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glibc/sysdeps/ieee754/dbl-64/e_atan2.c
Wilco Dijkstra 4e1a870b9a math: Remove slow paths from atan2 [BZ #15267] 
Remove slow paths from atan2. Add ULP annotations.

Reviewed-By: Paul Zimmermann <Paul.Zimmermann@inria.fr>
2021-03-11 14:26:36 +00:00

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/*
* IBM Accurate Mathematical Library
* written by International Business Machines Corp.
* Copyright (C) 2001-2021 Free Software Foundation, Inc.
*
* This program is free software; you can redistribute it and/or modify
* it under the terms of the GNU Lesser General Public License as published by
* the Free Software Foundation; either version 2.1 of the License, or
* (at your option) any later version.
*
* This program is distributed in the hope that it will be useful,
* but WITHOUT ANY WARRANTY; without even the implied warranty of
* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
* GNU Lesser General Public License for more details.
*
* You should have received a copy of the GNU Lesser General Public License
* along with this program; if not, see <https://www.gnu.org/licenses/>.
*/
/************************************************************************/
/* MODULE_NAME: atnat2.c */
/* */
/* FUNCTIONS: uatan2 */
/* signArctan2 */
/* */
/* FILES NEEDED: dla.h endian.h mydefs.h atnat2.h */
/* uatan.tbl */
/* */
/************************************************************************/
#include <dla.h>
#include "mydefs.h"
#include "uatan.tbl"
#include "atnat2.h"
#include <fenv.h>
#include <float.h>
#include <math.h>
#include <math-barriers.h>
#include <math_private.h>
#include <fenv_private.h>
#include <libm-alias-finite.h>
#ifndef SECTION
# define SECTION
#endif
#define TWO52 0x1.0p52
#define TWOM1022 0x1.0p-1022
/* Fix the sign and return after stage 1 or stage 2 */
static double
signArctan2 (double y, double z)
{
return copysign (z, y);
}
/* atan2 with max ULP of ~0.524 based on random sampling. */
double
SECTION
__ieee754_atan2 (double y, double x)
{
int i, de, ux, dx, uy, dy;
double ax, ay, u, du, v, vv, dv, t1, t2, t3,
z, zz, cor;
mynumber num;
static const int ep = 59768832, /* 57*16**5 */
em = -59768832; /* -57*16**5 */
/* x=NaN or y=NaN */
num.d = x;
ux = num.i[HIGH_HALF];
dx = num.i[LOW_HALF];
if ((ux & 0x7ff00000) == 0x7ff00000)
{
if (((ux & 0x000fffff) | dx) != 0x00000000)
return x + y;
}
num.d = y;
uy = num.i[HIGH_HALF];
dy = num.i[LOW_HALF];
if ((uy & 0x7ff00000) == 0x7ff00000)
{
if (((uy & 0x000fffff) | dy) != 0x00000000)
return y + y;
}
/* y=+-0 */
if (uy == 0x00000000)
{
if (dy == 0x00000000)
{
if ((ux & 0x80000000) == 0x00000000)
return 0;
else
return opi.d;
}
}
else if (uy == 0x80000000)
{
if (dy == 0x00000000)
{
if ((ux & 0x80000000) == 0x00000000)
return -0.0;
else
return mopi.d;
}
}
/* x=+-0 */
if (x == 0)
{
if ((uy & 0x80000000) == 0x00000000)
return hpi.d;
else
return mhpi.d;
}
/* x=+-INF */
if (ux == 0x7ff00000)
{
if (dx == 0x00000000)
{
if (uy == 0x7ff00000)
{
if (dy == 0x00000000)
return qpi.d;
}
else if (uy == 0xfff00000)
{
if (dy == 0x00000000)
return mqpi.d;
}
else
{
if ((uy & 0x80000000) == 0x00000000)
return 0;
else
return -0.0;
}
}
}
else if (ux == 0xfff00000)
{
if (dx == 0x00000000)
{
if (uy == 0x7ff00000)
{
if (dy == 0x00000000)
return tqpi.d;
}
else if (uy == 0xfff00000)
{
if (dy == 0x00000000)
return mtqpi.d;
}
else
{
if ((uy & 0x80000000) == 0x00000000)
return opi.d;
else
return mopi.d;
}
}
}
/* y=+-INF */
