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538 lines
17 KiB
C
538 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_j1(x), __ieee754_y1(x)
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* Bessel function of the first and second kinds of order zero.
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* Method -- j1(x):
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* 1. For tiny x, we use j1(x) = x/2 - x^3/16 + x^5/384 - ...
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* 2. Reduce x to |x| since j1(x)=-j1(-x), and
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* for x in (0,2)
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* j1(x) = x/2 + x*z*R0/S0, where z = x*x;
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* for x in (2,inf)
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* j1(x) = sqrt(2/(pi*x))*(p1(x)*cos(x1)-q1(x)*sin(x1))
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* y1(x) = sqrt(2/(pi*x))*(p1(x)*sin(x1)+q1(x)*cos(x1))
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* where x1 = x-3*pi/4. It is better to compute sin(x1),cos(x1)
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* as follow:
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* cos(x1) = cos(x)cos(3pi/4)+sin(x)sin(3pi/4)
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* = 1/sqrt(2) * (sin(x) - cos(x))
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* sin(x1) = 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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* 3 Special cases
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* j1(nan)= nan
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* j1(0) = 0
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* j1(inf) = 0
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*
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* Method -- y1(x):
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* 1. screen out x<=0 cases: y1(0)=-inf, y1(x<0)=NaN
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* 2. For x<2.
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* Since
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* y1(x) = 2/pi*(j1(x)*(ln(x/2)+Euler)-1/x-x/2+5/64*x^3-...)
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* therefore y1(x)-2/pi*j1(x)*ln(x)-1/x is an odd function.
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* We use the following function to approximate y1,
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* y1(x) = x*U(z)/V(z) + (2/pi)*(j1(x)*ln(x)-1/x), z= x^2
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* Note: For tiny x, 1/x dominate y1 and hence
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* y1(tiny) = -2/pi/tiny
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* 3. For x>=2.
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* y1(x) = sqrt(2/(pi*x))*(p1(x)*sin(x1)+q1(x)*cos(x1))
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* where x1 = x-3*pi/4. It is better to compute sin(x1),cos(x1)
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* by method mentioned above.
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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 pone (long double), qone (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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/* J1(x) = .5 x + x x^2 R(x^2) / S(x^2)
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0 <= x <= 2
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Peak relative error 4.5e-21 */
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R[5] = {
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-9.647406112428107954753770469290757756814E7L,
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2.686288565865230690166454005558203955564E6L,
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-3.689682683905671185891885948692283776081E4L,
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2.195031194229176602851429567792676658146E2L,
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-5.124499848728030297902028238597308971319E-1L,
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},
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S[4] =
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{
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1.543584977988497274437410333029029035089E9L,
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2.133542369567701244002565983150952549520E7L,
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1.394077011298227346483732156167414670520E5L,
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5.252401789085732428842871556112108446506E2L,
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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_j1l (long double x)
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{
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long double z, c, r, s, ss, cc, u, v, y;
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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;
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y = fabsl (x);
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if (ix >= 0x4000)
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{ /* |x| >= 2.0 */
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__sincosl (y, &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 y+y not overflow */
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z = __cosl (y + y);
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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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* j1(x) = 1/sqrt(pi) * (P(1,x)*cc - Q(1,x)*ss) / sqrt(x)
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* y1(x) = 1/sqrt(pi) * (P(1,x)*ss + Q(1,x)*cc) / sqrt(x)
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*/
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if (__builtin_expect (ix > 0x4080, 0))
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z = (invsqrtpi * cc) / __ieee754_sqrtl (y);
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else
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{
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u = pone (y);
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v = qone (y);
