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1 | @c -*-texinfo-*- |
2 | @c This is part of the GNU Emacs Lisp Reference Manual. | |
3 | @c Copyright (C) 1990, 1991, 1992, 1993, 1994 Free Software Foundation, Inc. | |
4 | @c See the file elisp.texi for copying conditions. | |
5 | @setfilename ../info/numbers | |
05fd2b65 | 6 | @node Numbers, Strings and Characters, Lisp Data Types, Top |
e6512bcf RS |
7 | @chapter Numbers |
8 | @cindex integers | |
9 | @cindex numbers | |
10 | ||
11 | GNU Emacs supports two numeric data types: @dfn{integers} and | |
12 | @dfn{floating point numbers}. Integers are whole numbers such as | |
13 | @minus{}3, 0, 7, 13, and 511. Their values are exact. Floating point | |
14 | numbers are numbers with fractional parts, such as @minus{}4.5, 0.0, or | |
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15 | 2.71828. They can also be expressed in exponential notation: |
16 | 1.5e2 equals 150; in this example, @samp{e2} stands for ten to the | |
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17 | second power, and is multiplied by 1.5. Floating point values are not |
18 | exact; they have a fixed, limited amount of precision. | |
19 | ||
20 | Support for floating point numbers is a new feature in Emacs 19, and it | |
21 | is controlled by a separate compilation option, so you may encounter a site | |
22 | where Emacs does not support them. | |
23 | ||
24 | @menu | |
25 | * Integer Basics:: Representation and range of integers. | |
26 | * Float Basics:: Representation and range of floating point. | |
27 | * Predicates on Numbers:: Testing for numbers. | |
28 | * Comparison of Numbers:: Equality and inequality predicates. | |
29 | * Numeric Conversions:: Converting float to integer and vice versa. | |
30 | * Arithmetic Operations:: How to add, subtract, multiply and divide. | |
31 | * Rounding Operations:: Explicitly rounding floating point numbers. | |
32 | * Bitwise Operations:: Logical and, or, not, shifting. | |
4ac771a0 | 33 | * Math Functions:: Trig, exponential and logarithmic functions. |
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34 | * Random Numbers:: Obtaining random integers, predictable or not. |
35 | @end menu | |
36 | ||
37 | @node Integer Basics | |
38 | @comment node-name, next, previous, up | |
39 | @section Integer Basics | |
40 | ||
41 | The range of values for an integer depends on the machine. The | |
94b00316 | 42 | minimum range is @minus{}134217728 to 134217727 (28 bits; i.e., |
e6512bcf | 43 | @ifinfo |
8be5e0fc | 44 | -2**27 |
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45 | @end ifinfo |
46 | @tex | |
94b00316 | 47 | $-2^{27}$ |
e6512bcf RS |
48 | @end tex |
49 | to | |
50 | @ifinfo | |
94b00316 | 51 | 2**27 - 1), |
e6512bcf RS |
52 | @end ifinfo |
53 | @tex | |
94b00316 | 54 | $2^{27}-1$), |
e6512bcf | 55 | @end tex |
94b00316 RS |
56 | but some machines may provide a wider range. Many examples in this |
57 | chapter assume an integer has 28 bits. | |
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58 | @cindex overflow |
59 | ||
60 | The Lisp reader reads an integer as a sequence of digits with optional | |
61 | initial sign and optional final period. | |
62 | ||
63 | @example | |
64 | 1 ; @r{The integer 1.} | |
65 | 1. ; @r{The integer 1.} | |
66 | +1 ; @r{Also the integer 1.} | |
67 | -1 ; @r{The integer @minus{}1.} | |
8be5e0fc | 68 | 268435457 ; @r{Also the integer 1, due to overflow.} |
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69 | 0 ; @r{The integer 0.} |
70 | -0 ; @r{The integer 0.} | |
71 | @end example | |
72 | ||
73 | To understand how various functions work on integers, especially the | |
74 | bitwise operators (@pxref{Bitwise Operations}), it is often helpful to | |
75 | view the numbers in their binary form. | |
76 | ||
8be5e0fc | 77 | In 28-bit binary, the decimal integer 5 looks like this: |
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78 | |
79 | @example | |
8be5e0fc | 80 | 0000 0000 0000 0000 0000 0000 0101 |
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81 | @end example |
82 | ||
83 | @noindent | |
84 | (We have inserted spaces between groups of 4 bits, and two spaces | |
85 | between groups of 8 bits, to make the binary integer easier to read.) | |
86 | ||
87 | The integer @minus{}1 looks like this: | |
88 | ||
89 | @example | |
8be5e0fc | 90 | 1111 1111 1111 1111 1111 1111 1111 |
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91 | @end example |
92 | ||
93 | @noindent | |
94 | @cindex two's complement | |
8be5e0fc | 95 | @minus{}1 is represented as 28 ones. (This is called @dfn{two's |
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96 | complement} notation.) |
97 | ||
98 | The negative integer, @minus{}5, is creating by subtracting 4 from | |
99 | @minus{}1. In binary, the decimal integer 4 is 100. Consequently, | |
100 | @minus{}5 looks like this: | |
101 | ||
102 | @example | |
8be5e0fc | 103 | 1111 1111 1111 1111 1111 1111 1011 |
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104 | @end example |
105 | ||
94b00316 | 106 | In this implementation, the largest 28-bit binary integer is the |
8be5e0fc | 107 | decimal integer 134,217,727. In binary, it looks like this: |
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108 | |
109 | @example | |
8be5e0fc | 110 | 0111 1111 1111 1111 1111 1111 1111 |
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111 | @end example |
112 | ||
113 | Since the arithmetic functions do not check whether integers go | |
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114 | outside their range, when you add 1 to 134,217,727, the value is the |
