Improve float range reduction accuracy near pi/2 (bug 21094).

Bug 21094 reports 3ulp errors of cosf and tanf for certain arguments
near pi/2 arising from the use of an insufficiently accurate range
reduction.  (To be clear, this is a quality-of-implementation issue
relating to the apparent intent of those particular cosf and tanf
implementations; 3ulp is within the general glibc accuracy goals, so
not inherently a bug.)

This patch fixes that error by making a wider range of cases use the
existing more accurate range reduction for arguments close to pi/2.
The wider range of values is still narrow enough for the "z -=
pio2_2;" in the more accurate case to be exact, as the code expects.

Tested for x86_64, x86 and mips64; no ulps updates needed (but at
least on mips64, the larger ulps were seen if the tests were added
without the substantive fix).

	[BZ #21094]
	* sysdeps/ieee754/flt-32/e_rem_pio2f.c (__ieee754_rem_pio2f): Use
	24+24+24-bit pi for wider range of values around pi/2.
	* math/auto-libm-test-in: Add more tests of cos and tan.
	* math/auto-libm-test-out-cos: Regenerated.
	* math/auto-libm-test-out-tan: Likewise.

5 files changed, 595 insertions, 2 deletions

diff --git a/ChangeLog b/ChangeLog
index f8b2e60bd6..f1f9ceb7ec 100644
--- a/ChangeLog
+++ b/ChangeLog
@@ -1,3 +1,12 @@
+2017-03-15  Joseph Myers  <joseph@codesourcery.com>
+
+	[BZ #21094]
+	* sysdeps/ieee754/flt-32/e_rem_pio2f.c (__ieee754_rem_pio2f): Use
+	24+24+24-bit pi for wider range of values around pi/2.
+	* math/auto-libm-test-in: Add more tests of cos and tan.
+	* math/auto-libm-test-out-cos: Regenerated.
+	* math/auto-libm-test-out-tan: Likewise.
+
 2017-03-15  John David Anglin  <danglin@gcc.gnu.org>
 
 	* sysdeps/hppa/dl-machine.h (DL_STACK_END): Define.
diff --git a/math/auto-libm-test-in b/math/auto-libm-test-in
index c8267aa822..6a47044f6b 100644
--- a/math/auto-libm-test-in
+++ b/math/auto-libm-test-in
@@ -4324,6 +4324,8 @@ cos -0x1.02e34cp+0
 cos 0xf.f0274p+4
 cos 0x3.042d88p+0
 cos 0x1.8475e5afd4481p+0
+cos 1.57079697
+cos -1.57079697
 
 cosh 0
 cosh -0
@@ -7236,6 +7238,8 @@ tan 0x2.091d68p+0
 tan -0x5.302ab9b18593264p+0
 tan 0x1.1ad374p+0
 tan -0x1.0d55b8p+0
+tan 1.57079697
+tan -1.57079697
 tan 0x1p-5
 tan 0x1p-10
 tan 0x1p-15
diff --git a/math/auto-libm-test-out-cos b/math/auto-libm-test-out-cos
index 329ae93719..072e04f157 100644
--- a/math/auto-libm-test-out-cos
+++ b/math/auto-libm-test-out-cos
@@ -2933,3 +2933,293 @@ cos 0x1.8475e5afd4481p+0
 = cos tonearest ibm128 0x1.8475e5afd4481p+0 : 0xd.a82683a33cbebfffffffa29668p-8 : inexact-ok
 = cos towardzero ibm128 0x1.8475e5afd4481p+0 : 0xd.a82683a33cbebfffffffa29668p-8 : inexact-ok
 = cos upward ibm128 0x1.8475e5afd4481p+0 : 0xd.a82683a33cbebfffffffa2966cp-8 : inexact-ok
+cos 1.57079697
+= cos downward binary32 0x1.921fc2p+0 : -0xc.bbbd3p-24 : inexact-ok
+= cos tonearest binary32 0x1.921fc2p+0 : -0xc.bbbd3p-24 : inexact-ok
+= cos towardzero binary32 0x1.921fc2p+0 : -0xc.bbbd2p-24 : inexact-ok
+= cos upward binary32 0x1.921fc2p+0 : -0xc.bbbd2p-24 : inexact-ok
+= cos downward binary64 0x1.921fc2p+0 : -0xc.bbbd2e7b951e8p-24 : inexact-ok
+= cos tonearest binary64 0x1.921fc2p+0 : -0xc.bbbd2e7b951e8p-24 : inexact-ok
+= cos towardzero binary64 0x1.921fc2p+0 : -0xc.bbbd2e7b951ep-24 : inexact-ok
+= cos upward binary64 0x1.921fc2p+0 : -0xc.bbbd2e7b951ep-24 : inexact-ok
+= cos downward intel96 0x1.921fc2p+0 : -0xc.bbbd2e7b951e5b2p-24 : inexact-ok
+= cos tonearest intel96 0x1.921fc2p+0 : -0xc.bbbd2e7b951e5b2p-24 : inexact-ok
+= cos towardzero intel96 0x1.921fc2p+0 : -0xc.bbbd2e7b951e5b1p-24 : inexact-ok
+= cos upward intel96 0x1.921fc2p+0 : -0xc.bbbd2e7b951e5b1p-24 : inexact-ok
+= cos downward m68k96 0x1.921fc2p+0 : -0xc.bbbd2e7b951e5b2p-24 : inexact-ok
