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Fix clog, clog10 inaccuracy (bug 19016).
For arguments with X^2 + Y^2 close to 1, clog and clog10 avoid large errors from log(hypot) by computing X^2 + Y^2 - 1 in a way that avoids cancellation error and then using log1p. However, the thresholds for using that approach still result in log being used on argument as large as sqrt(13/16) > 0.9, leading to significant errors, in some cases above the 9ulp maximum allowed in glibc libm. This patch arranges for the approach using log1p to be used in any cases where |X|, |Y| < 1 and X^2 + Y^2 >= 0.5 (with the existing allowance for cases where one of X and Y is very small), adjusting the __x2y2m1 functions to work with the wider range of inputs. This way, log only gets used on arguments below sqrt(1/2) (or substantially above 1), where the error involved is much less. Tested for x86_64, x86, mips64 and powerpc. For the ulps regeneration I removed the existing clog and clog10 ulps before regenerating to allow any reduced ulps to appear. Tests added include those found by random test generation to produce large ulps either before or after the patch, and some found by trying inputs close to the (0.75, 0.5) threshold where the potential errors from using log are largest. [BZ #19016] * sysdeps/generic/math_private.h (__x2y2m1f): Update comment to allow more cases with X^2 + Y^2 >= 0.5. * sysdeps/ieee754/dbl-64/x2y2m1.c (__x2y2m1): Likewise. Add -1 as normal element in sum instead of special-casing based on values of arguments. * sysdeps/ieee754/dbl-64/x2y2m1f.c (__x2y2m1f): Update comment. * sysdeps/ieee754/ldbl-128/x2y2m1l.c (__x2y2m1l): Likewise. Add -1 as normal element in sum instead of special-casing based on values of arguments. * sysdeps/ieee754/ldbl-128ibm/x2y2m1l.c (__x2y2m1l): Likewise. * sysdeps/ieee754/ldbl-96/x2y2m1.c [FLT_EVAL_METHOD != 0] (__x2y2m1): Update comment. * sysdeps/ieee754/ldbl-96/x2y2m1l.c (__x2y2m1l): Likewise. Add -1 as normal element in sum instead of special-casing based on values of arguments. * math/s_clog.c (__clog): Handle more cases using log1p without hypot. * math/s_clog10.c (__clog10): Likewise. * math/s_clog10f.c (__clog10f): Likewise. * math/s_clog10l.c (__clog10l): Likewise. * math/s_clogf.c (__clogf): Likewise. * math/s_clogl.c (__clogl): Likewise. * math/auto-libm-test-in: Add more tests of clog and clog10. * math/auto-libm-test-out: Regenerated. * sysdeps/i386/fpu/libm-test-ulps: Update. * sysdeps/x86_64/fpu/libm-test-ulps: Likewise.
This commit is contained in:
Joseph Myers
parent
60cf80f09d
commit
a5721ebc68
30
ChangeLog
30
ChangeLog
@ -1,3 +1,33 @@
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2015-09-28 Joseph Myers <joseph@codesourcery.com>
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[BZ #19016]
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* sysdeps/generic/math_private.h (__x2y2m1f): Update comment to
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allow more cases with X^2 + Y^2 >= 0.5.
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* sysdeps/ieee754/dbl-64/x2y2m1.c (__x2y2m1): Likewise. Add -1 as
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normal element in sum instead of special-casing based on values of
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arguments.
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* sysdeps/ieee754/dbl-64/x2y2m1f.c (__x2y2m1f): Update comment.
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* sysdeps/ieee754/ldbl-128/x2y2m1l.c (__x2y2m1l): Likewise. Add
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-1 as normal element in sum instead of special-casing based on
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values of arguments.
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* sysdeps/ieee754/ldbl-128ibm/x2y2m1l.c (__x2y2m1l): Likewise.
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* sysdeps/ieee754/ldbl-96/x2y2m1.c [FLT_EVAL_METHOD != 0]
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(__x2y2m1): Update comment.
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* sysdeps/ieee754/ldbl-96/x2y2m1l.c (__x2y2m1l): Likewise. Add -1
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as normal element in sum instead of special-casing based on values
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of arguments.
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* math/s_clog.c (__clog): Handle more cases using log1p without
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hypot.
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* math/s_clog10.c (__clog10): Likewise.
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* math/s_clog10f.c (__clog10f): Likewise.
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* math/s_clog10l.c (__clog10l): Likewise.
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* math/s_clogf.c (__clogf): Likewise.
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* math/s_clogl.c (__clogl): Likewise.
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* math/auto-libm-test-in: Add more tests of clog and clog10.
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* math/auto-libm-test-out: Regenerated.
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* sysdeps/i386/fpu/libm-test-ulps: Update.
