ad180676b8
A recent discussion in bug 14469 notes that a threshold in float Bessel function implementations, used to determine when to use a simpler implementation approach, results in substantially inaccurate results. As I discussed in <https://sourceware.org/ml/libc-alpha/2013-03/msg00345.html>, a heuristic argument suggests 2^(S+P) as the right order of magnitude for a suitable threshold, where S is the number of significand bits in the floating-point type and P is the number of significant bits in the representation of the floating-point type, and the float and ldbl-96 implementations use thresholds that are too small. Some threshold does need using, there or elsewhere in the implementation, to avoid spurious underflow and overflow for large arguments. This patch sets the thresholds in the affected implementations to more heuristically justifiable values. Results will still be inaccurate close to zeroes of the functions (thus this patch does *not* fix any of the bugs for Bessel function inaccuracy); fixing that would require a different implementation approach, likely along the lines described in <http://www.cl.cam.ac.uk/~jrh13/papers/bessel.ps.gz>. So the justification for a change such as this would be statistical rather than based on particular tests that had excessive errors and no longer do so (no doubt such tests could be found, but would probably be too fragile to add to the testsuite, as liable to give large errors again from very small implementation changes or even from compiler changes). See <https://sourceware.org/ml/libc-alpha/2020-02/msg00638.html> for such statistics of the resulting improvements for float functions. Tested (glibc testsuite) for x86_64.
534 lines
17 KiB
C
534 lines
17 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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/* Long double expansions 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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<https://www.gnu.org/licenses/>. */
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/* __ieee754_j0(x), __ieee754_y0(x)
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* Bessel function of the first and second kinds of order zero.
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* Method -- j0(x):
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* 1. For tiny x, we use j0(x) = 1 - x^2/4 + x^4/64 - ...
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* 2. Reduce x to |x| since j0(x)=j0(-x), and
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* for x in (0,2)
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* j0(x) = 1 - z/4 + z^2*R0/S0, where z = x*x;
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* for x in (2,inf)
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* j0(x) = sqrt(2/(pi*x))*(p0(x)*cos(x0)-q0(x)*sin(x0))
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* where x0 = x-pi/4. It is better to compute sin(x0),cos(x0)
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* as follow:
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* cos(x0) = cos(x)cos(pi/4)+sin(x)sin(pi/4)
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* = 1/sqrt(2) * (cos(x) + sin(x))
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* sin(x0) = sin(x)cos(pi/4)-cos(x)sin(pi/4)
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* = 1/sqrt(2) * (sin(x) - cos(x))
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* (To avoid cancellation, use
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* sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
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* to compute the worse one.)
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*
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* 3 Special cases
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* j0(nan)= nan
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* j0(0) = 1
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* j0(inf) = 0
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*
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* Method -- y0(x):
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* 1. For x<2.
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* Since
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* y0(x) = 2/pi*(j0(x)*(ln(x/2)+Euler) + x^2/4 - ...)
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* therefore y0(x)-2/pi*j0(x)*ln(x) is an even function.
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* We use the following function to approximate y0,
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* y0(x) = U(z)/V(z) + (2/pi)*(j0(x)*ln(x)), z= x^2
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*
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* Note: For tiny x, U/V = u0 and j0(x)~1, hence
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* y0(tiny) = u0 + (2/pi)*ln(tiny), (choose tiny<2**-27)
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* 2. For x>=2.
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* y0(x) = sqrt(2/(pi*x))*(p0(x)*cos(x0)+q0(x)*sin(x0))
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* where x0 = x-pi/4. It is better to compute sin(x0),cos(x0)
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* by the method mentioned above.
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* 3. Special cases: y0(0)=-inf, y0(x<0)=NaN, y0(inf)=0.
