428
+ − 1 @c -*-texinfo-*-
+ − 2 @c This is part of the XEmacs Lisp Reference Manual.
444
+ − 3 @c Copyright (C) 1990, 1991, 1992, 1993, 1994 Free Software Foundation, Inc.
428
+ − 4 @c See the file lispref.texi for copying conditions.
+ − 5 @setfilename ../../info/numbers.info
+ − 6 @node Numbers, Strings and Characters, Lisp Data Types, Top
+ − 7 @chapter Numbers
+ − 8 @cindex integers
+ − 9 @cindex numbers
+ − 10
+ − 11 XEmacs supports two numeric data types: @dfn{integers} and
+ − 12 @dfn{floating point numbers}. Integers are whole numbers such as
+ − 13 @minus{}3, 0, #b0111, #xFEED, #o744. Their values are exact. The
+ − 14 number prefixes `#b', `#o', and `#x' are supported to represent numbers
+ − 15 in binary, octal, and hexadecimal notation (or radix). Floating point
+ − 16 numbers are numbers with fractional parts, such as @minus{}4.5, 0.0, or
+ − 17 2.71828. They can also be expressed in exponential notation: 1.5e2
+ − 18 equals 150; in this example, @samp{e2} stands for ten to the second
+ − 19 power, and is multiplied by 1.5. Floating point values are not exact;
+ − 20 they have a fixed, limited amount of precision.
+ − 21
+ − 22 @menu
+ − 23 * Integer Basics:: Representation and range of integers.
+ − 24 * Float Basics:: Representation and range of floating point.
+ − 25 * Predicates on Numbers:: Testing for numbers.
+ − 26 * Comparison of Numbers:: Equality and inequality predicates.
+ − 27 * Numeric Conversions:: Converting float to integer and vice versa.
+ − 28 * Arithmetic Operations:: How to add, subtract, multiply and divide.
+ − 29 * Rounding Operations:: Explicitly rounding floating point numbers.
+ − 30 * Bitwise Operations:: Logical and, or, not, shifting.
+ − 31 * Math Functions:: Trig, exponential and logarithmic functions.
+ − 32 * Random Numbers:: Obtaining random integers, predictable or not.
+ − 33 @end menu
+ − 34
+ − 35 @node Integer Basics
+ − 36 @section Integer Basics
+ − 37
+ − 38 The range of values for an integer depends on the machine. The
+ − 39 minimum range is @minus{}134217728 to 134217727 (28 bits; i.e.,
444
+ − 40 @ifinfo
428
+ − 41 -2**27
+ − 42 @end ifinfo
444
+ − 43 @tex
428
+ − 44 $-2^{27}$
+ − 45 @end tex
444
+ − 46 to
+ − 47 @ifinfo
428
+ − 48 2**27 - 1),
+ − 49 @end ifinfo
444
+ − 50 @tex
428
+ − 51 $2^{27}-1$),
+ − 52 @end tex
+ − 53 but some machines may provide a wider range. Many examples in this
+ − 54 chapter assume an integer has 28 bits.
+ − 55 @cindex overflow
+ − 56
+ − 57 The Lisp reader reads an integer as a sequence of digits with optional
+ − 58 initial sign and optional final period.
+ − 59
+ − 60 @example
+ − 61 1 ; @r{The integer 1.}
+ − 62 1. ; @r{The integer 1.}
+ − 63 +1 ; @r{Also the integer 1.}
+ − 64 -1 ; @r{The integer @minus{}1.}
+ − 65 268435457 ; @r{Also the integer 1, due to overflow.}
+ − 66 0 ; @r{The integer 0.}
+ − 67 -0 ; @r{The integer 0.}
+ − 68 @end example
+ − 69
+ − 70 To understand how various functions work on integers, especially the
+ − 71 bitwise operators (@pxref{Bitwise Operations}), it is often helpful to
+ − 72 view the numbers in their binary form.
+ − 73
+ − 74 In 28-bit binary, the decimal integer 5 looks like this:
+ − 75
+ − 76 @example
+ − 77 0000 0000 0000 0000 0000 0000 0101
+ − 78 @end example
+ − 79
+ − 80 @noindent
+ − 81 (We have inserted spaces between groups of 4 bits, and two spaces
+ − 82 between groups of 8 bits, to make the binary integer easier to read.)
+ − 83
+ − 84 The integer @minus{}1 looks like this:
+ − 85
+ − 86 @example
+ − 87 1111 1111 1111 1111 1111 1111 1111
+ − 88 @end example
+ − 89
+ − 90 @noindent
+ − 91 @cindex two's complement
+ − 92 @minus{}1 is represented as 28 ones. (This is called @dfn{two's
+ − 93 complement} notation.)
+ − 94
+ − 95 The negative integer, @minus{}5, is creating by subtracting 4 from
+ − 96 @minus{}1. In binary, the decimal integer 4 is 100. Consequently,
+ − 97 @minus{}5 looks like this:
+ − 98
+ − 99 @example
+ − 100 1111 1111 1111 1111 1111 1111 1011
+ − 101 @end example
+ − 102
+ − 103 In this implementation, the largest 28-bit binary integer is the
+ − 104 decimal integer 134,217,727. In binary, it looks like this:
+ − 105
+ − 106 @example
+ − 107 0111 1111 1111 1111 1111 1111 1111
+ − 108 @end example
+ − 109
+ − 110 Since the arithmetic functions do not check whether integers go
+ − 111 outside their range, when you add 1 to 134,217,727, the value is the
+ − 112 negative integer @minus{}134,217,728:
+ − 113
+ − 114 @example
+ − 115 (+ 1 134217727)
+ − 116 @result{} -134217728
+ − 117 @result{} 1000 0000 0000 0000 0000 0000 0000
+ − 118 @end example
+ − 119
+ − 120 Many of the following functions accept markers for arguments as well
+ − 121 as integers. (@xref{Markers}.) More precisely, the actual arguments to
+ − 122 such functions may be either integers or markers, which is why we often
+ − 123 give these arguments the name @var{int-or-marker}. When the argument
+ − 124 value is a marker, its position value is used and its buffer is ignored.
+ − 125
+ − 126 @ignore
+ − 127 In version 19, except where @emph{integer} is specified as an
+ − 128 argument, all of the functions for markers and integers also work for
+ − 129 floating point numbers.
