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glibc/sysdeps/ieee754/dbl-64/e_atan2.c

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/*
* IBM Accurate Mathematical Library
* written by International Business Machines Corp.
* Copyright (C) 2001-2013 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 <http://www.gnu.org/licenses/>.
*/
/************************************************************************/
/* MODULE_NAME: atnat2.c */
/* */
/* FUNCTIONS: uatan2 */
/* atan2Mp */
/* signArctan2 */
/* normalized */
/* */
/* FILES NEEDED: dla.h endian.h mpa.h mydefs.h atnat2.h */
/* mpatan.c mpatan2.c mpsqrt.c */
/* uatan.tbl */
/* */
/* An ultimate atan2() routine. Given two IEEE double machine numbers y,*/
/* x it computes the correctly rounded (to nearest) value of atan2(y,x).*/
/* */
/* Assumption: Machine arithmetic operations are performed in */
/* round to nearest mode of IEEE 754 standard. */
/* */
/************************************************************************/
#include <dla.h>
#include "mpa.h"
#include "MathLib.h"
#include "uatan.tbl"
#include "atnat2.h"
#include <math_private.h>
#ifndef SECTION
# define SECTION
#endif
/************************************************************************/
/* An ultimate atan2 routine. Given two IEEE double machine numbers y,x */
/* it computes the correctly rounded (to nearest) value of atan2(y,x). */
/* Assumption: Machine arithmetic operations are performed in */
/* round to nearest mode of IEEE 754 standard. */
/************************************************************************/
static double atan2Mp(double ,double ,const int[]);
/* Fix the sign and return after stage 1 or stage 2 */
static double signArctan2(double y,double z)
{
return __copysign(z, y);
}
static double normalized(double ,double,double ,double);
void __mpatan2(mp_no *,mp_no *,mp_no *,int);
double
SECTION
__ieee754_atan2(double y,double x) {
int i,de,ux,dx,uy,dy;
#if 0
int p;
#endif
static const int pr[MM]={6,8,10,20,32};
double ax,ay,u,du,u9,ua,v,vv,dv,t1,t2,t3,t7,t8,
z,zz,cor,s1,ss1,s2,ss2;
#ifndef DLA_FMS
double t4,t5,t6;
#endif
#if 0
double z1,z2;
#endif
number num;
#if 0
mp_no mperr,mpt1,mpx,mpy,mpz,mpz1,mpz2;
#endif
