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422 lines
15 KiB
C
422 lines
15 KiB
C
/*
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* IBM Accurate Mathematical Library
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* written by International Business Machines Corp.
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* Copyright (C) 2001, 2011 Free Software Foundation
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*
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* This program is free software; you can redistribute it and/or modify
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* it under the terms of the GNU Lesser General Public License as published by
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* the Free Software Foundation; either version 2.1 of the License, or
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* (at your option) any later version.
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*
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* This program 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
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* GNU Lesser General Public License for more details.
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*
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* You should have received a copy of the GNU Lesser General Public License
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* along with this program; if not, see <http://www.gnu.org/licenses/>.
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*/
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/************************************************************************/
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/* MODULE_NAME: atnat2.c */
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/* */
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/* FUNCTIONS: uatan2 */
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/* atan2Mp */
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/* signArctan2 */
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/* normalized */
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/* */
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/* FILES NEEDED: dla.h endian.h mpa.h mydefs.h atnat2.h */
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/* mpatan.c mpatan2.c mpsqrt.c */
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/* uatan.tbl */
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/* */
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/* An ultimate atan2() routine. Given two IEEE double machine numbers y,*/
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/* x it computes the correctly rounded (to nearest) value of atan2(y,x).*/
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/* */
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/* Assumption: Machine arithmetic operations are performed in */
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/* round to nearest mode of IEEE 754 standard. */
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/* */
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/************************************************************************/
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#include <dla.h>
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#include "mpa.h"
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#include "MathLib.h"
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#include "uatan.tbl"
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#include "atnat2.h"
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#include <math_private.h>
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#ifndef SECTION
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# define SECTION
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#endif
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/************************************************************************/
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/* An ultimate atan2 routine. Given two IEEE double machine numbers y,x */
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/* it computes the correctly rounded (to nearest) value of atan2(y,x). */
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/* Assumption: Machine arithmetic operations are performed in */
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/* round to nearest mode of IEEE 754 standard. */
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/************************************************************************/
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static double atan2Mp(double ,double ,const int[]);
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/* Fix the sign and return after stage 1 or stage 2 */
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static double signArctan2(double y,double z)
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{
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return __copysign(z, y);
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}
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static double normalized(double ,double,double ,double);
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void __mpatan2(mp_no *,mp_no *,mp_no *,int);
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double
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SECTION
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__ieee754_atan2(double y,double x) {
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int i,de,ux,dx,uy,dy;
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#if 0
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int p;
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#endif
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static const int pr[MM]={6,8,10,20,32};
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double ax,ay,u,du,u9,ua,v,vv,dv,t1,t2,t3,t7,t8,
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z,zz,cor,s1,ss1,s2,ss2;
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#ifndef DLA_FMS
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double t4,t5,t6;
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#endif
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#if 0
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double z1,z2;
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#endif