+ − 130 @end ignore
+ − 131
+ − 132 @node Float Basics
+ − 133 @section Floating Point Basics
+ − 134
+ − 135 XEmacs supports floating point numbers. The precise range of floating
+ − 136 point numbers is machine-specific; it is the same as the range of the C
+ − 137 data type @code{double} on the machine in question.
+ − 138
+ − 139 The printed representation for floating point numbers requires either
+ − 140 a decimal point (with at least one digit following), an exponent, or
+ − 141 both. For example, @samp{1500.0}, @samp{15e2}, @samp{15.0e2},
+ − 142 @samp{1.5e3}, and @samp{.15e4} are five ways of writing a floating point
+ − 143 number whose value is 1500. They are all equivalent. You can also use
+ − 144 a minus sign to write negative floating point numbers, as in
+ − 145 @samp{-1.0}.
+ − 146
+ − 147 @cindex IEEE floating point
+ − 148 @cindex positive infinity
+ − 149 @cindex negative infinity
+ − 150 @cindex infinity
+ − 151 @cindex NaN
+ − 152 Most modern computers support the IEEE floating point standard, which
+ − 153 provides for positive infinity and negative infinity as floating point
+ − 154 values. It also provides for a class of values called NaN or
+ − 155 ``not-a-number''; numerical functions return such values in cases where
+ − 156 there is no correct answer. For example, @code{(sqrt -1.0)} returns a
+ − 157 NaN. For practical purposes, there's no significant difference between
+ − 158 different NaN values in XEmacs Lisp, and there's no rule for precisely
+ − 159 which NaN value should be used in a particular case, so this manual
+ − 160 doesn't try to distinguish them. XEmacs Lisp has no read syntax for NaNs
+ − 161 or infinities; perhaps we should create a syntax in the future.
+ − 162
+ − 163 You can use @code{logb} to extract the binary exponent of a floating
+ − 164 point number (or estimate the logarithm of an integer):
+ − 165
+ − 166 @defun logb number
+ − 167 This function returns the binary exponent of @var{number}. More
+ − 168 precisely, the value is the logarithm of @var{number} base 2, rounded
+ − 169 down to an integer.
+ − 170 @end defun
+ − 171
+ − 172 @node Predicates on Numbers
+ − 173 @section Type Predicates for Numbers
+ − 174
+ − 175 The functions in this section test whether the argument is a number or
+ − 176 whether it is a certain sort of number. The functions @code{integerp}
+ − 177 and @code{floatp} can take any type of Lisp object as argument (the
+ − 178 predicates would not be of much use otherwise); but the @code{zerop}
+ − 179 predicate requires a number as its argument. See also
+ − 180 @code{integer-or-marker-p}, @code{integer-char-or-marker-p},
+ − 181 @code{number-or-marker-p} and @code{number-char-or-marker-p}, in
+ − 182 @ref{Predicates on Markers}.
+ − 183
+ − 184 @defun floatp object
+ − 185 This predicate tests whether its argument is a floating point
+ − 186 number and returns @code{t} if so, @code{nil} otherwise.
+ − 187
+ − 188 @code{floatp} does not exist in Emacs versions 18 and earlier.
+ − 189 @end defun
+ − 190
+ − 191 @defun integerp object
+ − 192 This predicate tests whether its argument is an integer, and returns
+ − 193 @code{t} if so, @code{nil} otherwise.
+ − 194 @end defun
+ − 195
+ − 196 @defun numberp object
+ − 197 This predicate tests whether its argument is a number (either integer or
+ − 198 floating point), and returns @code{t} if so, @code{nil} otherwise.
+ − 199 @end defun
+ − 200
+ − 201 @defun natnump object
+ − 202 @cindex natural numbers
+ − 203 The @code{natnump} predicate (whose name comes from the phrase
+ − 204 ``natural-number-p'') tests to see whether its argument is a nonnegative
+ − 205 integer, and returns @code{t} if so, @code{nil} otherwise. 0 is
+ − 206 considered non-negative.
+ − 207 @end defun
+ − 208
+ − 209 @defun zerop number
+ − 210 This predicate tests whether its argument is zero, and returns @code{t}
+ − 211 if so, @code{nil} otherwise. The argument must be a number.
+ − 212
+ − 213 These two forms are equivalent: @code{(zerop x)} @equiv{} @code{(= x 0)}.
+ − 214 @end defun
+ − 215
+ − 216 @node Comparison of Numbers
+ − 217 @section Comparison of Numbers
+ − 218 @cindex number equality
+ − 219
+ − 220 To test numbers for numerical equality, you should normally use
+ − 221 @code{=}, not @code{eq}. There can be many distinct floating point
+ − 222 number objects with the same numeric value. If you use @code{eq} to
+ − 223 compare them, then you test whether two values are the same
+ − 224 @emph{object}. By contrast, @code{=} compares only the numeric values
+ − 225 of the objects.
+ − 226
+ − 227 At present, each integer value has a unique Lisp object in XEmacs Lisp.
+ − 228 Therefore, @code{eq} is equivalent to @code{=} where integers are
+ − 229 concerned. It is sometimes convenient to use @code{eq} for comparing an
+ − 230 unknown value with an integer, because @code{eq} does not report an
+ − 231 error if the unknown value is not a number---it accepts arguments of any
+ − 232 type. By contrast, @code{=} signals an error if the arguments are not
+ − 233 numbers or markers. However, it is a good idea to use @code{=} if you
+ − 234 can, even for comparing integers, just in case we change the
+ − 235 representation of integers in a future XEmacs version.
+ − 236
+ − 237 There is another wrinkle: because floating point arithmetic is not
+ − 238 exact, it is often a bad idea to check for equality of two floating
+ − 239 point values. Usually it is better to test for approximate equality.
