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

9 files changed, 777 insertions, 4 deletions

diff --git a/sysdeps/generic/math_private.h b/sysdeps/generic/math_private.h
index 5267ec327b..b375bc0c56 100644
--- a/sysdeps/generic/math_private.h
+++ b/sysdeps/generic/math_private.h
@@ -364,6 +364,13 @@ extern double __slowexp (double __x);
 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.  */
+extern float __x2y2m1f (float x, float y);
+extern double __x2y2m1 (double x, double y);
+extern long double __x2y2m1l (long double x, long double y);
+
 #ifndef math_opt_barrier
 # define math_opt_barrier(x) \
 ({ __typeof (x) __x = (x); __asm ("" : "+m" (__x)); __x; })
diff --git a/sysdeps/i386/fpu/libm-test-ulps b/sysdeps/i386/fpu/libm-test-ulps
index 71b633a3b4..239da44a79 100644
--- a/sysdeps/i386/fpu/libm-test-ulps
+++ b/sysdeps/i386/fpu/libm-test-ulps
@@ -880,6 +880,12 @@ float: 1
 ifloat: 1
 ildouble: 1
 ldouble: 1
+Test "Real part of: clog (0x0.ffffffffffffffffp0 + 0x0.ffffffffffffffffp-15000 i) == -5.421010862427522170184200798202494495630e-20 + 3.548665303440282824232502561095699343814e-4516 i":
+ildouble: 1
+ldouble: 1
+Test "Real part of: clog (0x0.ffffffp0 + 0x0.ffffffp-100 i) == -5.960464655174753498633255797994360530379e-8 + 7.888609052210118054117285652827862296732e-31 i":
+float: 1
+ifloat: 1
 Test "Real part of: clog (0x1.000566p0 + 0x1.234p-10 i) == 8.298731898331237038231468223024422855654e-5 + 1.110938609507128729312743251313024793990e-3 i":
 ildouble: 1
 ldouble: 1
@@ -904,6 +910,9 @@ ldouble: 1
 Test "Real part of: clog (0x1.fp+16383 - 0x1p-16445 i) == 11356.49165759582936919077408168801636572 - 0 i":
 ildouble: 1
 ldouble: 1
+Test "Imaginary part of: clog (0x1367a310575591p-54 + 0x3cfcc0a0541f60p-54 i) == 5.0844550531823026520677817684239496041087e-32 + 1.2627468605458094918919206628466016525397 i":
+ildouble: 1
+ldouble: 1
 Test "Real part of: clog (0x1p-1074 + 0x1p-1074 i) == -744.0934983311012896593986823853525458290 + pi/4 i":
 ildouble: 1
 ldouble: 1
@@ -922,6 +931,9 @@ ldouble: 1
 Test "Real part of: clog (0x1p-16445 - 0x1.fp+16383 i) == 11356.49165759582936919077408168801636572 - pi/2 i":
 ildouble: 1
 ldouble: 1
+Test "Real part of: clog (0x2dd46725bp-35 + 0x7783a1284p-35 i) == 4.4469229730850767799109418892826021157328e-20 + 1.2046235979300843056806465045930070146351 i":
+ildouble: 1
+ldouble: 1
 Test "Real part of: clog (1.0 + 0x1.234566p-10 i) == 6.172834701221959432440126967147726538097e-7 + 1.111110564353742042376451655136933182201e-3 i":
 float: 1
 ifloat: 1
@@ -1068,6 +1080,17 @@ idouble: 1
 ifloat: 1
 ildouble: 1
 ldouble: 1
+Test "Real part of: clog10 (0x0.fffffffffffff8p0 + 0x0.fffffffffffff8p-1000 i) == -4.821637332766435821255375046554377090472e-17 + 4.053112396770095089737411317782466262176e-302 i":
+ildouble: 1
+ldouble: 1
+Test "Real part of: clog10 (0x0.ffffffffffffffffp0 + 0x0.ffffffffffffffffp-15000 i) == -2.354315103889861110220423157644627849164e-20 + 1.541165759405643564697852372112893034397e-4516 i":
