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Re: Fix x86 acos near 1 (bug 13942)


On 04/30/2012 02:11 PM, Joseph S. Myers wrote:
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) */

I suggest to enhance the comment above with = atanl (sqrtl (((1-x) (1+x))/ x)

Other files have the same comment, please add it there as well.

Ok with those changes,

Andreas

    asm (	"fld	%%st\n"
-	"fmul	%%st(0)\n"		/* x^2 */
  	"fld1\n"
-	"fsubp\n"			/* 1 - x^2 */
+	"fsubp\n"
+	"fld1\n"
+	"fadd	%%st(2)\n"
+	"fmulp\n"			/* 1 - x^2 */
  	"fsqrt\n"			/* sqrtl (1 - x^2) */
  	"fabs\n"
  	"fxch	%%st(1)\n"
diff --git a/sysdeps/i386/fpu/e_asin.S b/sysdeps/i386/fpu/e_asin.S
index a17e922..b4d17b0 100644
--- a/sysdeps/i386/fpu/e_asin.S
+++ b/sysdeps/i386/fpu/e_asin.S
@@ -11,9 +11,11 @@ RCSID("$NetBSD: e_asin.S,v 1.4 1995/05/08 23:45:40 jtc Exp $")
  ENTRY(__ieee754_asin)
  	fldl	4(%esp)			/* x */
  	fld	%st
-	fmul	%st(0)			/* x^2 */
-	fld1
-	fsubp				/* 1 - x^2 */
+	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 */
  	fsqrt				/* sqrt (1 - x^2) */
  	fpatan
  	ret


--
 Andreas Jaeger aj@{suse.com,opensuse.org} Twitter/Identica: jaegerandi
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