+ − 240 Here's a function to do this:
+ − 241
+ − 242 @example
+ − 243 (defconst fuzz-factor 1.0e-6)
+ − 244 (defun approx-equal (x y)
+ − 245 (or (and (= x 0) (= y 0))
+ − 246 (< (/ (abs (- x y))
+ − 247 (max (abs x) (abs y)))
+ − 248 fuzz-factor)))
+ − 249 @end example
+ − 250
+ − 251 @cindex CL note---integers vrs @code{eq}
+ − 252 @quotation
+ − 253 @b{Common Lisp note:} Comparing numbers in Common Lisp always requires
+ − 254 @code{=} because Common Lisp implements multi-word integers, and two
+ − 255 distinct integer objects can have the same numeric value. XEmacs Lisp
+ − 256 can have just one integer object for any given value because it has a
+ − 257 limited range of integer values.
+ − 258 @end quotation
+ − 259
+ − 260 In addition to numbers, all of the following functions also accept
+ − 261 characters and markers as arguments, and treat them as their number
+ − 262 equivalents.
+ − 263
+ − 264 @defun = number &rest more-numbers
+ − 265 This function returns @code{t} if all of its arguments are numerically
+ − 266 equal, @code{nil} otherwise.
+ − 267
+ − 268 @example
+ − 269 (= 5)
+ − 270 @result{} t
+ − 271 (= 5 6)
+ − 272 @result{} nil
+ − 273 (= 5 5.0)
+ − 274 @result{} t
+ − 275 (= 5 5 6)
+ − 276 @result{} nil
+ − 277 @end example
+ − 278 @end defun
+ − 279
+ − 280 @defun /= number &rest more-numbers
+ − 281 This function returns @code{t} if no two arguments are numerically
+ − 282 equal, @code{nil} otherwise.
+ − 283
+ − 284 @example
+ − 285 (/= 5 6)
+ − 286 @result{} t
+ − 287 (/= 5 5 6)
+ − 288 @result{} nil
+ − 289 (/= 5 6 1)
+ − 290 @result{} t
+ − 291 @end example
+ − 292 @end defun
+ − 293
+ − 294 @defun < number &rest more-numbers
+ − 295 This function returns @code{t} if the sequence of its arguments is
+ − 296 monotonically increasing, @code{nil} otherwise.
+ − 297
+ − 298 @example
+ − 299 (< 5 6)
+ − 300 @result{} t
+ − 301 (< 5 6 6)
+ − 302 @result{} nil
+ − 303 (< 5 6 7)
+ − 304 @result{} t
+ − 305 @end example
+ − 306 @end defun
+ − 307
+ − 308 @defun <= number &rest more-numbers
+ − 309 This function returns @code{t} if the sequence of its arguments is
+ − 310 monotonically nondecreasing, @code{nil} otherwise.
+ − 311
+ − 312 @example
+ − 313 (<= 5 6)
+ − 314 @result{} t
+ − 315 (<= 5 6 6)
+ − 316 @result{} t
+ − 317 (<= 5 6 5)
+ − 318 @result{} nil
+ − 319 @end example
+ − 320 @end defun
+ − 321
+ − 322 @defun > number &rest more-numbers
+ − 323 This function returns @code{t} if the sequence of its arguments is
+ − 324 monotonically decreasing, @code{nil} otherwise.
+ − 325 @end defun
+ − 326
+ − 327 @defun >= number &rest more-numbers
+ − 328 This function returns @code{t} if the sequence of its arguments is
+ − 329 monotonically nonincreasing, @code{nil} otherwise.
+ − 330 @end defun
+ − 331
+ − 332 @defun max number &rest more-numbers
+ − 333 This function returns the largest of its arguments.
+ − 334
+ − 335 @example
+ − 336 (max 20)
+ − 337 @result{} 20
+ − 338 (max 1 2.5)
+ − 339 @result{} 2.5
+ − 340 (max 1 3 2.5)
+ − 341 @result{} 3
+ − 342 @end example
+ − 343 @end defun
+ − 344
+ − 345 @defun min number &rest more-numbers
+ − 346 This function returns the smallest of its arguments.
+ − 347
+ − 348 @example
+ − 349 (min -4 1)
+ − 350 @result{} -4
+ − 351 @end example
+ − 352 @end defun
+ − 353
+ − 354 @node Numeric Conversions
+ − 355 @section Numeric Conversions
+ − 356 @cindex rounding in conversions
+ − 357
+ − 358 To convert an integer to floating point, use the function @code{float}.
+ − 359
+ − 360 @defun float number
+ − 361 This returns @var{number} converted to floating point.
+ − 362 If @var{number} is already a floating point number, @code{float} returns
+ − 363 it unchanged.
+ − 364 @end defun
+ − 365
+ − 366 There are four functions to convert floating point numbers to integers;
+ − 367 they differ in how they round. These functions accept integer arguments
+ − 368 also, and return such arguments unchanged.
+ − 369
+ − 370 @defun truncate number
+ − 371 This returns @var{number}, converted to an integer by rounding towards
+ − 372 zero.
+ − 373 @end defun
+ − 374
+ − 375 @defun floor number &optional divisor
+ − 376 This returns @var{number}, converted to an integer by rounding downward
+ − 377 (towards negative infinity).
+ − 378
+ − 379 If @var{divisor} is specified, @var{number} is divided by @var{divisor}
+ − 380 before the floor is taken; this is the division operation that
+ − 381 corresponds to @code{mod}. An @code{arith-error} results if
+ − 382 @var{divisor} is 0.
+ − 383 @end defun
+ − 384
+ − 385 @defun ceiling number
+ − 386 This returns @var{number}, converted to an integer by rounding upward
+ − 387 (towards positive infinity).
+ − 388 @end defun
+ − 389
+ − 390 @defun round number
+ − 391 This returns @var{number}, converted to an integer by rounding towards the
+ − 392 nearest integer. Rounding a value equidistant between two integers
+ − 393 may choose the integer closer to zero, or it may prefer an even integer,
+ − 394 depending on your machine.
+ − 395 @end defun
+ − 396
+ − 397 @node Arithmetic Operations
+ − 398 @section Arithmetic Operations
+ − 399
+ − 400 XEmacs Lisp provides the traditional four arithmetic operations:
+ − 401 addition, subtraction, multiplication, and division. Remainder and modulus
+ − 402 functions supplement the division functions. The functions to
+ − 403 add or subtract 1 are provided because they are traditional in Lisp and
+ − 404 commonly used.
+ − 405
+ − 406 All of these functions except @code{%} return a floating point value
+ − 407 if any argument is floating.
