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6c4009 |
/*
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6c4009 |
* IBM Accurate Mathematical Library
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6c4009 |
* written by International Business Machines Corp.
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6c4009 |
* Copyright (C) 2001-2018 Free Software Foundation, Inc.
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6c4009 |
*
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6c4009 |
* This program is free software; you can redistribute it and/or modify
|
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Packit |
6c4009 |
* it under the terms of the GNU Lesser General Public License as published by
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|
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6c4009 |
* the Free Software Foundation; either version 2.1 of the License, or
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6c4009 |
* (at your option) any later version.
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6c4009 |
*
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6c4009 |
* This program is distributed in the hope that it will be useful,
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6c4009 |
* but WITHOUT ANY WARRANTY; without even the implied warranty of
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Packit |
6c4009 |
* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
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Packit |
6c4009 |
* GNU Lesser General Public License for more details.
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Packit |
6c4009 |
*
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6c4009 |
* You should have received a copy of the GNU Lesser General Public License
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6c4009 |
* along with this program; if not, see <http://www.gnu.org/licenses/>.
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6c4009 |
*/
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6c4009 |
/************************************************************************/
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6c4009 |
/* MODULE_NAME: atnat.c */
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6c4009 |
/* */
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6c4009 |
/* FUNCTIONS: uatan */
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6c4009 |
/* atanMp */
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6c4009 |
/* signArctan */
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6c4009 |
/* */
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|
Packit |
6c4009 |
/* */
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6c4009 |
/* FILES NEEDED: dla.h endian.h mpa.h mydefs.h atnat.h */
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6c4009 |
/* mpatan.c mpatan2.c mpsqrt.c */
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6c4009 |
/* uatan.tbl */
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6c4009 |
/* */
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6c4009 |
/* An ultimate atan() routine. Given an IEEE double machine number x */
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/* it computes the correctly rounded (to nearest) value of atan(x). */
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6c4009 |
/* */
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6c4009 |
/* Assumption: Machine arithmetic operations are performed in */
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/* round to nearest mode of IEEE 754 standard. */
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6c4009 |
/* */
