Improve the accuracy of tgamma (BZ #26983)

With this patch, the maximal known error for tgamma is now reduced to 9 ulps
for dbl-64, for all rounding modes. Since exhaustive testing is not possible
for dbl-64, it might be that there are still cases with an error larger than
9 ulps, but all known cases are fixed (intensive tests were done to find cases
with large errors).

Tested on x86_64 and powerpc (and by Adhemerval Zanella on aarch64, arm,
s390x, sparc, and i686).
Reviewed-by: Adhemerval Zanella  <adhemerval.zanella@linaro.org>

diff --git a/math/auto-libm-test-in b/math/auto-libm-test-in
index 9fbd0c626bd3c24d557da494dc897b4a0f5cee02..2045a0f8d1d3dbc1543435ad1d081374b9842e53 100644 (file)
--- a/math/auto-libm-test-in
+++ b/math/auto-libm-test-in
@@ -8251,8 +8251,12 @@ tgamma -0x6.ce9158p+0
 tgamma -0xd.cbf53d0e7d06p+0
 # the next value generates larger error bounds on x86_64 (binary32)
 tgamma -0x3.0aa534p+0
-# the next value generates larger error bounds on x86_64 (binary64)
+# the next values generate large error bounds on x86_64 (binary64)
 tgamma -0x1.62b8c36c7180bp+4
+tgamma -0x1.62c4d519e8677p+3
+tgamma -0x1.c033cc426752fp+2
+tgamma -0x1.62cfd0d34ade2p+3
+tgamma -0x1.8814da6eb7dbp+5
 
 y0 0.125
 y0 0.75

diff --git a/math/auto-libm-test-out-tgamma b/math/auto-libm-test-out-tgamma
index 032ad3a7d7b79cbd2a42aeae7ad73bd5cebb334d..20a9916afa45ff4b5e00878c905580538e3db1e1 100644 (file)
--- a/math/auto-libm-test-out-tgamma
+++ b/math/auto-libm-test-out-tgamma
@@ -27397,3 +27397,279 @@ tgamma -0x1.62b8c36c7180bp+4
 = tgamma tonearest ibm128 -0x1.62b8c36c7180bp+4 : -0xf.3fb73d327d6a801abb3039b104p-72 : inexact-ok
 = tgamma towardzero ibm128 -0x1.62b8c36c7180bp+4 : -0xf.3fb73d327d6a801abb3039b1p-72 : inexact-ok
 = tgamma upward ibm128 -0x1.62b8c36c7180bp+4 : -0xf.3fb73d327d6a801abb3039b1p-72 : inexact-ok
+tgamma -0x1.62c4d519e8677p+3
+= tgamma downward binary32 -0xb.1626ap+0 : 0x3.fac968p-24 : inexact-ok
+= tgamma tonearest binary32 -0xb.1626ap+0 : 0x3.fac96cp-24 : inexact-ok
+= tgamma towardzero binary32 -0xb.1626ap+0 : 0x3.fac968p-24 : inexact-ok
+= tgamma upward binary32 -0xb.1626ap+0 : 0x3.fac96cp-24 : inexact-ok
+= tgamma downward binary64 -0xb.1626ap+0 : 0x3.fac96a100e60ep-24 : inexact-ok
+= tgamma tonearest binary64 -0xb.1626ap+0 : 0x3.fac96a100e61p-24 : inexact-ok
+= tgamma towardzero binary64 -0xb.1626ap+0 : 0x3.fac96a100e60ep-24 : inexact-ok
+= tgamma upward binary64 -0xb.1626ap+0 : 0x3.fac96a100e61p-24 : inexact-ok
+= tgamma downward intel96 -0xb.1626ap+0 : 0x3.fac96a100e60f1f8p-24 : inexact-ok
+= tgamma tonearest intel96 -0xb.1626ap+0 : 0x3.fac96a100e60f1fcp-24 : inexact-ok
+= tgamma towardzero intel96 -0xb.1626ap+0 : 0x3.fac96a100e60f1f8p-24 : inexact-ok
+= tgamma upward intel96 -0xb.1626ap+0 : 0x3.fac96a100e60f1fcp-24 : inexact-ok
+= tgamma downward m68k96 -0xb.1626ap+0 : 0x3.fac96a100e60f1f8p-24 : inexact-ok
+= tgamma tonearest m68k96 -0xb.1626ap+0 : 0x3.fac96a100e60f1fcp-24 : inexact-ok
+= tgamma towardzero m68k96 -0xb.1626ap+0 : 0x3.fac96a100e60f1f8p-24 : inexact-ok
+= tgamma upward m68k96 -0xb.1626ap+0 : 0x3.fac96a100e60f1fcp-24 : inexact-ok
+= tgamma downward binary128 -0xb.1626ap+0 : 0x3.fac96a100e60f1fa454b8cdcdfb6p-24 : inexact-ok
+= tgamma tonearest binary128 -0xb.1626ap+0 : 0x3.fac96a100e60f1fa454b8cdcdfb8p-24 : inexact-ok
+= tgamma towardzero binary128 -0xb.1626ap+0 : 0x3.fac96a100e60f1fa454b8cdcdfb6p-24 : inexact-ok
+= tgamma upward binary128 -0xb.1626ap+0 : 0x3.fac96a100e60f1fa454b8cdcdfb8p-24 : inexact-ok
+= tgamma downward ibm128 -0xb.1626ap+0 : 0x3.fac96a100e60f1fa454b8cdcdfp-24 : inexact-ok
+= tgamma tonearest ibm128 -0xb.1626ap+0 : 0x3.fac96a100e60f1fa454b8cdcep-24 : inexact-ok
