aboutsummaryrefslogtreecommitdiff
path: root/sysdeps/ieee754/dbl-64/mpa.c
blob: f59ec6a9d80a8b5a2e8b18416b72cdcd32a58084 (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
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
795
796
797
798
799
800
801
802
803
804
805
806
807
808
809
810
811
812
813
814
815
816
817
818
819
820
821
822
823
824
825
826
827
828
829
830
831
832
833
834
835
836
837
838
839
840
841
842
843
844
845
846
847
848
849
850
851
852
853
854
855
856
857
858
859
860
861
862
863
864
865
866
867
868
869
870
871
872
873
874
875
876
877
878
879
880
881
882
883
884
885
886
887
888
889
890
891
892
893
894
895
896
897
898
899
900
901
902
903
904
905
906
/*
 * IBM Accurate Mathematical Library
 * written by International Business Machines Corp.
 * Copyright (C) 2001-2018 Free Software Foundation, Inc.
 *
 * 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 2.1 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 Lesser 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, see <http://www.gnu.org/licenses/>.
 */
/************************************************************************/
/*  MODULE_NAME: mpa.c                                                  */
/*                                                                      */
/*  FUNCTIONS:                                                          */
/*               mcr                                                    */
/*               acr                                                    */
/*               cpy                                                    */
/*               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 <sys/param.h>
#include <alloca.h>

#ifndef SECTION
# define SECTION
#endif

#ifndef NO__CONST
const mp_no __mpone = { 1, { 1.0, 1.0 } };
const mp_no __mptwo = { 1, { 1.0, 2.0 } };
#endif

#ifndef NO___ACR
/* Compare mantissa of two multiple precision numbers regardless of the sign
   and exponent of the numbers.  */
static int
mcr (const mp_no *x, const mp_no *y, int p)
{
  long i;
  long p2 = p;
  for (i = 1; i <= p2; i++)
    {
      if (X[i] == Y[i])
	continue;
      else if (X[i] > Y[i])
	return 1;
      else
	return -1;
    }
  return 0;
}

/* Compare the absolute values of two multiple precision numbers.  */
int
__acr (const mp_no *x, const mp_no *y, int p)
{
  long i;

  if (X[0] == 0)
    {
      if (Y[0] == 0)
	i = 0;
      else
	i = -1;
    }
  else if (Y[0] == 0)
    i = 1;
  else
    {
      if (EX > EY)
	i = 1;
      else if (EX < EY)
	i = -1;
      else
	i = mcr (x, y, p);
    }

  return i;
}
#endif

#ifndef NO___CPY
/* Copy multiple precision number X into Y.  They could be the same
   number.  */
void
__cpy (const mp_no *x, mp_no *y, int p)
{
  long i;

  EY = EX;
  for (i = 0; i <= p; i++)
    Y[i] = X[i];
}
#endif

#ifndef NO___MP_DBL
/* Convert a multiple precision number *X into a double precision
   number *Y, normalized case (|x| >= 2**(-1022))).  X has precision
   P, which is positive.  */
static void
norm (const mp_no *x, double *y, int p)
{
# define R RADIXI
  long i;
  double c;
  mantissa_t a, 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 /* p == 4.  */
	c = (X[1] + R * X[2]) + R * R * (X[3] + R * X[4]);
    }
  else
    {
      for (a = 1, z[1] = X[1]; z[1] < TWO23; )
	{
	  a *= 2;
	  z[1] *= 2;
	}

      for (i = 2; i < 5; i++)
	{
	  mantissa_store_t d, r;
	  d = X[i] * (mantissa_store_t) a;
	  DIV_RADIX (d, r);
	  z[i] = r;
	  z[i - 1] += d;
	}

      u = ALIGN_DOWN_TO (z[3], TWO19);
      v = z[3] - u;

      if (v == TWO18)
	{
	  if (z[4] == 0)
	    {
	      for (i = 5; i <= p; i++)
		{
		  if (X[i] == 0)
		    continue;
		  else
		    {
		      z[3] += 1;
		      break;
		    }
		}
	    }
	  else
	    z[3] += 1;
	}

