aboutsummaryrefslogtreecommitdiff
path: root/arch/arm/nwfpe/softfloat-macros
blob: cf2a6173149e3e5653618dba5f4c102950774329 (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
/*
===============================================================================

This C source fragment is part of the SoftFloat IEC/IEEE Floating-point
Arithmetic Package, Release 2.

Written by John R. Hauser.  This work was made possible in part by the
International Computer Science Institute, located at Suite 600, 1947 Center
Street, Berkeley, California 94704.  Funding was partially provided by the
National Science Foundation under grant MIP-9311980.  The original version
of this code was written as part of a project to build a fixed-point vector
processor in collaboration with the University of California at Berkeley,
overseen by Profs. Nelson Morgan and John Wawrzynek.  More information
is available through the web page
http://www.jhauser.us/arithmetic/SoftFloat-2b/SoftFloat-source.txt

THIS SOFTWARE IS DISTRIBUTED AS IS, FOR FREE.  Although reasonable effort
has been made to avoid it, THIS SOFTWARE MAY CONTAIN FAULTS THAT WILL AT
TIMES RESULT IN INCORRECT BEHAVIOR.  USE OF THIS SOFTWARE IS RESTRICTED TO
PERSONS AND ORGANIZATIONS WHO CAN AND WILL TAKE FULL RESPONSIBILITY FOR ANY
AND ALL LOSSES, COSTS, OR OTHER PROBLEMS ARISING FROM ITS USE.

Derivative works are acceptable, even for commercial purposes, so long as
(1) they include prominent notice that the work is derivative, and (2) they
include prominent notice akin to these three paragraphs for those parts of
this code that are retained.

===============================================================================
*/

/*
-------------------------------------------------------------------------------
Shifts `a' right by the number of bits given in `count'.  If any nonzero
bits are shifted off, they are ``jammed'' into the least significant bit of
the result by setting the least significant bit to 1.  The value of `count'
can be arbitrarily large; in particular, if `count' is greater than 32, the
result will be either 0 or 1, depending on whether `a' is zero or nonzero.
The result is stored in the location pointed to by `zPtr'.
-------------------------------------------------------------------------------
*/
INLINE void shift32RightJamming( bits32 a, int16 count, bits32 *zPtr )
{
    bits32 z;
    if ( count == 0 ) {
        z = a;
    }
    else if ( count < 32 ) {
        z = ( a>>count ) | ( ( a<<( ( - count ) & 31 ) ) != 0 );
    }
    else {
        z = ( a != 0 );
    }
    *zPtr = z;
}

/*
-------------------------------------------------------------------------------
Shifts `a' right by the number of bits given in `count'.  If any nonzero
bits are shifted off, they are ``jammed'' into the least significant bit of
the result by setting the least significant bit to 1.  The value of `count'
can be arbitrarily large; in particular, if `count' is greater than 64, the
result will be either 0 or 1, depending on whether `a' is zero or nonzero.
The result is stored in the location pointed to by `zPtr'.
-------------------------------------------------------------------------------
*/
INLINE void shift64RightJamming( bits64 a, int16 count, bits64 *zPtr )
{
    bits64 z;

 __asm__("@shift64RightJamming -- start");   
    if ( count == 0 ) {
        z = a;
    }
    else if ( count < 64 ) {
        z = ( a>>count ) | ( ( a<<( ( - count ) & 63 ) ) != 0 );
    }
    else {
        z = ( a != 0 );
    }
 __asm__("@shift64RightJamming -- end");   
    *zPtr = z;
}

/*
-------------------------------------------------------------------------------
Shifts the 128-bit value formed by concatenating `a0' and `a1' right by 64
_plus_ the number of bits given in `count'.  The shifted result is at most
64 nonzero bits; this is stored at the location pointed to by `z0Ptr'.  The
bits shifted off form a second 64-bit result as follows:  The _last_ bit
shifted off is the most-significant bit of the extra result, and the other
63 bits of the extra result are all zero if and only if _all_but_the_last_
bits shifted off were all zero.  This extra result is stored in the location
pointed to by `z1Ptr'.  The value of `count' can be arbitrarily large.
    (This routine makes more sense if `a0' and `a1' are considered to form a
fixed-point value with binary point between `a0' and `a1'.  This fixed-point
value is shifted right by the number of bits given in `count', and the
integer part of the result is returned at the location pointed to by
`z0Ptr'.  The fractional part of the result may be slightly corrupted as
described above, and is returned at the location pointed to by `z1Ptr'.)
-------------------------------------------------------------------------------
*/
INLINE void
 shift64ExtraRightJamming(
     bits64 a0, bits64 a1, int16 count, bits64 *z0Ptr, bits64 *z1Ptr )
{
    bits64 z0, z1;
    int8 negCount = ( - count ) & 63;

