Update.

* sysdeps/ieee754/ldbl-96/e_jnl.c: New file.
	Contributed by Stephen L. Moshier <moshier@na-net.ornl.gov>.

	* sysdeps/i386/fpu/libm-test-ulps: Update for jnl and ynl introduction.
	* sysdeps/ia64/fpu/libm-test-ulps: Likewise.

ChangeLog

@@ -1,5 +1,11 @@
 2001-02-26  Ulrich Drepper  <drepper@redhat.com>
 
+	* sysdeps/ieee754/ldbl-96/e_jnl.c: New file.
+	Contributed by Stephen L. Moshier <moshier@na-net.ornl.gov>.
+
+	* sysdeps/i386/fpu/libm-test-ulps: Update for jnl and ynl introduction.
+	* sysdeps/ia64/fpu/libm-test-ulps: Likewise.
+
 	* posix/wordexp-test.c (testit): Remove warnings.
 
 	* dlfcn/Makefile (distribute): Add modatexit.c and modcxaatexit.c.

linuxthreads/ChangeLog

@@ -1,3 +1,12 @@
+2001-02-20  Hans Boehm  <hans_boehm@hp.com>
+
+	* manager.c (manager_mask): Removed static vesion.  Now always local
+	to __pthread_manager().
+	(manager_mask_all): Removed completely.
+	(__pthread_manager): Remove manager_mask_all initialization.
+	(pthread_handle_create): Remove code to set and reset signal mask
+	around __clone2() calls.
+
 2001-02-17  Jakub Jelinek  <jakub@redhat.com>
 
 	* spinlock.c (__pthread_lock): Force lock->__status to be read from

linuxthreads/manager.c

@@ -88,13 +88,6 @@ static int main_thread_exiting;
 
 static pthread_t pthread_threads_counter;
 
-#ifdef NEED_SEPARATE_REGISTER_STACK
-/* Signal masks for the manager.  These have to be global only when clone2
-   is used since it's currently borken wrt signals in the child.  */
-static sigset_t manager_mask;		/* Manager normal signal mask	*/
-static sigset_t manager_mask_all;	/* All bits set.	*/
-#endif
-
 /* Forward declarations */
 
 static int pthread_handle_create(pthread_t *thread, const pthread_attr_t *attr,
@@ -113,9 +106,7 @@ int __pthread_manager(void *arg)
 {
   int reqfd = (int) (long int) arg;
   struct pollfd ufd;
-#ifndef NEED_SEPARATE_REGISTER_STACK
   sigset_t manager_mask;
-#endif
   int n;
   struct pthread_request request;
 
@@ -133,9 +124,6 @@ int __pthread_manager(void *arg)
   if (__pthread_threads_debug && __pthread_sig_debug > 0)
     sigdelset(&manager_mask, __pthread_sig_debug);
   sigprocmask(SIG_SETMASK, &manager_mask, NULL);
-#ifdef NEED_SEPARATE_REGISTER_STACK
-  sigfillset(&manager_mask_all);
-#endif
   /* Raise our priority to match that of main thread */
   __pthread_manager_adjust_prio(__pthread_main_thread->p_priority);
   /* Synchronize debugging of the thread manager */
@@ -583,17 +571,12 @@ static int pthread_handle_create(pthread_t *thread, const pthread_attr_t *attr,
 	   And there is some argument for changing the __clone2
 	   interface to pass sp and bsp instead, making it more IA64
 	   specific, but allowing stacks to grow outward from each
-	   other, to get less paging and fewer mmaps.  Clone2
+	   other, to get less paging and fewer mmaps.  */
-	   currently can't take signals in the child right after
-	   process creation.  Mask them in the child.  It resets the
-	   mask once it starts up.  */
-	  sigprocmask(SIG_SETMASK, &manager_mask_all, NULL);
 	  pid = __clone2(pthread_start_thread_event,
   		 (void **)new_thread_bottom,
 			 (char *)new_thread - new_thread_bottom,
 			 CLONE_VM | CLONE_FS | CLONE_FILES | CLONE_SIGHAND |
 			 __pthread_sig_cancel, new_thread);
-	  sigprocmask(SIG_SETMASK, &manager_mask, NULL);
 #else
 	  pid = __clone(pthread_start_thread_event, (void **) new_thread,
 			CLONE_VM | CLONE_FS | CLONE_FILES | CLONE_SIGHAND |
@@ -625,13 +608,11 @@ static int pthread_handle_create(pthread_t *thread, const pthread_attr_t *attr,
   if (pid == 0)
     {
 #ifdef NEED_SEPARATE_REGISTER_STACK
-      sigprocmask(SIG_SETMASK, &manager_mask_all, NULL);
       pid = __clone2(pthread_start_thread,
 		     (void **)new_thread_bottom,
                      (char *)new_thread - new_thread_bottom,
 		     CLONE_VM | CLONE_FS | CLONE_FILES | CLONE_SIGHAND |
 		     __pthread_sig_cancel, new_thread);
-      sigprocmask(SIG_SETMASK, &manager_mask, NULL);
 #else
       pid = __clone(pthread_start_thread, (void **) new_thread,
 		    CLONE_VM | CLONE_FS | CLONE_FILES | CLONE_SIGHAND |

