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/*
* IBM Accurate Mathematical Library
* written by International Business Machines Corp.
* Copyright (C) 2001, 2006 Free Software Foundation
*
* 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, write to the Free Software
* Foundation, Inc., 59 Temple Place - Suite 330, Boston, MA 02111-1307, USA.
*/
/************************************************************************/
/* MODULE_NAME: mpa.c */
/* */
/* FUNCTIONS: */
/* mcr */
/* acr */
/* cr */
/* cpy */
/* cpymn */
/* norm */
/* denorm */
/* mp_dbl */
/* dbl_mp */
/* add_magnitudes */
/* sub_magnitudes */
/* add */
/* sub */
/* mul */
/* inv */
/* dvd */
/* */
/* Arithmetic functions for multiple precision numbers. */
/* Relative errors are bounded */
/************************************************************************/
#include "endian.h"
#include "mpa.h"
#include "mpa2.h"
#include <sys/param.h> /* For MIN() */
/* mcr() compares the sizes of the mantissas of two multiple precision */
/* numbers. Mantissas are compared regardless of the signs of the */
/* numbers, even if x->d[0] or y->d[0] are zero. Exponents are also */
/* disregarded. */
static int mcr(const mp_no *x, const mp_no *y, int p) {
long i;
long p2 = p;
for (i=1; i<=p2; i++) {
if (X[i] == Y[i]) continue;
else if (X[i] > Y[i]) return 1;
else return -1; }
return 0;
}
/* acr() compares the absolute values of two multiple precision numbers */
int __acr(const mp_no *x, const mp_no *y, int p) {
long i;
if (X[0] == ZERO) {
if (Y[0] == ZERO) i= 0;
else i=-1;
}
else if (Y[0] == ZERO) i= 1;
else {
if (EX > EY) i= 1;
else if (EX < EY) i=-1;
else i= mcr(x,y,p);
}
return i;
}
/* cr90 compares the values of two multiple precision numbers */
int __cr(const mp_no *x, const mp_no *y, int p) {
int i;
if (X[0] > Y[0]) i= 1;
else if (X[0] < Y[0]) i=-1;
else if (X[0] < ZERO ) i= __acr(y,x,p);
else i= __acr(x,y,p);
return i;
}
/* Copy a multiple precision number. Set *y=*x. x=y is permissible. */
void __cpy(const mp_no *x, mp_no *y, int p) {
long i;
EY = EX;
for (i=0; i <= p; i++) Y[i] = X[i];
return;
}
/* Copy a multiple precision number x of precision m into a */
/* multiple precision number y of precision n. In case n>m, */
/* the digits of y beyond the m'th are set to zero. In case */
/* n<m, the digits of x beyond the n'th are ignored. */
/* x=y is permissible. */
void __cpymn(const mp_no *x, int m, mp_no *y, int n) {
long i,k;
long n2 = n;
long m2 = m;
EY = EX; k=MIN(m2,n2);
for (i=0; i <= k; i++) Y[i] = X[i];
for ( ; i <= n2; i++) Y[i] = ZERO;
return;
}
/* Convert a multiple precision number *x into a double precision */
/* number *y, normalized case (|x| >= 2**(-1022))) */
static void norm(const mp_no *x, double *y, int p)
{
#define R radixi.d
long i;
#if 0
int k;
#endif
double a,c,u,v,z[5];
if (p<5) {
if (p==1) c = X[1];
else if (p==2) c = X[1] + R* X[2];
else if (p==3) c = X[1] + R*(X[2] + R* X[3]);
else if (p==4) c =(X[1] + R* X[2]) + R*R*(X[3] + R*X[4]);
}
else {
for (a=ONE, z[1]=X[1]; z[1] < TWO23; )
{a *= TWO; z[1] *= TWO; }
for (i=2; i<5; i++) {
z[i] = X[i]*a;
u = (z[i] + CUTTER)-CUTTER;
if (u > z[i]) u -= RADIX;
z[i] -= u;
z[i-1] += u*RADIXI;
}
u = (z[3] + TWO71) - TWO71;
if (u > z[3]) u -= TWO19;
v = z[3]-u;
if (v == TWO18) {
if (z[4] == ZERO) {
for (i=5; i <= p; i++) {
if (X[i] == ZERO) continue;
else {z[3] += ONE; break; }
}
}
else z[3] += ONE;
}
c = (z[1] + R *(z[2] + R * z[3]))/a;
}
c *= X[0];
for (i=1; i<EX; i++) c *= RADIX;
for (i=1; i>EX; i--) c *= RADIXI;
*y = c;
return;
#undef R
}
/* Convert a multiple precision number *x into a double precision */