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* sysdeps/x86_64/fpu/libm-test-ulps: Likewise.
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2015-09-28 Martin Sebor <msebor@redhat.com>
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[BZ #18969]
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3
NEWS
3
NEWS
@ -16,7 +16,8 @@ Version 2.23
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18618, 18647, 18661, 18674, 18675, 18681, 18757, 18778, 18781, 18787,
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18789, 18790, 18795, 18796, 18803, 18820, 18823, 18824, 18825, 18857,
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18863, 18870, 18872, 18873, 18875, 18887, 18921, 18951, 18952, 18956,
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18961, 18966, 18967, 18969, 18970, 18977, 18980, 18981, 18985, 19003.
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18961, 18966, 18967, 18969, 18970, 18977, 18980, 18981, 18985, 19003,
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19016.
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* The obsolete header <regexp.h> has been removed. Programs that require
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this header must be updated to use <regex.h> instead.
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@ -660,6 +660,51 @@ clog -0xa.7ac41a0b417cb8fp-4 -0x6.c5a32eaeedd4p-4
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clog 0x3.c16p-136 0x8p-152
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clog -0x1.0a69de710590dp+0 -0x7.bc7e121e2b0d1088p-4
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clog -0x2.7bdep-4 0x5.ab7a4p-4
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clog -0xb.e1d3d0ff44358p-4 -0x7.54785e1b143f8p-4
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clog 0x3.ba473p+0 0x7.eea9ap-4
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clog 0x9.d02220baee4ep+36 0x2.b9a29cp+0
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clog -0x5.1a5cf8p-4 -0xb.73012p-4
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clog -0xa.ff292a609dbb8p-4 0x6.f73d4cp-4
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clog -0x5.1a5cfc2301114p-4 -0xb.730118p-4
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clog 0xb.ffffcp-4 0x7.ffff1p-4
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clog 0xb.ffffp-4 0x7.ffffap-4
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clog 0xb.ffffp-4 0x7.fffff8p-4
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clog 0xb.ffffp-4 0x7.ffffp-4
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clog 0xb.fffffp-4 0x7.ffff68p-4
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clog 0xb.fffffp-4 0x7.ffffp-4
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clog 0xb.ffff8p-4 0x7.ffffcp-4
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clog 0xb.ffffp-4 0x7.ffffcp-4
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clog 0xb.ffffp-4 0x7.ffffb8p-4
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clog 0xb.ffffp-4 0x7.ffff7p-4
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clog 0xb.ffffp-4 0x7.ffff5p-4
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clog 0xb.fffffffffff7p-4 0x7.fffff8p-4
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clog 0xb.fffffffffff08p-4 0x7.fffffffffffdp-4
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clog 0xb.fffffffffff08p-4 0x7.fffffffffff9p-4
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clog 0xb.fffffffffffp-4 0x7.fffffffffffdcp-4
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clog 0xb.fffffp-4 0x7.ffffffffffff4p-4
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clog 0xb.fffffffffffp-4 0x7.fffffffffffecp-4
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clog 0xb.fffffffffff8p-4 0x7.fffff8p-4
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clog 0x8p-152 -0x1.10233ap+0
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clog 0xa.03634p-4 -0x4.7bb918p-20
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clog -0x5.e23d2p-4 0x8.525df889c21ap-4
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clog 0x9.8ce58p-4 -0x8p-152
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clog 0x8p-152 0x9.2af75p-4
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clog 0x9.97a15de8e59d8p-4 -0
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clog -0x4.74556ec92eb4746p-4 0x1.1e7aa1d936f6efe6p+0
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clog 0x9.97a15de8e59d8p-4 -0
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clog -0x9.7f1d7p-64 0x9.db37dp-4
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clog -0x8.5efc4p-4 -0x5.40310cp-4
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clog -0x9.0b459p-4 0
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clog -0x6.a9419e9b30e68p-4 -0x6.262c7p-4
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clog 0x5.2767cdfdfbf2p-4 0x7.69ee98p-4
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clog -0x9.f5563cb3227d8p-4 0
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clog -0x9.5a284p-4 0x6.899578p-8
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clog 0xa.3e62bp-4 0x1.18c03p-100
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clog 0 -0x9.22a99p-4
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clog 0 0x9.7915bp-4
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clog 0x3.00d1ap-12 0x1.23ff6ap+0
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clog 0x1.fffffep+127 0x1.fffffep+127
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clog 0x1.fffffep+127 1.0
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clog 0x1p-149 0x1p-149
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@ -808,6 +853,51 @@ clog10 -0xa.7ac41a0b417cb8fp-4 -0x6.c5a32eaeedd4p-4
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clog10 0x3.c16p-136 0x8p-152
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clog10 -0x1.0a69de710590dp+0 -0x7.bc7e121e2b0d1088p-4
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clog10 -0x2.7bdep-4 0x5.ab7a4p-4
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clog10 -0xb.e1d3d0ff44358p-4 -0x7.54785e1b143f8p-4
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clog10 0x3.ba473p+0 0x7.eea9ap-4
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clog10 0x9.d02220baee4ep+36 0x2.b9a29cp+0
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clog10 -0x5.1a5cf8p-4 -0xb.73012p-4
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clog10 -0xa.ff292a609dbb8p-4 0x6.f73d4cp-4
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clog10 -0x5.1a5cfc2301114p-4 -0xb.730118p-4
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clog10 0xb.ffffcp-4 0x7.ffff1p-4