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/************************************************************************/
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#include <dla.h>
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6c4009 |
#include "mpa.h"
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6c4009 |
#include "MathLib.h"
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#include "uatan.tbl"
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#include "atnat.h"
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#include <fenv.h>
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#include <float.h>
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6c4009 |
#include <libm-alias-double.h>
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6c4009 |
#include <math.h>
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#include <math_private.h>
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#include <math-underflow.h>
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#include <stap-probe.h>
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void __mpatan (mp_no *, mp_no *, int); /* see definition in mpatan.c */
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static double atanMp (double, const int[]);
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/* Fix the sign of y and return */
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6c4009 |
static double
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__signArctan (double x, double y)
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6c4009 |
{
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6c4009 |
return __copysign (y, x);
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6c4009 |
}
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6c4009 |
|
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6c4009 |
|
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Packit |
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/* An ultimate atan() routine. Given an IEEE double machine number x, */
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/* routine computes the correctly rounded (to nearest) value of atan(x). */
|
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double
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__atan (double x)
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{
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double cor, s1, ss1, s2, ss2, t1, t2, t3, t7, t8, t9, t10, u, u2, u3,
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v, vv, w, ww, y, yy, z, zz;
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#ifndef DLA_FMS
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double t4, t5, t6;
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#endif
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int i, ux, dx;
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static const int pr[M] = { 6, 8, 10, 32 };
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6c4009 |
number num;
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6c4009 |
|
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6c4009 |
num.d = x;
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6c4009 |
ux = num.i[HIGH_HALF];
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6c4009 |
dx = num.i[LOW_HALF];
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6c4009 |
|
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Packit |
6c4009 |
/* x=NaN */
|
|
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6c4009 |
if (((ux & 0x7ff00000) == 0x7ff00000)
|
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6c4009 |
&& (((ux & 0x000fffff) | dx) != 0x00000000))
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6c4009 |
return x + x;
