Commit | Line | Data |
---|---|---|
e6512bcf RS |
1 | @c -*-texinfo-*- |
2 | @c This is part of the GNU Emacs Lisp Reference Manual. | |
e1e44180 | 3 | @c Copyright (C) 1990, 1991, 1992, 1993, 1994, 1995, 1998, 1999, 2003 |
177c0ea7 | 4 | @c Free Software Foundation, Inc. |
e6512bcf RS |
5 | @c See the file elisp.texi for copying conditions. |
6 | @setfilename ../info/numbers | |
05fd2b65 | 7 | @node Numbers, Strings and Characters, Lisp Data Types, Top |
e6512bcf RS |
8 | @chapter Numbers |
9 | @cindex integers | |
10 | @cindex numbers | |
11 | ||
12 | GNU Emacs supports two numeric data types: @dfn{integers} and | |
13 | @dfn{floating point numbers}. Integers are whole numbers such as | |
14 | @minus{}3, 0, 7, 13, and 511. Their values are exact. Floating point | |
15 | numbers are numbers with fractional parts, such as @minus{}4.5, 0.0, or | |
969fe9b5 RS |
16 | 2.71828. They can also be expressed in exponential notation: 1.5e2 |
17 | equals 150; in this example, @samp{e2} stands for ten to the second | |
18 | power, and that is multiplied by 1.5. Floating point values are not | |
e6512bcf RS |
19 | exact; they have a fixed, limited amount of precision. |
20 | ||
e6512bcf RS |
21 | @menu |
22 | * Integer Basics:: Representation and range of integers. | |
23 | * Float Basics:: Representation and range of floating point. | |
24 | * Predicates on Numbers:: Testing for numbers. | |
25 | * Comparison of Numbers:: Equality and inequality predicates. | |
26 | * Numeric Conversions:: Converting float to integer and vice versa. | |
27 | * Arithmetic Operations:: How to add, subtract, multiply and divide. | |
28 | * Rounding Operations:: Explicitly rounding floating point numbers. | |
29 | * Bitwise Operations:: Logical and, or, not, shifting. | |
4ac771a0 | 30 | * Math Functions:: Trig, exponential and logarithmic functions. |
e6512bcf RS |
31 | * Random Numbers:: Obtaining random integers, predictable or not. |
32 | @end menu | |
33 | ||
34 | @node Integer Basics | |
35 | @comment node-name, next, previous, up | |
36 | @section Integer Basics | |
37 | ||
38 | The range of values for an integer depends on the machine. The | |
e1e44180 | 39 | minimum range is @minus{}268435456 to 268435455 (29 bits; i.e., |
37680279 | 40 | @ifnottex |
e1e44180 | 41 | -2**28 |
37680279 | 42 | @end ifnottex |
177c0ea7 | 43 | @tex |
e1e44180 | 44 | @math{-2^{28}} |
e6512bcf | 45 | @end tex |
177c0ea7 | 46 | to |
37680279 | 47 | @ifnottex |
e1e44180 | 48 | 2**28 - 1), |
37680279 | 49 | @end ifnottex |
177c0ea7 | 50 | @tex |
e1e44180 | 51 | @math{2^{28}-1}), |
e6512bcf | 52 | @end tex |
94b00316 | 53 | but some machines may provide a wider range. Many examples in this |
e1e44180 | 54 | chapter assume an integer has 29 bits. |
e6512bcf RS |
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.} | |
b98ace27 | 65 | 536870913 ; @r{Also the integer 1, due to overflow.} |
e6512bcf RS |
66 | 0 ; @r{The integer 0.} |
67 | -0 ; @r{The integer 0.} | |
68 | @end example | |
69 | ||
d9940cbf EZ |
70 | @cindex integers in specific radix |
71 | @cindex radix for reading an integer | |
72 | @cindex base for reading an integer | |
75c65b20 EZ |
73 | @cindex hex numbers |
74 | @cindex octal numbers | |
75 | @cindex reading numbers in hex, octal, and binary | |
4182ecfc RS |
76 | The syntax for integers in bases other than 10 uses @samp{#} |
77 | followed by a letter that specifies the radix: @samp{b} for binary, | |
78 | @samp{o} for octal, @samp{x} for hex, or @samp{@var{radix}r} to | |
79 | specify radix @var{radix}. Case is not significant for the letter | |
80 | that specifies the radix. Thus, @samp{#b@var{integer}} reads | |
81 | @var{integer} in binary, and @samp{#@var{radix}r@var{integer}} reads | |
82 | @var{integer} in radix @var{radix}. Allowed values of @var{radix} run | |
83 | from 2 to 36. For example: | |
84 | ||
85 | @example | |
