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be25493251
This patch fixes bugs 16561 and 16562, bad results of yn in overflow cases in non-default rounding modes, both because an intermediate overflow in the recurrence does not get detected if the result is not an infinity and because an overflowing result may occur in the wrong sign. The fix is to set FE_TONEAREST mode internally for the parts of the function where such overflows can occur (which includes the call to y1 - where yn is used to compute a Bessel function of order -1, negating the result of y1 isn't correct for overflowing results in directed rounding modes) and then compute an overflowing value in the original rounding mode if the to-nearest result was an infinity. Tested x86_64 and x86 and ulps updated accordingly. Also tested for mips64 and powerpc32 to test the ldbl-128 and ldbl-128ibm changes. (The tests for these bugs were added in my previous y1 patch, so the only thing this patch has to do with the testsuite is enable yn testing in all rounding modes.) [BZ #16561] [BZ #16562] * sysdeps/ieee754/dbl-64/e_jn.c: Include <float.h>. (__ieee754_yn): Set FE_TONEAREST mode internally and then recompute overflowing results in original rounding mode. * sysdeps/ieee754/flt-32/e_jnf.c: Include <float.h>. (__ieee754_ynf): Set FE_TONEAREST mode internally and then recompute overflowing results in original rounding mode. * sysdeps/ieee754/ldbl-128/e_jnl.c: Include <float.h>. (__ieee754_ynl): Set FE_TONEAREST mode internally and then recompute overflowing results in original rounding mode. * sysdeps/ieee754/ldbl-128ibm/e_jnl.c: Include <float.h>. (__ieee754_ynl): Set FE_TONEAREST mode internally and then recompute overflowing results in original rounding mode. * sysdeps/ieee754/ldbl-96/e_jnl.c: Include <float.h>. (__ieee754_ynl): Set FE_TONEAREST mode internally and then recompute overflowing results in original rounding mode. * sysdeps/i386/fpu/fenv_private.h [!__SSE2_MATH__] (libc_feholdsetround_ctx): New macro. * math/libm-test.inc (yn_test): Use ALL_RM_TEST. * sysdeps/i386/fpu/libm-test-ulps: Update. * sysdeps/x86_64/fpu/libm-test-ulps : Likewise.
394 lines
10 KiB
C
394 lines
10 KiB
C
/*
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* ====================================================
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* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
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*
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* Developed at SunPro, a Sun Microsystems, Inc. business.
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* Permission to use, copy, modify, and distribute this
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* software is freely granted, provided that this notice
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* is preserved.
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* ====================================================
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*/
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/* Modifications for long double are
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Copyright (C) 2001 Stephen L. Moshier <moshier@na-net.ornl.gov>
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and are incorporated herein by permission of the author. The author
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reserves the right to distribute this material elsewhere under different
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copying permissions. These modifications are distributed here under
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the following terms:
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This library is free software; you can redistribute it and/or
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modify it under the terms of the GNU Lesser General Public
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License as published by the Free Software Foundation; either
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version 2.1 of the License, or (at your option) any later version.
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This library is distributed in the hope that it will be useful,
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but WITHOUT ANY WARRANTY; without even the implied warranty of
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MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
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Lesser General Public License for more details.
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You should have received a copy of the GNU Lesser General Public
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License along with this library; if not, see
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<http://www.gnu.org/licenses/>. */
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/*
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* __ieee754_jn(n, x), __ieee754_yn(n, x)
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* floating point Bessel's function of the 1st and 2nd kind
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* of order n
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*
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* Special cases:
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* y0(0)=y1(0)=yn(n,0) = -inf with overflow signal;
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* y0(-ve)=y1(-ve)=yn(n,-ve) are NaN with invalid signal.
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* Note 2. About jn(n,x), yn(n,x)
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* For n=0, j0(x) is called,
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* for n=1, j1(x) is called,
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* for n<x, forward recursion us used starting
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* from values of j0(x) and j1(x).
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* for n>x, a continued fraction approximation to
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* j(n,x)/j(n-1,x) is evaluated and then backward
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* recursion is used starting from a supposed value
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* for j(n,x). The resulting value of j(0,x) is
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* compared with the actual value to correct the
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* supposed value of j(n,x).
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*
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* yn(n,x) is similar in all respects, except
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* that forward recursion is used for all
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* values of n>1.