115 | negative integer @minus{}134,217,728: | |
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116 | |
117 | @example | |
8be5e0fc RS |
118 | (+ 1 134217727) |
119 | @result{} -134217728 | |
120 | @result{} 1000 0000 0000 0000 0000 0000 0000 | |
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121 | @end example |
122 | ||
123 | Many of the following functions accept markers for arguments as well | |
124 | as integers. (@xref{Markers}.) More precisely, the actual arguments to | |
125 | such functions may be either integers or markers, which is why we often | |
126 | give these arguments the name @var{int-or-marker}. When the argument | |
127 | value is a marker, its position value is used and its buffer is ignored. | |
128 | ||
129 | @ignore | |
130 | In version 19, except where @emph{integer} is specified as an | |
131 | argument, all of the functions for markers and integers also work for | |
132 | floating point numbers. | |
133 | @end ignore | |
134 | ||
135 | @node Float Basics | |
136 | @section Floating Point Basics | |
137 | ||
138 | @cindex @code{LISP_FLOAT_TYPE} configuration macro | |
139 | Emacs version 19 supports floating point numbers, if compiled with the | |
140 | macro @code{LISP_FLOAT_TYPE} defined. The precise range of floating | |
141 | point numbers is machine-specific; it is the same as the range of the C | |
142 | data type @code{double} on the machine in question. | |
143 | ||
144 | The printed representation for floating point numbers requires either | |
145 | a decimal point (with at least one digit following), an exponent, or | |
146 | both. For example, @samp{1500.0}, @samp{15e2}, @samp{15.0e2}, | |
147 | @samp{1.5e3}, and @samp{.15e4} are five ways of writing a floating point | |
148 | number whose value is 1500. They are all equivalent. You can also use | |
149 | a minus sign to write negative floating point numbers, as in | |
150 | @samp{-1.0}. | |
151 | ||
152 | @cindex IEEE floating point | |
153 | @cindex positive infinity | |
154 | @cindex negative infinity | |
155 | @cindex infinity | |
156 | @cindex NaN | |
157 | Most modern computers support the IEEE floating point standard, which | |
158 | provides for positive infinity and negative infinity as floating point | |
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159 | values. It also provides for a class of values called NaN or |
160 | ``not-a-number''; numerical functions return such values in cases where | |
161 | there is no correct answer. For example, @code{(sqrt -1.0)} returns a | |
162 | NaN. For practical purposes, there's no significant difference between | |
163 | different NaN values in Emacs Lisp, and there's no rule for precisely | |
164 | which NaN value should be used in a particular case, so this manual | |
165 | doesn't try to distinguish them. Emacs Lisp has no read syntax for NaNs | |
166 | or infinities; perhaps we should create a syntax in the future. | |
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167 | |
168 | You can use @code{logb} to extract the binary exponent of a floating | |
169 | point number (or estimate the logarithm of an integer): | |
170 | ||
171 | @defun logb number | |
172 | This function returns the binary exponent of @var{number}. More | |
173 | precisely, the value is the logarithm of @var{number} base 2, rounded | |
174 | down to an integer. | |
175 | @end defun | |
176 | ||
177 | @node Predicates on Numbers | |
178 | @section Type Predicates for Numbers | |
179 | ||
180 | The functions in this section test whether the argument is a number or | |
181 | whether it is a certain sort of number. The functions @code{integerp} | |
182 | and @code{floatp} can take any type of Lisp object as argument (the | |
183 | predicates would not be of much use otherwise); but the @code{zerop} | |
184 | predicate requires a number as its argument. See also | |
185 | @code{integer-or-marker-p} and @code{number-or-marker-p}, in | |
186 | @ref{Predicates on Markers}. | |
187 | ||
188 | @defun floatp object | |
189 | This predicate tests whether its argument is a floating point | |
190 | number and returns @code{t} if so, @code{nil} otherwise. | |
191 | ||
192 | @code{floatp} does not exist in Emacs versions 18 and earlier. | |
193 | @end defun | |
194 | ||
195 | @defun integerp object | |
196 | This predicate tests whether its argument is an integer, and returns | |
197 | @code{t} if so, @code{nil} otherwise. | |
198 | @end defun | |
199 | ||
200 | @defun numberp object | |
201 | This predicate tests whether its argument is a number (either integer or | |
202 | floating point), and returns @code{t} if so, @code{nil} otherwise. | |
203 | @end defun | |
204 | ||
05fd2b65 | 205 | @defun wholenump object |
e6512bcf | 206 | @cindex natural numbers |
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207 | The @code{wholenump} predicate (whose name comes from the phrase |
208 | ``whole-number-p'') tests to see whether its argument is a nonnegative | |
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209 | integer, and returns @code{t} if so, @code{nil} otherwise. 0 is |
210 | considered non-negative. | |
211 | ||
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212 | @findex natnump |
213 | @code{natnump} is an obsolete synonym for @code{wholenump}. | |
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214 | @end defun |
215 | ||
216 | @defun zerop number | |
217 | This predicate tests whether its argument is zero, and returns @code{t} | |
218 | if so, @code{nil} otherwise. The argument must be a number. | |
219 | ||
