Fix inaccuracy of clog, clog10 near |z| = 1 (bug 13629).

6 files changed, 532 insertions, 0 deletions

diff --git a/sysdeps/ieee754/dbl-64/x2y2m1.c b/sysdeps/ieee754/dbl-64/x2y2m1.c
new file mode 100644
index 0000000000..4badde3be4
--- /dev/null
+++ b/sysdeps/ieee754/dbl-64/x2y2m1.c
@@ -0,0 +1,111 @@
+/* Compute x^2 + y^2 - 1, without large cancellation error.
+   Copyright (C) 2012 Free Software Foundation, Inc.
+   This file is part of the GNU C Library.
+
+   The GNU C Library is free software; you can redistribute it and/or
+   modify it under the terms of the GNU Lesser General Public
+   License as published by the Free Software Foundation; either
+   version 2.1 of the License, or (at your option) any later version.
+
+   The GNU C Library is distributed in the hope that it will be useful,
+   but WITHOUT ANY WARRANTY; without even the implied warranty of
+   MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
+   Lesser General Public License for more details.
+
+   You should have received a copy of the GNU Lesser General Public
+   License along with the GNU C Library; if not, see
+   <http://www.gnu.org/licenses/>.  */
+
+#include <math.h>
+#include <math_private.h>
+#include <float.h>
+#include <stdlib.h>
+
+/* Calculate X + Y exactly and store the result in *HI + *LO.  It is
+   given that |X| >= |Y| and the values are small enough that no
+   overflow occurs.  */
+
+static inline void
+add_split (double *hi, double *lo, double x, double y)
+{
+  /* Apply Dekker's algorithm.  */
+  *hi = x + y;
+  *lo = (x - *hi) + y;
+}
+
+/* Calculate X * Y exactly and store the result in *HI + *LO.  It is
+   given that the values are small enough that no overflow occurs and
+   large enough (or zero) that no underflow occurs.  */
+
+static inline void
+mul_split (double *hi, double *lo, double x, double y)
+{
+#ifdef __FP_FAST_FMA
+  /* Fast built-in fused multiply-add.  */
+  *hi = x * y;
+  *lo = __builtin_fma (x, y, -*hi);
+#elif defined FP_FAST_FMA
+  /* Fast library fused multiply-add, compiler before GCC 4.6.  */
+  *hi = x * y;
+  *lo = __fma (x, y, -*hi);
+#else
+  /* Apply Dekker's algorithm.  */
+  *hi = x * y;
+# define C ((1 << (DBL_MANT_DIG + 1) / 2) + 1)
+  double x1 = x * C;
+  double y1 = y * C;
+# undef C
+  x1 = (x - x1) + x1;
+  y1 = (y - y1) + y1;
+  double x2 = x - x1;
+  double y2 = y - y1;
+  *lo = (((x1 * y1 - *hi) + x1 * y2) + x2 * y1) + x2 * y2;
+#endif
+}
+
+/* Compare absolute values of floating-point values pointed to by P
+   and Q for qsort.  */
+
+static int
+compare (const void *p, const void *q)
+{
+  double pd = fabs (*(const double *) p);
+  double qd = fabs (*(const double *) q);
+  if (pd < qd)
+    return -1;
+  else if (pd == qd)
+    return 0;
+  else
+    return 1;
+}
+
+/* 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.  */
+
+double
+__x2y2m1 (double x, double y)
+{
+  double vals[4];
+  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);
+  /* 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++)
+    {
+      add_split (&vals[i + 1], &vals[i], vals[i + 1], vals[i]);
+      qsort (vals + i + 1, 3 - i, sizeof (double), compare);
+    }
+  /* Now any error from this addition will be small.  */
+  return vals[3] + vals[2] + vals[1] + vals[0];
+}
diff --git a/sysdeps/ieee754/dbl-64/x2y2m1f.c b/sysdeps/ieee754/dbl-64/x2y2m1f.c
new file mode 100644
index 0000000000..fcaa70851c
--- /dev/null
+++ b/sysdeps/ieee754/dbl-64/x2y2m1f.c
@@ -0,0 +1,32 @@
+/* Compute x^2 + y^2 - 1, without large cancellation error.
+   Copyright (C) 2012 Free Software Foundation, Inc.
+   This file is part of the GNU C Library.
+
+   The GNU C Library is free software; you can redistribute it and/or
+   modify it under the terms of the GNU Lesser General Public
+   License as published by the Free Software Foundation; either
+   version 2.1 of the License, or (at your option) any later version.
+
+   The GNU C Library is distributed in the hope that it will be useful,
+   but WITHOUT ANY WARRANTY; without even the implied warranty of
+   MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
+   Lesser General Public License for more details.
