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
1001
1002
1003
1004
1005
1006
1007
1008
1009
1010
1011
1012
1013
1014
1015
1016
1017
1018
1019
1020
1021
1022
1023
1024
1025
1026
1027
1028
1029
1030
1031
1032
1033
1034
1035
1036
1037
1038
1039
1040
1041
1042
1043
1044
1045
1046
1047
1048
1049
1050
1051
1052
1053
1054
1055
1056
1057
1058
1059
1060
1061
1062
1063
1064
1065
1066
1067
1068
1069
1070
1071
1072
1073
1074
1075
1076
1077
1078
1079
1080
1081
1082
1083
1084
1085
1086
1087
1088
1089
1090
1091
1092
1093
1094
1095
1096
1097
1098
1099
1100
1101
1102
1103
1104
1105
1106
1107
1108
1109
1110
1111
1112
1113
1114
1115
1116
1117
1118
1119
1120
1121
1122
1123
1124
1125
1126
1127
1128
1129
1130
1131
1132
1133
1134
1135
1136
1137
1138
1139
1140
1141
1142
1143
1144
1145
1146
1147
1148
1149
1150
1151
1152
1153
1154
1155
1156
1157
1158
1159
1160
1161
1162
1163
1164
1165
1166
1167
1168
1169
1170
1171
1172
1173
1174
1175
1176
1177
1178
1179
1180
1181
1182
1183
1184
1185
1186
1187
1188
1189
1190
1191
1192
1193
1194
1195
1196
1197
1198
1199
1200
1201
1202
1203
1204
1205
1206
1207
1208
1209
1210
1211
1212
1213
1214
1215
1216
1217
1218
1219
1220
1221
1222
1223
1224
1225
1226
1227
1228
1229
1230
1231
1232
1233
1234
1235
1236
1237
1238
1239
1240
1241
1242
1243
1244
1245
1246
1247
1248
1249
1250
1251
1252
| | @c -*-texinfo-*-
@c This is part of the GNU Emacs Lisp Reference Manual.
@c Copyright (C) 1990-1995, 1998-1999, 2001-2019 Free Software
@c Foundation, Inc.
@c See the file elisp.texi for copying conditions.
@node Numbers
@chapter Numbers
@cindex integers
@cindex numbers
GNU Emacs supports two numeric data types: @dfn{integers} and
@dfn{floating-point numbers}. Integers are whole numbers such as
@minus{}3, 0, 7, 13, and 511. Floating-point numbers are numbers with
fractional parts, such as @minus{}4.5, 0.0, and 2.71828. They can
also be expressed in exponential notation: @samp{1.5e2} is the same as
@samp{150.0}; here, @samp{e2} stands for ten to the second power, and
that is multiplied by 1.5. Integer computations are exact.
Floating-point computations often involve rounding errors, as the
numbers have a fixed amount of precision.
@menu
* Integer Basics:: Representation and range of integers.
* Float Basics:: Representation and range of floating point.
* Predicates on Numbers:: Testing for numbers.
* Comparison of Numbers:: Equality and inequality predicates.
* Numeric Conversions:: Converting float to integer and vice versa.
* Arithmetic Operations:: How to add, subtract, multiply and divide.
* Rounding Operations:: Explicitly rounding floating-point numbers.
* Bitwise Operations:: Logical and, or, not, shifting.
* Math Functions:: Trig, exponential and logarithmic functions.
* Random Numbers:: Obtaining random integers, predictable or not.
@end menu
@node Integer Basics
@section Integer Basics
Integers in Emacs Lisp are not limited to the machine word size.
Under the hood, though, there are two kinds of integers: smaller
ones, called @dfn{fixnums}, and larger ones, called @dfn{bignums}.
Some functions in Emacs accept only fixnums. Also, while fixnums can
always be compared for numeric equality with @code{eq}, bignums
require more-heavyweight equality predicates like @code{eql}.
The range of values for bignums is limited by the amount of main
memory, by machine characteristics such as the size of the word used
to represent a bignum's exponent, and by the @code{integer-width}
variable. These limits are typically much more generous than the
limits for fixnums. A bignum is never numerically equal to a fixnum;
if Emacs computes an integer in fixnum range, it represents the
integer as a fixnum, not a bignum.
The range of values for a fixnum depends on the machine. The
minimum range is @minus{}536,870,912 to 536,870,911 (30 bits; i.e.,
@ifnottex
@minus{}2**29
@end ifnottex
@tex
@math{-2^{29}}
@end tex
to
@ifnottex
2**29 @minus{} 1),
@end ifnottex
@tex
@math{2^{29}-1}),
@end tex
but many machines provide a wider range.
The Lisp reader reads an integer as a nonempty sequence
of decimal digits with optional initial sign and optional
final period.