220 | These two forms are equivalent: @code{(zerop x)} @equiv{} @code{(= x 0)}. | |
221 | @end defun | |
222 | ||
223 | @node Comparison of Numbers | |
224 | @section Comparison of Numbers | |
225 | @cindex number equality | |
226 | ||
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227 | To test numbers for numerical equality, you should normally use |
228 | @code{=}, not @code{eq}. There can be many distinct floating point | |
229 | number objects with the same numeric value. If you use @code{eq} to | |
230 | compare them, then you test whether two values are the same | |
231 | @emph{object}. By contrast, @code{=} compares only the numeric values | |
232 | of the objects. | |
233 | ||
234 | At present, each integer value has a unique Lisp object in Emacs Lisp. | |
235 | Therefore, @code{eq} is equivalent @code{=} where integers are | |
236 | concerned. It is sometimes convenient to use @code{eq} for comparing an | |
237 | unknown value with an integer, because @code{eq} does not report an | |
238 | error if the unknown value is not a number---it accepts arguments of any | |
239 | type. By contrast, @code{=} signals an error if the arguments are not | |
240 | numbers or markers. However, it is a good idea to use @code{=} if you | |
241 | can, even for comparing integers, just in case we change the | |
242 | representation of integers in a future Emacs version. | |
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243 | |
244 | There is another wrinkle: because floating point arithmetic is not | |
245 | exact, it is often a bad idea to check for equality of two floating | |
246 | point values. Usually it is better to test for approximate equality. | |
247 | Here's a function to do this: | |
248 | ||
249 | @example | |
250 | (defvar fuzz-factor 1.0e-6) | |
251 | (defun approx-equal (x y) | |
bfe721d1 KH |
252 | (or (and (= x 0) (= y 0)) |
253 | (< (/ (abs (- x y)) | |
254 | (max (abs x) (abs y))) | |
255 | fuzz-factor))) | |
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256 | @end example |
257 | ||
258 | @cindex CL note---integers vrs @code{eq} | |
259 | @quotation | |
05fd2b65 | 260 | @b{Common Lisp note:} Comparing numbers in Common Lisp always requires |
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261 | @code{=} because Common Lisp implements multi-word integers, and two |
262 | distinct integer objects can have the same numeric value. Emacs Lisp | |
263 | can have just one integer object for any given value because it has a | |
264 | limited range of integer values. | |
265 | @end quotation | |
266 | ||
267 | @defun = number-or-marker1 number-or-marker2 | |
268 | This function tests whether its arguments are numerically equal, and | |
269 | returns @code{t} if so, @code{nil} otherwise. | |
270 | @end defun | |
271 | ||
272 | @defun /= number-or-marker1 number-or-marker2 | |
273 | This function tests whether its arguments are numerically equal, and | |
274 | returns @code{t} if they are not, and @code{nil} if they are. | |
275 | @end defun | |
276 | ||
277 | @defun < number-or-marker1 number-or-marker2 | |
278 | This function tests whether its first argument is strictly less than | |
279 | its second argument. It returns @code{t} if so, @code{nil} otherwise. | |
280 | @end defun | |
281 | ||
282 | @defun <= number-or-marker1 number-or-marker2 | |
283 | This function tests whether its first argument is less than or equal | |
284 | to its second argument. It returns @code{t} if so, @code{nil} | |
285 | otherwise. | |
286 | @end defun | |
287 | ||
288 | @defun > number-or-marker1 number-or-marker2 | |
289 | This function tests whether its first argument is strictly greater | |
290 | than its second argument. It returns @code{t} if so, @code{nil} | |
291 | otherwise. | |
292 | @end defun | |
293 | ||
294 | @defun >= number-or-marker1 number-or-marker2 | |
295 | This function tests whether its first argument is greater than or | |
296 | equal to its second argument. It returns @code{t} if so, @code{nil} | |
297 | otherwise. | |
298 | @end defun | |
299 | ||
300 | @defun max number-or-marker &rest numbers-or-markers | |
301 | This function returns the largest of its arguments. | |
302 | ||
303 | @example | |
304 | (max 20) | |
305 | @result{} 20 | |
306 | (max 1 2.5) | |
307 | @result{} 2.5 | |
308 | (max 1 3 2.5) | |
309 | @result{} 3 | |
310 | @end example | |
311 | @end defun | |
312 | ||
313 | @defun min number-or-marker &rest numbers-or-markers | |
314 | This function returns the smallest of its arguments. | |
315 | ||
316 | @example | |
317 | (min -4 1) | |
318 | @result{} -4 | |
319 | @end example | |
320 | @end defun | |
321 | ||
322 | @node Numeric Conversions | |
323 | @section Numeric Conversions | |
324 | @cindex rounding in conversions | |
325 | ||
326 | To convert an integer to floating point, use the function @code{float}. | |
327 | ||
328 | @defun float number | |
329 | This returns @var{number} converted to floating point. | |
330 | If @var{number} is already a floating point number, @code{float} returns | |
331 | it unchanged. | |
332 | @end defun | |
333 | ||
334 | There are four functions to convert floating point numbers to integers; | |
335 | they differ in how they round. These functions accept integer arguments | |
336 | also, and return such arguments unchanged. | |
337 | ||
338 | @defun truncate number | |
339 | This returns @var{number}, converted to an integer by rounding towards | |
340 | zero. | |
341 | @end defun | |
342 | ||
343 | @defun floor number &optional divisor | |
344 | This returns @var{number}, converted to an integer by rounding downward | |
