all messages for Emacs-related lists mirrored at yhetil.org
 help / color / mirror / code / Atom feed
blob 6c51b849d355b8c56cb31d282c5dd3998dbcdd2c 38048 bytes (raw)
name: doc/lispref/numbers.texi 	 # note: path name is non-authoritative(*)

   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
1253
1254
1255
1256
1257
1258
1259
1260
1261
1262
1263
1264
1265
1266
1267
1268
1269
1270
1271
1272
1273
1274
1275
1276
1277
1278
1279
1280
1281
1282
1283
1284
1285
1286
1287
1288
1289
1290
1291
 
@c -*-texinfo-*-
@c This is part of the GNU Emacs Lisp Reference Manual.
@c Copyright (C) 1990-1995, 1998-1999, 2001-2018 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, though
they may overflow.  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

  The range of values for an integer 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.  Many examples in this
chapter assume the minimum integer width of 30 bits.
@cindex overflow

  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

  If an integer is outside the Emacs range, the Lisp reader ordinarily
signals an overflow.  However, if a too-large plain integer ends in a
period, the Lisp reader treats it as a floating-point number instead.
This lets an Emacs Lisp program specify a large integer that is
quietly approximated by a floating-point number on machines with
limited word width.  For example, @samp{536870912.} is a
floating-point number if Emacs integers are only 30 bits wide and is
an integer otherwise.

  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 30-bit binary, the decimal integer 5 looks like this:

@example
0000...000101 (30 bits total)
@end example

@noindent
(The @samp{...} stands for enough bits to fill out a 30-bit word; in
this case, @samp{...} stands for twenty 0 bits.  Later examples also
use the @samp{...} notation to make binary integers easier to read.)

  The integer @minus{}1 looks like this:

@example
1111...111111 (30 bits total)
@end example

@noindent
@cindex two's complement
@minus{}1 is represented as 30 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
1111...111011 (30 bits total)
@end example

  In this implementation, the largest 30-bit binary integer is
536,870,911 in decimal.  In binary, it looks like this:

@example
0111...111111 (30 bits total)
@end example

  Since the arithmetic functions do not check whether integers go
outside their range, when you add 1 to 536,870,911, the value is the
negative integer @minus{}536,870,912:

@example
(+ 1 536870911)
     @result{} -536870912
     @result{} 1000...000000 (30 bits total)
@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 Lisp integer
@cindex maximum Lisp integer
@defvar most-positive-fixnum
The value of this variable is the largest 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 Lisp integer
@cindex minimum Lisp integer
@defvar most-negative-fixnum
The value of this variable is the smallest 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

  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 @code{equal} and @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
@findex eql
@findex sxhash-eql
  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 like
@code{eql} and @code{sxhash-eql} treat two NaNs as equal when their
signs and significands agree.  Significands of NaNs are
machine-dependent and are not directly visible to Emacs Lisp.

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, the value is the logarithm base 2 of @math{|x|}, rounded
down to an integer.

@example
(logb 10)
     @result{} 3
(logb 10.0e20)
     @result{} 69
@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 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{=}, not @code{eq}.  There can be many distinct floating-point
objects with the same numeric value.  If you use @code{eq} to
compare them, then you test whether two values are the same
@emph{object}.  By contrast, @code{=} compares only the numeric values
of the objects.

  In Emacs Lisp, each integer is a unique Lisp object.
Therefore, @code{eq} is equivalent to @code{=} where integers are
concerned.  It is sometimes convenient to use @code{eq} for comparing
an unknown value with an integer, 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{equal}, which
treats 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

@cindex CL note---integers vrs @code{eq}
@quotation
@b{Common Lisp note:} Comparing numbers in Common Lisp always requires
@code{=} because Common Lisp implements multi-word integers, and two
distinct integer objects can have the same numeric value.  Emacs Lisp
can have just one integer object for any given value because it has a
limited range of integers.
@end quotation

@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}.
@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.

  Emacs Lisp arithmetic functions do not check for integer overflow.
Thus @code{(1+ 536870911)} may evaluate to
@minus{}536870912, depending on your hardware.

@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).  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 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, @code{lsh} shifts zeros into the leftmost
(most-significant) bit, producing a positive result even if
@var{integer1} is negative.  Contrast this with @code{ash}, below.

Here are two examples of @code{lsh}, shifting a pattern of bits one
place to the left.  We show only the low-order eight bits of the binary
pattern; the rest are all zero.

@example
@group
(lsh 5 1)
     @result{} 10
;; @r{Decimal 5 becomes decimal 10.}
00000101 @result{} 00001010

(lsh 7 1)
     @result{} 14
;; @r{Decimal 7 becomes decimal 14.}
00000111 @result{} 00001110
@end group
@end example

@noindent
As the examples illustrate, shifting the pattern of bits one place to
the left produces a number that is twice the value of the previous
number.

Shifting a pattern of bits two places to the left produces results
like this (with 8-bit binary numbers):

@example
@group
(lsh 3 2)
     @result{} 12
;; @r{Decimal 3 becomes decimal 12.}
00000011 @result{} 00001100
@end group
@end example

On the other hand, shifting one place to the right looks like this:

@example
@group
(lsh 6 -1)
     @result{} 3
;; @r{Decimal 6 becomes decimal 3.}
00000110 @result{} 00000011
@end group

@group
(lsh 5 -1)
     @result{} 2
;; @r{Decimal 5 becomes decimal 2.}
00000101 @result{} 00000010
@end group
@end example

@noindent
As the example illustrates, shifting one place to the right divides the
value of a positive integer by two, rounding downward.

