-
Notifications
You must be signed in to change notification settings - Fork 0
/
numeric.html
1818 lines (1311 loc) · 76.1 KB
/
numeric.html
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
907
908
909
910
911
912
913
914
915
916
917
918
919
920
921
922
923
924
925
926
927
928
929
930
931
932
933
934
935
936
937
938
939
940
941
942
943
944
945
946
947
948
949
950
951
952
953
954
955
956
957
958
959
960
961
962
963
964
965
966
967
968
969
970
971
972
973
974
975
976
977
978
979
980
981
982
983
984
985
986
987
988
989
990
991
992
993
994
995
996
997
998
999
1000
<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN"
"http://www.w3.org/TR/html4/loose.dtd"><!-- DO NOT EDIT THIS FILE-->
<!-- Edit the .tex version instead-->
<html>
<head>
<title>Numeric Operations</title>
<link href="csug.css" rel="stylesheet" type="text/css">
</head>
<body>
<a name="g62"></a>
<a name="./numeric:h0"></a>
<h1>Chapter 8. Numeric Operations<a name="CHPTNUMERIC"></a></h1>
<p>
This chapter describes <i>Chez Scheme</i> extensions to the standard set of
operations on numbers.
See Chapter <a href="http://scheme.com/tspl4/./objects.html#g106">6</a> of <i>The Scheme Programming Language, 4th Edition</i> or the Revised<sup>6</sup> Report
on Scheme for a description of standard operations on numbers.
<p>
<i>Chez Scheme</i> supports the full set of Scheme numeric datatypes, including
exact and inexact integer, rational, real, and complex numbers.
A variety of representations are used to support these datatypes:
<p>
<dl compact>
<dt><a name="./numeric:s0"></a><i>Fixnums</i><dd> represent exact integers in the
fixnum range (see <tt>most-negative-fixnum</tt> and
<tt>most-positive-fixnum</tt>).
The length of a string, vector, or fxvector is constrained to be a fixnum.
<p>
<dt><a name="./numeric:s1"></a><i>Bignums</i><dd> represent arbitrary-precision
exact integers outside of the fixnum range.
<p>
<dt><a name="./numeric:s2"></a><i>Ratnums</i><dd> represent arbitrary-precision
exact rational numbers.
Each ratnum contains an exact integer (fixnum
or bignum) numerator and an exact integer denominator.
Ratios are always reduced to lowest terms and never have a denominator
of one or a numerator of zero.
<p>
<dt><a name="./numeric:s3"></a><i>Flonums</i><dd> represent inexact real numbers.
Flonums are IEEE 64-bit floating-point numbers.
(Since flonums cannot represent irrational numbers, all inexact real
numbers are actually rational, although they may approximate irrational
quantities.)
<p>
<dt><a name="./numeric:s4"></a><i>Exact complexnums</i><dd>
represent exact complex numbers.
Each exact complexnum contains an exact rational (fixnum, bignum, or
ratnum) real part and an exact rational imaginary part.
<p>
<dt><a name="./numeric:s5"></a><i>Inexact complexnums</i><dd>
represent inexact complex numbers.
Each inexact complexnum contains a flonum real part and a flonum imaginary part.
</dl>
<p>
Most numbers can be represented in only one way; however, real numbers
are sometimes represented as inexact complex numbers with imaginary
component equal to zero.
<p>
<i>Chez Scheme</i> extends the syntax of numbers with arbitrary radixes from
two through 36, nondecimal floating-point and scientific notation,
and printed representations for
IEEE infinities and NANs. (NAN stands for "not-a-number.")
<p>
Arbitrary radixes are specified with the prefix <tt>#<i>n</i>r</tt>, where
<tt><i>n</i></tt> ranges from 2 through 36.
Digits beyond 9 are specified with the letters (in either
upper or lower case) <tt>a</tt> through <tt>z</tt>.
For example, <tt>#2r101</tt> is 5<sub>10</sub>, and
<tt>#36rZ</tt> is 35<sub>10</sub>.
<p>
For higher radixes, an ambiguity arises between the interpretation of
certain letters, e.g., <tt>e</tt>, as digits or exponent specifiers; in
such cases, the letter is assumed to be a digit.
For example, the <tt>e</tt> in <tt>#x3.2e5</tt> is interpreted as a
digit, not as an exponent marker, whereas in <tt>3.2e5</tt> it is
treated as an exponent marker.
<p>
IEEE infinities are printed as <tt>+inf.0</tt> and <tt>-inf.0</tt>,
while IEEE NANs are printed as <tt>+nan.0</tt> or <tt>-nan.0</tt>.
(+nan.0 is used on output for all NANs.)
<p>
<p><tt>(/ 1.0 0.0) <img src="math/csug/0.gif" alt="<graphic>"> +inf.0<br>
(/ 1.0 -0.0) <img src="math/csug/0.gif" alt="<graphic>"> -inf.0<br>
(/ 0.0 0.0) <img src="math/csug/0.gif" alt="<graphic>"> +nan.0<br>
(/ +inf.0 -inf.0) <img src="math/csug/0.gif" alt="<graphic>"> +nan.0</tt>
<p>The first section of this chapter describes type-specific numeric type
predicates.
Sections <a href="./numeric.html#g64">8.2</a> through <a href="./numeric.html#g66">8.4</a>
describe fast, type-specific
numeric operations on fixnums, flonums, and inexact complex numbers
(flonums and/or inexact complexnums).
The fixnum-specific versions should be used only when the programmer
is certain that the operands and results (where appropriate) will be
fixnums, i.e., integers in the range <tt>(most-negative-fixnum)</tt> to
<tt>(most-positive-fixnum)</tt>, inclusive.
The flonum-specific versions should be used only when the
inputs and outputs (where appropriate) are certain to be flonums.
The mixed flonum/complexnum versions should be used only when the
inputs are certain to be either flonums or inexact complexnums.
Section <a href="./numeric.html#g67">8.5</a> describes operations, both
arbitrary precision and fixnum-specific, that allow
exact integers to be treated as sets or sequences of bits.
Random number generation is covered Section <a href="./numeric.html#g68">8.6</a>,
and miscellaneous numeric operations are covered in the
Section <a href="./numeric.html#g69">8.7</a>.
<p>
<h3><a name="g63"></a><a name="./numeric:h1"></a>Section 8.1. Numeric Type Predicates</h3>
<p>
<a name="./numeric:s6"></a><a name="./numeric:s7"></a><a name="./numeric:s8"></a><a name="./numeric:s9"></a><a name="./numeric:s10"></a>The Revised<sup>6</sup> Report distinguishes two types of special numeric objects:
fixnums and flonums.
