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+
+/*
+ * IBM Accurate Mathematical Library
+ * Copyright (c) International Business Machines Corp., 2001
+ *
+ * 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 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 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"
+/* 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) {
+ int i;
+ for (i=1; i<=p; 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) {
+ int 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) {
+ int 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) {
+
+ int i,k;
+
+ EY = EX; k=MIN(m,n);
+ for (i=0; i <= k; i++) Y[i] = X[i];
+ for ( ; i <= n; 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
+ int i,k;
+ 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)
+{
+ int i,k;
+ double a,c,u,v,z[5];
+
+#define R radixi.d
+ if (EX<-44 || (EX==-44 && X[1]<TWO5))
+ { *y=ZERO; return; }
+
+ if (p==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 (p==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 <= p; 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) {
+
+ int i,k;
+ double a,c,u,v,z[5];
+
+ 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) {
+
+ int i,n;
+ 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(p,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<=p; 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) {
+
+ int i,j,k;
+
+ EZ = EX;
+
+ i=p; j=p+ EY - EX; k=p+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<=p; 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) {
+
+ int i,j,k;
+
+ EZ = EX;
+
+ if (EX == EY) {
+ i=j=k=p;
+ Z[k] = Z[k+1] = ZERO; }
+ else {
+ j= EX - EY;
+ if (j > p) {cpy(x,z,p); return; }
+ else {
+ i=p; j=p+1-j; k=p;
+ 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 <= p+1; )
+ Z[k++] = Z[i++];
+ for (; k <= p; )
+ 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) {
+
+ int i, i1, i2, j, k, k2;
+ double u;
+
+ /* Is z=0? */
+ if (X[0]*Y[0]==ZERO)
+ { Z[0]=ZERO; return; }
+
+ /* Multiply, add and carry */
+ k2 = (p<3) ? p+p : p+3;
+ Z[k2]=ZERO;
+ for (k=k2; k>1; ) {
+ if (k > p) {i1=k-p; i2=p+1; }
+ else {i1=1; i2=k; }
+ for (i=i1,j=i2-1; i<i2; i++,j--) Z[k] += X[i]*Y[j];
+
+ u = (Z[k] + CUTTER)-CUTTER;
+ if (u > Z[k]) u -= RADIX;
+ Z[k] -= u;
+ Z[--k] = u*RADIXI;
+ }
+
+ /* Is there a carry beyond the most significant digit? */
+ if (Z[1] == ZERO) {
+ for (i=1; i<=p; 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) {
+ int i,l;
+ 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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