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*/
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#include <math.h>
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#include <math-barriers.h>
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#include <math_private.h>
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#include <libm-alias-finite.h>
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static long double pzero (long double), qzero (long double);
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static const long double
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huge = 1e4930L,
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one = 1.0L,
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invsqrtpi = 5.6418958354775628694807945156077258584405e-1L,
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tpi = 6.3661977236758134307553505349005744813784e-1L,
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/* J0(x) = 1 - x^2 / 4 + x^4 R0(x^2) / S0(x^2)
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0 <= x <= 2
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peak relative error 1.41e-22 */
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R[5] = {
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4.287176872744686992880841716723478740566E7L,
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-6.652058897474241627570911531740907185772E5L,
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7.011848381719789863458364584613651091175E3L,
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-3.168040850193372408702135490809516253693E1L,
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6.030778552661102450545394348845599300939E-2L,
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},
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S[4] = {
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2.743793198556599677955266341699130654342E9L,
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3.364330079384816249840086842058954076201E7L,
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1.924119649412510777584684927494642526573E5L,
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6.239282256012734914211715620088714856494E2L,
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/* 1.000000000000000000000000000000000000000E0L,*/
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};
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static const long double zero = 0.0;
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long double
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__ieee754_j0l (long double x)
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{
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long double z, s, c, ss, cc, r, u, v;
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int32_t ix;
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uint32_t se;
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GET_LDOUBLE_EXP (se, x);
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ix = se & 0x7fff;
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if (__glibc_unlikely (ix >= 0x7fff))
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return one / (x * x);
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x = fabsl (x);
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if (ix >= 0x4000) /* |x| >= 2.0 */
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{
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__sincosl (x, &s, &c);
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ss = s - c;
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cc = s + c;
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if (ix < 0x7ffe)
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{ /* make sure x+x not overflow */
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z = -__cosl (x + x);
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if ((s * c) < zero)
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cc = z / ss;
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else
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ss = z / cc;
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}
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/*
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* j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x)
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* y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x)
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*/
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if (__glibc_unlikely (ix > 0x408e)) /* 2^143 */
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z = (invsqrtpi * cc) / sqrtl (x);
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else
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{
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u = pzero (x);
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v = qzero (x);
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z = invsqrtpi * (u * cc - v * ss) / sqrtl (x);
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}
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return z;
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}
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if (__glibc_unlikely (ix < 0x3fef)) /* |x| < 2**-16 */
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{
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/* raise inexact if x != 0 */
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math_force_eval (huge + x);
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if (ix < 0x3fde) /* |x| < 2^-33 */
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return one;
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else
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return one - 0.25 * x * x;
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}
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z = x * x;
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r = z * (R[0] + z * (R[1] + z * (R[2] + z * (R[3] + z * R[4]))));
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s = S[0] + z * (S[1] + z * (S[2] + z * (S[3] + z)));
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if (ix < 0x3fff)
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{ /* |x| < 1.00 */
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return (one - 0.25 * z + z * (r / s));
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}
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else
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{
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u = 0.5 * x;
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return ((one + u) * (one - u) + z * (r / s));
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}
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}
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libm_alias_finite (__ieee754_j0l, __j0l)
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/* y0(x) = 2/pi ln(x) J0(x) + U(x^2)/V(x^2)
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0 < x <= 2
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peak relative error 1.7e-21 */
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static const long double
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U[6] = {
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-1.054912306975785573710813351985351350861E10L,
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2.520192609749295139432773849576523636127E10L,
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-1.856426071075602001239955451329519093395E9L,
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4.079209129698891442683267466276785956784E7L,
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-3.440684087134286610316661166492641011539E5L,
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1.005524356159130626192144663414848383774E3L,
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};
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static const long double
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V[5] = {
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1.429337283720789610137291929228082613676E11L,
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2.492593075325119157558811370165695013002E9L,
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2.186077620785925464237324417623665138376E7L,
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1.238407896366385175196515057064384929222E5L,
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4.693924035211032457494368947123233101664E2L,
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/* 1.000000000000000000000000000000000000000E0L */
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};
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long double
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__ieee754_y0l (long double x)
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{
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long double z, s, c, ss, cc, u, v;
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int32_t ix;
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uint32_t se, i0, i1;
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GET_LDOUBLE_WORDS (se, i0, i1, x);
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ix = se & 0x7fff;
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/* Y0(NaN) is NaN, y0(-inf) is Nan, y0(inf) is 0 */
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if (__glibc_unlikely (se & 0x8000))
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return zero / (zero * x);
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if (__glibc_unlikely (ix >= 0x7fff))
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return one / (x + x * x);
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if (__glibc_unlikely ((i0 | i1) == 0))
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return -HUGE_VALL + x; /* -inf and overflow exception. */
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if (ix >= 0x4000)
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{ /* |x| >= 2.0 */
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/* y0(x) = sqrt(2/(pi*x))*(p0(x)*sin(x0)+q0(x)*cos(x0))
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* where x0 = x-pi/4
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* Better formula:
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* cos(x0) = cos(x)cos(pi/4)+sin(x)sin(pi/4)
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* = 1/sqrt(2) * (sin(x) + cos(x))
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* sin(x0) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4)
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* = 1/sqrt(2) * (sin(x) - cos(x))
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* To avoid cancellation, use
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* sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
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* to compute the worse one.