<i>Chez Scheme</i> additionally distinguishes <i>bignums</i> (exact integers outside
of the bignum range) and <i>ratnums</i> (ratios of exact integers).
It also provides a predicate for recognizing <i>cflonums</i>, which are
flonums or inexact complex numbers.
<p>
<a name="./numeric:s11"></a><span class=formdef><b>procedure</b>: <tt>(bignum? <i>obj</i>)</tt></span>
<br>
<b>returns: </b><tt>#t</tt> if <tt><i>obj</i></tt> is a bignum, otherwise <tt>#f</tt>
<br>
<b>libraries: </b><tt>(chezscheme)</tt>
<p>
<p><tt>
(bignum? 0) <img src="math/csug/0.gif" alt="<graphic>"> #f<br>
(bignum? (most-positive-fixnum)) <img src="math/csug/0.gif" alt="<graphic>"> #f<br>
(bignum? (most-negative-fixnum)) <img src="math/csug/0.gif" alt="<graphic>"> #f<br>
(bignum? (* (most-positive-fixnum) 2)) <img src="math/csug/0.gif" alt="<graphic>"> #t<br>
(bignum? 3/4) <img src="math/csug/0.gif" alt="<graphic>"> #f<br>
(bignum? 'a) <img src="math/csug/0.gif" alt="<graphic>"> #f</tt>
<p><a name="./numeric:s12"></a><span class=formdef><b>procedure</b>: <tt>(ratnum? <i>obj</i>)</tt></span>
<br>
<b>returns: </b><tt>#t</tt> if <tt><i>obj</i></tt> is a ratnum, otherwise <tt>#f</tt>
<br>
<b>libraries: </b><tt>(chezscheme)</tt>
<p>
<p><tt>
(ratnum? 0) <img src="math/csug/0.gif" alt="<graphic>"> #f<br>
(ratnum? (* (most-positive-fixnum) 2)) <img src="math/csug/0.gif" alt="<graphic>"> #f<br>
(ratnum? 3/4) <img src="math/csug/0.gif" alt="<graphic>"> #t<br>
(ratnum? -10/2) <img src="math/csug/0.gif" alt="<graphic>"> #f<br>
(ratnum? -11/2) <img src="math/csug/0.gif" alt="<graphic>"> #t<br>
(ratnum? 'a) <img src="math/csug/0.gif" alt="<graphic>"> #f</tt>
<p><a name="./numeric:s13"></a><span class=formdef><b>procedure</b>: <tt>(cflonum? <i>obj</i>)</tt></span>
<br>
<b>returns: </b><tt>#t</tt> if <tt><i>obj</i></tt> is an inexact complexnum or flonum, otherwise <tt>#f</tt>
<br>
<b>libraries: </b><tt>(chezscheme)</tt>
<p>
<p><tt>
(cflonum? 0) <img src="math/csug/0.gif" alt="<graphic>"> #f<br>
(cflonum? 0.0) <img src="math/csug/0.gif" alt="<graphic>"> #t<br>
(cflonum? 3+4i) <img src="math/csug/0.gif" alt="<graphic>"> #f<br>
(cflonum? 3.0+4i) <img src="math/csug/0.gif" alt="<graphic>"> #t<br>
(cflonum? +i) <img src="math/csug/0.gif" alt="<graphic>"> #f<br>
(cflonum? +1.0i) <img src="math/csug/0.gif" alt="<graphic>"> #t</tt>
<p>
<h3><a name="g64"></a><a name="./numeric:h2"></a>Section 8.2. Fixnum Operations<a name="SECTNUMERICFIXNUM"></a></h3>
<p>
Fixnum-specific procedures normally check their inputs and outputs (where
appropriate), but at optimization level 3 the compiler generates, in most
cases, code that does not perform these checks.
<p>
<a name="./numeric:s14"></a><span class=formdef><b>procedure</b>: <tt>(most-positive-fixnum)</tt></span>
<br>
<b>returns: </b>the most negative fixnum supported by the system
<br><span class=formdef><b>procedure</b>: <tt>(most-negative-fixnum)</tt></span>
<br>
<b>returns: </b>the most positive fixnum supported by the system
<br>
<b>libraries: </b><tt>(chezscheme)</tt>
<p>These procedures are identical to the Revised<sup>6</sup> Report
<tt>greatest-fixnum</tt> and <tt>least-fixnum</tt> procedures.
<p>
<a name="./numeric:s15"></a><span class=formdef><b>procedure</b>: <tt>(fx= <i>fixnum<sub>1</sub></i> <i>fixnum<sub>2</sub></i> ...)</tt></span>
<br><span class=formdef><b>procedure</b>: <tt>(fx< <i>fixnum<sub>1</sub></i> <i>fixnum<sub>2</sub></i> ...)</tt></span>
<br><span class=formdef><b>procedure</b>: <tt>(fx> <i>fixnum<sub>1</sub></i> <i>fixnum<sub>2</sub></i> ...)</tt></span>
<br><span class=formdef><b>procedure</b>: <tt>(fx<= <i>fixnum<sub>1</sub></i> <i>fixnum<sub>2</sub></i> ...)</tt></span>
<br><span class=formdef><b>procedure</b>: <tt>(fx>= <i>fixnum<sub>1</sub></i> <i>fixnum<sub>2</sub></i> ...)</tt></span>
<br>
<b>returns: </b><tt>#t</tt> if the relation holds, <tt>#f</tt> otherwise
<br>
<b>libraries: </b><tt>(chezscheme)</tt>
<p>The predicate <tt>fx=</tt> returns <tt>#t</tt> if its arguments are equal.
The predicate <tt>fx<</tt> returns <tt>#t</tt> if its arguments are monotonically
increasing, i.e., each argument is greater than the preceding ones,
while <tt>fx></tt> returns <tt>#t</tt> if its arguments are monotonically decreasing.
The predicate <tt>fx<=</tt> returns <tt>#t</tt> if its arguments are monotonically
nondecreasing, i.e., each argument is not less than the preceding ones,
while <tt>fx>=</tt> returns <tt>#t</tt> if its arguments are monotonically nonincreasing.
When passed only one argument, each of these predicates returns <tt>#t</tt>.
<p>
These procedures are similar to the Revised<sup>6</sup> Report procedures
<tt>fx=?</tt>, <tt>fx<?</tt>, <tt>fx>?</tt>, <tt>fx<=?</tt>,
and <tt>fx>=?</tt> except that the Revised<sup>6</sup> Report procedures
require two or more arguments, and their names have the "<tt>?</tt>"
suffix.