+= cos tonearest m68k96 0x1.921fc2p+0 : -0xc.bbbd2e7b951e5b2p-24 : inexact-ok
+= cos towardzero m68k96 0x1.921fc2p+0 : -0xc.bbbd2e7b951e5b1p-24 : inexact-ok
+= cos upward m68k96 0x1.921fc2p+0 : -0xc.bbbd2e7b951e5b1p-24 : inexact-ok
+= cos downward binary128 0x1.921fc2p+0 : -0xc.bbbd2e7b951e5b1e4cc9f460a12p-24 : inexact-ok
+= cos tonearest binary128 0x1.921fc2p+0 : -0xc.bbbd2e7b951e5b1e4cc9f460a12p-24 : inexact-ok
+= cos towardzero binary128 0x1.921fc2p+0 : -0xc.bbbd2e7b951e5b1e4cc9f460a118p-24 : inexact-ok
+= cos upward binary128 0x1.921fc2p+0 : -0xc.bbbd2e7b951e5b1e4cc9f460a118p-24 : inexact-ok
+= cos downward ibm128 0x1.921fc2p+0 : -0xc.bbbd2e7b951e5b1e4cc9f460a4p-24 : inexact-ok
+= cos tonearest ibm128 0x1.921fc2p+0 : -0xc.bbbd2e7b951e5b1e4cc9f460ap-24 : inexact-ok
+= cos towardzero ibm128 0x1.921fc2p+0 : -0xc.bbbd2e7b951e5b1e4cc9f460ap-24 : inexact-ok
+= cos upward ibm128 0x1.921fc2p+0 : -0xc.bbbd2e7b951e5b1e4cc9f460ap-24 : inexact-ok
+= cos downward binary32 0x1.921fcp+0 : -0xa.bbbd3p-24 : inexact-ok
+= cos tonearest binary32 0x1.921fcp+0 : -0xa.bbbd3p-24 : inexact-ok
+= cos towardzero binary32 0x1.921fcp+0 : -0xa.bbbd2p-24 : inexact-ok
+= cos upward binary32 0x1.921fcp+0 : -0xa.bbbd2p-24 : inexact-ok
+= cos downward binary64 0x1.921fcp+0 : -0xa.bbbd2e7b95a88p-24 : inexact-ok
+= cos tonearest binary64 0x1.921fcp+0 : -0xa.bbbd2e7b95a88p-24 : inexact-ok
+= cos towardzero binary64 0x1.921fcp+0 : -0xa.bbbd2e7b95a8p-24 : inexact-ok
+= cos upward binary64 0x1.921fcp+0 : -0xa.bbbd2e7b95a8p-24 : inexact-ok
+= cos downward intel96 0x1.921fcp+0 : -0xa.bbbd2e7b95a85c7p-24 : inexact-ok
+= cos tonearest intel96 0x1.921fcp+0 : -0xa.bbbd2e7b95a85c6p-24 : inexact-ok
+= cos towardzero intel96 0x1.921fcp+0 : -0xa.bbbd2e7b95a85c6p-24 : inexact-ok
+= cos upward intel96 0x1.921fcp+0 : -0xa.bbbd2e7b95a85c6p-24 : inexact-ok
+= cos downward m68k96 0x1.921fcp+0 : -0xa.bbbd2e7b95a85c7p-24 : inexact-ok
+= cos tonearest m68k96 0x1.921fcp+0 : -0xa.bbbd2e7b95a85c6p-24 : inexact-ok
+= cos towardzero m68k96 0x1.921fcp+0 : -0xa.bbbd2e7b95a85c6p-24 : inexact-ok
+= cos upward m68k96 0x1.921fcp+0 : -0xa.bbbd2e7b95a85c6p-24 : inexact-ok
+= cos downward binary128 0x1.921fcp+0 : -0xa.bbbd2e7b95a85c638e746a5f4f68p-24 : inexact-ok
+= cos tonearest binary128 0x1.921fcp+0 : -0xa.bbbd2e7b95a85c638e746a5f4f6p-24 : inexact-ok
+= cos towardzero binary128 0x1.921fcp+0 : -0xa.bbbd2e7b95a85c638e746a5f4f6p-24 : inexact-ok
+= cos upward binary128 0x1.921fcp+0 : -0xa.bbbd2e7b95a85c638e746a5f4f6p-24 : inexact-ok
+= cos downward ibm128 0x1.921fcp+0 : -0xa.bbbd2e7b95a85c638e746a5f5p-24 : inexact-ok
+= cos tonearest ibm128 0x1.921fcp+0 : -0xa.bbbd2e7b95a85c638e746a5f5p-24 : inexact-ok
+= cos towardzero ibm128 0x1.921fcp+0 : -0xa.bbbd2e7b95a85c638e746a5f4cp-24 : inexact-ok
+= cos upward ibm128 0x1.921fcp+0 : -0xa.bbbd2e7b95a85c638e746a5f4cp-24 : inexact-ok
+= cos downward binary64 0x1.921fc00ece4f1p+0 : -0xa.ca8b7d8b95a58p-24 : inexact-ok
+= cos tonearest binary64 0x1.921fc00ece4f1p+0 : -0xa.ca8b7d8b95a5p-24 : inexact-ok
+= cos towardzero binary64 0x1.921fc00ece4f1p+0 : -0xa.ca8b7d8b95a5p-24 : inexact-ok
+= cos upward binary64 0x1.921fc00ece4f1p+0 : -0xa.ca8b7d8b95a5p-24 : inexact-ok
+= cos downward intel96 0x1.921fc00ece4f1p+0 : -0xa.ca8b7d8b95a502fp-24 : inexact-ok
+= cos tonearest intel96 0x1.921fc00ece4f1p+0 : -0xa.ca8b7d8b95a502fp-24 : inexact-ok
+= cos towardzero intel96 0x1.921fc00ece4f1p+0 : -0xa.ca8b7d8b95a502ep-24 : inexact-ok
+= cos upward intel96 0x1.921fc00ece4f1p+0 : -0xa.ca8b7d8b95a502ep-24 : inexact-ok
+= cos downward m68k96 0x1.921fc00ece4f1p+0 : -0xa.ca8b7d8b95a502fp-24 : inexact-ok
+= cos tonearest m68k96 0x1.921fc00ece4f1p+0 : -0xa.ca8b7d8b95a502fp-24 : inexact-ok