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*/
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__sincosl (x, &s, &c);
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ss = s - c;
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cc = s + c;
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/*
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* j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x)
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* y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x)
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*/
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if (ix < 0x7ffe)
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{ /* make sure x+x not overflow */
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z = -__cosl (x + x);
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if ((s * c) < zero)
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cc = z / ss;
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else
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ss = z / cc;
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}
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if (__glibc_unlikely (ix > 0x408e)) /* 2^143 */
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z = (invsqrtpi * ss) / sqrtl (x);
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else
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{
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u = pzero (x);
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v = qzero (x);
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z = invsqrtpi * (u * ss + v * cc) / sqrtl (x);
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}
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return z;
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}
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if (__glibc_unlikely (ix <= 0x3fde)) /* x < 2^-33 */
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{
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z = -7.380429510868722527629822444004602747322E-2L
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+ tpi * __ieee754_logl (x);
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return z;
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}
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z = x * x;
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u = U[0] + z * (U[1] + z * (U[2] + z * (U[3] + z * (U[4] + z * U[5]))));
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v = V[0] + z * (V[1] + z * (V[2] + z * (V[3] + z * (V[4] + z))));
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return (u / v + tpi * (__ieee754_j0l (x) * __ieee754_logl (x)));
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}
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libm_alias_finite (__ieee754_y0l, __y0l)
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/* The asymptotic expansions of pzero is
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* 1 - 9/128 s^2 + 11025/98304 s^4 - ..., where s = 1/x.
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* For x >= 2, We approximate pzero by
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* pzero(x) = 1 + s^2 R(s^2) / S(s^2)
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*/
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static const long double pR8[7] = {
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/* 8 <= x <= inf
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Peak relative error 4.62 */
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-4.094398895124198016684337960227780260127E-9L,
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-8.929643669432412640061946338524096893089E-7L,
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-6.281267456906136703868258380673108109256E-5L,
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-1.736902783620362966354814353559382399665E-3L,
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-1.831506216290984960532230842266070146847E-2L,
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-5.827178869301452892963280214772398135283E-2L,
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-2.087563267939546435460286895807046616992E-2L,
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};
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static const long double pS8[6] = {
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5.823145095287749230197031108839653988393E-8L,
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1.279281986035060320477759999428992730280E-5L,
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9.132668954726626677174825517150228961304E-4L,
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2.606019379433060585351880541545146252534E-2L,
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2.956262215119520464228467583516287175244E-1L,
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1.149498145388256448535563278632697465675E0L,
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/* 1.000000000000000000000000000000000000000E0L, */
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};
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static const long double pR5[7] = {
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/* 4.54541015625 <= x <= 8
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Peak relative error 6.51E-22 */
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-2.041226787870240954326915847282179737987E-7L,
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-2.255373879859413325570636768224534428156E-5L,
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-7.957485746440825353553537274569102059990E-4L,
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-1.093205102486816696940149222095559439425E-2L,