<p>
<p><tt>(fx= 0) <img src="math/csug/0.gif" alt="<graphic>"> #t<br>
(fx= 0 0) <img src="math/csug/0.gif" alt="<graphic>"> #t<br>
(fx< (most-negative-fixnum) 0 (most-positive-fixnum)) <img src="math/csug/0.gif" alt="<graphic>"> #t<br>
(let ([x 3]) (fx<= 0 x 9)) <img src="math/csug/0.gif" alt="<graphic>"> #t<br>
(fx<= 0 3 3) <img src="math/csug/0.gif" alt="<graphic>"> #t<br>
(fx>= 0 0 (most-negative-fixnum)) <img src="math/csug/0.gif" alt="<graphic>"> #t</tt>
<p><a name="./numeric:s16"></a><span class=formdef><b>procedure</b>: <tt>(fxnonpositive? <i>fixnum</i>)</tt></span>
<br>
<b>returns: </b><tt>#t</tt> if <tt><i>fixnum</i></tt> is not greater than zero, <tt>#f</tt> otherwise
<br><span class=formdef><b>procedure</b>: <tt>(fxnonnegative? <i>fixnum</i>)</tt></span>
<br>
<b>returns: </b><tt>#t</tt> if <tt><i>fixnum</i></tt> is not less than zero, <tt>#f</tt> otherwise
<br>
<b>libraries: </b><tt>(chezscheme)</tt>
<p><tt>fxnonpositive?</tt> is equivalent to <tt>(lambda (x) (fx<= x 0))</tt>,
and
<tt>fxnonnegative?</tt> is equivalent to <tt>(lambda (x) (fx>= x 0))</tt>.
<p>
<p><tt>(fxnonpositive? 128) <img src="math/csug/0.gif" alt="<graphic>"> #f<br>
(fxnonpositive? 0) <img src="math/csug/0.gif" alt="<graphic>"> #t<br>
(fxnonpositive? -1) <img src="math/csug/0.gif" alt="<graphic>"> #t
<br>
<br>
(fxnonnegative? -65) <img src="math/csug/0.gif" alt="<graphic>"> #f<br>
(fxnonnegative? 0) <img src="math/csug/0.gif" alt="<graphic>"> #t<br>
(fxnonnegative? 1) <img src="math/csug/0.gif" alt="<graphic>"> #t</tt>
<p><a name="./numeric:s17"></a><span class=formdef><b>procedure</b>: <tt>(fx+ <i>fixnum</i> ...)</tt></span>
<br>
<b>returns: </b>the sum of the arguments <tt><i>fixnum</i> ...</tt>
<br>
<b>libraries: </b><tt>(chezscheme)</tt>
<p>When called with no arguments, <tt>fx+</tt> returns <tt>0</tt>.
<p>
<p><tt>(fx+) <img src="math/csug/0.gif" alt="<graphic>"> 0<br>
(fx+ 1 2) <img src="math/csug/0.gif" alt="<graphic>"> 3<br>
(fx+ 3 4 5) <img src="math/csug/0.gif" alt="<graphic>"> 12<br>
(apply fx+ '(1 2 3 4 5)) <img src="math/csug/0.gif" alt="<graphic>"> 15</tt>
<p><a name="./numeric:s18"></a><span class=formdef><b>procedure</b>: <tt>(fx- <i>fixnum<sub>1</sub></i> <i>fixnum<sub>2</sub></i> ...)</tt></span>
<br>
<b>returns: </b>a fixnum
<br>
<b>libraries: </b><tt>(chezscheme)</tt>
<p>When called with one argument, <tt>fx-</tt> returns the negative of <tt><i>fixnum<sub>1</sub></i></tt>.
Thus, <tt>(fx- <i>fixnum<sub>1</sub></i>)</tt> is an idiom for <tt>(fx- 0 <i>fixnum<sub>1</sub></i>)</tt>.
<p>
When called with two or more arguments, <tt>fx-</tt> returns the result of
subtracting the sum of the numbers <tt><i>fixnum<sub>2</sub></i> ...</tt> from
<tt><i>fixnum<sub>1</sub></i></tt>.
<p>
<p><tt>(fx- 3) <img src="math/csug/0.gif" alt="<graphic>"> -3<br>
(fx- 4 3) <img src="math/csug/0.gif" alt="<graphic>"> 1<br>
(fx- 4 3 2 1) <img src="math/csug/0.gif" alt="<graphic>"> -2</tt>
<p><a name="./numeric:s19"></a><span class=formdef><b>procedure</b>: <tt>(fx* <i>fixnum</i> ...)</tt></span>
<br>
<b>returns: </b>the product of the arguments <tt><i>fixnum</i> ...</tt>
<br>
<b>libraries: </b><tt>(chezscheme)</tt>
<p>When called with no arguments, <tt>fx*</tt> returns <tt>1</tt>.
<p>
<p><tt>(fx*) <img src="math/csug/0.gif" alt="<graphic>"> 1<br>
(fx* 1 2) <img src="math/csug/0.gif" alt="<graphic>"> 2<br>
(fx* 3 -4 5) <img src="math/csug/0.gif" alt="<graphic>"> -60<br>
(apply fx* '(1 -2 3 -4 5)) <img src="math/csug/0.gif" alt="<graphic>"> 120</tt>
<p><a name="./numeric:s20"></a><span class=formdef><b>procedure</b>: <tt>(fx/ <i>fixnum<sub>1</sub></i> <i>fixnum<sub>2</sub></i> ...)</tt></span>
<br>
<b>returns: </b>see explanation
<br>
<b>libraries: </b><tt>(chezscheme)</tt>
<p>When called with one argument, <tt>fx/</tt> returns the reciprocal
of <tt><i>fixnum<sub>1</sub></i></tt>.
That is, <tt>(fx/ <i>fixnum<sub>1</sub></i>)</tt> is an idiom for
<tt>(fx/ 1 <i>fixnum<sub>1</sub></i>)</tt>.
<p>
When called with two or more arguments, <tt>fx/</tt> returns
the result of
dividing <tt><i>fixnum<sub>1</sub></i></tt> by the product of the remaining arguments
<tt><i>fixnum<sub>2</sub></i> ...</tt>.