@example
1 ; @r{The integer 1.}
1. ; @r{The integer 1.}
+1 ; @r{Also the integer 1.}
-1 ; @r{The integer @minus{}1.}
0 ; @r{The integer 0.}
-0 ; @r{The integer 0.}
@end example
@cindex integers in specific radix
@cindex radix for reading an integer
@cindex base for reading an integer
@cindex hex numbers
@cindex octal numbers
@cindex reading numbers in hex, octal, and binary
The syntax for integers in bases other than 10 consists of @samp{#}
followed by a radix indication followed by one or more digits. The
radix indications are @samp{b} for binary, @samp{o} for octal,
@samp{x} for hex, and @samp{@var{radix}r} for radix @var{radix}.
Thus, @samp{#b@var{integer}} reads
@var{integer} in binary, and @samp{#@var{radix}r@var{integer}} reads
@var{integer} in radix @var{radix}. Allowed values of @var{radix} run
from 2 to 36, and allowed digits are the first @var{radix} characters
taken from @samp{0}--@samp{9}, @samp{A}--@samp{Z}.
Letter case is ignored and there is no initial sign or final period.
For example:
@example
#b101100 @result{} 44
#o54 @result{} 44
#x2c @result{} 44
#24r1k @result{} 44
@end example
To understand how various functions work on integers, especially the
bitwise operators (@pxref{Bitwise Operations}), it is often helpful to
view the numbers in their binary form.
In binary, the decimal integer 5 looks like this:
@example
@dots{}000101
@end example
@noindent
(The ellipsis @samp{@dots{}} stands for a conceptually infinite number
of bits that match the leading bit; here, an infinite number of 0
bits. Later examples also use this @samp{@dots{}} notation.)
The integer @minus{}1 looks like this:
@example
@dots{}111111
@end example
@noindent
@cindex two's complement
@minus{}1 is represented as all ones. (This is called @dfn{two's
complement} notation.)
Subtracting 4 from @minus{}1 returns the negative integer @minus{}5.
In binary, the decimal integer 4 is 100. Consequently,
@minus{}5 looks like this:
@example
@dots{}111011
@end example
Many of the functions described in this chapter accept markers for
arguments in place of numbers. (@xref{Markers}.) Since the actual
arguments to such functions may be either numbers or markers, we often
give these arguments the name @var{number-or-marker}. When the argument
value is a marker, its position value is used and its buffer is ignored.
@cindex largest fixnum
@cindex maximum fixnum
@defvar most-positive-fixnum
The value of this variable is the greatest ``small'' integer that Emacs
Lisp can handle. Typical values are
@ifnottex
2**29 @minus{} 1
@end ifnottex
@tex
@math{2^{29}-1}
@end tex
on 32-bit and
@ifnottex
2**61 @minus{} 1
@end ifnottex
@tex
@math{2^{61}-1}
@end tex
on 64-bit platforms.
@end defvar
@cindex smallest fixnum
@cindex minimum fixnum
@defvar most-negative-fixnum
The value of this variable is the numerically least ``small'' integer
that Emacs Lisp can handle. It is negative. Typical values are
@ifnottex
@minus{}2**29
@end ifnottex
@tex
@math{-2^{29}}
@end tex
on 32-bit and
@ifnottex
@minus{}2**61
@end ifnottex
@tex
@math{-2^{61}}
@end tex
on 64-bit platforms.
@end defvar
@cindex bignum range
@cindex integer range
@cindex number of bignum bits, limit on
@defvar integer-width
The value of this variable is a nonnegative integer that controls
whether Emacs signals a range error when a large integer would be
calculated. Integers with absolute values less than
@ifnottex
2**@var{n},
@end ifnottex
@tex
@math{2^{n}},
@end tex
where @var{n} is this variable's value, do not signal a range error.
Attempts to create larger integers typically signal a range error,
although there might be no signal if a larger integer can be created cheaply.
Setting this variable to a large number can be costly if a computation
creates huge integers.
@end defvar
In Emacs Lisp, text characters are represented by integers. Any
integer between zero and the value of @code{(max-char)}, inclusive, is
considered to be valid as a character. @xref{Character Codes}.
@node Float Basics
@section Floating-Point Basics
@cindex @acronym{IEEE} floating point
Floating-point numbers are useful for representing numbers that are
not integral. The range of floating-point numbers is
the same as the range of the C data type @code{double} on the machine
you are using. On all computers currently supported by Emacs, this is
double-precision @acronym{IEEE} floating point.
The read syntax for floating-point numbers requires either a decimal
point, an exponent, or both. Optional signs (@samp{+} or @samp{-})
precede the number and its exponent. For example, @samp{1500.0},
@samp{+15e2}, @samp{15.0e+2}, @samp{+1500000e-3}, and @samp{.15e4} are
five ways of writing a floating-point number whose value is 1500.
They are all equivalent. Like Common Lisp, Emacs Lisp requires at
least one digit after any decimal point in a floating-point number;
@samp{1500.} is an integer, not a floating-point number.
Emacs Lisp treats @code{-0.0} as numerically equal to ordinary zero
with respect to numeric comparisons like @code{=}. This follows the
@acronym{IEEE} floating-point standard, which says @code{-0.0} and
@code{0.0} are numerically equal even though other operations can
distinguish them.