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6c4009 |
|
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6c4009 |
/* Regular values of x, including denormals +-0 and +-INF */
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SET_RESTORE_ROUND (FE_TONEAREST);
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u = (x < 0) ? -x : x;
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6c4009 |
if (u < C)
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6c4009 |
{
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6c4009 |
if (u < B)
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Packit |
6c4009 |
{
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Packit |
6c4009 |
if (u < A)
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Packit |
6c4009 |
{
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6c4009 |
math_check_force_underflow_nonneg (u);
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6c4009 |
return x;
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|
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6c4009 |
}
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6c4009 |
else
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6c4009 |
{ /* A <= u < B */
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6c4009 |
v = x * x;
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6c4009 |
yy = d11.d + v * d13.d;
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Packit |
6c4009 |
yy = d9.d + v * yy;
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|
Packit |
6c4009 |
yy = d7.d + v * yy;
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Packit |
6c4009 |
yy = d5.d + v * yy;
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Packit |
6c4009 |
yy = d3.d + v * yy;
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Packit |
6c4009 |
yy *= x * v;
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|
Packit |
6c4009 |
|
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Packit |
6c4009 |
if ((y = x + (yy - U1 * x)) == x + (yy + U1 * x))
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|
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6c4009 |
return y;
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|
Packit |
6c4009 |
|
|
Packit |
6c4009 |
EMULV (x, x, v, vv, t1, t2, t3, t4, t5); /* v+vv=x^2 */
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Packit |
6c4009 |
|
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Packit |
6c4009 |
s1 = f17.d + v * f19.d;
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Packit |
6c4009 |
s1 = f15.d + v * s1;
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Packit |
6c4009 |
s1 = f13.d + v * s1;
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|
Packit |
6c4009 |
s1 = f11.d + v * s1;
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|
Packit |
6c4009 |
s1 *= v;
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|
Packit |
6c4009 |
|
|
Packit |
6c4009 |
ADD2 (f9.d, ff9.d, s1, 0, s2, ss2, t1, t2);
|
|
Packit |
6c4009 |
MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2, t3, t4, t5, t6, t7, t8);
|
|
Packit |
6c4009 |
ADD2 (f7.d, ff7.d, s1, ss1, s2, ss2, t1, t2);
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|
Packit |
6c4009 |
MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2, t3, t4, t5, t6, t7, t8);
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6c4009 |
ADD2 (f5.d, ff5.d, s1, ss1, s2, ss2, t1, t2);
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Packit |
6c4009 |
MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2, t3, t4, t5, t6, t7, t8);
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6c4009 |
ADD2 (f3.d, ff3.d, s1, ss1, s2, ss2, t1, t2);
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Packit |