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+x; }
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 ZERO;
else return opi.d; } }
else if (uy==0x80000000) {
if (dy==0x00000000) {
if ((ux&0x80000000)==0x00000000) return MZERO;
else return mopi.d;} }
/* x=+-0 */
if (x==ZERO) {
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 ZERO;
else return MZERO; }
}
}
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; }
/* either x/y or y/x is very close to zero */
ax = (x<ZERO) ? -x : x; ay = (y<ZERO) ? -y : y;
de = (uy & 0x7ff00000) - (ux & 0x7ff00000);
if (de>=ep) { return ((y>ZERO) ? hpi.d : mhpi.d); }
else if (de<=em) {
if (x>ZERO) {
if ((z=ay/ax)<TWOM1022) return normalized(ax,ay,y,z);
else return signArctan2(y,z); }
else { return ((y>ZERO) ? 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,t1,t2,t3,t4,t5)
du=((ay-v)-vv)/ax; }
else {
u=ax/ay;
EMULV(ay,u,v,vv,t1,t2,t3,t4,t5)
du=((ax-v)-vv)/ay; }
if (x>ZERO) {
/* (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)))));
if ((z=u+(zz-u1.d*u)) == u+(zz+u1.d*u)) return signArctan2(y,z);
MUL2(u,du,u,du,v,vv,t1,t2,t3,t4,t5,t6,t7,t8)
s1=v*(f11.d+v*(f13.d+v*(f15.d+v*(f17.d+v*f19.d))));
ADD2(f9.d,ff9.d,s1,ZERO,s2,ss2,t1,t2)
MUL2(v,vv,s2,ss2,s1,ss1,t1,t2,t3,t4,t5,t6,t7,t8)
ADD2(f7.d,ff7.d,s1,ss1,s2,ss2,t1,t2)
MUL2(v,vv,s2,ss2,s1,ss1,t1,t2,t3,t4,t5,t6,t7,t8)
ADD2(f5.d,ff5.d,s1,ss1,s2,ss2,t1,t2)
MUL2(v,vv,s2,ss2,s1,ss1,t1,t2,t3,t4,t5,t6,t7,t8)
ADD2(f3.d,ff3.d,s1,ss1,s2,ss2,t1,t2)
MUL2(v,vv,s2,ss2,s1,ss1,t1,t2,t3,t4,t5,t6,t7,t8)
MUL2(u,du,s1,ss1,s2,ss2,t1,t2,t3,t4,t5,t6,t7,t8)
ADD2(u,du,s2,ss2,s1,ss1,t1,t2)
if ((z=s1+(ss1-u5.d*s1)) == s1+(ss1+u5.d*s1)) return signArctan2(y,z);
return atan2Mp(x,y,pr);
}
else {
i=(TWO52+TWO8*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))));
if (i<112) {
if (i<48) u9=u91.d; /* u < 1/4 */
else u9=u92.d; } /* 1/4 <= u < 1/2 */
else {
if (i<176) u9=u93.d; /* 1/2 <= u < 3/4 */
else u9=u94.d; } /* 3/4 <= u <= 1 */
if ((z=t1+(zz-u9*t1)) == t1+(zz+u9*t1)) return signArctan2(y,z);
t1=u-hij[i][0].d;
EADD(t1,du,v,vv)
s1=v*(hij[i][11].d+v*(hij[i][12].d+v*(hij[i][13].d+
v*(hij[i][14].d+v* hij[i][15].d))));
ADD2(hij[i][9].d,hij[i][10].d,s1,ZERO,s2,ss2,t1,t2)
MUL2(v,vv,s2,ss2,s1,ss1,t1,t2,t3,t4,t5,t6,t7,t8)