+ − 408
+ − 409 It is important to note that in XEmacs Lisp, arithmetic functions
+ − 410 do not check for overflow. Thus @code{(1+ 134217727)} may evaluate to
+ − 411 @minus{}134217728, depending on your hardware.
+ − 412
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+ − 413 @defun 1+ number
+ − 414 This function returns @var{number} plus one. @var{number} may be a
+ − 415 number, character or marker. Markers and characters are converted to
+ − 416 integers.
+ − 417
428
+ − 418 For example,
+ − 419
+ − 420 @example
+ − 421 (setq foo 4)
+ − 422 @result{} 4
+ − 423 (1+ foo)
+ − 424 @result{} 5
+ − 425 @end example
+ − 426
+ − 427 This function is not analogous to the C operator @code{++}---it does not
+ − 428 increment a variable. It just computes a sum. Thus, if we continue,
+ − 429
+ − 430 @example
+ − 431 foo
+ − 432 @result{} 4
+ − 433 @end example
+ − 434
+ − 435 If you want to increment the variable, you must use @code{setq},
+ − 436 like this:
+ − 437
+ − 438 @example
+ − 439 (setq foo (1+ foo))
+ − 440 @result{} 5
+ − 441 @end example
+ − 442
+ − 443 Now that the @code{cl} package is always available from lisp code, a
+ − 444 more convenient and natural way to increment a variable is
+ − 445 @w{@code{(incf foo)}}.
+ − 446 @end defun
+ − 447
444
+ − 448 @defun 1- number
+ − 449 This function returns @var{number} minus one. @var{number} may be a
+ − 450 number, character or marker. Markers and characters are converted to
+ − 451 integers.
428
+ − 452 @end defun
+ − 453
+ − 454 @defun abs number
+ − 455 This returns the absolute value of @var{number}.
+ − 456 @end defun
+ − 457
444
+ − 458 @defun + &rest numbers
428
+ − 459 This function adds its arguments together. When given no arguments,
+ − 460 @code{+} returns 0.
+ − 461
444
+ − 462 If any of the arguments are characters or markers, they are first
+ − 463 converted to integers.
+ − 464
428
+ − 465 @example
+ − 466 (+)
+ − 467 @result{} 0
+ − 468 (+ 1)
+ − 469 @result{} 1
+ − 470 (+ 1 2 3 4)
+ − 471 @result{} 10
+ − 472 @end example
+ − 473 @end defun
+ − 474
444
+ − 475 @defun - &optional number &rest other-numbers
428
+ − 476 The @code{-} function serves two purposes: negation and subtraction.
+ − 477 When @code{-} has a single argument, the value is the negative of the
+ − 478 argument. When there are multiple arguments, @code{-} subtracts each of
444
+ − 479 the @var{other-numbers} from @var{number}, cumulatively. If there are
+ − 480 no arguments, an error is signaled.
+ − 481
+ − 482 If any of the arguments are characters or markers, they are first
+ − 483 converted to integers.
428
+ − 484
+ − 485 @example
+ − 486 (- 10 1 2 3 4)
+ − 487 @result{} 0
+ − 488 (- 10)
+ − 489 @result{} -10
+ − 490 (-)
+ − 491 @result{} 0
+ − 492 @end example
+ − 493 @end defun
+ − 494
444
+ − 495 @defun * &rest numbers
428
+ − 496 This function multiplies its arguments together, and returns the
+ − 497 product. When given no arguments, @code{*} returns 1.
+ − 498
444
+ − 499 If any of the arguments are characters or markers, they are first
+ − 500 converted to integers.
+ − 501
428
+ − 502 @example
+ − 503 (*)
+ − 504 @result{} 1
+ − 505 (* 1)
+ − 506 @result{} 1
+ − 507 (* 1 2 3 4)
+ − 508 @result{} 24
+ − 509 @end example
+ − 510 @end defun
+ − 511
444
+ − 512 @defun / dividend &rest divisors
+ − 513 The @code{/} function serves two purposes: inversion and division. When
+ − 514 @code{/} has a single argument, the value is the inverse of the
+ − 515 argument. When there are multiple arguments, @code{/} divides
+ − 516 @var{dividend} by each of the @var{divisors}, cumulatively, returning
+ − 517 the quotient. If there are no arguments, an error is signaled.
428
+ − 518
444
+ − 519 If none of the arguments are floats, then the result is an integer.
428
+ − 520 This means the result has to be rounded. On most machines, the result
+ − 521 is rounded towards zero after each division, but some machines may round
+ − 522 differently with negative arguments. This is because the Lisp function
+ − 523 @code{/} is implemented using the C division operator, which also
+ − 524 permits machine-dependent rounding. As a practical matter, all known
+ − 525 machines round in the standard fashion.
+ − 526
444
+ − 527 If any of the arguments are characters or markers, they are first
+ − 528 converted to integers.
+ − 529
428
+ − 530 @cindex @code{arith-error} in division
+ − 531 If you divide by 0, an @code{arith-error} error is signaled.
+ − 532 (@xref{Errors}.)
+ − 533
+ − 534 @example
+ − 535 @group
+ − 536 (/ 6 2)
+ − 537 @result{} 3
+ − 538 @end group
+ − 539 (/ 5 2)
+ − 540 @result{} 2
+ − 541 (/ 25 3 2)
+ − 542 @result{} 4
444
+ − 543 (/ 3.0)
+ − 544 @result{} 0.3333333333333333
428
+ − 545 (/ -17 6)
+ − 546 @result{} -2
+ − 547 @end example
+ − 548
+ − 549 The result of @code{(/ -17 6)} could in principle be -3 on some
+ − 550 machines.
+ − 551 @end defun
+ − 552
+ − 553 @defun % dividend divisor
+ − 554 @cindex remainder
+ − 555 This function returns the integer remainder after division of @var{dividend}
+ − 556 by @var{divisor}. The arguments must be integers or markers.
+ − 557
+ − 558 For negative arguments, the remainder is in principle machine-dependent
+ − 559 since the quotient is; but in practice, all known machines behave alike.
+ − 560
+ − 561 An @code{arith-error} results if @var{divisor} is 0.