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-5.657957849316537477657603125260701114646E-2L,
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-8.641175552716402616180994954177818461588E-2L,
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-1.354654710097134007437166939230619726157E-2L,
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};
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static const long double pS5[6] = {
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2.903078099681108697057258628212823545290E-6L,
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3.253948449946735405975737677123673867321E-4L,
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1.181269751723085006534147920481582279979E-2L,
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1.719212057790143888884745200257619469363E-1L,
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1.006306498779212467670654535430694221924E0L,
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2.069568808688074324555596301126375951502E0L,
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/* 1.000000000000000000000000000000000000000E0L, */
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};
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static const long double pR3[7] = {
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/* 2.85711669921875 <= x <= 4.54541015625
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peak relative error 5.25e-21 */
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-5.755732156848468345557663552240816066802E-6L,
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-3.703675625855715998827966962258113034767E-4L,
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-7.390893350679637611641350096842846433236E-3L,
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-5.571922144490038765024591058478043873253E-2L,
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-1.531290690378157869291151002472627396088E-1L,
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-1.193350853469302941921647487062620011042E-1L,
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-8.567802507331578894302991505331963782905E-3L,
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};
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static const long double pS3[6] = {
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8.185931139070086158103309281525036712419E-5L,
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5.398016943778891093520574483111255476787E-3L,
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1.130589193590489566669164765853409621081E-1L,
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9.358652328786413274673192987670237145071E-1L,
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3.091711512598349056276917907005098085273E0L,
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3.594602474737921977972586821673124231111E0L,
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/* 1.000000000000000000000000000000000000000E0L, */
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};
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static const long double pR2[7] = {
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/* 2 <= x <= 2.85711669921875
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peak relative error 2.64e-21 */
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-1.219525235804532014243621104365384992623E-4L,
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-4.838597135805578919601088680065298763049E-3L,
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-5.732223181683569266223306197751407418301E-2L,
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-2.472947430526425064982909699406646503758E-1L,
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-3.753373645974077960207588073975976327695E-1L,
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-1.556241316844728872406672349347137975495E-1L,
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-5.355423239526452209595316733635519506958E-3L,
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};
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static const long double pS2[6] = {
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1.734442793664291412489066256138894953823E-3L,
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7.158111826468626405416300895617986926008E-2L,
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9.153839713992138340197264669867993552641E-1L,
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4.539209519433011393525841956702487797582E0L,
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8.868932430625331650266067101752626253644E0L,
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6.067161890196324146320763844772857713502E0L,
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/* 1.000000000000000000000000000000000000000E0L, */
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};
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static long double
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pzero (long double x)
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{
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const long double *p, *q;
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long double z, r, s;
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int32_t ix;
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uint32_t se, i0, i1;
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GET_LDOUBLE_WORDS (se, i0, i1, x);
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ix = se & 0x7fff;
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/* ix >= 0x4000 for all calls to this function. */
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if (ix >= 0x4002)
