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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 |
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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. | |
33 | * Transcendental Functions:: Trig, exponential and logarithmic functions. | |
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 | |
42 | range is @minus{}8388608 to 8388607 (24 bits; i.e., | |
43 | @ifinfo | |
44 | -2**23 | |
45 | @end ifinfo | |
46 | @tex | |
47 | $-2^{23}$ | |
48 | @end tex | |
49 | to | |
50 | @ifinfo | |
51 | 2**23 - 1) | |
52 | @end ifinfo | |
53 | @tex | |
54 | $2^{23}-1$) | |
55 | @end tex | |
56 | on most machines, but on others it is @minus{}16777216 to 16777215 (25 | |
57 | bits), or @minus{}33554432 to 33554431 (26 bits). Many examples in this | |
58 | chapter assume an integer has 24 bits. | |
59 | @cindex overflow | |
60 | ||
61 | The Lisp reader reads an integer as a sequence of digits with optional | |
62 | initial sign and optional final period. | |
63 | ||
64 | @example | |
65 | 1 ; @r{The integer 1.} | |
66 | 1. ; @r{The integer 1.} | |
67 | +1 ; @r{Also the integer 1.} | |
68 | -1 ; @r{The integer @minus{}1.} | |
69 | 16777217 ; @r{Also the integer 1, due to overflow.} | |
70 | 0 ; @r{The integer 0.} | |
71 | -0 ; @r{The integer 0.} | |
72 | @end example | |
73 | ||
74 | To understand how various functions work on integers, especially the | |
75 | bitwise operators (@pxref{Bitwise Operations}), it is often helpful to | |
76 | view the numbers in their binary form. | |
77 | ||
05fd2b65 | 78 | In 24-bit binary, the decimal integer 5 looks like this: |
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79 | |
80 | @example | |
81 | 0000 0000 0000 0000 0000 0101 | |
82 | @end example | |
83 | ||
84 | @noindent | |
85 | (We have inserted spaces between groups of 4 bits, and two spaces | |
86 | between groups of 8 bits, to make the binary integer easier to read.) | |
87 | ||
88 | The integer @minus{}1 looks like this: | |
89 | ||
90 | @example | |
91 | 1111 1111 1111 1111 1111 1111 | |
92 | @end example | |
93 | ||
94 | @noindent | |
95 | @cindex two's complement | |
96 | @minus{}1 is represented as 24 ones. (This is called @dfn{two's | |
97 | complement} notation.) | |
98 | ||
99 | The negative integer, @minus{}5, is creating by subtracting 4 from | |
100 | @minus{}1. In binary, the decimal integer 4 is 100. Consequently, | |
101 | @minus{}5 looks like this: | |
102 | ||
103 | @example | |
104 | 1111 1111 1111 1111 1111 1011 | |
105 | @end example | |
106 | ||
05fd2b65 | 107 | In this implementation, the largest 24-bit binary integer is the |
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108 | decimal integer 8,388,607. In binary, it looks like this: |
109 | ||
110 | @example | |
111 | 0111 1111 1111 1111 1111 1111 | |
112 | @end example | |
113 | ||
114 | Since the arithmetic functions do not check whether integers go | |
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115 | outside their range, when you add 1 to 8,388,607, the value is the |
116 | negative integer @minus{}8,388,608: | |
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117 | |
118 | @example | |
119 | (+ 1 8388607) | |
120 | @result{} -8388608 | |
121 | @result{} 1000 0000 0000 0000 0000 0000 | |
122 | @end example | |
123 | ||
124 | Many of the following functions accept markers for arguments as well | |
125 | as integers. (@xref{Markers}.) More precisely, the actual arguments to | |
126 | such functions may be either integers or markers, which is why we often | |
127 | give these arguments the name @var{int-or-marker}. When the argument | |
128 | value is a marker, its position value is used and its buffer is ignored. | |
129 | ||
130 | @ignore | |
131 | In version 19, except where @emph{integer} is specified as an | |
132 | argument, all of the functions for markers and integers also work for | |
133 | floating point numbers. | |
134 | @end ignore | |
135 | ||
136 | @node Float Basics | |
137 | @section Floating Point Basics | |
138 | ||
139 | @cindex @code{LISP_FLOAT_TYPE} configuration macro | |
140 | Emacs version 19 supports floating point numbers, if compiled with the | |
141 | macro @code{LISP_FLOAT_TYPE} defined. The precise range of floating | |
142 | point numbers is machine-specific; it is the same as the range of the C | |
