Fix spurious underflow exceptions for Bessel functions for double(from glibc bug 14155)

	This fix comes from glibc, from files which originated from
	the same place as the newlib files. Those files in glibc carry
	the same license as the newlib files.

Bug 14155 is spurious underflow exceptions from Bessel functions for
large arguments.  (The correct results for large x are roughly
constant * sin or cos (x + constant) / sqrt (x), so no underflow
exceptions should occur based on the final result.)

There are various places underflows may occur in the intermediate
calculations that cause the failures listed in that bug.  This patch
fixes problems for the double version where underflows occur in
calculating the intermediate functions P and Q (in particular, x**-12
gets computed while calculating Q).  Appropriate approximations are
used for P and Q for arguments at least 0x1p28 and above to avoid the
underflows.

For sufficiently large x - 0x1p129 and above - the code already has a
cut-off to avoid calculating P and Q at all, which means the
approximations -0.125 / x and 0.375 / x can't themselves cause
underflows calculating Q.  This cut-off is heuristically reasonable
for the point beyond which Q can be neglected (based on expecting
around 0x1p-64 to be the least absolute value of sin or cos for large
arguments representable in double).

The float versions use a cut-off 0x1p17, which is less heuristically
justifiable but should still only affect values near zeroes of the
Bessel functions where these implementations are intrinsically
inaccurate anyway (bugs 14469-14472), and should serve to avoid
underflows (the float underflow for jn in bug 14155 probably comes
from the recurrence to compute jn).  ldbl-96 uses 0x1p129, which may
not really be enough heuristically (0x1p143 or so might be safer - 143
= 64 + 79, number of mantissa bits plus total number of significant
bits in representation) but again should avoid underflows and only
affect values where the code is substantially inaccurate anyway.
ldbl-128 and ldbl-128ibm share a completely different implementation
with no such cut-off, which I propose to fix separately.