<p>
<p><tt>(fx/ 1) <img src="math/csug/0.gif" alt="<graphic>"> 1<br>
(fx/ -17) <img src="math/csug/0.gif" alt="<graphic>"> 0<br>
(fx/ 8 -2) <img src="math/csug/0.gif" alt="<graphic>"> -4<br>
(fx/ -9 2) <img src="math/csug/0.gif" alt="<graphic>"> -4<br>
(fx/ 60 5 3 2) <img src="math/csug/0.gif" alt="<graphic>"> 2</tt>
<p><a name="./numeric:s21"></a><span class=formdef><b>procedure</b>: <tt>(fx1+ <i>fixnum</i>)</tt></span>
<br><span class=formdef><b>procedure</b>: <tt>(fx1- <i>fixnum</i>)</tt></span>
<br>
<b>returns: </b><tt><i>fixnum</i></tt> plus 1 or <tt><i>fixnum</i></tt> minus 1
<br>
<b>libraries: </b><tt>(chezscheme)</tt>
<p>
<p><tt>(define fxplus<br>
(lambda (x y)<br>
(if (fxzero? x)<br>
y<br>
(fxplus (fx1- x) (fx1+ y)))))
<br>
<br>
(fxplus 7 8) <img src="math/csug/0.gif" alt="<graphic>"> 15</tt>
<p><tt>fx1+</tt> and <tt>fx1-</tt> can be defined as follows:
<p>
<p><tt>(define fx1+ (lambda (x) (fx+ x 1)))<br>
(define fx1- (lambda (x) (fx- x 1)))</tt>
<p><a name="./numeric:s22"></a><span class=formdef><b>procedure</b>: <tt>(fxquotient <i>fixnum<sub>1</sub></i> <i>fixnum<sub>2</sub></i> ...)</tt></span>
<br>
<b>returns: </b>see explanation
<br>
<b>libraries: </b><tt>(chezscheme)</tt>
<p><tt>fxquotient</tt> is identical to <tt>fx/</tt>.
See the description of <tt>fx/</tt> above.
<p>
<a name="./numeric:s23"></a><span class=formdef><b>procedure</b>: <tt>(fxremainder <i>fixnum<sub>1</sub></i> <i>fixnum<sub>2</sub></i>)</tt></span>
<br>
<b>returns: </b>the fixnum remainder of <tt><i>fixnum<sub>1</sub></i></tt> divided by <tt><i>fixnum<sub>2</sub></i></tt>
<br>
<b>libraries: </b><tt>(chezscheme)</tt>
<p>The result of <tt>fxremainder</tt> has the same sign as <tt><i>fixnum<sub>1</sub></i></tt>.
<p>
<p><tt>(fxremainder 16 4) <img src="math/csug/0.gif" alt="<graphic>"> 0<br>
(fxremainder 5 2) <img src="math/csug/0.gif" alt="<graphic>"> 1<br>
(fxremainder -45 7) <img src="math/csug/0.gif" alt="<graphic>"> -3<br>
(fxremainder 10 -3) <img src="math/csug/0.gif" alt="<graphic>"> 1<br>
(fxremainder -17 -9) <img src="math/csug/0.gif" alt="<graphic>"> -8</tt>
<p><a name="./numeric:s24"></a><span class=formdef><b>procedure</b>: <tt>(fxmodulo <i>fixnum<sub>1</sub></i> <i>fixnum<sub>2</sub></i>)</tt></span>
<br>
<b>returns: </b>the fixnum modulus of <tt><i>fixnum<sub>1</sub></i></tt> and <tt><i>fixnum<sub>2</sub></i></tt>
<br>
<b>libraries: </b><tt>(chezscheme)</tt>
<p>The result of <tt>fxmodulo</tt> has the same sign as <tt><i>fixnum<sub>2</sub></i></tt>.
<p>
<p><tt>(fxmodulo 16 4) <img src="math/csug/0.gif" alt="<graphic>"> 0<br>
(fxmodulo 5 2) <img src="math/csug/0.gif" alt="<graphic>"> 1<br>
(fxmodulo -45 7) <img src="math/csug/0.gif" alt="<graphic>"> 4<br>
(fxmodulo 10 -3) <img src="math/csug/0.gif" alt="<graphic>"> -2<br>
(fxmodulo -17 -9) <img src="math/csug/0.gif" alt="<graphic>"> -8</tt>
<p><a name="./numeric:s25"></a><span class=formdef><b>procedure</b>: <tt>(fxabs <i>fixnum</i>)</tt></span>
<br>
<b>returns: </b>the absolute value of <tt><i>fixnum</i></tt>
<br>
<b>libraries: </b><tt>(chezscheme)</tt>
<p>
<p><tt>
(fxabs 1) <img src="math/csug/0.gif" alt="<graphic>"> 1<br>
(fxabs -1) <img src="math/csug/0.gif" alt="<graphic>"> 1<br>
(fxabs 0) <img src="math/csug/0.gif" alt="<graphic>"> 0</tt>
<p>
<h3><a name="g65"></a><a name="./numeric:h3"></a>Section 8.3. Flonum Operations<a name="SECTNUMERICFLONUM"></a></h3>
<p>
Inexact real numbers are normally represented by <tt><i>flonums</i></tt>.
A flonum is a single 64-bit double-precision floating point
number.
This section describes operations on flonums, most of which accept
flonum arguments and return flonum values.
In most cases, the operations are inline-coded or coded as machine
language subroutines at optimize-level 3 with
no argument type checking; full type checking is performed at lower
optimize levels.
Flonum-specific procedure names begin with the prefix "<tt>fl</tt>" to
set them apart from their generic counterparts.
<p>
Inexact real numbers may also be represented by inexact complexnums
with imaginary parts equal to zero, which cannot be used as input
to the flonum-specific operators.
Such numbers are produced, however, only from operations involving
complex numbers with nonzero imaginary parts, by explicit calls
to <tt>fl-make-rectangular</tt>, <tt>make-rectangular</tt>, or
<tt>make-polar</tt>, or by numeric input in either polar or rectangular
format.
<p>
<a name="./numeric:s26"></a><span class=formdef><b>procedure</b>: <tt>(flonum->fixnum <i>flonum</i>)</tt></span>
<br>
<b>returns: </b>the fixnum representation of <tt><i>flonum</i></tt>, truncated
<br>
<b>libraries: </b><tt>(chezscheme)</tt>
<p>The truncated value of <tt><i>flonum</i></tt> must fall within the fixnum range.
<tt>flonum->fixnum</tt> is a restricted version of
<a name="./numeric:s27"></a><tt>exact</tt>,
which converts any numeric representation
to its exact equivalent.