/* number *y, denormalized case (|x| < 2**(-1022))) */
static void denorm(const mp_no *x, double *y, int p)
{
long i,k;
long p2 = p;
double c,u,z[5];
#if 0
double a,v;
#endif
#define R radixi.d
if (EX<-44 || (EX==-44 && X[1]<TWO5))
{ *y=ZERO; return; }
if (p2==1) {
if (EX==-42) {z[1]=X[1]+TWO10; z[2]=ZERO; z[3]=ZERO; k=3;}
else if (EX==-43) {z[1]= TWO10; z[2]=X[1]; z[3]=ZERO; k=2;}
else {z[1]= TWO10; z[2]=ZERO; z[3]=X[1]; k=1;}
}
else if (p2==2) {
if (EX==-42) {z[1]=X[1]+TWO10; z[2]=X[2]; z[3]=ZERO; k=3;}
else if (EX==-43) {z[1]= TWO10; z[2]=X[1]; z[3]=X[2]; k=2;}
else {z[1]= TWO10; z[2]=ZERO; z[3]=X[1]; k=1;}
}
else {
if (EX==-42) {z[1]=X[1]+TWO10; z[2]=X[2]; k=3;}
else if (EX==-43) {z[1]= TWO10; z[2]=X[1]; k=2;}
else {z[1]= TWO10; z[2]=ZERO; k=1;}
z[3] = X[k];
}
u = (z[3] + TWO57) - TWO57;
if (u > z[3]) u -= TWO5;
if (u==z[3]) {
for (i=k+1; i <= p2; i++) {
if (X[i] == ZERO) continue;
else {z[3] += ONE; break; }
}
}
c = X[0]*((z[1] + R*(z[2] + R*z[3])) - TWO10);
*y = c*TWOM1032;
return;
#undef R
}
/* Convert a multiple precision number *x into a double precision number *y. */
/* The result is correctly rounded to the nearest/even. *x is left unchanged */
void __mp_dbl(const mp_no *x, double *y, int p) {
#if 0
int i,k;
double a,c,u,v,z[5];
#endif
if (X[0] == ZERO) {*y = ZERO; return; }
if (EX> -42) norm(x,y,p);
else if (EX==-42 && X[1]>=TWO10) norm(x,y,p);
else denorm(x,y,p);
}
/* dbl_mp() converts a double precision number x into a multiple precision */
/* number *y. If the precision p is too small the result is truncated. x is */
/* left unchanged. */
void __dbl_mp(double x, mp_no *y, int p) {
long i,n;
long p2 = p;
double u;
/* Sign */
if (x == ZERO) {Y[0] = ZERO; return; }
else if (x > ZERO) Y[0] = ONE;
else {Y[0] = MONE; x=-x; }
/* Exponent */
for (EY=ONE; x >= RADIX; EY += ONE) x *= RADIXI;
for ( ; x < ONE; EY -= ONE) x *= RADIX;
/* Digits */
n=MIN(p2,4);
for (i=1; i<=n; i++) {
u = (x + TWO52) - TWO52;
if (u>x) u -= ONE;
Y[i] = u; x -= u; x *= RADIX; }
for ( ; i<=p2; i++) Y[i] = ZERO;
return;
}
/* add_magnitudes() adds the magnitudes of *x & *y assuming that */
/* abs(*x) >= abs(*y) > 0. */
/* The sign of the sum *z is undefined. x&y may overlap but not x&z or y&z. */
/* No guard digit is used. The result equals the exact sum, truncated. */
/* *x & *y are left unchanged. */
static void add_magnitudes(const mp_no *x, const mp_no *y, mp_no *z, int p) {
long i,j,k;
long p2 = p;
EZ = EX;
i=p2; j=p2+ EY - EX; k=p2+1;
if (j<1)
{__cpy(x,z,p); return; }
else Z[k] = ZERO;
for (; j>0; i--,j--) {
Z[k] += X[i] + Y[j];
if (Z[k] >= RADIX) {
Z[k] -= RADIX;
Z[--k] = ONE; }
else
Z[--k] = ZERO;
}
for (; i>0; i--) {
Z[k] += X[i];
if (Z[k] >= RADIX) {
Z[k] -= RADIX;
Z[--k] = ONE; }
else
Z[--k] = ZERO;
}
if (Z[1] == ZERO) {
for (i=1; i<=p2; i++) Z[i] = Z[i+1]; }
else EZ += ONE;
}
/* sub_magnitudes() subtracts the magnitudes of *x & *y assuming that */
/* abs(*x) > abs(*y) > 0. */
/* The sign of the difference *z is undefined. 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. */
static void sub_magnitudes(const mp_no *x, const mp_no *y, mp_no *z, int p) {
long i,j,k;
long p2 = p;
EZ = EX;
if (EX == EY) {
i=j=k=p2;
Z[k] = Z[k+1] = ZERO; }
else {
j= EX - EY;
if (j > p2) {__cpy(x,z,p); return; }
else {
i=p2; j=p2+1-j; k=p2;
if (Y[j] > ZERO) {
Z[k+1] = RADIX - Y[j--];
Z[k] = MONE; }
else {
Z[k+1] = ZERO;
Z[k] = ZERO; j--;}
}
}
for (; j>0; i--,j--) {
Z[k] += (X[i] - Y[j]);
if (Z[k] < ZERO) {
Z[k] += RADIX;
Z[--k] = MONE; }
else
Z[--k] = ZERO;
}
for (; i>0; i--) {
Z[k] += X[i];
if (Z[k] < ZERO) {
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;
}
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