6c4009 |
MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2, t3, t4, t5, t6, t7, t8);
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6c4009 |
MUL2 (x, 0, s1, ss1, s2, ss2, t1, t2, t3, t4, t5, t6, t7,
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6c4009 |
t8);
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6c4009 |
ADD2 (x, 0, s2, ss2, s1, ss1, t1, t2);
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Packit |
6c4009 |
if ((y = s1 + (ss1 - U5 * s1)) == s1 + (ss1 + U5 * s1))
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Packit |
6c4009 |
return y;
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Packit |
6c4009 |
|
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6c4009 |
return atanMp (x, pr);
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Packit |
6c4009 |
}
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Packit |
6c4009 |
}
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Packit |
6c4009 |
else
|
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Packit |
6c4009 |
{ /* B <= u < C */
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6c4009 |
i = (TWO52 + TWO8 * u) - TWO52;
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Packit |
6c4009 |
i -= 16;
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|
Packit |
6c4009 |
z = u - cij[i][0].d;
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Packit |
6c4009 |
yy = cij[i][5].d + z * cij[i][6].d;
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|
Packit |
6c4009 |
yy = cij[i][4].d + z * yy;
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Packit |
6c4009 |
yy = cij[i][3].d + z * yy;
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Packit |
6c4009 |
yy = cij[i][2].d + z * yy;
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Packit |
6c4009 |
yy *= z;
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Packit |
6c4009 |
|
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Packit |
6c4009 |
t1 = cij[i][1].d;
|
|
Packit |
6c4009 |
if (i < 112)
|
|
Packit |
6c4009 |
{
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|
Packit |
6c4009 |
if (i < 48)
|
|
Packit |
6c4009 |
u2 = U21; /* u < 1/4 */
|
|
Packit |
6c4009 |
else
|
|
Packit |
6c4009 |
u2 = U22;
|
|
Packit |
6c4009 |
} /* 1/4 <= u < 1/2 */
|
|
Packit |
6c4009 |
else
|
|
Packit |
6c4009 |
{
|
|
Packit |
6c4009 |
if (i < 176)
|
|
Packit |
6c4009 |
u2 = U23; /* 1/2 <= u < 3/4 */
|
|
Packit |
6c4009 |
else
|
|
Packit |
6c4009 |
u2 = U24;
|
|
Packit |
6c4009 |
} /* 3/4 <= u <= 1 */
|
|
Packit |
6c4009 |
if ((y = t1 + (yy - u2 * t1)) == t1 + (yy + u2 * t1))
|
|
Packit |
6c4009 |
return __signArctan (x, y);
|
|
Packit |
6c4009 |
|
|
Packit |
6c4009 |
z = u - hij[i][0].d;
|
|
Packit |
6c4009 |
|
|
Packit |
6c4009 |
s1 = hij[i][14].d + z * hij[i][15].d;
|
|
Packit |
6c4009 |
s1 = hij[i][13].d + z * s1;
|
|
Packit |
6c4009 |
s1 = hij[i][12].d + z * s1;
|
|
Packit |
6c4009 |
s1 = hij[i][11].d + z * s1;
|
|
Packit |
6c4009 |
s1 *= z;
|
|
Packit |
6c4009 |
|
|
Packit |
6c4009 |
ADD2 (hij[i][9].d, hij[i][10].d, s1, 0, s2, ss2, t1, t2);
|
|
Packit |
6c4009 |
MUL2 (z, 0, s2, ss2, s1, ss1, t1, t2, t3, t4, t5, t6, t7, t8);
|
|
Packit |
6c4009 |
ADD2 (hij[i][7].d, hij[i][8].d, s1, ss1, s2, ss2, t1, t2);
|
|
Packit |
6c4009 |
MUL2 (z, 0, s2, ss2, s1, ss1, t1, t2, t3, t4, t5, t6, t7, t8);
|
|
Packit |
6c4009 |
ADD2 (hij[i][5].d, hij[i][6].d, s1, ss1, s2, ss2, t1, t2);
|
|
Packit |
6c4009 |
MUL2 (z, 0, s2, ss2, s1, ss1, t1, t2, t3, t4, t5, t6, t7, t8);
|
|
Packit |
6c4009 |
ADD2 (hij[i][3].d, hij[i][4].d, s1, ss1, s2, ss2, t1, t2);
|
|
Packit |
6c4009 |
MUL2 (z, 0, s2, ss2, s1, ss1, t1, t2, t3, t4, t5, t6, t7, t8);
|
|
Packit |
6c4009 |
ADD2 (hij[i][1].d, hij[i][2].d, s1, ss1, s2, ss2, t1, t2);