+
+   You should have received a copy of the GNU Lesser General Public
+   License along with the GNU C Library; if not, see
+   <http://www.gnu.org/licenses/>.  */
+
+#include <math.h>
+#include <math_private.h>
+#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.  */
+
+float
+__x2y2m1f (float x, float y)
+{
+  double dx = x, dy = y;
+  return (float) ((dx - 1) * (dx + 1) + dy * dy);
+}
diff --git a/sysdeps/ieee754/ldbl-128/x2y2m1l.c b/sysdeps/ieee754/ldbl-128/x2y2m1l.c
new file mode 100644
index 0000000000..a249cd3479
--- /dev/null
+++ b/sysdeps/ieee754/ldbl-128/x2y2m1l.c
@@ -0,0 +1,111 @@
+/* Compute x^2 + y^2 - 1, without large cancellation error.
+   Copyright (C) 2012 Free Software Foundation, Inc.
+   This file is part of the GNU C Library.
+
+   The GNU C Library is free software; you can redistribute it and/or
+   modify it under the terms of the GNU Lesser General Public
+   License as published by the Free Software Foundation; either
+   version 2.1 of the License, or (at your option) any later version.
+
+   The GNU C Library is distributed in the hope that it will be useful,
+   but WITHOUT ANY WARRANTY; without even the implied warranty of
+   MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
+   Lesser General Public License for more details.
+
+   You should have received a copy of the GNU Lesser General Public
+   License along with the GNU C Library; if not, see
+   <http://www.gnu.org/licenses/>.  */
+
+#include <math.h>
+#include <math_private.h>
+#include <float.h>
+#include <stdlib.h>
+
+/* Calculate X + Y exactly and store the result in *HI + *LO.  It is
+   given that |X| >= |Y| and the values are small enough that no
+   overflow occurs.  */
+
+static inline void
+add_split (long double *hi, long double *lo, long double x, long double y)
+{
+  /* Apply Dekker's algorithm.  */
+  *hi = x + y;
+  *lo = (x - *hi) + y;
+}
+
+/* Calculate X * Y exactly and store the result in *HI + *LO.  It is
+   given that the values are small enough that no overflow occurs and
+   large enough (or zero) that no underflow occurs.  */
+
+static inline void
+mul_split (long double *hi, long double *lo, long double x, long double y)
+{
+#ifdef __FP_FAST_FMAL
+  /* Fast built-in fused multiply-add.  */
+  *hi = x * y;
+  *lo = __builtin_fmal (x, y, -*hi);
+#elif defined FP_FAST_FMAL
+  /* Fast library fused multiply-add, compiler before GCC 4.6.  */
+  *hi = x * y;
+  *lo = __fmal (x, y, -*hi);
+#else
+  /* Apply Dekker's algorithm.  */
+  *hi = x * y;
+# define C ((1LL << (LDBL_MANT_DIG + 1) / 2) + 1)
+  long double x1 = x * C;
+  long double y1 = y * C;
+# undef C
+  x1 = (x - x1) + x1;
+  y1 = (y - y1) + y1;
+  long double x2 = x - x1;
+  long double y2 = y - y1;
+  *lo = (((x1 * y1 - *hi) + x1 * y2) + x2 * y1) + x2 * y2;
+#endif
+}
+
+/* Compare absolute values of floating-point values pointed to by P
+   and Q for qsort.  */
+
+static int
+compare (const void *p, const void *q)
+{
+  long double pld = fabsl (*(const long double *) p);
+  long double qld = fabsl (*(const long double *) q);
+  if (pld < qld)
+    return -1;
+  else if (pld == qld)
+    return 0;
+  else
+    return 1;
+}
+
+/* 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.  */
+
+long double
+__x2y2m1l (long double x, long double y)
+{
+  long double vals[4];
+  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);
+  /* 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++)
+    {
+      add_split (&vals[i + 1], &vals[i], vals[i + 1], vals[i]);
+      qsort (vals + i + 1, 3 - i, sizeof (long double), compare);
+    }
+  /* Now any error from this addition will be small.  */
+  return vals[3] + vals[2] + vals[1] + vals[0];
+}
diff --git a/sysdeps/ieee754/ldbl-128ibm/x2y2m1l.c b/sysdeps/ieee754/ldbl-128ibm/x2y2m1l.c
new file mode 100644
index 0000000000..4379b68c3c
--- /dev/null
+++ b/sysdeps/ieee754/ldbl-128ibm/x2y2m1l.c
@@ -0,0 +1,128 @@
+/* Compute x^2 + y^2 - 1, without large cancellation error.
+   Copyright (C) 2012 Free Software Foundation, Inc.
+   This file is part of the GNU C Library.
