2007-06-03 23:20:06 +02:00
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/*
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* IBM Accurate Mathematical Library
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* written by International Business Machines Corp.
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2013-01-01 17:29:10 +01:00
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* Copyright (C) 2001-2012 Free Software Foundation, Inc.
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2007-06-03 23:20:06 +02:00
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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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2012-02-10 00:18:22 +01:00
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* along with this program; if not, see <http://www.gnu.org/licenses/>.
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2007-06-03 23:20:06 +02:00
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*/
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/************************************************************************/
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/* MODULE_NAME: mpa.c */
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/* */
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/* FUNCTIONS: */
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/* mcr */
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/* acr */
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/* cr */
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/* cpy */
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/* cpymn */
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/* norm */
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/* denorm */
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/* mp_dbl */
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/* dbl_mp */
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/* add_magnitudes */
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/* sub_magnitudes */
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/* add */
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/* sub */
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/* mul */
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/* inv */
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/* dvd */
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/* */
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/* Arithmetic functions for multiple precision numbers. */
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/* Relative errors are bounded */
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/************************************************************************/
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#include "endian.h"
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#include "mpa.h"
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#include "mpa2.h"
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#include <sys/param.h> /* For MIN() */
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/* mcr() compares the sizes of the mantissas of two multiple precision */
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/* numbers. Mantissas are compared regardless of the signs of the */
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/* numbers, even if x->d[0] or y->d[0] are zero. Exponents are also */
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/* disregarded. */
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static int mcr(const mp_no *x, const mp_no *y, int p) {
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long i;
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long p2 = p;
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for (i=1; i<=p2; i++) {
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if (X[i] == Y[i]) continue;
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else if (X[i] > Y[i]) return 1;
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else return -1; }
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return 0;
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}
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/* acr() compares the absolute values of two multiple precision numbers */
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int __acr(const mp_no *x, const mp_no *y, int p) {
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long i;
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if (X[0] == ZERO) {
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if (Y[0] == ZERO) i= 0;
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else i=-1;
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}
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else if (Y[0] == ZERO) i= 1;
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else {
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if (EX > EY) i= 1;
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else if (EX < EY) i=-1;
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else i= mcr(x,y,p);
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}
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return i;
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}
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/* cr90 compares the values of two multiple precision numbers */
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int __cr(const mp_no *x, const mp_no *y, int p) {
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int i;
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if (X[0] > Y[0]) i= 1;
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else if (X[0] < Y[0]) i=-1;