+ − 562
+ − 563 @example
+ − 564 (% 9 4)
+ − 565 @result{} 1
+ − 566 (% -9 4)
+ − 567 @result{} -1
+ − 568 (% 9 -4)
+ − 569 @result{} 1
+ − 570 (% -9 -4)
+ − 571 @result{} -1
+ − 572 @end example
+ − 573
+ − 574 For any two integers @var{dividend} and @var{divisor},
+ − 575
+ − 576 @example
+ − 577 @group
+ − 578 (+ (% @var{dividend} @var{divisor})
+ − 579 (* (/ @var{dividend} @var{divisor}) @var{divisor}))
+ − 580 @end group
+ − 581 @end example
+ − 582
+ − 583 @noindent
+ − 584 always equals @var{dividend}.
+ − 585 @end defun
+ − 586
+ − 587 @defun mod dividend divisor
+ − 588 @cindex modulus
+ − 589 This function returns the value of @var{dividend} modulo @var{divisor};
+ − 590 in other words, the remainder after division of @var{dividend}
+ − 591 by @var{divisor}, but with the same sign as @var{divisor}.
+ − 592 The arguments must be numbers or markers.
+ − 593
+ − 594 Unlike @code{%}, @code{mod} returns a well-defined result for negative
+ − 595 arguments. It also permits floating point arguments; it rounds the
+ − 596 quotient downward (towards minus infinity) to an integer, and uses that
+ − 597 quotient to compute the remainder.
+ − 598
+ − 599 An @code{arith-error} results if @var{divisor} is 0.
+ − 600
+ − 601 @example
+ − 602 @group
+ − 603 (mod 9 4)
+ − 604 @result{} 1
+ − 605 @end group
+ − 606 @group
+ − 607 (mod -9 4)
+ − 608 @result{} 3
+ − 609 @end group
+ − 610 @group
+ − 611 (mod 9 -4)
+ − 612 @result{} -3
+ − 613 @end group
+ − 614 @group
+ − 615 (mod -9 -4)
+ − 616 @result{} -1
+ − 617 @end group
+ − 618 @group
+ − 619 (mod 5.5 2.5)
+ − 620 @result{} .5
+ − 621 @end group
+ − 622 @end example
+ − 623
+ − 624 For any two numbers @var{dividend} and @var{divisor},
+ − 625
+ − 626 @example
+ − 627 @group
+ − 628 (+ (mod @var{dividend} @var{divisor})
+ − 629 (* (floor @var{dividend} @var{divisor}) @var{divisor}))
+ − 630 @end group
+ − 631 @end example
+ − 632
+ − 633 @noindent
+ − 634 always equals @var{dividend}, subject to rounding error if either
+ − 635 argument is floating point. For @code{floor}, see @ref{Numeric
+ − 636 Conversions}.
+ − 637 @end defun
+ − 638
+ − 639 @node Rounding Operations
+ − 640 @section Rounding Operations
+ − 641 @cindex rounding without conversion
+ − 642
+ − 643 The functions @code{ffloor}, @code{fceiling}, @code{fround} and
+ − 644 @code{ftruncate} take a floating point argument and return a floating
+ − 645 point result whose value is a nearby integer. @code{ffloor} returns the
+ − 646 nearest integer below; @code{fceiling}, the nearest integer above;
+ − 647 @code{ftruncate}, the nearest integer in the direction towards zero;
+ − 648 @code{fround}, the nearest integer.
+ − 649
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+ − 650 @defun ffloor number
+ − 651 This function rounds @var{number} to the next lower integral value, and
428
+ − 652 returns that value as a floating point number.
+ − 653 @end defun
+ − 654
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+ − 655 @defun fceiling number
+ − 656 This function rounds @var{number} to the next higher integral value, and
428
+ − 657 returns that value as a floating point number.
+ − 658 @end defun
+ − 659
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+ − 660 @defun ftruncate number
+ − 661 This function rounds @var{number} towards zero to an integral value, and
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+ − 662 returns that value as a floating point number.
+ − 663 @end defun
+ − 664
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+ − 665 @defun fround number
+ − 666 This function rounds @var{number} to the nearest integral value,
428
+ − 667 and returns that value as a floating point number.
+ − 668 @end defun
+ − 669
+ − 670 @node Bitwise Operations
+ − 671 @section Bitwise Operations on Integers
+ − 672
+ − 673 In a computer, an integer is represented as a binary number, a
+ − 674 sequence of @dfn{bits} (digits which are either zero or one). A bitwise
+ − 675 operation acts on the individual bits of such a sequence. For example,
+ − 676 @dfn{shifting} moves the whole sequence left or right one or more places,
+ − 677 reproducing the same pattern ``moved over''.
+ − 678
+ − 679 The bitwise operations in XEmacs Lisp apply only to integers.
+ − 680
+ − 681 @defun lsh integer1 count
+ − 682 @cindex logical shift
+ − 683 @code{lsh}, which is an abbreviation for @dfn{logical shift}, shifts the
+ − 684 bits in @var{integer1} to the left @var{count} places, or to the right
+ − 685 if @var{count} is negative, bringing zeros into the vacated bits. If
+ − 686 @var{count} is negative, @code{lsh} shifts zeros into the leftmost
+ − 687 (most-significant) bit, producing a positive result even if
+ − 688 @var{integer1} is negative. Contrast this with @code{ash}, below.
+ − 689
+ − 690 Here are two examples of @code{lsh}, shifting a pattern of bits one
+ − 691 place to the left. We show only the low-order eight bits of the binary
+ − 692 pattern; the rest are all zero.
+ − 693
+ − 694 @example
+ − 695 @group
+ − 696 (lsh 5 1)
+ − 697 @result{} 10
+ − 698 ;; @r{Decimal 5 becomes decimal 10.}
+ − 699 00000101 @result{} 00001010
+ − 700
+ − 701 (lsh 7 1)
+ − 702 @result{} 14
+ − 703 ;; @r{Decimal 7 becomes decimal 14.}
+ − 704 00000111 @result{} 00001110
+ − 705 @end group
+ − 706 @end example
+ − 707
+ − 708 @noindent
+ − 709 As the examples illustrate, shifting the pattern of bits one place to
+ − 710 the left produces a number that is twice the value of the previous
+ − 711 number.
