Fix catan, catanh inaccuracy from atan2 denominators near 0 (bug 15416).

7 files changed, 370 insertions, 12 deletions

diff --git a/math/libm-test.inc b/math/libm-test.inc
index b82c592..d131ffa6 100644
--- a/math/libm-test.inc
+++ b/math/libm-test.inc
@@ -4483,6 +4483,134 @@ catan_test (void)
   TEST_c_c (catan, -0x1.fp16383L, 0x1.fp16383L, -1.570796326794896619231321691639751442099L, 4.338197604015604524209906861060325938836e-4933L, UNDERFLOW_EXCEPTION);
   TEST_c_c (catan, -0x1.fp16383L, -0x1.fp16383L, -1.570796326794896619231321691639751442099L, -4.338197604015604524209906861060325938836e-4933L, UNDERFLOW_EXCEPTION);
 #endif
+  TEST_c_c (catan, 0x1p-13L, 1.0L, 7.854286809755354140031716771044626356262e-1L, 4.852030264850939738801379894163661227127L);
+  TEST_c_c (catan, 0x1p-13L, -1.0L, 7.854286809755354140031716771044626356262e-1L, -4.852030264850939738801379894163661227127L);
+  TEST_c_c (catan, -0x1p-13L, 1.0L, -7.854286809755354140031716771044626356262e-1L, 4.852030264850939738801379894163661227127L);
+  TEST_c_c (catan, -0x1p-13L, -1.0L, -7.854286809755354140031716771044626356262e-1L, -4.852030264850939738801379894163661227127L);
+  TEST_c_c (catan, 1.0L, 0x1p-13L, 7.853981671227386080775748393881580082970e-1L, 6.103515609841754902688560615027452023669e-5L);
+  TEST_c_c (catan, -1.0L, 0x1p-13L, -7.853981671227386080775748393881580082970e-1L, 6.103515609841754902688560615027452023669e-5L);
+  TEST_c_c (catan, 1.0L, -0x1p-13L, 7.853981671227386080775748393881580082970e-1L, -6.103515609841754902688560615027452023669e-5L);
+  TEST_c_c (catan, -1.0L, -0x1p-13L, -7.853981671227386080775748393881580082970e-1L, -6.103515609841754902688560615027452023669e-5L);
+  TEST_c_c (catan, 0x1p-27L, 1.0L, 7.853981652600934588466178684534110069553e-1L, 9.704060527839234335310696652368086117807L);
+  TEST_c_c (catan, 0x1p-27L, -1.0L, 7.853981652600934588466178684534110069553e-1L, -9.704060527839234335310696652368086117807L);
+  TEST_c_c (catan, -0x1p-27L, 1.0L, -7.853981652600934588466178684534110069553e-1L, 9.704060527839234335310696652368086117807L);
+  TEST_c_c (catan, -0x1p-27L, -1.0L, -7.853981652600934588466178684534110069553e-1L, -9.704060527839234335310696652368086117807L);
+  TEST_c_c (catan, 1.0L, 0x1p-27L, 7.853981633974483234934486536343324763447e-1L, 3.725290298461914028034141143623846306386e-9L);
+  TEST_c_c (catan, -1.0L, 0x1p-27L, -7.853981633974483234934486536343324763447e-1L, 3.725290298461914028034141143623846306386e-9L);
+  TEST_c_c (catan, 1.0L, -0x1p-27L, 7.853981633974483234934486536343324763447e-1L, -3.725290298461914028034141143623846306386e-9L);
+  TEST_c_c (catan, -1.0L, -0x1p-27L, -7.853981633974483234934486536343324763447e-1L, -3.725290298461914028034141143623846306386e-9L);
+  TEST_c_c (catan, 0x1p-33L, 1.0L, 7.853981634265521400723945494331241018449e-1L, 1.178350206951907026009379309773625595762e1L);
+  TEST_c_c (catan, 0x1p-33L, -1.0L, 7.853981634265521400723945494331241018449e-1L, -1.178350206951907026009379309773625595762e1L);
+  TEST_c_c (catan, -0x1p-33L, 1.0L, -7.853981634265521400723945494331241018449e-1L, 1.178350206951907026009379309773625595762e1L);
+  TEST_c_c (catan, -0x1p-33L, -1.0L, -7.853981634265521400723945494331241018449e-1L, -1.178350206951907026009379309773625595762e1L);
+  TEST_c_c (catan, 1.0L, 0x1p-33L, 7.853981633974483096190489776088929224056e-1L, 5.820766091346740722643102318246316469910e-11L);
+  TEST_c_c (catan, -1.0L, 0x1p-33L, -7.853981633974483096190489776088929224056e-1L, 5.820766091346740722643102318246316469910e-11L);
+  TEST_c_c (catan, 1.0L, -0x1p-33L, 7.853981633974483096190489776088929224056e-1L, -5.820766091346740722643102318246316469910e-11L);
+  TEST_c_c (catan, -1.0L, -0x1p-33L, -7.853981633974483096190489776088929224056e-1L, -5.820766091346740722643102318246316469910e-11L);
+  TEST_c_c (catan, 0x1p-54L, 1.0L, 7.853981633974483234934486536343324763447e-1L, 1.906154746539849600897388334009985581467e1L);
+  TEST_c_c (catan, 0x1p-54L, -1.0L, 7.853981633974483234934486536343324763447e-1L, -1.906154746539849600897388334009985581467e1L);
+  TEST_c_c (catan, -0x1p-54L, 1.0L, -7.853981633974483234934486536343324763447e-1L, 1.906154746539849600897388334009985581467e1L);
+  TEST_c_c (catan, -0x1p-54L, -1.0L, -7.853981633974483234934486536343324763447e-1L, -1.906154746539849600897388334009985581467e1L);
+  TEST_c_c (catan, 1.0L, 0x1p-54L, 7.853981633974483096156608458198764914213e-1L, 2.775557561562891351059079170227049355775e-17L);
+  TEST_c_c (catan, -1.0L, 0x1p-54L, -7.853981633974483096156608458198764914213e-1L, 2.775557561562891351059079170227049355775e-17L);
+  TEST_c_c (catan, 1.0L, -0x1p-54L, 7.853981633974483096156608458198764914213e-1L, -2.775557561562891351059079170227049355775e-17L);
+  TEST_c_c (catan, -1.0L, -0x1p-54L, -7.853981633974483096156608458198764914213e-1L, -2.775557561562891351059079170227049355775e-17L);
+  TEST_c_c (catan, 0x1p-57L, 1.0L, 7.853981633974483113503843217966828154612e-1L, 2.010126823623841397309973152228712047720e1L);
+  TEST_c_c (catan, 0x1p-57L, -1.0L, 7.853981633974483113503843217966828154612e-1L, -2.010126823623841397309973152228712047720e1L);
+  TEST_c_c (catan, -0x1p-57L, 1.0L, -7.853981633974483113503843217966828154612e-1L, 2.010126823623841397309973152228712047720e1L);
+  TEST_c_c (catan, -0x1p-57L, -1.0L, -7.853981633974483113503843217966828154612e-1L, -2.010126823623841397309973152228712047720e1L);
+  TEST_c_c (catan, 1.0L, 0x1p-57L, 7.853981633974483096156608458198757330864e-1L, 3.469446951953614188823848962783813448721e-18L);
+  TEST_c_c (catan, -1.0L, 0x1p-57L, -7.853981633974483096156608458198757330864e-1L, 3.469446951953614188823848962783813448721e-18L);
+  TEST_c_c (catan, 1.0L, -0x1p-57L, 7.853981633974483096156608458198757330864e-1L, -3.469446951953614188823848962783813448721e-18L);
+  TEST_c_c (catan, -1.0L, -0x1p-57L, -7.853981633974483096156608458198757330864e-1L, -3.469446951953614188823848962783813448721e-18L);
+  TEST_c_c (catan, 0x1p-13L, 0x1.000002p0L, 7.859169620684960844300240092596908675974e-1L, 4.852030056234795712498957387213592193975L);
+  TEST_c_c (catan, 0x1p-13L, -0x1.000002p0L, 7.859169620684960844300240092596908675974e-1L, -4.852030056234795712498957387213592193975L);