<p>
<p><tt>(flonum->fixnum 0.0) <img src="math/csug/0.gif" alt="<graphic>"> 0<br>
(flonum->fixnum 3.9) <img src="math/csug/0.gif" alt="<graphic>"> 3<br>
(flonum->fixnum -2.2) <img src="math/csug/0.gif" alt="<graphic>"> -2</tt>
<p><a name="./numeric:s28"></a><span class=formdef><b>procedure</b>: <tt>(fl= <i>flonum<sub>1</sub></i> <i>flonum<sub>2</sub></i> ...)</tt></span>
<br><span class=formdef><b>procedure</b>: <tt>(fl< <i>flonum<sub>1</sub></i> <i>flonum<sub>2</sub></i> ...)</tt></span>
<br><span class=formdef><b>procedure</b>: <tt>(fl> <i>flonum<sub>1</sub></i> <i>flonum<sub>2</sub></i> ...)</tt></span>
<br><span class=formdef><b>procedure</b>: <tt>(fl<= <i>flonum<sub>1</sub></i> <i>flonum<sub>2</sub></i> ...)</tt></span>
<br><span class=formdef><b>procedure</b>: <tt>(fl>= <i>flonum<sub>1</sub></i> <i>flonum<sub>2</sub></i> ...)</tt></span>
<br>
<b>returns: </b><tt>#t</tt> if the relation holds, <tt>#f</tt> otherwise
<br>
<b>libraries: </b><tt>(chezscheme)</tt>
<p>The predicate <tt>fl=</tt> returns <tt>#t</tt> if its arguments are equal.
The predicate <tt>fl<</tt> returns <tt>#t</tt> if its arguments are monotonically
increasing, i.e., each argument is greater than the preceding ones,
while <tt>fl></tt> returns <tt>#t</tt> if its arguments are monotonically decreasing.
The predicate <tt>fl<=</tt> returns <tt>#t</tt> if its arguments are monotonically
nondecreasing, i.e., each argument is not less than the preceding ones,
while <tt>fl>=</tt> returns <tt>#t</tt> if its arguments are monotonically nonincreasing.
When passed only one argument, each of these predicates returns <tt>#t</tt>.
<p>
IEEE NANs are not comparable, i.e., comparisons involving NANs always return
<tt>#f</tt>.
<p>
These procedures are similar to the Revised<sup>6</sup> Report procedures
<tt>fl=?</tt>, <tt>fl<?</tt>, <tt>fl>?</tt>, <tt>fl<=?</tt>,
and <tt>fl>=?</tt> except that the Revised<sup>6</sup> Report procedures
require two or more arguments, and their names have the "<tt>?</tt>"
suffix.
<p>
<p><tt>(fl= 0.0) <img src="math/csug/0.gif" alt="<graphic>"> #t<br>
(fl= 0.0 0.0) <img src="math/csug/0.gif" alt="<graphic>"> #t<br>
(fl< -1.0 0.0 1.0) <img src="math/csug/0.gif" alt="<graphic>"> #t<br>
(fl> -1.0 0.0 1.0) <img src="math/csug/0.gif" alt="<graphic>"> #f<br>
(fl<= 0.0 3.0 3.0) <img src="math/csug/0.gif" alt="<graphic>"> #t<br>
(fl>= 4.0 3.0 3.0) <img src="math/csug/0.gif" alt="<graphic>"> #t<br>
(fl< 7.0 +inf.0) <img src="math/csug/0.gif" alt="<graphic>"> #t<br>
(fl= +nan.0 0.0) <img src="math/csug/0.gif" alt="<graphic>"> #f<br>
(fl= +nan.0 +nan.0) <img src="math/csug/0.gif" alt="<graphic>"> #f<br>
(fl< +nan.0 +nan.0) <img src="math/csug/0.gif" alt="<graphic>"> #f<br>
(fl> +nan.0 +nan.0) <img src="math/csug/0.gif" alt="<graphic>"> #f</tt>
<p><a name="./numeric:s29"></a><span class=formdef><b>procedure</b>: <tt>(flnonpositive? <i>fl</i>)</tt></span>
<br>
<b>returns: </b><tt>#t</tt> if <tt><i>fl</i></tt> is not greater than zero, <tt>#f</tt> otherwise
<br><span class=formdef><b>procedure</b>: <tt>(flnonnegative? <i>fl</i>)</tt></span>
<br>
<b>returns: </b><tt>#t</tt> if <tt><i>fl</i></tt> is not less than zero, <tt>#f</tt> otherwise
<br>
<b>libraries: </b><tt>(chezscheme)</tt>
<p><tt>flnonpositive?</tt> is equivalent to <tt>(lambda (x) (fl<= x 0.0))</tt>,
and
<tt>flnonnegative?</tt> is equivalent to <tt>(lambda (x) (fl>= x 0.0))</tt>.
<p>
Even if the flonum representation distinguishes -0.0 from +0.0, both
are considered nonpositive and nonnegative.
<p>
<p><tt>(flnonpositive? 128.0) <img src="math/csug/0.gif" alt="<graphic>"> #f<br>
(flnonpositive? 0.0) <img src="math/csug/0.gif" alt="<graphic>"> #t<br>
(flnonpositive? -0.0) <img src="math/csug/0.gif" alt="<graphic>"> #t<br>
(flnonpositive? -1.0) <img src="math/csug/0.gif" alt="<graphic>"> #t
<br>
<br>
(flnonnegative? -65.0) <img src="math/csug/0.gif" alt="<graphic>"> #f<br>
(flnonnegative? 0.0) <img src="math/csug/0.gif" alt="<graphic>"> #t<br>
(flnonnegative? -0.0) <img src="math/csug/0.gif" alt="<graphic>"> #t<br>
(flnonnegative? 1.0) <img src="math/csug/0.gif" alt="<graphic>"> #t
<br>
<br>
(flnonnegative? +nan.0) <img src="math/csug/0.gif" alt="<graphic>"> #f<br>
(flnonpositive? +nan.0) <img src="math/csug/0.gif" alt="<graphic>"> #f
<br>
<br>
(flnonnegative? +inf.0) <img src="math/csug/0.gif" alt="<graphic>"> #t<br>
(flnonnegative? -inf.0) <img src="math/csug/0.gif" alt="<graphic>"> #f</tt>
<p><a name="./numeric:s30"></a><span class=formdef><b>procedure</b>: <tt>(decode-float <i>x</i>)</tt></span>
<br>
<b>returns: </b>see below
<br>
<b>libraries: </b><tt>(chezscheme)</tt>
<p><tt><i>x</i></tt> must be a flonum.