ADD2(hij[i][7].d,hij[i][8].d,s1,ss1,s2,ss2,t1,t2)
MUL2(v,vv,s2,ss2,s1,ss1,t1,t2,t3,t4,t5,t6,t7,t8)
ADD2(hij[i][5].d,hij[i][6].d,s1,ss1,s2,ss2,t1,t2)
MUL2(v,vv,s2,ss2,s1,ss1,t1,t2,t3,t4,t5,t6,t7,t8)
ADD2(hij[i][3].d,hij[i][4].d,s1,ss1,s2,ss2,t1,t2)
MUL2(v,vv,s2,ss2,s1,ss1,t1,t2,t3,t4,t5,t6,t7,t8)
ADD2(hij[i][1].d,hij[i][2].d,s1,ss1,s2,ss2,t1,t2)
if ((z=s2+(ss2-ub.d*s2)) == s2+(ss2+ub.d*s2)) return signArctan2(y,z);
return atan2Mp(x,y,pr);
}
}
/* (ii) x>0, abs(x)<=abs(y): pi/2-atan(ax/ay) */
else {
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;
if ((z=t2+(t3-u2.d)) == t2+(t3+u2.d)) return signArctan2(y,z);
MUL2(u,du,u,du,v,vv,t1,t2,t3,t4,t5,t6,t7,t8)
s1=v*(f11.d+v*(f13.d+v*(f15.d+v*(f17.d+v*f19.d))));
ADD2(f9.d,ff9.d,s1,ZERO,s2,ss2,t1,t2)
MUL2(v,vv,s2,ss2,s1,ss1,t1,t2,t3,t4,t5,t6,t7,t8)
ADD2(f7.d,ff7.d,s1,ss1,s2,ss2,t1,t2)
MUL2(v,vv,s2,ss2,s1,ss1,t1,t2,t3,t4,t5,t6,t7,t8)
ADD2(f5.d,ff5.d,s1,ss1,s2,ss2,t1,t2)
MUL2(v,vv,s2,ss2,s1,ss1,t1,t2,t3,t4,t5,t6,t7,t8)
ADD2(f3.d,ff3.d,s1,ss1,s2,ss2,t1,t2)
MUL2(v,vv,s2,ss2,s1,ss1,t1,t2,t3,t4,t5,t6,t7,t8)
MUL2(u,du,s1,ss1,s2,ss2,t1,t2,t3,t4,t5,t6,t7,t8)
ADD2(u,du,s2,ss2,s1,ss1,t1,t2)
SUB2(hpi.d,hpi1.d,s1,ss1,s2,ss2,t1,t2)
if ((z=s2+(ss2-u6.d)) == s2+(ss2+u6.d)) return signArctan2(y,z);
return atan2Mp(x,y,pr);
}
else {
i=(TWO52+TWO8*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;
if (i<112) ua=ua1.d; /* w < 1/2 */
else ua=ua2.d; /* w >= 1/2 */
if ((z=t1+(zz-ua)) == t1+(zz+ua)) return signArctan2(y,z);
t1=u-hij[i][0].d;
EADD(t1,du,v,vv)
s1=v*(hij[i][11].d+v*(hij[i][12].d+v*(hij[i][13].d+
v*(hij[i][14].d+v* hij[i][15].d))));
ADD2(hij[i][9].d,hij[i][10].d,s1,ZERO,s2,ss2,t1,t2)
MUL2(v,vv,s2,ss2,s1,ss1,t1,t2,t3,t4,t5,t6,t7,t8)
ADD2(hij[i][7].d,hij[i][8].d,s1,ss1,s2,ss2,t1,t2)
MUL2(v,vv,s2,ss2,s1,ss1,t1,t2,t3,t4,t5,t6,t7,t8)
ADD2(hij[i][5].d,hij[i][6].d,s1,ss1,s2,ss2,t1,t2)
MUL2(v,vv,s2,ss2,s1,ss1,t1,t2,t3,t4,t5,t6,t7,t8)
ADD2(hij[i][3].d,hij[i][4].d,s1,ss1,s2,ss2,t1,t2)
MUL2(v,vv,s2,ss2,s1,ss1,t1,t2,t3,t4,t5,t6,t7,t8)
ADD2(hij[i][1].d,hij[i][2].d,s1,ss1,s2,ss2,t1,t2)
SUB2(hpi.d,hpi1.d,s2,ss2,s1,ss1,t1,t2)
if ((z=s1+(ss1-uc.d)) == s1+(ss1+uc.d)) return signArctan2(y,z);
return atan2Mp(x,y,pr);
}
}
}
else {