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else if (X[0] < ZERO ) i= __acr(y,x,p);
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else i= __acr(x,y,p);
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return i;
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}
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/* Copy a multiple precision number. Set *y=*x. x=y is permissible. */
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void __cpy(const mp_no *x, mp_no *y, int p) {
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long i;
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EY = EX;
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for (i=0; i <= p; i++) Y[i] = X[i];
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return;
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}
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/* Copy a multiple precision number x of precision m into a */
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/* multiple precision number y of precision n. In case n>m, */
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/* the digits of y beyond the m'th are set to zero. In case */
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/* n<m, the digits of x beyond the n'th are ignored. */
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/* x=y is permissible. */
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void __cpymn(const mp_no *x, int m, mp_no *y, int n) {
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long i,k;
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long n2 = n;
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long m2 = m;
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EY = EX; k=MIN(m2,n2);
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for (i=0; i <= k; i++) Y[i] = X[i];
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for ( ; i <= n2; i++) Y[i] = ZERO;
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return;
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}
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/* Convert a multiple precision number *x into a double precision */
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/* number *y, normalized case (|x| >= 2**(-1022))) */
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static void norm(const mp_no *x, double *y, int p)
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{
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#define R radixi.d
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long i;
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#if 0
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int k;
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#endif
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double a,c,u,v,z[5];
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if (p<5) {
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if (p==1) c = X[1];
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else if (p==2) c = X[1] + R* X[2];
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else if (p==3) c = X[1] + R*(X[2] + R* X[3]);
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else if (p==4) c =(X[1] + R* X[2]) + R*R*(X[3] + R*X[4]);
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}
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else {
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for (a=ONE, z[1]=X[1]; z[1] < TWO23; )
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{a *= TWO; z[1] *= TWO; }
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for (i=2; i<5; i++) {
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z[i] = X[i]*a;
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u = (z[i] + CUTTER)-CUTTER;
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if (u > z[i]) u -= RADIX;
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z[i] -= u;
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z[i-1] += u*RADIXI;
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}
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u = (z[3] + TWO71) - TWO71;
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if (u > z[3]) u -= TWO19;
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v = z[3]-u;
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if (v == TWO18) {
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if (z[4] == ZERO) {
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for (i=5; i <= p; i++) {
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if (X[i] == ZERO) continue;
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else {z[3] += ONE; break; }
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}
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}
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else z[3] += ONE;
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}
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c = (z[1] + R *(z[2] + R * z[3]))/a;
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}
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c *= X[0];
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for (i=1; i<EX; i++) c *= RADIX;
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for (i=1; i>EX; i--) c *= RADIXI;
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*y = c;
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return;
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#undef R
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}
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/* Convert a multiple precision number *x into a double precision */
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/* number *y, denormalized case (|x| < 2**(-1022))) */