+= tgamma towardzero ibm128 -0xb.1626ap+0 : 0x3.fac96a100e60f1fa454b8cdcdfp-24 : inexact-ok
+= tgamma upward ibm128 -0xb.1626ap+0 : 0x3.fac96a100e60f1fa454b8cdcep-24 : inexact-ok
+= tgamma downward binary32 -0xb.1626bp+0 : 0x3.fac6p-24 : inexact-ok
+= tgamma tonearest binary32 -0xb.1626bp+0 : 0x3.fac6p-24 : inexact-ok
+= tgamma towardzero binary32 -0xb.1626bp+0 : 0x3.fac6p-24 : inexact-ok
+= tgamma upward binary32 -0xb.1626bp+0 : 0x3.fac604p-24 : inexact-ok
+= tgamma downward binary64 -0xb.1626bp+0 : 0x3.fac600633d74p-24 : inexact-ok
+= tgamma tonearest binary64 -0xb.1626bp+0 : 0x3.fac600633d742p-24 : inexact-ok
+= tgamma towardzero binary64 -0xb.1626bp+0 : 0x3.fac600633d74p-24 : inexact-ok
+= tgamma upward binary64 -0xb.1626bp+0 : 0x3.fac600633d742p-24 : inexact-ok
+= tgamma downward intel96 -0xb.1626bp+0 : 0x3.fac600633d7414fp-24 : inexact-ok
+= tgamma tonearest intel96 -0xb.1626bp+0 : 0x3.fac600633d7414f4p-24 : inexact-ok
+= tgamma towardzero intel96 -0xb.1626bp+0 : 0x3.fac600633d7414fp-24 : inexact-ok
+= tgamma upward intel96 -0xb.1626bp+0 : 0x3.fac600633d7414f4p-24 : inexact-ok
+= tgamma downward m68k96 -0xb.1626bp+0 : 0x3.fac600633d7414fp-24 : inexact-ok
+= tgamma tonearest m68k96 -0xb.1626bp+0 : 0x3.fac600633d7414f4p-24 : inexact-ok
+= tgamma towardzero m68k96 -0xb.1626bp+0 : 0x3.fac600633d7414fp-24 : inexact-ok
+= tgamma upward m68k96 -0xb.1626bp+0 : 0x3.fac600633d7414f4p-24 : inexact-ok
+= tgamma downward binary128 -0xb.1626bp+0 : 0x3.fac600633d7414f378d97364607cp-24 : inexact-ok
+= tgamma tonearest binary128 -0xb.1626bp+0 : 0x3.fac600633d7414f378d97364607cp-24 : inexact-ok
+= tgamma towardzero binary128 -0xb.1626bp+0 : 0x3.fac600633d7414f378d97364607cp-24 : inexact-ok
+= tgamma upward binary128 -0xb.1626bp+0 : 0x3.fac600633d7414f378d97364607ep-24 : inexact-ok
+= tgamma downward ibm128 -0xb.1626bp+0 : 0x3.fac600633d7414f378d973646p-24 : inexact-ok
+= tgamma tonearest ibm128 -0xb.1626bp+0 : 0x3.fac600633d7414f378d973646p-24 : inexact-ok
+= tgamma towardzero ibm128 -0xb.1626bp+0 : 0x3.fac600633d7414f378d973646p-24 : inexact-ok
+= tgamma upward ibm128 -0xb.1626bp+0 : 0x3.fac600633d7414f378d9736461p-24 : inexact-ok
+= tgamma downward binary64 -0xb.1626a8cf433b8p+0 : 0x3.fac7890382d7ep-24 : inexact-ok
+= tgamma tonearest binary64 -0xb.1626a8cf433b8p+0 : 0x3.fac7890382d7ep-24 : inexact-ok
+= tgamma towardzero binary64 -0xb.1626a8cf433b8p+0 : 0x3.fac7890382d7ep-24 : inexact-ok
+= tgamma upward binary64 -0xb.1626a8cf433b8p+0 : 0x3.fac7890382d8p-24 : inexact-ok
+= tgamma downward intel96 -0xb.1626a8cf433b8p+0 : 0x3.fac7890382d7e69p-24 : inexact-ok
+= tgamma tonearest intel96 -0xb.1626a8cf433b8p+0 : 0x3.fac7890382d7e694p-24 : inexact-ok
+= tgamma towardzero intel96 -0xb.1626a8cf433b8p+0 : 0x3.fac7890382d7e69p-24 : inexact-ok
+= tgamma upward intel96 -0xb.1626a8cf433b8p+0 : 0x3.fac7890382d7e694p-24 : inexact-ok
+= tgamma downward m68k96 -0xb.1626a8cf433b8p+0 : 0x3.fac7890382d7e69p-24 : inexact-ok
+= tgamma tonearest m68k96 -0xb.1626a8cf433b8p+0 : 0x3.fac7890382d7e694p-24 : inexact-ok
+= tgamma towardzero m68k96 -0xb.1626a8cf433b8p+0 : 0x3.fac7890382d7e69p-24 : inexact-ok
+= tgamma upward m68k96 -0xb.1626a8cf433b8p+0 : 0x3.fac7890382d7e694p-24 : inexact-ok
+= tgamma downward binary128 -0xb.1626a8cf433b8p+0 : 0x3.fac7890382d7e693c33b139429ecp-24 : inexact-ok
+= tgamma tonearest binary128 -0xb.1626a8cf433b8p+0 : 0x3.fac7890382d7e693c33b139429ecp-24 : inexact-ok
+= tgamma towardzero binary128 -0xb.1626a8cf433b8p+0 : 0x3.fac7890382d7e693c33b139429ecp-24 : inexact-ok
+= tgamma upward binary128 -0xb.1626a8cf433b8p+0 : 0x3.fac7890382d7e693c33b139429eep-24 : inexact-ok
+= tgamma downward ibm128 -0xb.1626a8cf433b8p+0 : 0x3.fac7890382d7e693c33b139429p-24 : inexact-ok