+= cos towardzero m68k96 0x1.921fc00ece4f1p+0 : -0xa.ca8b7d8b95a502ep-24 : inexact-ok
+= cos upward m68k96 0x1.921fc00ece4f1p+0 : -0xa.ca8b7d8b95a502ep-24 : inexact-ok
+= cos downward binary128 0x1.921fc00ece4f1p+0 : -0xa.ca8b7d8b95a502ede0fd607f3948p-24 : inexact-ok
+= cos tonearest binary128 0x1.921fc00ece4f1p+0 : -0xa.ca8b7d8b95a502ede0fd607f394p-24 : inexact-ok
+= cos towardzero binary128 0x1.921fc00ece4f1p+0 : -0xa.ca8b7d8b95a502ede0fd607f394p-24 : inexact-ok
+= cos upward binary128 0x1.921fc00ece4f1p+0 : -0xa.ca8b7d8b95a502ede0fd607f394p-24 : inexact-ok
+= cos downward ibm128 0x1.921fc00ece4f1p+0 : -0xa.ca8b7d8b95a502ede0fd607f3cp-24 : inexact-ok
+= cos tonearest ibm128 0x1.921fc00ece4f1p+0 : -0xa.ca8b7d8b95a502ede0fd607f38p-24 : inexact-ok
+= cos towardzero ibm128 0x1.921fc00ece4f1p+0 : -0xa.ca8b7d8b95a502ede0fd607f38p-24 : inexact-ok
+= cos upward ibm128 0x1.921fc00ece4f1p+0 : -0xa.ca8b7d8b95a502ede0fd607f38p-24 : inexact-ok
+= cos downward binary64 0x1.921fc00ece4fp+0 : -0xa.ca8b7d7b95a58p-24 : inexact-ok
+= cos tonearest binary64 0x1.921fc00ece4fp+0 : -0xa.ca8b7d7b95a5p-24 : inexact-ok
+= cos towardzero binary64 0x1.921fc00ece4fp+0 : -0xa.ca8b7d7b95a5p-24 : inexact-ok
+= cos upward binary64 0x1.921fc00ece4fp+0 : -0xa.ca8b7d7b95a5p-24 : inexact-ok
+= cos downward intel96 0x1.921fc00ece4fp+0 : -0xa.ca8b7d7b95a502fp-24 : inexact-ok
+= cos tonearest intel96 0x1.921fc00ece4fp+0 : -0xa.ca8b7d7b95a502fp-24 : inexact-ok
+= cos towardzero intel96 0x1.921fc00ece4fp+0 : -0xa.ca8b7d7b95a502ep-24 : inexact-ok
+= cos upward intel96 0x1.921fc00ece4fp+0 : -0xa.ca8b7d7b95a502ep-24 : inexact-ok
+= cos downward m68k96 0x1.921fc00ece4fp+0 : -0xa.ca8b7d7b95a502fp-24 : inexact-ok
+= cos tonearest m68k96 0x1.921fc00ece4fp+0 : -0xa.ca8b7d7b95a502fp-24 : inexact-ok
+= cos towardzero m68k96 0x1.921fc00ece4fp+0 : -0xa.ca8b7d7b95a502ep-24 : inexact-ok
+= cos upward m68k96 0x1.921fc00ece4fp+0 : -0xa.ca8b7d7b95a502ep-24 : inexact-ok
+= cos downward binary128 0x1.921fc00ece4fp+0 : -0xa.ca8b7d7b95a502ede4a0f9b11c3p-24 : inexact-ok
+= cos tonearest binary128 0x1.921fc00ece4fp+0 : -0xa.ca8b7d7b95a502ede4a0f9b11c28p-24 : inexact-ok
+= cos towardzero binary128 0x1.921fc00ece4fp+0 : -0xa.ca8b7d7b95a502ede4a0f9b11c28p-24 : inexact-ok
+= cos upward binary128 0x1.921fc00ece4fp+0 : -0xa.ca8b7d7b95a502ede4a0f9b11c28p-24 : inexact-ok
+= cos downward ibm128 0x1.921fc00ece4fp+0 : -0xa.ca8b7d7b95a502ede4a0f9b12p-24 : inexact-ok
+= cos tonearest ibm128 0x1.921fc00ece4fp+0 : -0xa.ca8b7d7b95a502ede4a0f9b11cp-24 : inexact-ok
+= cos towardzero ibm128 0x1.921fc00ece4fp+0 : -0xa.ca8b7d7b95a502ede4a0f9b11cp-24 : inexact-ok
+= cos upward ibm128 0x1.921fc00ece4fp+0 : -0xa.ca8b7d7b95a502ede4a0f9b11cp-24 : inexact-ok
+= cos downward intel96 0x1.921fc00ece4f02f4p+0 : -0xa.ca8b7d7e89a502fp-24 : inexact-ok
+= cos tonearest intel96 0x1.921fc00ece4f02f4p+0 : -0xa.ca8b7d7e89a502fp-24 : inexact-ok
+= cos towardzero intel96 0x1.921fc00ece4f02f4p+0 : -0xa.ca8b7d7e89a502ep-24 : inexact-ok
+= cos upward intel96 0x1.921fc00ece4f02f4p+0 : -0xa.ca8b7d7e89a502ep-24 : inexact-ok
+= cos downward m68k96 0x1.921fc00ece4f02f4p+0 : -0xa.ca8b7d7e89a502fp-24 : inexact-ok
+= cos tonearest m68k96 0x1.921fc00ece4f02f4p+0 : -0xa.ca8b7d7e89a502fp-24 : inexact-ok
+= cos towardzero m68k96 0x1.921fc00ece4f02f4p+0 : -0xa.ca8b7d7e89a502ep-24 : inexact-ok
+= cos upward m68k96 0x1.921fc00ece4f02f4p+0 : -0xa.ca8b7d7e89a502ep-24 : inexact-ok
+= cos downward binary128 0x1.921fc00ece4f02f4p+0 : -0xa.ca8b7d7e89a502ede3f507aaa7d8p-24 : inexact-ok
+= cos tonearest binary128 0x1.921fc00ece4f02f4p+0 : -0xa.ca8b7d7e89a502ede3f507aaa7d8p-24 : inexact-ok
+= cos towardzero binary128 0x1.921fc00ece4f02f4p+0 : -0xa.ca8b7d7e89a502ede3f507aaa7dp-24 : inexact-ok