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static void denorm(const mp_no *x, double *y, int p)
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{
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long i,k;
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long p2 = p;
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double c,u,z[5];
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#if 0
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double a,v;
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#endif
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#define R radixi.d
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if (EX<-44 || (EX==-44 && X[1]<TWO5))
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{ *y=ZERO; return; }
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if (p2==1) {
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if (EX==-42) {z[1]=X[1]+TWO10; z[2]=ZERO; z[3]=ZERO; k=3;}
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else if (EX==-43) {z[1]= TWO10; z[2]=X[1]; z[3]=ZERO; k=2;}
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else {z[1]= TWO10; z[2]=ZERO; z[3]=X[1]; k=1;}
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}
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else if (p2==2) {
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if (EX==-42) {z[1]=X[1]+TWO10; z[2]=X[2]; z[3]=ZERO; k=3;}
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else if (EX==-43) {z[1]= TWO10; z[2]=X[1]; z[3]=X[2]; k=2;}
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else {z[1]= TWO10; z[2]=ZERO; z[3]=X[1]; k=1;}
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}
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else {
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if (EX==-42) {z[1]=X[1]+TWO10; z[2]=X[2]; k=3;}
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else if (EX==-43) {z[1]= TWO10; z[2]=X[1]; k=2;}
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else {z[1]= TWO10; z[2]=ZERO; k=1;}
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z[3] = X[k];
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}
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u = (z[3] + TWO57) - TWO57;
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if (u > z[3]) u -= TWO5;
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if (u==z[3]) {
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for (i=k+1; i <= p2; i++) {
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if (X[i] == ZERO) continue;
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else {z[3] += ONE; break; }
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}
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}
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c = X[0]*((z[1] + R*(z[2] + R*z[3])) - TWO10);
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*y = c*TWOM1032;
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return;
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#undef R
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}
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/* Convert a multiple precision number *x into a double precision number *y. */
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/* The result is correctly rounded to the nearest/even. *x is left unchanged */
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void __mp_dbl(const mp_no *x, double *y, int p) {
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#if 0
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int i,k;
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double a,c,u,v,z[5];
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#endif
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if (X[0] == ZERO) {*y = ZERO; return; }
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if (EX> -42) norm(x,y,p);
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else if (EX==-42 && X[1]>=TWO10) norm(x,y,p);
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else denorm(x,y,p);
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}
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/* dbl_mp() converts a double precision number x into a multiple precision */
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/* number *y. If the precision p is too small the result is truncated. x is */
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/* left unchanged. */
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void __dbl_mp(double x, mp_no *y, int p) {
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long i,n;
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long p2 = p;
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double u;
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/* Sign */
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if (x == ZERO) {Y[0] = ZERO; return; }
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else if (x > ZERO) Y[0] = ONE;
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else {Y[0] = MONE; x=-x; }
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/* Exponent */
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for (EY=ONE; x >= RADIX; EY += ONE) x *= RADIXI;
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for ( ; x < ONE; EY -= ONE) x *= RADIX;
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/* Digits */
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n=MIN(p2,4);
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for (i=1; i<=n; i++) {
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u = (x + TWO52) - TWO52;
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if (u>x) u -= ONE;
