summaryrefslogtreecommitdiffstats
path: root/Objects/doubledigits.c
blob: 1f1c91cce78d8afea57d76606efcb6f7a43932a9 (plain)
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
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
/* Free-format floating point printer
 * 
 * Based on "Floating-Point Printer Sample Code", by Robert G. Burger,
 * http://www.cs.indiana.edu/~burger/fp/index.html
 */

#include "Python.h"

#if defined(__alpha) || defined(__i386) || defined(_M_IX86) || defined(_M_X64) || defined(_M_IA64)
#define LITTLE_ENDIAN_IEEE_DOUBLE
#elif !(defined(__ppc__) || defined(sparc) || defined(__sgi) || defined(_IBMR2) || defined(hpux))
#error unknown machine type
#endif

#if defined(_M_IX86)
#define UNSIGNED64 unsigned __int64
#elif defined(__alpha)
#define UNSIGNED64 unsigned long
#else
#define UNSIGNED64 unsigned long long
#endif

#ifndef U32
#define U32 unsigned int
#endif

/* exponent stored + 1024, hidden bit to left of decimal point */
#define bias 1023
#define bitstoright 52
#define m1mask 0xf
#define hidden_bit 0x100000
#ifdef LITTLE_ENDIAN_IEEE_DOUBLE
struct dblflt {
    unsigned int m4: 16;
    unsigned int m3: 16;
    unsigned int m2: 16;
    unsigned int m1: 4;
    unsigned int e: 11;
    unsigned int s: 1;
};
#else
/* Big Endian IEEE Double Floats */
struct dblflt {
    unsigned int s: 1;
    unsigned int e: 11;
    unsigned int m1: 4;
    unsigned int m2: 16;
    unsigned int m3: 16;
    unsigned int m4: 16;
};
#endif
#define float_radix 2.147483648e9


typedef UNSIGNED64 Bigit;
#define BIGSIZE 24
#define MIN_E -1074
#define MAX_FIVE 325
#define B_P1 ((Bigit)1 << 52)

typedef struct {
   int l;
   Bigit d[BIGSIZE];
} Bignum;

static Bignum R, S, MP, MM, five[MAX_FIVE];
static Bignum S2, S3, S4, S5, S6, S7, S8, S9;
static int ruf, k, s_n, use_mp, qr_shift, sl, slr;

static void mul10(Bignum *x);
static void big_short_mul(Bignum *x, Bigit y, Bignum *z);
/*
static void print_big(Bignum *x);
*/
static int estimate(int n);
static void one_shift_left(int y, Bignum *z);
static void short_shift_left(Bigit x, int y, Bignum *z);
static void big_shift_left(Bignum *x, int y, Bignum *z);
static int big_comp(Bignum *x, Bignum *y);
static int sub_big(Bignum *x, Bignum *y, Bignum *z);
static void add_big(Bignum *x, Bignum *y, Bignum *z);
static int add_cmp(void);
static int qr(void);

/*static int _PyFloat_Digits(char *buf, double v, int *signum);*/
/*static void _PyFloat_DigitsInit(void);*/

#define ADD(x, y, z, k) {\
   Bigit x_add, z_add;\
   x_add = (x);\
   if ((k))\
     z_add = x_add + (y) + 1, (k) = (z_add <= x_add);\
   else\
     z_add = x_add + (y), (k) = (z_add < x_add);\
   (z) = z_add;\
}

#define SUB(x, y, z, b) {\
   Bigit x_sub, y_sub;\
   x_sub = (x); y_sub = (y);\
   if ((b))\
     (z) = x_sub - y_sub - 1, b = (y_sub >= x_sub);\
   else\
     (z) = x_sub - y_sub, b = (y_sub > x_sub);\
}

#define MUL(x, y, z, k) {\
   Bigit x_mul, low, high;\
   x_mul = (x);\
   low = (x_mul & 0xffffffff) * (y) + (k);\
   high = (x_mul >> 32) * (y) + (low >> 32);\
   (k) = high >> 32;\
   (z) = (low & 0xffffffff) | (high << 32);\
}