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number num;
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#if 0
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mp_no mperr,mpt1,mpx,mpy,mpz,mpz1,mpz2;
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#endif
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static const int ep= 59768832, /* 57*16**5 */
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em=-59768832; /* -57*16**5 */
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/* x=NaN or y=NaN */
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num.d = x; ux = num.i[HIGH_HALF]; dx = num.i[LOW_HALF];
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if ((ux&0x7ff00000) ==0x7ff00000) {
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if (((ux&0x000fffff)|dx)!=0x00000000) return x+x; }
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num.d = y; uy = num.i[HIGH_HALF]; dy = num.i[LOW_HALF];
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if ((uy&0x7ff00000) ==0x7ff00000) {
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if (((uy&0x000fffff)|dy)!=0x00000000) return y+y; }
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/* y=+-0 */
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if (uy==0x00000000) {
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if (dy==0x00000000) {
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if ((ux&0x80000000)==0x00000000) return ZERO;
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else return opi.d; } }
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else if (uy==0x80000000) {
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if (dy==0x00000000) {
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if ((ux&0x80000000)==0x00000000) return MZERO;
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else return mopi.d;} }
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/* x=+-0 */
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if (x==ZERO) {
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if ((uy&0x80000000)==0x00000000) return hpi.d;
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else return mhpi.d; }
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/* x=+-INF */
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if (ux==0x7ff00000) {
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if (dx==0x00000000) {
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if (uy==0x7ff00000) {
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if (dy==0x00000000) return qpi.d; }
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else if (uy==0xfff00000) {
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if (dy==0x00000000) return mqpi.d; }
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else {
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if ((uy&0x80000000)==0x00000000) return ZERO;
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else return MZERO; }
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}
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}
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else if (ux==0xfff00000) {
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if (dx==0x00000000) {
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if (uy==0x7ff00000) {
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if (dy==0x00000000) return tqpi.d; }
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else if (uy==0xfff00000) {
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if (dy==0x00000000) return mtqpi.d; }
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else {
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if ((uy&0x80000000)==0x00000000) return opi.d;
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else return mopi.d; }
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}
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}
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/* y=+-INF */
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if (uy==0x7ff00000) {
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if (dy==0x00000000) return hpi.d; }
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else if (uy==0xfff00000) {
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if (dy==0x00000000) return mhpi.d; }
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/* either x/y or y/x is very close to zero */
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ax = (x<ZERO) ? -x : x; ay = (y<ZERO) ? -y : y;
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de = (uy & 0x7ff00000) - (ux & 0x7ff00000);
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if (de>=ep) { return ((y>ZERO) ? hpi.d : mhpi.d); }
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else if (de<=em) {
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if (x>ZERO) {
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if ((z=ay/ax)<TWOM1022) return normalized(ax,ay,y,z);
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else return signArctan2(y,z); }
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else { return ((y>ZERO) ? opi.d : mopi.d); } }
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/* if either x or y is extremely close to zero, scale abs(x), abs(y). */
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if (ax<twom500.d || ay<twom500.d) { ax*=two500.d; ay*=two500.d; }
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/* x,y which are neither special nor extreme */
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if (ay<ax) {
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u=ay/ax;
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EMULV(ax,u,v,vv,t1,t2,t3,t4,t5)
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du=((ay-v)-vv)/ax; }
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else {
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u=ax/ay;
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EMULV(ay,u,v,vv,t1,t2,t3,t4,t5)
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du=((ax-v)-vv)/ay; }
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if (x>ZERO) {
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/* (i) x>0, abs(y)< abs(x): atan(ay/ax) */
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if (ay<ax) {
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if (u<inv16.d) {