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clog10 0xb.ffffp-4 0x7.ffffap-4
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clog10 0xb.ffffp-4 0x7.fffff8p-4
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clog10 0xb.ffffp-4 0x7.ffffp-4
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clog10 0xb.fffffp-4 0x7.ffff68p-4
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clog10 0xb.fffffp-4 0x7.ffffp-4
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clog10 0xb.ffff8p-4 0x7.ffffcp-4
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clog10 0xb.ffffp-4 0x7.ffffcp-4
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clog10 0xb.ffffp-4 0x7.ffffb8p-4
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clog10 0xb.ffffp-4 0x7.ffff7p-4
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clog10 0xb.ffffp-4 0x7.ffff5p-4
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clog10 0xb.fffffffffff7p-4 0x7.fffff8p-4
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clog10 0xb.fffffffffff08p-4 0x7.fffffffffffdp-4
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clog10 0xb.fffffffffff08p-4 0x7.fffffffffff9p-4
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clog10 0xb.fffffffffffp-4 0x7.fffffffffffdcp-4
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clog10 0xb.fffffp-4 0x7.ffffffffffff4p-4
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clog10 0xb.fffffffffffp-4 0x7.fffffffffffecp-4
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clog10 0xb.fffffffffff8p-4 0x7.fffff8p-4
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clog10 0x8p-152 -0x1.10233ap+0
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clog10 0xa.03634p-4 -0x4.7bb918p-20
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clog10 -0x5.e23d2p-4 0x8.525df889c21ap-4
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clog10 0x9.8ce58p-4 -0x8p-152
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clog10 0x8p-152 0x9.2af75p-4
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clog10 0x9.97a15de8e59d8p-4 -0
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clog10 -0x4.74556ec92eb4746p-4 0x1.1e7aa1d936f6efe6p+0
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clog10 0x9.97a15de8e59d8p-4 -0
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clog10 -0x9.7f1d7p-64 0x9.db37dp-4
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clog10 -0x8.5efc4p-4 -0x5.40310cp-4
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clog10 -0x9.0b459p-4 0
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clog10 -0x6.a9419e9b30e68p-4 -0x6.262c7p-4
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clog10 0x5.2767cdfdfbf2p-4 0x7.69ee98p-4
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clog10 -0x9.f5563cb3227d8p-4 0
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clog10 -0x9.5a284p-4 0x6.899578p-8
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clog10 0xa.3e62bp-4 0x1.18c03p-100
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clog10 0 -0x9.22a99p-4
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clog10 0 0x9.7915bp-4
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clog10 0x3.00d1ap-12 0x1.23ff6ap+0
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clog10 0x1.fffffep+127 0x1.fffffep+127
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clog10 0x1.fffffep+127 1.0
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clog10 0x1p-149 0x1p-149
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File diff suppressed because it is too large
Load Diff
@ -76,14 +76,17 @@ __clog (__complex__ double x)
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__real__ result = __log1p (d2m1) / 2.0;
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}
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else if (absx < 1.0
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&& absx >= 0.75
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&& absx >= 0.5
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&& absy < DBL_EPSILON / 2.0
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&& scale == 0)
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{
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double d2m1 = (absx - 1.0) * (absx + 1.0);
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__real__ result = __log1p (d2m1) / 2.0;
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}
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else if (absx < 1.0 && (absx >= 0.75 || absy >= 0.5) && scale == 0)
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else if (absx < 1.0
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&& absx >= 0.5
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&& scale == 0
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&& absx * absx + absy * absy >= 0.5)
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{
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double d2m1 = __x2y2m1 (absx, absy);
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__real__ result = __log1p (d2m1) / 2.0;
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@ -82,14 +82,17 @@ __clog10 (__complex__ double x)
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__real__ result = __log1p (d2m1) * (M_LOG10E / 2.0);
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}
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else if (absx < 1.0
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&& absx >= 0.75
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&& absx >= 0.5
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&& absy < DBL_EPSILON / 2.0
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&& scale == 0)
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{
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double d2m1 = (absx - 1.0) * (absx + 1.0);
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__real__ result = __log1p (d2m1) * (M_LOG10E / 2.0);
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}
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else if (absx < 1.0 && (absx >= 0.75 || absy >= 0.5) && scale == 0)
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else if (absx < 1.0
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&& absx >= 0.5
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&& scale == 0
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&& absx * absx + absy * absy >= 0.5)
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{
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double d2m1 = __x2y2m1 (absx, absy);
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__real__ result = __log1p (d2m1) * (M_LOG10E / 2.0);
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@ -82,14 +82,17 @@ __clog10f (__complex__ float x)