+
+   The GNU C Library is free software; you can redistribute it and/or
+   modify it under the terms of the GNU Lesser General Public
+   License as published by the Free Software Foundation; either
+   version 2.1 of the License, or (at your option) any later version.
+
+   The GNU C Library is distributed in the hope that it will be useful,
+   but WITHOUT ANY WARRANTY; without even the implied warranty of
+   MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
+   Lesser General Public License for more details.
+
+   You should have received a copy of the GNU Lesser General Public
+   License along with the GNU C Library; if not, see
+   <http://www.gnu.org/licenses/>.  */
+
+#include <math.h>
+#include <math_private.h>
+#include <float.h>
+
+/* Calculate X + Y exactly and store the result in *HI + *LO.  It is
+   given that |X| >= |Y| and the values are small enough that no
+   overflow occurs.  */
+
+static inline void
+add_split (double *hi, double *lo, double x, double y)
+{
+  /* Apply Dekker's algorithm.  */
+  *hi = x + y;
+  *lo = (x - *hi) + y;
+}
+
+/* Calculate X * Y exactly and store the result in *HI + *LO.  It is
+   given that the values are small enough that no overflow occurs and
+   large enough (or zero) that no underflow occurs.  */
+
+static inline void
+mul_split (double *hi, double *lo, double x, double y)
+{
+#ifdef __FP_FAST_FMA
+  /* Fast built-in fused multiply-add.  */
+  *hi = x * y;
+  *lo = __builtin_fma (x, y, -*hi);
+#elif defined FP_FAST_FMA
+  /* Fast library fused multiply-add, compiler before GCC 4.6.  */
+  *hi = x * y;
+  *lo = __fma (x, y, -*hi);
+#else
+  /* Apply Dekker's algorithm.  */
+  *hi = x * y;
+# define C ((1 << (DBL_MANT_DIG + 1) / 2) + 1)
+  double x1 = x * C;
+  double y1 = y * C;
+# undef C
+  x1 = (x - x1) + x1;
+  y1 = (y - y1) + y1;
+  double x2 = x - x1;
+  double y2 = y - y1;
+  *lo = (((x1 * y1 - *hi) + x1 * y2) + x2 * y1) + x2 * y2;
+#endif
+}
+
+/* Compare absolute values of floating-point values pointed to by P
+   and Q for qsort.  */
+
+static int
+compare (const void *p, const void *q)
+{
+  double pd = fabs (*(const double *) p);
+  double qd = fabs (*(const double *) q);
+  if (pd < qd)
+    return -1;
+  else if (pd == qd)
+    return 0;
+  else
+    return 1;
+}
+
+/* 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.  */
+
+long double
+__x2y2m1l (long double x, long double y)
+{
+  double vals[12];
+  SET_RESTORE_ROUND (FE_TONEAREST);
+  union ibm_extended_long_double xu, yu;
+  xu.d = x;
+  yu.d = y;
+  if (fabs (xu.dd[1]) < 0x1p-500)
+    xu.dd[1] = 0.0;
+  if (fabs (yu.dd[1]) < 0x1p-500)
+    yu.dd[1] = 0.0;
+  mul_split (&vals[1], &vals[0], xu.dd[0], xu.dd[0]);
+  mul_split (&vals[3], &vals[2], xu.dd[0], xu.dd[1]);
+  vals[2] *= 2.0;
+  vals[3] *= 2.0;
+  mul_split (&vals[5], &vals[4], xu.dd[1], xu.dd[1]);
+  mul_split (&vals[7], &vals[6], yu.dd[0], yu.dd[0]);
+  mul_split (&vals[9], &vals[8], yu.dd[0], yu.dd[1]);
+  vals[8] *= 2.0;
+  vals[9] *= 2.0;
+  mul_split (&vals[11], &vals[10], yu.dd[1], yu.dd[1]);
+  if (xu.dd[0] >= 0.75)
+    vals[1] -= 1.0;
+  else
+    {
+      vals[1] -= 0.5;
+      vals[7] -= 0.5;
+    }
+  qsort (vals, 12, 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++)
+    {
+      add_split (&vals[i + 1], &vals[i], vals[i + 1], vals[i]);
+      qsort (vals + i + 1, 11 - 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--)
+    retval += (long double) vals[i];
+  return retval;
+}
diff --git a/sysdeps/ieee754/ldbl-96/x2y2m1.c b/sysdeps/ieee754/ldbl-96/x2y2m1.c
new file mode 100644
index 0000000000..3086fa2c80
--- /dev/null
+++ b/sysdeps/ieee754/ldbl-96/x2y2m1.c
@@ -0,0 +1,39 @@
+/* Compute x^2 + y^2 - 1, without large cancellation error.
+   Copyright (C) 2012 Free Software Foundation, Inc.