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*
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*/
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#include <errno.h>
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#include <float.h>
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#include <math.h>
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#include <math_private.h>
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static const long double
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invsqrtpi = 5.64189583547756286948079e-1L, two = 2.0e0L, one = 1.0e0L;
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static const long double zero = 0.0L;
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long double
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__ieee754_jnl (int n, long double x)
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{
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u_int32_t se, i0, i1;
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int32_t i, ix, sgn;
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long double a, b, temp, di;
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long double z, w;
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/* J(-n,x) = (-1)^n * J(n, x), J(n, -x) = (-1)^n * J(n, x)
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* Thus, J(-n,x) = J(n,-x)
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*/
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GET_LDOUBLE_WORDS (se, i0, i1, x);
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ix = se & 0x7fff;
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/* if J(n,NaN) is NaN */
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if (__glibc_unlikely ((ix == 0x7fff) && ((i0 & 0x7fffffff) != 0)))
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return x + x;
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if (n < 0)
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{
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n = -n;
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x = -x;
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se ^= 0x8000;
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}
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if (n == 0)
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return (__ieee754_j0l (x));
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if (n == 1)
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return (__ieee754_j1l (x));
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sgn = (n & 1) & (se >> 15); /* even n -- 0, odd n -- sign(x) */
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x = fabsl (x);
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if (__glibc_unlikely ((ix | i0 | i1) == 0 || ix >= 0x7fff))
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/* if x is 0 or inf */
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b = zero;
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else if ((long double) n <= x)
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{
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/* Safe to use J(n+1,x)=2n/x *J(n,x)-J(n-1,x) */
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if (ix >= 0x412D)
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{ /* x > 2**302 */
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/* ??? This might be a futile gesture.
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If x exceeds X_TLOSS anyway, the wrapper function
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will set the result to zero. */
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/* (x >> n**2)
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* Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
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* Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
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* Let s=sin(x), c=cos(x),
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* xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then
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*
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* n sin(xn)*sqt2 cos(xn)*sqt2
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* ----------------------------------
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* 0 s-c c+s
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* 1 -s-c -c+s
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* 2 -s+c -c-s
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* 3 s+c c-s
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*/
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long double s;
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long double c;
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__sincosl (x, &s, &c);
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switch (n & 3)
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{
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case 0:
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temp = c + s;
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break;
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case 1:
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temp = -c + s;
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break;
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case 2:
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temp = -c - s;
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break;
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case 3:
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temp = c - s;
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break;
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}
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b = invsqrtpi * temp / __ieee754_sqrtl (x);
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}
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else
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{
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a = __ieee754_j0l (x);
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b = __ieee754_j1l (x);
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for (i = 1; i < n; i++)
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{
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temp = b;
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b = b * ((long double) (i + i) / x) - a; /* avoid underflow */
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a = temp;
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}
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}
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}
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else
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{
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if (ix < 0x3fde)
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{ /* x < 2**-33 */
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/* x is tiny, return the first Taylor expansion of J(n,x)
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* J(n,x) = 1/n!*(x/2)^n - ...
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*/
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if (n >= 400) /* underflow, result < 10^-4952 */
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b = zero;
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else
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{
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temp = x * 0.5;
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b = temp;
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for (a = one, i = 2; i <= n; i++)
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{
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a *= (long double) i; /* a = n! */
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b *= temp; /* b = (x/2)^n */
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}
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b = b / a;
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}
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}
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else
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{
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/* use backward recurrence */
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/* x x^2 x^2
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* J(n,x)/J(n-1,x) = ---- ------ ------ .....
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* 2n - 2(n+1) - 2(n+2)
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*
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* 1 1 1
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* (for large x) = ---- ------ ------ .....
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* 2n 2(n+1) 2(n+2)
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* -- - ------ - ------ -
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* x x x
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*
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* Let w = 2n/x and h=2/x, then the above quotient
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* is equal to the continued fraction:
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* 1
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* = -----------------------
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* 1
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* w - -----------------
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* 1
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* w+h - ---------
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* w+2h - ...
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*
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* To determine how many terms needed, let
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* Q(0) = w, Q(1) = w(w+h) - 1,
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* Q(k) = (w+k*h)*Q(k-1) - Q(k-2),
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* When Q(k) > 1e4 good for single
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* When Q(k) > 1e9 good for double
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* When Q(k) > 1e17 good for quadruple
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*/
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/* determine k */
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long double t, v;
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long double q0, q1, h, tmp;
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int32_t k, m;
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w = (n + n) / (long double) x;
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h = 2.0L / (long double) x;
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q0 = w;
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z = w + h;
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q1 = w * z - 1.0L;
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k = 1;
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while (q1 < 1.0e11L)
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{
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k += 1;
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z += h;
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tmp = z * q1 - q0;
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q0 = q1;
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q1 = tmp;
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}
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m = n + n;
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for (t = zero, i = 2 * (n + k); i >= m; i -= 2)
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t = one / (i / x - t);
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a = t;
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b = one;
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/* estimate log((2/x)^n*n!) = n*log(2/x)+n*ln(n)
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* Hence, if n*(log(2n/x)) > ...