+  TEST_c_c (catan, -0x1p-13L, 0x1.000002p0L, -7.859169620684960844300240092596908675974e-1L, 4.852030056234795712498957387213592193975L);
+  TEST_c_c (catan, -0x1p-13L, -0x1.000002p0L, -7.859169620684960844300240092596908675974e-1L, -4.852030056234795712498957387213592193975L);
+  TEST_c_c (catan, 0x1.000002p0L, 0x1p-13L, 7.853982267273793866654490522673596014524e-1L, 6.103514882246036852433556327261700380577e-5L);
+  TEST_c_c (catan, -0x1.000002p0L, 0x1p-13L, -7.853982267273793866654490522673596014524e-1L, 6.103514882246036852433556327261700380577e-5L);
+  TEST_c_c (catan, 0x1.000002p0L, -0x1p-13L, 7.853982267273793866654490522673596014524e-1L, -6.103514882246036852433556327261700380577e-5L);
+  TEST_c_c (catan, -0x1.000002p0L, -0x1p-13L, -7.853982267273793866654490522673596014524e-1L, -6.103514882246036852433556327261700380577e-5L);
+  TEST_c_c (catan, 0x1p-13L, 0x0.ffffffp0L, 7.851845403708474595909269086711426246675e-1L, 4.852030190345140708455871037447717761868L);
+  TEST_c_c (catan, 0x1p-13L, -0x0.ffffffp0L, 7.851845403708474595909269086711426246675e-1L, -4.852030190345140708455871037447717761868L);
+  TEST_c_c (catan, -0x1p-13L, 0x0.ffffffp0L, -7.851845403708474595909269086711426246675e-1L, 4.852030190345140708455871037447717761868L);
+  TEST_c_c (catan, -0x1p-13L, -0x0.ffffffp0L, -7.851845403708474595909269086711426246675e-1L, -4.852030190345140708455871037447717761868L);
+  TEST_c_c (catan, 0x0.ffffffp0L, 0x1p-13L, 7.853981373204155542484315721351697277336e-1L, 6.103515973639646453881721999956617260502e-5L);
+  TEST_c_c (catan, -0x0.ffffffp0L, 0x1p-13L, -7.853981373204155542484315721351697277336e-1L, 6.103515973639646453881721999956617260502e-5L);
+  TEST_c_c (catan, 0x0.ffffffp0L, -0x1p-13L, 7.853981373204155542484315721351697277336e-1L, -6.103515973639646453881721999956617260502e-5L);
+  TEST_c_c (catan, -0x0.ffffffp0L, -0x1p-13L, -7.853981373204155542484315721351697277336e-1L, -6.103515973639646453881721999956617260502e-5L);
+#ifndef TEST_FLOAT
+  TEST_c_c (catan, 0x1p-27L, 0x1.0000000000001p0L, 7.853981801612546526942695000283242525531e-1L, 9.704060527839234168777242958594699810015L);
+  TEST_c_c (catan, 0x1p-27L, -0x1.0000000000001p0L, 7.853981801612546526942695000283242525531e-1L, -9.704060527839234168777242958594699810015L);
+  TEST_c_c (catan, -0x1p-27L, 0x1.0000000000001p0L, -7.853981801612546526942695000283242525531e-1L, 9.704060527839234168777242958594699810015L);
+  TEST_c_c (catan, -0x1p-27L, -0x1.0000000000001p0L, -7.853981801612546526942695000283242525531e-1L, -9.704060527839234168777242958594699810015L);
+  TEST_c_c (catan, 0x1.0000000000001p0L, 0x1p-27L, 7.853981633974484345157511161499711112683e-1L, 3.725290298461913200853528590596263270474e-9L);
+  TEST_c_c (catan, -0x1.0000000000001p0L, 0x1p-27L, -7.853981633974484345157511161499711112683e-1L, 3.725290298461913200853528590596263270474e-9L);
+  TEST_c_c (catan, 0x1.0000000000001p0L, -0x1p-27L, 7.853981633974484345157511161499711112683e-1L, -3.725290298461913200853528590596263270474e-9L);
+  TEST_c_c (catan, -0x1.0000000000001p0L, -0x1p-27L, -7.853981633974484345157511161499711112683e-1L, -3.725290298461913200853528590596263270474e-9L);
+  TEST_c_c (catan, 0x1p-27L, 0x0.fffffffffffff8p0L, 7.853981578095128619227903983047292781021e-1L, 9.704060527839234252043969805481351363824L);
+  TEST_c_c (catan, 0x1p-27L, -0x0.fffffffffffff8p0L, 7.853981578095128619227903983047292781021e-1L, -9.704060527839234252043969805481351363824L);
+  TEST_c_c (catan, -0x1p-27L, 0x0.fffffffffffff8p0L, -7.853981578095128619227903983047292781021e-1L, 9.704060527839234252043969805481351363824L);
+  TEST_c_c (catan, -0x1p-27L, -0x0.fffffffffffff8p0L, -7.853981578095128619227903983047292781021e-1L, -9.704060527839234252043969805481351363824L);
+  TEST_c_c (catan, 0x0.fffffffffffff8p0L, 0x1p-27L, 7.853981633974482679822974223765039144191e-1L, 3.725290298461914441624447420137706700965e-9L);
+  TEST_c_c (catan, -0x0.fffffffffffff8p0L, 0x1p-27L, -7.853981633974482679822974223765039144191e-1L, 3.725290298461914441624447420137706700965e-9L);
+  TEST_c_c (catan, 0x0.fffffffffffff8p0L, -0x1p-27L, 7.853981633974482679822974223765039144191e-1L, -3.725290298461914441624447420137706700965e-9L);
+  TEST_c_c (catan, -0x0.fffffffffffff8p0L, -0x1p-27L, -7.853981633974482679822974223765039144191e-1L, -3.725290298461914441624447420137706700965e-9L);
+#endif
+#if defined TEST_LDOUBLE && LDBL_MANT_DIG >= 64
+  TEST_c_c (catan, 0x1p-33L, 0x1.0000000000000002p0L, 7.853981638922134273801338071094141188767e-1L, 1.178350206951907025990405771755129268176e1L);
+  TEST_c_c (catan, 0x1p-33L, -0x1.0000000000000002p0L, 7.853981638922134273801338071094141188767e-1L, -1.178350206951907025990405771755129268176e1L);
+  TEST_c_c (catan, -0x1p-33L, 0x1.0000000000000002p0L, -7.853981638922134273801338071094141188767e-1L, 1.178350206951907025990405771755129268176e1L);
+  TEST_c_c (catan, -0x1p-33L, -0x1.0000000000000002p0L, -7.853981638922134273801338071094141188767e-1L, -1.178350206951907025990405771755129268176e1L);
+  TEST_c_c (catan, 0x1.0000000000000002p0L, 0x1p-33L, 7.853981633974483096732590862331681441026e-1L, 5.820766091346740722012013594069507025615e-11L);
+  TEST_c_c (catan, -0x1.0000000000000002p0L, 0x1p-33L, -7.853981633974483096732590862331681441026e-1L, 5.820766091346740722012013594069507025615e-11L);
+  TEST_c_c (catan, 0x1.0000000000000002p0L, -0x1p-33L, 7.853981633974483096732590862331681441026e-1L, -5.820766091346740722012013594069507025615e-11L);
+  TEST_c_c (catan, -0x1.0000000000000002p0L, -0x1p-33L, -7.853981633974483096732590862331681441026e-1L, -5.820766091346740722012013594069507025615e-11L);
+  TEST_c_c (catan, 0x1p-33L, 0x0.ffffffffffffffffp0L, 7.853981631937214964185249205444919953948e-1L, 1.178350206951907026002603046195591193050e1L);
+  TEST_c_c (catan, 0x1p-33L, -0x0.ffffffffffffffffp0L, 7.853981631937214964185249205444919953948e-1L, -1.178350206951907026002603046195591193050e1L);
+  TEST_c_c (catan, -0x1p-33L, 0x0.ffffffffffffffffp0L, -7.853981631937214964185249205444919953948e-1L, 1.178350206951907026002603046195591193050e1L);