sysdeps/i386/fpu/libm-test-ulps

@@ -676,6 +676,8 @@ float: 1
 ifloat: 1
 double: 2
 idouble: 2
+ldouble: 2
+ildouble: 2
 Test "jn (0, 2.0) == 0.22389077914123566805":
 float: 1
 ifloat: 1
@@ -684,46 +686,66 @@ idouble: 1
 Test "jn (0, 8.0) == 0.17165080713755390609":
 float: 1
 ifloat: 1
+ldouble: 1
+ildouble: 1
 Test "jn (1, 10.0) == 0.043472746168861436670":
 float: 1
 ifloat: 1
 double: 2
 idouble: 2
+ldouble: 2
+ildouble: 2
 Test "jn (1, 2.0) == 0.57672480775687338720":
 double: 1
 idouble: 1
 Test "jn (1, 8.0) == 0.23463634685391462438":
 float: 1
 ifloat: 1
+ldouble: 1
+ildouble: 1
 Test "jn (10, -1.0) == 0.26306151236874532070e-9":
 float: 1
 ifloat: 1
+ldouble: 1
+ildouble: 1
 Test "jn (10, 0.1) == 0.26905328954342155795e-19":
 double: 4
 float: 2
 idouble: 4
 ifloat: 2
+ldouble: 1
+ildouble: 1
 Test "jn (10, 0.7) == 0.75175911502153953928e-11":
 double: 4
 float: 1
 idouble: 4
 ifloat: 1
+ldouble: 2
+ildouble: 2
 Test "jn (10, 1.0) == 0.26306151236874532070e-9":
 float: 1
 ifloat: 1
+ldouble: 1
+ildouble: 1
 Test "jn (10, 2.0) == 0.25153862827167367096e-6":
 float: 1
 ifloat: 1
 double: 2
 idouble: 2
+ldouble: 1
+ildouble: 1
 Test "jn (10, 10.0) == 0.20748610663335885770":
 float: 2
 ifloat: 2
 double: 4
 idouble: 4
+ldouble: 2
+ildouble: 2
 Test "jn (3, 0.1) == 0.000020820315754756261429":
 double: 1
 idouble: 1
+ldouble: 1
+ildouble: 1
 Test "jn (3, 0.7) == 0.0069296548267508408077":
 double: 2
 idouble: 2
@@ -737,6 +759,14 @@ float: 1
 ifloat: 1
 double: 3
 idouble: 3
+ldouble: 1
+ildouble: 1
+Test "jn (3, -1.0) == -0.019563353982668405919":
+ldouble: 1
+ildouble: 1
+Test "jn (3, 1.0) == 0.019563353982668405919":
+ldouble: 1
+ildouble: 1
 