#define SLL(x, y, z, k) {\
   Bigit x_sll = (x);\
   (z) = (x_sll << (y)) | (k);\
   (k) = x_sll >> (64 - (y));\
}

static void
mul10(Bignum *x)
{
   int i, l;
   Bigit *p, k;

   l = x->l;
   for (i = l, p = &x->d[0], k = 0; i >= 0; i--)
      MUL(*p, 10, *p++, k);
   if (k != 0)
      *p = k, x->l = l+1;
}

static void
big_short_mul(Bignum *x, Bigit y, Bignum *z)
{
   int i, xl, zl;
   Bigit *xp, *zp, k;
   U32 high, low;

   xl = x->l;
   xp = &x->d[0];
   zl = xl;
   zp = &z->d[0];
   high = y >> 32;
   low = y & 0xffffffff;
   for (i = xl, k = 0; i >= 0; i--, xp++, zp++) {
      Bigit xlow, xhigh, z0, t, c, z1;
      xlow = *xp & 0xffffffff;
      xhigh = *xp >> 32;
      z0 = (xlow * low) + k; /* Cout is (z0 < k) */
      t = xhigh * low;
      z1 = (xlow * high) + t;
      c = (z1 < t);
      t = z0 >> 32;
      z1 += t;
      c += (z1 < t);
      *zp = (z1 << 32) | (z0 & 0xffffffff);
      k = (xhigh * high) + (c << 32) + (z1 >> 32) + (z0 < k);
   }
   if (k != 0)
      *zp = k, zl++;
   z->l = zl;
}

/*
static void
print_big(Bignum *x)
{
   int i;
   Bigit *p;

   printf("#x");
   i = x->l;
   p = &x->d[i];
   for (p = &x->d[i]; i >= 0; i--) {
      Bigit b = *p--;
      printf("%08x%08x", (int)(b >> 32), (int)(b & 0xffffffff));
   }
}
*/

static int
estimate(int n)
{
   if (n < 0)
      return (int)(n*0.3010299956639812);
   else
      return 1+(int)(n*0.3010299956639811);
}

static void
one_shift_left(int y, Bignum *z)
{
   int n, m, i;
   Bigit *zp;

   n = y / 64;
   m = y % 64;
   zp = &z->d[0];
   for (i = n; i > 0; i--) *zp++ = 0;
   *zp = (Bigit)1 << m;
   z->l = n;
}

static void
short_shift_left(Bigit x, int y, Bignum *z)
{
   int n, m, i, zl;
   Bigit *zp;
   
   n = y / 64;
   m = y % 64;
   zl = n;
   zp = &(z->d[0]);
   for (i = n; i > 0; i--) *zp++ = 0;
   if (m == 0)
      *zp = x;
   else {
      Bigit high = x >> (64 - m);
      *zp = x << m;
      if (high != 0)
         *++zp = high, zl++;
   }
   z->l = zl;
}

static void
big_shift_left(Bignum *x, int y, Bignum *z)
{
   int n, m, i, xl, zl;
   Bigit *xp, *zp, k;
   
   n = y / 64;
   m = y % 64;
   xl = x->l;
   xp = &(x->d[0]);
   zl = xl + n;
   zp = &(z->d[0]);
   for (i = n; i > 0; i--) *zp++ = 0;
   if (m == 0)
      for (i = xl; i >= 0; i--) *zp++ = *xp++;
   else {
      for (i = xl, k = 0; i >= 0; i--)
         SLL(*xp++, m, *zp++, k);
      if (k != 0)
         *zp = k, zl++;
   }
   z->l = zl;
}


static int
big_comp(Bignum *x, Bignum *y)
{
   int i, xl, yl;
   Bigit *xp, *yp;

   xl = x->l;
   yl = y->l;
   if (xl > yl) return 1;
   if (xl < yl) return -1;
   xp = &x->d[xl];
   yp = &y->d[xl];
   for (i = xl; i >= 0; i--, xp--, yp--) {
      Bigit a = *xp;
      Bigit b = *yp;

      if (a > b) return 1;
      else if (a < b) return -1;
   }
   return 0;
}

static int
sub_big(Bignum *x, Bignum *y, Bignum *z)
{
  int xl, yl, zl, b, i;
  Bigit *xp, *yp, *zp;

  xl = x->l;
  yl = y->l;
  if (yl > xl) return 1;
  xp = &x->d[0];
  yp = &y->d[0];
  zp = &z->d[0];
  