/* (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;
if ((z=t2+(t3-u3.d)) == t2+(t3+u3.d)) return signArctan2(y,z);
MUL2(u,du,u,du,v,vv,t1,t2,t3,t4,t5,t6,t7,t8)
s1=v*(f11.d+v*(f13.d+v*(f15.d+v*(f17.d+v*f19.d))));
ADD2(f9.d,ff9.d,s1,ZERO,s2,ss2,t1,t2)
MUL2(v,vv,s2,ss2,s1,ss1,t1,t2,t3,t4,t5,t6,t7,t8)
ADD2(f7.d,ff7.d,s1,ss1,s2,ss2,t1,t2)
MUL2(v,vv,s2,ss2,s1,ss1,t1,t2,t3,t4,t5,t6,t7,t8)
ADD2(f5.d,ff5.d,s1,ss1,s2,ss2,t1,t2)
MUL2(v,vv,s2,ss2,s1,ss1,t1,t2,t3,t4,t5,t6,t7,t8)
ADD2(f3.d,ff3.d,s1,ss1,s2,ss2,t1,t2)
MUL2(v,vv,s2,ss2,s1,ss1,t1,t2,t3,t4,t5,t6,t7,t8)
MUL2(u,du,s1,ss1,s2,ss2,t1,t2,t3,t4,t5,t6,t7,t8)
ADD2(u,du,s2,ss2,s1,ss1,t1,t2)
ADD2(hpi.d,hpi1.d,s1,ss1,s2,ss2,t1,t2)
if ((z=s2+(ss2-u7.d)) == s2+(ss2+u7.d)) return signArctan2(y,z);
return atan2Mp(x,y,pr);
}
else {
i=(TWO52+TWO8*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;
if (i<112) ua=ua1.d; /* w < 1/2 */
else ua=ua2.d; /* w >= 1/2 */
if ((z=t1+(zz-ua)) == t1+(zz+ua)) return signArctan2(y,z);
t1=u-hij[i][0].d;
EADD(t1,du,v,vv)
s1=v*(hij[i][11].d+v*(hij[i][12].d+v*(hij[i][13].d+
v*(hij[i][14].d+v* hij[i][15].d))));
ADD2(hij[i][9].d,hij[i][10].d,s1,ZERO,s2,ss2,t1,t2)
MUL2(v,vv,s2,ss2,s1,ss1,t1,t2,t3,t4,t5,t6,t7,t8)
ADD2(hij[i][7].d,hij[i][8].d,s1,ss1,s2,ss2,t1,t2)
MUL2(v,vv,s2,ss2,s1,ss1,t1,t2,t3,t4,t5,t6,t7,t8)
ADD2(hij[i][5].d,hij[i][6].d,s1,ss1,s2,ss2,t1,t2)
MUL2(v,vv,s2,ss2,s1,ss1,t1,t2,t3,t4,t5,t6,t7,t8)
ADD2(hij[i][3].d,hij[i][4].d,s1,ss1,s2,ss2,t1,t2)
MUL2(v,vv,s2,ss2,s1,ss1,t1,t2,t3,t4,t5,t6,t7,t8)
ADD2(hij[i][1].d,hij[i][2].d,s1,ss1,s2,ss2,t1,t2)
ADD2(hpi.d,hpi1.d,s2,ss2,s1,ss1,t1,t2)
if ((z=s1+(ss1-uc.d)) == s1+(ss1+uc.d)) return signArctan2(y,z);
return atan2Mp(x,y,pr);
}
}
/* (iv) x<0, abs(y)<=abs(x): pi-atan(ax/ay) */
else {
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;
if ((z=t2+(t3-u4.d)) == t2+(t3+u4.d)) return signArctan2(y,z);
MUL2(u,du,u,du,v,vv,t1,t2,t3,t4,t5,t6,t7,t8)
s1=v*(f11.d+v*(f13.d+v*(f15.d+v*(f17.d+v*f19.d))));
ADD2(f9.d,ff9.d,s1,ZERO,s2,ss2,t1,t2)
MUL2(v,vv,s2,ss2,s1,ss1,t1,t2,t3,t4,t5,t6,t7,t8)
ADD2(f7.d,ff7.d,s1,ss1,s2,ss2,t1,t2)
MUL2(v,vv,s2,ss2,s1,ss1,t1,t2,t3,t4,t5,t6,t7,t8)
ADD2(f5.d,ff5.d,s1,ss1,s2,ss2,t1,t2)
MUL2(v,vv,s2,ss2,s1,ss1,t1,t2,t3,t4,t5,t6,t7,t8)