    if ( count == 0 ) {
        z1 = a1;
        z0 = a0;
    }
    else if ( count < 64 ) {
        z1 = ( a0<<negCount ) | ( a1 != 0 );
        z0 = a0>>count;
    }
    else {
        if ( count == 64 ) {
            z1 = a0 | ( a1 != 0 );
        }
        else {
            z1 = ( ( a0 | a1 ) != 0 );
        }
        z0 = 0;
    }
    *z1Ptr = z1;
    *z0Ptr = z0;

}

/*
-------------------------------------------------------------------------------
Shifts the 128-bit value formed by concatenating `a0' and `a1' right by the
number of bits given in `count'.  Any bits shifted off are lost.  The value
of `count' can be arbitrarily large; in particular, if `count' is greater
than 128, the result will be 0.  The result is broken into two 64-bit pieces
which are stored at the locations pointed to by `z0Ptr' and `z1Ptr'.
-------------------------------------------------------------------------------
*/
INLINE void
 shift128Right(
     bits64 a0, bits64 a1, int16 count, bits64 *z0Ptr, bits64 *z1Ptr )
{
    bits64 z0, z1;
    int8 negCount = ( - count ) & 63;

    if ( count == 0 ) {
        z1 = a1;
        z0 = a0;
    }
    else if ( count < 64 ) {
        z1 = ( a0<<negCount ) | ( a1>>count );
        z0 = a0>>count;
    }
    else {
        z1 = ( count < 64 ) ? ( a0>>( count & 63 ) ) : 0;
        z0 = 0;
    }
    *z1Ptr = z1;
    *z0Ptr = z0;

}

/*
-------------------------------------------------------------------------------
Shifts the 128-bit value formed by concatenating `a0' and `a1' right by the
number of bits given in `count'.  If any nonzero bits are shifted off, they
are ``jammed'' into the least significant bit of the result by setting the
least significant bit to 1.  The value of `count' can be arbitrarily large;
in particular, if `count' is greater than 128, the result will be either 0
or 1, depending on whether the concatenation of `a0' and `a1' is zero or
nonzero.  The result is broken into two 64-bit pieces which are stored at
the locations pointed to by `z0Ptr' and `z1Ptr'.
-------------------------------------------------------------------------------
*/
INLINE void
 shift128RightJamming(
     bits64 a0, bits64 a1, int16 count, bits64 *z0Ptr, bits64 *z1Ptr )
{
    bits64 z0, z1;
    int8 negCount = ( - count ) & 63;

    if ( count == 0 ) {
        z1 = a1;
        z0 = a0;
    }
    else if ( count < 64 ) {
        z1 = ( a0<<negCount ) | ( a1>>count ) | ( ( a1<<negCount ) != 0 );
        z0 = a0>>count;
    }
    else {
        if ( count == 64 ) {
            z1 = a0 | ( a1 != 0 );
        }
        else if ( count < 128 ) {
            z1 = ( a0>>( count & 63 ) ) | ( ( ( a0<<negCount ) | a1 ) != 0 );
        }
        else {
            z1 = ( ( a0 | a1 ) != 0 );
        }
        z0 = 0;
    }
    *z1Ptr = z1;
    *z0Ptr = z0;

}

/*
-------------------------------------------------------------------------------
Shifts the 192-bit value formed by concatenating `a0', `a1', and `a2' right
by 64 _plus_ the number of bits given in `count'.  The shifted result is
at most 128 nonzero bits; these are broken into two 64-bit pieces which are
stored at the locations pointed to by `z0Ptr' and `z1Ptr'.  The bits shifted
off form a third 64-bit result as follows:  The _last_ bit shifted off is
the most-significant bit of the extra result, and the other 63 bits of the
extra result are all zero if and only if _all_but_the_last_ bits shifted off
were all zero.  This extra result is stored in the location pointed to by
`z2Ptr'.  The value of `count' can be arbitrarily large.
    (This routine makes more sense if `a0', `a1', and `a2' are considered
to form a fixed-point value with binary point between `a1' and `a2'.  This
fixed-point value is shifted right by the number of bits given in `count',
and the integer part of the result is returned at the locations pointed to
by `z0Ptr' and `z1Ptr'.  The fractional part of the result may be slightly
corrupted as described above, and is returned at the location pointed to by
`z2Ptr'.)
-------------------------------------------------------------------------------
*/
INLINE void
 shift128ExtraRightJamming(
     bits64 a0,
     bits64 a1,
     bits64 a2,
     int16 count,
     bits64 *z0Ptr,
     bits64 *z1Ptr,
     bits64 *z2Ptr
 )
{
    bits64 z0, z1, z2;
    int8 negCount = ( - count ) & 63;