@cindex positive infinity
@cindex negative infinity
@cindex infinity
@cindex NaN
The @acronym{IEEE} floating-point standard supports positive
infinity and negative infinity as floating-point values. It also
provides for a class of values called NaN, or ``not a number'';
numerical functions return such values in cases where there is no
correct answer. For example, @code{(/ 0.0 0.0)} returns a NaN@.
A NaN is never numerically equal to any value, not even to itself.
NaNs carry a sign and a significand, and non-numeric functions treat
two NaNs as equal when their
signs and significands agree. Significands of NaNs are
machine-dependent, as are the digits in their string representation.
When NaNs and signed zeros are involved, non-numeric functions like
@code{eql}, @code{equal}, @code{sxhash-eql}, @code{sxhash-equal} and
@code{gethash} determine whether values are indistinguishable, not
whether they are numerically equal. For example, when @var{x} and
@var{y} are the same NaN, @code{(equal x y)} returns @code{t} whereas
@code{(= x y)} uses numeric comparison and returns @code{nil};
conversely, @code{(equal 0.0 -0.0)} returns @code{nil} whereas
@code{(= 0.0 -0.0)} returns @code{t}.
Here are read syntaxes for these special floating-point values:
@table @asis
@item infinity
@samp{1.0e+INF} and @samp{-1.0e+INF}
@item not-a-number
@samp{0.0e+NaN} and @samp{-0.0e+NaN}
@end table
The following functions are specialized for handling floating-point
numbers:
@defun isnan x
This predicate returns @code{t} if its floating-point argument is a NaN,
@code{nil} otherwise.
@end defun
@defun frexp x
This function returns a cons cell @code{(@var{s} . @var{e})},
where @var{s} and @var{e} are respectively the significand and
exponent of the floating-point number @var{x}.
If @var{x} is finite, then @var{s} is a floating-point number between 0.5
(inclusive) and 1.0 (exclusive), @var{e} is an integer, and
@ifnottex
@var{x} = @var{s} * 2**@var{e}.
@end ifnottex
@tex
@math{x = s 2^e}.
@end tex
If @var{x} is zero or infinity, then @var{s} is the same as @var{x}.
If @var{x} is a NaN, then @var{s} is also a NaN@.
If @var{x} is zero, then @var{e} is 0.
@end defun
@defun ldexp s e
Given a numeric significand @var{s} and an integer exponent @var{e},
this function returns the floating point number
@ifnottex
@var{s} * 2**@var{e}.
@end ifnottex
@tex
@math{s 2^e}.
@end tex
@end defun
@defun copysign x1 x2
This function copies the sign of @var{x2} to the value of @var{x1},
and returns the result. @var{x1} and @var{x2} must be floating point.
@end defun
@defun logb x
This function returns the binary exponent of @var{x}. More
precisely, if @var{x} is finite and nonzero, the value is the
logarithm base 2 of @math{|x|}, rounded down to an integer.
If @var{x} is zero, infinite, or a NaN, the value is minus infinity,
plus infinity, or a NaN respectively.
@example
(logb 10)
@result{} 3
(logb 10.0e20)
@result{} 69
(logb 0)
@result{} -1.0e+INF
@end example
@end defun
@node Predicates on Numbers
@section Type Predicates for Numbers
@cindex predicates for numbers
The functions in this section test for numbers, or for a specific
type of number. The functions @code{integerp} and @code{floatp} can
take any type of Lisp object as argument (they would not be of much
use otherwise), but the @code{zerop} predicate requires a number as
its argument. See also @code{integer-or-marker-p} and
@code{number-or-marker-p}, in @ref{Predicates on Markers}.
@defun bignump object
This predicate tests whether its argument is a large integer, and
returns @code{t} if so, @code{nil} otherwise. Unlike small integers,
large integers can be @code{=} or @code{eql} even if they are not @code{eq}.
@end defun
@defun fixnump object
This predicate tests whether its argument is a small integer, and
returns @code{t} if so, @code{nil} otherwise. Small integers can be
compared with @code{eq}.
@end defun
@defun floatp object
This predicate tests whether its argument is floating point
and returns @code{t} if so, @code{nil} otherwise.
@end defun
@defun integerp object
This predicate tests whether its argument is an integer, and returns
@code{t} if so, @code{nil} otherwise.
@end defun
@defun numberp object
This predicate tests whether its argument is a number (either integer or
floating point), and returns @code{t} if so, @code{nil} otherwise.
@end defun
@defun natnump object
@cindex natural numbers
This predicate (whose name comes from the phrase ``natural number'')
tests to see whether its argument is a nonnegative integer, and
returns @code{t} if so, @code{nil} otherwise. 0 is considered
non-negative.
@findex wholenump
@code{wholenump} is a synonym for @code{natnump}.
@end defun
@defun zerop number
This predicate tests whether its argument is zero, and returns @code{t}
if so, @code{nil} otherwise. The argument must be a number.