+ildouble: 1
+ldouble: 1
+Test "Real part of: clog10 (0x0.ffffffp0 + 0x0.ffffffp-100 i) == -2.588596909321764128428416045209904492216e-8 + 3.425979381266895667295625489912064603415e-31 i":
+float: 1
+ifloat: 1
+ildouble: 1
+ldouble: 1
 Test "Real part of: clog10 (0x1.000566p0 + 0x1.234p-10 i) == 3.604093470239754109961125085078190708674e-5 + 4.824745078422174667425851670822596859720e-4 i":
 ildouble: 1
 ldouble: 1
@@ -1119,6 +1142,29 @@ ldouble: 1
 Test "Real part of: clog10 (0x1.fp+16383 - 0x1p-16445 i) == 4932.061660674182269085496060792589701158 - 0 i":
 ildouble: 1
 ldouble: 1
+Test "Real part of: clog10 (0x10673dd0f2481p-51 + 0x7ef1d17cefbd2p-51 i) == 1.3918041236396763648388478552321724382899e-29 + 0.6263795733790237053262025311642907438291 i":
+double: 1
+idouble: 1
+Test "Imaginary part of: clog10 (0x11682p-23 + 0x7ffed1p-23 i) == 5.0916490233953865181284669870035717560498e-13 + 0.6784968969384861816694467029319146542069 i":
+ildouble: 1
+ldouble: 1
+Test "Real part of: clog10 (0x1367a310575591p-54 + 0x3cfcc0a0541f60p-54 i) == 2.2081507730821788480616336165447731164865e-32 + 0.5484039935757001196548030312819898864760 i":
+double: 1
+idouble: 1
+Test "Imaginary part of: clog10 (0x1367a310575591p-54 + 0x3cfcc0a0541f60p-54 i) == 2.2081507730821788480616336165447731164865e-32 + 0.5484039935757001196548030312819898864760 i":
+double: 1
+idouble: 1
+ildouble: 1
+ldouble: 1
+Test "Imaginary part of: clog10 (0x164c74eea876p-45 + 0x16f393482f77p-45 i) == -1.3155760824064879362415202279780039150764e-26 + 0.3473590599762514228227328130640352044313 i":
+double: 1
+idouble: 1
+Test "Real part of: clog10 (0x1a6p-10 + 0x3a5p-10 i) == -6.2126412844802358329771948751248003038444e-07 + 0.4977135139537443711784513409096950995985 i":
+ildouble: 1
+ldouble: 1
+Test "Imaginary part of: clog10 (0x1a6p-10 + 0x3a5p-10 i) == -6.2126412844802358329771948751248003038444e-07 + 0.4977135139537443711784513409096950995985 i":
+double: 1
+idouble: 1
 Test "Imaginary part of: clog10 (0x1p-1073 + 0x1p-1073 i) == -322.8546703496198318667349645920187712089 + pi/4*log10(e) i":
 double: 1
 idouble: 1
@@ -1199,6 +1245,73 @@ ifloat: 1
 Test "Real part of: clog10 (0x1p-8190 + 1.0 i) == 2.920285685286322365786846845062520925172e-4932 + 6.821881769209206737428918127156778851051e-1 i":
 ildouble: 1
 ldouble: 1
+Test "Real part of: clog10 (0x2818p-15 + 0x798fp-15 i) == 6.6737261053986614395049481326819059203910e-09 + 0.5438241985991753781478398141908629586460 i":
+float: 1
+ifloat: 1
+Test "Imaginary part of: clog10 (0x2818p-15 + 0x798fp-15 i) == 6.6737261053986614395049481326819059203910e-09 + 0.5438241985991753781478398141908629586460 i":
+ildouble: 1
+ldouble: 1
+Test "Real part of: clog10 (0x2dd46725bp-35 + 0x7783a1284p-35 i) == 1.9312741086596516918394613098872836703188e-20 + 0.5231613813514771042838490538484014771862 i":
+double: 1
+idouble: 1
+Test "Imaginary part of: clog10 (0x2dd46725bp-35 + 0x7783a1284p-35 i) == 1.9312741086596516918394613098872836703188e-20 + 0.5231613813514771042838490538484014771862 i":