+= cos upward binary128 0x1.921fc00ece4f02f4p+0 : -0xa.ca8b7d7e89a502ede3f507aaa7dp-24 : inexact-ok
+= cos downward ibm128 0x1.921fc00ece4f02f4p+0 : -0xa.ca8b7d7e89a502ede3f507aaa8p-24 : inexact-ok
+= cos tonearest ibm128 0x1.921fc00ece4f02f4p+0 : -0xa.ca8b7d7e89a502ede3f507aaa8p-24 : inexact-ok
+= cos towardzero ibm128 0x1.921fc00ece4f02f4p+0 : -0xa.ca8b7d7e89a502ede3f507aaa4p-24 : inexact-ok
+= cos upward ibm128 0x1.921fc00ece4f02f4p+0 : -0xa.ca8b7d7e89a502ede3f507aaa4p-24 : inexact-ok
+= cos downward intel96 0x1.921fc00ece4f02f2p+0 : -0xa.ca8b7d7e87a502fp-24 : inexact-ok
+= cos tonearest intel96 0x1.921fc00ece4f02f2p+0 : -0xa.ca8b7d7e87a502fp-24 : inexact-ok
+= cos towardzero intel96 0x1.921fc00ece4f02f2p+0 : -0xa.ca8b7d7e87a502ep-24 : inexact-ok
+= cos upward intel96 0x1.921fc00ece4f02f2p+0 : -0xa.ca8b7d7e87a502ep-24 : inexact-ok
+= cos downward m68k96 0x1.921fc00ece4f02f2p+0 : -0xa.ca8b7d7e87a502fp-24 : inexact-ok
+= cos tonearest m68k96 0x1.921fc00ece4f02f2p+0 : -0xa.ca8b7d7e87a502fp-24 : inexact-ok
+= cos towardzero m68k96 0x1.921fc00ece4f02f2p+0 : -0xa.ca8b7d7e87a502ep-24 : inexact-ok
+= cos upward m68k96 0x1.921fc00ece4f02f2p+0 : -0xa.ca8b7d7e87a502ep-24 : inexact-ok
+= cos downward binary128 0x1.921fc00ece4f02f2p+0 : -0xa.ca8b7d7e87a502ede3f57c1dce18p-24 : inexact-ok
+= cos tonearest binary128 0x1.921fc00ece4f02f2p+0 : -0xa.ca8b7d7e87a502ede3f57c1dce1p-24 : inexact-ok
+= cos towardzero binary128 0x1.921fc00ece4f02f2p+0 : -0xa.ca8b7d7e87a502ede3f57c1dce1p-24 : inexact-ok
+= cos upward binary128 0x1.921fc00ece4f02f2p+0 : -0xa.ca8b7d7e87a502ede3f57c1dce1p-24 : inexact-ok
+= cos downward ibm128 0x1.921fc00ece4f02f2p+0 : -0xa.ca8b7d7e87a502ede3f57c1ddp-24 : inexact-ok
+= cos tonearest ibm128 0x1.921fc00ece4f02f2p+0 : -0xa.ca8b7d7e87a502ede3f57c1ddp-24 : inexact-ok
+= cos towardzero ibm128 0x1.921fc00ece4f02f2p+0 : -0xa.ca8b7d7e87a502ede3f57c1dccp-24 : inexact-ok
+= cos upward ibm128 0x1.921fc00ece4f02f2p+0 : -0xa.ca8b7d7e87a502ede3f57c1dccp-24 : inexact-ok
+= cos downward binary128 0x1.921fc00ece4f02f278ade6ad9e8ap+0 : -0xa.ca8b7d7e881db0d49193eaab43cp-24 : inexact-ok
+= cos tonearest binary128 0x1.921fc00ece4f02f278ade6ad9e8ap+0 : -0xa.ca8b7d7e881db0d49193eaab43cp-24 : inexact-ok
+= cos towardzero binary128 0x1.921fc00ece4f02f278ade6ad9e8ap+0 : -0xa.ca8b7d7e881db0d49193eaab43b8p-24 : inexact-ok
+= cos upward binary128 0x1.921fc00ece4f02f278ade6ad9e8ap+0 : -0xa.ca8b7d7e881db0d49193eaab43b8p-24 : inexact-ok
+= cos downward binary128 0x1.921fc00ece4f02f278ade6ad9e89p+0 : -0xa.ca8b7d7e881db0d49193e9ab43cp-24 : inexact-ok
+= cos tonearest binary128 0x1.921fc00ece4f02f278ade6ad9e89p+0 : -0xa.ca8b7d7e881db0d49193e9ab43cp-24 : inexact-ok
+= cos towardzero binary128 0x1.921fc00ece4f02f278ade6ad9e89p+0 : -0xa.ca8b7d7e881db0d49193e9ab43b8p-24 : inexact-ok
+= cos upward binary128 0x1.921fc00ece4f02f278ade6ad9e89p+0 : -0xa.ca8b7d7e881db0d49193e9ab43b8p-24 : inexact-ok
+= cos downward binary128 0x1.921fc00ece4f02f278ade6ad9fp+0 : -0xa.ca8b7d7e881db0d4919460ab43cp-24 : inexact-ok
+= cos tonearest binary128 0x1.921fc00ece4f02f278ade6ad9fp+0 : -0xa.ca8b7d7e881db0d4919460ab43cp-24 : inexact-ok
+= cos towardzero binary128 0x1.921fc00ece4f02f278ade6ad9fp+0 : -0xa.ca8b7d7e881db0d4919460ab43b8p-24 : inexact-ok
+= cos upward binary128 0x1.921fc00ece4f02f278ade6ad9fp+0 : -0xa.ca8b7d7e881db0d4919460ab43b8p-24 : inexact-ok
+= cos downward ibm128 0x1.921fc00ece4f02f278ade6ad9fp+0 : -0xa.ca8b7d7e881db0d4919460ab44p-24 : inexact-ok
+= cos tonearest ibm128 0x1.921fc00ece4f02f278ade6ad9fp+0 : -0xa.ca8b7d7e881db0d4919460ab44p-24 : inexact-ok
+= cos towardzero ibm128 0x1.921fc00ece4f02f278ade6ad9fp+0 : -0xa.ca8b7d7e881db0d4919460ab4p-24 : inexact-ok