 # lgamma
 Test "lgamma (-0.5) == log(2*sqrt(pi))":
@@ -972,6 +1002,8 @@ float: 1
 ifloat: 1
 double: 3
 idouble: 3
+ldouble: 2
+ildouble: 2
 Test "yn (0, 1.0) == 0.088256964215676957983":
 double: 2
 float: 1
@@ -990,16 +1022,22 @@ float: 1
 ifloat: 1
 double: 2
 idouble: 2
+ldouble: 1
+ildouble: 1
 Test "yn (0, 8.0) == 0.22352148938756622053":
 float: 1
 ifloat: 1
 double: 1
 idouble: 1
+ldouble: 1
+ildouble: 1
 Test "yn (1, 0.1) == -6.4589510947020269877":
 double: 1
 float: 1
 idouble: 1
 ifloat: 1
+ldouble: 1
+ildouble: 1
 Test "yn (1, 0.7) == -1.1032498719076333697":
 double: 1
 idouble: 1
@@ -1019,19 +1057,27 @@ double: 1
 float: 2
 idouble: 1
 ifloat: 2
+ldouble: 1
+ildouble: 1
 Test "yn (1, 8.0) == -0.15806046173124749426":
 float: 2
 ifloat: 2
+ldouble: 2
+ildouble: 2
 Test "yn (10, 0.1) == -0.11831335132045197885e19":
 double: 2
 float: 1
 idouble: 2
 ifloat: 1
+ldouble: 2
+ildouble: 2
 Test "yn (10, 0.7) == -0.42447194260703866924e10":
 double: 6
 float: 3
 idouble: 6
 ifloat: 3
+ldouble: 7
+ildouble: 7
 Test "yn (10, 1.0) == -0.12161801427868918929e9":
 double: 1
 float: 1
@@ -1047,14 +1093,20 @@ float: 1
 ifloat: 1
 double: 2
 idouble: 2
+ldouble: 1
+ildouble: 1
 Test "yn (3, 0.1) == -5099.3323786129048894":
 double: 1
 float: 1
 idouble: 1
 ifloat: 1
+ldouble: 2
+ildouble: 2
 Test "yn (3, 0.7) == -15.819479052819633505":
 double: 2
 idouble: 2
+ldouble: 2
+ildouble: 2
 Test "yn (3, 2.0) == -1.1277837768404277861":
 float: 1
 ifloat: 1
@@ -1413,6 +1465,8 @@ double: 4
 float: 2
 idouble: 4
 ifloat: 2
+ldouble: 2
+ildouble: 2
 
 Function: "lgamma":
 double: 1
@@ -1515,5 +1569,7 @@ double: 6
 float: 3
 idouble: 6
 ifloat: 3
+ldouble: 7
+ildouble: 7
 
 # end of automatic generation

sysdeps/ia64/fpu/libm-test-ulps

@@ -765,32 +765,46 @@ idouble: 1
 Test "jn (10, -1.0) == 0.26306151236874532070e-9":
 float: 1
 ifloat: 1
+ldouble: 1
+ildouble: 1
 Test "jn (10, 0.1) == 0.26905328954342155795e-19":
 float: 4
 ifloat: 4
 double: 6
 idouble: 6
+ldouble: 1
+ildouble: 1
 Test "jn (10, 0.7) == 0.75175911502153953928e-11":
 double: 4
 float: 1
 idouble: 4
 ifloat: 1
+ldouble: 2
+ildouble: 2
 Test "jn (10, 1.0) == 0.26306151236874532070e-9":
 float: 1
 ifloat: 1
+ldouble: 1
+ildouble: 1
 Test "jn (10, 2.0) == 0.25153862827167367096e-6":
 float: 3
 ifloat: 3
 double: 2
 idouble: 2
+ldouble: 1
+ildouble: 1
 Test "jn (10, 10.0) == 0.20748610663335885770":
 float: 2
 ifloat: 2
 double: 4
 idouble: 4
+ldouble: 2
+ildouble: 2
 Test "jn (3, 0.1) == 0.000020820315754756261429":
 double: 1
 idouble: 1
+ldouble: 1
+ildouble: 1
 Test "jn (3, 0.7) == 0.0069296548267508408077":
 float: 1
 ifloat: 1
@@ -806,6 +820,14 @@ float: 1
 ifloat: 1
 double: 3
 idouble: 3
+ldouble: 1
+ildouble: 1
+Test "jn (3, -1.0) == -0.019563353982668405919":
+ldouble: 1
+ildouble: 1
+Test "jn (3, 1.0) == 0.019563353982668405919":
+ldouble: 1
+ildouble: 1
 