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z = invsqrtpi * (u * cc - v * ss) / __ieee754_sqrtl (y);
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}
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if (se & 0x8000)
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return -z;
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else
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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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if (huge + x > one)
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return 0.5 * x; /* inexact if x!=0 necessary */
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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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r *= x;
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return (x * 0.5 + r / s);
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}
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strong_alias (__ieee754_j1l, __j1l_finite)
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/* Y1(x) = 2/pi * (log(x) * j1(x) - 1/x) + x R(x^2)
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0 <= x <= 2
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Peak relative error 2.3e-23 */
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static const long double U0[6] = {
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-5.908077186259914699178903164682444848615E10L,
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1.546219327181478013495975514375773435962E10L,
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-6.438303331169223128870035584107053228235E8L,
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9.708540045657182600665968063824819371216E6L,
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-6.138043997084355564619377183564196265471E4L,
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1.418503228220927321096904291501161800215E2L,
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};
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static const long double V0[5] = {
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3.013447341682896694781964795373783679861E11L,
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4.669546565705981649470005402243136124523E9L,
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3.595056091631351184676890179233695857260E7L,
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1.761554028569108722903944659933744317994E5L,
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5.668480419646516568875555062047234534863E2L,
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/* 1.000000000000000000000000000000000000000E0L, */
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};
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long double
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__ieee754_y1l (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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/* if Y1(NaN) is NaN, Y1(-inf) is NaN, Y1(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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__sincosl (x, &s, &c);
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ss = -s - c;
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cc = s - c;
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if (ix < 0x7fe00000)
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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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/* y1(x) = sqrt(2/(pi*x))*(p1(x)*sin(x0)+q1(x)*cos(x0))
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* where x0 = x-3pi/4
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* Better formula:
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* cos(x0) = cos(x)cos(3pi/4)+sin(x)sin(3pi/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) * (cos(x) + sin(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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if (__builtin_expect (ix > 0x4080, 0))
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z = (invsqrtpi * ss) / __ieee754_sqrtl (x);
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else
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{
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u = pone (x);
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v = qone (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 <= 0x3fbe, 0))
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{ /* x < 2**-65 */
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return (-tpi / x);
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}
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z = x * x;
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u = U0[0] + z * (U0[1] + z * (U0[2] + z * (U0[3] + z * (U0[4] + z * U0[5]))));
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v = V0[0] + z * (V0[1] + z * (V0[2] + z * (V0[3] + z * (V0[4] + z))));
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return (x * (u / v) +
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tpi * (__ieee754_j1l (x) * __ieee754_logl (x) - one / x));
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}
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strong_alias (__ieee754_y1l, __y1l_finite)
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/* For x >= 8, the asymptotic expansions of pone is
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* 1 + 15/128 s^2 - 4725/2^15 s^4 - ..., where s = 1/x.
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* We approximate pone by
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* pone(x) = 1 + (R/S)
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*/
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/* J1(x) cosX + Y1(x) sinX = sqrt( 2/(pi x)) P1(x)
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P1(x) = 1 + z^2 R(z^2), z=1/x
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8 <= x <= inf (0 <= z <= 0.125)
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Peak relative error 5.2e-22 */
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static const long double pr8[7] = {
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8.402048819032978959298664869941375143163E-9L,
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1.813743245316438056192649247507255996036E-6L,
