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
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
686
687
688
689
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
720
721
722
723
724
725
726
727
728
729
730
731
732
733
734
735
736
737
738
739
740
741
742
743
744
745
746
747
748
749
750
751
752
753
754
755
756
757
758
759
760
761
762
763
764
765
766
767
768
769
770
771
772
773
774
775
776
777
778
779
780
781
782
783
784
785
786
787
788
789
790
791
792
793
794
|
/* mpfr_div -- divide two floating-point numbers
Copyright 1999, 2001-2016 Free Software Foundation, Inc.
Contributed by the AriC and Caramel projects, INRIA.
This file is part of the GNU MPFR Library.
The GNU MPFR Library 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 3 of the License, or (at your
option) any later version.
The GNU MPFR Library 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 the GNU MPFR Library; see the file COPYING.LESSER. If not, see
http://www.gnu.org/licenses/ or write to the Free Software Foundation, Inc.,
51 Franklin St, Fifth Floor, Boston, MA 02110-1301, USA. */
/* References:
[1] Short Division of Long Integers, David Harvey and Paul Zimmermann,
Proceedings of the 20th Symposium on Computer Arithmetic (ARITH-20),
July 25-27, 2011, pages 7-14.
*/
#define MPFR_NEED_LONGLONG_H
#include "mpfr-impl.h"
#ifdef DEBUG2
#define mpfr_mpn_print(ap,n) mpfr_mpn_print3 (ap,n,MPFR_LIMB_ZERO)
static void
mpfr_mpn_print3 (mpfr_limb_ptr ap, mp_size_t n, mp_limb_t cy)
{
mp_size_t i;
for (i = 0; i < n; i++)
printf ("+%lu*2^%lu", (unsigned long) ap[i], (unsigned long)
(GMP_NUMB_BITS * i));
if (cy)
printf ("+2^%lu", (unsigned long) (GMP_NUMB_BITS * n));
printf ("\n");
}
#endif
/* check if {ap, an} is zero */
static int
mpfr_mpn_cmpzero (mpfr_limb_ptr ap, mp_size_t an)
{
while (an > 0)
if (MPFR_LIKELY(ap[--an] != MPFR_LIMB_ZERO))
return 1;
return 0;
}
/* compare {ap, an} and {bp, bn} >> extra,
aligned by the more significant limbs.
Takes into account bp[0] for extra=1.
*/
static int
mpfr_mpn_cmp_aux (mpfr_limb_ptr ap, mp_size_t an,
mpfr_limb_ptr bp, mp_size_t bn, int extra)
{
int cmp = 0;
mp_size_t k;
mp_limb_t bb;
if (an >= bn)
{
k = an - bn;
while (cmp == 0 && bn > 0)
{
bn --;
bb = (extra) ? ((bp[bn+1] << (GMP_NUMB_BITS - 1)) | (bp[bn] >> 1))
: bp[bn];
cmp = (ap[k + bn] > bb) ? 1 : ((ap[k + bn] < bb) ? -1 : 0);
}
bb = (extra) ? bp[0] << (GMP_NUMB_BITS - 1) : MPFR_LIMB_ZERO;
while (cmp == 0 && k > 0)
{
k--;
cmp = (ap[k] > bb) ? 1 : ((ap[k] < bb) ? -1 : 0);
bb = MPFR_LIMB_ZERO; /* ensure we consider only once bp[0] & 1 */
}
if (cmp == 0 && bb != MPFR_LIMB_ZERO)
cmp = -1;
}
else /* an < bn */
{
k = bn - an;
while (cmp == 0 && an > 0)
{
an --;
bb = (extra) ? ((bp[k+an+1] << (GMP_NUMB_BITS - 1)) | (bp[k+an] >> 1))
: bp[k+an];
if (ap[an] > bb)
cmp = 1;
else if (ap[an] < bb)
cmp = -1;
}
while (cmp == 0 && k > 0)
{
k--;
bb = (extra) ? ((bp[k+1] << (GMP_NUMB_BITS - 1)) | (bp[k] >> 1))
: bp[k];
cmp = (bb != MPFR_LIMB_ZERO) ? -1 : 0;
}
if (cmp == 0 && extra && (bp[0] & MPFR_LIMB_ONE))
cmp = -1;
}
return cmp;
}
/* {ap, n} <- {ap, n} - {bp, n} >> extra - cy, with cy = 0 or 1.