      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;
# undef R
}

/* Convert a multiple precision number *X into a double precision
   number *Y, Denormal case  (|x| < 2**(-1022))).  */
static void
denorm (const mp_no *x, double *y, int p)
{
  long i, k;
  long p2 = p;
  double c;
  mantissa_t u, z[5];

# define R RADIXI
  if (EX < -44 || (EX == -44 && X[1] < TWO5))
    {
      *y = 0;
      return;
    }

  if (p2 == 1)
    {
      if (EX == -42)
	{
	  z[1] = X[1] + TWO10;
	  z[2] = 0;
	  z[3] = 0;
	  k = 3;
	}
      else if (EX == -43)
	{
	  z[1] = TWO10;
	  z[2] = X[1];
	  z[3] = 0;
	  k = 2;
	}
      else
	{
	  z[1] = TWO10;
	  z[2] = 0;
	  z[3] = X[1];
	  k = 1;
	}
    }
  else if (p2 == 2)
    {
      if (EX == -42)
	{
	  z[1] = X[1] + TWO10;
	  z[2] = X[2];
	  z[3] = 0;
	  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] = 0;
	  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] = 0;
	  k = 1;
	}
      z[3] = X[k];
    }

  u = ALIGN_DOWN_TO (z[3], TWO5);

  if (u == z[3])
    {
      for (i = k + 1; i <= p2; i++)
	{
	  if (X[i] == 0)
	    continue;
	  else
	    {
	      z[3] += 1;
	      break;
	    }
	}
    }

  c = X[0] * ((z[1] + R * (z[2] + R * z[3])) - TWO10);

  *y = c * TWOM1032;
# undef R
}

/* Convert multiple precision number *X into double precision number *Y.  The
   result is correctly rounded to the nearest/even.  */
void
__mp_dbl (const mp_no *x, double *y, int p)
{
  if (X[0] == 0)
    {
      *y = 0;
      return;
    }

  if (__glibc_likely (EX > -42 || (EX == -42 && X[1] >= TWO10)))
    norm (x, y, p);
  else
    denorm (x, y, p);
}
#endif

/* Get the multiple precision equivalent of X into *Y.  If the precision is too
   small, the result is truncated.  */
void
SECTION
__dbl_mp (double x, mp_no *y, int p)
{
  long i, n;
  long p2 = p;

  /* Sign.  */
  if (x == 0)
    {
      Y[0] = 0;
      return;
    }
  else if (x > 0)
    Y[0] = 1;
  else
    {
      Y[0] = -1;
      x = -x;
    }

  /* Exponent.  */
  for (EY = 1; x >= RADIX; EY += 1)
    x *= RADIXI;
  for (; x < 1; EY -= 1)
    x *= RADIX;

  /* Digits.  */
  n = MIN (p2, 4);
  for (i = 1; i <= n; i++)
    {
      INTEGER_OF (x, Y[i]);
      x *= RADIX;
    }
  for (; i <= p2; i++)
    Y[i] = 0;
}

/* Add magnitudes of *X and *Y assuming that abs (*X) >= abs (*Y) > 0.  The
   sign of the sum *Z is not changed.  X and Y may overlap but not X and Z or
   Y and Z.  No guard digit is used.  The result equals the exact sum,
   truncated.  */
static void
SECTION
add_magnitudes (const mp_no *x, const mp_no *y, mp_no *z, int p)
{
  long i, j, k;
  long p2 = p;
  mantissa_t zk;

  EZ = EX;

  i = p2;
  j = p2 + EY - EX;
  k = p2 + 1;

  if (__glibc_unlikely (j < 1))
    {
      __cpy (x, z, p);
      return;
    }

  zk = 0;

  for (; j > 0; i--, j--)
    {
      zk += X[i] + Y[j];
      if (zk >= RADIX)
	{
	  Z[k--] = zk - RADIX;
	  zk = 1;
	}
      else
	{
	  Z[k--] = zk;
	  zk = 0;
	}
    }

  for (; i > 0; i--)
    {
      zk += X[i];
      if (zk >= RADIX)
	{
	  Z[k--] = zk - RADIX;
	  zk = 1;
	}
      else
	{
	  Z[k--] = zk;
	  zk = 0;
	}
    }

  if (zk == 0)
    {
      for (i = 1; i <= p2; i++)
	Z[i] = Z[i + 1];
    }
  else
    {
      Z[1] = zk;
      EZ += 1;
    }
}