+  TEST_c_c (catan, -0x1p-33L, -0x0.ffffffffffffffffp0L, -7.853981631937214964185249205444919953948e-1L, -1.178350206951907026002603046195591193050e1L);
+  TEST_c_c (catan, 0x0.ffffffffffffffffp0L, 0x1p-33L, 7.853981633974483095919439232967553115548e-1L, 5.820766091346740722958646680334721192083e-11L);
+  TEST_c_c (catan, -0x0.ffffffffffffffffp0L, 0x1p-33L, -7.853981633974483095919439232967553115548e-1L, 5.820766091346740722958646680334721192083e-11L);
+  TEST_c_c (catan, 0x0.ffffffffffffffffp0L, -0x1p-33L, 7.853981633974483095919439232967553115548e-1L, -5.820766091346740722958646680334721192083e-11L);
+  TEST_c_c (catan, -0x0.ffffffffffffffffp0L, -0x1p-33L, -7.853981633974483095919439232967553115548e-1L, -5.820766091346740722958646680334721192083e-11L);
+#endif
+#if defined TEST_LDOUBLE && LDBL_MANT_DIG >= 106
+  TEST_c_c (catan, 0x1p-54L, 0x1.000000000000000000000000008p0L, 7.853981633974485455380535786656405610710e-1L, 1.906154746539849600897388334009981267384e1L);
+  TEST_c_c (catan, 0x1p-54L, -0x1.000000000000000000000000008p0L, 7.853981633974485455380535786656405610710e-1L, -1.906154746539849600897388334009981267384e1L);
+  TEST_c_c (catan, -0x1p-54L, 0x1.000000000000000000000000008p0L, -7.853981633974485455380535786656405610710e-1L, 1.906154746539849600897388334009981267384e1L);
+  TEST_c_c (catan, -0x1p-54L, -0x1.000000000000000000000000008p0L, -7.853981633974485455380535786656405610710e-1L, -1.906154746539849600897388334009981267384e1L);
+  TEST_c_c (catan, 0x1.000000000000000000000000008p0L, 0x1p-54L, 7.853981633974483096156608458198888173729e-1L, 2.775557561562891351059079170226980932999e-17L);
+  TEST_c_c (catan, -0x1.000000000000000000000000008p0L, 0x1p-54L, -7.853981633974483096156608458198888173729e-1L, 2.775557561562891351059079170226980932999e-17L);
+  TEST_c_c (catan, 0x1.000000000000000000000000008p0L, -0x1p-54L, 7.853981633974483096156608458198888173729e-1L, -2.775557561562891351059079170226980932999e-17L);
+  TEST_c_c (catan, -0x1.000000000000000000000000008p0L, -0x1p-54L, -7.853981633974483096156608458198888173729e-1L, -2.775557561562891351059079170226980932999e-17L);
+  TEST_c_c (catan, 0x1p-54L, 0x0.ffffffffffffffffffffffffffcp0L, 7.853981633974482124711461911186784339815e-1L, 1.906154746539849600897388334009984040723e1L);
+  TEST_c_c (catan, 0x1p-54L, -0x0.ffffffffffffffffffffffffffcp0L, 7.853981633974482124711461911186784339815e-1L, -1.906154746539849600897388334009984040723e1L);
+  TEST_c_c (catan, -0x1p-54L, 0x0.ffffffffffffffffffffffffffcp0L, -7.853981633974482124711461911186784339815e-1L, 1.906154746539849600897388334009984040723e1L);
+  TEST_c_c (catan, -0x1p-54L, -0x0.ffffffffffffffffffffffffffcp0L, -7.853981633974482124711461911186784339815e-1L, -1.906154746539849600897388334009984040723e1L);
+  TEST_c_c (catan, 0x0.ffffffffffffffffffffffffffcp0L, 0x1p-54L, 7.853981633974483096156608458198703284454e-1L, 2.775557561562891351059079170227083567164e-17L);
+  TEST_c_c (catan, -0x0.ffffffffffffffffffffffffffcp0L, 0x1p-54L, -7.853981633974483096156608458198703284454e-1L, 2.775557561562891351059079170227083567164e-17L);
+  TEST_c_c (catan, 0x0.ffffffffffffffffffffffffffcp0L, -0x1p-54L, 7.853981633974483096156608458198703284454e-1L, -2.775557561562891351059079170227083567164e-17L);
+  TEST_c_c (catan, -0x0.ffffffffffffffffffffffffffcp0L, -0x1p-54L, -7.853981633974483096156608458198703284454e-1L, -2.775557561562891351059079170227083567164e-17L);
+#endif
+#if defined TEST_LDOUBLE && LDBL_MANT_DIG >= 113
+  TEST_c_c (catan, 0x1p-57L, 0x1.0000000000000000000000000001p0L, 7.853981633974483252281721296111395707566e-1L, 2.010126823623841397309973152228712033275e1L);
+  TEST_c_c (catan, 0x1p-57L, -0x1.0000000000000000000000000001p0L, 7.853981633974483252281721296111395707566e-1L, -2.010126823623841397309973152228712033275e1L);
+  TEST_c_c (catan, -0x1p-57L, 0x1.0000000000000000000000000001p0L, -7.853981633974483252281721296111395707566e-1L, 2.010126823623841397309973152228712033275e1L);
+  TEST_c_c (catan, -0x1p-57L, -0x1.0000000000000000000000000001p0L, -7.853981633974483252281721296111395707566e-1L, -2.010126823623841397309973152228712033275e1L);
+  TEST_c_c (catan, 0x1.0000000000000000000000000001p0L, 0x1p-57L, 7.853981633974483096156608458198758293829e-1L, 3.469446951953614188823848962783812780530e-18L);
+  TEST_c_c (catan, -0x1.0000000000000000000000000001p0L, 0x1p-57L, -7.853981633974483096156608458198758293829e-1L, 3.469446951953614188823848962783812780530e-18L);
+  TEST_c_c (catan, 0x1.0000000000000000000000000001p0L, -0x1p-57L, 7.853981633974483096156608458198758293829e-1L, -3.469446951953614188823848962783812780530e-18L);
+  TEST_c_c (catan, -0x1.0000000000000000000000000001p0L, -0x1p-57L, -7.853981633974483096156608458198758293829e-1L, -3.469446951953614188823848962783812780530e-18L);
+  TEST_c_c (catan, 0x1p-57L, 0x0.ffffffffffffffffffffffffffff8p0L, 7.853981633974483044114904178894544378135e-1L, 2.010126823623841397309973152228712040498e1L);
+  TEST_c_c (catan, 0x1p-57L, -0x0.ffffffffffffffffffffffffffff8p0L, 7.853981633974483044114904178894544378135e-1L, -2.010126823623841397309973152228712040498e1L);
+  TEST_c_c (catan, -0x1p-57L, 0x0.ffffffffffffffffffffffffffff8p0L, -7.853981633974483044114904178894544378135e-1L, 2.010126823623841397309973152228712040498e1L);
+  TEST_c_c (catan, -0x1p-57L, -0x0.ffffffffffffffffffffffffffff8p0L, -7.853981633974483044114904178894544378135e-1L, -2.010126823623841397309973152228712040498e1L);
+  TEST_c_c (catan, 0x0.ffffffffffffffffffffffffffff8p0L, 0x1p-57L, 7.853981633974483096156608458198756849381e-1L, 3.469446951953614188823848962783813782817e-18L);
+  TEST_c_c (catan, -0x0.ffffffffffffffffffffffffffff8p0L, 0x1p-57L, -7.853981633974483096156608458198756849381e-1L, 3.469446951953614188823848962783813782817e-18L);
+  TEST_c_c (catan, 0x0.ffffffffffffffffffffffffffff8p0L, -0x1p-57L, 7.853981633974483096156608458198756849381e-1L, -3.469446951953614188823848962783813782817e-18L);
+  TEST_c_c (catan, -0x0.ffffffffffffffffffffffffffff8p0L, -0x1p-57L, -7.853981633974483096156608458198756849381e-1L, -3.469446951953614188823848962783813782817e-18L);
+#endif
 
