mirror of
https://sourceware.org/git/glibc.git
synced 2024-11-23 13:30:06 +00:00
a8e2112ae3
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.
417 lines
10 KiB
C
417 lines
10 KiB
C
/*
|
|
* ====================================================
|
|
* 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 128-bit long double are
|
|
Copyright (C) 2001 Stephen L. Moshier <moshier@na-net.ornl.gov>
|
|
and are incorporated herein by permission of the author. The author
|
|
reserves the right to distribute this material elsewhere under different
|
|
copying permissions. These modifications are distributed here under
|
|
the following terms:
|
|
|
|
This library is free software; you can redistribute it and/or
|
|
modify it under the terms of the GNU Lesser General Public
|
|
License as published by the Free Software Foundation; either
|
|
version 2.1 of the License, or (at your option) any later version.
|
|
|
|
This library is distributed in the hope that it will be useful,
|
|
but WITHOUT ANY WARRANTY; without even the implied warranty of
|
|
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
|
|
Lesser General Public License for more details.
|
|
|
|
You should have received a copy of the GNU Lesser General Public
|
|
License along with this library; if not, see
|
|
<http://www.gnu.org/licenses/>. */
|
|
|
|
/*
|
|
* __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 <errno.h>
|
|
#include <float.h>
|
|
#include <math.h>
|
|
#include <math_private.h>
|
|
|
|
static const long double
|
|
invsqrtpi = 5.6418958354775628694807945156077258584405E-1L,
|
|
two = 2.0e0L,
|
|
one = 1.0e0L,
|
|
zero = 0.0L;
|
|
|
|
|
|
long double
|
|
__ieee754_jnl (int n, long double x)
|
|
{
|
|
uint32_t se, lx;
|
|
int32_t i, ix, sgn;
|
|
long double a, b, temp, di, ret;
|
|
long double z, w;
|
|
double xhi;
|
|
|
|
|
|
/* J(-n,x) = (-1)^n * J(n, x), J(n, -x) = (-1)^n * J(n, x)
|
|
* Thus, J(-n,x) = J(n,-x)
|
|
*/
|
|
|
|
xhi = ldbl_high (x);
|
|
EXTRACT_WORDS (se, lx, xhi);
|
|
ix = se & 0x7fffffff;
|
|
|
|
/* if J(n,NaN) is NaN */
|
|
if (ix >= 0x7ff00000)
|
|
{
|
|
if (((ix - 0x7ff00000) | lx) != 0)
|
|
return x + x;
|
|
}
|
|
|
|
if (n < 0)
|
|
{
|
|
n = -n;
|
|
x = -x;
|
|
se ^= 0x80000000;
|
|
}
|
|
if (n == 0)
|
|
return (__ieee754_j0l (x));
|
|
if (n == 1)
|
|
return (__ieee754_j1l (x));
|
|
sgn = (n & 1) & (se >> 31); /* even n -- 0, odd n -- sign(x) */
|
|
x = fabsl (x);
|
|
|
|
{
|
|
SET_RESTORE_ROUNDL (FE_TONEAREST);
|
|
if (x == 0.0L || ix >= 0x7ff00000) /* if x is 0 or inf */
|
|
return sgn == 1 ? -zero : 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 >= 0x52d00000)
|
|
{ /* x > 2**302 */
|
|
|
|
/* ??? Could use an expansion for large x here. */
|
|
|
|
/* (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 < 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, result < 10^-300 */
|
|
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 < 1.0e17L)
|
|
{
|
|
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;
|
|
}
|
|
}
|
|
}
|
|
/* 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_j0l (x);
|
|
w = __ieee754_j1l (x);
|
|
if (fabsl (z) >= fabsl (w))
|
|
b = (t * z / b);
|
|
else
|
|
b = (t * w / a);
|
|
}
|
|
}
|
|
if (sgn == 1)
|
|
ret = -b;
|
|
else
|
|
ret = b;
|
|
}
|
|
if (ret == 0)
|
|
ret = __copysignl (LDBL_MIN, ret) * LDBL_MIN;
|
|
return ret;
|
|
}
|
|
strong_alias (__ieee754_jnl, __jnl_finite)
|
|
|
|
long double
|
|
__ieee754_ynl (int n, long double x)
|
|
{
|
|
uint32_t se, lx;
|
|
int32_t i, ix;
|
|
int32_t sign;
|
|
long double a, b, temp, ret;
|
|
double xhi;
|
|
|
|
xhi = ldbl_high (x);
|
|
EXTRACT_WORDS (se, lx, xhi);
|
|
ix = se & 0x7fffffff;
|
|
|
|
/* if Y(n,NaN) is NaN */
|
|
if (ix >= 0x7ff00000)
|
|
{
|
|
if (((ix - 0x7ff00000) | lx) != 0)
|
|
return x + x;
|
|
}
|
|
if (x <= 0.0L)
|
|
{
|
|
if (x == 0.0L)
|
|
return ((n < 0 && (n & 1) != 0) ? 1.0L : -1.0L) / 0.0L;
|
|
if (se & 0x80000000)
|
|
return zero / (zero * x);
|
|
}
|
|
sign = 1;
|
|
if (n < 0)
|
|
{
|
|
n = -n;
|
|
sign = 1 - ((n & 1) << 1);
|
|
}
|
|
if (n == 0)
|
|
return (__ieee754_y0l (x));
|
|
{
|
|
SET_RESTORE_ROUNDL (FE_TONEAREST);
|
|
if (n == 1)
|
|
{
|
|
ret = sign * __ieee754_y1l (x);
|
|
goto out;
|
|
}
|
|
if (ix >= 0x7ff00000)
|
|
return zero;
|
|
if (ix >= 0x52D00000)
|
|
{ /* 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 */
|
|
xhi = ldbl_high (b);
|
|
GET_HIGH_WORD (se, xhi);
|
|
se &= 0xfff00000;
|
|
for (i = 1; i < n && se != 0xfff00000; i++)
|
|
{
|
|
temp = b;
|
|
b = ((long double) (i + i) / x) * b - a;
|
|
xhi = ldbl_high (b);
|
|
GET_HIGH_WORD (se, xhi);
|
|
se &= 0xfff00000;
|
|
a = temp;
|
|
}
|
|
}
|
|
/* If B is +-Inf, set up errno accordingly. */
|
|
if (! isfinite (b))
|
|
__set_errno (ERANGE);
|
|
if (sign > 0)
|
|
ret = b;
|
|
else
|
|
ret = -b;
|
|
}
|
|
out:
|
|
if (isinf (ret))
|
|
ret = __copysignl (LDBL_MAX, ret) * LDBL_MAX;
|
|
return ret;
|
|
}
|
|
strong_alias (__ieee754_ynl, __ynl_finite)
|