345 | (towards negative infinity). | |
346 | ||
347 | If @var{divisor} is specified, @var{number} is divided by @var{divisor} | |
348 | before the floor is taken; this is the division operation that | |
349 | corresponds to @code{mod}. An @code{arith-error} results if | |
350 | @var{divisor} is 0. | |
351 | @end defun | |
352 | ||
353 | @defun ceiling number | |
354 | This returns @var{number}, converted to an integer by rounding upward | |
355 | (towards positive infinity). | |
356 | @end defun | |
357 | ||
358 | @defun round number | |
359 | This returns @var{number}, converted to an integer by rounding towards the | |
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360 | nearest integer. Rounding a value equidistant between two integers |
361 | may choose the integer closer to zero, or it may prefer an even integer, | |
362 | depending on your machine. | |
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363 | @end defun |
364 | ||
365 | @node Arithmetic Operations | |
366 | @section Arithmetic Operations | |
367 | ||
368 | Emacs Lisp provides the traditional four arithmetic operations: | |
369 | addition, subtraction, multiplication, and division. Remainder and modulus | |
370 | functions supplement the division functions. The functions to | |
371 | add or subtract 1 are provided because they are traditional in Lisp and | |
372 | commonly used. | |
373 | ||
374 | All of these functions except @code{%} return a floating point value | |
375 | if any argument is floating. | |
376 | ||
377 | It is important to note that in GNU Emacs Lisp, arithmetic functions | |
22697dac KH |
378 | do not check for overflow. Thus @code{(1+ 134217727)} may evaluate to |
379 | @minus{}134217728, depending on your hardware. | |
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380 | |
381 | @defun 1+ number-or-marker | |
382 | This function returns @var{number-or-marker} plus 1. | |
383 | For example, | |
384 | ||
385 | @example | |
386 | (setq foo 4) | |
387 | @result{} 4 | |
388 | (1+ foo) | |
389 | @result{} 5 | |
390 | @end example | |
391 | ||
bfe721d1 KH |
392 | This function is not analogous to the C operator @code{++}---it does not |
393 | increment a variable. It just computes a sum. Thus, if we continue, | |
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394 | |
395 | @example | |
396 | foo | |
397 | @result{} 4 | |
398 | @end example | |
399 | ||
400 | If you want to increment the variable, you must use @code{setq}, | |
401 | like this: | |
402 | ||
403 | @example | |
404 | (setq foo (1+ foo)) | |
405 | @result{} 5 | |
406 | @end example | |
407 | @end defun | |
408 | ||
409 | @defun 1- number-or-marker | |
410 | This function returns @var{number-or-marker} minus 1. | |
411 | @end defun | |
412 | ||
413 | @defun abs number | |
414 | This returns the absolute value of @var{number}. | |
415 | @end defun | |
416 | ||
417 | @defun + &rest numbers-or-markers | |
418 | This function adds its arguments together. When given no arguments, | |
bfe721d1 | 419 | @code{+} returns 0. |
e6512bcf RS |
420 | |
421 | @example | |
422 | (+) | |
423 | @result{} 0 | |
424 | (+ 1) | |
425 | @result{} 1 | |
426 | (+ 1 2 3 4) | |
427 | @result{} 10 | |
428 | @end example | |
429 | @end defun | |
430 | ||
431 | @defun - &optional number-or-marker &rest other-numbers-or-markers | |
432 | The @code{-} function serves two purposes: negation and subtraction. | |
433 | When @code{-} has a single argument, the value is the negative of the | |
434 | argument. When there are multiple arguments, @code{-} subtracts each of | |
435 | the @var{other-numbers-or-markers} from @var{number-or-marker}, | |
bfe721d1 | 436 | cumulatively. If there are no arguments, the result is 0. |
e6512bcf RS |
437 | |
438 | @example | |
439 | (- 10 1 2 3 4) | |
440 | @result{} 0 | |
441 | (- 10) | |
442 | @result{} -10 | |
443 | (-) | |
444 | @result{} 0 | |
445 | @end example | |
446 | @end defun | |
447 | ||
448 | @defun * &rest numbers-or-markers | |
449 | This function multiplies its arguments together, and returns the | |
bfe721d1 | 450 | product. When given no arguments, @code{*} returns 1. |
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451 | |
452 | @example | |
453 | (*) | |
454 | @result{} 1 | |
455 | (* 1) | |
456 | @result{} 1 | |
457 | (* 1 2 3 4) | |
458 | @result{} 24 | |
459 | @end example | |
460 | @end defun | |
461 | ||
462 | @defun / dividend divisor &rest divisors | |
05fd2b65 | 463 | This function divides @var{dividend} by @var{divisor} and returns the |
e6512bcf RS |
464 | quotient. If there are additional arguments @var{divisors}, then it |
465 | divides @var{dividend} by each divisor in turn. Each argument may be a | |
466 | number or a marker. | |
467 | ||
468 | If all the arguments are integers, then the result is an integer too. | |
469 | This means the result has to be rounded. On most machines, the result | |
470 | is rounded towards zero after each division, but some machines may round | |
471 | differently with negative arguments. This is because the Lisp function | |
472 | @code{/} is implemented using the C division operator, which also | |
473 | permits machine-dependent rounding. As a practical matter, all known | |
474 | machines round in the standard fashion. | |
475 | ||
476 | @cindex @code{arith-error} in division | |
477 | If you divide by 0, an @code{arith-error} error is signaled. | |
478 | (@xref{Errors}.) | |
479 | ||
480 | @example | |
bda144f4 | 481 | @group |
e6512bcf RS |
482 | (/ 6 2) |
483 | @result{} 3 | |
bda144f4 | 484 | @end group |
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485 | (/ 5 2) |