   TEST_c_c (catan, 0.75L, 1.25L, 1.10714871779409050301706546017853704L, 0.549306144334054845697622618461262852L);
   TEST_c_c (catan, -2, -3, -1.4099210495965755225306193844604208L, -0.22907268296853876629588180294200276L);
@@ -4679,6 +4807,134 @@ catanh_test (void)
   TEST_c_c (catanh, -0x1.fp16383L, 0x1.fp16383L, -4.338197604015604524209906861060325938836e-4933L, 1.570796326794896619231321691639751442099L, UNDERFLOW_EXCEPTION);
   TEST_c_c (catanh, -0x1.fp16383L, -0x1.fp16383L, -4.338197604015604524209906861060325938836e-4933L, -1.570796326794896619231321691639751442099L, UNDERFLOW_EXCEPTION);
 #endif
+  TEST_c_c (catanh, 0x1p-13L, 1.0L, 6.103515609841754902688560615027452023669e-5L, 7.853981671227386080775748393881580082970e-1L);
+  TEST_c_c (catanh, 0x1p-13L, -1.0L, 6.103515609841754902688560615027452023669e-5L, -7.853981671227386080775748393881580082970e-1L);
+  TEST_c_c (catanh, -0x1p-13L, 1.0L, -6.103515609841754902688560615027452023669e-5L, 7.853981671227386080775748393881580082970e-1L);
+  TEST_c_c (catanh, -0x1p-13L, -1.0L, -6.103515609841754902688560615027452023669e-5L, -7.853981671227386080775748393881580082970e-1L);
+  TEST_c_c (catanh, 1.0L, 0x1p-13L, 4.852030264850939738801379894163661227127L, 7.854286809755354140031716771044626356262e-1L);
+  TEST_c_c (catanh, -1.0L, 0x1p-13L, -4.852030264850939738801379894163661227127L, 7.854286809755354140031716771044626356262e-1L);
+  TEST_c_c (catanh, 1.0L, -0x1p-13L, 4.852030264850939738801379894163661227127L, -7.854286809755354140031716771044626356262e-1L);
+  TEST_c_c (catanh, -1.0L, -0x1p-13L, -4.852030264850939738801379894163661227127L, -7.854286809755354140031716771044626356262e-1L);
+  TEST_c_c (catanh, 0x1p-27L, 1.0L, 3.725290298461914028034141143623846306386e-9L, 7.853981633974483234934486536343324763447e-1L);
+  TEST_c_c (catanh, 0x1p-27L, -1.0L, 3.725290298461914028034141143623846306386e-9L, -7.853981633974483234934486536343324763447e-1L);
+  TEST_c_c (catanh, -0x1p-27L, 1.0L, -3.725290298461914028034141143623846306386e-9L, 7.853981633974483234934486536343324763447e-1L);
+  TEST_c_c (catanh, -0x1p-27L, -1.0L, -3.725290298461914028034141143623846306386e-9L, -7.853981633974483234934486536343324763447e-1L);
+  TEST_c_c (catanh, 1.0L, 0x1p-27L, 9.704060527839234335310696652368086117807L, 7.853981652600934588466178684534110069553e-1L);
+  TEST_c_c (catanh, -1.0L, 0x1p-27L, -9.704060527839234335310696652368086117807L, 7.853981652600934588466178684534110069553e-1L);
+  TEST_c_c (catanh, 1.0L, -0x1p-27L, 9.704060527839234335310696652368086117807L, -7.853981652600934588466178684534110069553e-1L);
+  TEST_c_c (catanh, -1.0L, -0x1p-27L, -9.704060527839234335310696652368086117807L, -7.853981652600934588466178684534110069553e-1L);
+  TEST_c_c (catanh, 0x1p-33L, 1.0L, 5.820766091346740722643102318246316469910e-11L, 7.853981633974483096190489776088929224056e-1L);
+  TEST_c_c (catanh, 0x1p-33L, -1.0L, 5.820766091346740722643102318246316469910e-11L, -7.853981633974483096190489776088929224056e-1L);
+  TEST_c_c (catanh, -0x1p-33L, 1.0L, -5.820766091346740722643102318246316469910e-11L, 7.853981633974483096190489776088929224056e-1L);
+  TEST_c_c (catanh, -0x1p-33L, -1.0L, -5.820766091346740722643102318246316469910e-11L, -7.853981633974483096190489776088929224056e-1L);
+  TEST_c_c (catanh, 1.0L, 0x1p-33L, 1.178350206951907026009379309773625595762e1L, 7.853981634265521400723945494331241018449e-1L);
+  TEST_c_c (catanh, -1.0L, 0x1p-33L, -1.178350206951907026009379309773625595762e1L, 7.853981634265521400723945494331241018449e-1L);
+  TEST_c_c (catanh, 1.0L, -0x1p-33L, 1.178350206951907026009379309773625595762e1L, -7.853981634265521400723945494331241018449e-1L);
+  TEST_c_c (catanh, -1.0L, -0x1p-33L, -1.178350206951907026009379309773625595762e1L, -7.853981634265521400723945494331241018449e-1L);
+  TEST_c_c (catanh, 0x1p-54L, 1.0L, 2.775557561562891351059079170227049355775e-17L, 7.853981633974483096156608458198764914213e-1L);
+  TEST_c_c (catanh, 0x1p-54L, -1.0L, 2.775557561562891351059079170227049355775e-17L, -7.853981633974483096156608458198764914213e-1L);
+  TEST_c_c (catanh, -0x1p-54L, 1.0L, -2.775557561562891351059079170227049355775e-17L, 7.853981633974483096156608458198764914213e-1L);
+  TEST_c_c (catanh, -0x1p-54L, -1.0L, -2.775557561562891351059079170227049355775e-17L, -7.853981633974483096156608458198764914213e-1L);
+  TEST_c_c (catanh, 1.0L, 0x1p-54L, 1.906154746539849600897388334009985581467e1L, 7.853981633974483234934486536343324763447e-1L);
+  TEST_c_c (catanh, -1.0L, 0x1p-54L, -1.906154746539849600897388334009985581467e1L, 7.853981633974483234934486536343324763447e-1L);
+  TEST_c_c (catanh, 1.0L, -0x1p-54L, 1.906154746539849600897388334009985581467e1L, -7.853981633974483234934486536343324763447e-1L);
+  TEST_c_c (catanh, -1.0L, -0x1p-54L, -1.906154746539849600897388334009985581467e1L, -7.853981633974483234934486536343324763447e-1L);
+  TEST_c_c (catanh, 0x1p-57L, 1.0L, 3.469446951953614188823848962783813448721e-18L, 7.853981633974483096156608458198757330864e-1L);
+  TEST_c_c (catanh, 0x1p-57L, -1.0L, 3.469446951953614188823848962783813448721e-18L, -7.853981633974483096156608458198757330864e-1L);
+  TEST_c_c (catanh, -0x1p-57L, 1.0L, -3.469446951953614188823848962783813448721e-18L, 7.853981633974483096156608458198757330864e-1L);
+  TEST_c_c (catanh, -0x1p-57L, -1.0L, -3.469446951953614188823848962783813448721e-18L, -7.853981633974483096156608458198757330864e-1L);
+  TEST_c_c (catanh, 1.0L, 0x1p-57L, 2.010126823623841397309973152228712047720e1L, 7.853981633974483113503843217966828154612e-1L);
+  TEST_c_c (catanh, -1.0L, 0x1p-57L, -2.010126823623841397309973152228712047720e1L, 7.853981633974483113503843217966828154612e-1L);
+  TEST_c_c (catanh, 1.0L, -0x1p-57L, 2.010126823623841397309973152228712047720e1L, -7.853981633974483113503843217966828154612e-1L);
+  TEST_c_c (catanh, -1.0L, -0x1p-57L, -2.010126823623841397309973152228712047720e1L, -7.853981633974483113503843217966828154612e-1L);
+  TEST_c_c (catanh, 0x1p-13L, 0x1.000002p0L, 6.103514882246036852433556327261700380577e-5L, 7.853982267273793866654490522673596014524e-1L);
+  TEST_c_c (catanh, 0x1p-13L, -0x1.000002p0L, 6.103514882246036852433556327261700380577e-5L, -7.853982267273793866654490522673596014524e-1L);