    if ( count == 0 ) {
        z2 = a2;
        z1 = a1;
        z0 = a0;
    }
    else {
        if ( count < 64 ) {
            z2 = a1<<negCount;
            z1 = ( a0<<negCount ) | ( a1>>count );
            z0 = a0>>count;
        }
        else {
            if ( count == 64 ) {
                z2 = a1;
                z1 = a0;
            }
            else {
                a2 |= a1;
                if ( count < 128 ) {
                    z2 = a0<<negCount;
                    z1 = a0>>( count & 63 );
                }
                else {
                    z2 = ( count == 128 ) ? a0 : ( a0 != 0 );
                    z1 = 0;
                }
            }
            z0 = 0;
        }
        z2 |= ( a2 != 0 );
    }
    *z2Ptr = z2;
    *z1Ptr = z1;
    *z0Ptr = z0;

}

/*
-------------------------------------------------------------------------------
Shifts the 128-bit value formed by concatenating `a0' and `a1' left by the
number of bits given in `count'.  Any bits shifted off are lost.  The value
of `count' must be less than 64.  The result is broken into two 64-bit
pieces which are stored at the locations pointed to by `z0Ptr' and `z1Ptr'.
-------------------------------------------------------------------------------
*/
INLINE void
 shortShift128Left(
     bits64 a0, bits64 a1, int16 count, bits64 *z0Ptr, bits64 *z1Ptr )
{

    *z1Ptr = a1<<count;
    *z0Ptr =
        ( count == 0 ) ? a0 : ( a0<<count ) | ( a1>>( ( - count ) & 63 ) );

}

/*
-------------------------------------------------------------------------------
Shifts the 192-bit value formed by concatenating `a0', `a1', and `a2' left
by the number of bits given in `count'.  Any bits shifted off are lost.
The value of `count' must be less than 64.  The result is broken into three
64-bit pieces which are stored at the locations pointed to by `z0Ptr',
`z1Ptr', and `z2Ptr'.
-------------------------------------------------------------------------------
*/
INLINE void
 shortShift192Left(
     bits64 a0,
     bits64 a1,
     bits64 a2,
     int16 count,
     bits64 *z0Ptr,
     bits64 *z1Ptr,
     bits64 *z2Ptr
 )
{
    bits64 z0, z1, z2;
    int8 negCount;

    z2 = a2<<count;
    z1 = a1<<count;
    z0 = a0<<count;
    if ( 0 < count ) {
        negCount = ( ( - count ) & 63 );
        z1 |= a2>>negCount;
        z0 |= a1>>negCount;
    }
    *z2Ptr = z2;
    *z1Ptr = z1;
    *z0Ptr = z0;

}

/*
-------------------------------------------------------------------------------
Adds the 128-bit value formed by concatenating `a0' and `a1' to the 128-bit
value formed by concatenating `b0' and `b1'.  Addition is modulo 2^128, so
any carry out is lost.  The result is broken into two 64-bit pieces which
are stored at the locations pointed to by `z0Ptr' and `z1Ptr'.
-------------------------------------------------------------------------------
*/
INLINE void
 add128(
     bits64 a0, bits64 a1, bits64 b0, bits64 b1, bits64 *z0Ptr, bits64 *z1Ptr )
{
    bits64 z1;

    z1 = a1 + b1;
    *z1Ptr = z1;
    *z0Ptr = a0 + b0 + ( z1 < a1 );

