about summary refs log tree commit diff
path: root/sysdeps/ieee754/dbl-64/mpa.c
blob: cdd15a1602e945c88c73f889716274b57bbaea5b (plain) (blame)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497

/*
 * 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;
}