if (uy == 0x7ff00000)
{
if (dy == 0x00000000)
return hpi.d;
}
else if (uy == 0xfff00000)
{
if (dy == 0x00000000)
return mhpi.d;
}
SET_RESTORE_ROUND (FE_TONEAREST);
/* either x/y or y/x is very close to zero */
ax = (x < 0) ? -x : x;
ay = (y < 0) ? -y : y;
de = (uy & 0x7ff00000) - (ux & 0x7ff00000);
if (de >= ep)
{
return ((y > 0) ? hpi.d : mhpi.d);
}
else if (de <= em)
{
if (x > 0)
{
double ret;
z = ay / ax;
ret = signArctan2 (y, z);
if (fabs (ret) < DBL_MIN)
{
double vret = ret ? ret : DBL_MIN;
double force_underflow = vret * vret;
math_force_eval (force_underflow);
}
return ret;
}
else
{
return ((y > 0) ? opi.d : mopi.d);
}
}
/* if either x or y is extremely close to zero, scale abs(x), abs(y). */
if (ax < twom500.d || ay < twom500.d)
{
ax *= two500.d;
ay *= two500.d;
}
/* Likewise for large x and y. */
if (ax > two500.d || ay > two500.d)
{
ax *= twom500.d;
ay *= twom500.d;
}
/* x,y which are neither special nor extreme */
if (ay < ax)
{
u = ay / ax;
EMULV (ax, u, v, vv);
du = ((ay - v) - vv) / ax;
}
else
{
u = ax / ay;
EMULV (ay, u, v, vv);
du = ((ax - v) - vv) / ay;
}
if (x > 0)
{
/* (i) x>0, abs(y)< abs(x): atan(ay/ax) */
if (ay < ax)
{
if (u < inv16.d)
{
v = u * u;
zz = du + u * v * (d3.d
+ v * (d5.d
+ v * (d7.d
+ v * (d9.d
+ v * (d11.d
+ v * d13.d)))));
z = u + zz;
/* Max ULP is 0.504. */
return signArctan2 (y, z);
}
i = (TWO52 + 256 * u) - TWO52;
i -= 16;
t3 = u - cij[i][0].d;
EADD (t3, du, v, dv);
t1 = cij[i][1].d;
t2 = cij[i][2].d;
zz = v * t2 + (dv * t2
+ v * v * (cij[i][3].d
+ v * (cij[i][4].d
+ v * (cij[i][5].d
+ v * cij[i][6].d))));
z = t1 + zz;
/* Max ULP is 0.56. */
return signArctan2 (y, z);
}
/* (ii) x>0, abs(x)<=abs(y): pi/2-atan(ax/ay) */
if (u < inv16.d)
{
v = u * u;
zz = u * v * (d3.d
+ v * (d5.d
+ v * (d7.d
+ v * (d9.d
+ v * (d11.d
+ v * d13.d)))));
ESUB (hpi.d, u, t2, cor);
t3 = ((hpi1.d + cor) - du) - zz;
z = t2 + t3;
/* Max ULP is 0.501. */
return signArctan2 (y, z);
}
i = (TWO52 + 256 * u) - TWO52;
i -= 16;
v = (u - cij[i][0].d) + du;
zz = hpi1.d - v * (cij[i][2].d
+ v * (cij[i][3].d
+ v * (cij[i][4].d
+ v * (cij[i][5].d
+ v * cij[i][6].d))));
t1 = hpi.d - cij[i][1].d;
z = t1 + zz;
/* Max ULP is 0.503. */
return signArctan2 (y, z);
}
/* (iii) x<0, abs(x)< abs(y): pi/2+atan(ax/ay) */
if (ax < ay)
{
if (u < inv16.d)
{
v = u * u;
zz = u * v * (d3.d
+ v * (d5.d
+ v * (d7.d
+ v * (d9.d
+ v * (d11.d + v * d13.d)))));
EADD (hpi.d, u, t2, cor);
t3 = ((hpi1.d + cor) + du) + zz;
z = t2 + t3;
/* Max ULP is 0.501. */
return signArctan2 (y, z);
}
i = (TWO52 + 256 * u) - TWO52;
i -= 16;
v = (u - cij[i][0].d) + du;
zz = hpi1.d + v * (cij[i][2].d
+ v * (cij[i][3].d
+ v * (cij[i][4].d
+ v * (cij[i][5].d
+ v * cij[i][6].d))));
t1 = hpi.d + cij[i][1].d;
z = t1 + zz;
/* Max ULP is 0.503. */
return signArctan2 (y, z);
}
/* (iv) x<0, abs(y)<=abs(x): pi-atan(ax/ay) */
if (u < inv16.d)
{
v = u * u;
zz = u * v * (d3.d
+ v * (d5.d
+ v * (d7.d
+ v * (d9.d + v * (d11.d + v * d13.d)))));
ESUB (opi.d, u, t2, cor);
t3 = ((opi1.d + cor) - du) - zz;
z = t2 + t3;
/* Max ULP is 0.501. */
return signArctan2 (y, z);
}
i = (TWO52 + 256 * u) - TWO52;
i -= 16;
v = (u - cij[i][0].d) + du;
zz = opi1.d - v * (cij[i][2].d
+ v * (cij[i][3].d
+ v * (cij[i][4].d
+ v * (cij[i][5].d + v * cij[i][6].d))));
t1 = opi.d - cij[i][1].d;
z = t1 + zz;
/* Max ULP is 0.502. */
return signArctan2 (y, z);
}
#ifndef __ieee754_atan2
libm_alias_finite (__ieee754_atan2, __atan2)
#endif
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