Use round-to-nearest internally in jn, test with ALL_RM_TEST (bug 18602).

Some existing jn tests, if run in non-default rounding modes, produce
errors above those accepted in glibc, which causes problems for moving
tests of jn to use ALL_RM_TEST.  This patch makes jn set rounding
to-nearest internally, as was done for yn some time ago, then computes
the appropriate underflowing value for results that underflowed to
zero in to-nearest, and moves the tests to ALL_RM_TEST.  It does
nothing about the general inaccuracy of Bessel function
implementations in glibc, though it should make jn more accurate on
average in non-default rounding modes through reduced error
accumulation.  The recomputation of results that underflowed to zero
should as a side-effect fix some cases of bug 16559, where jn just
used an exact zero, but that is *not* the goal of this patch and other
cases of that bug remain unfixed.

(Most of the changes in the patch are reindentation to add new scopes
for SET_RESTORE_ROUND*.)

Tested for x86_64, x86, powerpc and mips64.

	[BZ #16559]
	[BZ #18602]
	* sysdeps/ieee754/dbl-64/e_jn.c (__ieee754_jn): Set
	round-to-nearest internally then recompute results that
	underflowed to zero in the original rounding mode.
	* sysdeps/ieee754/flt-32/e_jnf.c (__ieee754_jnf): Likewise.
	* sysdeps/ieee754/ldbl-128/e_jnl.c (__ieee754_jnl): Likewise.
	* sysdeps/ieee754/ldbl-128ibm/e_jnl.c (__ieee754_jnl): Likewise.
	* sysdeps/ieee754/ldbl-96/e_jnl.c (__ieee754_jnl): Likewise
	* math/libm-test.inc (jn_test): Use ALL_RM_TEST.
	* sysdeps/i386/fpu/libm-test-ulps: Update.
	* sysdeps/x86_64/fpu/libm-test-ulps: Likewise.

1 files changed, 159 insertions, 153 deletions

diff --git a/sysdeps/ieee754/dbl-64/e_jn.c b/sysdeps/ieee754/dbl-64/e_jn.c
index 900737c401..b0ddd5e841 100644
--- a/sysdeps/ieee754/dbl-64/e_jn.c
+++ b/sysdeps/ieee754/dbl-64/e_jn.c
@@ -52,7 +52,7 @@ double
 __ieee754_jn (int n, double x)
 {
   int32_t i, hx, ix, lx, sgn;
-  double a, b, temp, di;
+  double a, b, temp, di, ret;
   double z, w;
 