143 | data type @code{double} on the machine in question. | |
144 | ||
145 | The printed representation for floating point numbers requires either | |
146 | a decimal point (with at least one digit following), an exponent, or | |
147 | both. For example, @samp{1500.0}, @samp{15e2}, @samp{15.0e2}, | |
148 | @samp{1.5e3}, and @samp{.15e4} are five ways of writing a floating point | |
149 | number whose value is 1500. They are all equivalent. You can also use | |
150 | a minus sign to write negative floating point numbers, as in | |
151 | @samp{-1.0}. | |
152 | ||
153 | @cindex IEEE floating point | |
154 | @cindex positive infinity | |
155 | @cindex negative infinity | |
156 | @cindex infinity | |
157 | @cindex NaN | |
158 | Most modern computers support the IEEE floating point standard, which | |
159 | provides for positive infinity and negative infinity as floating point | |
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160 | values. It also provides for a class of values called NaN or |
161 | ``not-a-number''; numerical functions return such values in cases where | |
162 | there is no correct answer. For example, @code{(sqrt -1.0)} returns a | |
163 | NaN. For practical purposes, there's no significant difference between | |
164 | different NaN values in Emacs Lisp, and there's no rule for precisely | |
165 | which NaN value should be used in a particular case, so this manual | |
166 | doesn't try to distinguish them. Emacs Lisp has no read syntax for NaNs | |
167 | or infinities; perhaps we should create a syntax in the future. | |
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168 | |
169 | You can use @code{logb} to extract the binary exponent of a floating | |
170 | point number (or estimate the logarithm of an integer): | |
171 | ||
172 | @defun logb number | |
173 | This function returns the binary exponent of @var{number}. More | |
174 | precisely, the value is the logarithm of @var{number} base 2, rounded | |
175 | down to an integer. | |
176 | @end defun | |
177 | ||
178 | @node Predicates on Numbers | |
179 | @section Type Predicates for Numbers | |
180 | ||
181 | The functions in this section test whether the argument is a number or | |
182 | whether it is a certain sort of number. The functions @code{integerp} | |
183 | and @code{floatp} can take any type of Lisp object as argument (the | |
184 | predicates would not be of much use otherwise); but the @code{zerop} | |
185 | predicate requires a number as its argument. See also | |
186 | @code{integer-or-marker-p} and @code{number-or-marker-p}, in | |
187 | @ref{Predicates on Markers}. | |
188 | ||
189 | @defun floatp object | |
190 | This predicate tests whether its argument is a floating point | |
191 | number and returns @code{t} if so, @code{nil} otherwise. | |
192 | ||
193 | @code{floatp} does not exist in Emacs versions 18 and earlier. | |
194 | @end defun | |
195 | ||
196 | @defun integerp object | |
197 | This predicate tests whether its argument is an integer, and returns | |
198 | @code{t} if so, @code{nil} otherwise. | |
199 | @end defun | |
200 | ||
201 | @defun numberp object | |
202 | This predicate tests whether its argument is a number (either integer or | |
203 | floating point), and returns @code{t} if so, @code{nil} otherwise. | |
204 | @end defun | |
205 | ||
05fd2b65 | 206 | @defun wholenump object |
e6512bcf | 207 | @cindex natural numbers |
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208 | The @code{wholenump} predicate (whose name comes from the phrase |
209 | ``whole-number-p'') tests to see whether its argument is a nonnegative | |
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210 | integer, and returns @code{t} if so, @code{nil} otherwise. 0 is |
211 | considered non-negative. | |
212 | ||
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213 | @findex natnump |
214 | @code{natnump} is an obsolete synonym for @code{wholenump}. | |
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215 | @end defun |
216 | ||
217 | @defun zerop number | |
218 | This predicate tests whether its argument is zero, and returns @code{t} | |
219 | if so, @code{nil} otherwise. The argument must be a number. | |
220 | ||
221 | These two forms are equivalent: @code{(zerop x)} @equiv{} @code{(= x 0)}. | |
222 | @end defun | |
223 | ||
224 | @node Comparison of Numbers | |
225 | @section Comparison of Numbers | |
226 | @cindex number equality | |
