3 * IBM Accurate Mathematical Library
4 * written by International Business Machines Corp.
5 * Copyright (C) 2001 Free Software Foundation
7 * This program is free software; you can redistribute it and/or modify
8 * it under the terms of the GNU Lesser General Public License as published by
9 * the Free Software Foundation; either version 2.1 of the License, or
10 * (at your option) any later version.
12 * This program is distributed in the hope that it will be useful,
13 * but WITHOUT ANY WARRANTY; without even the implied warranty of
14 * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
15 * GNU Lesser General Public License for more details.
17 * You should have received a copy of the GNU Lesser General Public License
18 * along with this program; if not, write to the Free Software
19 * Foundation, Inc., 59 Temple Place - Suite 330, Boston, MA 02111-1307, USA.
21 /************************************************************************/
22 /* MODULE_NAME: mpa.c */
42 /* Arithmetic functions for multiple precision numbers. */
43 /* Relative errors are bounded */
44 /************************************************************************/
51 /* mcr() compares the sizes of the mantissas of two multiple precision */
52 /* numbers. Mantissas are compared regardless of the signs of the */
53 /* numbers, even if x->d[0] or y->d[0] are zero. Exponents are also */
55 static int mcr(const mp_no *x, const mp_no *y, int p) {
57 for (i=1; i<=p; i++) {
58 if (X[i] == Y[i]) continue;
59 else if (X[i] > Y[i]) return 1;
66 /* acr() compares the absolute values of two multiple precision numbers */
67 int __acr(const mp_no *x, const mp_no *y, int p) {
71 if (Y[0] == ZERO) i= 0;
74 else if (Y[0] == ZERO) i= 1;
77 else if (EX < EY) i=-1;
85 /* cr90 compares the values of two multiple precision numbers */
86 int __cr(const mp_no *x, const mp_no *y, int p) {
89 if (X[0] > Y[0]) i= 1;
90 else if (X[0] < Y[0]) i=-1;
91 else if (X[0] < ZERO ) i= __acr(y,x,p);
98 /* Copy a multiple precision number. Set *y=*x. x=y is permissible. */
99 void __cpy(const mp_no *x, mp_no *y, int p) {
103 for (i=0; i <= p; i++) Y[i] = X[i];
109 /* Copy a multiple precision number x of precision m into a */
110 /* multiple precision number y of precision n. In case n>m, */
111 /* the digits of y beyond the m'th are set to zero. In case */
112 /* n<m, the digits of x beyond the n'th are ignored. */
113 /* x=y is permissible. */
115 void __cpymn(const mp_no *x, int m, mp_no *y, int n) {
120 for (i=0; i <= k; i++) Y[i] = X[i];
121 for ( ; i <= n; i++) Y[i] = ZERO;
126 /* Convert a multiple precision number *x into a double precision */
127 /* number *y, normalized case (|x| >= 2**(-1022))) */
128 static void norm(const mp_no *x, double *y, int p)
135 double a,c=c,u,v,z[5];
138 else if (p==2) c = X[1] + R* X[2];
139 else if (p==3) c = X[1] + R*(X[2] + R* X[3]);
140 else if (p==4) c =(X[1] + R* X[2]) + R*R*(X[3] + R*X[4]);
143 for (a=ONE, z[1]=X[1]; z[1] < TWO23; )
144 {a *= TWO; z[1] *= TWO; }
146 for (i=2; i<5; i++) {
148 u = (z[i] + CUTTER)-CUTTER;
149 if (u > z[i]) u -= RADIX;
154 u = (z[3] + TWO71) - TWO71;
155 if (u > z[3]) u -= TWO19;
160 for (i=5; i <= p; i++) {