+ − 712
+ − 713 Shifting a pattern of bits two places to the left produces results
+ − 714 like this (with 8-bit binary numbers):
+ − 715
+ − 716 @example
+ − 717 @group
+ − 718 (lsh 3 2)
+ − 719 @result{} 12
+ − 720 ;; @r{Decimal 3 becomes decimal 12.}
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+ − 721 00000011 @result{} 00001100
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+ − 722 @end group
+ − 723 @end example
+ − 724
+ − 725 On the other hand, shifting one place to the right looks like this:
+ − 726
+ − 727 @example
+ − 728 @group
+ − 729 (lsh 6 -1)
+ − 730 @result{} 3
+ − 731 ;; @r{Decimal 6 becomes decimal 3.}
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+ − 732 00000110 @result{} 00000011
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+ − 733 @end group
+ − 734
+ − 735 @group
+ − 736 (lsh 5 -1)
+ − 737 @result{} 2
+ − 738 ;; @r{Decimal 5 becomes decimal 2.}
444
+ − 739 00000101 @result{} 00000010
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+ − 740 @end group
+ − 741 @end example
+ − 742
+ − 743 @noindent
+ − 744 As the example illustrates, shifting one place to the right divides the
+ − 745 value of a positive integer by two, rounding downward.
+ − 746
+ − 747 The function @code{lsh}, like all XEmacs Lisp arithmetic functions, does
+ − 748 not check for overflow, so shifting left can discard significant bits
+ − 749 and change the sign of the number. For example, left shifting
+ − 750 134,217,727 produces @minus{}2 on a 28-bit machine:
+ − 751
+ − 752 @example
+ − 753 (lsh 134217727 1) ; @r{left shift}
+ − 754 @result{} -2
+ − 755 @end example
+ − 756
+ − 757 In binary, in the 28-bit implementation, the argument looks like this:
+ − 758
+ − 759 @example
+ − 760 @group
+ − 761 ;; @r{Decimal 134,217,727}
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+ − 762 0111 1111 1111 1111 1111 1111 1111
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+ − 763 @end group
+ − 764 @end example
+ − 765
+ − 766 @noindent
+ − 767 which becomes the following when left shifted:
+ − 768
+ − 769 @example
+ − 770 @group
+ − 771 ;; @r{Decimal @minus{}2}
444
+ − 772 1111 1111 1111 1111 1111 1111 1110
428
+ − 773 @end group
+ − 774 @end example
+ − 775 @end defun
+ − 776
+ − 777 @defun ash integer1 count
+ − 778 @cindex arithmetic shift
+ − 779 @code{ash} (@dfn{arithmetic shift}) shifts the bits in @var{integer1}
+ − 780 to the left @var{count} places, or to the right if @var{count}
+ − 781 is negative.
+ − 782
+ − 783 @code{ash} gives the same results as @code{lsh} except when
+ − 784 @var{integer1} and @var{count} are both negative. In that case,
+ − 785 @code{ash} puts ones in the empty bit positions on the left, while
+ − 786 @code{lsh} puts zeros in those bit positions.
+ − 787
+ − 788 Thus, with @code{ash}, shifting the pattern of bits one place to the right
+ − 789 looks like this:
+ − 790
+ − 791 @example
+ − 792 @group
444
+ − 793 (ash -6 -1) @result{} -3
428
+ − 794 ;; @r{Decimal @minus{}6 becomes decimal @minus{}3.}
+ − 795 1111 1111 1111 1111 1111 1111 1010
444
+ − 796 @result{}
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+ − 797 1111 1111 1111 1111 1111 1111 1101
+ − 798 @end group
+ − 799 @end example
+ − 800
+ − 801 In contrast, shifting the pattern of bits one place to the right with
+ − 802 @code{lsh} looks like this:
+ − 803
+ − 804 @example
+ − 805 @group
+ − 806 (lsh -6 -1) @result{} 134217725
+ − 807 ;; @r{Decimal @minus{}6 becomes decimal 134,217,725.}
+ − 808 1111 1111 1111 1111 1111 1111 1010
444
+ − 809 @result{}
428
+ − 810 0111 1111 1111 1111 1111 1111 1101
+ − 811 @end group
+ − 812 @end example
+ − 813
+ − 814 Here are other examples:
+ − 815
+ − 816 @c !!! Check if lined up in smallbook format! XDVI shows problem
+ − 817 @c with smallbook but not with regular book! --rjc 16mar92
+ − 818 @smallexample
+ − 819 @group
+ − 820 ; @r{ 28-bit binary values}
+ − 821
+ − 822 (lsh 5 2) ; 5 = @r{0000 0000 0000 0000 0000 0000 0101}
+ − 823 @result{} 20 ; = @r{0000 0000 0000 0000 0000 0001 0100}
+ − 824 @end group
+ − 825 @group
+ − 826 (ash 5 2)
+ − 827 @result{} 20
+ − 828 (lsh -5 2) ; -5 = @r{1111 1111 1111 1111 1111 1111 1011}
+ − 829 @result{} -20 ; = @r{1111 1111 1111 1111 1111 1110 1100}
+ − 830 (ash -5 2)
+ − 831 @result{} -20
+ − 832 @end group
+ − 833 @group
+ − 834 (lsh 5 -2) ; 5 = @r{0000 0000 0000 0000 0000 0000 0101}
+ − 835 @result{} 1 ; = @r{0000 0000 0000 0000 0000 0000 0001}
+ − 836 @end group
+ − 837 @group
+ − 838 (ash 5 -2)
+ − 839 @result{} 1
+ − 840 @end group
+ − 841 @group
+ − 842 (lsh -5 -2) ; -5 = @r{1111 1111 1111 1111 1111 1111 1011}
+ − 843 @result{} 4194302 ; = @r{0011 1111 1111 1111 1111 1111 1110}
+ − 844 @end group
+ − 845 @group
+ − 846 (ash -5 -2) ; -5 = @r{1111 1111 1111 1111 1111 1111 1011}
+ − 847 @result{} -2 ; = @r{1111 1111 1111 1111 1111 1111 1110}
+ − 848 @end group
+ − 849 @end smallexample
+ − 850 @end defun
+ − 851
+ − 852 @defun logand &rest ints-or-markers
+ − 853 @cindex logical and
+ − 854 @cindex bitwise and
+ − 855 This function returns the ``logical and'' of the arguments: the
+ − 856 @var{n}th bit is set in the result if, and only if, the @var{n}th bit is
+ − 857 set in all the arguments. (``Set'' means that the value of the bit is 1
+ − 858 rather than 0.)