Return borrow out.
*/
static mp_limb_t
mpfr_mpn_sub_aux (mpfr_limb_ptr ap, mpfr_limb_ptr bp, mp_size_t n,
mp_limb_t cy, int extra)
{
mp_limb_t bb, rp;
MPFR_ASSERTD (cy <= 1);
while (n--)
{
bb = (extra) ? ((bp[1] << (GMP_NUMB_BITS-1)) | (bp[0] >> 1)) : bp[0];
rp = ap[0] - bb - cy;
cy = (ap[0] < bb) || (cy && ~rp == MPFR_LIMB_ZERO) ?
MPFR_LIMB_ONE : MPFR_LIMB_ZERO;
ap[0] = rp;
ap ++;
bp ++;
}
MPFR_ASSERTD (cy <= 1);
return cy;
}
int
mpfr_div (mpfr_ptr q, mpfr_srcptr u, mpfr_srcptr v, mpfr_rnd_t rnd_mode)
{
mp_size_t q0size = MPFR_LIMB_SIZE(q); /* number of limbs of destination */
mp_size_t usize = MPFR_LIMB_SIZE(u);
mp_size_t vsize = MPFR_LIMB_SIZE(v);
mp_size_t qsize; /* number of limbs wanted for the computed quotient */
mp_size_t qqsize;
mp_size_t k;
mpfr_limb_ptr q0p = MPFR_MANT(q), qp;
mpfr_limb_ptr up = MPFR_MANT(u);
mpfr_limb_ptr vp = MPFR_MANT(v);
mpfr_limb_ptr ap;
mpfr_limb_ptr bp;
mp_limb_t qh;
mp_limb_t sticky_u = MPFR_LIMB_ZERO;
mp_limb_t low_u;
mp_limb_t sticky_v = MPFR_LIMB_ZERO;
mp_limb_t sticky;
mp_limb_t sticky3;
mp_limb_t round_bit = MPFR_LIMB_ZERO;
mpfr_exp_t qexp;
int sign_quotient;
int extra_bit;
int sh, sh2;
int inex;
int like_rndz;
MPFR_TMP_DECL(marker);
MPFR_LOG_FUNC (
("u[%Pu]=%.*Rg v[%Pu]=%.*Rg rnd=%d",
mpfr_get_prec(u), mpfr_log_prec, u,
mpfr_get_prec (v),mpfr_log_prec, v, rnd_mode),
("q[%Pu]=%.*Rg inexact=%d", mpfr_get_prec(q), mpfr_log_prec, q, inex));
/**************************************************************************
* *
* This part of the code deals with special cases *
* *
**************************************************************************/
if (MPFR_UNLIKELY(MPFR_ARE_SINGULAR(u,v)))
{
if (MPFR_IS_NAN(u) || MPFR_IS_NAN(v))
{
MPFR_SET_NAN(q);
MPFR_RET_NAN;
}
sign_quotient = MPFR_MULT_SIGN( MPFR_SIGN(u) , MPFR_SIGN(v) );
MPFR_SET_SIGN(q, sign_quotient);
if (MPFR_IS_INF(u))
{
if (MPFR_IS_INF(v))
{
MPFR_SET_NAN(q);
MPFR_RET_NAN;
}
else
{
MPFR_SET_INF(q);
MPFR_RET(0);
}
}
else if (MPFR_IS_INF(v))
{
MPFR_SET_ZERO (q);
MPFR_RET (0);
}
else if (MPFR_IS_ZERO (v))
{
if (MPFR_IS_ZERO (u))
{
MPFR_SET_NAN(q);
MPFR_RET_NAN;
}
else
{
MPFR_ASSERTD (! MPFR_IS_INF (u));
MPFR_SET_INF(q);
mpfr_set_divby0 ();
MPFR_RET(0);
}
}
else
{
MPFR_ASSERTD (MPFR_IS_ZERO (u));
MPFR_SET_ZERO (q);
MPFR_RET (0);
}
}
/**************************************************************************
* *
* End of the part concerning special values. *
* *
**************************************************************************/
MPFR_TMP_MARK(marker);
/* set sign */
sign_quotient = MPFR_MULT_SIGN( MPFR_SIGN(u) , MPFR_SIGN(v) );
MPFR_SET_SIGN(q, sign_quotient);
/* determine if an extra bit comes from the division, i.e. if the
significand of u (as a fraction in [1/2, 1[) is larger than that
of v */
if (MPFR_LIKELY(up[usize - 1] != vp[vsize - 1]))
