Fix cosh spurious underflows from expm1 (bug 16354), inaccurate results near 0 (bug 17061).

This patch fixes bug 16354, spurious underflows from cosh when a tiny
argument is passed to expm1 and expm1 correctly underflows although
the final result of cosh should be 1.  As noted in that bug, some
cases are latent because of expm1 implementations not raising
underflow (bug 16353), but all the implementations are fixed
similarly.  They already contained checks for tiny arguments, but the
checks were too late to avoid underflow from expm1 (although they
would avoid underflow from subsequent squaring of the result of
expm1); they are moved before the expm1 calls.

The thresholds used for considering arguments tiny are not
particularly consistent in how they relate to the precision of the
floating-point format in question.  They are, however, all sufficient
to ensure that the round-to-nearest result of cosh is indeed 1 below
the threshold (although sometimes they are smaller than necessary).
But the previous logic did not return 1, but the previously computed 1
+ expm1(abs(x)) value.  And the thresholds in the ldbl-128 and
ldbl-128ibm code (0x1p-71L - I suspect 0x3f8b was intended in the code
instead of 0x3fb8 - and (roughly) 0x1p-55L) are not sufficient for
that value to be 1.  So by moving the test for tiny arguments, and
consequently returning 1 directly now the expm1 value hasn't been
computed by that point, this patch also fixes bug 17061, the (large
number of ulps) inaccuracy for small arguments in those
implementations.  Tests for that bug are duly added.

Tested x86_64 and x86 and ulps updated accordingly.  Also tested for
mips64 and powerpc32 to validate the ldbl-128 and ldbl-128ibm changes.

	[BZ #16354]
	[BZ #17061]
	* sysdeps/ieee754/dbl-64/e_cosh.c (__ieee754_cosh): Check for
	small arguments before calling __expm1.
	* sysdeps/ieee754/flt-32/e_coshf.c (__ieee754_coshf): Check for
	small arguments before calling __expm1f.
	* sysdeps/ieee754/ldbl-128/e_coshl.c (__ieee754_coshl): Check for
	small arguments before calling __expm1l.
	* sysdeps/ieee754/ldbl-128ibm/e_coshl.c (__ieee754_coshl):
	Likewise.
	* sysdeps/ieee754/ldbl-96/e_coshl.c (__ieee754_coshl): Likewise.
	* math/auto-libm-test-in: Add more cosh tests.  Do not allow
	spurious underflow for some cosh tests.
	* math/auto-libm-test-out: Regenerated.
	* sysdeps/i386/fpu/libm-test-ulps: Update.

5 files changed, 7 insertions, 7 deletions

diff --git a/sysdeps/ieee754/dbl-64/e_cosh.c b/sysdeps/ieee754/dbl-64/e_cosh.c
index 6caf943ed0..af3910dd6e 100644
--- a/sysdeps/ieee754/dbl-64/e_cosh.c
+++ b/sysdeps/ieee754/dbl-64/e_cosh.c
@@ -53,10 +53,10 @@ __ieee754_cosh (double x)
       /* |x| in [0,0.5*ln2], return 1+expm1(|x|)^2/(2*exp(|x|)) */
       if (ix < 0x3fd62e43)
 	{
+	  if (ix < 0x3c800000)
+	    return one;                                   /* cosh(tiny) = 1 */
 	  t = __expm1 (fabs (x));
 	  w = one + t;
-	  if (ix < 0x3c800000)
-	    return w;                                   /* cosh(tiny) = 1 */
 	  return one + (t * t) / (w + w);
 	}
 
diff --git a/sysdeps/ieee754/flt-32/e_coshf.c b/sysdeps/ieee754/flt-32/e_coshf.c
index 7eeea4d8da..dedda47c09 100644
--- a/sysdeps/ieee754/flt-32/e_coshf.c
+++ b/sysdeps/ieee754/flt-32/e_coshf.c
@@ -33,9 +33,9 @@ __ieee754_coshf (float x)
 	if (ix < 0x41b00000) {
 	    /* |x| in [0,0.5*ln2], return 1+expm1(|x|)^2/(2*exp(|x|)) */
 		if(ix<0x3eb17218) {
+		    if (ix<0x24000000) return one;	/* cosh(tiny) = 1 */
 		    t = __expm1f(fabsf(x));
 		    w = one+t;
-		    if (ix<0x24000000) return w;	/* cosh(tiny) = 1 */
 		    return one+(t*t)/(w+w);
 		}
 
diff --git a/sysdeps/ieee754/ldbl-128/e_coshl.c b/sysdeps/ieee754/ldbl-128/e_coshl.c
index 59f2030d54..488c318d27 100644
--- a/sysdeps/ieee754/ldbl-128/e_coshl.c
+++ b/sysdeps/ieee754/ldbl-128/e_coshl.c
@@ -77,10 +77,10 @@ __ieee754_coshl (long double x)
   /* |x| in [0,0.5*ln2], return 1+expm1l(|x|)^2/(2*expl(|x|)) */
   if (ex < 0x3ffd62e4) /* 0.3465728759765625 */
     {
+      if (ex < 0x3fb80000) /* |x| < 2^-116 */
+	return one;		/* cosh(tiny) = 1 */
       t = __expm1l (u.value);
       w = one + t;
-      if (ex < 0x3fb80000) /* |x| < 2^-116 */
-	return w;		/* cosh(tiny) = 1 */
 
       return one + (t * t) / (w + w);
     }
diff --git a/sysdeps/ieee754/ldbl-128ibm/e_coshl.c b/sysdeps/ieee754/ldbl-128ibm/e_coshl.c
index 92313e25e2..327b2ab960 100644
--- a/sysdeps/ieee754/ldbl-128ibm/e_coshl.c
+++ b/sysdeps/ieee754/ldbl-128ibm/e_coshl.c
@@ -53,9 +53,9 @@ __ieee754_coshl (long double x)
 
     /* |x| in [0,0.5*ln2], return 1+expm1(|x|)^2/(2*exp(|x|)) */
 	if(ix<0x3fd62e42fefa39efLL) {
+	    if (ix<0x3c80000000000000LL) return one;	/* cosh(tiny) = 1 */
 	    t = __expm1l(fabsl(x));
 	    w = one+t;
-	    if (ix<0x3c80000000000000LL) return w;	/* cosh(tiny) = 1 */
 	    return one+(t*t)/(w+w);
 	}
 
diff --git a/sysdeps/ieee754/ldbl-96/e_coshl.c b/sysdeps/ieee754/ldbl-96/e_coshl.c
index 995c9922cb..dd22cae363 100644
--- a/sysdeps/ieee754/ldbl-96/e_coshl.c
+++ b/sysdeps/ieee754/ldbl-96/e_coshl.c
@@ -54,9 +54,9 @@ __ieee754_coshl (long double x)
 	if (ex < 0x4003 || (ex == 0x4003 && mx < 0xb0000000u)) {
 	    /* |x| in [0,0.5*ln2], return 1+expm1l(|x|)^2/(2*expl(|x|)) */
 		if(ex < 0x3ffd || (ex == 0x3ffd && mx < 0xb17217f7u)) {
+		    if (ex<0x3fbc) return one;	/* cosh(tiny) = 1 */
 		    t = __expm1l(fabsl(x));
 		    w = one+t;
-		    if (ex<0x3fbc) return w;	/* cosh(tiny) = 1 */
 		    return one+(t*t)/(w+w);
 		}