227 | ||
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228 | To test numbers for numerical equality, you should normally use |
229 | @code{=}, not @code{eq}. There can be many distinct floating point | |
230 | number objects with the same numeric value. If you use @code{eq} to | |
231 | compare them, then you test whether two values are the same | |
232 | @emph{object}. By contrast, @code{=} compares only the numeric values | |
233 | of the objects. | |
234 | ||
235 | At present, each integer value has a unique Lisp object in Emacs Lisp. | |
236 | Therefore, @code{eq} is equivalent @code{=} where integers are | |
237 | concerned. It is sometimes convenient to use @code{eq} for comparing an | |
238 | unknown value with an integer, because @code{eq} does not report an | |
239 | error if the unknown value is not a number---it accepts arguments of any | |
240 | type. By contrast, @code{=} signals an error if the arguments are not | |
241 | numbers or markers. However, it is a good idea to use @code{=} if you | |
242 | can, even for comparing integers, just in case we change the | |
243 | representation of integers in a future Emacs version. | |
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244 | |
245 | There is another wrinkle: because floating point arithmetic is not | |
246 | exact, it is often a bad idea to check for equality of two floating | |
247 | point values. Usually it is better to test for approximate equality. | |
248 | Here's a function to do this: | |
249 | ||
250 | @example | |
251 | (defvar fuzz-factor 1.0e-6) | |
252 | (defun approx-equal (x y) | |
253 | (< (/ (abs (- x y)) | |
254 | (max (abs x) (abs y))) | |
255 | fuzz-factor)) | |
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 | |
360 | nearest integer. | |
361 | @end defun | |
362 | ||
363 | @node Arithmetic Operations | |
364 | @section Arithmetic Operations | |
365 | ||
366 | Emacs Lisp provides the traditional four arithmetic operations: | |
367 | addition, subtraction, multiplication, and division. Remainder and modulus | |
368 | functions supplement the division functions. The functions to | |
369 | add or subtract 1 are provided because they are traditional in Lisp and | |
370 | commonly used. | |
371 | ||
372 | All of these functions except @code{%} return a floating point value | |
373 | if any argument is floating. | |
374 | ||
375 | It is important to note that in GNU Emacs Lisp, arithmetic functions | |
376 | do not check for overflow. Thus @code{(1+ 8388607)} may evaluate to | |
377 | @minus{}8388608, depending on your hardware. | |
378 | ||
379 | @defun 1+ number-or-marker | |
380 | This function returns @var{number-or-marker} plus 1. | |
381 | For example, | |
382 | ||
383 | @example | |
384 | (setq foo 4) | |
385 | @result{} 4 | |
386 | (1+ foo) | |
387 | @result{} 5 | |
388 | @end example | |
389 | ||
390 | This function is not analogous to the C operator @code{++}---it does | |
391 | not increment a variable. It just computes a sum. Thus, | |
392 | ||
393 | @example | |
394 | foo | |
395 | @result{} 4 | |
396 | @end example | |
397 | ||
398 | If you want to increment the variable, you must use @code{setq}, | |
399 | like this: | |
400 | ||
401 | @example | |
402 | (setq foo (1+ foo)) | |
403 | @result{} 5 | |
404 | @end example | |
405 | @end defun | |
406 | ||
407 | @defun 1- number-or-marker | |
408 | This function returns @var{number-or-marker} minus 1. | |
409 | @end defun | |
410 | ||
411 | @defun abs number | |
412 | This returns the absolute value of @var{number}. | |
413 | @end defun | |
414 | ||
415 | @defun + &rest numbers-or-markers | |
416 | This function adds its arguments together. When given no arguments, | |
417 | @code{+} returns 0. It does not check for overflow. | |
418 | ||
419 | @example | |
420 | (+) | |
421 | @result{} 0 | |
422 | (+ 1) | |
423 | @result{} 1 | |
424 | (+ 1 2 3 4) | |
425 | @result{} 10 | |
426 | @end example | |
427 | @end defun | |
428 | ||
429 | @defun - &optional number-or-marker &rest other-numbers-or-markers | |
430 | The @code{-} function serves two purposes: negation and subtraction. | |
431 | When @code{-} has a single argument, the value is the negative of the | |
432 | argument. When there are multiple arguments, @code{-} subtracts each of | |
433 | the @var{other-numbers-or-markers} from @var{number-or-marker}, | |
