-
Notifications
You must be signed in to change notification settings - Fork 0
/
Copy paththeories-FloatingPoint.shtml
654 lines (483 loc) · 22.3 KB
/
theories-FloatingPoint.shtml
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
<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd">
<!--
Design by http://www.bluewebtemplates.com
Released for free under a Creative Commons Attribution 3.0 License
-->
<html xmlns="http://www.w3.org/1999/xhtml">
<head>
<title>SMT-LIB The Satisfiability Modulo Theories Library</title>
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8" />
<link href="style.css" rel="stylesheet" type="text/css" />
<!-- CuFon: Enables smooth pretty custom font rendering. 100% SEO friendly. To disable, remove this section -->
<script type="text/javascript" src="js/cufon-yui.js"></script>
<script type="text/javascript" src="js/arial.js"></script>
<script type="text/javascript" src="js/cuf_run.js"></script>
<!-- CuFon ends -->
<link href="code-prettify/prettify.css" type="text/css" rel="stylesheet" />
<script src="code-prettify/run_prettify.js?lang=smtlib&skin=desert"></script>
</head>
<body>
<div class="main">
<div class="header">
<div class="header_resize">
<div class="menu_nav">
<ul>
<li><a href="index.shtml">Home</a></li>
<li><a href="about.shtml">About</a></li>
<li><a href="news.shtml">News</a></li>
<li class="active"><a href="standard.shtml">Standard</a></li>
<li><a href="benchmarks.shtml">Benchmarks</a></li>
<li><a href="software.shtml">Software</a></li>
<li><a href="credits.shtml">Credits</a></li>
</ul>
</div>
<div class="clr"></div>
<div class="logo">
<h1><a href="index.shtml">SMT-LIB <br/>
<small>The Satisfiability Modulo Theories Library</small></a>
</h1>
</div>
</div>
</div>
<div class="content">
<div class="content_resize">
<div class="mainbar">
<h2>FloatingPoint</h2>
<pre class="prettyprint lang-smtlib">
(theory FloatingPoint
:smt-lib-version 2.7
:smt-lib-release "2024-07-21"
:written-by "Cesare Tinelli and Martin Brain"
:date "2014-05-27"
:last-updated "2024-07-21"
:update-history
"Note: history only accounts for content changes, not release changes.
2024-07-21 Updated to Version 2.7.
2015-04-25 Updated to Version 2.5.
Updated reference to tech report.
"
:notes
"This is a theory of floating point numbers largely based on the IEEE standard
754-2008 for floating-point arithmetic (http://grouper.ieee.org/groups/754/)
but restricted to the binary case only.
A major extension over 754-2008 is that the theory has a sort for every
possible exponent and significand length.
Version 1 of the theory was based on proposal by P. Ruemmer and T. Wahl [RW10].
[RW10] Philipp Ruemmer and Thomas Wahl.
An SMT-LIB Theory of Binary Floating-Point Arithmetic.
Proceedings of the 8th International Workshop on
Satisfiability Modulo Theories (SMT'10), Edinburgh, UK, July 2010.
(http://www.philipp.ruemmer.org/publications/smt-fpa.pdf)
Version 2 was written by C. Tinelli using community feedback.
Version 3, the current one, was written by C. Tinelli and M. Brain following
further discussion within the SMT-LIB community, and then relaborated with
P. Ruemmer and T. Wahl.
A more detailed description of this version together with the rationale of
several models decisions as well as a formal semantics of the theory can be
found in
[BTRW15] Martin Brain, Cesare Tinelli, Philipp Ruemmer, and Thomas Wahl.
An Automatable Formal Semantics for IEEE-754 Floating-Point Arithmetic
Technical Report, 2015.
(http://smt-lib.org/papers/BTRW15.pdf)
The following additional people provided substantial feedback and directions:
François Bobot, David Cok, Alberto Griggio, Florian Lapschies, Leonardo de
Moura, Gabriele Paganelli, Cody Roux, Christoph Wintersteiger.
"
;-------
; Sorts
;-------
:sorts ((RoundingMode 0) (Real 0))
; Bit vector sorts, indexed by vector size
:sorts_description "All sort symbols of the form
(_ BitVec m)
where m is a numeral greater than 0."