+= cos upward ibm128 0x1.921fc00ece4f02f278ade6ad9fp+0 : -0xa.ca8b7d7e881db0d4919460ab4p-24 : inexact-ok
+= cos downward binary128 0x1.921fc00ece4f02f278ade6ad9e8p+0 : -0xa.ca8b7d7e881db0d49193e0ab43cp-24 : inexact-ok
+= cos tonearest binary128 0x1.921fc00ece4f02f278ade6ad9e8p+0 : -0xa.ca8b7d7e881db0d49193e0ab43cp-24 : inexact-ok
+= cos towardzero binary128 0x1.921fc00ece4f02f278ade6ad9e8p+0 : -0xa.ca8b7d7e881db0d49193e0ab43b8p-24 : inexact-ok
+= cos upward binary128 0x1.921fc00ece4f02f278ade6ad9e8p+0 : -0xa.ca8b7d7e881db0d49193e0ab43b8p-24 : inexact-ok
+= cos downward ibm128 0x1.921fc00ece4f02f278ade6ad9e8p+0 : -0xa.ca8b7d7e881db0d49193e0ab44p-24 : inexact-ok
+= cos tonearest ibm128 0x1.921fc00ece4f02f278ade6ad9e8p+0 : -0xa.ca8b7d7e881db0d49193e0ab44p-24 : inexact-ok
+= cos towardzero ibm128 0x1.921fc00ece4f02f278ade6ad9e8p+0 : -0xa.ca8b7d7e881db0d49193e0ab4p-24 : inexact-ok
+= cos upward ibm128 0x1.921fc00ece4f02f278ade6ad9e8p+0 : -0xa.ca8b7d7e881db0d49193e0ab4p-24 : inexact-ok
+cos -1.57079697
+= cos downward binary32 -0x1.921fcp+0 : -0xa.bbbd3p-24 : inexact-ok
+= cos tonearest binary32 -0x1.921fcp+0 : -0xa.bbbd3p-24 : inexact-ok
+= cos towardzero binary32 -0x1.921fcp+0 : -0xa.bbbd2p-24 : inexact-ok
+= cos upward binary32 -0x1.921fcp+0 : -0xa.bbbd2p-24 : inexact-ok
+= cos downward binary64 -0x1.921fcp+0 : -0xa.bbbd2e7b95a88p-24 : inexact-ok
+= cos tonearest binary64 -0x1.921fcp+0 : -0xa.bbbd2e7b95a88p-24 : inexact-ok
+= cos towardzero binary64 -0x1.921fcp+0 : -0xa.bbbd2e7b95a8p-24 : inexact-ok
+= cos upward binary64 -0x1.921fcp+0 : -0xa.bbbd2e7b95a8p-24 : inexact-ok
+= cos downward intel96 -0x1.921fcp+0 : -0xa.bbbd2e7b95a85c7p-24 : inexact-ok
+= cos tonearest intel96 -0x1.921fcp+0 : -0xa.bbbd2e7b95a85c6p-24 : inexact-ok
+= cos towardzero intel96 -0x1.921fcp+0 : -0xa.bbbd2e7b95a85c6p-24 : inexact-ok
+= cos upward intel96 -0x1.921fcp+0 : -0xa.bbbd2e7b95a85c6p-24 : inexact-ok
+= cos downward m68k96 -0x1.921fcp+0 : -0xa.bbbd2e7b95a85c7p-24 : inexact-ok
+= cos tonearest m68k96 -0x1.921fcp+0 : -0xa.bbbd2e7b95a85c6p-24 : inexact-ok
+= cos towardzero m68k96 -0x1.921fcp+0 : -0xa.bbbd2e7b95a85c6p-24 : inexact-ok
+= cos upward m68k96 -0x1.921fcp+0 : -0xa.bbbd2e7b95a85c6p-24 : inexact-ok
+= cos downward binary128 -0x1.921fcp+0 : -0xa.bbbd2e7b95a85c638e746a5f4f68p-24 : inexact-ok
+= cos tonearest binary128 -0x1.921fcp+0 : -0xa.bbbd2e7b95a85c638e746a5f4f6p-24 : inexact-ok
+= cos towardzero binary128 -0x1.921fcp+0 : -0xa.bbbd2e7b95a85c638e746a5f4f6p-24 : inexact-ok
+= cos upward binary128 -0x1.921fcp+0 : -0xa.bbbd2e7b95a85c638e746a5f4f6p-24 : inexact-ok
+= cos downward ibm128 -0x1.921fcp+0 : -0xa.bbbd2e7b95a85c638e746a5f5p-24 : inexact-ok
+= cos tonearest ibm128 -0x1.921fcp+0 : -0xa.bbbd2e7b95a85c638e746a5f5p-24 : inexact-ok
+= cos towardzero ibm128 -0x1.921fcp+0 : -0xa.bbbd2e7b95a85c638e746a5f4cp-24 : inexact-ok
+= cos upward ibm128 -0x1.921fcp+0 : -0xa.bbbd2e7b95a85c638e746a5f4cp-24 : inexact-ok
+= cos downward binary32 -0x1.921fc2p+0 : -0xc.bbbd3p-24 : inexact-ok
+= cos tonearest binary32 -0x1.921fc2p+0 : -0xc.bbbd3p-24 : inexact-ok
+= cos towardzero binary32 -0x1.921fc2p+0 : -0xc.bbbd2p-24 : inexact-ok
+= cos upward binary32 -0x1.921fc2p+0 : -0xc.bbbd2p-24 : inexact-ok
+= cos downward binary64 -0x1.921fc2p+0 : -0xc.bbbd2e7b951e8p-24 : inexact-ok
+= cos tonearest binary64 -0x1.921fc2p+0 : -0xc.bbbd2e7b951e8p-24 : inexact-ok
+= cos towardzero binary64 -0x1.921fc2p+0 : -0xc.bbbd2e7b951ep-24 : inexact-ok
+= cos upward binary64 -0x1.921fc2p+0 : -0xc.bbbd2e7b951ep-24 : inexact-ok
+= cos downward intel96 -0x1.921fc2p+0 : -0xc.bbbd2e7b951e5b2p-24 : inexact-ok
+= cos tonearest intel96 -0x1.921fc2p+0 : -0xc.bbbd2e7b951e5b2p-24 : inexact-ok
+= cos towardzero intel96 -0x1.921fc2p+0 : -0xc.bbbd2e7b951e5b1p-24 : inexact-ok