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__real__ result = __log1pf (d2m1) * ((float) M_LOG10E / 2.0f);
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}
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else if (absx < 1.0f
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&& absx >= 0.75f
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&& absx >= 0.5f
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&& absy < FLT_EPSILON / 2.0f
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&& scale == 0)
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{
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float d2m1 = (absx - 1.0f) * (absx + 1.0f);
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__real__ result = __log1pf (d2m1) * ((float) M_LOG10E / 2.0f);
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}
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else if (absx < 1.0f && (absx >= 0.75f || absy >= 0.5f) && scale == 0)
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else if (absx < 1.0f
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&& absx >= 0.5f
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&& scale == 0
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&& absx * absx + absy * absy >= 0.5f)
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{
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float d2m1 = __x2y2m1f (absx, absy);
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__real__ result = __log1pf (d2m1) * ((float) M_LOG10E / 2.0f);
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@ -89,14 +89,17 @@ __clog10l (__complex__ long double x)
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__real__ result = __log1pl (d2m1) * (M_LOG10El / 2.0L);
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}
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else if (absx < 1.0L
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&& absx >= 0.75L
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&& absx >= 0.5L
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&& absy < LDBL_EPSILON / 2.0L
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&& scale == 0)
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{
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long double d2m1 = (absx - 1.0L) * (absx + 1.0L);
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__real__ result = __log1pl (d2m1) * (M_LOG10El / 2.0L);
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}
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else if (absx < 1.0L && (absx >= 0.75L || absy >= 0.5L) && scale == 0)
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else if (absx < 1.0L
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&& absx >= 0.5L
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&& scale == 0
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&& absx * absx + absy * absy >= 0.5L)
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{
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long double d2m1 = __x2y2m1l (absx, absy);
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__real__ result = __log1pl (d2m1) * (M_LOG10El / 2.0L);
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@ -76,14 +76,17 @@ __clogf (__complex__ float x)
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__real__ result = __log1pf (d2m1) / 2.0f;
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}
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else if (absx < 1.0f
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&& absx >= 0.75f
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&& absx >= 0.5f
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&& absy < FLT_EPSILON / 2.0f
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&& scale == 0)
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{
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float d2m1 = (absx - 1.0f) * (absx + 1.0f);
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__real__ result = __log1pf (d2m1) / 2.0f;
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}
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else if (absx < 1.0f && (absx >= 0.75f || absy >= 0.5f) && scale == 0)
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else if (absx < 1.0f
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&& absx >= 0.5f
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&& scale == 0
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&& absx * absx + absy * absy >= 0.5f)
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{
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float d2m1 = __x2y2m1f (absx, absy);
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__real__ result = __log1pf (d2m1) / 2.0f;
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@ -83,14 +83,17 @@ __clogl (__complex__ long double x)
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__real__ result = __log1pl (d2m1) / 2.0L;
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}
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else if (absx < 1.0L
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&& absx >= 0.75L
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&& absx >= 0.5L
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&& absy < LDBL_EPSILON / 2.0L
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&& scale == 0)
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{
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long double d2m1 = (absx - 1.0L) * (absx + 1.0L);
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__real__ result = __log1pl (d2m1) / 2.0L;
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}
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else if (absx < 1.0L && (absx >= 0.75L || absy >= 0.5L) && scale == 0)
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else if (absx < 1.0L
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&& absx >= 0.5L
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&& scale == 0
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&& absx * absx + absy * absy >= 0.5L)
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{
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long double d2m1 = __x2y2m1l (absx, absy);
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__real__ result = __log1pl (d2m1) / 2.0L;
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@ -365,8 +365,8 @@ extern double __slowpow (double __x, double __y, double __z);
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extern void __docos (double __x, double __dx, double __v[]);
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/* Return X^2 + Y^2 - 1, computed without large cancellation error.