; Floating point sort, indexed by the length of the exponent and significand
; components of the number
:sorts_description "All nullary sort symbols of the form
(_ FloatingPoint eb sb),
where eb and sb are numerals greater than 1."
:notes
"eb defines the number of bits in the exponent;
sb defines the number of bits in the significand, *including* the hidden bit.
"
; Short name for common floating point sorts
:sort ((Float16 0) (Float32 0) (Float64 0) (Float128 0))
:notes "
- Float16 is a synonym for (_ FloatingPoint 5 11)
- Float32 is a synonym for (_ FloatingPoint 8 24)
- Float64 is a synonym for (_ FloatingPoint 11 53)
- Float128 is a synonym for (_ FloatingPoint 15 113)
These correspond to the IEEE binary16, binary32, binary64 and binary128 formats.
"
;----------------
; Rounding modes
;----------------
; Constants for rounding modes, and their abbreviated version
:funs ((roundNearestTiesToEven RoundingMode) (RNE RoundingMode)
(roundNearestTiesToAway RoundingMode) (RNA RoundingMode)
(roundTowardPositive RoundingMode) (RTP RoundingMode)
(roundTowardNegative RoundingMode) (RTN RoundingMode)
(roundTowardZero RoundingMode) (RTZ RoundingMode)
)
;--------------------
; Value constructors
;--------------------
; Bitvector literals
:funs_description "
All binaries #bX of sort (_ BitVec m) where m is the number of digits in X.
All hexadecimals #xX of sort (_ BitVec m) where m is 4 times the number of
digits in X.
"
; FP literals as bit string triples, with the leading bit for the significand
; not represented (hidden bit)
:funs_description "All function symbols with declaration of the form
(fp (_ BitVec 1) (_ BitVec eb) (_ BitVec i) (_ FloatingPoint eb sb))
where eb and sb are numerals greater than 1 and i = sb - 1."
; Plus and minus infinity
:funs_description "All function symbols with declaration of the form
((_ +oo eb sb) (_ FloatingPoint eb sb))
((_ -oo eb sb) (_ FloatingPoint eb sb))
where eb and sb are numerals greater than 1."
:notes
"Semantically, for each eb and sb, there is exactly one +infinity value and
exactly one -infinity value in the set denoted by (_ FloatingPoint eb sb),
in agreement with the IEEE 754-2008 standard.
However, +/-infinity can have two representations in this theory.
E.g., +infinity for sort (_ FloatingPoint 2 3) is represented equivalently
by (_ +oo 2 3) and (fp #b0 #b11 #b00).
"
; Plus and minus zero
:funs_description "All function symbols with declaration of the form
((_ +zero eb sb) (_ FloatingPoint eb sb))
((_ -zero eb sb) (_ FloatingPoint eb sb))
where eb and sb are numerals greater than 1."
:note
"The +zero and -zero symbols are abbreviations for the corresponding fp literals.
E.g., (_ +zero 2 4) abbreviates (fp #b0 #b00 #b000)
(_ -zero 3 2) abbreviates (fp #b1 #b000 #b0)
"
; Non-numbers
:funs_description "All function symbols with declaration of the form
((_ NaN eb sb) (_ FloatingPoint eb sb))
where eb and sb are numerals greater than 1."
:notes
"For each eb and sb, there is exactly one NaN in the set denoted by
(_ FloatingPoint eb sb), in agreeement with Level 2 of IEEE 754-2008
(floating-point data). There is no distinction in this theory between
a ``quiet'' and a ``signaling'' NaN.
NaN has several representations, e.g.,(_ NaN eb sb) and any term of
the form (fp t #b1..1 s) where s is a binary containing at least a 1
and t is either #b0 or #b1.
"
;-----------
; Operators
;-----------
:funs_description "All function symbols with declarations of the form below
where eb and sb are numerals greater than 1.