+= cos upward intel96 -0x1.921fc2p+0 : -0xc.bbbd2e7b951e5b1p-24 : inexact-ok
+= cos downward m68k96 -0x1.921fc2p+0 : -0xc.bbbd2e7b951e5b2p-24 : inexact-ok
+= cos tonearest m68k96 -0x1.921fc2p+0 : -0xc.bbbd2e7b951e5b2p-24 : inexact-ok
+= cos towardzero m68k96 -0x1.921fc2p+0 : -0xc.bbbd2e7b951e5b1p-24 : inexact-ok
+= cos upward m68k96 -0x1.921fc2p+0 : -0xc.bbbd2e7b951e5b1p-24 : inexact-ok
+= cos downward binary128 -0x1.921fc2p+0 : -0xc.bbbd2e7b951e5b1e4cc9f460a12p-24 : inexact-ok
+= cos tonearest binary128 -0x1.921fc2p+0 : -0xc.bbbd2e7b951e5b1e4cc9f460a12p-24 : inexact-ok
+= cos towardzero binary128 -0x1.921fc2p+0 : -0xc.bbbd2e7b951e5b1e4cc9f460a118p-24 : inexact-ok
+= cos upward binary128 -0x1.921fc2p+0 : -0xc.bbbd2e7b951e5b1e4cc9f460a118p-24 : inexact-ok
+= cos downward ibm128 -0x1.921fc2p+0 : -0xc.bbbd2e7b951e5b1e4cc9f460a4p-24 : inexact-ok
+= cos tonearest ibm128 -0x1.921fc2p+0 : -0xc.bbbd2e7b951e5b1e4cc9f460ap-24 : inexact-ok
+= cos towardzero ibm128 -0x1.921fc2p+0 : -0xc.bbbd2e7b951e5b1e4cc9f460ap-24 : inexact-ok
+= cos upward ibm128 -0x1.921fc2p+0 : -0xc.bbbd2e7b951e5b1e4cc9f460ap-24 : inexact-ok
+= cos downward binary64 -0x1.921fc00ece4fp+0 : -0xa.ca8b7d7b95a58p-24 : inexact-ok
+= cos tonearest binary64 -0x1.921fc00ece4fp+0 : -0xa.ca8b7d7b95a5p-24 : inexact-ok
+= cos towardzero binary64 -0x1.921fc00ece4fp+0 : -0xa.ca8b7d7b95a5p-24 : inexact-ok
+= cos upward binary64 -0x1.921fc00ece4fp+0 : -0xa.ca8b7d7b95a5p-24 : inexact-ok
+= cos downward intel96 -0x1.921fc00ece4fp+0 : -0xa.ca8b7d7b95a502fp-24 : inexact-ok
+= cos tonearest intel96 -0x1.921fc00ece4fp+0 : -0xa.ca8b7d7b95a502fp-24 : inexact-ok
+= cos towardzero intel96 -0x1.921fc00ece4fp+0 : -0xa.ca8b7d7b95a502ep-24 : inexact-ok
+= cos upward intel96 -0x1.921fc00ece4fp+0 : -0xa.ca8b7d7b95a502ep-24 : inexact-ok
+= cos downward m68k96 -0x1.921fc00ece4fp+0 : -0xa.ca8b7d7b95a502fp-24 : inexact-ok
+= cos tonearest m68k96 -0x1.921fc00ece4fp+0 : -0xa.ca8b7d7b95a502fp-24 : inexact-ok
+= cos towardzero m68k96 -0x1.921fc00ece4fp+0 : -0xa.ca8b7d7b95a502ep-24 : inexact-ok
+= cos upward m68k96 -0x1.921fc00ece4fp+0 : -0xa.ca8b7d7b95a502ep-24 : inexact-ok
+= cos downward binary128 -0x1.921fc00ece4fp+0 : -0xa.ca8b7d7b95a502ede4a0f9b11c3p-24 : inexact-ok
+= cos tonearest binary128 -0x1.921fc00ece4fp+0 : -0xa.ca8b7d7b95a502ede4a0f9b11c28p-24 : inexact-ok
+= cos towardzero binary128 -0x1.921fc00ece4fp+0 : -0xa.ca8b7d7b95a502ede4a0f9b11c28p-24 : inexact-ok
+= cos upward binary128 -0x1.921fc00ece4fp+0 : -0xa.ca8b7d7b95a502ede4a0f9b11c28p-24 : inexact-ok
+= cos downward ibm128 -0x1.921fc00ece4fp+0 : -0xa.ca8b7d7b95a502ede4a0f9b12p-24 : inexact-ok
+= cos tonearest ibm128 -0x1.921fc00ece4fp+0 : -0xa.ca8b7d7b95a502ede4a0f9b11cp-24 : inexact-ok
+= cos towardzero ibm128 -0x1.921fc00ece4fp+0 : -0xa.ca8b7d7b95a502ede4a0f9b11cp-24 : inexact-ok
+= cos upward ibm128 -0x1.921fc00ece4fp+0 : -0xa.ca8b7d7b95a502ede4a0f9b11cp-24 : inexact-ok
+= cos downward binary64 -0x1.921fc00ece4f1p+0 : -0xa.ca8b7d8b95a58p-24 : inexact-ok
+= cos tonearest binary64 -0x1.921fc00ece4f1p+0 : -0xa.ca8b7d8b95a5p-24 : inexact-ok
+= cos towardzero binary64 -0x1.921fc00ece4f1p+0 : -0xa.ca8b7d8b95a5p-24 : inexact-ok
+= cos upward binary64 -0x1.921fc00ece4f1p+0 : -0xa.ca8b7d8b95a5p-24 : inexact-ok
+= cos downward intel96 -0x1.921fc00ece4f1p+0 : -0xa.ca8b7d8b95a502fp-24 : inexact-ok
+= cos tonearest intel96 -0x1.921fc00ece4f1p+0 : -0xa.ca8b7d8b95a502fp-24 : inexact-ok
+= cos towardzero intel96 -0x1.921fc00ece4f1p+0 : -0xa.ca8b7d8b95a502ep-24 : inexact-ok
+= cos upward intel96 -0x1.921fc00ece4f1p+0 : -0xa.ca8b7d8b95a502ep-24 : inexact-ok
+= cos downward m68k96 -0x1.921fc00ece4f1p+0 : -0xa.ca8b7d8b95a502fp-24 : inexact-ok