486 | @result{} 2 | |
487 | (/ 25 3 2) | |
488 | @result{} 4 | |
489 | (/ -17 6) | |
490 | @result{} -2 | |
491 | @end example | |
492 | ||
493 | The result of @code{(/ -17 6)} could in principle be -3 on some | |
494 | machines. | |
495 | @end defun | |
496 | ||
497 | @defun % dividend divisor | |
498 | @cindex remainder | |
499 | This function returns the integer remainder after division of @var{dividend} | |
500 | by @var{divisor}. The arguments must be integers or markers. | |
501 | ||
502 | For negative arguments, the remainder is in principle machine-dependent | |
503 | since the quotient is; but in practice, all known machines behave alike. | |
504 | ||
505 | An @code{arith-error} results if @var{divisor} is 0. | |
506 | ||
507 | @example | |
508 | (% 9 4) | |
509 | @result{} 1 | |
510 | (% -9 4) | |
511 | @result{} -1 | |
512 | (% 9 -4) | |
513 | @result{} 1 | |
514 | (% -9 -4) | |
515 | @result{} -1 | |
516 | @end example | |
517 | ||
518 | For any two integers @var{dividend} and @var{divisor}, | |
519 | ||
520 | @example | |
521 | @group | |
522 | (+ (% @var{dividend} @var{divisor}) | |
523 | (* (/ @var{dividend} @var{divisor}) @var{divisor})) | |
524 | @end group | |
525 | @end example | |
526 | ||
527 | @noindent | |
528 | always equals @var{dividend}. | |
529 | @end defun | |
530 | ||
531 | @defun mod dividend divisor | |
532 | @cindex modulus | |
533 | This function returns the value of @var{dividend} modulo @var{divisor}; | |
534 | in other words, the remainder after division of @var{dividend} | |
535 | by @var{divisor}, but with the same sign as @var{divisor}. | |
536 | The arguments must be numbers or markers. | |
537 | ||
538 | Unlike @code{%}, @code{mod} returns a well-defined result for negative | |
539 | arguments. It also permits floating point arguments; it rounds the | |
540 | quotient downward (towards minus infinity) to an integer, and uses that | |
541 | quotient to compute the remainder. | |
542 | ||
543 | An @code{arith-error} results if @var{divisor} is 0. | |
544 | ||
545 | @example | |
bda144f4 | 546 | @group |
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547 | (mod 9 4) |
548 | @result{} 1 | |
bda144f4 MW |
549 | @end group |
550 | @group | |
e6512bcf RS |
551 | (mod -9 4) |
552 | @result{} 3 | |
bda144f4 MW |
553 | @end group |
554 | @group | |
e6512bcf RS |
555 | (mod 9 -4) |
556 | @result{} -3 | |
bda144f4 MW |
557 | @end group |
558 | @group | |
e6512bcf RS |
559 | (mod -9 -4) |
560 | @result{} -1 | |
bda144f4 MW |
561 | @end group |
562 | @group | |
e6512bcf RS |
563 | (mod 5.5 2.5) |
564 | @result{} .5 | |
bda144f4 | 565 | @end group |
e6512bcf RS |
566 | @end example |
567 | ||
568 | For any two numbers @var{dividend} and @var{divisor}, | |
569 | ||
570 | @example | |
571 | @group | |
572 | (+ (mod @var{dividend} @var{divisor}) | |
573 | (* (floor @var{dividend} @var{divisor}) @var{divisor})) | |
574 | @end group | |
575 | @end example | |
576 | ||
577 | @noindent | |
bfe721d1 KH |
578 | always equals @var{dividend}, subject to rounding error if either |
579 | argument is floating point. For @code{floor}, see @ref{Numeric | |
580 | Conversions}. | |
e6512bcf RS |
581 | @end defun |
582 | ||
583 | @node Rounding Operations | |
584 | @section Rounding Operations | |
585 | @cindex rounding without conversion | |
586 | ||
bb61c6c6 | 587 | The functions @code{ffloor}, @code{fceiling}, @code{fround} and |
e6512bcf RS |
588 | @code{ftruncate} take a floating point argument and return a floating |
589 | point result whose value is a nearby integer. @code{ffloor} returns the | |
bb61c6c6 | 590 | nearest integer below; @code{fceiling}, the nearest integer above; |
05fd2b65 | 591 | @code{ftruncate}, the nearest integer in the direction towards zero; |
e6512bcf RS |
592 | @code{fround}, the nearest integer. |
593 | ||
594 | @defun ffloor float | |
595 | This function rounds @var{float} to the next lower integral value, and | |
596 | returns that value as a floating point number. | |
597 | @end defun | |
598 | ||
bb61c6c6 | 599 | @defun fceiling float |
e6512bcf RS |
600 | This function rounds @var{float} to the next higher integral value, and |
601 | returns that value as a floating point number. | |
602 | @end defun | |
603 | ||
05fd2b65 | 604 | @defun ftruncate float |
e6512bcf RS |
605 | This function rounds @var{float} towards zero to an integral value, and |
606 | returns that value as a floating point number. | |
607 | @end defun | |
608 | ||
609 | @defun fround float | |
610 | This function rounds @var{float} to the nearest integral value, | |
611 | and returns that value as a floating point number. | |
612 | @end defun | |
613 | ||
614 | @node Bitwise Operations | |
615 | @section Bitwise Operations on Integers | |
616 | ||
617 | In a computer, an integer is represented as a binary number, a | |
618 | sequence of @dfn{bits} (digits which are either zero or one). A bitwise | |
619 | operation acts on the individual bits of such a sequence. For example, | |
620 | @dfn{shifting} moves the whole sequence left or right one or more places, | |
621 | reproducing the same pattern ``moved over''. | |
622 | ||
623 | The bitwise operations in Emacs Lisp apply only to integers. | |
624 | ||
625 | @defun lsh integer1 count | |
626 | @cindex logical shift | |
627 | @code{lsh}, which is an abbreviation for @dfn{logical shift}, shifts the | |
05fd2b65 RS |
628 | bits in @var{integer1} to the left @var{count} places, or to the right |
629 | if @var{count} is negative, bringing zeros into the vacated bits. If | |
630 | @var{count} is negative, @code{lsh} shifts zeros into the leftmost | |