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Y[i] = u; x -= u; x *= RADIX; }
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for ( ; i<=p2; i++) Y[i] = ZERO;
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return;
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}
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/* add_magnitudes() adds the magnitudes of *x & *y assuming that */
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/* abs(*x) >= abs(*y) > 0. */
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/* The sign of the sum *z is undefined. x&y may overlap but not x&z or y&z. */
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/* No guard digit is used. The result equals the exact sum, truncated. */
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/* *x & *y are left unchanged. */
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static void add_magnitudes(const mp_no *x, const mp_no *y, mp_no *z, int p) {
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long i,j,k;
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long p2 = p;
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EZ = EX;
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i=p2; j=p2+ EY - EX; k=p2+1;
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if (j<1)
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{__cpy(x,z,p); return; }
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else Z[k] = ZERO;
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for (; j>0; i--,j--) {
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Z[k] += X[i] + Y[j];
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if (Z[k] >= RADIX) {
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Z[k] -= RADIX;
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Z[--k] = ONE; }
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else
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Z[--k] = ZERO;
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}
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for (; i>0; i--) {
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Z[k] += X[i];
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if (Z[k] >= RADIX) {
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Z[k] -= RADIX;
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Z[--k] = ONE; }
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else
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Z[--k] = ZERO;
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}
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if (Z[1] == ZERO) {
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for (i=1; i<=p2; i++) Z[i] = Z[i+1]; }
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else EZ += ONE;
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}
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/* sub_magnitudes() subtracts the magnitudes of *x & *y assuming that */
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/* abs(*x) > abs(*y) > 0. */
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/* The sign of the difference *z is undefined. x&y may overlap but not x&z */
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/* or y&z. One guard digit is used. The error is less than one ulp. */
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/* *x & *y are left unchanged. */
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static void sub_magnitudes(const mp_no *x, const mp_no *y, mp_no *z, int p) {
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long i,j,k;
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long p2 = p;
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EZ = EX;
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if (EX == EY) {
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i=j=k=p2;
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Z[k] = Z[k+1] = ZERO; }
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else {
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j= EX - EY;
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if (j > p2) {__cpy(x,z,p); return; }
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else {
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i=p2; j=p2+1-j; k=p2;
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if (Y[j] > ZERO) {
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Z[k+1] = RADIX - Y[j--];
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Z[k] = MONE; }
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else {
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Z[k+1] = ZERO;
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Z[k] = ZERO; j--;}
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}
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}
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for (; j>0; i--,j--) {
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Z[k] += (X[i] - Y[j]);
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if (Z[k] < ZERO) {
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Z[k] += RADIX;
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Z[--k] = MONE; }
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else
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Z[--k] = ZERO;
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}
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for (; i>0; i--) {
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Z[k] += X[i];
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if (Z[k] < ZERO) {
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Z[k] += RADIX;
|
|
|
|
Z[--k] = MONE; }
|
|
|
|
else
|
|
|
|
Z[--k] = ZERO;
|
|
|
|
}
|
|
|
|
|
|
|
|
for (i=1; Z[i] == ZERO; i++) ;
|
|
|
|
EZ = EZ - i + 1;