@code{(zerop x)} is equivalent to @code{(= x 0)}.
@end defun
@node Comparison of Numbers
@section Comparison of Numbers
@cindex number comparison
@cindex comparing numbers
To test numbers for numerical equality, you should normally use
@code{=} instead of non-numeric comparison predicates like @code{eq},
@code{eql} and @code{equal}. Distinct floating-point and large
integer objects can be numerically equal. If you use @code{eq} to
compare them, you test whether they are the same @emph{object}; if you
use @code{eql} or @code{equal}, you test whether their values are
@emph{indistinguishable}. In contrast, @code{=} uses numeric
comparison, and sometimes returns @code{t} when a non-numeric
comparison would return @code{nil} and vice versa. @xref{Float
Basics}.
In Emacs Lisp, if two fixnums are numerically equal, they are the
same Lisp object. That is, @code{eq} is equivalent to @code{=} on
fixnums. It is sometimes convenient to use @code{eq} for comparing
an unknown value with a fixnum, because @code{eq} does not report an
error if the unknown value is not a number---it accepts arguments of
any type. By contrast, @code{=} signals an error if the arguments are
not numbers or markers. However, it is better programming practice to
use @code{=} if you can, even for comparing integers.
Sometimes it is useful to compare numbers with @code{eql} or @code{equal},
which treat two numbers as equal if they have the same data type (both
integers, or both floating point) and the same value. By contrast,
@code{=} can treat an integer and a floating-point number as equal.
@xref{Equality Predicates}.
There is another wrinkle: because floating-point arithmetic is not
exact, it is often a bad idea to check for equality of floating-point
values. Usually it is better to test for approximate equality.
Here's a function to do this:
@example
(defvar fuzz-factor 1.0e-6)
(defun approx-equal (x y)
(or (= x y)
(< (/ (abs (- x y))
(max (abs x) (abs y)))
fuzz-factor)))
@end example
@defun = number-or-marker &rest number-or-markers
This function tests whether all its arguments are numerically equal,
and returns @code{t} if so, @code{nil} otherwise.
@end defun
@defun eql value1 value2
This function acts like @code{eq} except when both arguments are
numbers. It compares numbers by type and numeric value, so that
@code{(eql 1.0 1)} returns @code{nil}, but @code{(eql 1.0 1.0)} and
@code{(eql 1 1)} both return @code{t}. This can be used to compare
large integers as well as small ones.
@end defun
@defun /= number-or-marker1 number-or-marker2
This function tests whether its arguments are numerically equal, and
returns @code{t} if they are not, and @code{nil} if they are.
@end defun
@defun < number-or-marker &rest number-or-markers
This function tests whether each argument is strictly less than the
following argument. It returns @code{t} if so, @code{nil} otherwise.
@end defun
@defun <= number-or-marker &rest number-or-markers
This function tests whether each argument is less than or equal to
the following argument. It returns @code{t} if so, @code{nil} otherwise.
@end defun
@defun > number-or-marker &rest number-or-markers
This function tests whether each argument is strictly greater than
the following argument. It returns @code{t} if so, @code{nil} otherwise.
@end defun
@defun >= number-or-marker &rest number-or-markers
This function tests whether each argument is greater than or equal to
the following argument. It returns @code{t} if so, @code{nil} otherwise.
@end defun
@defun max number-or-marker &rest numbers-or-markers
This function returns the largest of its arguments.
@example
(max 20)
@result{} 20
(max 1 2.5)
@result{} 2.5
(max 1 3 2.5)
@result{} 3
@end example
@end defun
@defun min number-or-marker &rest numbers-or-markers
This function returns the smallest of its arguments.
@example
(min -4 1)
@result{} -4
@end example
@end defun
@defun abs number
This function returns the absolute value of @var{number}.
@end defun
@node Numeric Conversions
@section Numeric Conversions
@cindex rounding in conversions
@cindex number conversions
@cindex converting numbers
To convert an integer to floating point, use the function @code{float}.
@defun float number
This returns @var{number} converted to floating point.
If @var{number} is already floating point, @code{float} returns
it unchanged.
@end defun
There are four functions to convert floating-point numbers to
integers; they differ in how they round. All accept an argument
@var{number} and an optional argument @var{divisor}. Both arguments
may be integers or floating-point numbers. @var{divisor} may also be
@code{nil}. If @var{divisor} is @code{nil} or omitted, these
functions convert @var{number} to an integer, or return it unchanged
if it already is an integer. If @var{divisor} is non-@code{nil}, they
divide @var{number} by @var{divisor} and convert the result to an
integer. If @var{divisor} is zero (whether integer or
floating point), Emacs signals an @code{arith-error} error.
@defun truncate number &optional divisor
This returns @var{number}, converted to an integer by rounding towards
zero.