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1.260704554112906152344932388588243836276E-4L,
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3.439294839869103014614229832700986965110E-3L,
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3.576910849712074184504430254290179501209E-2L,
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1.131111483254318243139953003461511308672E-1L,
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4.480715825681029711521286449131671880953E-2L,
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};
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static const long double ps8[6] = {
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7.169748325574809484893888315707824924354E-8L,
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1.556549720596672576431813934184403614817E-5L,
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1.094540125521337139209062035774174565882E-3L,
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3.060978962596642798560894375281428805840E-2L,
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3.374146536087205506032643098619414507024E-1L,
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1.253830208588979001991901126393231302559E0L,
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/* 1.000000000000000000000000000000000000000E0L, */
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};
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/* J1(x) cosX + Y1(x) sinX = sqrt( 2/(pi x)) P1(x)
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P1(x) = 1 + z^2 R(z^2), z=1/x
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4.54541015625 <= x <= 8
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Peak relative error 7.7e-22 */
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static const long double pr5[7] = {
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4.318486887948814529950980396300969247900E-7L,
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4.715341880798817230333360497524173929315E-5L,
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1.642719430496086618401091544113220340094E-3L,
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2.228688005300803935928733750456396149104E-2L,
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1.142773760804150921573259605730018327162E-1L,
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1.755576530055079253910829652698703791957E-1L,
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3.218803858282095929559165965353784980613E-2L,
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};
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static const long double ps5[6] = {
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3.685108812227721334719884358034713967557E-6L,
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4.069102509511177498808856515005792027639E-4L,
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1.449728676496155025507893322405597039816E-2L,
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2.058869213229520086582695850441194363103E-1L,
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1.164890985918737148968424972072751066553E0L,
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2.274776933457009446573027260373361586841E0L,
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/* 1.000000000000000000000000000000000000000E0L,*/
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};
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/* J1(x) cosX + Y1(x) sinX = sqrt( 2/(pi x)) P1(x)
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P1(x) = 1 + z^2 R(z^2), z=1/x
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2.85711669921875 <= x <= 4.54541015625
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Peak relative error 6.5e-21 */
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static const long double pr3[7] = {
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1.265251153957366716825382654273326407972E-5L,
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8.031057269201324914127680782288352574567E-4L,
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1.581648121115028333661412169396282881035E-2L,
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1.179534658087796321928362981518645033967E-1L,
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3.227936912780465219246440724502790727866E-1L,
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2.559223765418386621748404398017602935764E-1L,
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2.277136933287817911091370397134882441046E-2L,
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};
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static const long double ps3[6] = {
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1.079681071833391818661952793568345057548E-4L,
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6.986017817100477138417481463810841529026E-3L,
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1.429403701146942509913198539100230540503E-1L,
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1.148392024337075609460312658938700765074E0L,
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3.643663015091248720208251490291968840882E0L,
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3.990702269032018282145100741746633960737E0L,
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/* 1.000000000000000000000000000000000000000E0L, */
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};
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/* J1(x) cosX + Y1(x) sinX = sqrt( 2/(pi x)) P1(x)
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P1(x) = 1 + z^2 R(z^2), z=1/x
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2 <= x <= 2.85711669921875
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Peak relative error 3.5e-21 */
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static const long double pr2[7] = {
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2.795623248568412225239401141338714516445E-4L,
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1.092578168441856711925254839815430061135E-2L,
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1.278024620468953761154963591853679640560E-1L,
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5.469680473691500673112904286228351988583E-1L,