+= tgamma tonearest ibm128 -0xb.1626a8cf433b8p+0 : 0x3.fac7890382d7e693c33b13942ap-24 : inexact-ok
+= tgamma towardzero ibm128 -0xb.1626a8cf433b8p+0 : 0x3.fac7890382d7e693c33b139429p-24 : inexact-ok
+= tgamma upward ibm128 -0xb.1626a8cf433b8p+0 : 0x3.fac7890382d7e693c33b13942ap-24 : inexact-ok
+tgamma -0x1.c033cc426752fp+2
+= tgamma downward binary32 -0x7.00cf3p+0 : 0xf.f7014p-8 : inexact-ok
+= tgamma tonearest binary32 -0x7.00cf3p+0 : 0xf.f7014p-8 : inexact-ok
+= tgamma towardzero binary32 -0x7.00cf3p+0 : 0xf.f7014p-8 : inexact-ok
+= tgamma upward binary32 -0x7.00cf3p+0 : 0xf.f7015p-8 : inexact-ok
+= tgamma downward binary64 -0x7.00cf3p+0 : 0xf.f7014642d9848p-8 : inexact-ok
+= tgamma tonearest binary64 -0x7.00cf3p+0 : 0xf.f7014642d9848p-8 : inexact-ok
+= tgamma towardzero binary64 -0x7.00cf3p+0 : 0xf.f7014642d9848p-8 : inexact-ok
+= tgamma upward binary64 -0x7.00cf3p+0 : 0xf.f7014642d985p-8 : inexact-ok
+= tgamma downward intel96 -0x7.00cf3p+0 : 0xf.f7014642d9849fcp-8 : inexact-ok
+= tgamma tonearest intel96 -0x7.00cf3p+0 : 0xf.f7014642d9849fcp-8 : inexact-ok
+= tgamma towardzero intel96 -0x7.00cf3p+0 : 0xf.f7014642d9849fcp-8 : inexact-ok
+= tgamma upward intel96 -0x7.00cf3p+0 : 0xf.f7014642d9849fdp-8 : inexact-ok
+= tgamma downward m68k96 -0x7.00cf3p+0 : 0xf.f7014642d9849fcp-8 : inexact-ok
+= tgamma tonearest m68k96 -0x7.00cf3p+0 : 0xf.f7014642d9849fcp-8 : inexact-ok
+= tgamma towardzero m68k96 -0x7.00cf3p+0 : 0xf.f7014642d9849fcp-8 : inexact-ok
+= tgamma upward m68k96 -0x7.00cf3p+0 : 0xf.f7014642d9849fdp-8 : inexact-ok
+= tgamma downward binary128 -0x7.00cf3p+0 : 0xf.f7014642d9849fc3f8a3bdea2ae8p-8 : inexact-ok
+= tgamma tonearest binary128 -0x7.00cf3p+0 : 0xf.f7014642d9849fc3f8a3bdea2afp-8 : inexact-ok
+= tgamma towardzero binary128 -0x7.00cf3p+0 : 0xf.f7014642d9849fc3f8a3bdea2ae8p-8 : inexact-ok
+= tgamma upward binary128 -0x7.00cf3p+0 : 0xf.f7014642d9849fc3f8a3bdea2afp-8 : inexact-ok
+= tgamma downward ibm128 -0x7.00cf3p+0 : 0xf.f7014642d9849fc3f8a3bdea28p-8 : inexact-ok
+= tgamma tonearest ibm128 -0x7.00cf3p+0 : 0xf.f7014642d9849fc3f8a3bdea2cp-8 : inexact-ok
+= tgamma towardzero ibm128 -0x7.00cf3p+0 : 0xf.f7014642d9849fc3f8a3bdea28p-8 : inexact-ok
+= tgamma upward ibm128 -0x7.00cf3p+0 : 0xf.f7014642d9849fc3f8a3bdea2cp-8 : inexact-ok
+= tgamma downward binary32 -0x7.00cf38p+0 : 0xf.f6627p-8 : inexact-ok
+= tgamma tonearest binary32 -0x7.00cf38p+0 : 0xf.f6628p-8 : inexact-ok
+= tgamma towardzero binary32 -0x7.00cf38p+0 : 0xf.f6627p-8 : inexact-ok
+= tgamma upward binary32 -0x7.00cf38p+0 : 0xf.f6628p-8 : inexact-ok
+= tgamma downward binary64 -0x7.00cf38p+0 : 0xf.f6627d248f8ep-8 : inexact-ok
+= tgamma tonearest binary64 -0x7.00cf38p+0 : 0xf.f6627d248f8ep-8 : inexact-ok
+= tgamma towardzero binary64 -0x7.00cf38p+0 : 0xf.f6627d248f8ep-8 : inexact-ok
+= tgamma upward binary64 -0x7.00cf38p+0 : 0xf.f6627d248f8e8p-8 : inexact-ok
+= tgamma downward intel96 -0x7.00cf38p+0 : 0xf.f6627d248f8e25dp-8 : inexact-ok
+= tgamma tonearest intel96 -0x7.00cf38p+0 : 0xf.f6627d248f8e25ep-8 : inexact-ok
+= tgamma towardzero intel96 -0x7.00cf38p+0 : 0xf.f6627d248f8e25dp-8 : inexact-ok
+= tgamma upward intel96 -0x7.00cf38p+0 : 0xf.f6627d248f8e25ep-8 : inexact-ok
+= tgamma downward m68k96 -0x7.00cf38p+0 : 0xf.f6627d248f8e25dp-8 : inexact-ok
+= tgamma tonearest m68k96 -0x7.00cf38p+0 : 0xf.f6627d248f8e25ep-8 : inexact-ok
+= tgamma towardzero m68k96 -0x7.00cf38p+0 : 0xf.f6627d248f8e25dp-8 : inexact-ok
+= tgamma upward m68k96 -0x7.00cf38p+0 : 0xf.f6627d248f8e25ep-8 : inexact-ok
+= tgamma downward binary128 -0x7.00cf38p+0 : 0xf.f6627d248f8e25ddceb81585fcbp-8 : inexact-ok