+ildouble: 1
+ldouble: 1
+Test "Real part of: clog10 (0x2ede88p-23 + 0x771c3fp-23 i) == -1.9440841725722970687903291200493082253766e-13 + 0.5193774116724956222518530053006822210323 i":
+float: 1
+ifloat: 1
+ildouble: 1
+ldouble: 1
+Test "Imaginary part of: clog10 (0x2ede88p-23 + 0x771c3fp-23 i) == -1.9440841725722970687903291200493082253766e-13 + 0.5193774116724956222518530053006822210323 i":
+double: 1
+idouble: 1
+ildouble: 1
+ldouble: 1
+Test "Real part of: clog10 (0x4447d7175p-35 + 0x6c445e00ap-35 i) == -6.4375803621988389731799033530075237868110e-21 + 0.4378257977686804492768642780897650927167 i":
+double: 1
+idouble: 1
+Test "Imaginary part of: clog10 (0x4447d7175p-35 + 0x6c445e00ap-35 i) == -6.4375803621988389731799033530075237868110e-21 + 0.4378257977686804492768642780897650927167 i":
+ildouble: 1
+ldouble: 1
+Test "Real part of: clog10 (0x4d4ep-15 + 0x6605p-15 i) == -7.0781945783414996953799915941870192015212e-09 + 0.4005747524909781155537088181659175147564 i":
+float: 1
+ifloat: 1
+ildouble: 1
+ldouble: 1
+Test "Imaginary part of: clog10 (0x5b06b680ea2ccp-52 + 0xef452b965da9fp-52 i) == 3.6079845358966994996207055940336690133424e-30 + 0.5243112258263349992771652393178033846555 i":
+double: 1
+idouble: 1
+ildouble: 1
+ldouble: 1
+Test "Imaginary part of: clog10 (0x659b70ab7971bp-53 + 0x1f5d111e08abecp-53 i) == -1.0893543813872082317104059174982092534059e-30 + 0.5954257879188711495921161433751775633232 i":
+ildouble: 1
+ldouble: 1
+Test "Imaginary part of: clog10 (0x659feap-24 + 0xeaf6f9p-24 i) == 1.6200701438094619117335617123525612051457e-14 + 0.5049027913635038013499728086604870749732 i":
+ildouble: 1
+ldouble: 1
+Test "Imaginary part of: clog10 (0x6b10b4f3520217b6p-64 + 0xe8893cbb449253a1p-64 i) == 1.0529283395205396881397407610630442563938e-37 + 0.4947949395762683446121140513971996916447 i":
+ildouble: 1
+ldouble: 1
+Test "Imaginary part of: clog10 (0x81b7efa81fc35ad1p-65 + 0x1ef4b835f1c79d812p-65 i) == -4.3074341162203896332989394770760901408798e-39 + 0.5709443672155660428417571212549720987784 i":
+ildouble: 1
+ldouble: 1
+Test "Real part of: clog10 (0x8ecbf810c4ae6p-52 + 0xd479468b09a37p-52 i) == -4.2289432987513243393180377141513840878196e-30 + 0.4252020027092323591068799049905597805296 i":
+ildouble: 1
+ldouble: 1
+Test "Imaginary part of: clog10 (0x9b57bp-20 + 0xcb7b4p-20 i) == -1.7182001068739620267773842120965071561416e-11 + 0.3990121149225253562859800593935899629087 i":
+double: 1
+idouble: 1
+Test "Real part of: clog10 (0xf2p-10 + 0x3e3p-10 i) == 2.6921240173351112953324592659528481616879e-06 + 0.5785726025799636431142862788413361783862 i":
+double: 1
+idouble: 1
+ildouble: 1
+ldouble: 1
+Test "Imaginary part of: clog10 (0xf2p-10 + 0x3e3p-10 i) == 2.6921240173351112953324592659528481616879e-06 + 0.5785726025799636431142862788413361783862 i":
+double: 1
+idouble: 1
+Test "Imaginary part of: clog10 (0xfe961079616p-45 + 0x1bc37e09e6d1p-45 i) == 2.3329549194675052736016290082882121135546e-26 + 0.4561756099441139182878993697611751382976 i":
