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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: mpa.c */
/* */
/* FUNCTIONS: */
/* mcr */
/* acr */
/* cpy */
/* 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>
const mp_no mpone = {1, {1.0, 1.0}};
const mp_no mptwo = {1, {1.0, 2.0}};
/* Compare mantissa of two multiple precision numbers regardless of the sign
and exponent of the numbers. */
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;
}
/* Compare 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;
}
/* Copy multiple precision number X into Y. They could be the same
number. */
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;
}
/* 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
long i;
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, Denormal 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];
#define R RADIXI
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 multiple precision number *X into double precision number *Y. The
result is correctly rounded to the nearest/even. */
void __mp_dbl(const mp_no *x, double *y, int p) {
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);
}
/* Get the multiple precision equivalent of X into *Y. If the precision is too
small, the result is truncated. */
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 of *X and *Y assuming that abs (*X) >= abs (*Y) > 0. The
sign of the sum *Z is not changed. X and Y may overlap but not X and Z or
Y and Z. No guard digit is used. The result equals the exact sum,
truncated. */
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;
}
/* Subtract the magnitudes of *X and *Y assuming that abs (*x) > abs (*y) > 0.
The sign of the difference *Z is not changed. X and Y may overlap but not X
and Z or Y and Z. One guard digit is used. The error is less than one
ULP. */
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 *X and *Y and store the result in *Z. X and Y may overlap, but not X
and Z or Y and Z. One guard digit is used. The error is less than one
ULP. */
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 *Y from *X and return the result in *Z. X and Y may overlap but
not X and Z or Y and Z. One guard digit is used. The error is less than
one ULP. */
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 *X and *Y and store result in *Z. X and Y may overlap but not X
and Z or Y and Z. For P in [1, 2, 3], the exact result is truncated to P
digits. In case P > 3 the error is bounded by 1.001 ULP. */
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
/* Rearrange this inner loop to allow the fmadd instructions to be
independent and execute in parallel on processors that have
dual symmetrical 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 = 0.0;
/* 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 original 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 *X and store in *Y. Relative error bound:
- For P = 2: 1.001 * R ^ (1 - P)
- For P = 3: 1.063 * R ^ (1 - P)
- For P > 3: 2.001 * R ^ (1 - P)
*X = 0 is not permissible. */
void __inv(const mp_no *x, mp_no *y, int p) {
long i;
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 *X by *Y and store result in *Z. X and Y may overlap but not X and Z
or Y and Z. Relative error bound:
- For P = 2: 2.001 * R ^ (1 - P)
- For P = 3: 2.063 * R ^ (1 - P)
- For P > 3: 3.001 * R ^ (1 - P)
*X = 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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