+= tgamma tonearest binary128 -0x7.00cf38p+0 : 0xf.f6627d248f8e25ddceb81585fcb8p-8 : inexact-ok
+= tgamma towardzero binary128 -0x7.00cf38p+0 : 0xf.f6627d248f8e25ddceb81585fcbp-8 : inexact-ok
+= tgamma upward binary128 -0x7.00cf38p+0 : 0xf.f6627d248f8e25ddceb81585fcb8p-8 : inexact-ok
+= tgamma downward ibm128 -0x7.00cf38p+0 : 0xf.f6627d248f8e25ddceb81585fcp-8 : inexact-ok
+= tgamma tonearest ibm128 -0x7.00cf38p+0 : 0xf.f6627d248f8e25ddceb81585fcp-8 : inexact-ok
+= tgamma towardzero ibm128 -0x7.00cf38p+0 : 0xf.f6627d248f8e25ddceb81585fcp-8 : inexact-ok
+= tgamma upward ibm128 -0x7.00cf38p+0 : 0xf.f6627d248f8e25ddceb81586p-8 : inexact-ok
+= tgamma downward binary64 -0x7.00cf31099d4bcp+0 : 0xf.f6ecad9999808p-8 : inexact-ok
+= tgamma tonearest binary64 -0x7.00cf31099d4bcp+0 : 0xf.f6ecad999981p-8 : inexact-ok
+= tgamma towardzero binary64 -0x7.00cf31099d4bcp+0 : 0xf.f6ecad9999808p-8 : inexact-ok
+= tgamma upward binary64 -0x7.00cf31099d4bcp+0 : 0xf.f6ecad999981p-8 : inexact-ok
+= tgamma downward intel96 -0x7.00cf31099d4bcp+0 : 0xf.f6ecad999980e4ap-8 : inexact-ok
+= tgamma tonearest intel96 -0x7.00cf31099d4bcp+0 : 0xf.f6ecad999980e4ap-8 : inexact-ok
+= tgamma towardzero intel96 -0x7.00cf31099d4bcp+0 : 0xf.f6ecad999980e4ap-8 : inexact-ok
+= tgamma upward intel96 -0x7.00cf31099d4bcp+0 : 0xf.f6ecad999980e4bp-8 : inexact-ok
+= tgamma downward m68k96 -0x7.00cf31099d4bcp+0 : 0xf.f6ecad999980e4ap-8 : inexact-ok
+= tgamma tonearest m68k96 -0x7.00cf31099d4bcp+0 : 0xf.f6ecad999980e4ap-8 : inexact-ok
+= tgamma towardzero m68k96 -0x7.00cf31099d4bcp+0 : 0xf.f6ecad999980e4ap-8 : inexact-ok
+= tgamma upward m68k96 -0x7.00cf31099d4bcp+0 : 0xf.f6ecad999980e4bp-8 : inexact-ok
+= tgamma downward binary128 -0x7.00cf31099d4bcp+0 : 0xf.f6ecad999980e4a4868608814ecp-8 : inexact-ok
+= tgamma tonearest binary128 -0x7.00cf31099d4bcp+0 : 0xf.f6ecad999980e4a4868608814ecp-8 : inexact-ok
+= tgamma towardzero binary128 -0x7.00cf31099d4bcp+0 : 0xf.f6ecad999980e4a4868608814ecp-8 : inexact-ok
+= tgamma upward binary128 -0x7.00cf31099d4bcp+0 : 0xf.f6ecad999980e4a4868608814ec8p-8 : inexact-ok
+= tgamma downward ibm128 -0x7.00cf31099d4bcp+0 : 0xf.f6ecad999980e4a4868608814cp-8 : inexact-ok
+= tgamma tonearest ibm128 -0x7.00cf31099d4bcp+0 : 0xf.f6ecad999980e4a4868608815p-8 : inexact-ok
+= tgamma towardzero ibm128 -0x7.00cf31099d4bcp+0 : 0xf.f6ecad999980e4a4868608814cp-8 : inexact-ok
+= tgamma upward ibm128 -0x7.00cf31099d4bcp+0 : 0xf.f6ecad999980e4a4868608815p-8 : inexact-ok
+tgamma -0x1.62cfd0d34ade2p+3
+= tgamma downward binary32 -0xb.167e8p+0 : 0x3.e85604p-24 : inexact-ok
+= tgamma tonearest binary32 -0xb.167e8p+0 : 0x3.e85608p-24 : inexact-ok
+= tgamma towardzero binary32 -0xb.167e8p+0 : 0x3.e85604p-24 : inexact-ok
+= tgamma upward binary32 -0xb.167e8p+0 : 0x3.e85608p-24 : inexact-ok
+= tgamma downward binary64 -0xb.167e8p+0 : 0x3.e85606c693e9ap-24 : inexact-ok
+= tgamma tonearest binary64 -0xb.167e8p+0 : 0x3.e85606c693e9cp-24 : inexact-ok
+= tgamma towardzero binary64 -0xb.167e8p+0 : 0x3.e85606c693e9ap-24 : inexact-ok
+= tgamma upward binary64 -0xb.167e8p+0 : 0x3.e85606c693e9cp-24 : inexact-ok
+= tgamma downward intel96 -0xb.167e8p+0 : 0x3.e85606c693e9bf24p-24 : inexact-ok
+= tgamma tonearest intel96 -0xb.167e8p+0 : 0x3.e85606c693e9bf24p-24 : inexact-ok
+= tgamma towardzero intel96 -0xb.167e8p+0 : 0x3.e85606c693e9bf24p-24 : inexact-ok
+= tgamma upward intel96 -0xb.167e8p+0 : 0x3.e85606c693e9bf28p-24 : inexact-ok
+= tgamma downward m68k96 -0xb.167e8p+0 : 0x3.e85606c693e9bf24p-24 : inexact-ok
+= tgamma tonearest m68k96 -0xb.167e8p+0 : 0x3.e85606c693e9bf24p-24 : inexact-ok
+= tgamma towardzero m68k96 -0xb.167e8p+0 : 0x3.e85606c693e9bf24p-24 : inexact-ok
+= tgamma upward m68k96 -0xb.167e8p+0 : 0x3.e85606c693e9bf28p-24 : inexact-ok