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It is given that 1 > X >= Y >= epsilon / 2, and that either X >=
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0.75 or Y >= 0.5. */
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It is given that 1 > X >= Y >= epsilon / 2, and that X^2 + Y^2 >=
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0.5. */
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extern float __x2y2m1f (float x, float y);
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extern double __x2y2m1 (double x, double y);
|
||||
extern long double __x2y2m1l (long double x, long double y);
|
||||
|
||||
@ -836,12 +836,12 @@ ildouble: 3
|
||||
ldouble: 3
|
||||
|
||||
Function: Real part of "clog":
|
||||
double: 3
|
||||
float: 2
|
||||
idouble: 3
|
||||
ifloat: 2
|
||||
ildouble: 4
|
||||
ldouble: 4
|
||||
double: 2
|
||||
float: 1
|
||||
idouble: 2
|
||||
ifloat: 1
|
||||
ildouble: 3
|
||||
ldouble: 3
|
||||
|
||||
Function: Imaginary part of "clog":
|
||||
double: 1
|
||||
@ -864,10 +864,10 @@ ildouble: 2
|
||||
ldouble: 2
|
||||
|
||||
Function: Real part of "clog10_downward":
|
||||
double: 5
|
||||
float: 4
|
||||
idouble: 5
|
||||
ifloat: 4
|
||||
double: 3
|
||||
float: 3
|
||||
idouble: 3
|
||||
ifloat: 3
|
||||
ildouble: 8
|
||||
ldouble: 8
|
||||
|
||||
@ -876,14 +876,14 @@ double: 1
|
||||
float: 1
|
||||
idouble: 1
|
||||
ifloat: 1
|
||||
ildouble: 2
|
||||
ldouble: 2
|
||||
ildouble: 3
|
||||
ldouble: 3
|
||||
|
||||
Function: Real part of "clog10_towardzero":
|
||||
double: 5
|
||||
float: 4
|
||||
idouble: 5
|
||||
ifloat: 4
|
||||
double: 3
|
||||
float: 3
|
||||
idouble: 3
|
||||
ifloat: 3
|
||||
ildouble: 8
|
||||
ldouble: 8
|
||||
|
||||
@ -896,12 +896,12 @@ ildouble: 3
|
||||
ldouble: 3
|
||||
|
||||
Function: Real part of "clog10_upward":
|
||||
double: 5
|
||||
float: 5
|
||||
idouble: 5
|
||||
ifloat: 5
|
||||
ildouble: 6
|
||||
ldouble: 6
|
||||
double: 3
|
||||
float: 3
|
||||
idouble: 3
|
||||
ifloat: 3
|
||||
ildouble: 7
|
||||
ldouble: 7
|
||||
|
||||
Function: Imaginary part of "clog10_upward":
|
||||
double: 1
|
||||
@ -912,12 +912,12 @@ ildouble: 3
|
||||
ldouble: 3
|
||||
|
||||
Function: Real part of "clog_downward":
|
||||
double: 5
|
||||
float: 5
|
||||
idouble: 5
|
||||
ifloat: 5
|
||||
ildouble: 7
|
||||
ldouble: 7
|
||||
double: 3
|
||||
float: 3
|
||||
idouble: 3
|
||||
ifloat: 3
|
||||
ildouble: 5
|
||||
ldouble: 5
|
||||
|
||||
Function: Imaginary part of "clog_downward":
|
||||
double: 1
|
||||
@ -928,12 +928,12 @@ ildouble: 1
|
||||
ldouble: 1
|
||||
|
||||
Function: Real part of "clog_towardzero":
|
||||
double: 5
|
||||
float: 5
|
||||
idouble: 5
|
||||
ifloat: 5
|
||||
ildouble: 8
|
||||
ldouble: 8
|
||||
double: 3
|
||||
float: 3
|
||||
idouble: 3
|
||||
ifloat: 3
|
||||
ildouble: 5
|
||||
ldouble: 5
|
||||
|
||||
Function: Imaginary part of "clog_towardzero":
|
||||
double: 1
|
||||
@ -944,12 +944,12 @@ ildouble: 1
|
||||
ldouble: 1
|
||||
|
||||
Function: Real part of "clog_upward":
|
||||
double: 5
|
||||
float: 5
|
||||
idouble: 5
|
||||
ifloat: 5
|
||||
ildouble: 6
|
||||
ldouble: 6
|
||||
double: 2
|
||||
float: 3
|
||||
idouble: 2
|
||||
ifloat: 3
|
||||
ildouble: 4
|
||||
ldouble: 4
|
||||
|
||||
Function: Imaginary part of "clog_upward":
|
||||
double: 1
|
||||
|
||||
@ -80,32 +80,26 @@ compare (const void *p, const void *q)
|
||||
}
|
||||
|
||||
/* Return X^2 + Y^2 - 1, computed without large cancellation error.