 # lgamma
 Test "lgamma (-0.5) == log(2*sqrt(pi))":
@@ -1043,6 +1065,8 @@ float: 1
 ifloat: 1
 double: 2
 idouble: 2
+ldouble: 2
+ildouble: 2
 Test "yn (0, 1.0) == 0.088256964215676957983":
 double: 2
 float: 1
@@ -1066,6 +1090,8 @@ float: 1
 ifloat: 1
 double: 1
 idouble: 1
+ldouble: 2
+ildouble: 2
 Test "yn (1, 0.1) == -6.4589510947020269877":
 double: 1
 float: 1
@@ -1092,21 +1118,29 @@ double: 1
 float: 2
 idouble: 1
 ifloat: 2
+ldouble: 2
+ildouble: 2
 Test "yn (1, 8.0) == -0.15806046173124749426":
 float: 2
 ifloat: 2
 double: 1
 idouble: 1
+ldouble: 2
+ildouble: 2
 Test "yn (10, 0.1) == -0.11831335132045197885e19":
 double: 2
 float: 1
 idouble: 2
 ifloat: 1
+ldouble: 1
+ildouble: 1
 Test "yn (10, 0.7) == -0.42447194260703866924e10":
 double: 6
 float: 3
 idouble: 6
 ifloat: 3
+ldouble: 7
+ildouble: 7
 Test "yn (10, 1.0) == -0.12161801427868918929e9":
 float: 2
 ifloat: 2
@@ -1122,6 +1156,8 @@ float: 1
 ifloat: 1
 double: 3
 idouble: 3
+ldouble: 1
+ildouble: 1
 Test "yn (3, 0.1) == -5099.3323786129048894":
 double: 1
 float: 1
@@ -1132,6 +1168,8 @@ float: 1
 ifloat: 1
 double: 2
 idouble: 2
+ldouble: 3
+ildouble: 3
 Test "yn (3, 2.0) == -1.1277837768404277861":
 float: 1
 ifloat: 1
@@ -1503,6 +1541,8 @@ float: 4
 ifloat: 4
 double: 6
 idouble: 6
+ldouble: 2
+ildouble: 2
 
 Function: "lgamma":
 double: 1
@@ -1611,5 +1651,7 @@ double: 6
 float: 3
 idouble: 6
 ifloat: 3
+ldouble: 7
+ildouble: 7
 