+= tgamma downward binary128 -0xb.167e8p+0 : 0x3.e85606c693e9bf25ff07132c8c3ap-24 : inexact-ok
+= tgamma tonearest binary128 -0xb.167e8p+0 : 0x3.e85606c693e9bf25ff07132c8c3cp-24 : inexact-ok
+= tgamma towardzero binary128 -0xb.167e8p+0 : 0x3.e85606c693e9bf25ff07132c8c3ap-24 : inexact-ok
+= tgamma upward binary128 -0xb.167e8p+0 : 0x3.e85606c693e9bf25ff07132c8c3cp-24 : inexact-ok
+= tgamma downward ibm128 -0xb.167e8p+0 : 0x3.e85606c693e9bf25ff07132c8cp-24 : inexact-ok
+= tgamma tonearest ibm128 -0xb.167e8p+0 : 0x3.e85606c693e9bf25ff07132c8cp-24 : inexact-ok
+= tgamma towardzero ibm128 -0xb.167e8p+0 : 0x3.e85606c693e9bf25ff07132c8cp-24 : inexact-ok
+= tgamma upward ibm128 -0xb.167e8p+0 : 0x3.e85606c693e9bf25ff07132c8dp-24 : inexact-ok
+= tgamma downward binary32 -0xb.167e9p+0 : 0x3.e852b8p-24 : inexact-ok
+= tgamma tonearest binary32 -0xb.167e9p+0 : 0x3.e852b8p-24 : inexact-ok
+= tgamma towardzero binary32 -0xb.167e9p+0 : 0x3.e852b8p-24 : inexact-ok
+= tgamma upward binary32 -0xb.167e9p+0 : 0x3.e852bcp-24 : inexact-ok
+= tgamma downward binary64 -0xb.167e9p+0 : 0x3.e852b838ddf22p-24 : inexact-ok
+= tgamma tonearest binary64 -0xb.167e9p+0 : 0x3.e852b838ddf24p-24 : inexact-ok
+= tgamma towardzero binary64 -0xb.167e9p+0 : 0x3.e852b838ddf22p-24 : inexact-ok
+= tgamma upward binary64 -0xb.167e9p+0 : 0x3.e852b838ddf24p-24 : inexact-ok
+= tgamma downward intel96 -0xb.167e9p+0 : 0x3.e852b838ddf2348cp-24 : inexact-ok
+= tgamma tonearest intel96 -0xb.167e9p+0 : 0x3.e852b838ddf2349p-24 : inexact-ok
+= tgamma towardzero intel96 -0xb.167e9p+0 : 0x3.e852b838ddf2348cp-24 : inexact-ok
+= tgamma upward intel96 -0xb.167e9p+0 : 0x3.e852b838ddf2349p-24 : inexact-ok
+= tgamma downward m68k96 -0xb.167e9p+0 : 0x3.e852b838ddf2348cp-24 : inexact-ok
+= tgamma tonearest m68k96 -0xb.167e9p+0 : 0x3.e852b838ddf2349p-24 : inexact-ok
+= tgamma towardzero m68k96 -0xb.167e9p+0 : 0x3.e852b838ddf2348cp-24 : inexact-ok
+= tgamma upward m68k96 -0xb.167e9p+0 : 0x3.e852b838ddf2349p-24 : inexact-ok
+= tgamma downward binary128 -0xb.167e9p+0 : 0x3.e852b838ddf2348f185dd745531ep-24 : inexact-ok
+= tgamma tonearest binary128 -0xb.167e9p+0 : 0x3.e852b838ddf2348f185dd745531ep-24 : inexact-ok
+= tgamma towardzero binary128 -0xb.167e9p+0 : 0x3.e852b838ddf2348f185dd745531ep-24 : inexact-ok
+= tgamma upward binary128 -0xb.167e9p+0 : 0x3.e852b838ddf2348f185dd745532p-24 : inexact-ok
+= tgamma downward ibm128 -0xb.167e9p+0 : 0x3.e852b838ddf2348f185dd74553p-24 : inexact-ok
+= tgamma tonearest ibm128 -0xb.167e9p+0 : 0x3.e852b838ddf2348f185dd74553p-24 : inexact-ok
+= tgamma towardzero ibm128 -0xb.167e9p+0 : 0x3.e852b838ddf2348f185dd74553p-24 : inexact-ok
+= tgamma upward ibm128 -0xb.167e9p+0 : 0x3.e852b838ddf2348f185dd74554p-24 : inexact-ok
+= tgamma downward binary64 -0xb.167e869a56f1p+0 : 0x3.e854a96acdfa6p-24 : inexact-ok
+= tgamma tonearest binary64 -0xb.167e869a56f1p+0 : 0x3.e854a96acdfa8p-24 : inexact-ok
+= tgamma towardzero binary64 -0xb.167e869a56f1p+0 : 0x3.e854a96acdfa6p-24 : inexact-ok
+= tgamma upward binary64 -0xb.167e869a56f1p+0 : 0x3.e854a96acdfa8p-24 : inexact-ok
+= tgamma downward intel96 -0xb.167e869a56f1p+0 : 0x3.e854a96acdfa7e84p-24 : inexact-ok
+= tgamma tonearest intel96 -0xb.167e869a56f1p+0 : 0x3.e854a96acdfa7e84p-24 : inexact-ok
+= tgamma towardzero intel96 -0xb.167e869a56f1p+0 : 0x3.e854a96acdfa7e84p-24 : inexact-ok
+= tgamma upward intel96 -0xb.167e869a56f1p+0 : 0x3.e854a96acdfa7e88p-24 : inexact-ok
+= tgamma downward m68k96 -0xb.167e869a56f1p+0 : 0x3.e854a96acdfa7e84p-24 : inexact-ok
+= tgamma tonearest m68k96 -0xb.167e869a56f1p+0 : 0x3.e854a96acdfa7e84p-24 : inexact-ok