}

/*
-------------------------------------------------------------------------------
Adds the 192-bit value formed by concatenating `a0', `a1', and `a2' to the
192-bit value formed by concatenating `b0', `b1', and `b2'.  Addition is
modulo 2^192, so any carry out is lost.  The result is broken into three
64-bit pieces which are stored at the locations pointed to by `z0Ptr',
`z1Ptr', and `z2Ptr'.
-------------------------------------------------------------------------------
*/
INLINE void
 add192(
     bits64 a0,
     bits64 a1,
     bits64 a2,
     bits64 b0,
     bits64 b1,
     bits64 b2,
     bits64 *z0Ptr,
     bits64 *z1Ptr,
     bits64 *z2Ptr
 )
{
    bits64 z0, z1, z2;
    int8 carry0, carry1;

    z2 = a2 + b2;
    carry1 = ( z2 < a2 );
    z1 = a1 + b1;
    carry0 = ( z1 < a1 );
    z0 = a0 + b0;
    z1 += carry1;
    z0 += ( z1 < carry1 );
    z0 += carry0;
    *z2Ptr = z2;
    *z1Ptr = z1;
    *z0Ptr = z0;

}

/*
-------------------------------------------------------------------------------
Subtracts the 128-bit value formed by concatenating `b0' and `b1' from the
128-bit value formed by concatenating `a0' and `a1'.  Subtraction is modulo
2^128, so any borrow out (carry out) is lost.  The result is broken into two
64-bit pieces which are stored at the locations pointed to by `z0Ptr' and
`z1Ptr'.
-------------------------------------------------------------------------------
*/
INLINE void
 sub128(
     bits64 a0, bits64 a1, bits64 b0, bits64 b1, bits64 *z0Ptr, bits64 *z1Ptr )
{

    *z1Ptr = a1 - b1;
    *z0Ptr = a0 - b0 - ( a1 < b1 );

}

/*
-------------------------------------------------------------------------------
Subtracts the 192-bit value formed by concatenating `b0', `b1', and `b2'
from the 192-bit value formed by concatenating `a0', `a1', and `a2'.
Subtraction is modulo 2^192, so any borrow out (carry out) is lost.  The
result is broken into three 64-bit pieces which are stored at the locations
pointed to by `z0Ptr', `z1Ptr', and `z2Ptr'.
-------------------------------------------------------------------------------
*/
INLINE void
 sub192(
     bits64 a0,
     bits64 a1,
     bits64 a2,
     bits64 b0,
     bits64 b1,
     bits64 b2,
     bits64 *z0Ptr,
     bits64 *z1Ptr,
     bits64 *z2Ptr
 )
{
    bits64 z0, z1, z2;
    int8 borrow0, borrow1;

    z2 = a2 - b2;
    borrow1 = ( a2 < b2 );
    z1 = a1 - b1;
    borrow0 = ( a1 < b1 );
    z0 = a0 - b0;
    z0 -= ( z1 < borrow1 );
    z1 -= borrow1;
    z0 -= borrow0;
    *z2Ptr = z2;
    *z1Ptr = z1;
    *z0Ptr = z0;

}

/*
-------------------------------------------------------------------------------
Multiplies `a' by `b' to obtain a 128-bit product.  The product is broken
into two 64-bit pieces which are stored at the locations pointed to by
`z0Ptr' and `z1Ptr'.
-------------------------------------------------------------------------------
*/
INLINE void mul64To128( bits64 a, bits64 b, bits64 *z0Ptr, bits64 *z1Ptr )
{
    bits32 aHigh, aLow, bHigh, bLow;
    bits64 z0, zMiddleA, zMiddleB, z1;

    aLow = a;
    aHigh = a>>32;
    bLow = b;
    bHigh = b>>32;
    z1 = ( (bits64) aLow ) * bLow;
    zMiddleA = ( (bits64) aLow ) * bHigh;
    zMiddleB = ( (bits64) aHigh ) * bLow;
    z0 = ( (bits64) aHigh ) * bHigh;
    zMiddleA += zMiddleB;
    z0 += ( ( (bits64) ( zMiddleA < zMiddleB ) )<<32 ) + ( zMiddleA>>32 );
    zMiddleA <<= 32;
    z1 += zMiddleA;
    z0 += ( z1 < zMiddleA );
    *z1Ptr = z1;
    *z0Ptr = z0;

}

/*
-------------------------------------------------------------------------------
Multiplies the 128-bit value formed by concatenating `a0' and `a1' by `b' to
obtain a 192-bit product.  The product is broken into three 64-bit pieces
which are stored at the locations pointed to by `z0Ptr', `z1Ptr', and
`z2Ptr'.
-------------------------------------------------------------------------------
*/
INLINE void
 mul128By64To192(
     bits64 a0,
     bits64 a1,
     bits64 b,
     bits64 *z0Ptr,
     bits64 *z1Ptr,
     bits64 *z2Ptr
 )
{
    bits64 z0, z1, z2, more1;

    mul64To128( a1, b, &z1, &z2 );
    mul64To128( a0, b, &z0, &more1 );
    add128( z0, more1, 0, z1, &z0, &z1 );
    *z2Ptr = z2;
    *z1Ptr = z1;
    *z0Ptr = z0;