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v=u*u; zz=du+u*v*(d3.d+v*(d5.d+v*(d7.d+v*(d9.d+v*(d11.d+v*d13.d)))));
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if ((z=u+(zz-u1.d*u)) == u+(zz+u1.d*u)) return signArctan2(y,z);
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MUL2(u,du,u,du,v,vv,t1,t2,t3,t4,t5,t6,t7,t8)
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s1=v*(f11.d+v*(f13.d+v*(f15.d+v*(f17.d+v*f19.d))));
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ADD2(f9.d,ff9.d,s1,ZERO,s2,ss2,t1,t2)
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MUL2(v,vv,s2,ss2,s1,ss1,t1,t2,t3,t4,t5,t6,t7,t8)
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ADD2(f7.d,ff7.d,s1,ss1,s2,ss2,t1,t2)
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MUL2(v,vv,s2,ss2,s1,ss1,t1,t2,t3,t4,t5,t6,t7,t8)
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ADD2(f5.d,ff5.d,s1,ss1,s2,ss2,t1,t2)
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MUL2(v,vv,s2,ss2,s1,ss1,t1,t2,t3,t4,t5,t6,t7,t8)
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ADD2(f3.d,ff3.d,s1,ss1,s2,ss2,t1,t2)
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MUL2(v,vv,s2,ss2,s1,ss1,t1,t2,t3,t4,t5,t6,t7,t8)
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MUL2(u,du,s1,ss1,s2,ss2,t1,t2,t3,t4,t5,t6,t7,t8)
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ADD2(u,du,s2,ss2,s1,ss1,t1,t2)
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if ((z=s1+(ss1-u5.d*s1)) == s1+(ss1+u5.d*s1)) return signArctan2(y,z);
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return atan2Mp(x,y,pr);
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}
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else {
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i=(TWO52+TWO8*u)-TWO52; i-=16;
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t3=u-cij[i][0].d;
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EADD(t3,du,v,dv)
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t1=cij[i][1].d; t2=cij[i][2].d;
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zz=v*t2+(dv*t2+v*v*(cij[i][3].d+v*(cij[i][4].d+
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v*(cij[i][5].d+v* cij[i][6].d))));
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if (i<112) {
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if (i<48) u9=u91.d; /* u < 1/4 */
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else u9=u92.d; } /* 1/4 <= u < 1/2 */
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else {
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if (i<176) u9=u93.d; /* 1/2 <= u < 3/4 */
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else u9=u94.d; } /* 3/4 <= u <= 1 */
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if ((z=t1+(zz-u9*t1)) == t1+(zz+u9*t1)) return signArctan2(y,z);
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t1=u-hij[i][0].d;
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EADD(t1,du,v,vv)
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s1=v*(hij[i][11].d+v*(hij[i][12].d+v*(hij[i][13].d+
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v*(hij[i][14].d+v* hij[i][15].d))));
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ADD2(hij[i][9].d,hij[i][10].d,s1,ZERO,s2,ss2,t1,t2)
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MUL2(v,vv,s2,ss2,s1,ss1,t1,t2,t3,t4,t5,t6,t7,t8)
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ADD2(hij[i][7].d,hij[i][8].d,s1,ss1,s2,ss2,t1,t2)
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MUL2(v,vv,s2,ss2,s1,ss1,t1,t2,t3,t4,t5,t6,t7,t8)
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ADD2(hij[i][5].d,hij[i][6].d,s1,ss1,s2,ss2,t1,t2)
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MUL2(v,vv,s2,ss2,s1,ss1,t1,t2,t3,t4,t5,t6,t7,t8)
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ADD2(hij[i][3].d,hij[i][4].d,s1,ss1,s2,ss2,t1,t2)
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MUL2(v,vv,s2,ss2,s1,ss1,t1,t2,t3,t4,t5,t6,t7,t8)
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ADD2(hij[i][1].d,hij[i][2].d,s1,ss1,s2,ss2,t1,t2)
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if ((z=s2+(ss2-ub.d*s2)) == s2+(ss2+ub.d*s2)) return signArctan2(y,z);
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return atan2Mp(x,y,pr);
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}
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}
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/* (ii) x>0, abs(x)<=abs(y): pi/2-atan(ax/ay) */
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else {
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if (u<inv16.d) {
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v=u*u;
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zz=u*v*(d3.d+v*(d5.d+v*(d7.d+v*(d9.d+v*(d11.d+v*d13.d)))));
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ESUB(hpi.d,u,t2,cor)
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t3=((hpi1.d+cor)-du)-zz;
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if ((z=t2+(t3-u2.d)) == t2+(t3+u2.d)) return signArctan2(y,z);
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MUL2(u,du,u,du,v,vv,t1,t2,t3,t4,t5,t6,t7,t8)
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s1=v*(f11.d+v*(f13.d+v*(f15.d+v*(f17.d+v*f19.d))));
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ADD2(f9.d,ff9.d,s1,ZERO,s2,ss2,t1,t2)
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MUL2(v,vv,s2,ss2,s1,ss1,t1,t2,t3,t4,t5,t6,t7,t8)
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ADD2(f7.d,ff7.d,s1,ss1,s2,ss2,t1,t2)
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MUL2(v,vv,s2,ss2,s1,ss1,t1,t2,t3,t4,t5,t6,t7,t8)
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ADD2(f5.d,ff5.d,s1,ss1,s2,ss2,t1,t2)
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MUL2(v,vv,s2,ss2,s1,ss1,t1,t2,t3,t4,t5,t6,t7,t8)
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ADD2(f3.d,ff3.d,s1,ss1,s2,ss2,t1,t2)
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MUL2(v,vv,s2,ss2,s1,ss1,t1,t2,t3,t4,t5,t6,t7,t8)
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MUL2(u,du,s1,ss1,s2,ss2,t1,t2,t3,t4,t5,t6,t7,t8)
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ADD2(u,du,s2,ss2,s1,ss1,t1,t2)
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SUB2(hpi.d,hpi1.d,s1,ss1,s2,ss2,t1,t2)
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if ((z=s2+(ss2-u6.d)) == s2+(ss2+u6.d)) return signArctan2(y,z);