; absolute value
(fp.abs (_ FloatingPoint eb sb) (_ FloatingPoint eb sb))
; negation (no rounding needed)
(fp.neg (_ FloatingPoint eb sb) (_ FloatingPoint eb sb))
; addition
(fp.add RoundingMode (_ FloatingPoint eb sb) (_ FloatingPoint eb sb)
(_ FloatingPoint eb sb))
; subtraction
(fp.sub RoundingMode (_ FloatingPoint eb sb) (_ FloatingPoint eb sb)
(_ FloatingPoint eb sb))
; multiplication
(fp.mul RoundingMode (_ FloatingPoint eb sb) (_ FloatingPoint eb sb)
(_ FloatingPoint eb sb))
; division
(fp.div RoundingMode (_ FloatingPoint eb sb) (_ FloatingPoint eb sb)
(_ FloatingPoint eb sb))
; fused multiplication and addition; (x * y) + z
(fp.fma RoundingMode (_ FloatingPoint eb sb) (_ FloatingPoint eb sb) (_ FloatingPoint eb sb)
(_ FloatingPoint eb sb))
; square root
(fp.sqrt RoundingMode (_ FloatingPoint eb sb) (_ FloatingPoint eb sb))
; remainder: x - y * n, where n in Z is nearest to x/y
(fp.rem (_ FloatingPoint eb sb) (_ FloatingPoint eb sb) (_ FloatingPoint eb sb))
; rounding to integral
(fp.roundToIntegral RoundingMode (_ FloatingPoint eb sb) (_ FloatingPoint eb sb))
; minimum and maximum
(fp.min (_ FloatingPoint eb sb) (_ FloatingPoint eb sb) (_ FloatingPoint eb sb))
(fp.max (_ FloatingPoint eb sb) (_ FloatingPoint eb sb) (_ FloatingPoint eb sb))
; comparison operators
; Note that all comparisons evaluate to false if either argument is NaN
(fp.leq (_ FloatingPoint eb sb) (_ FloatingPoint eb sb) Bool :chainable)
(fp.lt (_ FloatingPoint eb sb) (_ FloatingPoint eb sb) Bool :chainable)
(fp.geq (_ FloatingPoint eb sb) (_ FloatingPoint eb sb) Bool :chainable)
(fp.gt (_ FloatingPoint eb sb) (_ FloatingPoint eb sb) Bool :chainable)
; IEEE 754-2008 equality (as opposed to SMT-LIB =)
(fp.eq (_ FloatingPoint eb sb) (_ FloatingPoint eb sb) Bool :chainable)
; Classification of numbers
(fp.isNormal (_ FloatingPoint eb sb) Bool)
(fp.isSubnormal (_ FloatingPoint eb sb) Bool)
(fp.isZero (_ FloatingPoint eb sb) Bool)
(fp.isInfinite (_ FloatingPoint eb sb) Bool)
(fp.isNaN (_ FloatingPoint eb sb) Bool)
(fp.isNegative (_ FloatingPoint eb sb) Bool)
(fp.isPositive (_ FloatingPoint eb sb) Bool)
"
:note
"(fp.eq x y) evaluates to true if x evaluates to -zero and y to +zero, or vice versa.
fp.eq and all the other comparison operators evaluate to false if one of their
arguments is NaN.
"
;------------------------------
; Conversions from other sorts
;------------------------------
:funs_description "All function symbols with declarations of the form below
where m is a numerals greater than 0 and eb, sb, mb and nb are numerals
greater than 1.
; from single bitstring representation in IEEE 754-2008 interchange format,
; with m = eb + sb
((_ to_fp eb sb) (_ BitVec m) (_ FloatingPoint eb sb))
; from another floating point sort
((_ to_fp eb sb) RoundingMode (_ FloatingPoint mb nb) (_ FloatingPoint eb sb))
; from real
((_ to_fp eb sb) RoundingMode Real (_ FloatingPoint eb sb))
; from signed machine integer, represented as a 2's complement bit vector
((_ to_fp eb sb) RoundingMode (_ BitVec m) (_ FloatingPoint eb sb))
; from unsigned machine integer, represented as bit vector
((_ to_fp_unsigned eb sb) RoundingMode (_ BitVec m) (_ FloatingPoint eb sb))
"
;----------------------------
; Conversions to other sorts
;----------------------------
:funs_description "All function symbols with declarations of the form below
where m is a numeral greater than 0 and eb and sb are numerals greater than 1.