}

/*
-------------------------------------------------------------------------------
Multiplies the 128-bit value formed by concatenating `a0' and `a1' to the
128-bit value formed by concatenating `b0' and `b1' to obtain a 256-bit
product.  The product is broken into four 64-bit pieces which are stored at
the locations pointed to by `z0Ptr', `z1Ptr', `z2Ptr', and `z3Ptr'.
-------------------------------------------------------------------------------
*/
INLINE void
 mul128To256(
     bits64 a0,
     bits64 a1,
     bits64 b0,
     bits64 b1,
     bits64 *z0Ptr,
     bits64 *z1Ptr,
     bits64 *z2Ptr,
     bits64 *z3Ptr
 )
{
    bits64 z0, z1, z2, z3;
    bits64 more1, more2;

    mul64To128( a1, b1, &z2, &z3 );
    mul64To128( a1, b0, &z1, &more2 );
    add128( z1, more2, 0, z2, &z1, &z2 );
    mul64To128( a0, b0, &z0, &more1 );
    add128( z0, more1, 0, z1, &z0, &z1 );
    mul64To128( a0, b1, &more1, &more2 );
    add128( more1, more2, 0, z2, &more1, &z2 );
    add128( z0, z1, 0, more1, &z0, &z1 );
    *z3Ptr = z3;
    *z2Ptr = z2;
    *z1Ptr = z1;
    *z0Ptr = z0;

}

/*
-------------------------------------------------------------------------------
Returns an approximation to the 64-bit integer quotient obtained by dividing
`b' into the 128-bit value formed by concatenating `a0' and `a1'.  The
divisor `b' must be at least 2^63.  If q is the exact quotient truncated
toward zero, the approximation returned lies between q and q + 2 inclusive.
If the exact quotient q is larger than 64 bits, the maximum positive 64-bit
unsigned integer is returned.
-------------------------------------------------------------------------------
*/
static bits64 estimateDiv128To64( bits64 a0, bits64 a1, bits64 b )
{
    bits64 b0, b1;
    bits64 rem0, rem1, term0, term1;
    bits64 z;
    if ( b <= a0 ) return LIT64( 0xFFFFFFFFFFFFFFFF );
    b0 = b>>32;  /* hence b0 is 32 bits wide now */
    if ( b0<<32 <= a0 ) {
        z = LIT64( 0xFFFFFFFF00000000 );
    }  else {
        z = a0;
        do_div( z, b0 );
        z <<= 32;
    }
    mul64To128( b, z, &term0, &term1 );
    sub128( a0, a1, term0, term1, &rem0, &rem1 );
    while ( ( (sbits64) rem0 ) < 0 ) {
        z -= LIT64( 0x100000000 );
        b1 = b<<32;
        add128( rem0, rem1, b0, b1, &rem0, &rem1 );
    }
    rem0 = ( rem0<<32 ) | ( rem1>>32 );
    if ( b0<<32 <= rem0 ) {
        z |= 0xFFFFFFFF;
    } else {
        do_div( rem0, b0 );
        z |= rem0;
    }
    return z;

}

/*
-------------------------------------------------------------------------------
Returns an approximation to the square root of the 32-bit significand given
by `a'.  Considered as an integer, `a' must be at least 2^31.  If bit 0 of
`aExp' (the least significant bit) is 1, the integer returned approximates
2^31*sqrt(`a'/2^31), where `a' is considered an integer.  If bit 0 of `aExp'
is 0, the integer returned approximates 2^31*sqrt(`a'/2^30).  In either
case, the approximation returned lies strictly within +/-2 of the exact
value.
-------------------------------------------------------------------------------
*/
static bits32 estimateSqrt32( int16 aExp, bits32 a )
{
    static const bits16 sqrtOddAdjustments[] = {
        0x0004, 0x0022, 0x005D, 0x00B1, 0x011D, 0x019F, 0x0236, 0x02E0,
        0x039C, 0x0468, 0x0545, 0x0631, 0x072B, 0x0832, 0x0946, 0x0A67
    };
    static const bits16 sqrtEvenAdjustments[] = {
        0x0A2D, 0x08AF, 0x075A, 0x0629, 0x051A, 0x0429, 0x0356, 0x029E,
        0x0200, 0x0179, 0x0109, 0x00AF, 0x0068, 0x0034, 0x0012, 0x0002
    };
    int8 index;
    bits32 z;
    bits64 A;