@example
(truncate 1.2)
@result{} 1
(truncate 1.7)
@result{} 1
(truncate -1.2)
@result{} -1
(truncate -1.7)
@result{} -1
@end example
@end defun
@defun floor number &optional divisor
This returns @var{number}, converted to an integer by rounding downward
(towards negative infinity).
If @var{divisor} is specified, this uses the kind of division
operation that corresponds to @code{mod}, rounding downward.
@example
(floor 1.2)
@result{} 1
(floor 1.7)
@result{} 1
(floor -1.2)
@result{} -2
(floor -1.7)
@result{} -2
(floor 5.99 3)
@result{} 1
@end example
@end defun
@defun ceiling number &optional divisor
This returns @var{number}, converted to an integer by rounding upward
(towards positive infinity).
@example
(ceiling 1.2)
@result{} 2
(ceiling 1.7)
@result{} 2
(ceiling -1.2)
@result{} -1
(ceiling -1.7)
@result{} -1
@end example
@end defun
@defun round number &optional divisor
This returns @var{number}, converted to an integer by rounding towards the
nearest integer. Rounding a value equidistant between two integers
returns the even integer.
@example
(round 1.2)
@result{} 1
(round 1.7)
@result{} 2
(round -1.2)
@result{} -1
(round -1.7)
@result{} -2
@end example
@end defun
@node Arithmetic Operations
@section Arithmetic Operations
@cindex arithmetic operations
Emacs Lisp provides the traditional four arithmetic operations
(addition, subtraction, multiplication, and division), as well as
remainder and modulus functions, and functions to add or subtract 1.
Except for @code{%}, each of these functions accepts both integer and
floating-point arguments, and returns a floating-point number if any
argument is floating point.
@defun 1+ number-or-marker
This function returns @var{number-or-marker} plus 1.
For example,
@example
(setq foo 4)
@result{} 4
(1+ foo)
@result{} 5
@end example
This function is not analogous to the C operator @code{++}---it does not
increment a variable. It just computes a sum. Thus, if we continue,
@example
foo
@result{} 4
@end example
If you want to increment the variable, you must use @code{setq},
like this:
@example
(setq foo (1+ foo))
@result{} 5
@end example
@end defun
@defun 1- number-or-marker
This function returns @var{number-or-marker} minus 1.
@end defun
@defun + &rest numbers-or-markers
This function adds its arguments together. When given no arguments,
@code{+} returns 0.
@example
(+)
@result{} 0
(+ 1)
@result{} 1
(+ 1 2 3 4)
@result{} 10
@end example
@end defun
@defun - &optional number-or-marker &rest more-numbers-or-markers
The @code{-} function serves two purposes: negation and subtraction.
When @code{-} has a single argument, the value is the negative of the
argument. When there are multiple arguments, @code{-} subtracts each of
the @var{more-numbers-or-markers} from @var{number-or-marker},
cumulatively. If there are no arguments, the result is 0.
@example
(- 10 1 2 3 4)
@result{} 0
(- 10)
@result{} -10
(-)
@result{} 0
@end example
@end defun
@defun * &rest numbers-or-markers
This function multiplies its arguments together, and returns the
product. When given no arguments, @code{*} returns 1.
@example
(*)
@result{} 1
(* 1)
@result{} 1
(* 1 2 3 4)
@result{} 24
@end example
@end defun
@defun / number &rest divisors
With one or more @var{divisors}, this function divides @var{number}
by each divisor in @var{divisors} in turn, and returns the quotient.
With no @var{divisors}, this function returns 1/@var{number}, i.e.,
the multiplicative inverse of @var{number}. Each argument may be a
number or a marker.
If all the arguments are integers, the result is an integer, obtained
by rounding the quotient towards zero after each division.
@example
@group
(/ 6 2)
@result{} 3
@end group
@group
(/ 5 2)
@result{} 2
@end group
@group
(/ 5.0 2)
@result{} 2.5
@end group
@group
(/ 5 2.0)
@result{} 2.5
@end group
@group
(/ 5.0 2.0)
@result{} 2.5
@end group
@group
(/ 4.0)
@result{} 0.25
@end group
@group
(/ 4)
@result{} 0
@end group
@group
(/ 25 3 2)
@result{} 4
@end group
@group
(/ -17 6)
@result{} -2
@end group
@end example
@cindex @code{arith-error} in division
If you divide an integer by the integer 0, Emacs signals an
@code{arith-error} error (@pxref{Errors}). Floating-point division of
a nonzero number by zero yields either positive or negative infinity
(@pxref{Float Basics}).
@end defun
@defun % dividend divisor
@cindex remainder
This function returns the integer remainder after division of @var{dividend}
by @var{divisor}. The arguments must be integers or markers.
For any two integers @var{dividend} and @var{divisor},
@example
@group
(+ (% @var{dividend} @var{divisor})
(* (/ @var{dividend} @var{divisor}) @var{divisor}))
@end group
@end example
@noindent
always equals @var{dividend} if @var{divisor} is nonzero.