+double: 1
+idouble: 1
 Test "Imaginary part of: clog10 (1.0 + 0x1.234566p-10 i) == 2.680828048441605163181684680300513080769e-7 + 4.825491868832381486767558728169977751564e-4 i":
 double: 1
 idouble: 1
@@ -3271,6 +3384,10 @@ ifloat: 1
 ildouble: 1
 ldouble: 1
 
+Function: Imaginary part of "clog":
+ildouble: 1
+ldouble: 1
+
 Function: Real part of "clog10":
 double: 1
 float: 1
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];
+}
diff --git a/sysdeps/x86_64/fpu/libm-test-ulps b/sysdeps/x86_64/fpu/libm-test-ulps
index d79578ce41..714b6b633c 100644
--- a/sysdeps/x86_64/fpu/libm-test-ulps
+++ b/sysdeps/x86_64/fpu/libm-test-ulps
@@ -812,6 +812,12 @@ float: 1
 ifloat: 1
 ildouble: 1
 ldouble: 1
+Test "Real part of: clog (0x0.ffffffffffffffffp0 + 0x0.ffffffffffffffffp-15000 i) == -5.421010862427522170184200798202494495630e-20 + 3.548665303440282824232502561095699343814e-4516 i":
+ildouble: 1
+ldouble: 1
+Test "Real part of: clog (0x0.ffffffp0 + 0x0.ffffffp-100 i) == -5.960464655174753498633255797994360530379e-8 + 7.888609052210118054117285652827862296732e-31 i":
+float: 1
+ifloat: 1
 Test "Real part of: clog (0x1.000566p0 + 0x1.234p-10 i) == 8.298731898331237038231468223024422855654e-5 + 1.110938609507128729312743251313024793990e-3 i":
 float: 1
 ifloat: 1
@@ -868,6 +874,12 @@ ldouble: 1
 Test "Real part of: clog (0x1p-16445 - 0x1.fp+16383 i) == 11356.49165759582936919077408168801636572 - pi/2 i":
 ildouble: 1
 ldouble: 1
+Test "Imaginary part of: clog (0x2818p-15 + 0x798fp-15 i) == 1.5366822245016167178749091974664853785194e-08 + 1.2522014929038946066987318471922169174157 i":
+float: 1
+ifloat: 1
+Test "Real part of: clog (0x2dd46725bp-35 + 0x7783a1284p-35 i) == 4.4469229730850767799109418892826021157328e-20 + 1.2046235979300843056806465045930070146351 i":
+ildouble: 1
+ldouble: 1
 Test "Real part of: clog (1.0 + 0x1.234566p-10 i) == 6.172834701221959432440126967147726538097e-7 + 1.111110564353742042376451655136933182201e-3 i":
 float: 1
 ifloat: 1
@@ -1009,6 +1021,21 @@ idouble: 1
 ifloat: 1
 ildouble: 1
 ldouble: 1
+Test "Real part of: clog10 (0x0.fffffffffffff8p0 + 0x0.fffffffffffff8p-1000 i) == -4.821637332766435821255375046554377090472e-17 + 4.053112396770095089737411317782466262176e-302 i":
+double: 1
+idouble: 1
+ildouble: 1
+ldouble: 1
+Test "Real part of: clog10 (0x0.ffffffffffffffffp0 + 0x0.ffffffffffffffffp-15000 i) == -2.354315103889861110220423157644627849164e-20 + 1.541165759405643564697852372112893034397e-4516 i":
+ildouble: 1
+ldouble: 1
+Test "Real part of: clog10 (0x0.ffffffp0 + 0x0.ffffffp-100 i) == -2.588596909321764128428416045209904492216e-8 + 3.425979381266895667295625489912064603415e-31 i":
+double: 1
+float: 2
+idouble: 1
+ifloat: 2
+ildouble: 1
+ldouble: 1
 Test "Real part of: clog10 (0x1.000566p0 + 0x1.234p-10 i) == 3.604093470239754109961125085078190708674e-5 + 4.824745078422174667425851670822596859720e-4 i":
 float: 1
 ifloat: 1
@@ -1068,6 +1095,27 @@ ldouble: 1
 Test "Real part of: clog10 (0x1.fp+16383 - 0x1p-16445 i) == 4932.061660674182269085496060792589701158 - 0 i":