86 | #b101100 @result{} 44 | |
87 | #o54 @result{} 44 | |
88 | #x2c @result{} 44 | |
89 | #24r1k @result{} 44 | |
90 | @end example | |
d9940cbf | 91 | |
e6512bcf RS |
92 | To understand how various functions work on integers, especially the |
93 | bitwise operators (@pxref{Bitwise Operations}), it is often helpful to | |
94 | view the numbers in their binary form. | |
95 | ||
e1e44180 | 96 | In 29-bit binary, the decimal integer 5 looks like this: |
e6512bcf RS |
97 | |
98 | @example | |
e1e44180 | 99 | 0 0000 0000 0000 0000 0000 0000 0101 |
e6512bcf RS |
100 | @end example |
101 | ||
102 | @noindent | |
103 | (We have inserted spaces between groups of 4 bits, and two spaces | |
104 | between groups of 8 bits, to make the binary integer easier to read.) | |
105 | ||
106 | The integer @minus{}1 looks like this: | |
107 | ||
108 | @example | |
e1e44180 | 109 | 1 1111 1111 1111 1111 1111 1111 1111 |
e6512bcf RS |
110 | @end example |
111 | ||
112 | @noindent | |
113 | @cindex two's complement | |
e1e44180 | 114 | @minus{}1 is represented as 29 ones. (This is called @dfn{two's |
e6512bcf RS |
115 | complement} notation.) |
116 | ||
117 | The negative integer, @minus{}5, is creating by subtracting 4 from | |
118 | @minus{}1. In binary, the decimal integer 4 is 100. Consequently, | |
119 | @minus{}5 looks like this: | |
120 | ||
121 | @example | |
e1e44180 | 122 | 1 1111 1111 1111 1111 1111 1111 1011 |
e6512bcf RS |
123 | @end example |
124 | ||
e1e44180 SM |
125 | In this implementation, the largest 29-bit binary integer value is |
126 | 268,435,455 in decimal. In binary, it looks like this: | |
e6512bcf RS |
127 | |
128 | @example | |
e1e44180 | 129 | 0 1111 1111 1111 1111 1111 1111 1111 |
e6512bcf RS |
130 | @end example |
131 | ||
132 | Since the arithmetic functions do not check whether integers go | |
e1e44180 SM |
133 | outside their range, when you add 1 to 268,435,455, the value is the |
134 | negative integer @minus{}268,435,456: | |
e6512bcf RS |
135 | |
136 | @example | |
e1e44180 SM |
137 | (+ 1 268435455) |
138 | @result{} -268435456 | |
139 | @result{} 1 0000 0000 0000 0000 0000 0000 0000 | |
e6512bcf RS |
140 | @end example |
141 | ||
f9f59935 RS |
142 | Many of the functions described in this chapter accept markers for |
143 | arguments in place of numbers. (@xref{Markers}.) Since the actual | |
144 | arguments to such functions may be either numbers or markers, we often | |
145 | give these arguments the name @var{number-or-marker}. When the argument | |
e6512bcf RS |
146 | value is a marker, its position value is used and its buffer is ignored. |
147 | ||
24fc2020 RS |
148 | @defvar most-positive-fixnum |
149 | The value of this variable is the largest integer that Emacs Lisp | |
150 | can handle. | |
151 | @end defvar | |
152 | ||
153 | @defvar most-negative-fixnum | |
154 | The value of this variable is the smallest integer that Emacs Lisp can | |
155 | handle. It is negative. | |
156 | @end defvar | |
157 | ||
e6512bcf RS |
158 | @node Float Basics |
159 | @section Floating Point Basics | |
160 | ||
f9f59935 RS |
161 | Floating point numbers are useful for representing numbers that are |
162 | not integral. The precise range of floating point numbers is | |
163 | machine-specific; it is the same as the range of the C data type | |
164 | @code{double} on the machine you are using. | |
e6512bcf | 165 | |
f9f59935 RS |
166 | The read-syntax for floating point numbers requires either a decimal |
167 | point (with at least one digit following), an exponent, or both. For | |
168 | example, @samp{1500.0}, @samp{15e2}, @samp{15.0e2}, @samp{1.5e3}, and | |
169 | @samp{.15e4} are five ways of writing a floating point number whose | |
170 | value is 1500. They are all equivalent. You can also use a minus sign | |
171 | to write negative floating point numbers, as in @samp{-1.0}. | |
e6512bcf | 172 | |
