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530 lines
17 KiB
C
530 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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<http://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_private.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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u_int32_t se;
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GET_LDOUBLE_EXP (se, x);
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ix = se & 0x7fff;
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if (__builtin_expect (ix >= 0x7fff, 0))
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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 (__builtin_expect (ix > 0x4080, 0)) /* 2^129 */
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z = (invsqrtpi * cc) / __ieee754_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) / __ieee754_sqrtl (x);
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}
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return z;
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}
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if (__builtin_expect (ix < 0x3fef, 0)) /* |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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strong_alias (__ieee754_j0l, __j0l_finite)
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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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u_int32_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 (__builtin_expect (se & 0x8000, 0))
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return zero / (zero * x);
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if (__builtin_expect (ix >= 0x7fff, 0))
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return one / (x + x * x);
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if (__builtin_expect ((i0 | i1) == 0, 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 (__builtin_expect (ix > 0x4080, 0)) /* 1e39 */
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z = (invsqrtpi * ss) / __ieee754_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) / __ieee754_sqrtl (x);
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}
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return z;
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}
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if (__builtin_expect (ix <= 0x3fde, 0)) /* 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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strong_alias (__ieee754_y0l, __y0l_finite)
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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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u_int32_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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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 */
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{
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p = pR3;
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q = pS3;
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}
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else if (ix >= 0x4000) /* x better be >= 2 */
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{
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p = pR2;
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q = pS2;
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}
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}
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z = one / (x * x);
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r =
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p[0] + z * (p[1] +
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z * (p[2] + z * (p[3] + z * (p[4] + z * (p[5] + z * p[6])))));
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s =
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q[0] + z * (q[1] + z * (q[2] + z * (q[3] + z * (q[4] + z * (q[5] + z)))));
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return (one + z * r / s);
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}
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/* For x >= 8, the asymptotic expansions of qzero is
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* -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;
|
|
u_int32_t se, i0, i1;
|
|
|
|
GET_LDOUBLE_WORDS (se, i0, i1, x);
|
|
ix = se & 0x7fff;
|
|
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 if (ix >= 0x4000) /* x better be >= 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;
|
|
}
|