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8.313769490922351300461498619045639016059E-1L,
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3.544176317308370086415403567097130611468E-1L,
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1.604142674802373041247957048801599740644E-2L,
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};
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static const long double ps2[6] = {
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2.385605161555183386205027000675875235980E-3L,
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9.616778294482695283928617708206967248579E-2L,
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1.195215570959693572089824415393951258510E0L,
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5.718412857897054829999458736064922974662E0L,
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1.065626298505499086386584642761602177568E1L,
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6.809140730053382188468983548092322151791E0L,
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/* 1.000000000000000000000000000000000000000E0L, */
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};
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static long double
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pone (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) /* x >= 8 */
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{
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p = pr8;
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q = ps8;
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}
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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 = 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 = 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 qone is
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* 3/8 s - 105/1024 s^3 - ..., where s = 1/x.
|
|
* We approximate pone by
|
|
* qone(x) = s*(0.375 + (R/S))
|
|
*/
|
|
|
|
/* Y1(x)cosX - J1(x)sinX = sqrt( 2/(pi x)) Q1(x),
|
|
Q1(x) = z(.375 + z^2 R(z^2)), z=1/x
|
|
8 <= x <= inf
|
|
Peak relative error 8.3e-22 */
|
|
|
|
static const long double qr8[7] = {
|
|
-5.691925079044209246015366919809404457380E-10L,
|
|
-1.632587664706999307871963065396218379137E-7L,
|
|
-1.577424682764651970003637263552027114600E-5L,
|
|
-6.377627959241053914770158336842725291713E-4L,
|
|
-1.087408516779972735197277149494929568768E-2L,
|
|
-6.854943629378084419631926076882330494217E-2L,
|
|
-1.055448290469180032312893377152490183203E-1L,
|
|
};
|
|
static const long double qs8[7] = {
|
|
5.550982172325019811119223916998393907513E-9L,
|
|
1.607188366646736068460131091130644192244E-6L,
|
|
1.580792530091386496626494138334505893599E-4L,
|
|
6.617859900815747303032860443855006056595E-3L,
|
|
1.212840547336984859952597488863037659161E-1L,
|
|
9.017885953937234900458186716154005541075E-1L,
|
|
2.201114489712243262000939120146436167178E0L,
|
|
/* 1.000000000000000000000000000000000000000E0L, */
|
|
};
|
|
|
|
/* Y1(x)cosX - J1(x)sinX = sqrt( 2/(pi x)) Q1(x),
|
|
Q1(x) = z(.375 + z^2 R(z^2)), z=1/x
|
|
4.54541015625 <= x <= 8
|
|
Peak relative error 4.1e-22 */
|
|
static const long double qr5[7] = {
|
|
-6.719134139179190546324213696633564965983E-8L,
|
|
-9.467871458774950479909851595678622044140E-6L,
|
|
-4.429341875348286176950914275723051452838E-4L,
|
|
-8.539898021757342531563866270278505014487E-3L,
|
|
-6.818691805848737010422337101409276287170E-2L,
|
|
-1.964432669771684034858848142418228214855E-1L,
|
|
-1.333896496989238600119596538299938520726E-1L,
|
|
};
|
|
static const long double qs5[7] = {
|
|
6.552755584474634766937589285426911075101E-7L,
|
|
9.410814032118155978663509073200494000589E-5L,
|
|
4.561677087286518359461609153655021253238E-3L,
|
|
9.397742096177905170800336715661091535805E-2L,
|
|
8.518538116671013902180962914473967738771E-1L,
|
|
3.177729183645800174212539541058292579009E0L,
|
|
4.006745668510308096259753538973038902990E0L,
|
|
/* 1.000000000000000000000000000000000000000E0L, */
|
|
};
|
|
|
|
/* Y1(x)cosX - J1(x)sinX = sqrt( 2/(pi x)) Q1(x),
|
|
Q1(x) = z(.375 + z^2 R(z^2)), z=1/x
|
|
2.85711669921875 <= x <= 4.54541015625
|
|
Peak relative error 2.2e-21 */
|
|
static const long double qr3[7] = {
|
|
-3.618746299358445926506719188614570588404E-6L,
|
|
-2.951146018465419674063882650970344502798E-4L,
|
|
-7.728518171262562194043409753656506795258E-3L,
|
|
-8.058010968753999435006488158237984014883E-2L,
|
|
-3.356232856677966691703904770937143483472E-1L,
|
|
-4.858192581793118040782557808823460276452E-1L,
|
|
-1.592399251246473643510898335746432479373E-1L,
|
|
};
|
|
static const long double qs3[7] = {
|
|
3.529139957987837084554591421329876744262E-5L,
|
|
2.973602667215766676998703687065066180115E-3L,
|
|
8.273534546240864308494062287908662592100E-2L,
|
|
9.613359842126507198241321110649974032726E-1L,
|
|
4.853923697093974370118387947065402707519E0L,
|
|
1.002671608961669247462020977417828796933E1L,
|
|
7.028927383922483728931327850683151410267E0L,
|
|
/* 1.000000000000000000000000000000000000000E0L, */
|
|
};
|
|
|
|
/* Y1(x)cosX - J1(x)sinX = sqrt( 2/(pi x)) Q1(x),
|
|
Q1(x) = z(.375 + z^2 R(z^2)), z=1/x
|
|
2 <= x <= 2.85711669921875
|
|
Peak relative error 6.9e-22 */
|
|
static const long double qr2[7] = {
|
|
-1.372751603025230017220666013816502528318E-4L,
|
|
-6.879190253347766576229143006767218972834E-3L,
|
|
-1.061253572090925414598304855316280077828E-1L,
|
|
-6.262164224345471241219408329354943337214E-1L,
|
|
-1.423149636514768476376254324731437473915E0L,
|
|
-1.087955310491078933531734062917489870754E0L,
|
|
-1.826821119773182847861406108689273719137E-1L,
|
|
};
|
|
static const long double qs2[7] = {
|
|
1.338768933634451601814048220627185324007E-3L,
|
|
7.071099998918497559736318523932241901810E-2L,
|
|
1.200511429784048632105295629933382142221E0L,
|
|
8.327301713640367079030141077172031825276E0L,
|
|
2.468479301872299311658145549931764426840E1L,
|
|
2.961179686096262083509383820557051621644E1L,
|
|
1.201402313144305153005639494661767354977E1L,
|
|
/* 1.000000000000000000000000000000000000000E0L, */
|
|
};
|
|
|
|
|
|
static long double
|
|
qone (long double x)
|
|
{
|
|
const long double *p, *q;
|
|
static 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 (.375 + z * r / s) / x;
|
|
}
|