ad800164 | 173 | @cindex @acronym{IEEE} floating point |
e6512bcf RS |
174 | @cindex positive infinity |
175 | @cindex negative infinity | |
176 | @cindex infinity | |
177 | @cindex NaN | |
fa091c04 LT |
178 | Most modern computers support the @acronym{IEEE} floating point standard, |
179 | which provides for positive infinity and negative infinity as floating point | |
05fd2b65 RS |
180 | values. It also provides for a class of values called NaN or |
181 | ``not-a-number''; numerical functions return such values in cases where | |
903a1a91 | 182 | there is no correct answer. For example, @code{(/ 0.0 0.0)} returns a |
05fd2b65 RS |
183 | NaN. For practical purposes, there's no significant difference between |
184 | different NaN values in Emacs Lisp, and there's no rule for precisely | |
969fe9b5 | 185 | which NaN value should be used in a particular case, so Emacs Lisp |
f9f59935 | 186 | doesn't try to distinguish them. Here are the read syntaxes for |
969fe9b5 | 187 | these special floating point values: |
f9f59935 RS |
188 | |
189 | @table @asis | |
190 | @item positive infinity | |
191 | @samp{1.0e+INF} | |
192 | @item negative infinity | |
193 | @samp{-1.0e+INF} | |
194 | @item Not-a-number | |
195 | @samp{0.0e+NaN}. | |
196 | @end table | |
e6512bcf | 197 | |
969fe9b5 | 198 | In addition, the value @code{-0.0} is distinguishable from ordinary |
fa091c04 LT |
199 | zero in @acronym{IEEE} floating point (although @code{equal} and |
200 | @code{=} consider them equal values). | |
969fe9b5 | 201 | |
e6512bcf RS |
202 | You can use @code{logb} to extract the binary exponent of a floating |
203 | point number (or estimate the logarithm of an integer): | |
204 | ||
205 | @defun logb number | |
206 | This function returns the binary exponent of @var{number}. More | |
207 | precisely, the value is the logarithm of @var{number} base 2, rounded | |
208 | down to an integer. | |
f9f59935 RS |
209 | |
210 | @example | |
211 | (logb 10) | |
212 | @result{} 3 | |
213 | (logb 10.0e20) | |
214 | @result{} 69 | |
215 | @end example | |
e6512bcf RS |
216 | @end defun |
217 | ||
218 | @node Predicates on Numbers | |
219 | @section Type Predicates for Numbers | |
220 | ||
4182ecfc RS |
221 | The functions in this section test for numbers, or for a specific |
222 | type of number. The functions @code{integerp} and @code{floatp} can | |
223 | take any type of Lisp object as argument (they would not be of much | |
224 | use otherwise), but the @code{zerop} predicate requires a number as | |
225 | its argument. See also @code{integer-or-marker-p} and | |
226 | @code{number-or-marker-p}, in @ref{Predicates on Markers}. | |
e6512bcf RS |
227 | |
228 | @defun floatp object | |
229 | This predicate tests whether its argument is a floating point | |
230 | number and returns @code{t} if so, @code{nil} otherwise. | |
231 | ||
232 | @code{floatp} does not exist in Emacs versions 18 and earlier. | |
233 | @end defun | |
234 | ||
235 | @defun integerp object | |
236 | This predicate tests whether its argument is an integer, and returns | |
237 | @code{t} if so, @code{nil} otherwise. | |
238 | @end defun | |
239 | ||
240 | @defun numberp object | |
241 | This predicate tests whether its argument is a number (either integer or | |
242 | floating point), and returns @code{t} if so, @code{nil} otherwise. | |
243 | @end defun | |
244 | ||
05fd2b65 | 245 | @defun wholenump object |
e6512bcf | 246 | @cindex natural numbers |
05fd2b65 RS |
247 | The @code{wholenump} predicate (whose name comes from the phrase |
248 | ``whole-number-p'') tests to see whether its argument is a nonnegative | |
e6512bcf RS |
249 | integer, and returns @code{t} if so, @code{nil} otherwise. 0 is |
250 | considered non-negative. | |
251 | ||
05fd2b65 RS |
252 | @findex natnump |
253 | @code{natnump} is an obsolete synonym for @code{wholenump}. | |
e6512bcf RS |
254 | @end defun |