631 | (most-significant) bit, producing a positive result even if | |
632 | @var{integer1} is negative. Contrast this with @code{ash}, below. | |
633 | ||
634 | Here are two examples of @code{lsh}, shifting a pattern of bits one | |
635 | place to the left. We show only the low-order eight bits of the binary | |
636 | pattern; the rest are all zero. | |
e6512bcf RS |
637 | |
638 | @example | |
639 | @group | |
640 | (lsh 5 1) | |
641 | @result{} 10 | |
642 | ;; @r{Decimal 5 becomes decimal 10.} | |
643 | 00000101 @result{} 00001010 | |
644 | ||
645 | (lsh 7 1) | |
646 | @result{} 14 | |
647 | ;; @r{Decimal 7 becomes decimal 14.} | |
648 | 00000111 @result{} 00001110 | |
649 | @end group | |
650 | @end example | |
651 | ||
652 | @noindent | |
653 | As the examples illustrate, shifting the pattern of bits one place to | |
654 | the left produces a number that is twice the value of the previous | |
655 | number. | |
656 | ||
bfe721d1 | 657 | Shifting a pattern of bits two places to the left produces results |
e6512bcf RS |
658 | like this (with 8-bit binary numbers): |
659 | ||
660 | @example | |
661 | @group | |
662 | (lsh 3 2) | |
663 | @result{} 12 | |
664 | ;; @r{Decimal 3 becomes decimal 12.} | |
665 | 00000011 @result{} 00001100 | |
666 | @end group | |
667 | @end example | |
668 | ||
bfe721d1 | 669 | On the other hand, shifting one place to the right looks like this: |
e6512bcf RS |
670 | |
671 | @example | |
672 | @group | |
673 | (lsh 6 -1) | |
674 | @result{} 3 | |
675 | ;; @r{Decimal 6 becomes decimal 3.} | |
676 | 00000110 @result{} 00000011 | |
677 | @end group | |
678 | ||
679 | @group | |
680 | (lsh 5 -1) | |
681 | @result{} 2 | |
682 | ;; @r{Decimal 5 becomes decimal 2.} | |
683 | 00000101 @result{} 00000010 | |
684 | @end group | |
685 | @end example | |
686 | ||
687 | @noindent | |
bfe721d1 KH |
688 | As the example illustrates, shifting one place to the right divides the |
689 | value of a positive integer by two, rounding downward. | |
690 | ||
691 | The function @code{lsh}, like all Emacs Lisp arithmetic functions, does | |
692 | not check for overflow, so shifting left can discard significant bits | |
693 | and change the sign of the number. For example, left shifting | |
694 | 134,217,727 produces @minus{}2 on a 28-bit machine: | |
695 | ||
696 | @example | |
697 | (lsh 134217727 1) ; @r{left shift} | |
698 | @result{} -2 | |
699 | @end example | |
700 | ||
701 | In binary, in the 28-bit implementation, the argument looks like this: | |
702 | ||
703 | @example | |
704 | @group | |
cc8c51f1 | 705 | ;; @r{Decimal 134,217,727} |
bfe721d1 KH |
706 | 0111 1111 1111 1111 1111 1111 1111 |
707 | @end group | |
708 | @end example | |
709 | ||
710 | @noindent | |
711 | which becomes the following when left shifted: | |
712 | ||
713 | @example | |
714 | @group | |
715 | ;; @r{Decimal @minus{}2} | |
716 | 1111 1111 1111 1111 1111 1111 1110 | |
717 | @end group | |
718 | @end example | |
e6512bcf RS |
719 | @end defun |
720 | ||
721 | @defun ash integer1 count | |
722 | @cindex arithmetic shift | |
723 | @code{ash} (@dfn{arithmetic shift}) shifts the bits in @var{integer1} | |
724 | to the left @var{count} places, or to the right if @var{count} | |
725 | is negative. | |
726 | ||
727 | @code{ash} gives the same results as @code{lsh} except when | |
728 | @var{integer1} and @var{count} are both negative. In that case, | |
bfe721d1 KH |
729 | @code{ash} puts ones in the empty bit positions on the left, while |
730 | @code{lsh} puts zeros in those bit positions. | |
e6512bcf RS |
731 | |
732 | Thus, with @code{ash}, shifting the pattern of bits one place to the right | |
733 | looks like this: | |
734 | ||
735 | @example | |
736 | @group | |
737 | (ash -6 -1) @result{} -3 | |
738 | ;; @r{Decimal @minus{}6 becomes decimal @minus{}3.} | |
8be5e0fc | 739 | 1111 1111 1111 1111 1111 1111 1010 |
e6512bcf | 740 | @result{} |
8be5e0fc | 741 | 1111 1111 1111 1111 1111 1111 1101 |
e6512bcf RS |
742 | @end group |
743 | @end example | |
744 | ||
745 | In contrast, shifting the pattern of bits one place to the right with | |
746 | @code{lsh} looks like this: | |
747 | ||
748 | @example | |
749 | @group | |
8be5e0fc RS |
750 | (lsh -6 -1) @result{} 134217725 |
751 | ;; @r{Decimal @minus{}6 becomes decimal 134,217,725.} | |
752 | 1111 1111 1111 1111 1111 1111 1010 | |
e6512bcf | 753 | @result{} |
8be5e0fc | 754 | 0111 1111 1111 1111 1111 1111 1101 |
e6512bcf RS |
755 | @end group |
756 | @end example | |
757 | ||
e6512bcf RS |
758 | Here are other examples: |
759 | ||
760 | @c !!! Check if lined up in smallbook format! XDVI shows problem | |
761 | @c with smallbook but not with regular book! --rjc 16mar92 | |
762 | @smallexample | |
763 | @group | |
8be5e0fc | 764 | ; @r{ 28-bit binary values} |
e6512bcf | 765 | |
8be5e0fc RS |
766 | (lsh 5 2) ; 5 = @r{0000 0000 0000 0000 0000 0000 0101} |
767 | @result{} 20 ; = @r{0000 0000 0000 0000 0000 0001 0100} | |
e6512bcf RS |
768 | @end group |
769 | @group | |
770 | (ash 5 2) | |
771 | @result{} 20 | |
8be5e0fc RS |
772 | (lsh -5 2) ; -5 = @r{1111 1111 1111 1111 1111 1111 1011} |
773 | @result{} -20 ; = @r{1111 1111 1111 1111 1111 1110 1100} | |
e6512bcf RS |
774 | (ash -5 2) |
775 | @result{} -20 | |
776 | @end group | |
777 | @group | |
8be5e0fc RS |
778 | (lsh 5 -2) ; 5 = @r{0000 0000 0000 0000 0000 0000 0101} |
779 | @result{} 1 ; = @r{0000 0000 0000 0000 0000 0000 0001} | |
e6512bcf RS |
780 | @end group |
781 | @group | |
782 | (ash 5 -2) | |
783 | @result{} 1 | |
784 | @end group | |
785 | @group | |
8be5e0fc RS |