<tt>decode-float</tt> returns a vector with three integer elements,
<tt><i>m</i></tt>, <tt><i>e</i></tt>, and <tt><i>s</i></tt>, such that
<i>x</i> = <i>sm</i>2<sup>e</sup>.
It is useful primarily in the printing of floating-point numbers.
<p>
<p><tt>(decode-float 1.0) <img src="math/csug/0.gif" alt="<graphic>"> #(4503599627370496 -52 1)<br>
(decode-float -1.0) <img src="math/csug/0.gif" alt="<graphic>"> #(4503599627370496 -52 -1)
<br>
<br>
(define slow-identity<br>
(lambda (x)<br>
(inexact<br>
(let ([v (decode-float x)])<br>
(let ([m (vector-ref v 0)]<br>
[e (vector-ref v 1)]<br>
[s (vector-ref v 2)])<br>
(* s m (expt 2 e)))))))
<br>
<br>
(slow-identity 1.0) <img src="math/csug/0.gif" alt="<graphic>"> 1.0<br>
(slow-identity -1e20) <img src="math/csug/0.gif" alt="<graphic>"> -1e20</tt>
<p><a name="./numeric:s31"></a><span class=formdef><b>procedure</b>: <tt>(fllp <i>flonum</i>)</tt></span>
<br>
<b>returns: </b>see below
<br>
<b>libraries: </b><tt>(chezscheme)</tt>
<p><tt>fllp</tt> returns the 12-bit integer consisting of the exponent
plus highest order represented bit of a flonum (ieee 64-bit
floating-point number).
It can be used to compute a fast approximation of the logarithm of
the number.
<p>
<p><tt>(fllp 0.0) <img src="math/csug/0.gif" alt="<graphic>"> 0<br>
(fllp 1.0) <img src="math/csug/0.gif" alt="<graphic>"> 2046<br>
(fllp -1.0) <img src="math/csug/0.gif" alt="<graphic>"> 2046
<br>
<br>
(fllp 1.5) <img src="math/csug/0.gif" alt="<graphic>"> 2047
<br>
<br>
(fllp +inf.0) <img src="math/csug/0.gif" alt="<graphic>"> 4094<br>
(fllp -inf.0) <img src="math/csug/0.gif" alt="<graphic>"> 4094
<br>
<br>
(fllp #b1.0e-1111111111) <img src="math/csug/0.gif" alt="<graphic>"> 1<br>
(fllp #b1.0e-10000000000) <img src="math/csug/0.gif" alt="<graphic>"> 0</tt>
<p>
<h3><a name="g66"></a><a name="./numeric:h4"></a>Section 8.4. Inexact Complex Operations<a name="SECTNUMERICCOMPLEXNUM"></a></h3>
<p>
The procedures described in this section provide mechanisms for
creating and operating on inexact complex numbers.
Inexact complex numbers with nonzero imaginary parts are represented as
<i>inexact complexnums</i>.
An inexact complexnum contains two 64-bit double-precision floating point
numbers.
Inexact <a name="./numeric:s32"></a><a name="./numeric:s33"></a>complex numbers
with imaginary parts equal to zero (in other words, inexact real numbers)
may be represented as either inexact complexnums or flonums.
The operations described in this section accept any mix of
inexact complexnum and flonum arguments
(collectively, "<a name="./numeric:s34"></a>cflonums").
<p>
In most cases, the operations are performed with minimal type checking
at optimize-level 3; full type checking is performed at lower optimize
levels.
Inexact complex procedure names begin with the prefix "<tt>cfl</tt>"
to set them apart from their generic counterparts.
<p>
<a name="./numeric:s35"></a><span class=formdef><b>procedure</b>: <tt>(fl-make-rectangular <i>flonum<sub>1</sub></i> <i>flonum<sub>2</sub></i>)</tt></span>
<br>
<b>returns: </b>an inexact complexnum
<br>
<b>libraries: </b><tt>(chezscheme)</tt>
<p>The inexact complexnum produced by fl-make-rectangular has real part equal
to <tt><i>flonum<sub>1</sub></i></tt> and imaginary part equal to <tt><i>flonum<sub>2</sub></i></tt>.
<p>
<p><tt>(fl-make-rectangular 2.0 -3.0) <img src="math/csug/0.gif" alt="<graphic>"> 2.0-3.0i<br>
(fl-make-rectangular 2.0 0.0) <img src="math/csug/0.gif" alt="<graphic>"> 2.0+0.0i<br>
(fl-make-rectangular 2.0 -0.0) <img src="math/csug/0.gif" alt="<graphic>"> 2.0-0.0i</tt>
<p><a name="./numeric:s36"></a><span class=formdef><b>procedure</b>: <tt>(cfl-real-part <i>cflonum</i>)</tt></span>
<br>
<b>returns: </b>the real part of <tt><i>cflonum</i></tt>
<br><span class=formdef><b>procedure</b>: <tt>(cfl-imag-part <i>cflonum</i>)</tt></span>
<br>
<b>returns: </b>the imaginary part of <tt><i>cflonum</i></tt>
<br>
<b>libraries: </b><tt>(chezscheme)</tt>
<p>
<p><tt>(cfl-real-part 2.0-3.0i) <img src="math/csug/0.gif" alt="<graphic>"> 2.0<br>
(cfl-imag-part 2.0-3.0i) <img src="math/csug/0.gif" alt="<graphic>"> -3.0<br>
(cfl-imag-part 2.0-0.0i) <img src="math/csug/0.gif" alt="<graphic>"> -0.0<br>
(cfl-imag-part 2.0-inf.0i) <img src="math/csug/0.gif" alt="<graphic>"> -inf.0</tt>
<p><a name="./numeric:s37"></a><span class=formdef><b>procedure</b>: <tt>(cfl= <i>cflonum</i> ...)</tt></span>
<br>
<b>returns: </b><tt>#t</tt> if its arguments are equal, <tt>#f</tt> otherwise
<br>
<b>libraries: </b><tt>(chezscheme)</tt>
<p>
<p><tt>
(cfl= 7.0+0.0i 7.0) <img src="math/csug/0.gif" alt="<graphic>"> #t<br>
(cfl= 1.0+2.0i 1.0+2.0i) <img src="math/csug/0.gif" alt="<graphic>"> #t<br>
(cfl= 1.0+2.0i 1.0-2.0i) <img src="math/csug/0.gif" alt="<graphic>"> #f</tt>
<p><a name="./numeric:s38"></a><span class=formdef><b>procedure</b>: <tt>(cfl+ <i>cflonum</i> ...)</tt></span>
<br><span class=formdef><b>procedure</b>: <tt>(cfl* <i>cflonum</i> ...)</tt></span>
<br><span class=formdef><b>procedure</b>: <tt>(cfl- <i>cflonum<sub>1</sub></i> <i>cflonum<sub>2</sub></i> ...)</tt></span>
<br><span class=formdef><b>procedure</b>: <tt>(cfl/ <i>cflonum<sub>1</sub></i> <i>cflonum<sub>2</sub></i> ...)</tt></span>
<br>
<b>returns: </b>a cflonum
<br>
<b>libraries: </b><tt>(chezscheme)</tt>
<p>These procedures compute the sum, difference, product, or quotient
of inexact complex quantities, whether these quantities are represented
by flonums or inexact complexnums.
