Fix the inaccuracy of j0f/j1f/y0f/y1f [BZ #14469, #14470, #14471, #14472]
For j0f/j1f/y0f/y1f, the largest error for all binary32
inputs is reduced to at most 9 ulps for all rounding modes.
The new code is enabled only when there is a cancellation at the very end of
the j0f/j1f/y0f/y1f computation, or for very large inputs, thus should not
give any visible slowdown on average. Two different algorithms are used:
* around the first 64 zeros of j0/j1/y0/y1, approximation polynomials of
degree 3 are used, computed using the Sollya tool (https://www.sollya.org/)
* for large inputs, an asymptotic formula from [1] is used
[1] Fast and Accurate Bessel Function Computation,
John Harrison, Proceedings of Arith 19, 2009.
Inputs yielding the new largest errors are added to auto-libm-test-in,
and ulps are regenerated for various targets (thanks Adhemerval Zanella).
Tested on x86_64 with --disable-multi-arch and on powerpc64le-linux-gnu.
Reviewed-by: Adhemerval Zanella <adhemerval.zanella@linaro.org>
2021-04-01 06:14:10 +00:00
|
|
|
/* Auxiliary routine for the Bessel functions (j0f, y0f, j1f, y1f).
|
2023-01-06 21:08:04 +00:00
|
|
|
Copyright (C) 2021-2023 Free Software Foundation, Inc.
|
Fix the inaccuracy of j0f/j1f/y0f/y1f [BZ #14469, #14470, #14471, #14472]
For j0f/j1f/y0f/y1f, the largest error for all binary32
inputs is reduced to at most 9 ulps for all rounding modes.
The new code is enabled only when there is a cancellation at the very end of
the j0f/j1f/y0f/y1f computation, or for very large inputs, thus should not
give any visible slowdown on average. Two different algorithms are used:
* around the first 64 zeros of j0/j1/y0/y1, approximation polynomials of
degree 3 are used, computed using the Sollya tool (https://www.sollya.org/)
* for large inputs, an asymptotic formula from [1] is used
[1] Fast and Accurate Bessel Function Computation,
John Harrison, Proceedings of Arith 19, 2009.
Inputs yielding the new largest errors are added to auto-libm-test-in,
and ulps are regenerated for various targets (thanks Adhemerval Zanella).
Tested on x86_64 with --disable-multi-arch and on powerpc64le-linux-gnu.
Reviewed-by: Adhemerval Zanella <adhemerval.zanella@linaro.org>
2021-04-01 06:14:10 +00:00
|
|
|
This file is part of the GNU C Library.
|
|
|
|
|
|
|
|
|
|
The GNU C 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.
|
|
|
|
|
|
|
|
|
|
The GNU C 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 the GNU C Library; if not, see
|
|
|
|
|
<https://www.gnu.org/licenses/>. */
|
|
|
|
|
|
|
|
|
|
#ifndef _MATH_REDUCE_AUX_H
|
|
|
|
|
#define _MATH_REDUCE_AUX_H
|
|
|
|
|
|
|
|
|
|
#include <math.h>
|
|
|
|
|
#include <math_private.h>
|
|
|
|
|
#include <s_sincosf.h>
|
|
|
|
|
|
|
|
|
|
/* Return h and update n such that:
|
|
|
|
|
Now x - pi/4 - alpha = h + n*pi/2 mod (2*pi). */
|
|
|
|
|
static inline double
|
|
|
|
|
reduce_aux (float x, int *n, double alpha)
|
|
|
|
|
{
|
|
|
|
|
double h;
|
|
|
|
|
h = reduce_large (asuint (x), n);
|
|
|
|
|
/* Now |x| = h+n*pi/2 mod 2*pi. */
|
|
|
|
|
/* Recover sign. */
|
|
|
|
|
if (x < 0)
|
|
|
|
|
{
|
|
|
|
|
h = -h;
|
|
|
|
|
*n = -*n;
|
|
|
|
|
}
|
|
|
|
|
/* Subtract pi/4. */
|
|
|
|
|
double piover2 = 0xc.90fdaa22168cp-3;
|
|
|
|
|
if (h >= 0)
|
|
|
|
|
h -= piover2 / 2;
|
|
|
|
|
else
|
|
|
|
|
{
|
|
|
|
|
h += piover2 / 2;
|
|
|
|
|
(*n) --;
|
|
|
|
|
}
|
|
|
|
|
/* Subtract alpha and reduce if needed mod pi/2. */
|
|
|
|
|
h -= alpha;
|
|
|
|
|
if (h > piover2)
|
|
|
|
|
{
|
|
|
|
|
h -= piover2;
|
|
|
|
|
(*n) ++;
|
|
|
|
|
}
|
|
|
|
|
else if (h < -piover2)
|
|
|
|
|
{
|
|
|
|
|
h += piover2;
|
|
|
|
|
(*n) --;
|
|
|
|
|
}
|
|
|
|
|
return h;
|
|
|
|
|
}
|
|
|
|
|
|
|
|
|
|
#endif
|