 ildouble: 1
 ldouble: 1
+Test "Real part of: clog10 (0x10673dd0f2481p-51 + 0x7ef1d17cefbd2p-51 i) == 1.3918041236396763648388478552321724382899e-29 + 0.6263795733790237053262025311642907438291 i":
+double: 1
+idouble: 1
+Test "Imaginary part of: clog10 (0x11682p-23 + 0x7ffed1p-23 i) == 5.0916490233953865181284669870035717560498e-13 + 0.6784968969384861816694467029319146542069 i":
+ildouble: 1
+ldouble: 1
+Test "Real part of: clog10 (0x1367a310575591p-54 + 0x3cfcc0a0541f60p-54 i) == 2.2081507730821788480616336165447731164865e-32 + 0.5484039935757001196548030312819898864760 i":
+double: 1
+idouble: 1
+Test "Imaginary part of: clog10 (0x1367a310575591p-54 + 0x3cfcc0a0541f60p-54 i) == 2.2081507730821788480616336165447731164865e-32 + 0.5484039935757001196548030312819898864760 i":
+double: 1
+idouble: 1
+Test "Imaginary part of: clog10 (0x164c74eea876p-45 + 0x16f393482f77p-45 i) == -1.3155760824064879362415202279780039150764e-26 + 0.3473590599762514228227328130640352044313 i":
+double: 1
+idouble: 1
+Test "Real part of: clog10 (0x1a6p-10 + 0x3a5p-10 i) == -6.2126412844802358329771948751248003038444e-07 + 0.4977135139537443711784513409096950995985 i":
+ildouble: 1
+ldouble: 1
+Test "Imaginary part of: clog10 (0x1a6p-10 + 0x3a5p-10 i) == -6.2126412844802358329771948751248003038444e-07 + 0.4977135139537443711784513409096950995985 i":
+double: 1
+idouble: 1
 Test "Imaginary part of: clog10 (0x1p-1073 + 0x1p-1073 i) == -322.8546703496198318667349645920187712089 + pi/4*log10(e) i":
 double: 1
 idouble: 1
@@ -1151,6 +1199,75 @@ ifloat: 1
 Test "Real part of: clog10 (0x1p-8190 + 1.0 i) == 2.920285685286322365786846845062520925172e-4932 + 6.821881769209206737428918127156778851051e-1 i":
 ildouble: 1
 ldouble: 1
+Test "Real part of: clog10 (0x2818p-15 + 0x798fp-15 i) == 6.6737261053986614395049481326819059203910e-09 + 0.5438241985991753781478398141908629586460 i":
+double: 1
+float: 1
+idouble: 1
+ifloat: 1
+Test "Imaginary part of: clog10 (0x2818p-15 + 0x798fp-15 i) == 6.6737261053986614395049481326819059203910e-09 + 0.5438241985991753781478398141908629586460 i":
+float: 1
+ifloat: 1
+ildouble: 1
+ldouble: 1
+Test "Real part of: clog10 (0x2dd46725bp-35 + 0x7783a1284p-35 i) == 1.9312741086596516918394613098872836703188e-20 + 0.5231613813514771042838490538484014771862 i":
+double: 1
+idouble: 1
+Test "Imaginary part of: clog10 (0x2dd46725bp-35 + 0x7783a1284p-35 i) == 1.9312741086596516918394613098872836703188e-20 + 0.5231613813514771042838490538484014771862 i":
+ildouble: 1
+ldouble: 1
+Test "Real part of: clog10 (0x2ede88p-23 + 0x771c3fp-23 i) == -1.9440841725722970687903291200493082253766e-13 + 0.5193774116724956222518530053006822210323 i":
+float: 1
+ifloat: 1
+ildouble: 1
+ldouble: 1
+Test "Imaginary part of: clog10 (0x2ede88p-23 + 0x771c3fp-23 i) == -1.9440841725722970687903291200493082253766e-13 + 0.5193774116724956222518530053006822210323 i":
+double: 1
+idouble: 1
+ildouble: 1
+ldouble: 1