   /* J(-n,x) = (-1)^n * J(n, x), J(n, -x) = (-1)^n * J(n, x)
@@ -75,14 +75,16 @@ __ieee754_jn (int n, double x)
     return (__ieee754_j1 (x));
   sgn = (n & 1) & (hx >> 31);   /* even n -- 0, odd n -- sign(x) */
   x = fabs (x);
-  if (__glibc_unlikely ((ix | lx) == 0 || ix >= 0x7ff00000))
-    /* if x is 0 or inf */
-    b = zero;
-  else if ((double) n <= x)
-    {
-      /* Safe to use J(n+1,x)=2n/x *J(n,x)-J(n-1,x) */
-      if (ix >= 0x52D00000)      /* x > 2**302 */
-	{ /* (x >> n**2)
+  {
+    SET_RESTORE_ROUND (FE_TONEAREST);
+    if (__glibc_unlikely ((ix | lx) == 0 || ix >= 0x7ff00000))
+      /* if x is 0 or inf */
+      return sgn == 1 ? -zero : zero;
+    else if ((double) n <= x)
+      {
+	/* Safe to use J(n+1,x)=2n/x *J(n,x)-J(n-1,x) */
+	if (ix >= 0x52D00000)      /* x > 2**302 */
+	  { /* (x >> n**2)
 			 *	    Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
 			 *	    Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
 			 *	    Let s=sin(x), c=cos(x),
@@ -95,152 +97,156 @@ __ieee754_jn (int n, double x)
 			 *		   2	-s+c		-c-s
 			 *		   3	 s+c		 c-s
 			 */
-	  double s;
-	  double c;
-	  __sincos (x, &s, &c);
-	  switch (n & 3)
-	    {
-	    case 0: temp = c + s; break;
-	    case 1: temp = -c + s; break;
-	    case 2: temp = -c - s; break;
-	    case 3: temp = c - s; break;
-	    }
-	  b = invsqrtpi * temp / __ieee754_sqrt (x);
-	}
-      else
-	{
-	  a = __ieee754_j0 (x);
-	  b = __ieee754_j1 (x);
-	  for (i = 1; i < n; i++)
-	    {
-	      temp = b;
-	      b = b * ((double) (i + i) / x) - a; /* avoid underflow */
-	      a = temp;
-	    }
-	}
-    }
-  else
-    {
-      if (ix < 0x3e100000)      /* x < 2**-29 */
-	{ /* x is tiny, return the first Taylor expansion of J(n,x)
+	    double s;
+	    double c;
+	    __sincos (x, &s, &c);
+	    switch (n & 3)
+	      {
+	      case 0: temp = c + s; break;
+	      case 1: temp = -c + s; break;
+	      case 2: temp = -c - s; break;
+	      case 3: temp = c - s; break;
+	      }
+	    b = invsqrtpi * temp / __ieee754_sqrt (x);
+	  }
+	else
+	  {
+	    a = __ieee754_j0 (x);
+	    b = __ieee754_j1 (x);
+	    for (i = 1; i < n; i++)
+	      {
+		temp = b;
+		b = b * ((double) (i + i) / x) - a; /* avoid underflow */
+		a = temp;
+	      }
+	  }
+      }
+    else
+      {
+	if (ix < 0x3e100000)      /* x < 2**-29 */
+	  { /* x is tiny, return the first Taylor expansion of J(n,x)
 			 * J(n,x) = 1/n!*(x/2)^n  - ...
 			 */
-	  if (n > 33)           /* underflow */
-	    b = zero;
-	  else
-	    {
-	      temp = x * 0.5; b = temp;
-	      for (a = one, i = 2; i <= n; i++)
-		{
-		  a *= (double) i;              /* a = n! */
-		  b *= temp;                    /* b = (x/2)^n */
-		}
-	      b = b / a;
-	    }
-	}
-      else
-	{
-	  /* use backward recurrence */
-	  /*			x      x^2      x^2
-	   *  J(n,x)/J(n-1,x) =  ----   ------   ------   .....
-	   *			2n  - 2(n+1) - 2(n+2)
-	   *
-	   *			1      1        1
-	   *  (for large x)   =  ----  ------   ------   .....
-	   *			2n   2(n+1)   2(n+2)
-	   *			-- - ------ - ------ -
-	   *			 x     x         x
-	   *
-	   * Let w = 2n/x and h=2/x, then the above quotient
-	   * is equal to the continued fraction:
-	   *		    1
-	   *	= -----------------------
-	   *		       1
-	   *	   w - -----------------
-	   *			  1
-	   *		w+h - ---------
-	   *		       w+2h - ...
-	   *
-	   * To determine how many terms needed, let
-	   * Q(0) = w, Q(1) = w(w+h) - 1,
-	   * Q(k) = (w+k*h)*Q(k-1) - Q(k-2),
-	   * When Q(k) > 1e4	good for single
-	   * When Q(k) > 1e9	good for double
-	   * When Q(k) > 1e17	good for quadruple
-	   */
-	  /* determine k */
-	  double t, v;
-	  double q0, q1, h, tmp; int32_t k, m;
-	  w = (n + n) / (double) x; h = 2.0 / (double) x;
-	  q0 = w;  z = w + h; q1 = w * z - 1.0; k = 1;
-	  while (q1 < 1.0e9)
-	    {
-	      k += 1; z += h;
-	      tmp = z * q1 - q0;
-	      q0 = q1;
-	      q1 = tmp;
-	    }
-	  m = n + n;
-	  for (t = zero, i = 2 * (n + k); i >= m; i -= 2)
-	    t = one / (i / x - t);
-	  a = t;
-	  b = one;
-	  /*  estimate log((2/x)^n*n!) = n*log(2/x)+n*ln(n)
-	   *  Hence, if n*(log(2n/x)) > ...
-	   *  single 8.8722839355e+01
-	   *  double 7.09782712893383973096e+02
-	   *  long double 1.1356523406294143949491931077970765006170e+04