161 if (X[i] == ZERO) continue;
162 else {z[3] += ONE; break; }
168 c = (z[1] + R *(z[2] + R * z[3]))/a;
173 for (i=1; i<EX; i++) c *= RADIX;
174 for (i=1; i>EX; i--) c *= RADIXI;
181 /* Convert a multiple precision number *x into a double precision */
182 /* number *y, denormalized case (|x| < 2**(-1022))) */
183 static void denorm(const mp_no *x, double *y, int p)
192 if (EX<-44 || (EX==-44 && X[1]<TWO5))
196 if (EX==-42) {z[1]=X[1]+TWO10; z[2]=ZERO; z[3]=ZERO; k=3;}
197 else if (EX==-43) {z[1]= TWO10; z[2]=X[1]; z[3]=ZERO; k=2;}
198 else {z[1]= TWO10; z[2]=ZERO; z[3]=X[1]; k=1;}
201 if (EX==-42) {z[1]=X[1]+TWO10; z[2]=X[2]; z[3]=ZERO; k=3;}
202 else if (EX==-43) {z[1]= TWO10; z[2]=X[1]; z[3]=X[2]; k=2;}
203 else {z[1]= TWO10; z[2]=ZERO; z[3]=X[1]; k=1;}
206 if (EX==-42) {z[1]=X[1]+TWO10; z[2]=X[2]; k=3;}
207 else if (EX==-43) {z[1]= TWO10; z[2]=X[1]; k=2;}
208 else {z[1]= TWO10; z[2]=ZERO; k=1;}
212 u = (z[3] + TWO57) - TWO57;
213 if (u > z[3]) u -= TWO5;
216 for (i=k+1; i <= p; i++) {
217 if (X[i] == ZERO) continue;
218 else {z[3] += ONE; break; }
222 c = X[0]*((z[1] + R*(z[2] + R*z[3])) - TWO10);
230 /* Convert a multiple precision number *x into a double precision number *y. */
231 /* The result is correctly rounded to the nearest/even. *x is left unchanged */
233 void __mp_dbl(const mp_no *x, double *y, int p) {
239 if (X[0] == ZERO) {*y = ZERO; return; }
241 if (EX> -42) norm(x,y,p);
242 else if (EX==-42 && X[1]>=TWO10) norm(x,y,p);
247 /* dbl_mp() converts a double precision number x into a multiple precision */
248 /* number *y. If the precision p is too small the result is truncated. x is */
249 /* left unchanged. */
251 void __dbl_mp(double x, mp_no *y, int p) {
257 if (x == ZERO) {Y[0] = ZERO; return; }
258 else if (x > ZERO) Y[0] = ONE;
259 else {Y[0] = MONE; x=-x; }
262 for (EY=ONE; x >= RADIX; EY += ONE) x *= RADIXI;
263 for ( ; x < ONE; EY -= ONE) x *= RADIX;
267 for (i=1; i<=n; i++) {
268 u = (x + TWO52) - TWO52;
270 Y[i] = u; x -= u; x *= RADIX; }
271 for ( ; i<=p; i++) Y[i] = ZERO;
276 /* add_magnitudes() adds the magnitudes of *x & *y assuming that */
277 /* abs(*x) >= abs(*y) > 0. */
278 /* The sign of the sum *z is undefined. x&y may overlap but not x&z or y&z. */
279 /* No guard digit is used. The result equals the exact sum, truncated. */
280 /* *x & *y are left unchanged. */
282 static void add_magnitudes(const mp_no *x, const mp_no *y, mp_no *z, int p) {
288 i=p; j=p+ EY - EX; k=p+1;
291 {__cpy(x,z,p); return; }
294 for (; j>0; i--,j--) {
313 for (i=1; i<=p; i++) Z[i] = Z[i+1]; }
318 /* sub_magnitudes() subtracts the magnitudes of *x & *y assuming that */
319 /* abs(*x) > abs(*y) > 0. */
320 /* The sign of the difference *z is undefined. x&y may overlap but not x&z */
321 /* or y&z. One guard digit is used. The error is less than one ulp. */
322 /* *x & *y are left unchanged. */
324 static void sub_magnitudes(const mp_no *x, const mp_no *y, mp_no *z, int p) {
332 Z[k] = Z[k+1] = ZERO; }
335 if (j > p) {__cpy(x,z,p); return; }
339 Z[k+1] = RADIX - Y[j--];
347 for (; j>0; i--,j--) {
348 Z[k] += (X[i] - Y[j]);
365 for (i=1; Z[i] == ZERO; i++) ;
367 for (k=1; i <= p+1; )