@example
(% 9 4)
@result{} 1
(% -9 4)
@result{} -1
(% 9 -4)
@result{} 1
(% -9 -4)
@result{} -1
@end example
@end defun
@defun mod dividend divisor
@cindex modulus
This function returns the value of @var{dividend} modulo @var{divisor};
in other words, the remainder after division of @var{dividend}
by @var{divisor}, but with the same sign as @var{divisor}.
The arguments must be numbers or markers.
Unlike @code{%}, @code{mod} permits floating-point arguments; it
rounds the quotient downward (towards minus infinity) to an integer,
and uses that quotient to compute the remainder.
If @var{divisor} is zero, @code{mod} signals an @code{arith-error}
error if both arguments are integers, and returns a NaN otherwise.
@example
@group
(mod 9 4)
@result{} 1
@end group
@group
(mod -9 4)
@result{} 3
@end group
@group
(mod 9 -4)
@result{} -3
@end group
@group
(mod -9 -4)
@result{} -1
@end group
@group
(mod 5.5 2.5)
@result{} .5
@end group
@end example
For any two numbers @var{dividend} and @var{divisor},
@example
@group
(+ (mod @var{dividend} @var{divisor})
(* (floor @var{dividend} @var{divisor}) @var{divisor}))
@end group
@end example
@noindent
always equals @var{dividend}, subject to rounding error if either
argument is floating point and to an @code{arith-error} if @var{dividend} is an
integer and @var{divisor} is 0. For @code{floor}, see @ref{Numeric
Conversions}.
@end defun
@node Rounding Operations
@section Rounding Operations
@cindex rounding without conversion
The functions @code{ffloor}, @code{fceiling}, @code{fround}, and
@code{ftruncate} take a floating-point argument and return a floating-point
result whose value is a nearby integer. @code{ffloor} returns the
nearest integer below; @code{fceiling}, the nearest integer above;
@code{ftruncate}, the nearest integer in the direction towards zero;
@code{fround}, the nearest integer.
@defun ffloor float
This function rounds @var{float} to the next lower integral value, and
returns that value as a floating-point number.
@end defun
@defun fceiling float
This function rounds @var{float} to the next higher integral value, and
returns that value as a floating-point number.
@end defun
@defun ftruncate float
This function rounds @var{float} towards zero to an integral value, and
returns that value as a floating-point number.
@end defun
@defun fround float
This function rounds @var{float} to the nearest integral value,
and returns that value as a floating-point number.
Rounding a value equidistant between two integers returns the even integer.
@end defun
@node Bitwise Operations
@section Bitwise Operations on Integers
@cindex bitwise arithmetic
@cindex logical arithmetic
In a computer, an integer is represented as a binary number, a
sequence of @dfn{bits} (digits which are either zero or one).
Conceptually the bit sequence is infinite on the left, with the
most-significant bits being all zeros or all ones. A bitwise
operation acts on the individual bits of such a sequence. For example,
@dfn{shifting} moves the whole sequence left or right one or more places,
reproducing the same pattern moved over.
The bitwise operations in Emacs Lisp apply only to integers.
@defun ash integer1 count
@cindex arithmetic shift
@code{ash} (@dfn{arithmetic shift}) shifts the bits in @var{integer1}
to the left @var{count} places, or to the right if @var{count} is
negative. Left shifts introduce zero bits on the right; right shifts
discard the rightmost bits. Considered as an integer operation,
@code{ash} multiplies @var{integer1} by
@ifnottex
2**@var{count},
@end ifnottex
@tex
@math{2^{count}},
@end tex
and then converts the result to an integer by rounding downward, toward
minus infinity.
Here are examples of @code{ash}, shifting a pattern of bits one place
to the left and to the right. These examples show only the low-order
bits of the binary pattern; leading bits all agree with the
highest-order bit shown. As you can see, shifting left by one is
equivalent to multiplying by two, whereas shifting right by one is
equivalent to dividing by two and then rounding toward minus infinity.
@example
@group
(ash 7 1) @result{} 14
;; @r{Decimal 7 becomes decimal 14.}
@dots{}000111
@result{}
@dots{}001110
@end group
@group
(ash 7 -1) @result{} 3
@dots{}000111
@result{}
@dots{}000011
@end group
@group
(ash -7 1) @result{} -14
@dots{}111001
@result{}
@dots{}110010
@end group
@group
(ash -7 -1) @result{} -4
@dots{}111001
@result{}
@dots{}111100
@end group
@end example
Here are examples of shifting left or right by two bits:
@smallexample
@group
; @r{ binary values}
(ash 5 2) ; 5 = @r{@dots{}000101}
@result{} 20 ; = @r{@dots{}010100}
(ash -5 2) ; -5 = @r{@dots{}111011}
@result{} -20 ; = @r{@dots{}101100}
@end group
@group
(ash 5 -2)
@result{} 1 ; = @r{@dots{}000001}
@end group
@group
(ash -5 -2)
@result{} -2 ; = @r{@dots{}111110}
@end group
@end smallexample
@end defun
@defun lsh integer1 count
@cindex logical shift
@code{lsh}, which is an abbreviation for @dfn{logical shift}, shifts the
bits in @var{integer1} to the left @var{count} places, or to the right
if @var{count} is negative, bringing zeros into the vacated bits. If
@var{count} is negative, then @var{integer1} must be either a fixnum
or a positive bignum, and @code{lsh} treats a negative fixnum as if it
were unsigned by subtracting twice @code{most-negative-fixnum} before
shifting, producing a nonnegative result. This quirky behavior dates
back to when Emacs supported only fixnums; nowadays @code{ash} is a
better choice.