786 | (lsh -5 -2) ; -5 = @r{1111 1111 1111 1111 1111 1111 1011} |
787 | @result{} 4194302 ; = @r{0011 1111 1111 1111 1111 1111 1110} | |
e6512bcf RS |
788 | @end group |
789 | @group | |
8be5e0fc RS |
790 | (ash -5 -2) ; -5 = @r{1111 1111 1111 1111 1111 1111 1011} |
791 | @result{} -2 ; = @r{1111 1111 1111 1111 1111 1111 1110} | |
e6512bcf RS |
792 | @end group |
793 | @end smallexample | |
794 | @end defun | |
795 | ||
796 | @defun logand &rest ints-or-markers | |
797 | @cindex logical and | |
798 | @cindex bitwise and | |
799 | This function returns the ``logical and'' of the arguments: the | |
800 | @var{n}th bit is set in the result if, and only if, the @var{n}th bit is | |
801 | set in all the arguments. (``Set'' means that the value of the bit is 1 | |
802 | rather than 0.) | |
803 | ||
804 | For example, using 4-bit binary numbers, the ``logical and'' of 13 and | |
805 | 12 is 12: 1101 combined with 1100 produces 1100. | |
e6512bcf RS |
806 | In both the binary numbers, the leftmost two bits are set (i.e., they |
807 | are 1's), so the leftmost two bits of the returned value are set. | |
808 | However, for the rightmost two bits, each is zero in at least one of | |
809 | the arguments, so the rightmost two bits of the returned value are 0's. | |
810 | ||
811 | @noindent | |
812 | Therefore, | |
813 | ||
814 | @example | |
815 | @group | |
816 | (logand 13 12) | |
817 | @result{} 12 | |
818 | @end group | |
819 | @end example | |
820 | ||
821 | If @code{logand} is not passed any argument, it returns a value of | |
822 | @minus{}1. This number is an identity element for @code{logand} | |
823 | because its binary representation consists entirely of ones. If | |
824 | @code{logand} is passed just one argument, it returns that argument. | |
825 | ||
826 | @smallexample | |
827 | @group | |
8be5e0fc | 828 | ; @r{ 28-bit binary values} |
e6512bcf | 829 | |
8be5e0fc RS |
830 | (logand 14 13) ; 14 = @r{0000 0000 0000 0000 0000 0000 1110} |
831 | ; 13 = @r{0000 0000 0000 0000 0000 0000 1101} | |
832 | @result{} 12 ; 12 = @r{0000 0000 0000 0000 0000 0000 1100} | |
e6512bcf RS |
833 | @end group |
834 | ||
835 | @group | |
8be5e0fc RS |
836 | (logand 14 13 4) ; 14 = @r{0000 0000 0000 0000 0000 0000 1110} |
837 | ; 13 = @r{0000 0000 0000 0000 0000 0000 1101} | |
838 | ; 4 = @r{0000 0000 0000 0000 0000 0000 0100} | |
839 | @result{} 4 ; 4 = @r{0000 0000 0000 0000 0000 0000 0100} | |
e6512bcf RS |
840 | @end group |
841 | ||
842 | @group | |
843 | (logand) | |
8be5e0fc | 844 | @result{} -1 ; -1 = @r{1111 1111 1111 1111 1111 1111 1111} |
e6512bcf RS |
845 | @end group |
846 | @end smallexample | |
847 | @end defun | |
848 | ||
849 | @defun logior &rest ints-or-markers | |
850 | @cindex logical inclusive or | |
851 | @cindex bitwise or | |
852 | This function returns the ``inclusive or'' of its arguments: the @var{n}th bit | |
853 | is set in the result if, and only if, the @var{n}th bit is set in at least | |
854 | one of the arguments. If there are no arguments, the result is zero, | |
855 | which is an identity element for this operation. If @code{logior} is | |
856 | passed just one argument, it returns that argument. | |
857 | ||
858 | @smallexample | |
859 | @group | |
8be5e0fc | 860 | ; @r{ 28-bit binary values} |
e6512bcf | 861 | |
8be5e0fc RS |
862 | (logior 12 5) ; 12 = @r{0000 0000 0000 0000 0000 0000 1100} |
863 | ; 5 = @r{0000 0000 0000 0000 0000 0000 0101} | |
864 | @result{} 13 ; 13 = @r{0000 0000 0000 0000 0000 0000 1101} | |
e6512bcf RS |
865 | @end group |
866 | ||
867 | @group | |
8be5e0fc RS |
868 | (logior 12 5 7) ; 12 = @r{0000 0000 0000 0000 0000 0000 1100} |
869 | ; 5 = @r{0000 0000 0000 0000 0000 0000 0101} | |
870 | ; 7 = @r{0000 0000 0000 0000 0000 0000 0111} | |
871 | @result{} 15 ; 15 = @r{0000 0000 0000 0000 0000 0000 1111} | |
e6512bcf RS |
872 | @end group |
873 | @end smallexample | |
874 | @end defun | |
875 | ||
876 | @defun logxor &rest ints-or-markers | |
877 | @cindex bitwise exclusive or | |
878 | @cindex logical exclusive or | |
879 | This function returns the ``exclusive or'' of its arguments: the | |
05fd2b65 RS |
880 | @var{n}th bit is set in the result if, and only if, the @var{n}th bit is |
881 | set in an odd number of the arguments. If there are no arguments, the | |
882 | result is 0, which is an identity element for this operation. If | |
883 | @code{logxor} is passed just one argument, it returns that argument. | |
e6512bcf RS |
884 | |
885 | @smallexample | |
886 | @group | |
8be5e0fc | 887 | ; @r{ 28-bit binary values} |
e6512bcf | 888 | |
8be5e0fc RS |
889 | (logxor 12 5) ; 12 = @r{0000 0000 0000 0000 0000 0000 1100} |
890 | ; 5 = @r{0000 0000 0000 0000 0000 0000 0101} | |
891 | @result{} 9 ; 9 = @r{0000 0000 0000 0000 0000 0000 1001} | |
e6512bcf RS |
892 | @end group |
893 | ||
894 | @group | |
8be5e0fc RS |
895 | (logxor 12 5 7) ; 12 = @r{0000 0000 0000 0000 0000 0000 1100} |
896 | ; 5 = @r{0000 0000 0000 0000 0000 0000 0101} | |
897 | ; 7 = @r{0000 0000 0000 0000 0000 0000 0111} | |
898 | @result{} 14 ; 14 = @r{0000 0000 0000 0000 0000 0000 1110} | |
e6512bcf RS |
899 | @end group |
900 | @end smallexample | |
901 | @end defun | |
902 | ||
903 | @defun lognot integer | |
904 | @cindex logical not | |
905 | @cindex bitwise not | |
906 | This function returns the logical complement of its argument: the @var{n}th | |
907 | bit is one in the result if, and only if, the @var{n}th bit is zero in | |
908 | @var{integer}, and vice-versa. | |
909 | ||
910 | @example | |
911 | (lognot 5) | |
912 | @result{} -6 | |
8be5e0fc | 913 | ;; 5 = @r{0000 0000 0000 0000 0000 0000 0101} |