extra_bit = (up[usize - 1] > vp[vsize - 1]) ? 1 : 0;
else /* most significant limbs are equal, must look at further limbs */
{
mp_size_t l;
k = usize - 1;
l = vsize - 1;
while (k != 0 && l != 0 && up[--k] == vp[--l]);
/* now k=0 or l=0 or up[k] != vp[l] */
if (up[k] > vp[l])
extra_bit = 1;
else if (up[k] < vp[l])
extra_bit = 0;
/* now up[k] = vp[l], thus either k=0 or l=0 */
else if (l == 0) /* no more divisor limb */
extra_bit = 1;
else /* k=0: no more dividend limb */
extra_bit = mpfr_mpn_cmpzero (vp, l) == 0;
}
#ifdef DEBUG
printf ("extra_bit=%d\n", extra_bit);
#endif
/* set exponent */
qexp = MPFR_GET_EXP (u) - MPFR_GET_EXP (v) + extra_bit;
/* sh is the number of zero bits in the low limb of the quotient */
MPFR_UNSIGNED_MINUS_MODULO(sh, MPFR_PREC(q));
like_rndz = rnd_mode == MPFR_RNDZ ||
rnd_mode == (sign_quotient < 0 ? MPFR_RNDU : MPFR_RNDD);
/**************************************************************************
* *
* We first try Mulders' short division (for large operands) *
* *
**************************************************************************/
if (MPFR_UNLIKELY(q0size >= MPFR_DIV_THRESHOLD &&
vsize >= MPFR_DIV_THRESHOLD))
{
mp_size_t n = q0size + 1; /* we will perform a short (2n)/n division */
mpfr_limb_ptr ap, bp, qp;
mpfr_prec_t p;
/* since Mulders' short division clobbers the dividend, we have to
copy it */
ap = MPFR_TMP_LIMBS_ALLOC (n + n);
if (usize >= n + n) /* truncate the dividend */
MPN_COPY(ap, up + usize - (n + n), n + n);
else /* zero-pad the dividend */
{
MPN_COPY(ap + (n + n) - usize, up, usize);
MPN_ZERO(ap, (n + n) - usize);
}
if (vsize >= n) /* truncate the divisor */
bp = vp + vsize - n;
else /* zero-pad the divisor */
{
bp = MPFR_TMP_LIMBS_ALLOC (n);
MPN_COPY(bp + n - vsize, vp, vsize);
MPN_ZERO(bp, n - vsize);
}
qp = MPFR_TMP_LIMBS_ALLOC (n);
qh = mpfr_divhigh_n (qp, ap, bp, n);
/* in all cases, the error is at most (2n+2) ulps on qh*B^n+{qp,n},
cf algorithms.tex */
p = n * GMP_NUMB_BITS - MPFR_INT_CEIL_LOG2 (2 * n + 2);
/* if qh is 1, then we need only PREC(q)-1 bits of {qp,n},
if rnd=RNDN, we need to be able to round with a directed rounding
and one more bit */
if (MPFR_LIKELY (mpfr_round_p (qp, n, p,
MPFR_PREC(q) + (rnd_mode == MPFR_RNDN) - qh)))
{
/* we can round correctly whatever the rounding mode */
if (qh == 0)
MPN_COPY (q0p, qp + 1, q0size);
else
{
mpn_rshift (q0p, qp + 1, q0size, 1);
q0p[q0size - 1] ^= MPFR_LIMB_HIGHBIT;
}
q0p[0] &= ~MPFR_LIMB_MASK(sh); /* put to zero low sh bits */
if (rnd_mode == MPFR_RNDN) /* round to nearest */
{
/* we know we can round, thus we are never in the even rule case:
if the round bit is 0, we truncate
if the round bit is 1, we add 1 */
if (qh == 0)
{
if (sh > 0)
round_bit = (qp[1] >> (sh - 1)) & 1;
else
round_bit = qp[0] >> (GMP_NUMB_BITS - 1);
}
else /* qh = 1 */