The function @code{lsh}, like all Emacs Lisp arithmetic functions, does
not check for overflow, so shifting left can discard significant bits
and change the sign of the number.  For example, left shifting
536,870,911 produces @minus{}2 in the 30-bit implementation:

@example
(lsh 536870911 1)          ; @r{left shift}
     @result{} -2
@end example

In binary, the argument looks like this:

@example
@group
;; @r{Decimal 536,870,911}
0111...111111 (30 bits total)
@end group
@end example

@noindent
which becomes the following when left shifted:

@example
@group
;; @r{Decimal @minus{}2}
1111...111110 (30 bits total)
@end group
@end example
@end defun

@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.

@code{ash} gives the same results as @code{lsh} except when
@var{integer1} and @var{count} are both negative.  In that case,
@code{ash} puts ones in the empty bit positions on the left, while
@code{lsh} puts zeros in those bit positions.

Thus, with @code{ash}, shifting the pattern of bits one place to the right
looks like this:

@example
@group
(ash -6 -1) @result{} -3
;; @r{Decimal @minus{}6 becomes decimal @minus{}3.}
1111...111010 (30 bits total)
     @result{}
1111...111101 (30 bits total)
@end group
@end example

In contrast, shifting the pattern of bits one place to the right with
@code{lsh} looks like this:

@example
@group
(lsh -6 -1) @result{} 536870909
;; @r{Decimal @minus{}6 becomes decimal 536,870,909.}
1111...111010 (30 bits total)
     @result{}
0111...111101 (30 bits total)
@end group
@end example

Here are other examples:

@c !!! Check if lined up in smallbook format!  XDVI shows problem
@c     with smallbook but not with regular book! --rjc 16mar92
@smallexample
@group
                   ;  @r{       30-bit binary values}

(lsh 5 2)          ;   5  =  @r{0000...000101}
     @result{} 20         ;      =  @r{0000...010100}
@end group
@group
(ash 5 2)
     @result{} 20
(lsh -5 2)         ;  -5  =  @r{1111...111011}
     @result{} -20        ;      =  @r{1111...101100}
(ash -5 2)
     @result{} -20
@end group
@group
(lsh 5 -2)         ;   5  =  @r{0000...000101}
     @result{} 1          ;      =  @r{0000...000001}
@end group
@group
(ash 5 -2)
     @result{} 1
@end group
@group
(lsh -5 -2)        ;  -5  =  @r{1111...111011}
     @result{} 268435454
                   ;      =  @r{0011...111110}
@end group
@group
(ash -5 -2)        ;  -5  =  @r{1111...111011}
     @result{} -2         ;      =  @r{1111...111110}
@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{       30-bit binary values}

(logand 14 13)     ; 14  =  @r{0000...001110}
                   ; 13  =  @r{0000...001101}
     @result{} 12         ; 12  =  @r{0000...001100}
@end group

@group
(logand 14 13 4)   ; 14  =  @r{0000...001110}
                   ; 13  =  @r{0000...001101}
                   ;  4  =  @r{0000...000100}
     @result{} 4          ;  4  =  @r{0000...000100}
@end group

@group
(logand)
     @result{} -1         ; -1  =  @r{1111...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{       30-bit binary values}

(logior 12 5)      ; 12  =  @r{0000...001100}
                   ;  5  =  @r{0000...000101}
     @result{} 13         ; 13  =  @r{0000...001101}
@end group

@group
(logior 12 5 7)    ; 12  =  @r{0000...001100}
                   ;  5  =  @r{0000...000101}
                   ;  7  =  @r{0000...000111}
     @result{} 15         ; 15  =  @r{0000...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{       30-bit binary values}

(logxor 12 5)      ; 12  =  @r{0000...001100}
                   ;  5  =  @r{0000...000101}
     @result{} 9          ;  9  =  @r{0000...001001}
@end group

@group
(logxor 12 5 7)    ; 12  =  @r{0000...001100}
                   ;  5  =  @r{0000...000101}
                   ;  7  =  @r{0000...000111}
     @result{} 14         ; 14  =  @r{0000...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.

@example
(lognot 5)
     @result{} -6
;;  5  =  @r{0000...000101} (30 bits total)
;; @r{becomes}
;; -6  =  @r{1111...111010} (30 bits total)
@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 = #b101011
     @result{} 4
(logcount -43)    ; -43 = #b111...1010101
     @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 positive, the result is an
integer; in this case, overflow causes truncation, 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 integer, the value is chosen to be
nonnegative and less than @var{limit}.  Otherwise, the value might be
any integer representable in Lisp, i.e., an integer between
@code{most-negative-fixnum} and @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

debug log:

solving 6c51b84 ...
found 6c51b84 in https://git.savannah.gnu.org/cgit/emacs.git

(*) Git path names are given by the tree(s) the blob belongs to.
    Blobs themselves have no identifier aside from the hash of its contents.^

Code repositories for project(s) associated with this external index

	https://git.savannah.gnu.org/cgit/emacs.git
	https://git.savannah.gnu.org/cgit/emacs/org-mode.git

This is an external index of several public inboxes,
see mirroring instructions on how to clone and mirror
all data and code used by this external index.