 # end of automatic generation

sysdeps/ieee754/ldbl-96/e_jnl.c

@@ -0,0 +1,368 @@
+/*
+ * ====================================================
+ * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
+ *
+ * Developed at SunPro, a Sun Microsystems, Inc. business.
+ * Permission to use, copy, modify, and distribute this
+ * software is freely granted, provided that this notice
+ * is preserved.
+ * ====================================================
+ */
+
+/* Modifications for long double contributed by
+   Stephen L. Moshier <moshier@na-net.ornl.gov>  */
+
+/*
+ * __ieee754_jn(n, x), __ieee754_yn(n, x)
+ * floating point Bessel's function of the 1st and 2nd kind
+ * of order n
+ *
+ * Special cases:
+ *	y0(0)=y1(0)=yn(n,0) = -inf with division by zero signal;
+ *	y0(-ve)=y1(-ve)=yn(n,-ve) are NaN with invalid signal.
+ * Note 2. About jn(n,x), yn(n,x)
+ *	For n=0, j0(x) is called,
+ *	for n=1, j1(x) is called,
+ *	for n<x, forward recursion us used starting
+ *	from values of j0(x) and j1(x).
+ *	for n>x, a continued fraction approximation to
+ *	j(n,x)/j(n-1,x) is evaluated and then backward
+ *	recursion is used starting from a supposed value
+ *	for j(n,x). The resulting value of j(0,x) is
+ *	compared with the actual value to correct the
+ *	supposed value of j(n,x).
+ *
+ *	yn(n,x) is similar in all respects, except
+ *	that forward recursion is used for all
+ *	values of n>1.
+ *
+ */
+
+#include "math.h"
+#include "math_private.h"
+
+#ifdef __STDC__
+static const long double
+#else
+static long double
+#endif
+  invsqrtpi = 5.64189583547756286948079e-1L, two = 2.0e0L, one = 1.0e0L;
+
+#ifdef __STDC__
+static const long double zero = 0.0L;
+#else
+static long double zero = 0.0L;
+#endif
+
+#ifdef __STDC__
+long double
+__ieee754_jnl (int n, long double x)
+#else
+long double
+__ieee754_jnl (n, x)
+     int n;
+     long double x;
+#endif
+{
+  u_int32_t se, i0, i1;
+  int32_t i, ix, sgn;
+  long double a, b, temp, di;
+  long double z, w;
+
+  /* J(-n,x) = (-1)^n * J(n, x), J(n, -x) = (-1)^n * J(n, x)
+   * Thus, J(-n,x) = J(n,-x)
+   */
+
+  GET_LDOUBLE_WORDS (se, i0, i1, x);
+  ix = se & 0x7fff;
+
+  /* if J(n,NaN) is NaN */
+  if ((ix == 0x7fff) && ((i0 & 0x7fffffff) != 0))
+    return x + x;
+  if (n < 0)
+    {
+      n = -n;
+      x = -x;
+      se ^= 0x8000;
+    }
+  if (n == 0)
+    return (__ieee754_j0l (x));
+  if (n == 1)
+    return (__ieee754_j1l (x));
+  sgn = (n & 1) & (se >> 15);	/* even n -- 0, odd n -- sign(x) */
+  x = fabsl (x);
+  if ((ix | i0 | i1) == 0 || ix >= 0x7fff)	/* if x is 0 or inf */
+    b = zero;
+  else if ((long double) n <= x)
+    {
+      /* Safe to use J(n+1,x)=2n/x *J(n,x)-J(n-1,x) */
+      if (ix >= 0x412D)
+	{			/* x > 2**302 */
+
+	  /* ??? This might be a futile gesture.
+	     If x exceeds X_TLOSS anyway, the wrapper function
+	     will set the result to zero. */
+
+	  /* (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),
+	   *          xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then
+	   *
+	   *             n    sin(xn)*sqt2    cos(xn)*sqt2
+	   *          ----------------------------------
+	   *             0     s-c             c+s
+	   *             1    -s-c            -c+s
+	   *             2    -s+c            -c-s
+	   *             3     s+c             c-s
+	   */
+	  long double s;
+	  long double c;
+	  __sincosl (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_sqrtl (x);
+	}
+      else
+	{
+	  a = __ieee754_j0l (x);
