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newlib-cygwin/newlib/libm/math/ef_j0.c
Joseph S. Myers 5e24839658 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>
2020-03-26 12:21:33 +01:00

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/* ef_j0.c -- float version of e_j0.c.
* Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com.
*/
/*
* ====================================================
* 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.
* ====================================================
*/
#include "fdlibm.h"
#ifdef __STDC__
static float pzerof(float), qzerof(float);
#else
static float pzerof(), qzerof();
#endif
#ifdef __STDC__
static const float
#else
static float
#endif
huge = 1e30,
one = 1.0,
invsqrtpi= 5.6418961287e-01, /* 0x3f106ebb */
tpi = 6.3661974669e-01, /* 0x3f22f983 */
/* R0/S0 on [0, 2.00] */
R02 = 1.5625000000e-02, /* 0x3c800000 */
R03 = -1.8997929874e-04, /* 0xb947352e */
R04 = 1.8295404516e-06, /* 0x35f58e88 */
R05 = -4.6183270541e-09, /* 0xb19eaf3c */
S01 = 1.5619102865e-02, /* 0x3c7fe744 */
S02 = 1.1692678527e-04, /* 0x38f53697 */
S03 = 5.1354652442e-07, /* 0x3509daa6 */
S04 = 1.1661400734e-09; /* 0x30a045e8 */
#ifdef __STDC__
static const float zero = 0.0;
#else
static float zero = 0.0;
#endif
#ifdef __STDC__
float __ieee754_j0f(float x)
#else
float __ieee754_j0f(x)
float x;
#endif
{
float z, s,c,ss,cc,r,u,v;
__int32_t hx,ix;
GET_FLOAT_WORD(hx,x);
ix = hx&0x7fffffff;
if(!FLT_UWORD_IS_FINITE(ix)) return one/(x*x);
x = fabsf(x);
if(ix >= 0x40000000) { /* |x| >= 2.0 */
s = sinf(x);
c = cosf(x);
ss = s-c;
cc = s+c;
if(ix<=FLT_UWORD_HALF_MAX) { /* make sure x+x not overflow */
z = -cosf(x+x);
if ((s*c)<zero) cc = z/ss;
else ss = z/cc;
}
/*
* 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>0x5c000000) z = (invsqrtpi*cc)/__ieee754_sqrtf(x);
else {
u = pzerof(x); v = qzerof(x);
z = invsqrtpi*(u*cc-v*ss)/__ieee754_sqrtf(x);
}
return z;
}
if(ix<0x39000000) { /* |x| < 2**-13 */
if(huge+x>one) { /* raise inexact if x != 0 */
if(ix<0x32000000) return one; /* |x|<2**-27 */
else return one - (float)0.25*x*x;
}
}
z = x*x;
r = z*(R02+z*(R03+z*(R04+z*R05)));
s = one+z*(S01+z*(S02+z*(S03+z*S04)));
if(ix < 0x3F800000) { /* |x| < 1.00 */
return one + z*((float)-0.25+(r/s));
} else {
u = (float)0.5*x;
return((one+u)*(one-u)+z*(r/s));
}
}
#ifdef __STDC__
static const float
#else
static float
#endif
u00 = -7.3804296553e-02, /* 0xbd9726b5 */
u01 = 1.7666645348e-01, /* 0x3e34e80d */
u02 = -1.3818567619e-02, /* 0xbc626746 */
u03 = 3.4745343146e-04, /* 0x39b62a69 */
u04 = -3.8140706238e-06, /* 0xb67ff53c */
u05 = 1.9559013964e-08, /* 0x32a802ba */
u06 = -3.9820518410e-11, /* 0xae2f21eb */
v01 = 1.2730483897e-02, /* 0x3c509385 */
v02 = 7.6006865129e-05, /* 0x389f65e0 */
v03 = 2.5915085189e-07, /* 0x348b216c */
v04 = 4.4111031494e-10; /* 0x2ff280c2 */
#ifdef __STDC__
float __ieee754_y0f(float x)
#else
float __ieee754_y0f(x)
float x;
#endif
{
float z, s,c,ss,cc,u,v;
__int32_t hx,ix;
GET_FLOAT_WORD(hx,x);
ix = 0x7fffffff&hx;
/* Y0(NaN) is NaN, y0(-inf) is Nan, y0(inf) is 0 */
if(!FLT_UWORD_IS_FINITE(ix)) return one/(x+x*x);
if(FLT_UWORD_IS_ZERO(ix)) return -one/zero;
if(hx<0) return zero/zero;
if(ix >= 0x40000000) { /* |x| >= 2.0 */
/* y0(x) = sqrt(2/(pi*x))*(p0(x)*sin(x0)+q0(x)*cos(x0))
* where x0 = x-pi/4
* Better formula:
* cos(x0) = cos(x)cos(pi/4)+sin(x)sin(pi/4)
* = 1/sqrt(2) * (sin(x) + cos(x))
* sin(x0) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4)
* = 1/sqrt(2) * (sin(x) - cos(x))
* To avoid cancellation, use
* sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
* to compute the worse one.