+= tgamma towardzero m68k96 -0xb.167e869a56f1p+0 : 0x3.e854a96acdfa7e84p-24 : inexact-ok
+= tgamma upward m68k96 -0xb.167e869a56f1p+0 : 0x3.e854a96acdfa7e88p-24 : inexact-ok
+= tgamma downward binary128 -0xb.167e869a56f1p+0 : 0x3.e854a96acdfa7e84ba5e7829836ap-24 : inexact-ok
+= tgamma tonearest binary128 -0xb.167e869a56f1p+0 : 0x3.e854a96acdfa7e84ba5e7829836ap-24 : inexact-ok
+= tgamma towardzero binary128 -0xb.167e869a56f1p+0 : 0x3.e854a96acdfa7e84ba5e7829836ap-24 : inexact-ok
+= tgamma upward binary128 -0xb.167e869a56f1p+0 : 0x3.e854a96acdfa7e84ba5e7829836cp-24 : inexact-ok
+= tgamma downward ibm128 -0xb.167e869a56f1p+0 : 0x3.e854a96acdfa7e84ba5e782983p-24 : inexact-ok
+= tgamma tonearest ibm128 -0xb.167e869a56f1p+0 : 0x3.e854a96acdfa7e84ba5e782983p-24 : inexact-ok
+= tgamma towardzero ibm128 -0xb.167e869a56f1p+0 : 0x3.e854a96acdfa7e84ba5e782983p-24 : inexact-ok
+= tgamma upward ibm128 -0xb.167e869a56f1p+0 : 0x3.e854a96acdfa7e84ba5e782984p-24 : inexact-ok
+tgamma -0x1.8814da6eb7dbp+5
+= tgamma downward binary32 -0x3.1029b4p+4 : 0x0p+0 : inexact-ok underflow errno-erange
+= tgamma tonearest binary32 -0x3.1029b4p+4 : 0x0p+0 : inexact-ok underflow errno-erange
+= tgamma towardzero binary32 -0x3.1029b4p+4 : 0x0p+0 : inexact-ok underflow errno-erange
+= tgamma upward binary32 -0x3.1029b4p+4 : 0x8p-152 : inexact-ok underflow errno-erange-ok
+= tgamma downward binary64 -0x3.1029b4p+4 : 0x3.fd9063a22653ep-204 : inexact-ok
+= tgamma tonearest binary64 -0x3.1029b4p+4 : 0x3.fd9063a22653ep-204 : inexact-ok
+= tgamma towardzero binary64 -0x3.1029b4p+4 : 0x3.fd9063a22653ep-204 : inexact-ok
+= tgamma upward binary64 -0x3.1029b4p+4 : 0x3.fd9063a22654p-204 : inexact-ok
+= tgamma downward intel96 -0x3.1029b4p+4 : 0x3.fd9063a22653e8e4p-204 : inexact-ok
+= tgamma tonearest intel96 -0x3.1029b4p+4 : 0x3.fd9063a22653e8e4p-204 : inexact-ok
+= tgamma towardzero intel96 -0x3.1029b4p+4 : 0x3.fd9063a22653e8e4p-204 : inexact-ok
+= tgamma upward intel96 -0x3.1029b4p+4 : 0x3.fd9063a22653e8e8p-204 : inexact-ok
+= tgamma downward m68k96 -0x3.1029b4p+4 : 0x3.fd9063a22653e8e4p-204 : inexact-ok
+= tgamma tonearest m68k96 -0x3.1029b4p+4 : 0x3.fd9063a22653e8e4p-204 : inexact-ok
+= tgamma towardzero m68k96 -0x3.1029b4p+4 : 0x3.fd9063a22653e8e4p-204 : inexact-ok
+= tgamma upward m68k96 -0x3.1029b4p+4 : 0x3.fd9063a22653e8e8p-204 : inexact-ok
+= tgamma downward binary128 -0x3.1029b4p+4 : 0x3.fd9063a22653e8e5a73bb2d5a5bcp-204 : inexact-ok
+= tgamma tonearest binary128 -0x3.1029b4p+4 : 0x3.fd9063a22653e8e5a73bb2d5a5bcp-204 : inexact-ok
+= tgamma towardzero binary128 -0x3.1029b4p+4 : 0x3.fd9063a22653e8e5a73bb2d5a5bcp-204 : inexact-ok
+= tgamma upward binary128 -0x3.1029b4p+4 : 0x3.fd9063a22653e8e5a73bb2d5a5bep-204 : inexact-ok
+= tgamma downward ibm128 -0x3.1029b4p+4 : 0x3.fd9063a22653e8e5a73bb2d5a5p-204 : inexact-ok
+= tgamma tonearest ibm128 -0x3.1029b4p+4 : 0x3.fd9063a22653e8e5a73bb2d5a6p-204 : inexact-ok
+= tgamma towardzero ibm128 -0x3.1029b4p+4 : 0x3.fd9063a22653e8e5a73bb2d5a5p-204 : inexact-ok
+= tgamma upward ibm128 -0x3.1029b4p+4 : 0x3.fd9063a22653e8e5a73bb2d5a6p-204 : inexact-ok
+= tgamma downward binary32 -0x3.1029b8p+4 : 0x0p+0 : inexact-ok underflow errno-erange
+= tgamma tonearest binary32 -0x3.1029b8p+4 : 0x0p+0 : inexact-ok underflow errno-erange
+= tgamma towardzero binary32 -0x3.1029b8p+4 : 0x0p+0 : inexact-ok underflow errno-erange
+= tgamma upward binary32 -0x3.1029b8p+4 : 0x8p-152 : inexact-ok underflow errno-erange-ok
+= tgamma downward binary64 -0x3.1029b8p+4 : 0x3.fd2a955e20f56p-204 : inexact-ok
+= tgamma tonearest binary64 -0x3.1029b8p+4 : 0x3.fd2a955e20f58p-204 : inexact-ok
+= tgamma towardzero binary64 -0x3.1029b8p+4 : 0x3.fd2a955e20f56p-204 : inexact-ok