For example, if <tt>cfl+</tt> receives two flonum arguments <i>a</i> and <i>b</i>, it
returns the sum <i>a</i> + <i>b</i>; in this case, it behaves the same as <tt>fl+</tt>.
With two inexact complexnum arguments <i>a</i> + <i>bi</i> and <i>c</i> + <i>di</i>, it returns
the sum (<i>a</i> + <i>c</i>) + (<i>b</i> + <i>d</i>)<i>i</i>.
If one argument is a flonum <i>a</i> and the other an inexact complexnum
<i>c</i> + <i>di</i>, <tt>cfl+</tt> returns (<i>a</i> + <i>c</i>) + <i>di</i>.
<p>
When passed zero arguments, <tt>cfl+</tt> returns 0.0 and
<tt>cfl*</tt> returns 1.0.
When passed one argument, <tt>cfl-</tt> returns the additive inverse
of the argument, and <tt>cfl/</tt> returns the multiplicative inverse
of the argument.
When passed three or more arguments, <tt>cfl-</tt> returns the
difference between its first and the sum of its remaining arguments,
and <tt>cfl/</tt> returns the quotient of its first and the product
of its remaining arguments.
<p>
<p><tt>(cfl+) <img src="math/csug/0.gif" alt="<graphic>"> 0.0<br>
(cfl*) <img src="math/csug/0.gif" alt="<graphic>"> 1.0<br>
(cfl- 5.0+1.0i) <img src="math/csug/0.gif" alt="<graphic>"> -5.0-1.0i<br>
(cfl/ 2.0+2.0i) <img src="math/csug/0.gif" alt="<graphic>"> 0.25-0.25i
<br>
<br>
(cfl+ 1.0+2.2i -3.7+5.3i) <img src="math/csug/0.gif" alt="<graphic>"> -2.7+7.5i<br>
(cfl+ 1.0 -5.3) <img src="math/csug/0.gif" alt="<graphic>"> -4.3<br>
(cfl+ 1.0 2.0 -5.3i) <img src="math/csug/0.gif" alt="<graphic>"> 3.0-5.3i<br>
(cfl- 1.0+2.5i -3.7) <img src="math/csug/0.gif" alt="<graphic>"> 4.7+2.5i<br>
(cfl* 1.0+2.0i 3.0+4.0i) <img src="math/csug/0.gif" alt="<graphic>"> -5.0+10.0i<br>
(cfl/ -5.0+10.0i 1.0+2.0i 2.0) <img src="math/csug/0.gif" alt="<graphic>"> 1.5+2.0i</tt>
<p><a name="./numeric:s39"></a><span class=formdef><b>procedure</b>: <tt>(cfl-conjugate <i>cflonum</i>)</tt></span>
<br>
<b>returns: </b>complex conjugate of <tt><i>cflonum</i></tt>
<br>
<b>libraries: </b><tt>(chezscheme)</tt>
<p>The procedure <tt>cfl-conjugate</tt>, when passed an inexact complex argument
<i>a</i> + <i>bi</i>, returns its complex conjugate <i>a</i> + (-<i>b</i>)<i>i</i>.
<p>
See also <a name="./numeric:s40"></a><tt>conjugate</tt>, which is a generic
version of this operator that returns the complex conjugate of any
valid representation for a complex number.
<p>
<p><tt>(cfl-conjugate 3.0) <img src="math/csug/0.gif" alt="<graphic>"> 3.0<br>
(cfl-conjugate 3.0+4.0i) <img src="math/csug/0.gif" alt="<graphic>"> 3.0-4.0i<br>
(cfl-conjugate 1e-20-2e-30i) <img src="math/csug/0.gif" alt="<graphic>"> 1e-20+2e-30i</tt>
<p><a name="./numeric:s41"></a><span class=formdef><b>procedure</b>: <tt>(cfl-magnitude-squared <i>cflonum</i>)</tt></span>
<br>
<b>returns: </b>magnitude of <tt><i>cflonum</i></tt> squared
<br>
<b>libraries: </b><tt>(chezscheme)</tt>
<p>The procedure <tt>cfl-magnitude-squared</tt>, when passed an inexact complex
argument <i>a</i> + <i>bi</i> returns a flonum representing the magnitude of the
argument squared, i.e., <i>a</i><sup>2</sup> + <i>b</i><sup>2</sup>.
<p>
See also <a name="./numeric:s42"></a><tt>magnitude-squared</tt>,
which is a generic version of this
operator that returns the magnitude squared of any valid representation
for a complex number.
Both operations are similar to the <a name="./numeric:s43"></a><tt>magnitude</tt> procedure,
which returns the magnitude, <i>sqrt</i>(<i>a</i><sup>2</sup> + <i>b</i><sup>2</sup>), of its generic complex
argument.
<p>
<p><tt>(cfl-magnitude-squared 3.0) <img src="math/csug/0.gif" alt="<graphic>"> 9.0<br>
(cfl-magnitude-squared 3.0-4.0i) <img src="math/csug/0.gif" alt="<graphic>"> 25.0</tt>
<p>
<h3><a name="g67"></a><a name="./numeric:h5"></a>Section 8.5. Bitwise and Logical Operators<a name="SECTNUMERICLOGICAL"></a></h3>
<p>
<i>Chez Scheme</i> provides a set of logical operators that allow exact
integers (fixnums and bignums) to be treated as sets or sequences
of bits.