*/
s = sinf(x);
c = cosf(x);
ss = s-c;
cc = s+c;
/*
* 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<=FLT_UWORD_HALF_MAX) { /* make sure x+x not overflow */
z = -cosf(x+x);
if ((s*c)<zero) cc = z/ss;
else ss = z/cc;
}
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<=0x39800000) { /* x < 2**-27 */
return(u00 + tpi*__ieee754_logf(x));
}
z = x*x;
u = u00+z*(u01+z*(u02+z*(u03+z*(u04+z*(u05+z*u06)))));
v = one+z*(v01+z*(v02+z*(v03+z*v04)));
return(u/v + tpi*(__ieee754_j0f(x)*__ieee754_logf(x)));
}
/* The asymptotic expansions of pzero is
* 1 - 9/128 s^2 + 11025/98304 s^4 - ..., where s = 1/x.
* For x >= 2, We approximate pzero by
* pzero(x) = 1 + (R/S)
* where R = pR0 + pR1*s^2 + pR2*s^4 + ... + pR5*s^10
* S = 1 + pS0*s^2 + ... + pS4*s^10
* and
* | pzero(x)-1-R/S | <= 2 ** ( -60.26)
*/
#ifdef __STDC__
static const float pR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
#else
static float pR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
#endif
0.0000000000e+00, /* 0x00000000 */
-7.0312500000e-02, /* 0xbd900000 */
-8.0816707611e+00, /* 0xc1014e86 */
-2.5706311035e+02, /* 0xc3808814 */
-2.4852163086e+03, /* 0xc51b5376 */
-5.2530439453e+03, /* 0xc5a4285a */
};
#ifdef __STDC__
static const float pS8[5] = {
#else
static float pS8[5] = {
#endif
1.1653436279e+02, /* 0x42e91198 */
3.8337448730e+03, /* 0x456f9beb */
4.0597855469e+04, /* 0x471e95db */
1.1675296875e+05, /* 0x47e4087c */
4.7627726562e+04, /* 0x473a0bba */
};
#ifdef __STDC__
static const float pR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
#else
static float pR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
#endif
-1.1412546255e-11, /* 0xad48c58a */
-7.0312492549e-02, /* 0xbd8fffff */
-4.1596107483e+00, /* 0xc0851b88 */
-6.7674766541e+01, /* 0xc287597b */
-3.3123129272e+02, /* 0xc3a59d9b */
-3.4643338013e+02, /* 0xc3ad3779 */
};
#ifdef __STDC__
static const float pS5[5] = {
#else
static float pS5[5] = {
#endif
6.0753936768e+01, /* 0x42730408 */
1.0512523193e+03, /* 0x44836813 */
5.9789707031e+03, /* 0x45bad7c4 */
9.6254453125e+03, /* 0x461665c8 */
2.4060581055e+03, /* 0x451660ee */
};
#ifdef __STDC__
static const float pR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
#else
static float pR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
#endif
-2.5470459075e-09, /* 0xb12f081b */
-7.0311963558e-02, /* 0xbd8fffb8 */
-2.4090321064e+00, /* 0xc01a2d95 */
-2.1965976715e+01, /* 0xc1afba52 */
-5.8079170227e+01, /* 0xc2685112 */
-3.1447946548e+01, /* 0xc1fb9565 */
};
#ifdef __STDC__
static const float pS3[5] = {
#else
static float pS3[5] = {
#endif
3.5856033325e+01, /* 0x420f6c94 */
3.6151397705e+02, /* 0x43b4c1ca */
1.1936077881e+03, /* 0x44953373 */
1.1279968262e+03, /* 0x448cffe6 */
1.7358093262e+02, /* 0x432d94b8 */
};
#ifdef __STDC__
static const float pR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
#else
static float pR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
#endif
-8.8753431271e-08, /* 0xb3be98b7 */
-7.0303097367e-02, /* 0xbd8ffb12 */
-1.4507384300e+00, /* 0xbfb9b1cc */
-7.6356959343e+00, /* 0xc0f4579f */
-1.1193166733e+01, /* 0xc1331736 */
-3.2336456776e+00, /* 0xc04ef40d */
};
#ifdef __STDC__
static const float pS2[5] = {
#else
static float pS2[5] = {
#endif
2.2220300674e+01, /* 0x41b1c32d */
1.3620678711e+02, /* 0x430834f0 */
2.7047027588e+02, /* 0x43873c32 */
1.5387539673e+02, /* 0x4319e01a */
1.4657617569e+01, /* 0x416a859a */
};
#ifdef __STDC__
static float pzerof(float x)
#else
static float pzerof(x)
float x;
#endif
{
#ifdef __STDC__
const float *p,*q;
#else
float *p,*q;
#endif
float z,r,s;
__int32_t ix;
GET_FLOAT_WORD(ix,x);
ix &= 0x7fffffff;
if(ix>=0x41000000) {p = pR8; q= pS8;}
else if(ix>=0x40f71c58){p = pR5; q= pS5;}
else if(ix>=0x4036db68){p = pR3; q= pS3;}
else {p = pR2; q= pS2;}
z = one/(x*x);
r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*q[4]))));
return one+ r/s;
}
/* For x >= 8, the asymptotic expansions of qzero is
* -1/8 s + 75/1024 s^3 - ..., where s = 1/x.