As @code{lsh} behaves like @code{ash} except when @var{integer1} and
@var{count1} are both negative, the following examples focus on these
exceptional cases. These examples assume 30-bit fixnums.
@smallexample
@group
; @r{ binary values}
(ash -7 -1) ; -7 = @r{@dots{}111111111111111111111111111001}
@result{} -4 ; = @r{@dots{}111111111111111111111111111100}
(lsh -7 -1)
@result{} 536870908 ; = @r{@dots{}011111111111111111111111111100}
@end group
@group
(ash -5 -2) ; -5 = @r{@dots{}111111111111111111111111111011}
@result{} -2 ; = @r{@dots{}111111111111111111111111111110}
(lsh -5 -2)
@result{} 268435454 ; = @r{@dots{}001111111111111111111111111110}
@end group
@end smallexample
@end defun
@defun logand &rest ints-or-markers
This function returns the bitwise AND of the arguments: the @var{n}th
bit is 1 in the result if, and only if, the @var{n}th bit is 1 in all
the arguments.
For example, using 4-bit binary numbers, the bitwise AND of 13 and
12 is 12: 1101 combined with 1100 produces 1100.
In both the binary numbers, the leftmost two bits are both 1
so the leftmost two bits of the returned value are both 1.
However, for the rightmost two bits, each is 0 in at least one of
the arguments, so the rightmost two bits of the returned value are both 0.
@noindent
Therefore,
@example
@group
(logand 13 12)
@result{} 12
@end group
@end example
If @code{logand} is not passed any argument, it returns a value of
@minus{}1. This number is an identity element for @code{logand}
because its binary representation consists entirely of ones. If
@code{logand} is passed just one argument, it returns that argument.
@smallexample
@group
; @r{ binary values}
(logand 14 13) ; 14 = @r{@dots{}001110}
; 13 = @r{@dots{}001101}
@result{} 12 ; 12 = @r{@dots{}001100}
@end group
@group
(logand 14 13 4) ; 14 = @r{@dots{}001110}
; 13 = @r{@dots{}001101}
; 4 = @r{@dots{}000100}
@result{} 4 ; 4 = @r{@dots{}000100}
@end group
@group
(logand)
@result{} -1 ; -1 = @r{@dots{}111111}
@end group
@end smallexample
@end defun
@defun logior &rest ints-or-markers
This function returns the bitwise inclusive OR of its arguments: the @var{n}th
bit is 1 in the result if, and only if, the @var{n}th bit is 1 in at
least one of the arguments. If there are no arguments, the result is 0,
which is an identity element for this operation. If @code{logior} is
passed just one argument, it returns that argument.
@smallexample
@group
; @r{ binary values}
(logior 12 5) ; 12 = @r{@dots{}001100}
; 5 = @r{@dots{}000101}
@result{} 13 ; 13 = @r{@dots{}001101}
@end group
@group
(logior 12 5 7) ; 12 = @r{@dots{}001100}
; 5 = @r{@dots{}000101}
; 7 = @r{@dots{}000111}
@result{} 15 ; 15 = @r{@dots{}001111}
@end group
@end smallexample
@end defun
@defun logxor &rest ints-or-markers
This function returns the bitwise exclusive OR of its arguments: the
@var{n}th bit is 1 in the result if, and only if, the @var{n}th bit is
1 in an odd number of the arguments. If there are no arguments, the
result is 0, which is an identity element for this operation. If
@code{logxor} is passed just one argument, it returns that argument.
@smallexample
@group
; @r{ binary values}
(logxor 12 5) ; 12 = @r{@dots{}001100}
; 5 = @r{@dots{}000101}
@result{} 9 ; 9 = @r{@dots{}001001}
@end group
@group
(logxor 12 5 7) ; 12 = @r{@dots{}001100}
; 5 = @r{@dots{}000101}
; 7 = @r{@dots{}000111}
@result{} 14 ; 14 = @r{@dots{}001110}
@end group
@end smallexample
@end defun
@defun lognot integer
This function returns the bitwise complement of its argument: the @var{n}th
bit is one in the result if, and only if, the @var{n}th bit is zero in
@var{integer}, and vice-versa. The result equals @minus{}1 @minus{}
@var{integer}.