|
|
|
|
for (k=1; i <= p2+1; )
|
|
|
|
Z[k++] = Z[i++];
|
|
|
|
for (; k <= p2; )
|
|
|
|
Z[k++] = ZERO;
|
|
|
|
|
|
|
|
return;
|
|
|
|
}
|
|
|
|
|
|
|
|
|
|
|
|
/* Add two multiple precision numbers. Set *z = *x + *y. x&y may overlap */
|
|
|
|
/* but not x&z or y&z. One guard digit is used. The error is less than */
|
|
|
|
/* one ulp. *x & *y are left unchanged. */
|
|
|
|
|
|
|
|
void __add(const mp_no *x, const mp_no *y, mp_no *z, int p) {
|
|
|
|
|
|
|
|
int n;
|
|
|
|
|
|
|
|
if (X[0] == ZERO) {__cpy(y,z,p); return; }
|
|
|
|
else if (Y[0] == ZERO) {__cpy(x,z,p); return; }
|
|
|
|
|
|
|
|
if (X[0] == Y[0]) {
|
|
|
|
if (__acr(x,y,p) > 0) {add_magnitudes(x,y,z,p); Z[0] = X[0]; }
|
|
|
|
else {add_magnitudes(y,x,z,p); Z[0] = Y[0]; }
|
|
|
|
}
|
|
|
|
else {
|
|
|
|
if ((n=__acr(x,y,p)) == 1) {sub_magnitudes(x,y,z,p); Z[0] = X[0]; }
|
|
|
|
else if (n == -1) {sub_magnitudes(y,x,z,p); Z[0] = Y[0]; }
|
|
|
|
else Z[0] = ZERO;
|
|
|
|
}
|
|
|
|
return;
|
|
|
|
}
|
|
|
|
|
|
|
|
|
|
|
|
/* Subtract two multiple precision numbers. *z is set to *x - *y. x&y may */
|
|
|
|
/* overlap but not x&z or y&z. One guard digit is used. The error is */
|
|
|
|
/* less than one ulp. *x & *y are left unchanged. */
|
|
|
|
|
|
|
|
void __sub(const mp_no *x, const mp_no *y, mp_no *z, int p) {
|
|
|
|
|
|
|
|
int n;
|
|
|
|
|
|
|
|
if (X[0] == ZERO) {__cpy(y,z,p); Z[0] = -Z[0]; return; }
|
|
|
|
else if (Y[0] == ZERO) {__cpy(x,z,p); return; }
|
|
|
|
|
|
|
|
if (X[0] != Y[0]) {
|
|
|
|
if (__acr(x,y,p) > 0) {add_magnitudes(x,y,z,p); Z[0] = X[0]; }
|
|
|
|
else {add_magnitudes(y,x,z,p); Z[0] = -Y[0]; }
|
|
|
|
}
|
|
|
|
else {
|
|
|
|
if ((n=__acr(x,y,p)) == 1) {sub_magnitudes(x,y,z,p); Z[0] = X[0]; }
|
|
|
|
else if (n == -1) {sub_magnitudes(y,x,z,p); Z[0] = -Y[0]; }
|
|
|
|
else Z[0] = ZERO;
|
|
|
|
}
|
|
|
|
return;
|
|
|
|
}
|
|
|
|
|
|
|
|
|
|
|
|
/* Multiply two multiple precision numbers. *z is set to *x * *y. x&y */
|
|
|
|
/* may overlap but not x&z or y&z. In case p=1,2,3 the exact result is */
|
|
|
|
/* truncated to p digits. In case p>3 the error is bounded by 1.001 ulp. */
|
|
|
|
/* *x & *y are left unchanged. */
|
|
|
|
|
|
|
|
void __mul(const mp_no *x, const mp_no *y, mp_no *z, int p) {
|
|
|
|
|
|
|
|
long i, i1, i2, j, k, k2;
|
|
|
|
long p2 = p;
|
|
|
|
double u, zk, zk2;
|
|
|
|
|
|
|
|
/* Is z=0? */
|
|
|
|
if (X[0]*Y[0]==ZERO)
|
|
|
|
{ Z[0]=ZERO; return; }
|
|
|
|
|
|
|
|
/* Multiply, add and carry */
|
|
|
|
k2 = (p2<3) ? p2+p2 : p2+3;
|
|
|
|
zk = Z[k2]=ZERO;
|
|
|
|
for (k=k2; k>1; ) {
|
|
|
|
if (k > p2) {i1=k-p2; i2=p2+1; }
|
|
|
|
else {i1=1; i2=k; }
|
|
|
|
#if 1
|
|
|
|
/* rearange this inner loop to allow the fmadd instructions to be
|
|
|
|
independent and execute in parallel on processors that have
|
|
|
|
dual symetrical FP pipelines. */
|
|
|
|
if (i1 < (i2-1))
|
|
|
|
{
|
|
|
|
/* make sure we have at least 2 iterations */
|
|
|
|
if (((i2 - i1) & 1L) == 1L)
|
|
|
|
{
|
|
|
|
/* Handle the odd iterations case. */
|
|
|
|
zk2 = x->d[i2-1]*y->d[i1];
|
|
|
|
}
|
|
|
|
else
|
|
|
|
zk2 = zero.d;
|
|
|
|
/* Do two multiply/adds per loop iteration, using independent
|
|
|
|
accumulators; zk and zk2. */
|
|
|
|
for (i=i1,j=i2-1; i<i2-1; i+=2,j-=2)
|
|
|
|
{
|
|
|
|
zk += x->d[i]*y->d[j];
|
|
|
|
zk2 += x->d[i+1]*y->d[j-1];
|
|
|
|
}
|
|
|
|
zk += zk2; /* final sum. */
|
|
|
|
}
|
|
|
|
else
|
|
|
|
{
|
|
|
|
/* Special case when iterations is 1. */
|
|
|
|
zk += x->d[i1]*y->d[i1];
|
|
|
|
}
|
|
|
|
#else
|
|
|
|
/* The orginal code. */
|
|
|
|
for (i=i1,j=i2-1; i<i2; i++,j--) zk += X[i]*Y[j];
|
|
|
|
#endif
|
|
|
|
|
|
|
|
u = (zk + CUTTER)-CUTTER;
|
|
|
|
if (u > zk) u -= RADIX;
|
|
|
|
Z[k] = zk - u;
|
|
|
|
zk = u*RADIXI;
|
|
|
|
--k;
|
|
|
|
}
|
|
|
|
Z[k] = zk;
|
|
|
|
|
|
|
|
/* Is there a carry beyond the most significant digit? */
|
|
|
|
if (Z[1] == ZERO) {
|
|
|
|
for (i=1; i<=p2; i++) Z[i]=Z[i+1];
|
|
|
|
EZ = EX + EY - 1; }
|
|
|
|
else
|
|
|
|
EZ = EX + EY;
|
|
|
|
|
|
|
|
Z[0] = X[0] * Y[0];
|
|
|
|
return;
|
|
|
|
}
|
|
|
|
|
|
|
|
|
|
|
|
/* Invert a multiple precision number. Set *y = 1 / *x. */
|
|
|
|
/* Relative error bound = 1.001*r**(1-p) for p=2, 1.063*r**(1-p) for p=3, */
|
|
|
|
/* 2.001*r**(1-p) for p>3. */
|
|
|
|
/* *x=0 is not permissible. *x is left unchanged. */
|
|
|
|
|
|
|
|
void __inv(const mp_no *x, mp_no *y, int p) {
|
|
|
|
long i;
|
|
|
|
#if 0
|
|
|
|
int l;
|
|
|
|
#endif
|
|
|
|
double t;
|
|
|
|
mp_no z,w;
|
|
|
|
static const int np1[] = {0,0,0,0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,
|
|
|
|
4,4,4,4,4,4,4,4,4,4,4,4,4,4,4};
|
|
|
|
const mp_no mptwo = {1,{1.0,2.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,
|
|
|
|
0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,
|
|
|
|
0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,
|
|
|
|
0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0}};
|
|
|
|
|
|
|
|
__cpy(x,&z,p); z.e=0; __mp_dbl(&z,&t,p);
|
|
|
|
t=ONE/t; __dbl_mp(t,y,p); EY -= EX;
|
|
|
|
|
|
|
|
for (i=0; i<np1[p]; i++) {
|
|
|
|
__cpy(y,&w,p);
|
|
|
|
__mul(x,&w,y,p);
|
|
|
|
__sub(&mptwo,y,&z,p);
|
|
|
|
__mul(&w,&z,y,p);
|
|
|
|
}
|
|
|
|
return;
|
|
|
|
}
|
|
|
|
|
|
|
|
|
|
|
|
/* Divide one multiple precision number by another.Set *z = *x / *y. *x & *y */
|
|
|
|
/* are left unchanged. x&y may overlap but not x&z or y&z. */
|
|
|
|
/* Relative error bound = 2.001*r**(1-p) for p=2, 2.063*r**(1-p) for p=3 */
|
|
|
|
/* and 3.001*r**(1-p) for p>3. *y=0 is not permissible. */
|
|
|
|
|
|
|
|
void __dvd(const mp_no *x, const mp_no *y, mp_no *z, int p) {
|
|
|
|
|
|
|
|
mp_no w;
|
|
|
|
|
|
|
|
if (X[0] == ZERO) Z[0] = ZERO;
|
|
|
|
else {__inv(y,&w,p); __mul(x,&w,z,p);}
|
|
|
|
return;
|
|
|
|
}
|