255 | ||
256 | @defun zerop number | |
257 | This predicate tests whether its argument is zero, and returns @code{t} | |
258 | if so, @code{nil} otherwise. The argument must be a number. | |
259 | ||
4182ecfc | 260 | @code{(zerop x)} is equivalent to @code{(= x 0)}. |
e6512bcf RS |
261 | @end defun |
262 | ||
263 | @node Comparison of Numbers | |
264 | @section Comparison of Numbers | |
265 | @cindex number equality | |
266 | ||
05fd2b65 RS |
267 | To test numbers for numerical equality, you should normally use |
268 | @code{=}, not @code{eq}. There can be many distinct floating point | |
269 | number objects with the same numeric value. If you use @code{eq} to | |
270 | compare them, then you test whether two values are the same | |
271 | @emph{object}. By contrast, @code{=} compares only the numeric values | |
272 | of the objects. | |
273 | ||
274 | At present, each integer value has a unique Lisp object in Emacs Lisp. | |
f9f59935 | 275 | Therefore, @code{eq} is equivalent to @code{=} where integers are |
05fd2b65 RS |
276 | concerned. It is sometimes convenient to use @code{eq} for comparing an |
277 | unknown value with an integer, because @code{eq} does not report an | |
278 | error if the unknown value is not a number---it accepts arguments of any | |
279 | type. By contrast, @code{=} signals an error if the arguments are not | |
280 | numbers or markers. However, it is a good idea to use @code{=} if you | |
281 | can, even for comparing integers, just in case we change the | |
282 | representation of integers in a future Emacs version. | |
e6512bcf | 283 | |
4182ecfc RS |
284 | Sometimes it is useful to compare numbers with @code{equal}; it |
285 | treats two numbers as equal if they have the same data type (both | |
286 | integers, or both floating point) and the same value. By contrast, | |
287 | @code{=} can treat an integer and a floating point number as equal. | |
288 | @xref{Equality Predicates}. | |
a9f0a989 | 289 | |
e6512bcf RS |
290 | There is another wrinkle: because floating point arithmetic is not |
291 | exact, it is often a bad idea to check for equality of two floating | |
292 | point values. Usually it is better to test for approximate equality. | |
293 | Here's a function to do this: | |
294 | ||
295 | @example | |
296 | (defvar fuzz-factor 1.0e-6) | |
297 | (defun approx-equal (x y) | |
bfe721d1 KH |
298 | (or (and (= x 0) (= y 0)) |
299 | (< (/ (abs (- x y)) | |
300 | (max (abs x) (abs y))) | |
301 | fuzz-factor))) | |
e6512bcf RS |
302 | @end example |
303 | ||
304 | @cindex CL note---integers vrs @code{eq} | |
305 | @quotation | |
05fd2b65 | 306 | @b{Common Lisp note:} Comparing numbers in Common Lisp always requires |
e6512bcf RS |
307 | @code{=} because Common Lisp implements multi-word integers, and two |
308 | distinct integer objects can have the same numeric value. Emacs Lisp | |
309 | can have just one integer object for any given value because it has a | |
310 | limited range of integer values. | |
311 | @end quotation | |
312 | ||
313 | @defun = number-or-marker1 number-or-marker2 | |
314 | This function tests whether its arguments are numerically equal, and | |
315 | returns @code{t} if so, @code{nil} otherwise. | |
316 | @end defun | |
317 | ||
5e859db9 | 318 | @defun eql value1 value2 |
4182ecfc RS |
319 | This function acts like @code{eq} except when both arguments are |
320 | numbers. It compares numbers by type and numberic value, so that | |
321 | @code{(eql 1.0 1)} returns @code{nil}, but @code{(eql 1.0 1.0)} and | |
322 | @code{(eql 1 1)} both return @code{t}. | |
5e859db9 RS |
323 | @end defun |
324 | ||
e6512bcf RS |
325 | @defun /= number-or-marker1 number-or-marker2 |
326 | This function tests whether its arguments are numerically equal, and | |
327 | returns @code{t} if they are not, and @code{nil} if they are. | |
328 | @end defun | |
329 | ||