  for (i = yl, b = 0; i >= 0; i--)
    SUB(*xp++, *yp++, *zp++, b);
  for (i = xl-yl; b && i > 0; i--) {
    Bigit x_sub;
    x_sub = *xp++;
    *zp++ = x_sub - 1;
    b = (x_sub == 0);
  }
  for (; i > 0; i--) *zp++ = *xp++;
  if (b) return 1;
  zl = xl;
  while (*--zp == 0) zl--;
  z->l = zl;
  return 0;
}

static void
add_big(Bignum *x, Bignum *y, Bignum *z)
{
  int xl, yl, k, i;
  Bigit *xp, *yp, *zp;

  xl = x->l;
  yl = y->l;
  if (yl > xl) {
    int tl;
    Bignum *tn;
    tl = xl; xl = yl; yl = tl;
    tn = x; x = y; y = tn;
  }

  xp = &x->d[0];
  yp = &y->d[0];
  zp = &z->d[0];
  
  for (i = yl, k = 0; i >= 0; i--)
    ADD(*xp++, *yp++, *zp++, k);
  for (i = xl-yl; k && i > 0; i--) {
    Bigit z_add;
    z_add = *xp++ + 1;
    k = (z_add == 0);
    *zp++ = z_add;
  }
  for (; i > 0; i--) *zp++ = *xp++;
  if (k)
    *zp = 1, z->l = xl+1;
  else
    z->l = xl;
}

static int
add_cmp()
{
   int rl, ml, sl, suml;
   static Bignum sum;

   rl = R.l;
   ml = (use_mp ? MP.l : MM.l);
   sl = S.l;
   
   suml = rl >= ml ? rl : ml;
   if ((sl > suml+1) || ((sl == suml+1) && (S.d[sl] > 1))) return -1;
   if (sl < suml) return 1;

   add_big(&R, (use_mp ? &MP : &MM), &sum);
   return big_comp(&sum, &S);
}

static int
qr()
{
  if (big_comp(&R, &S5) < 0)
    if (big_comp(&R, &S2) < 0)
      if (big_comp(&R, &S) < 0)
        return 0;
      else {
        sub_big(&R, &S, &R);
        return 1;
      }
    else if (big_comp(&R, &S3) < 0) {
      sub_big(&R, &S2, &R);
      return 2;
    }
    else if (big_comp(&R, &S4) < 0) {
      sub_big(&R, &S3, &R);
      return 3;
    }
    else {
      sub_big(&R, &S4, &R);
      return 4;
    }
  else if (big_comp(&R, &S7) < 0)
    if (big_comp(&R, &S6) < 0) {
      sub_big(&R, &S5, &R);
      return 5;
    }
    else {
      sub_big(&R, &S6, &R);
      return 6;
    }
  else if (big_comp(&R, &S9) < 0)
    if (big_comp(&R, &S8) < 0) {
      sub_big(&R, &S7, &R);
      return 7;
    }
    else {
      sub_big(&R, &S8, &R);
      return 8;
    }
  else {
    sub_big(&R, &S9, &R);
    return 9;
  }
}

#define OUTDIG(d) { *buf++ = (d) + '0'; *buf = 0; return k; }

int
_PyFloat_Digits(char *buf, double v, int *signum)
{
   struct dblflt *x;
   int sign, e, f_n, m_n, i, d, tc1, tc2;
   Bigit f;

   /* decompose float into sign, mantissa & exponent */
   x = (struct dblflt *)&v;
   sign = x->s;
   e = x->e; 
   f = (Bigit)(x->m1 << 16 | x->m2) << 32 | (U32)(x->m3 << 16 | x->m4);
   if (e != 0) {
      e = e - bias - bitstoright;
      f |= (Bigit)hidden_bit << 32;
   }
   else if (f != 0)
      /* denormalized */
      e = 1 - bias - bitstoright;

   *signum = sign;
   if (f == 0) {
     *buf++ = '0';
     *buf = 0;
     return 0;
   }
   
   ruf = !(f & 1); /* ruf = (even? f) */

   /* Compute the scaling factor estimate, k */
   if (e > MIN_E)
      k = estimate(e+52);
   else {
      int n;
      Bigit y;
      
      for (n = e+52, y = (Bigit)1 << 52; f < y; n--) y >>= 1;
      k = estimate(n);
   }

   if (e >= 0)
      if (f != B_P1)
         use_mp = 0, f_n = e+1, s_n = 1, m_n = e;
      else
         use_mp = 1, f_n = e+2, s_n = 2, m_n = e;
   else
      if ((e == MIN_E) || (f != B_P1))
         use_mp = 0, f_n = 1, s_n = 1-e, m_n = 0;
      else
         use_mp = 1, f_n = 2, s_n = 2-e, m_n = 0;
   