ADD2(f3.d,ff3.d,s1,ss1,s2,ss2,t1,t2)
MUL2(v,vv,s2,ss2,s1,ss1,t1,t2,t3,t4,t5,t6,t7,t8)
MUL2(u,du,s1,ss1,s2,ss2,t1,t2,t3,t4,t5,t6,t7,t8)
ADD2(u,du,s2,ss2,s1,ss1,t1,t2)
SUB2(opi.d,opi1.d,s1,ss1,s2,ss2,t1,t2)
if ((z=s2+(ss2-u8.d)) == s2+(ss2+u8.d)) return signArctan2(y,z);
return atan2Mp(x,y,pr);
}
else {
i=(TWO52+TWO8*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;
if (i<112) ua=ua1.d; /* w < 1/2 */
else ua=ua2.d; /* w >= 1/2 */
if ((z=t1+(zz-ua)) == t1+(zz+ua)) return signArctan2(y,z);
t1=u-hij[i][0].d;
EADD(t1,du,v,vv)
s1=v*(hij[i][11].d+v*(hij[i][12].d+v*(hij[i][13].d+
v*(hij[i][14].d+v* hij[i][15].d))));
ADD2(hij[i][9].d,hij[i][10].d,s1,ZERO,s2,ss2,t1,t2)
MUL2(v,vv,s2,ss2,s1,ss1,t1,t2,t3,t4,t5,t6,t7,t8)
ADD2(hij[i][7].d,hij[i][8].d,s1,ss1,s2,ss2,t1,t2)
MUL2(v,vv,s2,ss2,s1,ss1,t1,t2,t3,t4,t5,t6,t7,t8)
ADD2(hij[i][5].d,hij[i][6].d,s1,ss1,s2,ss2,t1,t2)
MUL2(v,vv,s2,ss2,s1,ss1,t1,t2,t3,t4,t5,t6,t7,t8)
ADD2(hij[i][3].d,hij[i][4].d,s1,ss1,s2,ss2,t1,t2)
MUL2(v,vv,s2,ss2,s1,ss1,t1,t2,t3,t4,t5,t6,t7,t8)
ADD2(hij[i][1].d,hij[i][2].d,s1,ss1,s2,ss2,t1,t2)
SUB2(opi.d,opi1.d,s2,ss2,s1,ss1,t1,t2)
if ((z=s1+(ss1-uc.d)) == s1+(ss1+uc.d)) return signArctan2(y,z);
return atan2Mp(x,y,pr);
}
}
}
}
#ifndef __ieee754_atan2
strong_alias (__ieee754_atan2, __atan2_finite)
#endif
/* Treat the Denormalized case */
static double
SECTION
normalized(double ax,double ay,double y, double z)
{ int p;
mp_no mpx,mpy,mpz,mperr,mpz2,mpt1;
p=6;
__dbl_mp(ax,&mpx,p); __dbl_mp(ay,&mpy,p); __dvd(&mpy,&mpx,&mpz,p);
__dbl_mp(ue.d,&mpt1,p); __mul(&mpz,&mpt1,&mperr,p);
__sub(&mpz,&mperr,&mpz2,p); __mp_dbl(&mpz2,&z,p);
return signArctan2(y,z);
}
/* Stage 3: Perform a multi-Precision computation */
static double
SECTION
atan2Mp(double x,double y,const int pr[])
{
double z1,z2;
int i,p;
mp_no mpx,mpy,mpz,mpz1,mpz2,mperr,mpt1;
for (i=0; i<MM; i++) {
p = pr[i];
__dbl_mp(x,&mpx,p); __dbl_mp(y,&mpy,p);
__mpatan2(&mpy,&mpx,&mpz,p);
__dbl_mp(ud[i].d,&mpt1,p); __mul(&mpz,&mpt1,&mperr,p);
__add(&mpz,&mperr,&mpz1,p); __sub(&mpz,&mperr,&mpz2,p);
__mp_dbl(&mpz1,&z1,p); __mp_dbl(&mpz2,&z2,p);
if (z1==z2) return z1;
}
return z1; /*if unpossible to do exact computing */
}
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