330 | @defun < number-or-marker1 number-or-marker2 | |
331 | This function tests whether its first argument is strictly less than | |
332 | its second argument. It returns @code{t} if so, @code{nil} otherwise. | |
333 | @end defun | |
334 | ||
335 | @defun <= number-or-marker1 number-or-marker2 | |
336 | This function tests whether its first argument is less than or equal | |
337 | to its second argument. It returns @code{t} if so, @code{nil} | |
338 | otherwise. | |
339 | @end defun | |
340 | ||
341 | @defun > number-or-marker1 number-or-marker2 | |
342 | This function tests whether its first argument is strictly greater | |
343 | than its second argument. It returns @code{t} if so, @code{nil} | |
344 | otherwise. | |
345 | @end defun | |
346 | ||
347 | @defun >= number-or-marker1 number-or-marker2 | |
348 | This function tests whether its first argument is greater than or | |
349 | equal to its second argument. It returns @code{t} if so, @code{nil} | |
350 | otherwise. | |
351 | @end defun | |
352 | ||
353 | @defun max number-or-marker &rest numbers-or-markers | |
354 | This function returns the largest of its arguments. | |
4182ecfc | 355 | If any of the arguments is floating-point, the value is returned |
8241495d | 356 | as floating point, even if it was given as an integer. |
e6512bcf RS |
357 | |
358 | @example | |
359 | (max 20) | |
360 | @result{} 20 | |
361 | (max 1 2.5) | |
362 | @result{} 2.5 | |
363 | (max 1 3 2.5) | |
8241495d | 364 | @result{} 3.0 |
e6512bcf RS |
365 | @end example |
366 | @end defun | |
367 | ||
368 | @defun min number-or-marker &rest numbers-or-markers | |
369 | This function returns the smallest of its arguments. | |
4182ecfc | 370 | If any of the arguments is floating-point, the value is returned |
8241495d | 371 | as floating point, even if it was given as an integer. |
e6512bcf RS |
372 | |
373 | @example | |
374 | (min -4 1) | |
375 | @result{} -4 | |
376 | @end example | |
377 | @end defun | |
378 | ||
969fe9b5 | 379 | @defun abs number |
a9f0a989 | 380 | This function returns the absolute value of @var{number}. |
969fe9b5 RS |
381 | @end defun |
382 | ||
e6512bcf RS |
383 | @node Numeric Conversions |
384 | @section Numeric Conversions | |
385 | @cindex rounding in conversions | |
386 | ||
387 | To convert an integer to floating point, use the function @code{float}. | |
388 | ||
389 | @defun float number | |
390 | This returns @var{number} converted to floating point. | |
391 | If @var{number} is already a floating point number, @code{float} returns | |
392 | it unchanged. | |
393 | @end defun | |
394 | ||
395 | There are four functions to convert floating point numbers to integers; | |
fa091c04 LT |
396 | they differ in how they round. All accept an argument @var{number} |
397 | and an optional argument @var{divisor}. Both arguments may be | |
398 | integers or floating point numbers. @var{divisor} may also be | |
399 | @code{nil}. If @var{divisor} is @code{nil} or omitted, these | |
400 | functions convert @var{number} to an integer, or return it unchanged | |
401 | if it already is an integer. If @var{divisor} is non-@code{nil}, they | |
402 | divide @var{number} by @var{divisor} and convert the result to an | |
403 | integer. An @code{arith-error} results if @var{divisor} is 0. | |
404 | ||
405 | @defun truncate number &optional divisor | |
e6512bcf RS |
406 | This returns @var{number}, converted to an integer by rounding towards |
407 | zero. | |
b7a2fc9b RS |
408 | |
409 | @example | |
410 | (truncate 1.2) | |
411 | @result{} 1 | |
412 | (truncate 1.7) | |
413 | @result{} 1 | |
414 | (truncate -1.2) | |
415 | @result{} -1 | |
416 | (truncate -1.7) | |
417 | @result{} -1 | |
418 | @end example | |
e6512bcf RS |
419 | @end defun |
420 | ||
421 | @defun floor number &optional divisor | |
422 | This returns @var{number}, converted to an integer by rounding downward | |