e6512bcf | 914 | ;; @r{becomes} |
8be5e0fc | 915 | ;; -6 = @r{1111 1111 1111 1111 1111 1111 1010} |
e6512bcf RS |
916 | @end example |
917 | @end defun | |
918 | ||
4ac771a0 RS |
919 | @node Math Functions |
920 | @section Standard Mathematical Functions | |
e6512bcf RS |
921 | @cindex transcendental functions |
922 | @cindex mathematical functions | |
923 | ||
924 | These mathematical functions are available if floating point is | |
925 | supported. They allow integers as well as floating point numbers | |
926 | as arguments. | |
927 | ||
928 | @defun sin arg | |
929 | @defunx cos arg | |
930 | @defunx tan arg | |
931 | These are the ordinary trigonometric functions, with argument measured | |
932 | in radians. | |
933 | @end defun | |
934 | ||
935 | @defun asin arg | |
05fd2b65 RS |
936 | The value of @code{(asin @var{arg})} is a number between @minus{}pi/2 |
937 | and pi/2 (inclusive) whose sine is @var{arg}; if, however, @var{arg} | |
e6512bcf RS |
938 | is out of range (outside [-1, 1]), then the result is a NaN. |
939 | @end defun | |
940 | ||
941 | @defun acos arg | |
942 | The value of @code{(acos @var{arg})} is a number between 0 and pi | |
943 | (inclusive) whose cosine is @var{arg}; if, however, @var{arg} | |
944 | is out of range (outside [-1, 1]), then the result is a NaN. | |
945 | @end defun | |
946 | ||
947 | @defun atan arg | |
05fd2b65 RS |
948 | The value of @code{(atan @var{arg})} is a number between @minus{}pi/2 |
949 | and pi/2 (exclusive) whose tangent is @var{arg}. | |
e6512bcf RS |
950 | @end defun |
951 | ||
952 | @defun exp arg | |
953 | This is the exponential function; it returns @i{e} to the power | |
954 | @var{arg}. @i{e} is a fundamental mathematical constant also called the | |
955 | base of natural logarithms. | |
956 | @end defun | |
957 | ||
958 | @defun log arg &optional base | |
959 | This function returns the logarithm of @var{arg}, with base @var{base}. | |
960 | If you don't specify @var{base}, the base @var{e} is used. If @var{arg} | |
961 | is negative, the result is a NaN. | |
962 | @end defun | |
963 | ||
964 | @ignore | |
965 | @defun expm1 arg | |
966 | This function returns @code{(1- (exp @var{arg}))}, but it is more | |
967 | accurate than that when @var{arg} is negative and @code{(exp @var{arg})} | |
968 | is close to 1. | |
969 | @end defun | |
970 | ||
971 | @defun log1p arg | |
972 | This function returns @code{(log (1+ @var{arg}))}, but it is more | |
973 | accurate than that when @var{arg} is so small that adding 1 to it would | |
974 | lose accuracy. | |
975 | @end defun | |
976 | @end ignore | |
977 | ||
978 | @defun log10 arg | |
979 | This function returns the logarithm of @var{arg}, with base 10. If | |
05fd2b65 RS |
980 | @var{arg} is negative, the result is a NaN. @code{(log10 @var{x})} |
981 | @equiv{} @code{(log @var{x} 10)}, at least approximately. | |
e6512bcf RS |
982 | @end defun |
983 | ||
984 | @defun expt x y | |
8be5e0fc RS |
985 | This function returns @var{x} raised to power @var{y}. If both |
986 | arguments are integers and @var{y} is positive, the result is an | |
987 | integer; in this case, it is truncated to fit the range of possible | |
988 | integer values. | |
e6512bcf RS |
989 | @end defun |
990 | ||
991 | @defun sqrt arg | |
992 | This returns the square root of @var{arg}. If @var{arg} is negative, | |
993 | the value is a NaN. | |
994 | @end defun | |
995 | ||
996 | @node Random Numbers | |
997 | @section Random Numbers | |
998 | @cindex random numbers | |
999 | ||
1000 | A deterministic computer program cannot generate true random numbers. | |
1001 | For most purposes, @dfn{pseudo-random numbers} suffice. A series of | |
1002 | pseudo-random numbers is generated in a deterministic fashion. The | |
1003 | numbers are not truly random, but they have certain properties that | |
1004 | mimic a random series. For example, all possible values occur equally | |
1005 | often in a pseudo-random series. | |
1006 | ||
1007 | In Emacs, pseudo-random numbers are generated from a ``seed'' number. | |
1008 | Starting from any given seed, the @code{random} function always | |
1009 | generates the same sequence of numbers. Emacs always starts with the | |
1010 | same seed value, so the sequence of values of @code{random} is actually | |
1011 | the same in each Emacs run! For example, in one operating system, the | |
1012 | first call to @code{(random)} after you start Emacs always returns | |
1013 | -1457731, and the second one always returns -7692030. This | |
1014 | repeatability is helpful for debugging. | |
1015 | ||
1016 | If you want truly unpredictable random numbers, execute @code{(random | |
1017 | t)}. This chooses a new seed based on the current time of day and on | |
1018 | Emacs's process @sc{id} number. | |
1019 | ||
1020 | @defun random &optional limit | |
1021 | This function returns a pseudo-random integer. Repeated calls return a | |
1022 | series of pseudo-random integers. | |
1023 | ||
22697dac | 1024 | If @var{limit} is a positive integer, the value is chosen to be |
bfe721d1 | 1025 | nonnegative and less than @var{limit}. |
e6512bcf RS |
1026 | |
1027 | If @var{limit} is @code{t}, it means to choose a new seed based on the | |
1028 | current time of day and on Emacs's process @sc{id} number. | |
1029 | @c "Emacs'" is incorrect usage! | |
1030 | ||
1031 | On some machines, any integer representable in Lisp may be the result | |
1032 | of @code{random}. On other machines, the result can never be larger | |
1033 | than a certain maximum or less than a certain (negative) minimum. | |
1034 | @end defun |