+= tgamma upward binary64 -0x3.1029b8p+4 : 0x3.fd2a955e20f58p-204 : inexact-ok
+= tgamma downward intel96 -0x3.1029b8p+4 : 0x3.fd2a955e20f57628p-204 : inexact-ok
+= tgamma tonearest intel96 -0x3.1029b8p+4 : 0x3.fd2a955e20f5762cp-204 : inexact-ok
+= tgamma towardzero intel96 -0x3.1029b8p+4 : 0x3.fd2a955e20f57628p-204 : inexact-ok
+= tgamma upward intel96 -0x3.1029b8p+4 : 0x3.fd2a955e20f5762cp-204 : inexact-ok
+= tgamma downward m68k96 -0x3.1029b8p+4 : 0x3.fd2a955e20f57628p-204 : inexact-ok
+= tgamma tonearest m68k96 -0x3.1029b8p+4 : 0x3.fd2a955e20f5762cp-204 : inexact-ok
+= tgamma towardzero m68k96 -0x3.1029b8p+4 : 0x3.fd2a955e20f57628p-204 : inexact-ok
+= tgamma upward m68k96 -0x3.1029b8p+4 : 0x3.fd2a955e20f5762cp-204 : inexact-ok
+= tgamma downward binary128 -0x3.1029b8p+4 : 0x3.fd2a955e20f5762a464af7f01716p-204 : inexact-ok
+= tgamma tonearest binary128 -0x3.1029b8p+4 : 0x3.fd2a955e20f5762a464af7f01718p-204 : inexact-ok
+= tgamma towardzero binary128 -0x3.1029b8p+4 : 0x3.fd2a955e20f5762a464af7f01716p-204 : inexact-ok
+= tgamma upward binary128 -0x3.1029b8p+4 : 0x3.fd2a955e20f5762a464af7f01718p-204 : inexact-ok
+= tgamma downward ibm128 -0x3.1029b8p+4 : 0x3.fd2a955e20f5762a464af7f017p-204 : inexact-ok
+= tgamma tonearest ibm128 -0x3.1029b8p+4 : 0x3.fd2a955e20f5762a464af7f017p-204 : inexact-ok
+= tgamma towardzero ibm128 -0x3.1029b8p+4 : 0x3.fd2a955e20f5762a464af7f017p-204 : inexact-ok
+= tgamma upward ibm128 -0x3.1029b8p+4 : 0x3.fd2a955e20f5762a464af7f018p-204 : inexact-ok
+= tgamma downward binary64 -0x3.1029b4dd6fb6p+4 : 0x3.fd7a5e1abda5cp-204 : inexact-ok
+= tgamma tonearest binary64 -0x3.1029b4dd6fb6p+4 : 0x3.fd7a5e1abda5cp-204 : inexact-ok
+= tgamma towardzero binary64 -0x3.1029b4dd6fb6p+4 : 0x3.fd7a5e1abda5cp-204 : inexact-ok
+= tgamma upward binary64 -0x3.1029b4dd6fb6p+4 : 0x3.fd7a5e1abda5ep-204 : inexact-ok
+= tgamma downward intel96 -0x3.1029b4dd6fb6p+4 : 0x3.fd7a5e1abda5c6d4p-204 : inexact-ok
+= tgamma tonearest intel96 -0x3.1029b4dd6fb6p+4 : 0x3.fd7a5e1abda5c6d8p-204 : inexact-ok
+= tgamma towardzero intel96 -0x3.1029b4dd6fb6p+4 : 0x3.fd7a5e1abda5c6d4p-204 : inexact-ok
+= tgamma upward intel96 -0x3.1029b4dd6fb6p+4 : 0x3.fd7a5e1abda5c6d8p-204 : inexact-ok
+= tgamma downward m68k96 -0x3.1029b4dd6fb6p+4 : 0x3.fd7a5e1abda5c6d4p-204 : inexact-ok
+= tgamma tonearest m68k96 -0x3.1029b4dd6fb6p+4 : 0x3.fd7a5e1abda5c6d8p-204 : inexact-ok
+= tgamma towardzero m68k96 -0x3.1029b4dd6fb6p+4 : 0x3.fd7a5e1abda5c6d4p-204 : inexact-ok
+= tgamma upward m68k96 -0x3.1029b4dd6fb6p+4 : 0x3.fd7a5e1abda5c6d8p-204 : inexact-ok
+= tgamma downward binary128 -0x3.1029b4dd6fb6p+4 : 0x3.fd7a5e1abda5c6d62103dbf6d82ep-204 : inexact-ok
+= tgamma tonearest binary128 -0x3.1029b4dd6fb6p+4 : 0x3.fd7a5e1abda5c6d62103dbf6d82ep-204 : inexact-ok
+= tgamma towardzero binary128 -0x3.1029b4dd6fb6p+4 : 0x3.fd7a5e1abda5c6d62103dbf6d82ep-204 : inexact-ok
+= tgamma upward binary128 -0x3.1029b4dd6fb6p+4 : 0x3.fd7a5e1abda5c6d62103dbf6d83p-204 : inexact-ok
+= tgamma downward ibm128 -0x3.1029b4dd6fb6p+4 : 0x3.fd7a5e1abda5c6d62103dbf6d8p-204 : inexact-ok
+= tgamma tonearest ibm128 -0x3.1029b4dd6fb6p+4 : 0x3.fd7a5e1abda5c6d62103dbf6d8p-204 : inexact-ok
+= tgamma towardzero ibm128 -0x3.1029b4dd6fb6p+4 : 0x3.fd7a5e1abda5c6d62103dbf6d8p-204 : inexact-ok
+= tgamma upward ibm128 -0x3.1029b4dd6fb6p+4 : 0x3.fd7a5e1abda5c6d62103dbf6d9p-204 : inexact-ok

diff --git a/math/mul_split.h b/math/mul_split.h
index 8cc60ec630761472664ee147add681066c7af9b4..af614fb8b4ba5844031c46bfb4f79f2780b81ab6 100644 (file)
--- a/math/mul_split.h
+++ b/math/mul_split.h
@@ -47,4 +47,69 @@ mul_split (double *hi, double *lo, double x, double y)
 #endif
 }
 
+/* Add a + b exactly, such that *hi + *lo = a + b.