; to unsigned machine integer, represented as a bit vector
((_ fp.to_ubv m) RoundingMode (_ FloatingPoint eb sb) (_ BitVec m))
; to signed machine integer, represented as a 2's complement bit vector
((_ fp.to_sbv m) RoundingMode (_ FloatingPoint eb sb) (_ BitVec m))
; to real
(fp.to_real (_ FloatingPoint eb sb) Real)
"
:notes
"All fp.to_* functions are unspecified for NaN and infinity input values.
In addition, fp.to_ubv and fp.to_sbv are unspecified for finite number inputs
that are out of range (which includes all negative numbers for fp.to_ubv).
This means for instance that the formula
(= (fp.to_real (_ NaN 8 24)) (fp.to_real (fp c1 c2 c3)))
is satisfiable in this theory for all binary constants c1, c2, and c3
(of the proper sort).
"
:note
"There is no function for converting from (_ FloatingPoint eb sb) to the
corresponding IEEE 754-2008 binary format, as a bit vector (_ BitVec m) with
m = eb + sb, because (_ NaN eb sb) has multiple, well-defined representations.
Instead, an encoding of the kind below is recommended, where f is a term
of sort (_ FloatingPoint eb sb):
(declare-fun b () (_ BitVec m))
(assert (= ((_ to_fp eb sb) b) f))
"
;--------
; Values
;--------
:values "For all m,n > 1, the values of sort (_ FloatingPoint m n) are
- (_ +oo m n)
- (_ -oo m n)
- (_ NaN m n)
- all terms of the form (fp c1 c2 c3) where
- c1 is the binary #b0 or #b1
- c2 is a binary of size m other than #b1...1 (all 1s)
- c3 is a binary of size n-1
The set of values for RoundingMode is {RNE, RNA, RTP, RTN, RTZ}.
"
:notes
"No values are specified for the sorts Real and (_ BitVec n) in this theory.
They are specified in the theory declarations Reals and FixedSizeBitVectors,
respectively.
"
;-----------
; Semantics
;-----------
:note
"The semantics of this theory is described somewhat informally here.
A rigorous, self-contained specification can be found in [BTRW14]:
'An Automatable Formal Semantics for IEEE-754 Floating-Point Arithmetic'
and it takes precedence in the case of any (unintended) inconsistencies.
"
:definition
"For every expanded signature Sigma, the instance of FloatingPoints with
that signature is the theory consisting of all Sigma-models that satisfy
the constraints detailed below.
We use [[ _ ]] to denote the meaning of a sort or function symbol in
a given Sigma-model.
o (_ FloatingPoint eb sb)
[[(_ FloatingPoint eb sb)]] is the set of all the binary floating point
numbers with eb bits for the exponent and sb bits for the significand,
as defined by IEEE 754-2008.
Technically, [[(_ FloatingPoint eb sb)]] is the union of the set
{not_a_number} with four sets N, S, Z, I of bit-vector triples
(corresponding to normal numbers, subnormal numbers, zeros and
infinities) of the form (s, e, m) where s, e, and m correspond
respectively to the sign, the exponent and the significand (see
the paper for more details).
Note that the (semantic) value not_a_number is shared across all
[[(_ FloatingPoint eb sb)]].
o (_ BitVec m), binary and hexadecimal constants
These are interpreted as in the theory FixedSizeBitVectors.
o Real
[[Real]] is the set of real numbers.
o RoundingMode
[[RoundingMode]] is the set of the 5 rounding modes defined by IEEE 754-2008.
o (roundNearestTiesToEven RoundingMode), (roundNearestTiesToAway RoundingMode), ...
[[roundNearestTiesToEven]], [[roundNearestTiesToAway]], [[roundTowardPositive]],
[[roundTowardNegative]], and [[roundTowardZero]] are the 5 distinct elements
of [[RoundingMode]], and each corresponds to the rounding mode suggested by
the symbol's name.
o (RNE RoundingMode), (RNA RoundingMode), ...
[[RNE]] = [[roundNearestTiesToEven]];
[[RNA]] = [[roundNearestTiesToAway]];
[[RTP]] = [[roundTowardPositive]];
[[RTN]] = [[roundTowardNegative]];
[[RTZ]] = [[roundTowardZero]].
o (fp (_ BitVec 1) (_ BitVec eb) (_ BitVec i) (_ FloatingPoint eb sb))
[[fp]] returns the element of [[(_ FloatingPoint eb sb)]] whose IEEE 754-2008
binary encoding matches the input bit strings:
for all bitvectors
b1 in [[(_ BitVec 1)]], b2 in [[(_ BitVec eb)]] and b3 in [[(_ BitVec i)]],
[[fp]](b1, b2 ,b3) is the binary floating point number encoded in the IEEE
754-2008 standard with sign bit b1, exponent bits b2, and significant bit b3
(with 1 hidden bit).