-	   *  then recurrent value may overflow and the result is
-	   *  likely underflow to zero
-	   */
-	  tmp = n;
-	  v = two / x;
-	  tmp = tmp * __ieee754_log (fabs (v * tmp));
-	  if (tmp < 7.09782712893383973096e+02)
-	    {
-	      for (i = n - 1, di = (double) (i + i); i > 0; i--)
-		{
-		  temp = b;
-		  b *= di;
-		  b = b / x - a;
-		  a = temp;
-		  di -= two;
-		}
-	    }
-	  else
-	    {
-	      for (i = n - 1, di = (double) (i + i); i > 0; i--)
-		{
-		  temp = b;
-		  b *= di;
-		  b = b / x - a;
-		  a = temp;
-		  di -= two;
-		  /* scale b to avoid spurious overflow */
-		  if (b > 1e100)
-		    {
-		      a /= b;
-		      t /= b;
-		      b = one;
-		    }
-		}
-	    }
-	  /* j0() and j1() suffer enormous loss of precision at and
-	   * near zero; however, we know that their zero points never
-	   * coincide, so just choose the one further away from zero.
-	   */
-	  z = __ieee754_j0 (x);
-	  w = __ieee754_j1 (x);
-	  if (fabs (z) >= fabs (w))
-	    b = (t * z / b);
-	  else
-	    b = (t * w / a);
-	}
-    }
-  if (sgn == 1)
-    return -b;
-  else
-    return b;
+	    if (n > 33)           /* underflow */
+	      b = zero;
+	    else
+	      {
+		temp = x * 0.5; b = temp;
+		for (a = one, i = 2; i <= n; i++)
+		  {
+		    a *= (double) i;              /* a = n! */
+		    b *= temp;                    /* b = (x/2)^n */
+		  }
+		b = b / a;
+	      }
+	  }
+	else
+	  {
+	    /* use backward recurrence */
+	    /*			x      x^2      x^2
+	     *  J(n,x)/J(n-1,x) =  ----   ------   ------   .....
+	     *			2n  - 2(n+1) - 2(n+2)
+	     *
+	     *			1      1        1
+	     *  (for large x)   =  ----  ------   ------   .....
+	     *			2n   2(n+1)   2(n+2)
+	     *			-- - ------ - ------ -
+	     *			 x     x         x
+	     *
+	     * Let w = 2n/x and h=2/x, then the above quotient
+	     * is equal to the continued fraction:
+	     *		    1
+	     *	= -----------------------
+	     *		       1
+	     *	   w - -----------------
+	     *			  1
+	     *		w+h - ---------
+	     *		       w+2h - ...
+	     *
+	     * To determine how many terms needed, let
+	     * Q(0) = w, Q(1) = w(w+h) - 1,
+	     * Q(k) = (w+k*h)*Q(k-1) - Q(k-2),
+	     * When Q(k) > 1e4	good for single
+	     * When Q(k) > 1e9	good for double
+	     * When Q(k) > 1e17	good for quadruple
+	     */
+	    /* determine k */
+	    double t, v;
+	    double q0, q1, h, tmp; int32_t k, m;
+	    w = (n + n) / (double) x; h = 2.0 / (double) x;
+	    q0 = w;  z = w + h; q1 = w * z - 1.0; k = 1;
+	    while (q1 < 1.0e9)
+	      {
+		k += 1; z += h;
+		tmp = z * q1 - q0;
+		q0 = q1;
+		q1 = tmp;
+	      }
+	    m = n + n;
+	    for (t = zero, i = 2 * (n + k); i >= m; i -= 2)
+	      t = one / (i / x - t);
+	    a = t;
+	    b = one;
+	    /*  estimate log((2/x)^n*n!) = n*log(2/x)+n*ln(n)
+	     *  Hence, if n*(log(2n/x)) > ...
+	     *  single 8.8722839355e+01
+	     *  double 7.09782712893383973096e+02
+	     *  long double 1.1356523406294143949491931077970765006170e+04
+	     *  then recurrent value may overflow and the result is
+	     *  likely underflow to zero
+	     */
+	    tmp = n;
+	    v = two / x;
+	    tmp = tmp * __ieee754_log (fabs (v * tmp));
+	    if (tmp < 7.09782712893383973096e+02)
+	      {
+		for (i = n - 1, di = (double) (i + i); i > 0; i--)
+		  {
+		    temp = b;
+		    b *= di;
+		    b = b / x - a;
+		    a = temp;
+		    di -= two;
+		  }
+	      }
+	    else
+	      {
+		for (i = n - 1, di = (double) (i + i); i > 0; i--)
+		  {
+		    temp = b;
+		    b *= di;
+		    b = b / x - a;
+		    a = temp;
+		    di -= two;
+		    /* scale b to avoid spurious overflow */
+		    if (b > 1e100)
+		      {
+			a /= b;
+			t /= b;
+			b = one;
+		      }
+		  }
+	      }
+	    /* j0() and j1() suffer enormous loss of precision at and
+	     * near zero; however, we know that their zero points never
+	     * coincide, so just choose the one further away from zero.
+	     */
+	    z = __ieee754_j0 (x);
+	    w = __ieee754_j1 (x);
+	    if (fabs (z) >= fabs (w))
+	      b = (t * z / b);
+	    else
+	      b = (t * w / a);
+	  }
+      }
+    if (sgn == 1)
+      ret = -b;
+    else
+      ret = b;
+  }
+  if (ret == 0)
+    ret = __copysign (DBL_MIN, ret) * DBL_MIN;
+  return ret;
 }
 strong_alias (__ieee754_jn, __jn_finite)