   /* Scale it! */
   if (k == 0) {
      short_shift_left(f, f_n, &R);
      one_shift_left(s_n, &S);
      one_shift_left(m_n, &MM);
      if (use_mp) one_shift_left(m_n+1, &MP);
      qr_shift = 1;
   }
   else if (k > 0) {
      s_n += k;
      if (m_n >= s_n)
         f_n -= s_n, m_n -= s_n, s_n = 0;
      else 
         f_n -= m_n, s_n -= m_n, m_n = 0;
      short_shift_left(f, f_n, &R);
      big_shift_left(&five[k-1], s_n, &S);
      one_shift_left(m_n, &MM);
      if (use_mp) one_shift_left(m_n+1, &MP);
      qr_shift = 0;
   }
   else {
      Bignum *power = &five[-k-1];

      s_n += k;
      big_short_mul(power, f, &S);
      big_shift_left(&S, f_n, &R);
      one_shift_left(s_n, &S);
      big_shift_left(power, m_n, &MM);
      if (use_mp) big_shift_left(power, m_n+1, &MP);
      qr_shift = 1;
   }

   /* fixup */
   if (add_cmp() <= -ruf) {
     k--;
     mul10(&R);
     mul10(&MM);
     if (use_mp) mul10(&MP);
   }

   /*
   printf("\nk = %d\n", k);
   printf("R = "); print_big(&R);
   printf("\nS = "); print_big(&S);
   printf("\nM- = "); print_big(&MM);
   if (use_mp) printf("\nM+ = "), print_big(&MP);
   putchar('\n');
   fflush(0);
   */
   
   if (qr_shift) {
     sl = s_n / 64;
     slr = s_n % 64;
   }
   else {
     big_shift_left(&S, 1, &S2);
     add_big(&S2, &S, &S3);
     big_shift_left(&S2, 1, &S4);
     add_big(&S4, &S, &S5);
     add_big(&S4, &S2, &S6);
     add_big(&S4, &S3, &S7);
     big_shift_left(&S4, 1, &S8);
     add_big(&S8, &S, &S9);
   }

again:
   if (qr_shift) { /* Take advantage of the fact that S = (ash 1 s_n) */
      if (R.l < sl)
         d = 0;
      else if (R.l == sl) {
            Bigit *p;

            p = &R.d[sl];
            d = *p >> slr;
            *p &= ((Bigit)1 << slr) - 1;
            for (i = sl; (i > 0) && (*p == 0); i--) p--;
            R.l = i;
         }
      else {
         Bigit *p;
         
         p = &R.d[sl+1];
         d = *p << (64 - slr) | *(p-1) >> slr;
         p--;
         *p &= ((Bigit)1 << slr) - 1;
         for (i = sl; (i > 0) && (*p == 0); i--) p--;
         R.l = i;
      }
   }
   else /* We need to do quotient-remainder */
     d = qr();

   tc1 = big_comp(&R, &MM) < ruf;
   tc2 = add_cmp() > -ruf;
   if (!tc1)
      if (!tc2) {
         mul10(&R);
         mul10(&MM);
         if (use_mp) mul10(&MP);
         *buf++ = d + '0';
         goto again;
      }
      else
        OUTDIG(d+1)
   else
      if (!tc2)
         OUTDIG(d)
      else {
         big_shift_left(&R, 1, &MM);
         if (big_comp(&MM, &S) == -1)
            OUTDIG(d)
         else
            OUTDIG(d+1)
      }
}

void
_PyFloat_DigitsInit()
{
   int n, i, l;
   Bignum *b;
   Bigit *xp, *zp, k;

   five[0].l = l = 0;
   five[0].d[0] = 5;
   for (n = MAX_FIVE-1, b = &five[0]; n > 0; n--) {
      xp = &b->d[0];
      b++;
      zp = &b->d[0];
      for (i = l, k = 0; i >= 0; i--)
         MUL(*xp++, 5, *zp++, k);
      if (k != 0)
         *zp = k, l++;
      b->l = l;
   }

   /*
   for (n = 1, b = &five[0]; n <= MAX_FIVE; n++) {
      big_shift_left(b++, n, &R);
      print_big(&R);
      putchar('\n');
   }
   fflush(0);
   */
}