|
||||
It is given that 1 > X >= Y >= epsilon / 2, and that either X >=
|
||||
0.75 or Y >= 0.5. */
|
||||
It is given that 1 > X >= Y >= epsilon / 2, and that X^2 + Y^2 >=
|
||||
0.5. */
|
||||
|
||||
double
|
||||
__x2y2m1 (double x, double y)
|
||||
{
|
||||
double vals[4];
|
||||
double vals[5];
|
||||
SET_RESTORE_ROUND (FE_TONEAREST);
|
||||
mul_split (&vals[1], &vals[0], x, x);
|
||||
mul_split (&vals[3], &vals[2], y, y);
|
||||
if (x >= 0.75)
|
||||
vals[1] -= 1.0;
|
||||
else
|
||||
{
|
||||
vals[1] -= 0.5;
|
||||
vals[3] -= 0.5;
|
||||
}
|
||||
qsort (vals, 4, sizeof (double), compare);
|
||||
vals[4] = -1.0;
|
||||
qsort (vals, 5, sizeof (double), compare);
|
||||
/* Add up the values so that each element of VALS has absolute value
|
||||
at most equal to the last set bit of the next nonzero
|
||||
element. */
|
||||
for (size_t i = 0; i <= 2; i++)
|
||||
for (size_t i = 0; i <= 3; i++)
|
||||
{
|
||||
add_split (&vals[i + 1], &vals[i], vals[i + 1], vals[i]);
|
||||
qsort (vals + i + 1, 3 - i, sizeof (double), compare);
|
||||
qsort (vals + i + 1, 4 - i, sizeof (double), compare);
|
||||
}
|
||||
/* Now any error from this addition will be small. */
|
||||
return vals[3] + vals[2] + vals[1] + vals[0];
|
||||
return vals[4] + vals[3] + vals[2] + vals[1] + vals[0];
|
||||
}
|
||||
|
||||
@ -21,8 +21,8 @@
|
||||
#include <float.h>
|
||||
|
||||
/* Return X^2 + Y^2 - 1, computed without large cancellation error.
|
||||
It is given that 1 > X >= Y >= epsilon / 2, and that either X >=
|
||||
0.75 or Y >= 0.5. */
|
||||
It is given that 1 > X >= Y >= epsilon / 2, and that X^2 + Y^2 >=
|
||||
0.5. */
|
||||
|
||||
float
|
||||
__x2y2m1f (float x, float y)
|
||||
|
||||
@ -80,32 +80,26 @@ compare (const void *p, const void *q)
|
||||
}
|
||||
|
||||
/* Return X^2 + Y^2 - 1, computed without large cancellation error.
|
||||
It is given that 1 > X >= Y >= epsilon / 2, and that either X >=
|
||||
0.75 or Y >= 0.5. */
|
||||
It is given that 1 > X >= Y >= epsilon / 2, and that X^2 + Y^2 >=
|
||||
0.5. */
|
||||
|
||||
long double
|
||||
__x2y2m1l (long double x, long double y)
|
||||
{
|
||||
long double vals[4];
|
||||
long double vals[5];
|
||||
SET_RESTORE_ROUNDL (FE_TONEAREST);
|
||||
mul_split (&vals[1], &vals[0], x, x);
|
||||
mul_split (&vals[3], &vals[2], y, y);
|
||||
if (x >= 0.75L)
|
||||
vals[1] -= 1.0L;
|
||||
else
|
||||
{
|
||||
vals[1] -= 0.5L;
|
||||
vals[3] -= 0.5L;
|
||||
}
|
||||
qsort (vals, 4, sizeof (long double), compare);
|
||||
vals[4] = -1.0L;
|
||||
qsort (vals, 5, sizeof (long double), compare);
|
||||
/* Add up the values so that each element of VALS has absolute value
|
||||
at most equal to the last set bit of the next nonzero
|
||||
element. */
|
||||
for (size_t i = 0; i <= 2; i++)
|
||||
for (size_t i = 0; i <= 3; i++)
|
||||
{
|
||||
add_split (&vals[i + 1], &vals[i], vals[i + 1], vals[i]);
|
||||
qsort (vals + i + 1, 3 - i, sizeof (long double), compare);
|
||||
qsort (vals + i + 1, 4 - i, sizeof (long double), compare);
|
||||
}
|
||||
/* Now any error from this addition will be small. */
|
||||
return vals[3] + vals[2] + vals[1] + vals[0];
|
||||
return vals[4] + vals[3] + vals[2] + vals[1] + vals[0];
|
||||
}
|
||||
|
||||
@ -80,13 +80,13 @@ compare (const void *p, const void *q)
|
||||
}
|
||||
|
||||
/* Return X^2 + Y^2 - 1, computed without large cancellation error.