423 | (towards negative infinity). | |
424 | ||
fa091c04 LT |
425 | If @var{divisor} is specified, this uses the kind of division |
426 | operation that corresponds to @code{mod}, rounding downward. | |
b7a2fc9b RS |
427 | |
428 | @example | |
429 | (floor 1.2) | |
430 | @result{} 1 | |
431 | (floor 1.7) | |
432 | @result{} 1 | |
433 | (floor -1.2) | |
434 | @result{} -2 | |
435 | (floor -1.7) | |
436 | @result{} -2 | |
437 | (floor 5.99 3) | |
438 | @result{} 1 | |
439 | @end example | |
e6512bcf RS |
440 | @end defun |
441 | ||
fa091c04 | 442 | @defun ceiling number &optional divisor |
e6512bcf RS |
443 | This returns @var{number}, converted to an integer by rounding upward |
444 | (towards positive infinity). | |
b7a2fc9b RS |
445 | |
446 | @example | |
447 | (ceiling 1.2) | |
448 | @result{} 2 | |
449 | (ceiling 1.7) | |
450 | @result{} 2 | |
451 | (ceiling -1.2) | |
452 | @result{} -1 | |
453 | (ceiling -1.7) | |
454 | @result{} -1 | |
455 | @end example | |
e6512bcf RS |
456 | @end defun |
457 | ||
fa091c04 | 458 | @defun round number &optional divisor |
e6512bcf | 459 | This returns @var{number}, converted to an integer by rounding towards the |
bfe721d1 KH |
460 | nearest integer. Rounding a value equidistant between two integers |
461 | may choose the integer closer to zero, or it may prefer an even integer, | |
462 | depending on your machine. | |
b7a2fc9b RS |
463 | |
464 | @example | |
465 | (round 1.2) | |
466 | @result{} 1 | |
467 | (round 1.7) | |
468 | @result{} 2 | |
469 | (round -1.2) | |
470 | @result{} -1 | |
471 | (round -1.7) | |
472 | @result{} -2 | |
473 | @end example | |
e6512bcf RS |
474 | @end defun |
475 | ||
476 | @node Arithmetic Operations | |
477 | @section Arithmetic Operations | |
478 | ||
479 | Emacs Lisp provides the traditional four arithmetic operations: | |
480 | addition, subtraction, multiplication, and division. Remainder and modulus | |
481 | functions supplement the division functions. The functions to | |
482 | add or subtract 1 are provided because they are traditional in Lisp and | |
483 | commonly used. | |
484 | ||
485 | All of these functions except @code{%} return a floating point value | |
486 | if any argument is floating. | |
487 | ||
969fe9b5 | 488 | It is important to note that in Emacs Lisp, arithmetic functions |
e1e44180 SM |
489 | do not check for overflow. Thus @code{(1+ 268435455)} may evaluate to |
490 | @minus{}268435456, depending on your hardware. | |
e6512bcf RS |
491 | |
492 | @defun 1+ number-or-marker | |
493 | This function returns @var{number-or-marker} plus 1. | |
494 | For example, | |
495 | ||
496 | @example | |
497 | (setq foo 4) | |
498 | @result{} 4 | |
499 | (1+ foo) | |
500 | @result{} 5 | |
501 | @end example | |
502 | ||
bfe721d1 KH |
503 | This function is not analogous to the C operator @code{++}---it does not |
504 | increment a variable. It just computes a sum. Thus, if we continue, | |
e6512bcf RS |
505 | |
506 | @example | |
507 | foo | |
508 | @result{} 4 | |
509 | @end example | |
510 | ||
511 | If you want to increment the variable, you must use @code{setq}, | |
512 | like this: | |
513 | ||
514 | @example | |
515 | (setq foo (1+ foo)) | |
516 | @result{} 5 | |
517 | @end example | |
518 | @end defun | |
519 | ||
520 | @defun 1- number-or-marker | |
521 | This function returns @var{number-or-marker} minus 1. | |
522 | @end defun | |
523 | ||
e6512bcf RS |
524 | @defun + &rest numbers-or-markers |
525 | This function adds its arguments together. When given no arguments, | |
bfe721d1 | 526 | @code{+} returns 0. |
e6512bcf RS |
527 | |
528 | @example | |
529 | (+) | |
530 | @result{} 0 | |
531 | (+ 1) | |
532 | @result{} 1 | |
533 | (+ 1 2 3 4) | |
534 | @result{} 10 | |
535 | @end example | |
536 | @end defun | |
537 | ||
f9f59935 | 538 | @defun - &optional number-or-marker &rest more-numbers-or-markers |
e6512bcf RS |