Note that not_a_number can be denoted with fp terms as well. For instance, in
(_ FloatingPoint 2 2),
[[(_ NaN 2 2)]] = [[fp]]([[#b0]], [[#b11]], [[#b1]])
= [[fp]]([[#b1]], [[#b11]], [[#b1]])
Similarly,
[[(_ +oo 2 2)]] = [[fp]]([[#b0]], [[#b11]], [[#b0]])
[[(_ -oo 2 2)]] = [[fp]]([[#b1]], [[#b11]], [[#b0]])
o ((_ +oo eb sb) (_ FloatingPoint eb sb))
((_ -oo eb sb) (_ FloatingPoint eb sb))
((_ NaN eb sb) (_ FloatingPoint eb sb))
((_ +zero eb sb) (_ FloatingPoint eb sb))
((_ -zero eb sb) (_ FloatingPoint eb sb))
[[(_ +oo eb sb)]] is +infinity
[[(_ -oo eb sb)]] is -infinity
[[(_ NaN eb sb)]] is not_a_number
[[(_ +zero eb sb)]] is [[fp]]([[#b0]], [[#b0..0]], [[#b0..0]]) where
the first bitvector literal has eb 0s and
the second has sb - 1 0s
[[(_ -zero eb sb)]] is [[fp]]([[#b1]], [[#b0..0]], [[#b0..0]]) where
the first bitvector literal has eb 0s and
the second has sb - 1 0s
o ((_ to_fp eb sb) (_ BitVec m) (_ FloatingPoint eb sb))
[[(_ to_fp eb sb)]](b) = [[fp]](b[m-1:m-1], b[eb+sb-1:sb], b[sb-1:0])
where b[p:q] denotes the subvector of bitvector b between positions p and q.
o (fp.to_real (_ FloatingPoint eb sb) Real)
[[fp.to_real]](x) is the real number represented by x if x is not in
{-infinity, -infinity, not_a_number}. Otherwise, it is unspecified.
o ((_ to_fp eb sb) RoundingMode (_ FloatingPoint m n) (_ FloatingPoint eb sb))
[[(_ to_fp eb sb)]](r, x) = x if x in {+infinity, -infinity, not_a_number}.
[[(_ to_fp eb sb)]](r, x) = +/-infinity if [[fp.to_real]](x) is too large/too
small to be represented as a finite number of [[(_ FloatingPoint eb sb)]];
[[(_ to_fp eb sb)]](r, x) = y otherwise, where y is the finite number
such that [[fp.to_real]](y) is closest to [[fp.to_real]](x) according to
rounding mode r.
o ((_ to_fp eb sb) RoundingMode Real (_ FloatingPoint eb sb))
[[(_ to_fp eb sb)]](r, x) = +/-infinity if x is too large/too small
to be represented as a finite number of [[(_ FloatingPoint eb sb)]];
[[(_ to_fp eb sb)]](r, x) = y otherwise, where y is the finite number
such that [[fp.to_real]](y) is closest to x according to rounding mode r.
o ((_ to_fp eb sb) RoundingMode (_ BitVec m) (_ FloatingPoint eb sb))
Let b in [[(_ BitVec m)]] and let n be the signed integer represented by b
(in 2's complement format).
[[(_ to_fp eb sb)]](r, b) = +/-infinity if n is too large/too small to be
represented as a finite number of [[(_ FloatingPoint eb sb)]];
[[(_ to_fp eb sb)]](r, x) = y otherwise, where y is the finite number
such that [[fp.to_real]](y) is closest to n according to rounding mode r.
o ((_ to_fp_unsigned eb sb) RoundingMode (_ BitVec m) (_ FloatingPoint eb sb))
Let b in [[(_ BitVec m)]] and let n be the unsigned integer represented by b.