|
|
Packit |
6c4009 |
if ((y = s2 + (ss2 - U6 * s2)) == s2 + (ss2 + U6 * s2))
|
|
Packit |
6c4009 |
return __signArctan (x, y);
|
|
Packit |
6c4009 |
|
|
Packit |
6c4009 |
return atanMp (x, pr);
|
|
Packit |
6c4009 |
}
|
|
Packit |
6c4009 |
}
|
|
Packit |
6c4009 |
else
|
|
Packit |
6c4009 |
{
|
|
Packit |
6c4009 |
if (u < D)
|
|
Packit |
6c4009 |
{ /* C <= u < D */
|
|
Packit |
6c4009 |
w = 1 / u;
|
|
Packit |
6c4009 |
EMULV (w, u, t1, t2, t3, t4, t5, t6, t7);
|
|
Packit |
6c4009 |
ww = w * ((1 - t1) - t2);
|
|
Packit |
6c4009 |
i = (TWO52 + TWO8 * w) - TWO52;
|
|
Packit |
6c4009 |
i -= 16;
|
|
Packit |
6c4009 |
z = (w - cij[i][0].d) + ww;
|
|
Packit |
6c4009 |
|
|
Packit |
6c4009 |
yy = cij[i][5].d + z * cij[i][6].d;
|
|
Packit |
6c4009 |
yy = cij[i][4].d + z * yy;
|
|
Packit |
6c4009 |
yy = cij[i][3].d + z * yy;
|
|
Packit |
6c4009 |
yy = cij[i][2].d + z * yy;
|
|
Packit |
6c4009 |
yy = HPI1 - z * yy;
|
|
Packit |
6c4009 |
|
|
Packit |
6c4009 |
t1 = HPI - cij[i][1].d;
|
|
Packit |
6c4009 |
if (i < 112)
|
|
Packit |
6c4009 |
u3 = U31; /* w < 1/2 */
|
|
Packit |
6c4009 |
else
|
|
Packit |
6c4009 |
u3 = U32; /* w >= 1/2 */
|
|
Packit |
6c4009 |
if ((y = t1 + (yy - u3)) == t1 + (yy + u3))
|
|
Packit |
6c4009 |
return __signArctan (x, y);
|
|
Packit |
6c4009 |
|
|
Packit |
6c4009 |
DIV2 (1, 0, u, 0, w, ww, t1, t2, t3, t4, t5, t6, t7, t8, t9,
|
|
Packit |
6c4009 |
t10);
|
|
Packit |
6c4009 |
t1 = w - hij[i][0].d;
|
|
Packit |
6c4009 |
EADD (t1, ww, z, zz);
|
|
Packit |
6c4009 |
|
|
Packit |
6c4009 |
s1 = hij[i][14].d + z * hij[i][15].d;
|
|
Packit |
6c4009 |
s1 = hij[i][13].d + z * s1;
|
|
Packit |
6c4009 |
s1 = hij[i][12].d + z * s1;
|
|
Packit |
6c4009 |
s1 = hij[i][11].d + z * s1;
|
|
Packit |
6c4009 |
s1 *= z;
|
|
Packit |
6c4009 |
|
|
Packit |
6c4009 |
ADD2 (hij[i][9].d, hij[i][10].d, s1, 0, s2, ss2, t1, t2);
|
|
Packit |
6c4009 |
MUL2 (z, zz, s2, ss2, s1, ss1, t1, t2, t3, t4, t5, t6, t7, t8);
|
|
Packit |
6c4009 |
ADD2 (hij[i][7].d, hij[i][8].d, s1, ss1, s2, ss2, t1, t2);
|
|
Packit |
6c4009 |
MUL2 (z, zz, s2, ss2, s1, ss1, t1, t2, t3, t4, t5, t6, t7, t8);
|
|
Packit |
6c4009 |
ADD2 (hij[i][5].d, hij[i][6].d, s1, ss1, s2, ss2, t1, t2);
|
|
Packit |
6c4009 |
MUL2 (z, zz, s2, ss2, s1, ss1, t1, t2, t3, t4, t5, t6, t7, t8);
|
|
Packit |
6c4009 |
ADD2 (hij[i][3].d, hij[i][4].d, s1, ss1, s2, ss2, t1, t2);
|
|
Packit |
6c4009 |
MUL2 (z, zz, s2, ss2, s1, ss1, t1, t2, t3, t4, t5, t6, t7, t8);
|
|
Packit |
6c4009 |
ADD2 (hij[i][1].d, hij[i][2].d, s1, ss1, s2, ss2, t1, t2);
|
|
Packit |
6c4009 |
SUB2 (HPI, HPI1, s2, ss2, s1, ss1, t1, t2);
|
|
Packit |
6c4009 |
if ((y = s1 + (ss1 - U7)) == s1 + (ss1 + U7))
|
|
Packit |
6c4009 |
return __signArctan (x, y);
|
|
Packit |
6c4009 |
|
|
Packit |
6c4009 |
return atanMp (x, pr);
|
|
Packit |
6c4009 |
}
|
|
Packit |
6c4009 |
else
|
|
Packit |
6c4009 |
{
|
|
Packit |
6c4009 |
if (u < E)
|
|
Packit |
6c4009 |
{ /* D <= u < E */
|
|
Packit |
6c4009 |
w = 1 / u;
|
|
Packit |
6c4009 |
v = w * w;
|
|
Packit |
6c4009 |
EMULV (w, u, t1, t2, t3, t4, t5, t6, t7);
|
|
Packit |
6c4009 |
|
|
Packit |
6c4009 |
yy = d11.d + v * d13.d;
|
|
Packit |
6c4009 |
yy = d9.d + v * yy;
|
|
Packit |
6c4009 |
yy = d7.d + v * yy;
|
|
Packit |
6c4009 |
yy = d5.d + v * yy;
|
|
Packit |
6c4009 |
yy = d3.d + v * yy;
|
|
Packit |
6c4009 |
yy *= w * v;
|
|
Packit |
6c4009 |
|
|
Packit |
6c4009 |
ww = w * ((1 - t1) - t2);
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Packit |
6c4009 |
ESUB (HPI, w, t3, cor);
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Packit |
6c4009 |
yy = ((HPI1 + cor) - ww) - yy;
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Packit |
6c4009 |
if ((y = t3 + (yy - U4)) == t3 + (yy + U4))