These operators include
<tt>logand</tt> (bitwise logical <tt>and</tt>),
<tt>logior</tt> (bitwise logical <tt>or</tt>),
<tt>logxor</tt> (bitwise logical exclusive <tt>or</tt>),
<tt>lognot</tt> (bitwise logical <tt>not</tt>),
<tt>logtest</tt> (test multiple bits),
<tt>logbit?</tt> (test single bit),
<tt>logbit0</tt> (reset single bit),
<tt>logbit1</tt> (set single bit),
and <tt>ash</tt> (arithmetic shift).
Each of these operators treats its arguments as two's complement integers,
regardless of the underlying representation.
This treatment can be exploited to represent infinite sets:
a negative number represents an infinite number of one bits beyond the
leftmost zero, and a nonnegative number represents an infinite number of zero
bits beyond the leftmost one bit.
<p>
Fixnum equivalents of the logical operators are provided, as
<tt>fxlogand</tt>, <tt>fxlogior</tt>, <tt>fxlogxor</tt>,
<tt>fxlognot</tt>, <tt>fxlogtest</tt>, <tt>fxlogbit?</tt>,
<tt>fxlogbit0</tt>, and <tt>fxlogbit1</tt>.
Three separate fixnum operators are provided for shifting:
<tt>fxsll</tt> (shift-left logical),
<tt>fxsrl</tt> (shift-right logical),
<tt>fxsra</tt> (shift-right arithmetic).
Logical and arithmetic shifts differ only for right shifts.
Shift-right logical shifts in zero bits on the left end, and shift-right
arithmetic replicates the sign bit.
<p>
Logical shifts do not make sense for arbitrary-precision integers,
since these have no "left end" into which bits must be shifted.
<p>
<a name="./numeric:s44"></a><span class=formdef><b>procedure</b>: <tt>(logand <i>int</i> ...)</tt></span>
<br>
<b>returns: </b>the logical "and" of the arguments <tt><i>int</i> ...</tt>
<br>
<b>libraries: </b><tt>(chezscheme)</tt>
<p>The arguments must be exact integers (fixnums or bignums) and are treated
as two's complement integers, regardless of the underlying representation.
With no arguments, <tt>logand</tt> returns -1, i.e., all bits set.
<p>
<p><tt>(logand) <img src="math/csug/0.gif" alt="<graphic>"> -1<br>
(logand 15) <img src="math/csug/0.gif" alt="<graphic>"> 15<br>
(logand -1 -1) <img src="math/csug/0.gif" alt="<graphic>"> -1<br>
(logand -1 0) <img src="math/csug/0.gif" alt="<graphic>"> 0<br>
(logand 5 3) <img src="math/csug/0.gif" alt="<graphic>"> 1<br>
(logand #x173C8D95 7) <img src="math/csug/0.gif" alt="<graphic>"> 5<br>
(logand #x173C8D95 -8) <img src="math/csug/0.gif" alt="<graphic>"> #x173C8D90<br>
(logand #b1100 #b1111 #b1101) <img src="math/csug/0.gif" alt="<graphic>"> #b1100</tt>
<p><a name="./numeric:s45"></a><span class=formdef><b>procedure</b>: <tt>(logior <i>int</i> ...)</tt></span>
<br><span class=formdef><b>procedure</b>: <tt>(logor <i>int</i> ...)</tt></span>
<br>
<b>returns: </b>the logical "or" of the arguments <tt><i>int</i> ...</tt>
<br>
<b>libraries: </b><tt>(chezscheme)</tt>
<p>The arguments must be exact integers (fixnums or bignums) and are treated
as two's complement integers, regardless of the underlying representation.
With no arguments, <tt>logior</tt> returns 0, i.e., all bits reset.
<p>
<p><tt>(logior) <img src="math/csug/0.gif" alt="<graphic>"> 0<br>
(logior 15) <img src="math/csug/0.gif" alt="<graphic>"> 15<br>
(logior -1 -1) <img src="math/csug/0.gif" alt="<graphic>"> -1<br>
(logior -1 0) <img src="math/csug/0.gif" alt="<graphic>"> -1<br>
(logior 5 3) <img src="math/csug/0.gif" alt="<graphic>"> 7<br>
(logior #b111000 #b101010) <img src="math/csug/0.gif" alt="<graphic>"> #b111010<br>
(logior #b1000 #b0100 #b0010) <img src="math/csug/0.gif" alt="<graphic>"> #b1110<br>
(apply logior '(1 2 4 8 16)) <img src="math/csug/0.gif" alt="<graphic>"> 31</tt>
<p><a name="./numeric:s46"></a><span class=formdef><b>procedure</b>: <tt>(logxor <i>int</i> ...)</tt></span>
<br>
<b>returns: </b>the logical "exclusive or" of the arguments <tt><i>int</i> ...</tt>
<br>
<b>libraries: </b><tt>(chezscheme)</tt>
<p>The arguments must be exact integers (fixnums or bignums) and are treated
as two's complement integers, regardless of the underlying representation.
With no arguments, <tt>logxor</tt> returns 0, i.e., all bits reset.
<p>
<p><tt>(logxor) <img src="math/csug/0.gif" alt="<graphic>"> 0<br>
(logxor 15) <img src="math/csug/0.gif" alt="<graphic>"> 15<br>
(logxor -1 -1) <img src="math/csug/0.gif" alt="<graphic>"> 0<br>
(logxor -1 0) <img src="math/csug/0.gif" alt="<graphic>"> -1<br>
(logxor 5 3) <img src="math/csug/0.gif" alt="<graphic>"> 6<br>
(logxor #b111000 #b101010) <img src="math/csug/0.gif" alt="<graphic>"> #b010010<br>
(logxor #b1100 #b0100 #b0110) <img src="math/csug/0.gif" alt="<graphic>"> #b1110</tt>
<p><a name="./numeric:s47"></a><span class=formdef><b>procedure</b>: <tt>(lognot <i>int</i>)</tt></span>
<br>
<b>returns: </b>the logical "not" of <tt><i>int</i></tt>
<br>
<b>libraries: </b><tt>(chezscheme)</tt>
<p>The argument must be an exact integer (fixnum or bignum) and is treated
as a two's complement integer, regardless of the underlying representation.
<p>
<p><tt>(lognot -1) <img src="math/csug/0.gif" alt="<graphic>"> 0<br>
(lognot 0) <img src="math/csug/0.gif" alt="<graphic>"> -1<br>
(lognot 7) <img src="math/csug/0.gif" alt="<graphic>"> -8<br>
(lognot -8) <img src="math/csug/0.gif" alt="<graphic>"> 7</tt>
<p><a name="./numeric:s48"></a><span class=formdef><b>procedure</b>: <tt>(logbit? <i>index</i> <i>int</i>)</tt></span>
<br>