+   This file is part of the GNU C Library.
+
+   The GNU C Library is free software; you can redistribute it and/or
+   modify it under the terms of the GNU Lesser General Public
+   License as published by the Free Software Foundation; either
+   version 2.1 of the License, or (at your option) any later version.
+
+   The GNU C Library is distributed in the hope that it will be useful,
+   but WITHOUT ANY WARRANTY; without even the implied warranty of
+   MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
+   Lesser General Public License for more details.
+
+   You should have received a copy of the GNU Lesser General Public
+   License along with the GNU C Library; if not, see
+   <http://www.gnu.org/licenses/>.  */
+
+#include <math.h>
+#include <math_private.h>
+#include <float.h>
+
+#if FLT_EVAL_METHOD == 0
+
+# include <sysdeps/ieee754/dbl-64/x2y2m1.c>
+
+#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.  */
+
+double
+__x2y2m1 (double x, double y)
+{
+  return (double) __x2y2m1l (x, y);
+}
+
+#endif
diff --git a/sysdeps/ieee754/ldbl-96/x2y2m1l.c b/sysdeps/ieee754/ldbl-96/x2y2m1l.c
new file mode 100644
index 0000000000..a249cd3479
--- /dev/null
+++ b/sysdeps/ieee754/ldbl-96/x2y2m1l.c
@@ -0,0 +1,111 @@
+/* Compute x^2 + y^2 - 1, without large cancellation error.
+   Copyright (C) 2012 Free Software Foundation, Inc.
+   This file is part of the GNU C Library.
+
+   The GNU C Library is free software; you can redistribute it and/or
+   modify it under the terms of the GNU Lesser General Public
+   License as published by the Free Software Foundation; either
+   version 2.1 of the License, or (at your option) any later version.
+
+   The GNU C Library is distributed in the hope that it will be useful,
+   but WITHOUT ANY WARRANTY; without even the implied warranty of
+   MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
+   Lesser General Public License for more details.
+
+   You should have received a copy of the GNU Lesser General Public
+   License along with the GNU C Library; if not, see
+   <http://www.gnu.org/licenses/>.  */
+
+#include <math.h>
+#include <math_private.h>
+#include <float.h>
+#include <stdlib.h>
+
+/* Calculate X + Y exactly and store the result in *HI + *LO.  It is
+   given that |X| >= |Y| and the values are small enough that no
+   overflow occurs.  */
+
+static inline void
+add_split (long double *hi, long double *lo, long double x, long double y)
+{
+  /* Apply Dekker's algorithm.  */
+  *hi = x + y;
+  *lo = (x - *hi) + y;
+}
+
+/* Calculate X * Y exactly and store the result in *HI + *LO.  It is
+   given that the values are small enough that no overflow occurs and
+   large enough (or zero) that no underflow occurs.  */
+
+static inline void
+mul_split (long double *hi, long double *lo, long double x, long double y)
+{
+#ifdef __FP_FAST_FMAL
+  /* Fast built-in fused multiply-add.  */
+  *hi = x * y;
+  *lo = __builtin_fmal (x, y, -*hi);
+#elif defined FP_FAST_FMAL
+  /* Fast library fused multiply-add, compiler before GCC 4.6.  */
+  *hi = x * y;
+  *lo = __fmal (x, y, -*hi);
+#else
+  /* Apply Dekker's algorithm.  */
+  *hi = x * y;
+# define C ((1LL << (LDBL_MANT_DIG + 1) / 2) + 1)
+  long double x1 = x * C;
+  long double y1 = y * C;
+# undef C
+  x1 = (x - x1) + x1;
+  y1 = (y - y1) + y1;
+  long double x2 = x - x1;
+  long double y2 = y - y1;
+  *lo = (((x1 * y1 - *hi) + x1 * y2) + x2 * y1) + x2 * y2;
+#endif
+}
+
+/* Compare absolute values of floating-point values pointed to by P
+   and Q for qsort.  */
+
+static int
+compare (const void *p, const void *q)
+{
+  long double pld = fabsl (*(const long double *) p);
+  long double qld = fabsl (*(const long double *) q);
+  if (pld < qld)
+    return -1;
+  else if (pld == qld)
+    return 0;
+  else
+    return 1;
+}
+
+/* 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.  */
+
+long double
+__x2y2m1l (long double x, long double y)
+{
+  long double vals[4];
+  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);
+  /* 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++)
+    {
+      add_split (&vals[i + 1], &vals[i], vals[i + 1], vals[i]);
+      qsort (vals + i + 1, 3 - i, sizeof (long double), compare);
+    }
+  /* Now any error from this addition will be small.  */
+  return vals[3] + vals[2] + vals[1] + vals[0];
+}