Signed-off-by: Keith Packard <keithp@keithp.com>

4 files changed, 13 insertions, 9 deletions

diff --git a/newlib/libm/math/e_j0.c b/newlib/libm/math/e_j0.c
index 13773cb..d3af9d3 100644
--- a/newlib/libm/math/e_j0.c
+++ b/newlib/libm/math/e_j0.c
@@ -338,7 +338,8 @@ static double pS2[5] = {
 	__int32_t ix;
 	GET_HIGH_WORD(ix,x);
 	ix &= 0x7fffffff;
-	if(ix>=0x40200000)     {p = pR8; q= pS8;}
+	if (ix>=0x41b00000)    {return one;}
+	else if(ix>=0x40200000){p = pR8; q= pS8;}
 	else if(ix>=0x40122E8B){p = pR5; q= pS5;}
 	else if(ix>=0x4006DB6D){p = pR3; q= pS3;}
       else {p = pR2; q= pS2;}
@@ -474,7 +475,8 @@ static double qS2[6] = {
 	__int32_t ix;
 	GET_HIGH_WORD(ix,x);
 	ix &= 0x7fffffff;
-	if(ix>=0x40200000)     {p = qR8; q= qS8;}
+	if (ix>=0x41b00000)    {return -.125/x;}
+	else if(ix>=0x40200000){p = qR8; q= qS8;}
 	else if(ix>=0x40122E8B){p = qR5; q= qS5;}
 	else if(ix>=0x4006DB6D){p = qR3; q= qS3;}
       else {p = qR2; q= qS2;}
diff --git a/newlib/libm/math/e_j1.c b/newlib/libm/math/e_j1.c
index 098eb56..72855e3 100644
--- a/newlib/libm/math/e_j1.c
+++ b/newlib/libm/math/e_j1.c
@@ -336,7 +336,8 @@ static double ps2[5] = {
         __int32_t ix;
 	GET_HIGH_WORD(ix,x);
 	ix &= 0x7fffffff;
-        if(ix>=0x40200000)     {p = pr8; q= ps8;}
+	if (ix>=0x41b00000)    {return one;}
+	else if(ix>=0x40200000){p = pr8; q= ps8;}
         else if(ix>=0x40122E8B){p = pr5; q= ps5;}
         else if(ix>=0x4006DB6D){p = pr3; q= ps3;}
         else {p = pr2; q= ps2;}
@@ -473,7 +474,8 @@ static double qs2[6] = {
 	__int32_t ix;
 	GET_HIGH_WORD(ix,x);
 	ix &= 0x7fffffff;
-	if(ix>=0x40200000)     {p = qr8; q= qs8;}
+	if (ix>=0x41b00000)    {return .375/x;}
+	else if(ix>=0x40200000){p = qr8; q= qs8;}
 	else if(ix>=0x40122E8B){p = qr5; q= qs5;}
 	else if(ix>=0x4006DB6D){p = qr3; q= qs3;}
       else {p = qr2; q= qs2;}
diff --git a/newlib/libm/math/ef_j0.c b/newlib/libm/math/ef_j0.c
index 866cfcf..854801f 100644
--- a/newlib/libm/math/ef_j0.c
+++ b/newlib/libm/math/ef_j0.c
@@ -74,7 +74,7 @@ static float zero = 0.0;
 	 * j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x)
 	 * y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x)
 	 */
-		if(ix>0x80000000) z = (invsqrtpi*cc)/__ieee754_sqrtf(x);
+		if(ix>0x5c000000) z = (invsqrtpi*cc)/__ieee754_sqrtf(x);
 		else {
 		    u = pzerof(x); v = qzerof(x);
 		    z = invsqrtpi*(u*cc-v*ss)/__ieee754_sqrtf(x);
@@ -156,14 +156,14 @@ v04  =  4.4111031494e-10; /* 0x2ff280c2 */
                     if ((s*c)<zero) cc = z/ss;
                     else            ss = z/cc;
                 }
-                if(ix>0x80000000) z = (invsqrtpi*ss)/__ieee754_sqrtf(x);
+                if(ix>0x5c000000) z = (invsqrtpi*ss)/__ieee754_sqrtf(x);
                 else {
                     u = pzerof(x); v = qzerof(x);
                     z = invsqrtpi*(u*ss+v*cc)/__ieee754_sqrtf(x);
                 }
                 return z;
 	}
-	if(ix<=0x32000000) {	/* x < 2**-27 */
+	if(ix<=0x39800000) {	/* x < 2**-27 */
 	    return(u00 + tpi*__ieee754_logf(x));
 	}
 	z = x*x;
diff --git a/newlib/libm/math/ef_j1.c b/newlib/libm/math/ef_j1.c
index 01bd24c..f4c9c9d 100644
--- a/newlib/libm/math/ef_j1.c
+++ b/newlib/libm/math/ef_j1.c
@@ -75,7 +75,7 @@ static float zero    = 0.0;
 	 * j1(x) = 1/sqrt(pi) * (P(1,x)*cc - Q(1,x)*ss) / sqrt(x)
 	 * y1(x) = 1/sqrt(pi) * (P(1,x)*ss + Q(1,x)*cc) / sqrt(x)
 	 */
-		if(ix>0x80000000) z = (invsqrtpi*cc)/__ieee754_sqrtf(y);
+		if(ix>0x5c000000) z = (invsqrtpi*cc)/__ieee754_sqrtf(y);
 		else {
 		    u = ponef(y); v = qonef(y);
 		    z = invsqrtpi*(u*cc-v*ss)/__ieee754_sqrtf(y);
@@ -153,7 +153,7 @@ static float V0[5] = {
          *              sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
          * to compute the worse one.
          */
-                if(ix>0x48000000) z = (invsqrtpi*ss)/__ieee754_sqrtf(x);
+                if(ix>0x5c000000) z = (invsqrtpi*ss)/__ieee754_sqrtf(x);
                 else {
                     u = ponef(x); v = qonef(x);
                     z = invsqrtpi*(u*ss+v*cc)/__ieee754_sqrtf(x);