* We approximate qzero by
* qzero(x) = s*(-1.25 + (R/S))
* where R = qR0 + qR1*s^2 + qR2*s^4 + ... + qR5*s^10
* S = 1 + qS0*s^2 + ... + qS5*s^12
* and
* | qzero(x)/s +1.25-R/S | <= 2 ** ( -61.22)
*/
#ifdef __STDC__
static const float qR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
#else
static float qR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
#endif
0.0000000000e+00, /* 0x00000000 */
7.3242187500e-02, /* 0x3d960000 */
1.1768206596e+01, /* 0x413c4a93 */
5.5767340088e+02, /* 0x440b6b19 */
8.8591972656e+03, /* 0x460a6cca */
3.7014625000e+04, /* 0x471096a0 */
};
#ifdef __STDC__
static const float qS8[6] = {
#else
static float qS8[6] = {
#endif
1.6377603149e+02, /* 0x4323c6aa */
8.0983447266e+03, /* 0x45fd12c2 */
1.4253829688e+05, /* 0x480b3293 */
8.0330925000e+05, /* 0x49441ed4 */
8.4050156250e+05, /* 0x494d3359 */
-3.4389928125e+05, /* 0xc8a7eb69 */
};
#ifdef __STDC__
static const float qR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
#else
static float qR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
#endif
1.8408595828e-11, /* 0x2da1ec79 */
7.3242180049e-02, /* 0x3d95ffff */
5.8356351852e+00, /* 0x40babd86 */
1.3511157227e+02, /* 0x43071c90 */
1.0272437744e+03, /* 0x448067cd */
1.9899779053e+03, /* 0x44f8bf4b */
};
#ifdef __STDC__
static const float qS5[6] = {
#else
static float qS5[6] = {
#endif
8.2776611328e+01, /* 0x42a58da0 */
2.0778142090e+03, /* 0x4501dd07 */
1.8847289062e+04, /* 0x46933e94 */
5.6751113281e+04, /* 0x475daf1d */
3.5976753906e+04, /* 0x470c88c1 */
-5.3543427734e+03, /* 0xc5a752be */
};
#ifdef __STDC__
static const float qR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
#else
static float qR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
#endif
4.3774099900e-09, /* 0x3196681b */
7.3241114616e-02, /* 0x3d95ff70 */
3.3442313671e+00, /* 0x405607e3 */
4.2621845245e+01, /* 0x422a7cc5 */
1.7080809021e+02, /* 0x432acedf */
1.6673394775e+02, /* 0x4326bbe4 */
};
#ifdef __STDC__
static const float qS3[6] = {
#else
static float qS3[6] = {
#endif
4.8758872986e+01, /* 0x42430916 */
7.0968920898e+02, /* 0x44316c1c */
3.7041481934e+03, /* 0x4567825f */
6.4604252930e+03, /* 0x45c9e367 */
2.5163337402e+03, /* 0x451d4557 */
-1.4924745178e+02, /* 0xc3153f59 */
};
#ifdef __STDC__
static const float qR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
#else
static float qR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
#endif
1.5044444979e-07, /* 0x342189db */
7.3223426938e-02, /* 0x3d95f62a */
1.9981917143e+00, /* 0x3fffc4bf */
1.4495602608e+01, /* 0x4167edfd */
3.1666231155e+01, /* 0x41fd5471 */
1.6252708435e+01, /* 0x4182058c */
};
#ifdef __STDC__
static const float qS2[6] = {
#else
static float qS2[6] = {
#endif
3.0365585327e+01, /* 0x41f2ecb8 */
2.6934811401e+02, /* 0x4386ac8f */
8.4478375244e+02, /* 0x44533229 */
8.8293585205e+02, /* 0x445cbbe5 */
2.1266638184e+02, /* 0x4354aa98 */
-5.3109550476e+00, /* 0xc0a9f358 */
};
#ifdef __STDC__
static float qzerof(float x)
#else
static float qzerof(x)
float x;
#endif
{
#ifdef __STDC__
const float *p,*q;
#else
float *p,*q;
#endif
float s,r,z;
__int32_t ix;
GET_FLOAT_WORD(ix,x);
ix &= 0x7fffffff;
if(ix>=0x41000000) {p = qR8; q= qS8;}
else if(ix>=0x40f71c58){p = qR5; q= qS5;}
else if(ix>=0x4036db68){p = qR3; q= qS3;}
else {p = qR2; q= qS2;}
z = one/(x*x);
r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*(q[4]+z*q[5])))));
return (-(float).125 + r/s)/x;
}
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