+	  b = __ieee754_j1l (x);
+	  for (i = 1; i < n; i++)
+	    {
+	      temp = b;
+	      b = b * ((long double) (i + i) / x) - a;	/* avoid underflow */
+	      a = temp;
+	    }
+	}
+    }
+  else
+    {
+      if (ix < 0x3fde)
+	{			/* x < 2**-33 */
+	  /* x is tiny, return the first Taylor expansion of J(n,x)
+	   * J(n,x) = 1/n!*(x/2)^n  - ...
+	   */
+	  if (n >= 400)		/* underflow, result < 10^-4952 */
+	    b = zero;
+	  else
+	    {
+	      temp = x * 0.5;
+	      b = temp;
+	      for (a = one, i = 2; i <= n; i++)
+		{
+		  a *= (long 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 */
+	  long double t, v;
+	  long double q0, q1, h, tmp;
+	  int32_t k, m;
+	  w = (n + n) / (long double) x;
+	  h = 2.0L / (long double) x;
+	  q0 = w;
+	  z = w + h;
+	  q1 = w * z - 1.0L;
+	  k = 1;
+	  while (q1 < 1.0e11L)
+	    {
+	      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_logl (fabsl (v * tmp));
+
+	  if (tmp < 1.1356523406294143949491931077970765006170e+04L)
+	    {
+	      for (i = n - 1, di = (long double) (i + i); i > 0; i--)
+		{
+		  temp = b;
+		  b *= di;
+		  b = b / x - a;
+		  a = temp;
+		  di -= two;
+		}
+	    }
+	  else
+	    {
+	      for (i = n - 1, di = (long 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 > 1e100L)
+		    {
+		      a /= b;
+		      t /= b;
+		      b = one;
+		    }
+		}
+	    }
+	  b = (t * __ieee754_j0l (x) / b);
+	}
+    }
+  if (sgn == 1)
+    return -b;
+  else
+    return b;
+}
+
+#ifdef __STDC__
+long double
+__ieee754_ynl (int n, long double x)
+#else
+long double
+__ieee754_ynl (n, x)
+     int n;
+     long double x;
+#endif
+{
+  u_int32_t se, i0, i1;
+  int32_t i, ix;
+  int32_t sign;
+  long double a, b, temp;
+
+
+  GET_LDOUBLE_WORDS (se, i0, i1, x);
+  ix = se & 0x7fff;
+  /* if Y(n,NaN) is NaN */
+  if ((ix == 0x7fff) && ((i0 & 0x7fffffff) != 0))
+    return x + x;
+  if ((ix | i0 | i1) == 0)
+    return -one / zero;
+  if (se & 0x8000)
+    return zero / zero;
+  sign = 1;
+  if (n < 0)
+    {
+      n = -n;
+      sign = 1 - ((n & 1) << 1);
+    }
+  if (n == 0)
+    return (__ieee754_y0l (x));
+  if (n == 1)
+    return (sign * __ieee754_y1l (x));
+  if (ix == 0x7fff)
+    return zero;
+  if (ix >= 0x412D)
+    {				/* x > 2**302 */
+
+      /* ??? See comment above on the possible futility of this.  */
+
+      /* (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),
+       *          xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then
+       *
+       *             n    sin(xn)*sqt2    cos(xn)*sqt2
+       *          ----------------------------------
+       *             0     s-c             c+s
+       *             1    -s-c            -c+s
+       *             2    -s+c            -c-s
+       *             3     s+c             c-s
+       */
+      long double s;
+      long double c;
+      __sincosl (x, &s, &c);
+      switch (n & 3)
+	{
+	case 0:
+	  temp = s - c;
+	  break;
+	case 1:
+	  temp = -s - c;
+	  break;
+	case 2:
+	  temp = -s + c;
+	  break;
+	case 3:
+	  temp = s + c;
+	  break;
+	}
+      b = invsqrtpi * temp / __ieee754_sqrtl (x);
+    }
+  else
+    {
+      a = __ieee754_y0l (x);
+      b = __ieee754_y1l (x);
+      /* quit if b is -inf */
+      GET_LDOUBLE_WORDS (se, i0, i1, b);
+      for (i = 1; i < n && se != 0xffff; i++)
+	{
+	  temp = b;
+	  b = ((long double) (i + i) / x) * b - a;
+	  GET_LDOUBLE_WORDS (se, i0, i1, b);
+	  a = temp;
+	}
+    }
+  if (sign > 0)
+    return b;
+  else
+    return -b;
+}