434 | cumulatively. If there are no arguments, the result is 0. This | |
435 | function does not check for overflow. | |
436 | ||
437 | @example | |
438 | (- 10 1 2 3 4) | |
439 | @result{} 0 | |
440 | (- 10) | |
441 | @result{} -10 | |
442 | (-) | |
443 | @result{} 0 | |
444 | @end example | |
445 | @end defun | |
446 | ||
447 | @defun * &rest numbers-or-markers | |
448 | This function multiplies its arguments together, and returns the | |
449 | product. When given no arguments, @code{*} returns 1. It does | |
450 | not check for overflow. | |
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 |
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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 | |
481 | (/ 6 2) | |
482 | @result{} 3 | |
483 | (/ 5 2) | |
484 | @result{} 2 | |
485 | (/ 25 3 2) | |
486 | @result{} 4 | |
487 | (/ -17 6) | |
488 | @result{} -2 | |
489 | @end example | |
490 | ||
491 | The result of @code{(/ -17 6)} could in principle be -3 on some | |
492 | machines. | |
493 | @end defun | |
494 | ||
495 | @defun % dividend divisor | |
496 | @cindex remainder | |
497 | This function returns the integer remainder after division of @var{dividend} | |
498 | by @var{divisor}. The arguments must be integers or markers. | |
499 | ||
500 | For negative arguments, the remainder is in principle machine-dependent | |
501 | since the quotient is; but in practice, all known machines behave alike. | |
502 | ||
503 | An @code{arith-error} results if @var{divisor} is 0. | |
504 | ||
505 | @example | |
506 | (% 9 4) | |
507 | @result{} 1 | |
508 | (% -9 4) | |
509 | @result{} -1 | |
510 | (% 9 -4) | |
511 | @result{} 1 | |
512 | (% -9 -4) | |
513 | @result{} -1 | |
514 | @end example | |
515 | ||
516 | For any two integers @var{dividend} and @var{divisor}, | |
517 | ||
518 | @example | |
519 | @group | |
520 | (+ (% @var{dividend} @var{divisor}) | |
521 | (* (/ @var{dividend} @var{divisor}) @var{divisor})) | |
522 | @end group | |
523 | @end example | |
524 | ||
525 | @noindent | |
526 | always equals @var{dividend}. | |
527 | @end defun | |
528 | ||
529 | @defun mod dividend divisor | |
530 | @cindex modulus | |
531 | This function returns the value of @var{dividend} modulo @var{divisor}; | |
532 | in other words, the remainder after division of @var{dividend} | |
533 | by @var{divisor}, but with the same sign as @var{divisor}. | |
534 | The arguments must be numbers or markers. | |
535 | ||
536 | Unlike @code{%}, @code{mod} returns a well-defined result for negative | |
537 | arguments. It also permits floating point arguments; it rounds the | |
538 | quotient downward (towards minus infinity) to an integer, and uses that | |
539 | quotient to compute the remainder. | |
540 | ||
541 | An @code{arith-error} results if @var{divisor} is 0. | |
542 | ||
543 | @example | |
544 | (mod 9 4) | |
545 | @result{} 1 | |
546 | (mod -9 4) | |
547 | @result{} 3 | |
548 | (mod 9 -4) | |
549 | @result{} -3 | |
550 | (mod -9 -4) | |
551 | @result{} -1 | |
552 | (mod 5.5 2.5) | |
553 | @result{} .5 | |
554 | @end example | |
555 | ||
556 | For any two numbers @var{dividend} and @var{divisor}, | |
557 | ||
558 | @example | |
559 | @group | |
560 | (+ (mod @var{dividend} @var{divisor}) | |
561 | (* (floor @var{dividend} @var{divisor}) @var{divisor})) | |
562 | @end group | |
563 | @end example | |
564 | ||
565 | @noindent | |
566 | always equals @var{dividend}, subject to rounding error if | |
567 | either argument is floating point. | |
568 | @end defun | |
569 | ||
570 | @node Rounding Operations | |
571 | @section Rounding Operations | |
572 | @cindex rounding without conversion | |
573 | ||
bb61c6c6 | 574 | The functions @code{ffloor}, @code{fceiling}, @code{fround} and |
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575 | @code{ftruncate} take a floating point argument and return a floating |
576 | point result whose value is a nearby integer. @code{ffloor} returns the | |
bb61c6c6 | 577 | nearest integer below; @code{fceiling}, the nearest integer above; |
05fd2b65 | 578 | @code{ftruncate}, the nearest integer in the direction towards zero; |
e6512bcf RS |
579 | @code{fround}, the nearest integer. |
580 | ||
581 | @defun ffloor float | |
582 | This function rounds @var{float} to the next lower integral value, and | |
583 | returns that value as a floating point number. | |
584 | @end defun | |
585 | ||
bb61c6c6 | 586 | @defun fceiling float |
