 Function: Real part of "clog_towardzero":
-double: 7
-float: 5
-idouble: 7
-ifloat: 5
-ildouble: 8
-ldouble: 8
+double: 4
+float: 4
+idouble: 4
+ifloat: 4
+ildouble: 5
+ldouble: 5
 
 Function: Imaginary part of "clog_towardzero":
 double: 1
-float: 2
+float: 3
 idouble: 1
-ifloat: 2
+ifloat: 3
 ildouble: 1
 ldouble: 1
 
 Function: Real part of "clog_upward":
-double: 8
-float: 5
-idouble: 8
-ifloat: 5
-ildouble: 6
-ldouble: 6
+double: 4
+float: 3
+idouble: 4
+ifloat: 3
+ildouble: 4
+ldouble: 4
 
 Function: Imaginary part of "clog_upward":
 double: 1