+  TEST_c_c (catanh, -0x1p-13L, 0x1.000002p0L, -6.103514882246036852433556327261700380577e-5L, 7.853982267273793866654490522673596014524e-1L);
+  TEST_c_c (catanh, -0x1p-13L, -0x1.000002p0L, -6.103514882246036852433556327261700380577e-5L, -7.853982267273793866654490522673596014524e-1L);
+  TEST_c_c (catanh, 0x1.000002p0L, 0x1p-13L, 4.852030056234795712498957387213592193975L, 7.859169620684960844300240092596908675974e-1L);
+  TEST_c_c (catanh, -0x1.000002p0L, 0x1p-13L, -4.852030056234795712498957387213592193975L, 7.859169620684960844300240092596908675974e-1L);
+  TEST_c_c (catanh, 0x1.000002p0L, -0x1p-13L, 4.852030056234795712498957387213592193975L, -7.859169620684960844300240092596908675974e-1L);
+  TEST_c_c (catanh, -0x1.000002p0L, -0x1p-13L, -4.852030056234795712498957387213592193975L, -7.859169620684960844300240092596908675974e-1L);
+  TEST_c_c (catanh, 0x1p-13L, 0x0.ffffffp0L, 6.103515973639646453881721999956617260502e-5L, 7.853981373204155542484315721351697277336e-1L);
+  TEST_c_c (catanh, 0x1p-13L, -0x0.ffffffp0L, 6.103515973639646453881721999956617260502e-5L, -7.853981373204155542484315721351697277336e-1L);
+  TEST_c_c (catanh, -0x1p-13L, 0x0.ffffffp0L, -6.103515973639646453881721999956617260502e-5L, 7.853981373204155542484315721351697277336e-1L);
+  TEST_c_c (catanh, -0x1p-13L, -0x0.ffffffp0L, -6.103515973639646453881721999956617260502e-5L, -7.853981373204155542484315721351697277336e-1L);
+  TEST_c_c (catanh, 0x0.ffffffp0L, 0x1p-13L, 4.852030190345140708455871037447717761868L, 7.851845403708474595909269086711426246675e-1L);
+  TEST_c_c (catanh, -0x0.ffffffp0L, 0x1p-13L, -4.852030190345140708455871037447717761868L, 7.851845403708474595909269086711426246675e-1L);
+  TEST_c_c (catanh, 0x0.ffffffp0L, -0x1p-13L, 4.852030190345140708455871037447717761868L, -7.851845403708474595909269086711426246675e-1L);
+  TEST_c_c (catanh, -0x0.ffffffp0L, -0x1p-13L, -4.852030190345140708455871037447717761868L, -7.851845403708474595909269086711426246675e-1L);
+#ifndef TEST_FLOAT
+  TEST_c_c (catanh, 0x1p-27L, 0x1.0000000000001p0L, 3.725290298461913200853528590596263270474e-9L, 7.853981633974484345157511161499711112683e-1L);
+  TEST_c_c (catanh, 0x1p-27L, -0x1.0000000000001p0L, 3.725290298461913200853528590596263270474e-9L, -7.853981633974484345157511161499711112683e-1L);
+  TEST_c_c (catanh, -0x1p-27L, 0x1.0000000000001p0L, -3.725290298461913200853528590596263270474e-9L, 7.853981633974484345157511161499711112683e-1L);
+  TEST_c_c (catanh, -0x1p-27L, -0x1.0000000000001p0L, -3.725290298461913200853528590596263270474e-9L, -7.853981633974484345157511161499711112683e-1L);
+  TEST_c_c (catanh, 0x1.0000000000001p0L, 0x1p-27L, 9.704060527839234168777242958594699810015L, 7.853981801612546526942695000283242525531e-1L);
+  TEST_c_c (catanh, -0x1.0000000000001p0L, 0x1p-27L, -9.704060527839234168777242958594699810015L, 7.853981801612546526942695000283242525531e-1L);
+  TEST_c_c (catanh, 0x1.0000000000001p0L, -0x1p-27L, 9.704060527839234168777242958594699810015L, -7.853981801612546526942695000283242525531e-1L);
+  TEST_c_c (catanh, -0x1.0000000000001p0L, -0x1p-27L, -9.704060527839234168777242958594699810015L, -7.853981801612546526942695000283242525531e-1L);
+  TEST_c_c (catanh, 0x1p-27L, 0x0.fffffffffffff8p0L, 3.725290298461914441624447420137706700965e-9L, 7.853981633974482679822974223765039144191e-1L);
+  TEST_c_c (catanh, 0x1p-27L, -0x0.fffffffffffff8p0L, 3.725290298461914441624447420137706700965e-9L, -7.853981633974482679822974223765039144191e-1L);
+  TEST_c_c (catanh, -0x1p-27L, 0x0.fffffffffffff8p0L, -3.725290298461914441624447420137706700965e-9L, 7.853981633974482679822974223765039144191e-1L);
+  TEST_c_c (catanh, -0x1p-27L, -0x0.fffffffffffff8p0L, -3.725290298461914441624447420137706700965e-9L, -7.853981633974482679822974223765039144191e-1L);
+  TEST_c_c (catanh, 0x0.fffffffffffff8p0L, 0x1p-27L, 9.704060527839234252043969805481351363824L, 7.853981578095128619227903983047292781021e-1L);
+  TEST_c_c (catanh, -0x0.fffffffffffff8p0L, 0x1p-27L, -9.704060527839234252043969805481351363824L, 7.853981578095128619227903983047292781021e-1L);
+  TEST_c_c (catanh, 0x0.fffffffffffff8p0L, -0x1p-27L, 9.704060527839234252043969805481351363824L, -7.853981578095128619227903983047292781021e-1L);
+  TEST_c_c (catanh, -0x0.fffffffffffff8p0L, -0x1p-27L, -9.704060527839234252043969805481351363824L, -7.853981578095128619227903983047292781021e-1L);
+#endif
+#if defined TEST_LDOUBLE && LDBL_MANT_DIG >= 64
+  TEST_c_c (catanh, 0x1p-33L, 0x1.0000000000000002p0L, 5.820766091346740722012013594069507025615e-11L, 7.853981633974483096732590862331681441026e-1L);
+  TEST_c_c (catanh, 0x1p-33L, -0x1.0000000000000002p0L, 5.820766091346740722012013594069507025615e-11L, -7.853981633974483096732590862331681441026e-1L);
+  TEST_c_c (catanh, -0x1p-33L, 0x1.0000000000000002p0L, -5.820766091346740722012013594069507025615e-11L, 7.853981633974483096732590862331681441026e-1L);
+  TEST_c_c (catanh, -0x1p-33L, -0x1.0000000000000002p0L, -5.820766091346740722012013594069507025615e-11L, -7.853981633974483096732590862331681441026e-1L);
+  TEST_c_c (catanh, 0x1.0000000000000002p0L, 0x1p-33L, 1.178350206951907025990405771755129268176e1L, 7.853981638922134273801338071094141188767e-1L);
+  TEST_c_c (catanh, -0x1.0000000000000002p0L, 0x1p-33L, -1.178350206951907025990405771755129268176e1L, 7.853981638922134273801338071094141188767e-1L);
+  TEST_c_c (catanh, 0x1.0000000000000002p0L, -0x1p-33L, 1.178350206951907025990405771755129268176e1L, -7.853981638922134273801338071094141188767e-1L);
+  TEST_c_c (catanh, -0x1.0000000000000002p0L, -0x1p-33L, -1.178350206951907025990405771755129268176e1L, -7.853981638922134273801338071094141188767e-1L);
+  TEST_c_c (catanh, 0x1p-33L, 0x0.ffffffffffffffffp0L, 5.820766091346740722958646680334721192083e-11L, 7.853981633974483095919439232967553115548e-1L);
+  TEST_c_c (catanh, 0x1p-33L, -0x0.ffffffffffffffffp0L, 5.820766091346740722958646680334721192083e-11L, -7.853981633974483095919439232967553115548e-1L);
+  TEST_c_c (catanh, -0x1p-33L, 0x0.ffffffffffffffffp0L, -5.820766091346740722958646680334721192083e-11L, 7.853981633974483095919439232967553115548e-1L);