e6512bcf RS |
587 | This function rounds @var{float} to the next higher integral value, and |
588 | returns that value as a floating point number. | |
589 | @end defun | |
590 | ||
05fd2b65 | 591 | @defun ftruncate float |
e6512bcf RS |
592 | This function rounds @var{float} towards zero to an integral value, and |
593 | returns that value as a floating point number. | |
594 | @end defun | |
595 | ||
596 | @defun fround float | |
597 | This function rounds @var{float} to the nearest integral value, | |
598 | and returns that value as a floating point number. | |
599 | @end defun | |
600 | ||
601 | @node Bitwise Operations | |
602 | @section Bitwise Operations on Integers | |
603 | ||
604 | In a computer, an integer is represented as a binary number, a | |
605 | sequence of @dfn{bits} (digits which are either zero or one). A bitwise | |
606 | operation acts on the individual bits of such a sequence. For example, | |
607 | @dfn{shifting} moves the whole sequence left or right one or more places, | |
608 | reproducing the same pattern ``moved over''. | |
609 | ||
610 | The bitwise operations in Emacs Lisp apply only to integers. | |
611 | ||
612 | @defun lsh integer1 count | |
613 | @cindex logical shift | |
614 | @code{lsh}, which is an abbreviation for @dfn{logical shift}, shifts the | |
05fd2b65 RS |
615 | bits in @var{integer1} to the left @var{count} places, or to the right |
616 | if @var{count} is negative, bringing zeros into the vacated bits. If | |
617 | @var{count} is negative, @code{lsh} shifts zeros into the leftmost | |
618 | (most-significant) bit, producing a positive result even if | |
619 | @var{integer1} is negative. Contrast this with @code{ash}, below. | |
620 | ||
621 | Here are two examples of @code{lsh}, shifting a pattern of bits one | |
622 | place to the left. We show only the low-order eight bits of the binary | |
623 | pattern; the rest are all zero. | |
e6512bcf RS |
624 | |
625 | @example | |
626 | @group | |
627 | (lsh 5 1) | |
628 | @result{} 10 | |
629 | ;; @r{Decimal 5 becomes decimal 10.} | |
630 | 00000101 @result{} 00001010 | |
631 | ||
632 | (lsh 7 1) | |
633 | @result{} 14 | |
634 | ;; @r{Decimal 7 becomes decimal 14.} | |
635 | 00000111 @result{} 00001110 | |
636 | @end group | |
637 | @end example | |
638 | ||
639 | @noindent | |
640 | As the examples illustrate, shifting the pattern of bits one place to | |
641 | the left produces a number that is twice the value of the previous | |
642 | number. | |
643 | ||
05fd2b65 RS |
644 | The function @code{lsh}, like all Emacs Lisp arithmetic functions, does |
645 | not check for overflow, so shifting left can discard significant bits | |
646 | and change the sign of the number. For example, left shifting 8,388,607 | |
647 | produces @minus{}2 on a 24-bit machine: | |
e6512bcf RS |
648 | |
649 | @example | |
650 | (lsh 8388607 1) ; @r{left shift} | |
651 | @result{} -2 | |
652 | @end example | |
653 | ||
05fd2b65 | 654 | In binary, in the 24-bit implementation, the argument looks like this: |
e6512bcf RS |
655 | |
656 | @example | |
657 | @group | |
658 | ;; @r{Decimal 8,388,607} | |
659 | 0111 1111 1111 1111 1111 1111 | |
660 | @end group | |
661 | @end example | |
662 | ||
663 | @noindent | |
664 | which becomes the following when left shifted: | |
665 | ||
666 | @example | |
667 | @group | |
668 | ;; @r{Decimal @minus{}2} | |
669 | 1111 1111 1111 1111 1111 1110 | |
670 | @end group | |
671 | @end example | |
672 | ||
673 | Shifting the pattern of bits two places to the left produces results | |
674 | like this (with 8-bit binary numbers): | |
675 | ||
676 | @example | |
677 | @group | |
678 | (lsh 3 2) | |
679 | @result{} 12 | |
680 | ;; @r{Decimal 3 becomes decimal 12.} | |
681 | 00000011 @result{} 00001100 | |
682 | @end group | |
683 | @end example | |
684 | ||
685 | On the other hand, shifting the pattern of bits one place to the right | |
686 | looks like this: | |
687 | ||
688 | @example | |
689 | @group | |
690 | (lsh 6 -1) | |
691 | @result{} 3 | |
692 | ;; @r{Decimal 6 becomes decimal 3.} | |
693 | 00000110 @result{} 00000011 | |
694 | @end group | |
695 | ||
696 | @group | |
697 | (lsh 5 -1) | |
698 | @result{} 2 | |
699 | ;; @r{Decimal 5 becomes decimal 2.} | |
700 | 00000101 @result{} 00000010 | |
701 | @end group | |
702 | @end example | |
703 | ||
704 | @noindent | |