/* Subtract the magnitudes of *X and *Y assuming that abs (*x) > abs (*y) > 0.
   The sign of the difference *Z is not changed.  X and Y may overlap but not X
   and Z or Y and Z.  One guard digit is used.  The error is less than one
   ULP.  */
static void
SECTION
sub_magnitudes (const mp_no *x, const mp_no *y, mp_no *z, int p)
{
  long i, j, k;
  long p2 = p;
  mantissa_t zk;

  EZ = EX;
  i = p2;
  j = p2 + EY - EX;
  k = p2;

  /* Y is too small compared to X, copy X over to the result.  */
  if (__glibc_unlikely (j < 1))
    {
      __cpy (x, z, p);
      return;
    }

  /* The relevant least significant digit in Y is non-zero, so we factor it in
     to enhance accuracy.  */
  if (j < p2 && Y[j + 1] > 0)
    {
      Z[k + 1] = RADIX - Y[j + 1];
      zk = -1;
    }
  else
    zk = Z[k + 1] = 0;

  /* Subtract and borrow.  */
  for (; j > 0; i--, j--)
    {
      zk += (X[i] - Y[j]);
      if (zk < 0)
	{
	  Z[k--] = zk + RADIX;
	  zk = -1;
	}
      else
	{
	  Z[k--] = zk;
	  zk = 0;
	}
    }

  /* We're done with digits from Y, so it's just digits in X.  */
  for (; i > 0; i--)
    {
      zk += X[i];
      if (zk < 0)
	{
	  Z[k--] = zk + RADIX;
	  zk = -1;
	}
      else
	{
	  Z[k--] = zk;
	  zk = 0;
	}
    }

  /* Normalize.  */
  for (i = 1; Z[i] == 0; i++)
    ;
  EZ = EZ - i + 1;
  for (k = 1; i <= p2 + 1; )
    Z[k++] = Z[i++];
  for (; k <= p2; )
    Z[k++] = 0;
}

/* Add *X and *Y and store the result in *Z.  X and Y may overlap, but not X
   and Z or Y and Z.  One guard digit is used.  The error is less than one
   ULP.  */
void
SECTION
__add (const mp_no *x, const mp_no *y, mp_no *z, int p)
{
  int n;

  if (X[0] == 0)
    {
      __cpy (y, z, p);
      return;
    }
  else if (Y[0] == 0)
    {
      __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] = 0;
    }
}

/* Subtract *Y from *X and return the result in *Z.  X and Y may overlap but
   not X and Z or Y and Z.  One guard digit is used.  The error is less than
   one ULP.  */
void
SECTION
__sub (const mp_no *x, const mp_no *y, mp_no *z, int p)
{
  int n;

  if (X[0] == 0)
    {
      __cpy (y, z, p);
      Z[0] = -Z[0];
      return;
    }
  else if (Y[0] == 0)
    {
      __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] = 0;
    }
}

#ifndef NO__MUL
/* Multiply *X and *Y and store result in *Z.  X and Y may overlap but not X
   and Z or Y and Z.  For P in [1, 2, 3], the exact result is truncated to P
   digits.  In case P > 3 the error is bounded by 1.001 ULP.  */
void
SECTION
__mul (const mp_no *x, const mp_no *y, mp_no *z, int p)
{
  long i, j, k, ip, ip2;
  long p2 = p;
  mantissa_store_t zk;
  const mp_no *a;
  mantissa_store_t *diag;

  /* Is z=0?  */
  if (__glibc_unlikely (X[0] * Y[0] == 0))
    {
      Z[0] = 0;
      return;
    }

  /* We need not iterate through all X's and Y's since it's pointless to
     multiply zeroes.  Here, both are zero...  */
  for (ip2 = p2; ip2 > 0; ip2--)
    if (X[ip2] != 0 || Y[ip2] != 0)
      break;

  a = X[ip2] != 0 ? y : x;