+  TEST_c_c (catanh, -0x1p-33L, -0x0.ffffffffffffffffp0L, -5.820766091346740722958646680334721192083e-11L, -7.853981633974483095919439232967553115548e-1L);
+  TEST_c_c (catanh, 0x0.ffffffffffffffffp0L, 0x1p-33L, 1.178350206951907026002603046195591193050e1L, 7.853981631937214964185249205444919953948e-1L);
+  TEST_c_c (catanh, -0x0.ffffffffffffffffp0L, 0x1p-33L, -1.178350206951907026002603046195591193050e1L, 7.853981631937214964185249205444919953948e-1L);
+  TEST_c_c (catanh, 0x0.ffffffffffffffffp0L, -0x1p-33L, 1.178350206951907026002603046195591193050e1L, -7.853981631937214964185249205444919953948e-1L);
+  TEST_c_c (catanh, -0x0.ffffffffffffffffp0L, -0x1p-33L, -1.178350206951907026002603046195591193050e1L, -7.853981631937214964185249205444919953948e-1L);
+#endif
+#if defined TEST_LDOUBLE && LDBL_MANT_DIG >= 106
+  TEST_c_c (catanh, 0x1p-54L, 0x1.000000000000000000000000008p0L, 2.775557561562891351059079170226980932999e-17L, 7.853981633974483096156608458198888173729e-1L);
+  TEST_c_c (catanh, 0x1p-54L, -0x1.000000000000000000000000008p0L, 2.775557561562891351059079170226980932999e-17L, -7.853981633974483096156608458198888173729e-1L);
+  TEST_c_c (catanh, -0x1p-54L, 0x1.000000000000000000000000008p0L, -2.775557561562891351059079170226980932999e-17L, 7.853981633974483096156608458198888173729e-1L);
+  TEST_c_c (catanh, -0x1p-54L, -0x1.000000000000000000000000008p0L, -2.775557561562891351059079170226980932999e-17L, -7.853981633974483096156608458198888173729e-1L);
+  TEST_c_c (catanh, 0x1.000000000000000000000000008p0L, 0x1p-54L, 1.906154746539849600897388334009981267384e1L, 7.853981633974485455380535786656405610710e-1L);
+  TEST_c_c (catanh, -0x1.000000000000000000000000008p0L, 0x1p-54L, -1.906154746539849600897388334009981267384e1L, 7.853981633974485455380535786656405610710e-1L);
+  TEST_c_c (catanh, 0x1.000000000000000000000000008p0L, -0x1p-54L, 1.906154746539849600897388334009981267384e1L, -7.853981633974485455380535786656405610710e-1L);
+  TEST_c_c (catanh, -0x1.000000000000000000000000008p0L, -0x1p-54L, -1.906154746539849600897388334009981267384e1L, -7.853981633974485455380535786656405610710e-1L);
+  TEST_c_c (catanh, 0x1p-54L, 0x0.ffffffffffffffffffffffffffcp0L, 2.775557561562891351059079170227083567164e-17L, 7.853981633974483096156608458198703284454e-1L);
+  TEST_c_c (catanh, 0x1p-54L, -0x0.ffffffffffffffffffffffffffcp0L, 2.775557561562891351059079170227083567164e-17L, -7.853981633974483096156608458198703284454e-1L);
+  TEST_c_c (catanh, -0x1p-54L, 0x0.ffffffffffffffffffffffffffcp0L, -2.775557561562891351059079170227083567164e-17L, 7.853981633974483096156608458198703284454e-1L);
+  TEST_c_c (catanh, -0x1p-54L, -0x0.ffffffffffffffffffffffffffcp0L, -2.775557561562891351059079170227083567164e-17L, -7.853981633974483096156608458198703284454e-1L);
+  TEST_c_c (catanh, 0x0.ffffffffffffffffffffffffffcp0L, 0x1p-54L, 1.906154746539849600897388334009984040723e1L, 7.853981633974482124711461911186784339815e-1L);
+  TEST_c_c (catanh, -0x0.ffffffffffffffffffffffffffcp0L, 0x1p-54L, -1.906154746539849600897388334009984040723e1L, 7.853981633974482124711461911186784339815e-1L);
+  TEST_c_c (catanh, 0x0.ffffffffffffffffffffffffffcp0L, -0x1p-54L, 1.906154746539849600897388334009984040723e1L, -7.853981633974482124711461911186784339815e-1L);
+  TEST_c_c (catanh, -0x0.ffffffffffffffffffffffffffcp0L, -0x1p-54L, -1.906154746539849600897388334009984040723e1L, -7.853981633974482124711461911186784339815e-1L);
+#endif
+#if defined TEST_LDOUBLE && LDBL_MANT_DIG >= 113
+  TEST_c_c (catanh, 0x1p-57L, 0x1.0000000000000000000000000001p0L, 3.469446951953614188823848962783812780530e-18L, 7.853981633974483096156608458198758293829e-1L);
+  TEST_c_c (catanh, 0x1p-57L, -0x1.0000000000000000000000000001p0L, 3.469446951953614188823848962783812780530e-18L, -7.853981633974483096156608458198758293829e-1L);
+  TEST_c_c (catanh, -0x1p-57L, 0x1.0000000000000000000000000001p0L, -3.469446951953614188823848962783812780530e-18L, 7.853981633974483096156608458198758293829e-1L);
+  TEST_c_c (catanh, -0x1p-57L, -0x1.0000000000000000000000000001p0L, -3.469446951953614188823848962783812780530e-18L, -7.853981633974483096156608458198758293829e-1L);
+  TEST_c_c (catanh, 0x1.0000000000000000000000000001p0L, 0x1p-57L, 2.010126823623841397309973152228712033275e1L, 7.853981633974483252281721296111395707566e-1L);
+  TEST_c_c (catanh, -0x1.0000000000000000000000000001p0L, 0x1p-57L, -2.010126823623841397309973152228712033275e1L, 7.853981633974483252281721296111395707566e-1L);
+  TEST_c_c (catanh, 0x1.0000000000000000000000000001p0L, -0x1p-57L, 2.010126823623841397309973152228712033275e1L, -7.853981633974483252281721296111395707566e-1L);
+  TEST_c_c (catanh, -0x1.0000000000000000000000000001p0L, -0x1p-57L, -2.010126823623841397309973152228712033275e1L, -7.853981633974483252281721296111395707566e-1L);
+  TEST_c_c (catanh, 0x1p-57L, 0x0.ffffffffffffffffffffffffffff8p0L, 3.469446951953614188823848962783813782817e-18L, 7.853981633974483096156608458198756849381e-1L);
+  TEST_c_c (catanh, 0x1p-57L, -0x0.ffffffffffffffffffffffffffff8p0L, 3.469446951953614188823848962783813782817e-18L, -7.853981633974483096156608458198756849381e-1L);
+  TEST_c_c (catanh, -0x1p-57L, 0x0.ffffffffffffffffffffffffffff8p0L, -3.469446951953614188823848962783813782817e-18L, 7.853981633974483096156608458198756849381e-1L);
+  TEST_c_c (catanh, -0x1p-57L, -0x0.ffffffffffffffffffffffffffff8p0L, -3.469446951953614188823848962783813782817e-18L, -7.853981633974483096156608458198756849381e-1L);
+  TEST_c_c (catanh, 0x0.ffffffffffffffffffffffffffff8p0L, 0x1p-57L, 2.010126823623841397309973152228712040498e1L, 7.853981633974483044114904178894544378135e-1L);
+  TEST_c_c (catanh, -0x0.ffffffffffffffffffffffffffff8p0L, 0x1p-57L, -2.010126823623841397309973152228712040498e1L, 7.853981633974483044114904178894544378135e-1L);
+  TEST_c_c (catanh, 0x0.ffffffffffffffffffffffffffff8p0L, -0x1p-57L, 2.010126823623841397309973152228712040498e1L, -7.853981633974483044114904178894544378135e-1L);
+  TEST_c_c (catanh, -0x0.ffffffffffffffffffffffffffff8p0L, -0x1p-57L, -2.010126823623841397309973152228712040498e1L, -7.853981633974483044114904178894544378135e-1L);
+#endif
 