[[(_ to_fp_unsigned eb sb)]](r, x) = +infinity if n is too large to be
represented as a finite number of [[(_ FloatingPoint eb sb)]];
[[(_ to_fp_unsigned eb sb)]](r, x) = y otherwise, where y is the finite number
such that [[fp.to_real]](y) is closest to n according to rounding mode r.
o ((_ fp.to_ubv m) RoundingMode (_ FloatingPoint eb sb) (_ BitVec m))
[[(_ fp.to_ubv m)]](r, x) = b if the unsigned integer n represented by b is
the closest integer according to rounding mode r to the real number
represented by x, and n is in the range [0, 2^m - 1].
[[(_ fp.to_ubv m)]](r, x) is unspecified in all other cases (including when
x is in {-infinity, -infinity, not_a_number}).
o ((_ fp.to_sbv m) RoundingMode (_ FloatingPoint eb sb) (_ BitVec m))
[[(_ fp.to_sbv m)]](r, x) = b if the signed integer n represented by b
(in 2's complement format) is the closest integer according to rounding mode
r to the real number represented by x, and n is in the range
[-2^{m-1}, 2^{m-1} - 1].
[[(_ fp.to_sbv m)]](r, x) is unspecified in all other cases (including when
x is in {-infinity, -infinity, not_a_number}).
o (fp.isNormal (_ FloatingPoint eb sb) Bool)
[[fp.isNormal]](x) = true iff x is a normal number.
o (fp.isSubnormal (_ FloatingPoint eb sb) Bool)
[[fp.isSubnormal]](x) = true iff x is a subnormal number.
o (fp.isZero (_ FloatingPoint eb sb) Bool)
[[fp.isZero]](x) = true iff x is positive or negative zero.
o (fp.isInfinite (_ FloatingPoint eb sb) Bool)
[[fp.isInfinite]](x) = true iff x is +infinity or -infinity.
o (fp.isNaN (_ FloatingPoint eb sb) Bool)
[[fp.isNaN]](x) = true iff x = not_a_number.
o (fp.isNegative (_ FloatingPoint eb sb) Bool)
[[fp.isNegative]](x) = true iff x is [[-zero]] or [[fp.lt]](x, [[-zero]]) holds.
o (fp.isPositive (_ FloatingPoint eb sb) Bool)
[[fp.isPositive]](x) = true iff x is [[+zero]] or [[fp.lt]]([[+zero]], x) holds.
o all the other function symbols are interpreted as described in [BTRW15].
"
)
</pre>
(<a href="Theories/FloatingPoint.smt2">raw file</a>)
<br>
<br>
</div>
<div class="sidebar">
<div class="gadget">
<ul class="sb_menu">
<li><a href="index.shtml">Home</a></li>
<li><a href="about.shtml">About</a></li>
<li><a href="news.shtml">News</a></li>
<li>Standard
<ul class="ex_menu">
<li><a href="language.shtml">Language</a>
<li><a href="theories.shtml">Theories</a>
<li><a href="logics.shtml">Logics</a>
<li><a href="examples.shtml">Examples</a>
</ul>
</li>
<li><a href="benchmarks.shtml">Benchmarks</a></li>
<li>Software
<ul class="ex_menu">
<li><a href="solvers.shtml">Solvers</a></li>
<li><a href="utilities.shtml">Utilities</a></li>
</ul>
</li>
<li><a href="contact.shtml">Contact</a></li>
<li><a href="related.shtml">Related</a></li>
<li><a href="credits.shtml">Credits</a></li>
</ul>
</div>
</div>
</div>
</div>
<div class="clr"></div>
<div class="footer">
<div class="footer_resize">
<p class="lf">
© Copyright The SMT-LIB Initiative <br>
Based on a design by
Blue <a href="http://www.bluewebtemplates.com">Web Templates</a>
</p>
<ul class="fmenu">
<li><a href="index.shtml">Home</a></li>
<li><a href="about.shtml">About</a></li>
<li><a href="news.shtml">News</a></li>
<li class="active"><a href="standard.shtml">Standard</a></li>
<li><a href="benchmarks.shtml">Benchmarks</a></li>
<li><a href="software.shtml">Software</a></li>
<li><a href="credits.shtml">Credits</a></li>
</ul>
<div class="clr"></div>
</div>
</div>
</div>
</body>
</html>