539 | The @code{-} function serves two purposes: negation and subtraction. |
540 | When @code{-} has a single argument, the value is the negative of the | |
541 | argument. When there are multiple arguments, @code{-} subtracts each of | |
f9f59935 | 542 | the @var{more-numbers-or-markers} from @var{number-or-marker}, |
bfe721d1 | 543 | cumulatively. If there are no arguments, the result is 0. |
e6512bcf RS |
544 | |
545 | @example | |
546 | (- 10 1 2 3 4) | |
547 | @result{} 0 | |
548 | (- 10) | |
549 | @result{} -10 | |
550 | (-) | |
551 | @result{} 0 | |
552 | @end example | |
553 | @end defun | |
554 | ||
555 | @defun * &rest numbers-or-markers | |
556 | This function multiplies its arguments together, and returns the | |
bfe721d1 | 557 | product. When given no arguments, @code{*} returns 1. |
e6512bcf RS |
558 | |
559 | @example | |
560 | (*) | |
561 | @result{} 1 | |
562 | (* 1) | |
563 | @result{} 1 | |
564 | (* 1 2 3 4) | |
565 | @result{} 24 | |
566 | @end example | |
567 | @end defun | |
568 | ||
569 | @defun / dividend divisor &rest divisors | |
05fd2b65 | 570 | This function divides @var{dividend} by @var{divisor} and returns the |
e6512bcf RS |
571 | quotient. If there are additional arguments @var{divisors}, then it |
572 | divides @var{dividend} by each divisor in turn. Each argument may be a | |
573 | number or a marker. | |
574 | ||
575 | If all the arguments are integers, then the result is an integer too. | |
576 | This means the result has to be rounded. On most machines, the result | |
577 | is rounded towards zero after each division, but some machines may round | |
578 | differently with negative arguments. This is because the Lisp function | |
579 | @code{/} is implemented using the C division operator, which also | |
580 | permits machine-dependent rounding. As a practical matter, all known | |
581 | machines round in the standard fashion. | |
582 | ||
583 | @cindex @code{arith-error} in division | |
969fe9b5 RS |
584 | If you divide an integer by 0, an @code{arith-error} error is signaled. |
585 | (@xref{Errors}.) Floating point division by zero returns either | |
ad800164 | 586 | infinity or a NaN if your machine supports @acronym{IEEE} floating point; |
969fe9b5 | 587 | otherwise, it signals an @code{arith-error} error. |
e6512bcf RS |
588 | |
589 | @example | |
bda144f4 | 590 | @group |
e6512bcf RS |
591 | (/ 6 2) |
592 | @result{} 3 | |
bda144f4 | 593 | @end group |
e6512bcf RS |
594 | (/ 5 2) |
595 | @result{} 2 | |
969fe9b5 RS |
596 | (/ 5.0 2) |
597 | @result{} 2.5 | |
598 | (/ 5 2.0) | |
599 | @result{} 2.5 | |
600 | (/ 5.0 2.0) | |
601 | @result{} 2.5 | |
e6512bcf RS |
602 | (/ 25 3 2) |
603 | @result{} 4 | |
604 | (/ -17 6) | |
605 | @result{} -2 | |
606 | @end example | |
607 | ||
608 | The result of @code{(/ -17 6)} could in principle be -3 on some | |
609 | machines. | |
610 | @end defun | |
611 | ||
612 | @defun % dividend divisor | |
613 | @cindex remainder | |
614 | This function returns the integer remainder after division of @var{dividend} | |
615 | by @var{divisor}. The arguments must be integers or markers. | |
616 | ||
617 | For negative arguments, the remainder is in principle machine-dependent | |
618 | since the quotient is; but in practice, all known machines behave alike. | |
619 | ||
620 | An @code{arith-error} results if @var{divisor} is 0. | |
621 | ||
622 | @example | |
623 | (% 9 4) | |
624 | @result{} 1 | |
625 | (% -9 4) | |
626 | @result{} -1 | |
627 | (% 9 -4) | |
628 | @result{} 1 | |
629 | (% -9 -4) | |
630 | @result{} -1 | |
631 | @end example | |
632 | ||
633 | For any two integers @var{dividend} and @var{divisor}, | |
634 | ||
635 | @example | |
636 | @group | |
637 | (+ (% @var{dividend} @var{divisor}) | |
638 | (* (/ @var{dividend} @var{divisor}) @var{divisor})) | |
639 | @end group | |
640 | @end example | |
641 | ||
642 | @noindent | |