376 /* Add two multiple precision numbers. Set *z = *x + *y. x&y may overlap */
377 /* but not x&z or y&z. One guard digit is used. The error is less than */
378 /* one ulp. *x & *y are left unchanged. */
380 void __add(const mp_no *x, const mp_no *y, mp_no *z, int p) {
384 if (X[0] == ZERO) {__cpy(y,z,p); return; }
385 else if (Y[0] == ZERO) {__cpy(x,z,p); return; }
388 if (__acr(x,y,p) > 0) {add_magnitudes(x,y,z,p); Z[0] = X[0]; }
389 else {add_magnitudes(y,x,z,p); Z[0] = Y[0]; }
392 if ((n=__acr(x,y,p)) == 1) {sub_magnitudes(x,y,z,p); Z[0] = X[0]; }
393 else if (n == -1) {sub_magnitudes(y,x,z,p); Z[0] = Y[0]; }
400 /* Subtract two multiple precision numbers. *z is set to *x - *y. x&y may */
401 /* overlap but not x&z or y&z. One guard digit is used. The error is */
402 /* less than one ulp. *x & *y are left unchanged. */
404 void __sub(const mp_no *x, const mp_no *y, mp_no *z, int p) {
408 if (X[0] == ZERO) {__cpy(y,z,p); Z[0] = -Z[0]; return; }
409 else if (Y[0] == ZERO) {__cpy(x,z,p); return; }
412 if (__acr(x,y,p) > 0) {add_magnitudes(x,y,z,p); Z[0] = X[0]; }
413 else {add_magnitudes(y,x,z,p); Z[0] = -Y[0]; }
416 if ((n=__acr(x,y,p)) == 1) {sub_magnitudes(x,y,z,p); Z[0] = X[0]; }
417 else if (n == -1) {sub_magnitudes(y,x,z,p); Z[0] = -Y[0]; }
424 /* Multiply two multiple precision numbers. *z is set to *x * *y. x&y */
425 /* may overlap but not x&z or y&z. In case p=1,2,3 the exact result is */
426 /* truncated to p digits. In case p>3 the error is bounded by 1.001 ulp. */
427 /* *x & *y are left unchanged. */
429 void __mul(const mp_no *x, const mp_no *y, mp_no *z, int p) {
431 int i, i1, i2, j, k, k2;
436 { Z[0]=ZERO; return; }
438 /* Multiply, add and carry */
439 k2 = (p<3) ? p+p : p+3;
442 if (k > p) {i1=k-p; i2=p+1; }
444 for (i=i1,j=i2-1; i<i2; i++,j--) Z[k] += X[i]*Y[j];
446 u = (Z[k] + CUTTER)-CUTTER;
447 if (u > Z[k]) u -= RADIX;
452 /* Is there a carry beyond the most significant digit? */
454 for (i=1; i<=p; i++) Z[i]=Z[i+1];
464 /* Invert a multiple precision number. Set *y = 1 / *x. */
465 /* Relative error bound = 1.001*r**(1-p) for p=2, 1.063*r**(1-p) for p=3, */
466 /* 2.001*r**(1-p) for p>3. */
467 /* *x=0 is not permissible. *x is left unchanged. */
469 void __inv(const mp_no *x, mp_no *y, int p) {
476 static const int np1[] = {0,0,0,0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,
477 4,4,4,4,4,4,4,4,4,4,4,4,4,4,4};
478 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,
479 0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,
480 0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,
481 0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0}};
483 __cpy(x,&z,p); z.e=0; __mp_dbl(&z,&t,p);
484 t=ONE/t; __dbl_mp(t,y,p); EY -= EX;
486 for (i=0; i<np1[p]; i++) {
489 __sub(&mptwo,y,&z,p);
496 /* Divide one multiple precision number by another.Set *z = *x / *y. *x & *y */
497 /* are left unchanged. x&y may overlap but not x&z or y&z. */
498 /* Relative error bound = 2.001*r**(1-p) for p=2, 2.063*r**(1-p) for p=3 */
499 /* and 3.001*r**(1-p) for p>3. *y=0 is not permissible. */
501 void __dvd(const mp_no *x, const mp_no *y, mp_no *z, int p) {
505 if (X[0] == ZERO) Z[0] = ZERO;
506 else {__inv(y,&w,p); __mul(x,&w,z,p);}