round_bit = (qp[1] >> sh) & 1;
if (round_bit == 0)
{
inex = -1;
goto truncate;
}
else /* round_bit = 1 */
goto add_one_ulp;
}
else if (like_rndz == 0) /* round away */
goto add_one_ulp;
/* else round to zero: nothing to do */
else
{
inex = -1;
goto truncate;
}
}
}
/**************************************************************************
* *
* Mulders' short division failed: we revert to integer division *
* *
**************************************************************************/
if (MPFR_UNLIKELY(rnd_mode == MPFR_RNDN && sh == 0))
{ /* we compute the quotient with one more limb, in order to get
the round bit in the quotient, and the remainder only contains
sticky bits */
qsize = q0size + 1;
/* need to allocate memory for the quotient */
qp = MPFR_TMP_LIMBS_ALLOC (qsize);
}
else
{
qsize = q0size;
qp = q0p; /* directly put the quotient in the destination */
}
qqsize = qsize + qsize;
/* prepare the dividend */
ap = MPFR_TMP_LIMBS_ALLOC (qqsize);
if (MPFR_LIKELY(qqsize > usize)) /* use the full dividend */
{
k = qqsize - usize; /* k > 0 */
MPN_ZERO(ap, k);
if (extra_bit)
ap[k - 1] = mpn_rshift (ap + k, up, usize, 1);
else
MPN_COPY(ap + k, up, usize);
}
else /* truncate the dividend */
{
k = usize - qqsize;
if (extra_bit)
sticky_u = mpn_rshift (ap, up + k, qqsize, 1);
else
MPN_COPY(ap, up + k, qqsize);
sticky_u = sticky_u || mpfr_mpn_cmpzero (up, k);
}
low_u = sticky_u;
/* now sticky_u is non-zero iff the truncated part of u is non-zero */
/* prepare the divisor */
if (MPFR_LIKELY(vsize >= qsize))
{
k = vsize - qsize;
if (qp != vp)
bp = vp + k; /* avoid copying the divisor */
else /* need to copy, since mpn_divrem doesn't allow overlap
between quotient and divisor, necessarily k = 0
since quotient and divisor are the same mpfr variable */
{
bp = MPFR_TMP_LIMBS_ALLOC (qsize);
MPN_COPY(bp, vp, vsize);
}
sticky_v = sticky_v || mpfr_mpn_cmpzero (vp, k);
k = 0;
}
else /* vsize < qsize: small divisor case */
{
bp = vp;
k = qsize - vsize;
}
/**************************************************************************
* *
* Here we perform the real division of {ap+k,qqsize-k} by {bp,qsize-k} *
* *
**************************************************************************/
/* if Mulders' short division failed, we revert to division with remainder */
qh = mpn_divrem (qp, 0, ap + k, qqsize - k, bp, qsize - k);
/* warning: qh may be 1 if u1 == v1, but u < v */
#ifdef DEBUG2
printf ("q="); mpfr_mpn_print (qp, qsize);
printf ("r="); mpfr_mpn_print (ap, qsize);
#endif
k = qsize;
sticky_u = sticky_u || mpfr_mpn_cmpzero (ap, k);
sticky = sticky_u | sticky_v;
/* now sticky is non-zero iff one of the following holds:
(a) the truncated part of u is non-zero
(b) the truncated part of v is non-zero
(c) the remainder from division is non-zero */
if (MPFR_LIKELY(qsize == q0size))
{