  /* ... and here, at least one of them is still zero.  */
  for (ip = ip2; ip > 0; ip--)
    if (a->d[ip] != 0)
      break;

  /* The product looks like this for p = 3 (as an example):


				a1    a2    a3
		 x		b1    b2    b3
		 -----------------------------
			     a1*b3 a2*b3 a3*b3
		       a1*b2 a2*b2 a3*b2
		 a1*b1 a2*b1 a3*b1

     So our K needs to ideally be P*2, but we're limiting ourselves to P + 3
     for P >= 3.  We compute the above digits in two parts; the last P-1
     digits and then the first P digits.  The last P-1 digits are a sum of
     products of the input digits from P to P-k where K is 0 for the least
     significant digit and increases as we go towards the left.  The product
     term is of the form X[k]*X[P-k] as can be seen in the above example.

     The first P digits are also a sum of products with the same product term,
     except that the sum is from 1 to k.  This is also evident from the above
     example.

     Another thing that becomes evident is that only the most significant
     ip+ip2 digits of the result are non-zero, where ip and ip2 are the
     'internal precision' of the input numbers, i.e. digits after ip and ip2
     are all 0.  */

  k = (__glibc_unlikely (p2 < 3)) ? p2 + p2 : p2 + 3;

  while (k > ip + ip2 + 1)
    Z[k--] = 0;

  zk = 0;

  /* Precompute sums of diagonal elements so that we can directly use them
     later.  See the next comment to know we why need them.  */
  diag = alloca (k * sizeof (mantissa_store_t));
  mantissa_store_t d = 0;
  for (i = 1; i <= ip; i++)
    {
      d += X[i] * (mantissa_store_t) Y[i];
      diag[i] = d;
    }
  while (i < k)
    diag[i++] = d;

  while (k > p2)
    {
      long lim = k / 2;

      if (k % 2 == 0)
	/* We want to add this only once, but since we subtract it in the sum
	   of products above, we add twice.  */
	zk += 2 * X[lim] * (mantissa_store_t) Y[lim];

      for (i = k - p2, j = p2; i < j; i++, j--)
	zk += (X[i] + X[j]) * (mantissa_store_t) (Y[i] + Y[j]);

      zk -= diag[k - 1];

      DIV_RADIX (zk, Z[k]);
      k--;
    }

  /* The real deal.  Mantissa digit Z[k] is the sum of all X[i] * Y[j] where i
     goes from 1 -> k - 1 and j goes the same range in reverse.  To reduce the
     number of multiplications, we halve the range and if k is an even number,
     add the diagonal element X[k/2]Y[k/2].  Through the half range, we compute
     X[i] * Y[j] as (X[i] + X[j]) * (Y[i] + Y[j]) - X[i] * Y[i] - X[j] * Y[j].

     This reduction tells us that we're summing two things, the first term
     through the half range and the negative of the sum of the product of all
     terms of X and Y in the full range.  i.e.

     SUM(X[i] * Y[i]) for k terms.  This is precalculated above for each k in
     a single loop so that it completes in O(n) time and can hence be directly
     used in the loop below.  */
  while (k > 1)
    {
      long lim = k / 2;

      if (k % 2 == 0)
	/* We want to add this only once, but since we subtract it in the sum
	   of products above, we add twice.  */
        zk += 2 * X[lim] * (mantissa_store_t) Y[lim];

      for (i = 1, j = k - 1; i < j; i++, j--)
	zk += (X[i] + X[j]) * (mantissa_store_t) (Y[i] + Y[j]);

      zk -= diag[k - 1];

      DIV_RADIX (zk, Z[k]);
      k--;
    }
  Z[k] = zk;

  /* Get the exponent sum into an intermediate variable.  This is a subtle
     optimization, where given enough registers, all operations on the exponent
     happen in registers and the result is written out only once into EZ.  */
  int e = EX + EY;

  /* Is there a carry beyond the most significant digit?  */
  if (__glibc_unlikely (Z[1] == 0))
    {
      for (i = 1; i <= p2; i++)
	Z[i] = Z[i + 1];
      e--;
    }