+ − 859
+ − 860 For example, using 4-bit binary numbers, the ``logical and'' of 13 and
+ − 861 12 is 12: 1101 combined with 1100 produces 1100.
+ − 862 In both the binary numbers, the leftmost two bits are set (i.e., they
+ − 863 are 1's), so the leftmost two bits of the returned value are set.
+ − 864 However, for the rightmost two bits, each is zero in at least one of
+ − 865 the arguments, so the rightmost two bits of the returned value are 0's.
+ − 866
+ − 867 @noindent
+ − 868 Therefore,
+ − 869
+ − 870 @example
+ − 871 @group
+ − 872 (logand 13 12)
+ − 873 @result{} 12
+ − 874 @end group
+ − 875 @end example
+ − 876
+ − 877 If @code{logand} is not passed any argument, it returns a value of
+ − 878 @minus{}1. This number is an identity element for @code{logand}
+ − 879 because its binary representation consists entirely of ones. If
+ − 880 @code{logand} is passed just one argument, it returns that argument.
+ − 881
+ − 882 @smallexample
+ − 883 @group
+ − 884 ; @r{ 28-bit binary values}
+ − 885
+ − 886 (logand 14 13) ; 14 = @r{0000 0000 0000 0000 0000 0000 1110}
+ − 887 ; 13 = @r{0000 0000 0000 0000 0000 0000 1101}
+ − 888 @result{} 12 ; 12 = @r{0000 0000 0000 0000 0000 0000 1100}
+ − 889 @end group
+ − 890
+ − 891 @group
+ − 892 (logand 14 13 4) ; 14 = @r{0000 0000 0000 0000 0000 0000 1110}
+ − 893 ; 13 = @r{0000 0000 0000 0000 0000 0000 1101}
+ − 894 ; 4 = @r{0000 0000 0000 0000 0000 0000 0100}
+ − 895 @result{} 4 ; 4 = @r{0000 0000 0000 0000 0000 0000 0100}
+ − 896 @end group
+ − 897
+ − 898 @group
+ − 899 (logand)
+ − 900 @result{} -1 ; -1 = @r{1111 1111 1111 1111 1111 1111 1111}
+ − 901 @end group
+ − 902 @end smallexample
+ − 903 @end defun
+ − 904
+ − 905 @defun logior &rest ints-or-markers
+ − 906 @cindex logical inclusive or
+ − 907 @cindex bitwise or
+ − 908 This function returns the ``inclusive or'' of its arguments: the @var{n}th bit
+ − 909 is set in the result if, and only if, the @var{n}th bit is set in at least
+ − 910 one of the arguments. If there are no arguments, the result is zero,
+ − 911 which is an identity element for this operation. If @code{logior} is
+ − 912 passed just one argument, it returns that argument.
+ − 913
+ − 914 @smallexample
+ − 915 @group
+ − 916 ; @r{ 28-bit binary values}
+ − 917
+ − 918 (logior 12 5) ; 12 = @r{0000 0000 0000 0000 0000 0000 1100}
+ − 919 ; 5 = @r{0000 0000 0000 0000 0000 0000 0101}
+ − 920 @result{} 13 ; 13 = @r{0000 0000 0000 0000 0000 0000 1101}
+ − 921 @end group
+ − 922
+ − 923 @group
+ − 924 (logior 12 5 7) ; 12 = @r{0000 0000 0000 0000 0000 0000 1100}
+ − 925 ; 5 = @r{0000 0000 0000 0000 0000 0000 0101}
+ − 926 ; 7 = @r{0000 0000 0000 0000 0000 0000 0111}
+ − 927 @result{} 15 ; 15 = @r{0000 0000 0000 0000 0000 0000 1111}
+ − 928 @end group
+ − 929 @end smallexample
+ − 930 @end defun
+ − 931
+ − 932 @defun logxor &rest ints-or-markers
+ − 933 @cindex bitwise exclusive or
+ − 934 @cindex logical exclusive or
+ − 935 This function returns the ``exclusive or'' of its arguments: the
+ − 936 @var{n}th bit is set in the result if, and only if, the @var{n}th bit is
+ − 937 set in an odd number of the arguments. If there are no arguments, the
+ − 938 result is 0, which is an identity element for this operation. If
+ − 939 @code{logxor} is passed just one argument, it returns that argument.
+ − 940
+ − 941 @smallexample
+ − 942 @group
+ − 943 ; @r{ 28-bit binary values}
+ − 944
+ − 945 (logxor 12 5) ; 12 = @r{0000 0000 0000 0000 0000 0000 1100}
+ − 946 ; 5 = @r{0000 0000 0000 0000 0000 0000 0101}
+ − 947 @result{} 9 ; 9 = @r{0000 0000 0000 0000 0000 0000 1001}
+ − 948 @end group
+ − 949
+ − 950 @group
+ − 951 (logxor 12 5 7) ; 12 = @r{0000 0000 0000 0000 0000 0000 1100}
+ − 952 ; 5 = @r{0000 0000 0000 0000 0000 0000 0101}
+ − 953 ; 7 = @r{0000 0000 0000 0000 0000 0000 0111}
+ − 954 @result{} 14 ; 14 = @r{0000 0000 0000 0000 0000 0000 1110}
+ − 955 @end group
+ − 956 @end smallexample
+ − 957 @end defun
+ − 958
+ − 959 @defun lognot integer
+ − 960 @cindex logical not
+ − 961 @cindex bitwise not
+ − 962 This function returns the logical complement of its argument: the @var{n}th
+ − 963 bit is one in the result if, and only if, the @var{n}th bit is zero in
+ − 964 @var{integer}, and vice-versa.