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|
Packit |
6c4009 |
return __signArctan (x, y);
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|
Packit |
6c4009 |
|
|
Packit |
6c4009 |
DIV2 (1, 0, u, 0, w, ww, t1, t2, t3, t4, t5, t6, t7, t8,
|
|
Packit |
6c4009 |
t9, t10);
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|
Packit |
6c4009 |
MUL2 (w, ww, w, ww, v, vv, t1, t2, t3, t4, t5, t6, t7, t8);
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|
Packit |
6c4009 |
|
|
Packit |
6c4009 |
s1 = f17.d + v * f19.d;
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|
Packit |
6c4009 |
s1 = f15.d + v * s1;
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|
Packit |
6c4009 |
s1 = f13.d + v * s1;
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|
Packit |
6c4009 |
s1 = f11.d + v * s1;
|
|
Packit |
6c4009 |
s1 *= v;
|
|
Packit |
6c4009 |
|
|
Packit |
6c4009 |
ADD2 (f9.d, ff9.d, s1, 0, s2, ss2, t1, t2);
|
|
Packit |
6c4009 |
MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2, t3, t4, t5, t6, t7, t8);
|
|
Packit |
6c4009 |
ADD2 (f7.d, ff7.d, s1, ss1, s2, ss2, t1, t2);
|
|
Packit |
6c4009 |
MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2, t3, t4, t5, t6, t7, t8);
|
|
Packit |
6c4009 |
ADD2 (f5.d, ff5.d, s1, ss1, s2, ss2, t1, t2);
|
|
Packit |
6c4009 |
MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2, t3, t4, t5, t6, t7, t8);
|
|
Packit |
6c4009 |
ADD2 (f3.d, ff3.d, s1, ss1, s2, ss2, t1, t2);
|
|
Packit |
6c4009 |
MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2, t3, t4, t5, t6, t7, t8);
|
|
Packit |
6c4009 |
MUL2 (w, ww, s1, ss1, s2, ss2, t1, t2, t3, t4, t5, t6, t7, t8);
|
|
Packit |
6c4009 |
ADD2 (w, ww, s2, ss2, s1, ss1, t1, t2);
|
|
Packit |
6c4009 |
SUB2 (HPI, HPI1, s1, ss1, s2, ss2, t1, t2);
|
|
Packit |
6c4009 |
|
|
Packit |
6c4009 |
if ((y = s2 + (ss2 - U8)) == s2 + (ss2 + U8))
|
|
Packit |
6c4009 |
return __signArctan (x, y);
|
|
Packit |
6c4009 |
|
|
Packit |
6c4009 |
return atanMp (x, pr);
|
|
Packit |
6c4009 |
}
|
|
Packit |
6c4009 |
else
|
|
Packit |
6c4009 |
{
|
|
Packit |
6c4009 |
/* u >= E */
|
|
Packit |
6c4009 |
if (x > 0)
|
|
Packit |
6c4009 |
return HPI;
|
|
Packit |
6c4009 |
else
|
|
Packit |
6c4009 |
return MHPI;
|
|
Packit |
6c4009 |
}
|
|
Packit |
6c4009 |
}
|
|
Packit |
6c4009 |
}
|
|
Packit |
6c4009 |
}
|
|
Packit |
6c4009 |
|
|
Packit |
6c4009 |
/* Final stages. Compute atan(x) by multiple precision arithmetic */
|
|
Packit |
6c4009 |
static double
|
|
Packit |
6c4009 |
atanMp (double x, const int pr[])
|
|
Packit |
6c4009 |
{
|
|
Packit |
6c4009 |
mp_no mpx, mpy, mpy2, mperr, mpt1, mpy1;
|
|
Packit |
6c4009 |
double y1, y2;
|
|
Packit |
6c4009 |
int i, p;
|
|
Packit |
6c4009 |
|
|
Packit |
6c4009 |
for (i = 0; i < M; i++)
|
|
Packit |
6c4009 |
{
|
|
Packit |
6c4009 |
p = pr[i];
|
|
Packit |
6c4009 |
__dbl_mp (x, &mpx, p);
|
|
Packit |
6c4009 |
__mpatan (&mpx, &mpy, p);
|
|
Packit |
6c4009 |
__dbl_mp (u9[i].d, &mpt1, p);
|
|
Packit |
6c4009 |
__mul (&mpy, &mpt1, &mperr, p);
|
|
Packit |
6c4009 |
__add (&mpy, &mperr, &mpy1, p);
|
|
Packit |
6c4009 |
__sub (&mpy, &mperr, &mpy2, p);
|
|
Packit |
6c4009 |
__mp_dbl (&mpy1, &y1, p);
|
|
Packit |
6c4009 |
__mp_dbl (&mpy2, &y2, p);
|
|
Packit |
6c4009 |
if (y1 == y2)
|
|
Packit |
6c4009 |
{
|
|
Packit |
6c4009 |
LIBC_PROBE (slowatan, 3, &p, &x, &y1;;
|
|
Packit |
6c4009 |
return y1;
|
|
Packit |
6c4009 |
}
|
|
Packit |
6c4009 |
}
|
|
Packit |
6c4009 |
LIBC_PROBE (slowatan_inexact, 3, &p, &x, &y1;;
|
|
Packit |
6c4009 |
return y1; /*if impossible to do exact computing */
|
|
Packit |
6c4009 |
}
|
|
Packit |
6c4009 |
|
|
Packit |
6c4009 |
#ifndef __atan
|
|
Packit |
6c4009 |
libm_alias_double (__atan, atan)
|
|
Packit |
6c4009 |
#endif
|