sticky3 = qp[0] & MPFR_LIMB_MASK(sh); /* does nothing when sh=0 */
sh2 = sh;
}
else /* qsize = q0size + 1: only happens when rnd_mode=MPFR_RNDN and sh=0 */
{
MPN_COPY (q0p, qp + 1, q0size);
sticky3 = qp[0];
sh2 = GMP_NUMB_BITS;
}
qp[0] ^= sticky3;
/* sticky3 contains the truncated bits from the quotient,
including the round bit, and 1 <= sh2 <= GMP_NUMB_BITS
is the number of bits in sticky3 */
inex = (sticky != MPFR_LIMB_ZERO) || (sticky3 != MPFR_LIMB_ZERO);
#ifdef DEBUG
printf ("sticky=%lu sticky3=%lu inex=%d\n",
(unsigned long) sticky, (unsigned long) sticky3, inex);
#endif
/* to round, we distinguish two cases:
(a) vsize <= qsize: we used the full divisor
(b) vsize > qsize: the divisor was truncated
*/
#ifdef DEBUG
printf ("vsize=%lu qsize=%lu\n",
(unsigned long) vsize, (unsigned long) qsize);
#endif
if (MPFR_LIKELY(vsize <= qsize)) /* use the full divisor */
{
if (MPFR_LIKELY(rnd_mode == MPFR_RNDN))
{
round_bit = sticky3 & (MPFR_LIMB_ONE << (sh2 - 1));
sticky = (sticky3 ^ round_bit) | sticky_u;
}
else if (like_rndz || inex == 0)
sticky = (inex == 0) ? MPFR_LIMB_ZERO : MPFR_LIMB_ONE;
else /* round away from zero */
sticky = MPFR_LIMB_ONE;
goto case_1;
}
else /* vsize > qsize: need to truncate the divisor */
{
if (inex == 0)
goto truncate;
else
{
/* We know the estimated quotient is an upper bound of the exact
quotient (with rounding toward zero), with a difference of at
most 2 in qp[0].
Thus we can round except when sticky3 is 000...000 or 000...001
for directed rounding, and 100...000 or 100...001 for rounding
to nearest. (For rounding to nearest, we cannot determine the
inexact flag for 000...000 or 000...001.)
*/
mp_limb_t sticky3orig = sticky3;
if (rnd_mode == MPFR_RNDN)
{
round_bit = sticky3 & (MPFR_LIMB_ONE << (sh2 - 1));
sticky3 = sticky3 ^ round_bit;
#ifdef DEBUG
printf ("rb=%lu sb=%lu\n",
(unsigned long) round_bit, (unsigned long) sticky3);
#endif
}
if (sticky3 != MPFR_LIMB_ZERO && sticky3 != MPFR_LIMB_ONE)
{
sticky = sticky3;
goto case_1;
}
else /* hard case: we have to compare q1 * v0 and r + low(u),
where q1 * v0 has qsize + (vsize-qsize) = vsize limbs, and
r + low(u) has qsize + (usize-2*qsize) = usize-qsize limbs */
{
mp_size_t l;
mpfr_limb_ptr sp;
int cmp_s_r;
mp_limb_t qh2;
sp = MPFR_TMP_LIMBS_ALLOC (vsize);
k = vsize - qsize;
/* sp <- {qp, qsize} * {vp, vsize-qsize} */
qp[0] ^= sticky3orig; /* restore original quotient */
if (qsize >= k)
mpn_mul (sp, qp, qsize, vp, k);
else
mpn_mul (sp, vp, k, qp, qsize);
if (qh)
qh2 = mpn_add_n (sp + qsize, sp + qsize, vp, k);
else
qh2 = (mp_limb_t) 0;
qp[0] ^= sticky3orig; /* restore truncated quotient */
/* compare qh2 + {sp, k + qsize} to {ap, qsize} + low(u) */
cmp_s_r = (qh2 != 0) ? 1 : mpn_cmp (sp + k, ap, qsize);
if (cmp_s_r == 0) /* compare {sp, k} and low(u) */
{
cmp_s_r = (usize >= qqsize) ?