   TEST_c_c (catanh, 0.75L, 1.25L, 0.261492138795671927078652057366532140L, 0.996825126463918666098902241310446708L);
   TEST_c_c (catanh, -2, -3, -0.14694666622552975204743278515471595L, -1.3389725222944935611241935759091443L);
diff --git a/math/s_catan.c b/math/s_catan.c
index 4b7fbad..3a041f2 100644
--- a/math/s_catan.c
+++ b/math/s_catan.c
@@ -77,14 +77,30 @@ __catan (__complex__ double x)
 	}
       else
 	{
-	  double r2, num, den, f;
+	  double r2, num, den, f, absx, absy;
 
-	  r2 = __real__ x * __real__ x;
+	  absx = fabs (__real__ x);
+	  absy = fabs (__imag__ x);
+	  if (absx < absy)
+	    {
+	      double t = absx;
+	      absx = absy;
+	      absy = t;
+	    }
 
-	  den = 1 - r2 - __imag__ x * __imag__ x;
+	  if (absx >= 1.0)
+	    den = (1.0 - absx) * (1.0 + absx) - absy * absy;
+	  else if (absx >= 0.75 && absy < DBL_EPSILON / 2.0)
+	    den = (1.0 - absx) * (1.0 + absx);
+	  else if (absx >= 0.75 || absy >= 0.5)
+	    den = -__x2y2m1 (absx, absy);
+	  else
+	    den = (1.0 - absx) * (1.0 + absx) - absy * absy;
 