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* single 8.8722839355e+01
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* double 7.09782712893383973096e+02
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* long double 1.1356523406294143949491931077970765006170e+04
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* then recurrent value may overflow and the result is
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* likely underflow to zero
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*/
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tmp = n;
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v = two / x;
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tmp = tmp * __ieee754_logl (fabsl (v * tmp));
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if (tmp < 1.1356523406294143949491931077970765006170e+04L)
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{
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for (i = n - 1, di = (long double) (i + i); i > 0; i--)
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{
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temp = b;
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b *= di;
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b = b / x - a;
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a = temp;
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di -= two;
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}
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}
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else
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{
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for (i = n - 1, di = (long double) (i + i); i > 0; i--)
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{
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temp = b;
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b *= di;
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b = b / x - a;
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a = temp;
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di -= two;
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/* scale b to avoid spurious overflow */
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if (b > 1e100L)
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{
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a /= b;
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t /= b;
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b = one;
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}
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}
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}
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/* j0() and j1() suffer enormous loss of precision at and
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* near zero; however, we know that their zero points never
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* coincide, so just choose the one further away from zero.
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*/
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z = __ieee754_j0l (x);
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w = __ieee754_j1l (x);
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if (fabsl (z) >= fabsl (w))
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b = (t * z / b);
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else
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b = (t * w / a);
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}
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}
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if (sgn == 1)
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return -b;
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else
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return b;
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}
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strong_alias (__ieee754_jnl, __jnl_finite)
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long double
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__ieee754_ynl (int n, long double x)
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{
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u_int32_t se, i0, i1;
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int32_t i, ix;
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int32_t sign;
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long double a, b, temp, ret;
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GET_LDOUBLE_WORDS (se, i0, i1, x);
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ix = se & 0x7fff;
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/* if Y(n,NaN) is NaN */
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if (__builtin_expect ((ix == 0x7fff) && ((i0 & 0x7fffffff) != 0), 0))
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return x + x;
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if (__builtin_expect ((ix | i0 | i1) == 0, 0))
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/* -inf or inf and divide-by-zero exception. */
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return ((n < 0 && (n & 1) != 0) ? 1.0L : -1.0L) / 0.0L;
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if (__builtin_expect (se & 0x8000, 0))
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return zero / (zero * x);
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sign = 1;
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if (n < 0)
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{
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n = -n;
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sign = 1 - ((n & 1) << 1);
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}
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if (n == 0)
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return (__ieee754_y0l (x));
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{
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SET_RESTORE_ROUNDL (FE_TONEAREST);
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if (n == 1)
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{
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ret = sign * __ieee754_y1l (x);
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goto out;
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}
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if (__glibc_unlikely (ix == 0x7fff))
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return zero;
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if (ix >= 0x412D)
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{ /* x > 2**302 */
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/* ??? See comment above on the possible futility of this. */
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/* (x >> n**2)
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* Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
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* Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
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* Let s=sin(x), c=cos(x),
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* xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then
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*
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* n sin(xn)*sqt2 cos(xn)*sqt2
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* ----------------------------------
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* 0 s-c c+s
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* 1 -s-c -c+s
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* 2 -s+c -c-s
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* 3 s+c c-s
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*/
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long double s;
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long double c;
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__sincosl (x, &s, &c);
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switch (n & 3)
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{
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case 0:
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temp = s - c;
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break;
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case 1:
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temp = -s - c;
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break;
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case 2:
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temp = -s + c;
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break;
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case 3:
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temp = s + c;
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break;
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}
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b = invsqrtpi * temp / __ieee754_sqrtl (x);
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}
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else
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{
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a = __ieee754_y0l (x);
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b = __ieee754_y1l (x);
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/* quit if b is -inf */
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GET_LDOUBLE_WORDS (se, i0, i1, b);
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/* Use 0xffffffff since GET_LDOUBLE_WORDS sign-extends SE. */
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for (i = 1; i < n && se != 0xffffffff; i++)
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{
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temp = b;
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b = ((long double) (i + i) / x) * b - a;
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GET_LDOUBLE_WORDS (se, i0, i1, b);
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a = temp;
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}
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}
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/* If B is +-Inf, set up errno accordingly. */
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if (! __finitel (b))
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__set_errno (ERANGE);
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if (sign > 0)
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ret = b;
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else
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ret = -b;
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}
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out:
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if (__isinfl (ret))
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ret = __copysignl (LDBL_MAX, ret) * LDBL_MAX;
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return ret;
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}
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strong_alias (__ieee754_ynl, __ynl_finite)
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