+Test "Real part of: clog10 (0x4447d7175p-35 + 0x6c445e00ap-35 i) == -6.4375803621988389731799033530075237868110e-21 + 0.4378257977686804492768642780897650927167 i":
+double: 1
+idouble: 1
+Test "Imaginary part of: clog10 (0x4447d7175p-35 + 0x6c445e00ap-35 i) == -6.4375803621988389731799033530075237868110e-21 + 0.4378257977686804492768642780897650927167 i":
+ildouble: 1
+ldouble: 1
+Test "Real part of: clog10 (0x4d4ep-15 + 0x6605p-15 i) == -7.0781945783414996953799915941870192015212e-09 + 0.4005747524909781155537088181659175147564 i":
+ildouble: 1
+ldouble: 1
+Test "Imaginary part of: clog10 (0x5b06b680ea2ccp-52 + 0xef452b965da9fp-52 i) == 3.6079845358966994996207055940336690133424e-30 + 0.5243112258263349992771652393178033846555 i":
+double: 1
+idouble: 1
+ildouble: 1
+ldouble: 1
+Test "Imaginary part of: clog10 (0x659b70ab7971bp-53 + 0x1f5d111e08abecp-53 i) == -1.0893543813872082317104059174982092534059e-30 + 0.5954257879188711495921161433751775633232 i":
+ildouble: 1
+ldouble: 1
+Test "Imaginary part of: clog10 (0x659feap-24 + 0xeaf6f9p-24 i) == 1.6200701438094619117335617123525612051457e-14 + 0.5049027913635038013499728086604870749732 i":
+ildouble: 1
+ldouble: 1
+Test "Imaginary part of: clog10 (0x6b10b4f3520217b6p-64 + 0xe8893cbb449253a1p-64 i) == 1.0529283395205396881397407610630442563938e-37 + 0.4947949395762683446121140513971996916447 i":
+ildouble: 1
+ldouble: 1
+Test "Imaginary part of: clog10 (0x81b7efa81fc35ad1p-65 + 0x1ef4b835f1c79d812p-65 i) == -4.3074341162203896332989394770760901408798e-39 + 0.5709443672155660428417571212549720987784 i":
+ildouble: 1
+ldouble: 1
+Test "Real part of: clog10 (0x8ecbf810c4ae6p-52 + 0xd479468b09a37p-52 i) == -4.2289432987513243393180377141513840878196e-30 + 0.4252020027092323591068799049905597805296 i":
+ildouble: 1
+ldouble: 1
+Test "Imaginary part of: clog10 (0x9b57bp-20 + 0xcb7b4p-20 i) == -1.7182001068739620267773842120965071561416e-11 + 0.3990121149225253562859800593935899629087 i":
+double: 1
+idouble: 1
+Test "Real part of: clog10 (0xf2p-10 + 0x3e3p-10 i) == 2.6921240173351112953324592659528481616879e-06 + 0.5785726025799636431142862788413361783862 i":
+double: 1
+idouble: 1
+ildouble: 1
+ldouble: 1
+Test "Imaginary part of: clog10 (0xf2p-10 + 0x3e3p-10 i) == 2.6921240173351112953324592659528481616879e-06 + 0.5785726025799636431142862788413361783862 i":
+double: 1
+idouble: 1
+Test "Imaginary part of: clog10 (0xfe961079616p-45 + 0x1bc37e09e6d1p-45 i) == 2.3329549194675052736016290082882121135546e-26 + 0.4561756099441139182878993697611751382976 i":
+double: 1
+idouble: 1
 Test "Imaginary part of: clog10 (1.0 + 0x1.234566p-10 i) == 2.680828048441605163181684680300513080769e-7 + 4.825491868832381486767558728169977751564e-4 i":
 double: 1
 idouble: 1
@@ -3050,17 +3167,17 @@ ifloat: 3
 
 Function: Real part of "clog10":
 double: 2
-float: 1
+float: 2
 idouble: 2
-ifloat: 1
+ifloat: 2
 ildouble: 1
 ldouble: 1
 
 Function: Imaginary part of "clog10":
 double: 1
-float: 5
+float: 1
 idouble: 1
-ifloat: 5
+ifloat: 1
 ildouble: 1
 ldouble: 1