|
||||
It is given that 1 > X >= Y >= epsilon / 2, and that either X >=
|
||||
0.75 or Y >= 0.5. */
|
||||
It is given that 1 > X >= Y >= epsilon / 2, and that X^2 + Y^2 >=
|
||||
0.5. */
|
||||
|
||||
long double
|
||||
__x2y2m1l (long double x, long double y)
|
||||
{
|
||||
double vals[12];
|
||||
double vals[13];
|
||||
SET_RESTORE_ROUND (FE_TONEAREST);
|
||||
union ibm_extended_long_double xu, yu;
|
||||
xu.ld = x;
|
||||
@ -105,25 +105,19 @@ __x2y2m1l (long double x, long double y)
|
||||
vals[8] *= 2.0;
|
||||
vals[9] *= 2.0;
|
||||
mul_split (&vals[11], &vals[10], yu.d[1].d, yu.d[1].d);
|
||||
if (xu.d[0].d >= 0.75)
|
||||
vals[1] -= 1.0;
|
||||
else
|
||||
{
|
||||
vals[1] -= 0.5;
|
||||
vals[7] -= 0.5;
|
||||
}
|
||||
qsort (vals, 12, sizeof (double), compare);
|
||||
vals[12] = -1.0;
|
||||
qsort (vals, 13, sizeof (double), compare);
|
||||
/* Add up the values so that each element of VALS has absolute value
|
||||
at most equal to the last set bit of the next nonzero
|
||||
element. */
|
||||
for (size_t i = 0; i <= 10; i++)
|
||||
for (size_t i = 0; i <= 11; i++)
|
||||
{
|
||||
add_split (&vals[i + 1], &vals[i], vals[i + 1], vals[i]);
|
||||
qsort (vals + i + 1, 11 - i, sizeof (double), compare);
|
||||
qsort (vals + i + 1, 12 - i, sizeof (double), compare);
|
||||
}
|
||||
/* Now any error from this addition will be small. */
|
||||
long double retval = (long double) vals[11];
|
||||
for (size_t i = 10; i != (size_t) -1; i--)
|
||||
long double retval = (long double) vals[12];
|
||||
for (size_t i = 11; i != (size_t) -1; i--)
|
||||
retval += (long double) vals[i];
|
||||
return retval;
|
||||
}
|
||||
|
||||
@ -27,8 +27,8 @@
|
||||
#else
|
||||
|
||||
/* Return X^2 + Y^2 - 1, computed without large cancellation error.
|
||||
It is given that 1 > X >= Y >= epsilon / 2, and that either X >=
|
||||
0.75 or Y >= 0.5. */
|
||||
It is given that 1 > X >= Y >= epsilon / 2, and that X^2 + Y^2 >=
|
||||
0.5. */
|
||||
|
||||
double
|
||||
__x2y2m1 (double x, double y)
|
||||
|
||||
@ -80,32 +80,26 @@ compare (const void *p, const void *q)
|
||||
}
|
||||
|
||||
/* Return X^2 + Y^2 - 1, computed without large cancellation error.
|
||||
It is given that 1 > X >= Y >= epsilon / 2, and that either X >=
|
||||
0.75 or Y >= 0.5. */
|
||||
It is given that 1 > X >= Y >= epsilon / 2, and that X^2 + Y^2 >=
|
||||
0.5. */
|
||||
|
||||
long double
|
||||
__x2y2m1l (long double x, long double y)
|
||||
{
|
||||
long double vals[4];
|
||||
long double vals[5];
|
||||
SET_RESTORE_ROUNDL (FE_TONEAREST);
|
||||
mul_split (&vals[1], &vals[0], x, x);
|
||||
mul_split (&vals[3], &vals[2], y, y);
|
||||
if (x >= 0.75L)
|
||||
vals[1] -= 1.0L;
|
||||
else
|
||||
{
|
||||
vals[1] -= 0.5L;
|
||||
vals[3] -= 0.5L;
|
||||
}
|
||||
qsort (vals, 4, sizeof (long double), compare);
|
||||
vals[4] = -1.0L;
|
||||
qsort (vals, 5, sizeof (long double), compare);
|
||||
/* Add up the values so that each element of VALS has absolute value
|
||||
at most equal to the last set bit of the next nonzero
|
||||
element. */
|
||||
for (size_t i = 0; i <= 2; i++)
|
||||
for (size_t i = 0; i <= 3; i++)
|
||||
{
|
||||
add_split (&vals[i + 1], &vals[i], vals[i + 1], vals[i]);
|
||||
qsort (vals + i + 1, 3 - i, sizeof (long double), compare);