+   Assumes |a| >= |b| and rounding to nearest.  */
+static inline void
+fast_two_sum (double *hi, double *lo, double a, double b)
+{
+  double e;
+
+  *hi = a + b;
+  e = *hi - a; /* exact  */
+  *lo = b - e; /* exact  */
+  /* Now *hi + *lo = a + b exactly.  */
+}
+
+/* Multiplication of two floating-point expansions: *hi + *lo is an
+   approximation of (h1+l1)*(h2+l2), assuming |l1| <= 1/2*ulp(h1)
+   and |l2| <= 1/2*ulp(h2) and rounding to nearest.  */
+static inline void
+mul_expansion (double *hi, double *lo, double h1, double l1,
+              double h2, double l2)
+{
+  double r, e;
+
+  mul_split (hi, lo, h1, h2);
+  r = h1 * l2 + h2 * l1;
+  /* Now add r to (hi,lo) using fast two-sum, where we know |r| < |hi|.  */
+  fast_two_sum (hi, &e, *hi, r);
+  *lo -= e;
+}
+
+/* Calculate X / Y and store the approximate result in *HI + *LO.  It is
+   assumed that Y is not zero, that no overflow nor underflow occurs, and
+   rounding is to nearest.  */
+static inline void
+div_split (double *hi, double *lo, double x, double y)
+{
+  double a, b;
+
+  *hi = x / y;
+  mul_split (&a, &b, *hi, y);
+  /* a + b = hi*y, which should be near x.  */
+  a = x - a; /* huge cancellation  */
+  a = a - b;
+  /* Now x ~ hi*y + a thus x/y ~ hi + a/y.  */
+  *lo = a / y;
+}
+
+/* Division of two floating-point expansions: *hi + *lo is an
+   approximation of (h1+l1)/(h2+l2), assuming |l1| <= 1/2*ulp(h1)
+   and |l2| <= 1/2*ulp(h2), h2+l2 is not zero, and rounding to nearest.  */
+static inline void
+div_expansion (double *hi, double *lo, double h1, double l1,
+              double h2, double l2)
+{
+  double r, e;
+
+  div_split (hi, lo, h1, h2);
+  /* (h1+l1)/(h2+l2) ~ h1/h2 + (l1*h2 - l2*h1)/h2^2  */
+  r = (l1 * h2 - l2 * h1) / (h2 * h2);
+  /* Now add r to (hi,lo) using fast two-sum, where we know |r| < |hi|.  */
+  fast_two_sum (hi, &e, *hi, r);
+  *lo += e;
+  /* Renormalize since |lo| might be larger than 0.5 ulp(hi).  */
+  fast_two_sum (hi, lo, *hi, *lo);
+}
+
 #endif /* _MUL_SPLIT_H */

diff --git a/sysdeps/ieee754/dbl-64/e_gamma_r.c b/sysdeps/ieee754/dbl-64/e_gamma_r.c
index fe69920c761316c71824f45048726d739d494329..eb36a9f4644fc79ebb00a002bafec5e4c84dd64c 100644 (file)
--- a/sysdeps/ieee754/dbl-64/e_gamma_r.c
+++ b/sysdeps/ieee754/dbl-64/e_gamma_r.c
@@ -24,6 +24,7 @@
 #include <math-underflow.h>
 #include <float.h>
 #include <libm-alias-finite.h>
+#include <mul_split.h>
 
 /* Coefficients B_2k / 2k(2k-1) of x^-(2k-1) inside exp in Stirling's
    approximation to gamma function.  */
@@ -88,7 +89,6 @@ gamma_positive (double x, int *exp2_adj)
         Compute gamma (X_ADJ + X_EPS) using Stirling's approximation,
         starting by computing pow (X_ADJ, X_ADJ) with a power of 2
         factored out.  */
-      double exp_adj = -eps;
       double x_adj_int = round (x_adj);
       double x_adj_frac = x_adj - x_adj_int;
       int x_adj_log2;
@@ -99,18 +99,22 @@ gamma_positive (double x, int *exp2_adj)
          x_adj_mant *= 2.0;
        }
       *exp2_adj = x_adj_log2 * (int) x_adj_int;
-      double ret = (__ieee754_pow (x_adj_mant, x_adj)
-                   * __ieee754_exp2 (x_adj_log2 * x_adj_frac)
-                   * __ieee754_exp (-x_adj)
-                   * sqrt (2 * M_PI / x_adj)
-                   / prod);
-      exp_adj += x_eps * __ieee754_log (x_adj);
+      double h1, l1, h2, l2;
+      mul_split (&h1, &l1, __ieee754_pow (x_adj_mant, x_adj),
+                          __ieee754_exp2 (x_adj_log2 * x_adj_frac));
+      mul_split (&h2, &l2, __ieee754_exp (-x_adj), sqrt (2 * M_PI / x_adj));
+      mul_expansion (&h1, &l1, h1, l1, h2, l2);
+      /* Divide by prod * (1 + eps).  */
+      div_expansion (&h1, &l1, h1, l1, prod, prod * eps);
+      double exp_adj = x_eps * __ieee754_log (x_adj);
       double bsum = gamma_coeff[NCOEFF - 1];
       double x_adj2 = x_adj * x_adj;
       for (size_t i = 1; i <= NCOEFF - 1; i++)
        bsum = bsum / x_adj2 + gamma_coeff[NCOEFF - 1 - i];
       exp_adj += bsum / x_adj;
-      return ret + ret * __expm1 (exp_adj);
+      /* Now return (h1+l1) * exp(exp_adj), where exp_adj is small.  */
+      l1 += h1 * __expm1 (exp_adj);
+      return h1 + l1;
     }
 }
 
@@ -188,9 +192,20 @@ __ieee754_gamma_r (double x, int *signgamp)
                               ? __sin (M_PI * frac)
                               : __cos (M_PI * (0.5 - frac)));
              int exp2_adj;
-             double tret = M_PI / (-x * sinpix
-                                   * gamma_positive (-x, &exp2_adj));
-             ret = __scalbn (tret, -exp2_adj);
+             double h1, l1, h2, l2;
+             h2 = gamma_positive (-x, &exp2_adj);
+             mul_split (&h1, &l1, sinpix, h2);
+             /* sinpix*gamma_positive(.) = h1 + l1 */
+             mul_split (&h2, &l2, h1, x);
+             /* h1*x = h2 + l2 */
+             /* (h1 + l1) * x = h1*x + l1*x = h2 + l2 + l1*x */
+             l2 += l1 * x;
+             /* x*sinpix*gamma_positive(.) ~ h2 + l2 */
+             h1 = 0x3.243f6a8885a3p+0;   /* binary64 approximation of Pi */
+             l1 = 0x8.d313198a2e038p-56; /* |h1+l1-Pi| < 3e-33 */
+             /* Now we divide h1 + l1 by h2 + l2.  */
+             div_expansion (&h1, &l1, h1, l1, h2, l2);
+             ret = __scalbn (-h1, -exp2_adj);
              math_check_force_underflow_nonneg (ret);
            }
        }

diff --git a/sysdeps/x86_64/fpu/libm-test-ulps b/sysdeps/x86_64/fpu/libm-test-ulps
index 0edb95e14c6c7686541722ce900b2fc9c499c891..c621a0326875acc66bdd9e5cd8da5bba2e1f0dab 100644 (file)
--- a/sysdeps/x86_64/fpu/libm-test-ulps
+++ b/sysdeps/x86_64/fpu/libm-test-ulps
@@ -1734,16 +1734,16 @@ float128: 4
 ldouble: 5
 
 Function: "tgamma_downward":
-double: 8
+double: 9
 float: 7
 float128: 5
-ldouble: 5
+ldouble: 6
 
 Function: "tgamma_towardzero":
 double: 9
 float: 7
 float128: 5
-ldouble: 5
+ldouble: 6
 
 Function: "tgamma_upward":
 double: 9