@example
(lognot 5)
@result{} -6
;; 5 = @r{@dots{}000101}
;; @r{becomes}
;; -6 = @r{@dots{}111010}
@end example
@end defun
@cindex popcount
@cindex Hamming weight
@cindex counting set bits
@defun logcount integer
This function returns the @dfn{Hamming weight} of @var{integer}: the
number of ones in the binary representation of @var{integer}.
If @var{integer} is negative, it returns the number of zero bits in
its two's complement binary representation. The result is always
nonnegative.
@example
(logcount 43) ; 43 = @r{@dots{}000101011}
@result{} 4
(logcount -43) ; -43 = @r{@dots{}111010101}
@result{} 3
@end example
@end defun
@node Math Functions
@section Standard Mathematical Functions
@cindex transcendental functions
@cindex mathematical functions
@cindex floating-point functions
These mathematical functions allow integers as well as floating-point
numbers as arguments.
@defun sin arg
@defunx cos arg
@defunx tan arg
These are the basic trigonometric functions, with argument @var{arg}
measured in radians.
@end defun
@defun asin arg
The value of @code{(asin @var{arg})} is a number between
@ifnottex
@minus{}pi/2
@end ifnottex
@tex
@math{-\pi/2}
@end tex
and
@ifnottex
pi/2
@end ifnottex
@tex
@math{\pi/2}
@end tex
(inclusive) whose sine is @var{arg}. If @var{arg} is out of range
(outside [@minus{}1, 1]), @code{asin} returns a NaN.
@end defun
@defun acos arg
The value of @code{(acos @var{arg})} is a number between 0 and
@ifnottex
pi
@end ifnottex
@tex
@math{\pi}
@end tex
(inclusive) whose cosine is @var{arg}. If @var{arg} is out of range
(outside [@minus{}1, 1]), @code{acos} returns a NaN.
@end defun
@defun atan y &optional x
The value of @code{(atan @var{y})} is a number between
@ifnottex
@minus{}pi/2
@end ifnottex
@tex
@math{-\pi/2}
@end tex
and
@ifnottex
pi/2
@end ifnottex
@tex
@math{\pi/2}
@end tex
(exclusive) whose tangent is @var{y}. If the optional second
argument @var{x} is given, the value of @code{(atan y x)} is the
angle in radians between the vector @code{[@var{x}, @var{y}]} and the
@code{X} axis.
@end defun
@defun exp arg
This is the exponential function; it returns @math{e} to the power
@var{arg}.
@end defun
@defun log arg &optional base
This function returns the logarithm of @var{arg}, with base
@var{base}. If you don't specify @var{base}, the natural base
@math{e} is used. If @var{arg} or @var{base} is negative, @code{log}
returns a NaN.
@end defun
@defun expt x y
This function returns @var{x} raised to power @var{y}. If both
arguments are integers and @var{y} is nonnegative, the result is an
integer; in this case, overflow signals an error, so watch out.
If @var{x} is a finite negative number and @var{y} is a finite
non-integer, @code{expt} returns a NaN.
@end defun
@defun sqrt arg
This returns the square root of @var{arg}. If @var{arg} is finite
and less than zero, @code{sqrt} returns a NaN.
@end defun
In addition, Emacs defines the following common mathematical
constants:
@defvar float-e
The mathematical constant @math{e} (2.71828@dots{}).
@end defvar
@defvar float-pi
The mathematical constant @math{pi} (3.14159@dots{}).
@end defvar
@node Random Numbers
@section Random Numbers
@cindex random numbers
A deterministic computer program cannot generate true random
numbers. For most purposes, @dfn{pseudo-random numbers} suffice. A
series of pseudo-random numbers is generated in a deterministic
fashion. The numbers are not truly random, but they have certain
properties that mimic a random series. For example, all possible
values occur equally often in a pseudo-random series.
@cindex seed, for random number generation
Pseudo-random numbers are generated from a @dfn{seed value}. Starting from
any given seed, the @code{random} function always generates the same
sequence of numbers. By default, Emacs initializes the random seed at
startup, in such a way that the sequence of values of @code{random}
(with overwhelming likelihood) differs in each Emacs run.
Sometimes you want the random number sequence to be repeatable. For
example, when debugging a program whose behavior depends on the random
number sequence, it is helpful to get the same behavior in each
program run. To make the sequence repeat, execute @code{(random "")}.
This sets the seed to a constant value for your particular Emacs
executable (though it may differ for other Emacs builds). You can use
other strings to choose various seed values.
@defun random &optional limit
This function returns a pseudo-random integer. Repeated calls return a
series of pseudo-random integers.
If @var{limit} is a positive fixnum, the value is chosen to be
nonnegative and less than @var{limit}. Otherwise, the value might be
any fixnum, i.e., any integer from @code{most-negative-fixnum} through
@code{most-positive-fixnum} (@pxref{Integer Basics}).
If @var{limit} is @code{t}, it means to choose a new seed as if Emacs
were restarting, typically from the system entropy. On systems
lacking entropy pools, choose the seed from less-random volatile data
such as the current time.
If @var{limit} is a string, it means to choose a new seed based on the
string's contents.
@end defun
|