705 | As the example illustrates, shifting the pattern of bits one place to | |
706 | the right divides the value of the binary number by two, rounding downward. | |
707 | @end defun | |
708 | ||
709 | @defun ash integer1 count | |
710 | @cindex arithmetic shift | |
711 | @code{ash} (@dfn{arithmetic shift}) shifts the bits in @var{integer1} | |
712 | to the left @var{count} places, or to the right if @var{count} | |
713 | is negative. | |
714 | ||
715 | @code{ash} gives the same results as @code{lsh} except when | |
716 | @var{integer1} and @var{count} are both negative. In that case, | |
717 | @code{ash} puts a one in the leftmost position, while @code{lsh} puts | |
718 | a zero in the leftmost position. | |
719 | ||
720 | Thus, with @code{ash}, shifting the pattern of bits one place to the right | |
721 | looks like this: | |
722 | ||
723 | @example | |
724 | @group | |
725 | (ash -6 -1) @result{} -3 | |
726 | ;; @r{Decimal @minus{}6 becomes decimal @minus{}3.} | |
727 | 1111 1111 1111 1111 1111 1010 | |
728 | @result{} | |
729 | 1111 1111 1111 1111 1111 1101 | |
730 | @end group | |
731 | @end example | |
732 | ||
733 | In contrast, shifting the pattern of bits one place to the right with | |
734 | @code{lsh} looks like this: | |
735 | ||
736 | @example | |
737 | @group | |
738 | (lsh -6 -1) @result{} 8388605 | |
739 | ;; @r{Decimal @minus{}6 becomes decimal 8,388,605.} | |
740 | 1111 1111 1111 1111 1111 1010 | |
741 | @result{} | |
742 | 0111 1111 1111 1111 1111 1101 | |
743 | @end group | |
744 | @end example | |
745 | ||
e6512bcf RS |
746 | Here are other examples: |
747 | ||
748 | @c !!! Check if lined up in smallbook format! XDVI shows problem | |
749 | @c with smallbook but not with regular book! --rjc 16mar92 | |
750 | @smallexample | |
751 | @group | |
752 | ; @r{ 24-bit binary values} | |
753 | ||
754 | (lsh 5 2) ; 5 = @r{0000 0000 0000 0000 0000 0101} | |
05fd2b65 | 755 | @result{} 20 ; = @r{0000 0000 0000 0000 0001 0100} |
e6512bcf RS |
756 | @end group |
757 | @group | |
758 | (ash 5 2) | |
759 | @result{} 20 | |
760 | (lsh -5 2) ; -5 = @r{1111 1111 1111 1111 1111 1011} | |
05fd2b65 | 761 | @result{} -20 ; = @r{1111 1111 1111 1111 1110 1100} |
e6512bcf RS |
762 | (ash -5 2) |
763 | @result{} -20 | |
764 | @end group | |
765 | @group | |
766 | (lsh 5 -2) ; 5 = @r{0000 0000 0000 0000 0000 0101} | |
05fd2b65 | 767 | @result{} 1 ; = @r{0000 0000 0000 0000 0000 0001} |
e6512bcf RS |
768 | @end group |
769 | @group | |
770 | (ash 5 -2) | |
771 | @result{} 1 | |
772 | @end group | |
773 | @group | |
774 | (lsh -5 -2) ; -5 = @r{1111 1111 1111 1111 1111 1011} | |
05fd2b65 | 775 | @result{} 4194302 ; = @r{0011 1111 1111 1111 1111 1110} |
e6512bcf RS |
776 | @end group |
777 | @group | |
778 | (ash -5 -2) ; -5 = @r{1111 1111 1111 1111 1111 1011} | |
05fd2b65 | 779 | @result{} -2 ; = @r{1111 1111 1111 1111 1111 1110} |
e6512bcf RS |
780 | @end group |
781 | @end smallexample | |
782 | @end defun | |
783 | ||
784 | @defun logand &rest ints-or-markers | |
785 | @cindex logical and | |
786 | @cindex bitwise and | |
787 | This function returns the ``logical and'' of the arguments: the | |
788 | @var{n}th bit is set in the result if, and only if, the @var{n}th bit is | |
789 | set in all the arguments. (``Set'' means that the value of the bit is 1 | |
790 | rather than 0.) | |
791 | ||
792 | For example, using 4-bit binary numbers, the ``logical and'' of 13 and | |
793 | 12 is 12: 1101 combined with 1100 produces 1100. | |
e6512bcf RS |
794 | In both the binary numbers, the leftmost two bits are set (i.e., they |
795 | are 1's), so the leftmost two bits of the returned value are set. | |
796 | However, for the rightmost two bits, each is zero in at least one of | |
797 | the arguments, so the rightmost two bits of the returned value are 0's. | |
798 | ||
799 | @noindent | |
800 | Therefore, | |
801 | ||
802 | @example | |
803 | @group | |
804 | (logand 13 12) | |
805 | @result{} 12 | |
806 | @end group | |
807 | @end example | |
808 | ||
809 | If @code{logand} is not passed any argument, it returns a value of | |
810 | @minus{}1. This number is an identity element for @code{logand} | |
811 | because its binary representation consists entirely of ones. If | |
812 | @code{logand} is passed just one argument, it returns that argument. | |
813 | ||
814 | @smallexample | |
815 | @group | |
816 | ; @r{ 24-bit binary values} | |
817 | ||