  EZ = e;
  Z[0] = X[0] * Y[0];
}
#endif

#ifndef NO__SQR
/* Square *X and store result in *Y.  X and Y may not overlap.  For P in
   [1, 2, 3], the exact result is truncated to P digits.  In case P > 3 the
   error is bounded by 1.001 ULP.  This is a faster special case of
   multiplication.  */
void
SECTION
__sqr (const mp_no *x, mp_no *y, int p)
{
  long i, j, k, ip;
  mantissa_store_t yk;

  /* Is z=0?  */
  if (__glibc_unlikely (X[0] == 0))
    {
      Y[0] = 0;
      return;
    }

  /* We need not iterate through all X's since it's pointless to
     multiply zeroes.  */
  for (ip = p; ip > 0; ip--)
    if (X[ip] != 0)
      break;

  k = (__glibc_unlikely (p < 3)) ? p + p : p + 3;

  while (k > 2 * ip + 1)
    Y[k--] = 0;

  yk = 0;

  while (k > p)
    {
      mantissa_store_t yk2 = 0;
      long lim = k / 2;

      if (k % 2 == 0)
	yk += X[lim] * (mantissa_store_t) X[lim];

      /* In __mul, this loop (and the one within the next while loop) run
         between a range to calculate the mantissa as follows:

         Z[k] = X[k] * Y[n] + X[k+1] * Y[n-1] ... + X[n-1] * Y[k+1]
		+ X[n] * Y[k]

         For X == Y, we can get away with summing halfway and doubling the
	 result.  For cases where the range size is even, the mid-point needs
	 to be added separately (above).  */
      for (i = k - p, j = p; i < j; i++, j--)
	yk2 += X[i] * (mantissa_store_t) X[j];

      yk += 2 * yk2;

      DIV_RADIX (yk, Y[k]);
      k--;
    }

  while (k > 1)
    {
      mantissa_store_t yk2 = 0;
      long lim = k / 2;

      if (k % 2 == 0)
	yk += X[lim] * (mantissa_store_t) X[lim];

      /* Likewise for this loop.  */
      for (i = 1, j = k - 1; i < j; i++, j--)
	yk2 += X[i] * (mantissa_store_t) X[j];

      yk += 2 * yk2;

      DIV_RADIX (yk, Y[k]);
      k--;
    }
  Y[k] = yk;

  /* Squares are always positive.  */
  Y[0] = 1;

  /* Get the exponent sum into an intermediate variable.  This is a subtle
     optimization, where given enough registers, all operations on the exponent
     happen in registers and the result is written out only once into EZ.  */
  int e = EX * 2;

  /* Is there a carry beyond the most significant digit?  */
  if (__glibc_unlikely (Y[1] == 0))
    {
      for (i = 1; i <= p; i++)
	Y[i] = Y[i + 1];
      e--;
    }

  EY = e;
}
#endif

/* Invert *X and store in *Y.  Relative error bound:
   - For P = 2: 1.001 * R ^ (1 - P)
   - For P = 3: 1.063 * R ^ (1 - P)
   - For P > 3: 2.001 * R ^ (1 - P)

   *X = 0 is not permissible.  */
static void
SECTION
__inv (const mp_no *x, mp_no *y, int p)
{
  long i;
  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
  };

  __cpy (x, &z, p);
  z.e = 0;
  __mp_dbl (&z, &t, p);
  t = 1 / 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);
    }
}

/* Divide *X by *Y and store result in *Z.  X and Y may overlap but not X and Z
   or Y and Z.  Relative error bound:
   - For P = 2: 2.001 * R ^ (1 - P)
   - For P = 3: 2.063 * R ^ (1 - P)
   - For P > 3: 3.001 * R ^ (1 - P)

   *X = 0 is not permissible.  */
void
SECTION
__dvd (const mp_no *x, const mp_no *y, mp_no *z, int p)
{
  mp_no w;

  if (X[0] == 0)
    Z[0] = 0;
  else
    {
      __inv (y, &w, p);
      __mul (x, &w, z, p);
    }
}