+ − 965
+ − 966 @example
444
+ − 967 (lognot 5)
428
+ − 968 @result{} -6
+ − 969 ;; 5 = @r{0000 0000 0000 0000 0000 0000 0101}
+ − 970 ;; @r{becomes}
+ − 971 ;; -6 = @r{1111 1111 1111 1111 1111 1111 1010}
+ − 972 @end example
+ − 973 @end defun
+ − 974
+ − 975 @node Math Functions
+ − 976 @section Standard Mathematical Functions
+ − 977 @cindex transcendental functions
+ − 978 @cindex mathematical functions
+ − 979
+ − 980 These mathematical functions are available if floating point is
+ − 981 supported (which is the normal state of affairs). They allow integers
+ − 982 as well as floating point numbers as arguments.
+ − 983
444
+ − 984 @defun sin number
+ − 985 @defunx cos number
+ − 986 @defunx tan number
428
+ − 987 These are the ordinary trigonometric functions, with argument measured
+ − 988 in radians.
+ − 989 @end defun
+ − 990
444
+ − 991 @defun asin number
+ − 992 The value of @code{(asin @var{number})} is a number between @minus{}pi/2
+ − 993 and pi/2 (inclusive) whose sine is @var{number}; if, however, @var{number}
428
+ − 994 is out of range (outside [-1, 1]), then the result is a NaN.
+ − 995 @end defun
+ − 996
444
+ − 997 @defun acos number
+ − 998 The value of @code{(acos @var{number})} is a number between 0 and pi
+ − 999 (inclusive) whose cosine is @var{number}; if, however, @var{number}
428
+ − 1000 is out of range (outside [-1, 1]), then the result is a NaN.
+ − 1001 @end defun
+ − 1002
444
+ − 1003 @defun atan number &optional number2
+ − 1004 The value of @code{(atan @var{number})} is a number between @minus{}pi/2
+ − 1005 and pi/2 (exclusive) whose tangent is @var{number}.
+ − 1006
+ − 1007 If optional argument @var{number2} is supplied, the function returns
+ − 1008 @code{atan2(@var{number},@var{number2})}.
428
+ − 1009 @end defun
+ − 1010
444
+ − 1011 @defun sinh number
+ − 1012 @defunx cosh number
+ − 1013 @defunx tanh number
428
+ − 1014 These are the ordinary hyperbolic trigonometric functions.
+ − 1015 @end defun
+ − 1016
444
+ − 1017 @defun asinh number
+ − 1018 @defunx acosh number
+ − 1019 @defunx atanh number
428
+ − 1020 These are the inverse hyperbolic trigonometric functions.
+ − 1021 @end defun
+ − 1022
444
+ − 1023 @defun exp number
428
+ − 1024 This is the exponential function; it returns @i{e} to the power
444
+ − 1025 @var{number}. @i{e} is a fundamental mathematical constant also called the
428
+ − 1026 base of natural logarithms.
+ − 1027 @end defun
+ − 1028
444
+ − 1029 @defun log number &optional base
+ − 1030 This function returns the logarithm of @var{number}, with base @var{base}.
+ − 1031 If you don't specify @var{base}, the base @var{e} is used. If @var{number}
428
+ − 1032 is negative, the result is a NaN.
+ − 1033 @end defun
+ − 1034
444
+ − 1035 @defun log10 number
+ − 1036 This function returns the logarithm of @var{number}, with base 10. If
+ − 1037 @var{number} is negative, the result is a NaN. @code{(log10 @var{x})}
428
+ − 1038 @equiv{} @code{(log @var{x} 10)}, at least approximately.
+ − 1039 @end defun
+ − 1040
+ − 1041 @defun expt x y
+ − 1042 This function returns @var{x} raised to power @var{y}. If both
+ − 1043 arguments are integers and @var{y} is positive, the result is an
+ − 1044 integer; in this case, it is truncated to fit the range of possible
+ − 1045 integer values.
+ − 1046 @end defun
+ − 1047
444
+ − 1048 @defun sqrt number
+ − 1049 This returns the square root of @var{number}. If @var{number} is negative,
428
+ − 1050 the value is a NaN.
+ − 1051 @end defun
+ − 1052
444
+ − 1053 @defun cube-root number
+ − 1054 This returns the cube root of @var{number}.
428
+ − 1055 @end defun
+ − 1056
+ − 1057 @node Random Numbers
+ − 1058 @section Random Numbers
+ − 1059 @cindex random numbers
+ − 1060
+ − 1061 A deterministic computer program cannot generate true random numbers.
+ − 1062 For most purposes, @dfn{pseudo-random numbers} suffice. A series of
+ − 1063 pseudo-random numbers is generated in a deterministic fashion. The
+ − 1064 numbers are not truly random, but they have certain properties that
+ − 1065 mimic a random series. For example, all possible values occur equally
+ − 1066 often in a pseudo-random series.
+ − 1067
+ − 1068 In XEmacs, pseudo-random numbers are generated from a ``seed'' number.
+ − 1069 Starting from any given seed, the @code{random} function always
+ − 1070 generates the same sequence of numbers. XEmacs always starts with the
+ − 1071 same seed value, so the sequence of values of @code{random} is actually
+ − 1072 the same in each XEmacs run! For example, in one operating system, the
+ − 1073 first call to @code{(random)} after you start XEmacs always returns
+ − 1074 -1457731, and the second one always returns -7692030. This
+ − 1075 repeatability is helpful for debugging.
+ − 1076
+ − 1077 If you want truly unpredictable random numbers, execute @code{(random
+ − 1078 t)}. This chooses a new seed based on the current time of day and on
+ − 1079 XEmacs's process @sc{id} number.
+ − 1080
+ − 1081 @defun random &optional limit
+ − 1082 This function returns a pseudo-random integer. Repeated calls return a
+ − 1083 series of pseudo-random integers.
+ − 1084
+ − 1085 If @var{limit} is a positive integer, the value is chosen to be
+ − 1086 nonnegative and less than @var{limit}.
+ − 1087
+ − 1088 If @var{limit} is @code{t}, it means to choose a new seed based on the
+ − 1089 current time of day and on XEmacs's process @sc{id} number.
+ − 1090 @c "XEmacs'" is incorrect usage!
+ − 1091
+ − 1092 On some machines, any integer representable in Lisp may be the result
+ − 1093 of @code{random}. On other machines, the result can never be larger
+ − 1094 than a certain maximum or less than a certain (negative) minimum.
+ − 1095 @end defun