818 | (logand 14 13) ; 14 = @r{0000 0000 0000 0000 0000 1110} | |
819 | ; 13 = @r{0000 0000 0000 0000 0000 1101} | |
820 | @result{} 12 ; 12 = @r{0000 0000 0000 0000 0000 1100} | |
821 | @end group | |
822 | ||
823 | @group | |
824 | (logand 14 13 4) ; 14 = @r{0000 0000 0000 0000 0000 1110} | |
825 | ; 13 = @r{0000 0000 0000 0000 0000 1101} | |
826 | ; 4 = @r{0000 0000 0000 0000 0000 0100} | |
827 | @result{} 4 ; 4 = @r{0000 0000 0000 0000 0000 0100} | |
828 | @end group | |
829 | ||
830 | @group | |
831 | (logand) | |
832 | @result{} -1 ; -1 = @r{1111 1111 1111 1111 1111 1111} | |
833 | @end group | |
834 | @end smallexample | |
835 | @end defun | |
836 | ||
837 | @defun logior &rest ints-or-markers | |
838 | @cindex logical inclusive or | |
839 | @cindex bitwise or | |
840 | This function returns the ``inclusive or'' of its arguments: the @var{n}th bit | |
841 | is set in the result if, and only if, the @var{n}th bit is set in at least | |
842 | one of the arguments. If there are no arguments, the result is zero, | |
843 | which is an identity element for this operation. If @code{logior} is | |
844 | passed just one argument, it returns that argument. | |
845 | ||
846 | @smallexample | |
847 | @group | |
848 | ; @r{ 24-bit binary values} | |
849 | ||
850 | (logior 12 5) ; 12 = @r{0000 0000 0000 0000 0000 1100} | |
851 | ; 5 = @r{0000 0000 0000 0000 0000 0101} | |
852 | @result{} 13 ; 13 = @r{0000 0000 0000 0000 0000 1101} | |
853 | @end group | |
854 | ||
855 | @group | |
856 | (logior 12 5 7) ; 12 = @r{0000 0000 0000 0000 0000 1100} | |
857 | ; 5 = @r{0000 0000 0000 0000 0000 0101} | |
858 | ; 7 = @r{0000 0000 0000 0000 0000 0111} | |
859 | @result{} 15 ; 15 = @r{0000 0000 0000 0000 0000 1111} | |
860 | @end group | |
861 | @end smallexample | |
862 | @end defun | |
863 | ||
864 | @defun logxor &rest ints-or-markers | |
865 | @cindex bitwise exclusive or | |
866 | @cindex logical exclusive or | |
867 | This function returns the ``exclusive or'' of its arguments: the | |
05fd2b65 RS |
868 | @var{n}th bit is set in the result if, and only if, the @var{n}th bit is |
869 | set in an odd number of the arguments. If there are no arguments, the | |
870 | result is 0, which is an identity element for this operation. If | |
871 | @code{logxor} is passed just one argument, it returns that argument. | |
e6512bcf RS |
872 | |
873 | @smallexample | |
874 | @group | |
875 | ; @r{ 24-bit binary values} | |
876 | ||
877 | (logxor 12 5) ; 12 = @r{0000 0000 0000 0000 0000 1100} | |
878 | ; 5 = @r{0000 0000 0000 0000 0000 0101} | |
879 | @result{} 9 ; 9 = @r{0000 0000 0000 0000 0000 1001} | |
880 | @end group | |
881 | ||
882 | @group | |
883 | (logxor 12 5 7) ; 12 = @r{0000 0000 0000 0000 0000 1100} | |
884 | ; 5 = @r{0000 0000 0000 0000 0000 0101} | |
885 | ; 7 = @r{0000 0000 0000 0000 0000 0111} | |
886 | @result{} 14 ; 14 = @r{0000 0000 0000 0000 0000 1110} | |
887 | @end group | |
888 | @end smallexample | |
889 | @end defun | |
890 | ||
891 | @defun lognot integer | |
892 | @cindex logical not | |
893 | @cindex bitwise not | |
894 | This function returns the logical complement of its argument: the @var{n}th | |
895 | bit is one in the result if, and only if, the @var{n}th bit is zero in | |
896 | @var{integer}, and vice-versa. | |
897 | ||
898 | @example | |
899 | (lognot 5) | |
900 | @result{} -6 | |
901 | ;; 5 = @r{0000 0000 0000 0000 0000 0101} | |
902 | ;; @r{becomes} | |
903 | ;; -6 = @r{1111 1111 1111 1111 1111 1010} | |
904 | @end example | |
905 | @end defun | |
906 | ||
907 | @node Transcendental Functions | |
908 | @section Transcendental Functions | |
909 | @cindex transcendental functions | |
910 | @cindex mathematical functions | |
911 | ||
912 | These mathematical functions are available if floating point is | |
913 | supported. They allow integers as well as floating point numbers | |
914 | as arguments. | |
915 | ||
916 | @defun sin arg | |
917 | @defunx cos arg | |
918 | @defunx tan arg | |
919 | These are the ordinary trigonometric functions, with argument measured | |
920 | in radians. | |
921 | @end defun | |
922 | ||
923 | @defun asin arg | |
05fd2b65 RS |
924 | The value of @code{(asin @var{arg})} is a number between @minus{}pi/2 |
925 | and pi/2 (inclusive) whose sine is @var{arg}; if, however, @var{arg} | |
e6512bcf RS |
926 | is out of range (outside [-1, 1]), then the result is a NaN. |
927 | @end defun | |