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{
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p = pR8;
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q = pS8;
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} /* x >= 8 */
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else
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{
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i1 = (ix << 16) | (i0 >> 16);
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if (i1 >= 0x40019174) /* x >= 4.54541015625 */
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{
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p = pR5;
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q = pS5;
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}
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else if (i1 >= 0x4000b6db) /* x >= 2.85711669921875 */
|
|
{
|
|
p = pR3;
|
|
q = pS3;
|
|
}
|
|
else /* x >= 2 */
|
|
{
|
|
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] + z * p[6])))));
|
|
s =
|
|
q[0] + z * (q[1] + z * (q[2] + z * (q[3] + z * (q[4] + z * (q[5] + z)))));
|
|
return (one + z * 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*(-.125 + R(s^2) / S(s^2))
|
|
*/
|
|
static const long double qR8[7] = {
|
|
/* 8 <= x <= inf
|
|
peak relative error 2.23e-21 */
|
|
3.001267180483191397885272640777189348008E-10L,
|
|
8.693186311430836495238494289942413810121E-8L,
|
|
8.496875536711266039522937037850596580686E-6L,
|
|
3.482702869915288984296602449543513958409E-4L,
|
|
6.036378380706107692863811938221290851352E-3L,
|
|
3.881970028476167836382607922840452192636E-2L,
|
|
6.132191514516237371140841765561219149638E-2L,
|
|
};
|
|
static const long double qS8[7] = {
|
|
4.097730123753051126914971174076227600212E-9L,
|
|
1.199615869122646109596153392152131139306E-6L,
|
|
1.196337580514532207793107149088168946451E-4L,
|
|
5.099074440112045094341500497767181211104E-3L,
|
|
9.577420799632372483249761659674764460583E-2L,
|
|
7.385243015344292267061953461563695918646E-1L,
|
|
1.917266424391428937962682301561699055943E0L,
|
|
/* 1.000000000000000000000000000000000000000E0L, */
|
|
};
|
|
|
|
static const long double qR5[7] = {
|
|
/* 4.54541015625 <= x <= 8
|
|
peak relative error 1.03e-21 */
|
|
3.406256556438974327309660241748106352137E-8L,
|
|
4.855492710552705436943630087976121021980E-6L,
|
|
2.301011739663737780613356017352912281980E-4L,
|
|
4.500470249273129953870234803596619899226E-3L,
|
|
3.651376459725695502726921248173637054828E-2L,
|
|
1.071578819056574524416060138514508609805E-1L,
|
|
7.458950172851611673015774675225656063757E-2L,
|
|
};
|
|
static const long double qS5[7] = {
|
|
4.650675622764245276538207123618745150785E-7L,
|
|
6.773573292521412265840260065635377164455E-5L,
|
|
3.340711249876192721980146877577806687714E-3L,
|
|
7.036218046856839214741678375536970613501E-2L,
|
|
6.569599559163872573895171876511377891143E-1L,
|
|
2.557525022583599204591036677199171155186E0L,
|
|
3.457237396120935674982927714210361269133E0L,
|
|
/* 1.000000000000000000000000000000000000000E0L,*/
|
|
};
|
|
|
|
static const long double qR3[7] = {
|
|
/* 2.85711669921875 <= x <= 4.54541015625
|
|
peak relative error 5.24e-21 */
|
|
1.749459596550816915639829017724249805242E-6L,
|
|
1.446252487543383683621692672078376929437E-4L,
|
|
3.842084087362410664036704812125005761859E-3L,
|
|
4.066369994699462547896426554180954233581E-2L,
|
|
1.721093619117980251295234795188992722447E-1L,
|
|
2.538595333972857367655146949093055405072E-1L,
|
|
8.560591367256769038905328596020118877936E-2L,
|
|
};
|
|
static const long double qS3[7] = {
|
|
2.388596091707517488372313710647510488042E-5L,
|
|
2.048679968058758616370095132104333998147E-3L,
|
|
5.824663198201417760864458765259945181513E-2L,
|
|
6.953906394693328750931617748038994763958E-1L,
|
|
3.638186936390881159685868764832961092476E0L,
|
|
7.900169524705757837298990558459547842607E0L,
|
|
5.992718532451026507552820701127504582907E0L,
|
|
/* 1.000000000000000000000000000000000000000E0L, */
|
|
};
|
|
|
|
static const long double qR2[7] = {
|
|
/* 2 <= x <= 2.85711669921875
|
|
peak relative error 1.58e-21 */
|
|
6.306524405520048545426928892276696949540E-5L,
|
|
3.209606155709930950935893996591576624054E-3L,
|
|
5.027828775702022732912321378866797059604E-2L,
|
|
3.012705561838718956481911477587757845163E-1L,
|
|
6.960544893905752937420734884995688523815E-1L,
|
|
5.431871999743531634887107835372232030655E-1L,
|
|
9.447736151202905471899259026430157211949E-2L,
|
|
};
|
|
static const long double qS2[7] = {
|
|
8.610579901936193494609755345106129102676E-4L,
|
|
4.649054352710496997203474853066665869047E-2L,
|
|
8.104282924459837407218042945106320388339E-1L,
|
|
5.807730930825886427048038146088828206852E0L,
|
|
1.795310145936848873627710102199881642939E1L,
|
|
2.281313316875375733663657188888110605044E1L,
|
|
1.011242067883822301487154844458322200143E1L,
|
|
/* 1.000000000000000000000000000000000000000E0L, */
|
|
};
|
|
|
|
static long double
|
|
qzero (long double x)
|
|
{
|
|
const long double *p, *q;
|
|
long double s, r, z;
|
|
int32_t ix;
|
|
uint32_t se, i0, i1;
|
|
|
|
GET_LDOUBLE_WORDS (se, i0, i1, x);
|
|
ix = se & 0x7fff;
|
|
/* ix >= 0x4000 for all calls to this function. */
|
|
if (ix >= 0x4002) /* x >= 8 */
|
|
{
|
|
p = qR8;
|
|
q = qS8;
|
|
}
|
|
else
|
|
{
|
|
i1 = (ix << 16) | (i0 >> 16);
|
|
if (i1 >= 0x40019174) /* x >= 4.54541015625 */
|
|
{
|
|
p = qR5;
|
|
q = qS5;
|
|
}
|
|
else if (i1 >= 0x4000b6db) /* x >= 2.85711669921875 */
|
|
{
|
|
p = qR3;
|
|
q = qS3;
|
|
}
|
|
else /* x >= 2 */
|
|
{
|
|
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] + z * p[6])))));
|
|
s =
|
|
q[0] + z * (q[1] +
|
|
z * (q[2] +
|
|
z * (q[3] + z * (q[4] + z * (q[5] + z * (q[6] + z))))));
|
|
return (-.125 + z * r / s) / x;
|
|
}
|