+= cos tonearest m68k96 -0x1.921fc00ece4f1p+0 : -0xa.ca8b7d8b95a502fp-24 : inexact-ok
+= cos towardzero m68k96 -0x1.921fc00ece4f1p+0 : -0xa.ca8b7d8b95a502ep-24 : inexact-ok
+= cos upward m68k96 -0x1.921fc00ece4f1p+0 : -0xa.ca8b7d8b95a502ep-24 : inexact-ok
+= cos downward binary128 -0x1.921fc00ece4f1p+0 : -0xa.ca8b7d8b95a502ede0fd607f3948p-24 : inexact-ok
+= cos tonearest binary128 -0x1.921fc00ece4f1p+0 : -0xa.ca8b7d8b95a502ede0fd607f394p-24 : inexact-ok
+= cos towardzero binary128 -0x1.921fc00ece4f1p+0 : -0xa.ca8b7d8b95a502ede0fd607f394p-24 : inexact-ok
+= cos upward binary128 -0x1.921fc00ece4f1p+0 : -0xa.ca8b7d8b95a502ede0fd607f394p-24 : inexact-ok
+= cos downward ibm128 -0x1.921fc00ece4f1p+0 : -0xa.ca8b7d8b95a502ede0fd607f3cp-24 : inexact-ok
+= cos tonearest ibm128 -0x1.921fc00ece4f1p+0 : -0xa.ca8b7d8b95a502ede0fd607f38p-24 : inexact-ok
+= cos towardzero ibm128 -0x1.921fc00ece4f1p+0 : -0xa.ca8b7d8b95a502ede0fd607f38p-24 : inexact-ok
+= cos upward ibm128 -0x1.921fc00ece4f1p+0 : -0xa.ca8b7d8b95a502ede0fd607f38p-24 : inexact-ok
+= cos downward intel96 -0x1.921fc00ece4f02f2p+0 : -0xa.ca8b7d7e87a502fp-24 : inexact-ok
+= cos tonearest intel96 -0x1.921fc00ece4f02f2p+0 : -0xa.ca8b7d7e87a502fp-24 : inexact-ok
+= cos towardzero intel96 -0x1.921fc00ece4f02f2p+0 : -0xa.ca8b7d7e87a502ep-24 : inexact-ok
+= cos upward intel96 -0x1.921fc00ece4f02f2p+0 : -0xa.ca8b7d7e87a502ep-24 : inexact-ok
+= cos downward m68k96 -0x1.921fc00ece4f02f2p+0 : -0xa.ca8b7d7e87a502fp-24 : inexact-ok
+= cos tonearest m68k96 -0x1.921fc00ece4f02f2p+0 : -0xa.ca8b7d7e87a502fp-24 : inexact-ok
+= cos towardzero m68k96 -0x1.921fc00ece4f02f2p+0 : -0xa.ca8b7d7e87a502ep-24 : inexact-ok
+= cos upward m68k96 -0x1.921fc00ece4f02f2p+0 : -0xa.ca8b7d7e87a502ep-24 : inexact-ok
+= cos downward binary128 -0x1.921fc00ece4f02f2p+0 : -0xa.ca8b7d7e87a502ede3f57c1dce18p-24 : inexact-ok
+= cos tonearest binary128 -0x1.921fc00ece4f02f2p+0 : -0xa.ca8b7d7e87a502ede3f57c1dce1p-24 : inexact-ok
+= cos towardzero binary128 -0x1.921fc00ece4f02f2p+0 : -0xa.ca8b7d7e87a502ede3f57c1dce1p-24 : inexact-ok
+= cos upward binary128 -0x1.921fc00ece4f02f2p+0 : -0xa.ca8b7d7e87a502ede3f57c1dce1p-24 : inexact-ok
+= cos downward ibm128 -0x1.921fc00ece4f02f2p+0 : -0xa.ca8b7d7e87a502ede3f57c1ddp-24 : inexact-ok
+= cos tonearest ibm128 -0x1.921fc00ece4f02f2p+0 : -0xa.ca8b7d7e87a502ede3f57c1ddp-24 : inexact-ok
+= cos towardzero ibm128 -0x1.921fc00ece4f02f2p+0 : -0xa.ca8b7d7e87a502ede3f57c1dccp-24 : inexact-ok
+= cos upward ibm128 -0x1.921fc00ece4f02f2p+0 : -0xa.ca8b7d7e87a502ede3f57c1dccp-24 : inexact-ok
+= cos downward intel96 -0x1.921fc00ece4f02f4p+0 : -0xa.ca8b7d7e89a502fp-24 : inexact-ok
+= cos tonearest intel96 -0x1.921fc00ece4f02f4p+0 : -0xa.ca8b7d7e89a502fp-24 : inexact-ok
+= cos towardzero intel96 -0x1.921fc00ece4f02f4p+0 : -0xa.ca8b7d7e89a502ep-24 : inexact-ok
+= cos upward intel96 -0x1.921fc00ece4f02f4p+0 : -0xa.ca8b7d7e89a502ep-24 : inexact-ok
+= cos downward m68k96 -0x1.921fc00ece4f02f4p+0 : -0xa.ca8b7d7e89a502fp-24 : inexact-ok
+= cos tonearest m68k96 -0x1.921fc00ece4f02f4p+0 : -0xa.ca8b7d7e89a502fp-24 : inexact-ok
+= cos towardzero m68k96 -0x1.921fc00ece4f02f4p+0 : -0xa.ca8b7d7e89a502ep-24 : inexact-ok
+= cos upward m68k96 -0x1.921fc00ece4f02f4p+0 : -0xa.ca8b7d7e89a502ep-24 : inexact-ok
+= cos downward binary128 -0x1.921fc00ece4f02f4p+0 : -0xa.ca8b7d7e89a502ede3f507aaa7d8p-24 : inexact-ok
+= cos tonearest binary128 -0x1.921fc00ece4f02f4p+0 : -0xa.ca8b7d7e89a502ede3f507aaa7d8p-24 : inexact-ok
+= cos towardzero binary128 -0x1.921fc00ece4f02f4p+0 : -0xa.ca8b7d7e89a502ede3f507aaa7dp-24 : inexact-ok
+= cos upward binary128 -0x1.921fc00ece4f02f4p+0 : -0xa.ca8b7d7e89a502ede3f507aaa7dp-24 : inexact-ok
+= cos downward ibm128 -0x1.921fc00ece4f02f4p+0 : -0xa.ca8b7d7e89a502ede3f507aaa8p-24 : inexact-ok