928 | ||
929 | @defun acos arg | |
930 | The value of @code{(acos @var{arg})} is a number between 0 and pi | |
931 | (inclusive) whose cosine is @var{arg}; if, however, @var{arg} | |
932 | is out of range (outside [-1, 1]), then the result is a NaN. | |
933 | @end defun | |
934 | ||
935 | @defun atan arg | |
05fd2b65 RS |
936 | The value of @code{(atan @var{arg})} is a number between @minus{}pi/2 |
937 | and pi/2 (exclusive) whose tangent is @var{arg}. | |
e6512bcf RS |
938 | @end defun |
939 | ||
940 | @defun exp arg | |
941 | This is the exponential function; it returns @i{e} to the power | |
942 | @var{arg}. @i{e} is a fundamental mathematical constant also called the | |
943 | base of natural logarithms. | |
944 | @end defun | |
945 | ||
946 | @defun log arg &optional base | |
947 | This function returns the logarithm of @var{arg}, with base @var{base}. | |
948 | If you don't specify @var{base}, the base @var{e} is used. If @var{arg} | |
949 | is negative, the result is a NaN. | |
950 | @end defun | |
951 | ||
952 | @ignore | |
953 | @defun expm1 arg | |
954 | This function returns @code{(1- (exp @var{arg}))}, but it is more | |
955 | accurate than that when @var{arg} is negative and @code{(exp @var{arg})} | |
956 | is close to 1. | |
957 | @end defun | |
958 | ||
959 | @defun log1p arg | |
960 | This function returns @code{(log (1+ @var{arg}))}, but it is more | |
961 | accurate than that when @var{arg} is so small that adding 1 to it would | |
962 | lose accuracy. | |
963 | @end defun | |
964 | @end ignore | |
965 | ||
966 | @defun log10 arg | |
967 | This function returns the logarithm of @var{arg}, with base 10. If | |
05fd2b65 RS |
968 | @var{arg} is negative, the result is a NaN. @code{(log10 @var{x})} |
969 | @equiv{} @code{(log @var{x} 10)}, at least approximately. | |
e6512bcf RS |
970 | @end defun |
971 | ||
972 | @defun expt x y | |
973 | This function returns @var{x} raised to power @var{y}. | |
974 | @end defun | |
975 | ||
976 | @defun sqrt arg | |
977 | This returns the square root of @var{arg}. If @var{arg} is negative, | |
978 | the value is a NaN. | |
979 | @end defun | |
980 | ||
981 | @node Random Numbers | |
982 | @section Random Numbers | |
983 | @cindex random numbers | |
984 | ||
985 | A deterministic computer program cannot generate true random numbers. | |
986 | For most purposes, @dfn{pseudo-random numbers} suffice. A series of | |
987 | pseudo-random numbers is generated in a deterministic fashion. The | |
988 | numbers are not truly random, but they have certain properties that | |
989 | mimic a random series. For example, all possible values occur equally | |
990 | often in a pseudo-random series. | |
991 | ||
992 | In Emacs, pseudo-random numbers are generated from a ``seed'' number. | |
993 | Starting from any given seed, the @code{random} function always | |
994 | generates the same sequence of numbers. Emacs always starts with the | |
995 | same seed value, so the sequence of values of @code{random} is actually | |
996 | the same in each Emacs run! For example, in one operating system, the | |
997 | first call to @code{(random)} after you start Emacs always returns | |
998 | -1457731, and the second one always returns -7692030. This | |
999 | repeatability is helpful for debugging. | |
1000 | ||
1001 | If you want truly unpredictable random numbers, execute @code{(random | |
1002 | t)}. This chooses a new seed based on the current time of day and on | |
1003 | Emacs's process @sc{id} number. | |
1004 | ||
1005 | @defun random &optional limit | |
1006 | This function returns a pseudo-random integer. Repeated calls return a | |
1007 | series of pseudo-random integers. | |
1008 | ||
1009 | If @var{limit} is @code{nil}, then the value may in principle be any | |
1010 | integer. If @var{limit} is a positive integer, the value is chosen to | |
1011 | be nonnegative and less than @var{limit} (only in Emacs 19). | |
1012 | ||
1013 | If @var{limit} is @code{t}, it means to choose a new seed based on the | |
1014 | current time of day and on Emacs's process @sc{id} number. | |
1015 | @c "Emacs'" is incorrect usage! | |
1016 | ||
1017 | On some machines, any integer representable in Lisp may be the result | |
1018 | of @code{random}. On other machines, the result can never be larger | |
1019 | than a certain maximum or less than a certain (negative) minimum. | |
1020 | @end defun |