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return atan2Mp(x,y,pr);
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}
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else {
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i=(TWO52+TWO8*u)-TWO52; i-=16;
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v=(u-cij[i][0].d)+du;
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zz=hpi1.d-v*(cij[i][2].d+v*(cij[i][3].d+v*(cij[i][4].d+
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v*(cij[i][5].d+v* cij[i][6].d))));
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t1=hpi.d-cij[i][1].d;
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if (i<112) ua=ua1.d; /* w < 1/2 */
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else ua=ua2.d; /* w >= 1/2 */
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if ((z=t1+(zz-ua)) == t1+(zz+ua)) return signArctan2(y,z);
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t1=u-hij[i][0].d;
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EADD(t1,du,v,vv)
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s1=v*(hij[i][11].d+v*(hij[i][12].d+v*(hij[i][13].d+
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v*(hij[i][14].d+v* hij[i][15].d))));
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ADD2(hij[i][9].d,hij[i][10].d,s1,ZERO,s2,ss2,t1,t2)
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MUL2(v,vv,s2,ss2,s1,ss1,t1,t2,t3,t4,t5,t6,t7,t8)
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ADD2(hij[i][7].d,hij[i][8].d,s1,ss1,s2,ss2,t1,t2)
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MUL2(v,vv,s2,ss2,s1,ss1,t1,t2,t3,t4,t5,t6,t7,t8)
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ADD2(hij[i][5].d,hij[i][6].d,s1,ss1,s2,ss2,t1,t2)
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MUL2(v,vv,s2,ss2,s1,ss1,t1,t2,t3,t4,t5,t6,t7,t8)
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ADD2(hij[i][3].d,hij[i][4].d,s1,ss1,s2,ss2,t1,t2)
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MUL2(v,vv,s2,ss2,s1,ss1,t1,t2,t3,t4,t5,t6,t7,t8)
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ADD2(hij[i][1].d,hij[i][2].d,s1,ss1,s2,ss2,t1,t2)
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SUB2(hpi.d,hpi1.d,s2,ss2,s1,ss1,t1,t2)
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if ((z=s1+(ss1-uc.d)) == s1+(ss1+uc.d)) return signArctan2(y,z);
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return atan2Mp(x,y,pr);
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}
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}
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}
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else {
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/* (iii) x<0, abs(x)< abs(y): pi/2+atan(ax/ay) */
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if (ax<ay) {
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if (u<inv16.d) {
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v=u*u;
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zz=u*v*(d3.d+v*(d5.d+v*(d7.d+v*(d9.d+v*(d11.d+v*d13.d)))));
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EADD(hpi.d,u,t2,cor)
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t3=((hpi1.d+cor)+du)+zz;
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if ((z=t2+(t3-u3.d)) == t2+(t3+u3.d)) return signArctan2(y,z);
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MUL2(u,du,u,du,v,vv,t1,t2,t3,t4,t5,t6,t7,t8)
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s1=v*(f11.d+v*(f13.d+v*(f15.d+v*(f17.d+v*f19.d))));
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ADD2(f9.d,ff9.d,s1,ZERO,s2,ss2,t1,t2)
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MUL2(v,vv,s2,ss2,s1,ss1,t1,t2,t3,t4,t5,t6,t7,t8)
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ADD2(f7.d,ff7.d,s1,ss1,s2,ss2,t1,t2)
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MUL2(v,vv,s2,ss2,s1,ss1,t1,t2,t3,t4,t5,t6,t7,t8)
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ADD2(f5.d,ff5.d,s1,ss1,s2,ss2,t1,t2)
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MUL2(v,vv,s2,ss2,s1,ss1,t1,t2,t3,t4,t5,t6,t7,t8)
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ADD2(f3.d,ff3.d,s1,ss1,s2,ss2,t1,t2)
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MUL2(v,vv,s2,ss2,s1,ss1,t1,t2,t3,t4,t5,t6,t7,t8)
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MUL2(u,du,s1,ss1,s2,ss2,t1,t2,t3,t4,t5,t6,t7,t8)
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ADD2(u,du,s2,ss2,s1,ss1,t1,t2)
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ADD2(hpi.d,hpi1.d,s1,ss1,s2,ss2,t1,t2)
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if ((z=s2+(ss2-u7.d)) == s2+(ss2+u7.d)) return signArctan2(y,z);
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return atan2Mp(x,y,pr);
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}
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else {
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i=(TWO52+TWO8*u)-TWO52; i-=16;
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v=(u-cij[i][0].d)+du;
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zz=hpi1.d+v*(cij[i][2].d+v*(cij[i][3].d+v*(cij[i][4].d+
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v*(cij[i][5].d+v* cij[i][6].d))));
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t1=hpi.d+cij[i][1].d;
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if (i<112) ua=ua1.d; /* w < 1/2 */
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else ua=ua2.d; /* w >= 1/2 */
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if ((z=t1+(zz-ua)) == t1+(zz+ua)) return signArctan2(y,z);
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t1=u-hij[i][0].d;
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EADD(t1,du,v,vv)
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s1=v*(hij[i][11].d+v*(hij[i][12].d+v*(hij[i][13].d+
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v*(hij[i][14].d+v* hij[i][15].d))));
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ADD2(hij[i][9].d,hij[i][10].d,s1,ZERO,s2,ss2,t1,t2)
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MUL2(v,vv,s2,ss2,s1,ss1,t1,t2,t3,t4,t5,t6,t7,t8)
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ADD2(hij[i][7].d,hij[i][8].d,s1,ss1,s2,ss2,t1,t2)
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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 */
|
|
}
|