mpfr_mpn_cmp_aux (sp, k, up, usize - qqsize, extra_bit) :
mpfr_mpn_cmpzero (sp, k);
}
#ifdef DEBUG
printf ("cmp(q*v0,r+u0)=%d\n", cmp_s_r);
#endif
/* now cmp_s_r > 0 if {sp, vsize} > {ap, qsize} + low(u)
cmp_s_r = 0 if {sp, vsize} = {ap, qsize} + low(u)
cmp_s_r < 0 if {sp, vsize} < {ap, qsize} + low(u) */
if (cmp_s_r <= 0) /* quotient is in [q1, q1+1) */
{
sticky = (cmp_s_r == 0) ? sticky3 : MPFR_LIMB_ONE;
goto case_1;
}
else /* cmp_s_r > 0, quotient is < q1: to determine if it is
in [q1-2,q1-1] or in [q1-1,q1], we need to subtract
the low part u0 of the dividend u0 from q*v0 */
{
mp_limb_t cy = MPFR_LIMB_ZERO;
/* subtract low(u)>>extra_bit if non-zero */
if (qh2 != 0) /* whatever the value of {up, m + k}, it
will be smaller than qh2 + {sp, k} */
cmp_s_r = 1;
else
{
if (low_u != MPFR_LIMB_ZERO)
{
mp_size_t m;
l = usize - qqsize; /* number of low limbs in u */
m = (l > k) ? l - k : 0;
cy = (extra_bit) ?
(up[m] & MPFR_LIMB_ONE) : MPFR_LIMB_ZERO;
if (l >= k) /* u0 has more limbs than s:
first look if {up, m} is not zero,
and compare {sp, k} and {up + m, k} */
{
cy = cy || mpfr_mpn_cmpzero (up, m);
low_u = cy;
cy = mpfr_mpn_sub_aux (sp, up + m, k,
cy, extra_bit);
}
else /* l < k: s has more limbs than u0 */
{
low_u = MPFR_LIMB_ZERO;
if (cy != MPFR_LIMB_ZERO)
cy = mpn_sub_1 (sp + k - l - 1, sp + k - l - 1,
1, MPFR_LIMB_HIGHBIT);
cy = mpfr_mpn_sub_aux (sp + k - l, up, l,
cy, extra_bit);
}
}
MPFR_ASSERTD (cy <= 1);
cy = mpn_sub_1 (sp + k, sp + k, qsize, cy);
/* subtract r */
cy += mpn_sub_n (sp + k, sp + k, ap, qsize);
MPFR_ASSERTD (cy <= 1);
/* now compare {sp, ssize} to v */
cmp_s_r = mpn_cmp (sp, vp, vsize);
if (cmp_s_r == 0 && low_u != MPFR_LIMB_ZERO)
cmp_s_r = 1; /* since in fact we subtracted
less than 1 */
}
#ifdef DEBUG
printf ("cmp(q*v0-(r+u0),v)=%d\n", cmp_s_r);
#endif
if (cmp_s_r <= 0) /* q1-1 <= u/v < q1 */
{
if (sticky3 == MPFR_LIMB_ONE)
{ /* q1-1 is either representable (directed rounding),
or the middle of two numbers (nearest) */
sticky = (cmp_s_r) ? MPFR_LIMB_ONE : MPFR_LIMB_ZERO;
goto case_1;
}
/* now necessarily sticky3=0 */
else if (round_bit == MPFR_LIMB_ZERO)
{ /* round_bit=0, sticky3=0: q1-1 is exact only
when sh=0 */
inex = (cmp_s_r || sh) ? -1 : 0;
if (rnd_mode == MPFR_RNDN ||
(! like_rndz && inex != 0))
{
inex = 1;
goto truncate_check_qh;
}
else /* round down */
goto sub_one_ulp;
}
else /* sticky3=0, round_bit=1 ==> rounding to nearest */
{
inex = cmp_s_r;
goto truncate;
}
}
else /* q1-2 < u/v < q1-1 */
{
/* if rnd=MPFR_RNDN, the result is q1 when
q1-2 >= q1-2^(sh-1), i.e. sh >= 2,
otherwise (sh=1) it is q1-2 */
if (rnd_mode == MPFR_RNDN) /* sh > 0 */
{
/* Case sh=1: sb=0 always, and q1-rb is exactly
representable, like q1-rb-2.