    index = ( a>>27 ) & 15;
    if ( aExp & 1 ) {
        z = 0x4000 + ( a>>17 ) - sqrtOddAdjustments[ index ];
        z = ( ( a / z )<<14 ) + ( z<<15 );
        a >>= 1;
    }
    else {
        z = 0x8000 + ( a>>17 ) - sqrtEvenAdjustments[ index ];
        z = a / z + z;
        z = ( 0x20000 <= z ) ? 0xFFFF8000 : ( z<<15 );
        if ( z <= a ) return (bits32) ( ( (sbits32) a )>>1 );
    }
    A = ( (bits64) a )<<31;
    do_div( A, z );
    return ( (bits32) A ) + ( z>>1 );

}

/*
-------------------------------------------------------------------------------
Returns the number of leading 0 bits before the most-significant 1 bit
of `a'.  If `a' is zero, 32 is returned.
-------------------------------------------------------------------------------
*/
static int8 countLeadingZeros32( bits32 a )
{
    static const int8 countLeadingZerosHigh[] = {
        8, 7, 6, 6, 5, 5, 5, 5, 4, 4, 4, 4, 4, 4, 4, 4,
        3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3,
        2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2,
        2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2,
        1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
        1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
        1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
        1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
        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, 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
    };
    int8 shiftCount;

    shiftCount = 0;
    if ( a < 0x10000 ) {
        shiftCount += 16;
        a <<= 16;
    }
    if ( a < 0x1000000 ) {
        shiftCount += 8;
        a <<= 8;
    }
    shiftCount += countLeadingZerosHigh[ a>>24 ];
    return shiftCount;

}

/*
-------------------------------------------------------------------------------
Returns the number of leading 0 bits before the most-significant 1 bit
of `a'.  If `a' is zero, 64 is returned.
-------------------------------------------------------------------------------
*/
static int8 countLeadingZeros64( bits64 a )
{
    int8 shiftCount;

    shiftCount = 0;
    if ( a < ( (bits64) 1 )<<32 ) {
        shiftCount += 32;
    }
    else {
        a >>= 32;
    }
    shiftCount += countLeadingZeros32( a );
    return shiftCount;

}

/*
-------------------------------------------------------------------------------
Returns 1 if the 128-bit value formed by concatenating `a0' and `a1'
is equal to the 128-bit value formed by concatenating `b0' and `b1'.
Otherwise, returns 0.
-------------------------------------------------------------------------------
*/
INLINE flag eq128( bits64 a0, bits64 a1, bits64 b0, bits64 b1 )
{

    return ( a0 == b0 ) && ( a1 == b1 );

}

/*
-------------------------------------------------------------------------------
Returns 1 if the 128-bit value formed by concatenating `a0' and `a1' is less
than or equal to the 128-bit value formed by concatenating `b0' and `b1'.
Otherwise, returns 0.
-------------------------------------------------------------------------------
*/
INLINE flag le128( bits64 a0, bits64 a1, bits64 b0, bits64 b1 )
{

    return ( a0 < b0 ) || ( ( a0 == b0 ) && ( a1 <= b1 ) );

}

/*
-------------------------------------------------------------------------------
Returns 1 if the 128-bit value formed by concatenating `a0' and `a1' is less
than the 128-bit value formed by concatenating `b0' and `b1'.  Otherwise,
returns 0.
-------------------------------------------------------------------------------
*/
INLINE flag lt128( bits64 a0, bits64 a1, bits64 b0, bits64 b1 )
{

    return ( a0 < b0 ) || ( ( a0 == b0 ) && ( a1 < b1 ) );

}

/*
-------------------------------------------------------------------------------
Returns 1 if the 128-bit value formed by concatenating `a0' and `a1' is
not equal to the 128-bit value formed by concatenating `b0' and `b1'.
Otherwise, returns 0.
-------------------------------------------------------------------------------
*/
INLINE flag ne128( bits64 a0, bits64 a1, bits64 b0, bits64 b1 )
{

    return ( a0 != b0 ) || ( a1 != b1 );

}