 	  __real__ res = 0.5 * __ieee754_atan2 (2.0 * __real__ x, den);
 
+	  r2 = __real__ x * __real__ x;
+
 	  num = __imag__ x + 1.0;
 	  num = r2 + num * num;
 
diff --git a/math/s_catanf.c b/math/s_catanf.c
index aa71e5e..698e2af 100644
--- a/math/s_catanf.c
+++ b/math/s_catanf.c
@@ -78,14 +78,30 @@ __catanf (__complex__ float x)
 	}
       else
 	{
-	  float r2, num, den, f;
+	  float r2, num, den, f, absx, absy;
 
-	  r2 = __real__ x * __real__ x;
+	  absx = fabsf (__real__ x);
+	  absy = fabsf (__imag__ x);
+	  if (absx < absy)
+	    {
+	      float t = absx;
+	      absx = absy;
+	      absy = t;
+	    }
 
-	  den = 1 - r2 - __imag__ x * __imag__ x;
+	  if (absx >= 1.0f)
+	    den = (1.0f - absx) * (1.0f + absx) - absy * absy;
+	  else if (absx >= 0.75f && absy < FLT_EPSILON / 2.0f)
+	    den = (1.0f - absx) * (1.0f + absx);
+	  else if (absx >= 0.75f || absy >= 0.5f)
+	    den = -__x2y2m1f (absx, absy);
+	  else
+	    den = (1.0f - absx) * (1.0f + absx) - absy * absy;
 
 	  __real__ res = 0.5f * __ieee754_atan2f (2.0f * __real__ x, den);
 
+	  r2 = __real__ x * __real__ x;
+
 	  num = __imag__ x + 1.0f;
 	  num = r2 + num * num;
 
diff --git a/math/s_catanh.c b/math/s_catanh.c
index 55c5628..5248cbc 100644
--- a/math/s_catanh.c
+++ b/math/s_catanh.c
@@ -89,7 +89,25 @@ __catanh (__complex__ double x)
 	      __real__ res = 0.25 * __log1p (num / den);
 	    }
 
-	  den = 1 - __real__ x * __real__ x - i2;
+	  double absx, absy;
+
+	  absx = fabs (__real__ x);
+	  absy = fabs (__imag__ x);
+	  if (absx < absy)
+	    {
+	      double t = absx;
+	      absx = absy;
+	      absy = t;
+	    }
+
+	  if (absx >= 1.0)
+	    den = (1.0 - absx) * (1.0 + absx) - absy * absy;
+	  else if (absx >= 0.75 && absy < DBL_EPSILON / 2.0)
+	    den = (1.0 - absx) * (1.0 + absx);
+	  else if (absx >= 0.75 || absy >= 0.5)
+	    den = -__x2y2m1 (absx, absy);
+	  else
+	    den = (1.0 - absx) * (1.0 + absx) - absy * absy;
 
 	  __imag__ res = 0.5 * __ieee754_atan2 (2.0 * __imag__ x, den);
 	}
diff --git a/math/s_catanhf.c b/math/s_catanhf.c
index 3e87942..249deb5 100644
--- a/math/s_catanhf.c
+++ b/math/s_catanhf.c
@@ -90,7 +90,25 @@ __catanhf (__complex__ float x)
 	      __real__ res = 0.25f * __log1pf (num / den);
 	    }
 
-	  den = 1 - __real__ x * __real__ x - i2;
+	  float absx, absy;
+
+	  absx = fabsf (__real__ x);
+	  absy = fabsf (__imag__ x);
+	  if (absx < absy)
+	    {
+	      float t = absx;
+	      absx = absy;
+	      absy = t;
+	    }
+
+	  if (absx >= 1.0f)
+	    den = (1.0f - absx) * (1.0f + absx) - absy * absy;
+	  else if (absx >= 0.75f && absy < FLT_EPSILON / 2.0f)
+	    den = (1.0f - absx) * (1.0f + absx);
+	  else if (absx >= 0.75f || absy >= 0.5f)
+	    den = -__x2y2m1f (absx, absy);
+	  else
+	    den = (1.0f - absx) * (1.0f + absx) - absy * absy;
 
 	  __imag__ res = 0.5f * __ieee754_atan2f (2.0f * __imag__ x, den);
 	}
diff --git a/math/s_catanhl.c b/math/s_catanhl.c
index 64c30b5..ab75b80 100644
--- a/math/s_catanhl.c
+++ b/math/s_catanhl.c
@@ -97,7 +97,25 @@ __catanhl (__complex__ long double x)
 	      __real__ res = 0.25L * __log1pl (num / den);
 	    }
 
-	  den = 1 - __real__ x * __real__ x - i2;
+	  long double absx, absy;
+
+	  absx = fabsl (__real__ x);
+	  absy = fabsl (__imag__ x);
+	  if (absx < absy)
+	    {
+	      long double t = absx;
+	      absx = absy;
+	      absy = t;
+	    }
+
+	  if (absx >= 1.0L)
+	    den = (1.0L - absx) * (1.0L + absx) - absy * absy;
+	  else if (absx >= 0.75 && absy < LDBL_EPSILON / 2.0L)
+	    den = (1.0L - absx) * (1.0L + absx);
+	  else if (absx >= 0.75L || absy >= 0.5L)
+	    den = -__x2y2m1l (absx, absy);
+	  else
+	    den = (1.0L - absx) * (1.0L + absx) - absy * absy;
 
 	  __imag__ res = 0.5L * __ieee754_atan2l (2.0L * __imag__ x, den);
 	}
diff --git a/math/s_catanl.c b/math/s_catanl.c
index 17ab62d..1a815ad 100644
--- a/math/s_catanl.c
+++ b/math/s_catanl.c
@@ -85,14 +85,30 @@ __catanl (__complex__ long double x)
 	}
       else
 	{
-	  long double r2, num, den, f;
+	  long double r2, num, den, f, absx, absy;
 
-	  r2 = __real__ x * __real__ x;
+	  absx = fabsl (__real__ x);
+	  absy = fabsl (__imag__ x);
+	  if (absx < absy)
+	    {
+	      long double t = absx;
+	      absx = absy;
+	      absy = t;
+	    }
 
-	  den = 1 - r2 - __imag__ x * __imag__ x;
+	  if (absx >= 1.0L)
+	    den = (1.0L - absx) * (1.0L + absx) - absy * absy;
+	  else if (absx >= 0.75L && absy < LDBL_EPSILON / 2.0L)
+	    den = (1.0L - absx) * (1.0L + absx);
+	  else if (absx >= 0.75L || absy >= 0.5L)
+	    den = -__x2y2m1l (absx, absy);
+	  else
+	    den = (1.0L - absx) * (1.0L + absx) - absy * absy;
 
 	  __real__ res = 0.5L * __ieee754_atan2l (2.0L * __real__ x, den);
 
+	  r2 = __real__ x * __real__ x;
+
 	  num = __imag__ x + 1.0L;
 	  num = r2 + num * num;