Bug 13942 is the problem of acos on x86, and acosl on x86/x86_64,
being very inaccurate for arguments near 1 because of a naive
calculation of 1 - x*x, which can be calculated more accurately as (1
- x) * (1 + x).
I propose this patch to fix this. I also fixed the corresponding
calculation in asin, although that only makes a small accuracy
improvement there (2ulp error changes to 0ulp). acosf and asinf are
unaffected, since the squaring is exact there, and asinl uses a
different (C) implementation, not x87 assembly. Note that the tests
added would suffice to catch any such problems for the wider long
double implementations, since squaring a 64-bit value is substantially
inaccurate even for 113-bit mantissa, so there is no need for further
tests specifically for wider long double formats.
Tested x86 and x86_64 and ulps updated accordingly.
2012-04-30 Joseph Myers<joseph@codesourcery.com>
[BZ #13942]
* sysdeps/i386/fpu/e_acos.S (__ieee754_acos): Calculate 1 - x^2 as
(1 - x) * (1 + x).
* sysdeps/i386/fpu/e_acosl.c (__ieee754_acosl): Likewise.
* sysdeps/i386/fpu/e_asin.S (__ieee754_asin): Likewise.
* math/libm-test.inc (acos_test): Add more tests.
(asin_test): Likewise.
* sysdeps/i386/fpu/libm-test-ulps: Update.
* sysdeps/x86_64/fpu/libm-test-ulps: Likewise.
diff --git a/math/libm-test.inc b/math/libm-test.inc
index bedff09..d643bad 100644
--- a/math/libm-test.inc
+++ b/math/libm-test.inc
@@ -758,6 +758,18 @@ acos_test (void)
TEST_f_f (acos, 0.75L, 0.722734247813415611178377352641333362L);
TEST_f_f (acos, 2e-17L, 1.57079632679489659923132169163975144L);
TEST_f_f (acos, 0.0625L, 1.50825556499840522843072005474337068L);
+ TEST_f_f (acos, 0x0.ffffffp0L, 3.4526698471620358760324948263873649728491e-4L);
+ TEST_f_f (acos, -0x0.ffffffp0L, 3.1412473866050770348750401337968641476999L);
+#ifndef TEST_FLOAT
+ TEST_f_f (acos, 0x0.ffffffff8p0L, 1.5258789062648029736620564947844627548516e-5L);
+ TEST_f_f (acos, -0x0.ffffffff8p0L, 3.1415773948007305904329067627145550395696L);
+ TEST_f_f (acos, 0x0.ffffffffffffp0L, 8.4293697021788088529885473244391795127130e-8L);
+ TEST_f_f (acos, -0x0.ffffffffffffp0L, 3.1415925692960962166745548533940296398054L);
+#endif
+#if defined TEST_LDOUBLE&& LDBL_MANT_DIG>= 64
+ TEST_f_f (acos, 0x0.ffffffffffffffffp0L, 3.2927225399135962333718255320079907245059e-10L);
+ TEST_f_f (acos, -0x0.ffffffffffffffffp0L, 3.1415926532605209844712837599423203309964L);
+#endif
END (acos);
}
@@ -933,6 +945,18 @@ asin_test (void)
TEST_f_f (asin, 1.0, M_PI_2l);
TEST_f_f (asin, -1.0, -M_PI_2l);
TEST_f_f (asin, 0.75L, 0.848062078981481008052944338998418080L);
+ TEST_f_f (asin, 0x0.ffffffp0L, 1.5704510598101804156437184421571127056013L);
+ TEST_f_f (asin, -0x0.ffffffp0L, -1.5704510598101804156437184421571127056013L);
+#ifndef TEST_FLOAT
+ TEST_f_f (asin, 0x0.ffffffff8p0L, 1.5707810680058339712015850710748035974710L);
+ TEST_f_f (asin, -0x0.ffffffff8p0L, -1.5707810680058339712015850710748035974710L);
+ TEST_f_f (asin, 0x0.ffffffffffffp0L, 1.5707962425011995974432331617542781977068L);
+ TEST_f_f (asin, -0x0.ffffffffffffp0L, -1.5707962425011995974432331617542781977068L);
+#endif
+#if defined TEST_LDOUBLE&& LDBL_MANT_DIG>= 64
+ TEST_f_f (asin, 0x0.ffffffffffffffffp0L, 1.5707963264656243652399620683025688888978L);
+ TEST_f_f (asin, -0x0.ffffffffffffffffp0L, -1.5707963264656243652399620683025688888978L);
+#endif
END (asin);
}
diff --git a/sysdeps/i386/fpu/e_acos.S b/sysdeps/i386/fpu/e_acos.S
index d10a054..fc680ee 100644
--- a/sysdeps/i386/fpu/e_acos.S
+++ b/sysdeps/i386/fpu/e_acos.S
@@ -11,9 +11,11 @@ RCSID("$NetBSD: e_acos.S,v 1.4 1995/05/08 23:44:37 jtc Exp $")
ENTRY(__ieee754_acos)
fldl 4(%esp) /* x */
fld %st /* x : x */
- fmul %st(0) /* x^2 : x */
- fld1 /* 1 : x^2 : x */
- fsubp /* 1 - x^2 : x */
+ fld1 /* 1 : x : x */
+ fsubp /* 1 - x : x */
+ fld1 /* 1 : 1 - x : x */
+ fadd %st(2) /* 1 + x : 1 - x : x */
+ fmulp /* 1 - x^2 : x */
fsqrt /* sqrt (1 - x^2) : x */
fabs
fxch %st(1) /* x : sqrt (1 - x^2) */
diff --git a/sysdeps/i386/fpu/e_acosl.c b/sysdeps/i386/fpu/e_acosl.c
index d249d5a..42c5f01 100644
--- a/sysdeps/i386/fpu/e_acosl.c
+++ b/sysdeps/i386/fpu/e_acosl.c
@@ -14,9 +14,11 @@ __ieee754_acosl (long double x)
/* acosl = atanl (sqrtl(1 - x^2) / x) */