|
||||
qsort (vals + i + 1, 4 - i, sizeof (long double), compare);
|
||||
}
|
||||
/* Now any error from this addition will be small. */
|
||||
return vals[3] + vals[2] + vals[1] + vals[0];
|
||||
return vals[4] + vals[3] + vals[2] + vals[1] + vals[0];
|
||||
}
|
||||
|
||||
@ -869,11 +869,11 @@ ldouble: 3
|
||||
|
||||
Function: Real part of "clog":
|
||||
double: 3
|
||||
float: 2
|
||||
float: 3
|
||||
idouble: 3
|
||||
ifloat: 2
|
||||
ildouble: 4
|
||||
ldouble: 4
|
||||
ifloat: 3
|
||||
ildouble: 3
|
||||
ldouble: 3
|
||||
|
||||
Function: Imaginary part of "clog":
|
||||
float: 1
|
||||
@ -883,9 +883,9 @@ ldouble: 1
|
||||
|
||||
Function: Real part of "clog10":
|
||||
double: 3
|
||||
float: 3
|
||||
float: 4
|
||||
idouble: 3
|
||||
ifloat: 3
|
||||
ifloat: 4
|
||||
ildouble: 4
|
||||
ldouble: 4
|
||||
|
||||
@ -898,10 +898,10 @@ ildouble: 2
|
||||
ldouble: 2
|
||||
|
||||
Function: Real part of "clog10_downward":
|
||||
double: 6
|
||||
float: 6
|
||||
idouble: 6
|
||||
ifloat: 6
|
||||
double: 5
|
||||
float: 4
|
||||
idouble: 5
|
||||
ifloat: 4
|
||||
ildouble: 8
|
||||
ldouble: 8
|
||||
|
||||
@ -910,14 +910,14 @@ double: 2
|
||||
float: 4
|
||||
idouble: 2
|
||||
ifloat: 4
|
||||
ildouble: 2
|
||||
ldouble: 2
|
||||
ildouble: 3
|
||||
ldouble: 3
|
||||
|
||||
Function: Real part of "clog10_towardzero":
|
||||
double: 5
|
||||
float: 4
|
||||
float: 5
|
||||
idouble: 5
|
||||
ifloat: 4
|
||||
ifloat: 5
|
||||
ildouble: 8
|
||||
ldouble: 8
|
||||
|
||||
@ -930,28 +930,28 @@ ildouble: 3
|
||||
ldouble: 3
|
||||
|
||||
Function: Real part of "clog10_upward":
|
||||
double: 8
|
||||
double: 6
|
||||
float: 5
|
||||
idouble: 8
|
||||
idouble: 6
|
||||
ifloat: 5
|
||||
ildouble: 6
|
||||
ldouble: 6
|
||||
ildouble: 7
|
||||
ldouble: 7
|
||||
|
||||
Function: Imaginary part of "clog10_upward":
|
||||
double: 2
|
||||
float: 3
|
||||
float: 4
|
||||
idouble: 2
|
||||
ifloat: 3
|
||||
ifloat: 4
|
||||
ildouble: 3
|
||||
ldouble: 3
|
||||
|
||||
Function: Real part of "clog_downward":
|
||||
double: 7
|
||||
float: 5
|
||||
idouble: 7
|
||||
ifloat: 5
|
||||
ildouble: 7
|
||||
ldouble: 7
|
||||
double: 4
|
||||
float: 3
|
||||
idouble: 4
|
||||
ifloat: 3
|
||||
ildouble: 5
|
||||
ldouble: 5
|
||||
|
||||
Function: Imaginary part of "clog_downward":
|
||||
double: 1
|
||||
@ -962,28 +962,28 @@ ildouble: 1
|
||||
ldouble: 1
|
||||
|
||||
Function: Real part of "clog_towardzero":
|
||||
double: 7
|
||||
float: 5
|
||||
idouble: 7
|
||||
ifloat: 5
|
||||
ildouble: 8
|
||||
ldouble: 8
|
||||
double: 4
|
||||
float: 4
|
||||
idouble: 4
|
||||
ifloat: 4
|
||||
ildouble: 5
|
||||
ldouble: 5
|
||||
|
||||
Function: Imaginary part of "clog_towardzero":
|
||||
double: 1
|
||||
float: 2
|
||||
float: 3
|
||||
idouble: 1
|
||||
ifloat: 2
|
||||
ifloat: 3
|
||||
ildouble: 1
|
||||
ldouble: 1
|
||||
|
||||
Function: Real part of "clog_upward":
|
||||
double: 8
|
||||
float: 5
|
||||
idouble: 8
|
||||
ifloat: 5
|
||||
ildouble: 6
|
||||
ldouble: 6
|
||||
double: 4
|
||||
float: 3
|
||||
idouble: 4
|
||||
ifloat: 3
|
||||
ildouble: 4
|
||||
ldouble: 4
|
||||
|
||||
Function: Imaginary part of "clog_upward":
|
||||
double: 1
|
||||
|
||||
Loading…
Reference in New Issue
Block a user