rb action
0 subtract two ulps, inex=-1
1 truncate, inex=1
Case sh>1: one ulp is 2^(sh-1) >= 2
rb sb action
0 0 truncate, inex=1
0 1 truncate, inex=1
1 x truncate, inex=-1
*/
if (sh == 1)
{
if (round_bit == MPFR_LIMB_ZERO)
{
inex = -1;
sh = 0;
goto sub_two_ulp;
}
else
{
inex = 1;
goto truncate_check_qh;
}
}
else /* sh > 1 */
{
inex = (round_bit == MPFR_LIMB_ZERO) ? 1 : -1;
goto truncate_check_qh;
}
}
else if (like_rndz)
{
/* the result is down(q1-2), i.e. subtract one
ulp if sh > 0, and two ulps if sh=0 */
inex = -1;
if (sh > 0)
goto sub_one_ulp;
else
goto sub_two_ulp;
}
/* if round away from zero, the result is up(q1-1),
which is q1 unless sh = 0, where it is q1-1 */
else
{
inex = 1;
if (sh > 0)
goto truncate_check_qh;
else /* sh = 0 */
goto sub_one_ulp;
}
}
}
}
}
}
case_1: /* quotient is in [q1, q1+1),
round_bit is the round_bit (0 for directed rounding),
sticky the sticky bit */
if (like_rndz || (round_bit == MPFR_LIMB_ZERO && sticky == MPFR_LIMB_ZERO))
{
inex = round_bit == MPFR_LIMB_ZERO && sticky == MPFR_LIMB_ZERO ? 0 : -1;
goto truncate;
}
else if (rnd_mode == MPFR_RNDN) /* sticky <> 0 or round <> 0 */
{
if (round_bit == MPFR_LIMB_ZERO) /* necessarily sticky <> 0 */
{
inex = -1;
goto truncate;
}
/* round_bit = 1 */
else if (sticky != MPFR_LIMB_ZERO)
goto add_one_ulp; /* inex=1 */
else /* round_bit=1, sticky=0 */
goto even_rule;
}
else /* round away from zero, sticky <> 0 */
goto add_one_ulp; /* with inex=1 */
sub_two_ulp:
/* we cannot subtract MPFR_LIMB_MPFR_LIMB_ONE << (sh+1) since this is
undefined for sh = GMP_NUMB_BITS */
qh -= mpn_sub_1 (q0p, q0p, q0size, MPFR_LIMB_ONE << sh);
/* go through */
sub_one_ulp:
qh -= mpn_sub_1 (q0p, q0p, q0size, MPFR_LIMB_ONE << sh);
/* go through truncate_check_qh */
truncate_check_qh:
if (qh)
{
if (MPFR_LIKELY (qexp < MPFR_EXP_MAX))
qexp ++;
/* else qexp is now incorrect, but one will still get an overflow */
q0p[q0size - 1] = MPFR_LIMB_HIGHBIT;
}
goto truncate;
even_rule: /* has to set inex */
inex = (q0p[0] & (MPFR_LIMB_ONE << sh)) ? 1 : -1;
if (inex < 0)
goto truncate;
/* else go through add_one_ulp */
add_one_ulp:
inex = 1; /* always here */
if (mpn_add_1 (q0p, q0p, q0size, MPFR_LIMB_ONE << sh))
{
if (MPFR_LIKELY (qexp < MPFR_EXP_MAX))
qexp ++;
/* else qexp is now incorrect, but one will still get an overflow */
q0p[q0size - 1] = MPFR_LIMB_HIGHBIT;
}
truncate: /* inex already set */
MPFR_TMP_FREE(marker);
/* check for underflow/overflow */
if (MPFR_UNLIKELY(qexp > __gmpfr_emax))
return mpfr_overflow (q, rnd_mode, sign_quotient);
else if (MPFR_UNLIKELY(qexp < __gmpfr_emin))
{
if (rnd_mode == MPFR_RNDN && ((qexp < __gmpfr_emin - 1) ||
(inex >= 0 && mpfr_powerof2_raw (q))))
rnd_mode = MPFR_RNDZ;
return mpfr_underflow (q, rnd_mode, sign_quotient);
}
MPFR_SET_EXP(q, qexp);
inex *= sign_quotient;
MPFR_RET (inex);
}
|