+= cos tonearest ibm128 -0x1.921fc00ece4f02f4p+0 : -0xa.ca8b7d7e89a502ede3f507aaa8p-24 : inexact-ok
+= cos towardzero ibm128 -0x1.921fc00ece4f02f4p+0 : -0xa.ca8b7d7e89a502ede3f507aaa4p-24 : inexact-ok
+= cos upward ibm128 -0x1.921fc00ece4f02f4p+0 : -0xa.ca8b7d7e89a502ede3f507aaa4p-24 : inexact-ok
+= cos downward binary128 -0x1.921fc00ece4f02f278ade6ad9e89p+0 : -0xa.ca8b7d7e881db0d49193e9ab43cp-24 : inexact-ok
+= cos tonearest binary128 -0x1.921fc00ece4f02f278ade6ad9e89p+0 : -0xa.ca8b7d7e881db0d49193e9ab43cp-24 : inexact-ok
+= cos towardzero binary128 -0x1.921fc00ece4f02f278ade6ad9e89p+0 : -0xa.ca8b7d7e881db0d49193e9ab43b8p-24 : inexact-ok
+= cos upward binary128 -0x1.921fc00ece4f02f278ade6ad9e89p+0 : -0xa.ca8b7d7e881db0d49193e9ab43b8p-24 : inexact-ok
+= cos downward binary128 -0x1.921fc00ece4f02f278ade6ad9e8ap+0 : -0xa.ca8b7d7e881db0d49193eaab43cp-24 : inexact-ok
+= cos tonearest binary128 -0x1.921fc00ece4f02f278ade6ad9e8ap+0 : -0xa.ca8b7d7e881db0d49193eaab43cp-24 : inexact-ok
+= cos towardzero binary128 -0x1.921fc00ece4f02f278ade6ad9e8ap+0 : -0xa.ca8b7d7e881db0d49193eaab43b8p-24 : inexact-ok
+= cos upward binary128 -0x1.921fc00ece4f02f278ade6ad9e8ap+0 : -0xa.ca8b7d7e881db0d49193eaab43b8p-24 : inexact-ok
+= cos downward binary128 -0x1.921fc00ece4f02f278ade6ad9e8p+0 : -0xa.ca8b7d7e881db0d49193e0ab43cp-24 : inexact-ok
+= cos tonearest binary128 -0x1.921fc00ece4f02f278ade6ad9e8p+0 : -0xa.ca8b7d7e881db0d49193e0ab43cp-24 : inexact-ok
+= cos towardzero binary128 -0x1.921fc00ece4f02f278ade6ad9e8p+0 : -0xa.ca8b7d7e881db0d49193e0ab43b8p-24 : inexact-ok
+= cos upward binary128 -0x1.921fc00ece4f02f278ade6ad9e8p+0 : -0xa.ca8b7d7e881db0d49193e0ab43b8p-24 : inexact-ok
+= cos downward ibm128 -0x1.921fc00ece4f02f278ade6ad9e8p+0 : -0xa.ca8b7d7e881db0d49193e0ab44p-24 : inexact-ok
+= cos tonearest ibm128 -0x1.921fc00ece4f02f278ade6ad9e8p+0 : -0xa.ca8b7d7e881db0d49193e0ab44p-24 : inexact-ok
+= cos towardzero ibm128 -0x1.921fc00ece4f02f278ade6ad9e8p+0 : -0xa.ca8b7d7e881db0d49193e0ab4p-24 : inexact-ok
+= cos upward ibm128 -0x1.921fc00ece4f02f278ade6ad9e8p+0 : -0xa.ca8b7d7e881db0d49193e0ab4p-24 : inexact-ok
+= cos downward binary128 -0x1.921fc00ece4f02f278ade6ad9fp+0 : -0xa.ca8b7d7e881db0d4919460ab43cp-24 : inexact-ok
+= cos tonearest binary128 -0x1.921fc00ece4f02f278ade6ad9fp+0 : -0xa.ca8b7d7e881db0d4919460ab43cp-24 : inexact-ok
+= cos towardzero binary128 -0x1.921fc00ece4f02f278ade6ad9fp+0 : -0xa.ca8b7d7e881db0d4919460ab43b8p-24 : inexact-ok
+= cos upward binary128 -0x1.921fc00ece4f02f278ade6ad9fp+0 : -0xa.ca8b7d7e881db0d4919460ab43b8p-24 : inexact-ok
+= cos downward ibm128 -0x1.921fc00ece4f02f278ade6ad9fp+0 : -0xa.ca8b7d7e881db0d4919460ab44p-24 : inexact-ok
+= cos tonearest ibm128 -0x1.921fc00ece4f02f278ade6ad9fp+0 : -0xa.ca8b7d7e881db0d4919460ab44p-24 : inexact-ok
+= cos towardzero ibm128 -0x1.921fc00ece4f02f278ade6ad9fp+0 : -0xa.ca8b7d7e881db0d4919460ab4p-24 : inexact-ok
+= cos upward ibm128 -0x1.921fc00ece4f02f278ade6ad9fp+0 : -0xa.ca8b7d7e881db0d4919460ab4p-24 : inexact-ok
diff --git a/math/auto-libm-test-out-tan b/math/auto-libm-test-out-tan
index 9f3e6cbce0..7d00d03e1d 100644
--- a/math/auto-libm-test-out-tan
+++ b/math/auto-libm-test-out-tan
@@ -2242,6 +2242,296 @@ tan -0x1.0d55b8p+0
 = tan tonearest ibm128 -0x1.0d55b8p+0 : -0x1.c074f83e72236f1900dbba65f78p+0 : inexact-ok
 = tan towardzero ibm128 -0x1.0d55b8p+0 : -0x1.c074f83e72236f1900dbba65f78p+0 : inexact-ok
 = tan upward ibm128 -0x1.0d55b8p+0 : -0x1.c074f83e72236f1900dbba65f78p+0 : inexact-ok
+tan 1.57079697
+= tan downward binary32 0x1.921fc2p+0 : -0x1.41acc2p+20 : inexact-ok
+= tan tonearest binary32 0x1.921fc2p+0 : -0x1.41acc2p+20 : inexact-ok
+= tan towardzero binary