643 | always equals @var{dividend}. | |
644 | @end defun | |
645 | ||
646 | @defun mod dividend divisor | |
647 | @cindex modulus | |
648 | This function returns the value of @var{dividend} modulo @var{divisor}; | |
649 | in other words, the remainder after division of @var{dividend} | |
650 | by @var{divisor}, but with the same sign as @var{divisor}. | |
651 | The arguments must be numbers or markers. | |
652 | ||
653 | Unlike @code{%}, @code{mod} returns a well-defined result for negative | |
654 | arguments. It also permits floating point arguments; it rounds the | |
655 | quotient downward (towards minus infinity) to an integer, and uses that | |
656 | quotient to compute the remainder. | |
657 | ||
658 | An @code{arith-error} results if @var{divisor} is 0. | |
659 | ||
660 | @example | |
bda144f4 | 661 | @group |
e6512bcf RS |
662 | (mod 9 4) |
663 | @result{} 1 | |
bda144f4 MW |
664 | @end group |
665 | @group | |
e6512bcf RS |
666 | (mod -9 4) |
667 | @result{} 3 | |
bda144f4 MW |
668 | @end group |
669 | @group | |
e6512bcf RS |
670 | (mod 9 -4) |
671 | @result{} -3 | |
bda144f4 MW |
672 | @end group |
673 | @group | |
e6512bcf RS |
674 | (mod -9 -4) |
675 | @result{} -1 | |
bda144f4 MW |
676 | @end group |
677 | @group | |
e6512bcf RS |
678 | (mod 5.5 2.5) |
679 | @result{} .5 | |
bda144f4 | 680 | @end group |
e6512bcf RS |
681 | @end example |
682 | ||
683 | For any two numbers @var{dividend} and @var{divisor}, | |
684 | ||
685 | @example | |
686 | @group | |
687 | (+ (mod @var{dividend} @var{divisor}) | |
688 | (* (floor @var{dividend} @var{divisor}) @var{divisor})) | |
689 | @end group | |
690 | @end example | |
691 | ||
692 | @noindent | |
bfe721d1 KH |
693 | always equals @var{dividend}, subject to rounding error if either |
694 | argument is floating point. For @code{floor}, see @ref{Numeric | |
695 | Conversions}. | |
e6512bcf RS |
696 | @end defun |
697 | ||
698 | @node Rounding Operations | |
699 | @section Rounding Operations | |
700 | @cindex rounding without conversion | |
701 | ||
a9f0a989 | 702 | The functions @code{ffloor}, @code{fceiling}, @code{fround}, and |
e6512bcf RS |
703 | @code{ftruncate} take a floating point argument and return a floating |
704 | point result whose value is a nearby integer. @code{ffloor} returns the | |
bb61c6c6 | 705 | nearest integer below; @code{fceiling}, the nearest integer above; |
05fd2b65 | 706 | @code{ftruncate}, the nearest integer in the direction towards zero; |
e6512bcf RS |
707 | @code{fround}, the nearest integer. |
708 | ||
709 | @defun ffloor float | |
710 | This function rounds @var{float} to the next lower integral value, and | |
711 | returns that value as a floating point number. | |
712 | @end defun | |
713 | ||
bb61c6c6 | 714 | @defun fceiling float |
e6512bcf RS |
715 | This function rounds @var{float} to the next higher integral value, and |
716 | returns that value as a floating point number. | |
717 | @end defun | |
718 | ||
05fd2b65 | 719 | @defun ftruncate float |
e6512bcf RS |
720 | This function rounds @var{float} towards zero to an integral value, and |
721 | returns that value as a floating point number. | |
722 | @end defun | |
723 | ||
724 | @defun fround float | |
725 | This function rounds @var{float} to the nearest integral value, | |
726 | and returns that value as a floating point number. | |
727 | @end defun | |
728 | ||
729 | @node Bitwise Operations | |
730 | @section Bitwise Operations on Integers | |
731 | ||
732 | In a computer, an integer is represented as a binary number, a | |
733 | sequence of @dfn{bits} (digits which are either zero or one). A bitwise | |
734 | operation acts on the individual bits of such a sequence. For example, | |
735 | @dfn{shifting} moves the whole sequence left or right one or more places, | |
736 | reproducing the same pattern ``moved over''. | |
737 | ||
738 | The bitwise operations in Emacs Lisp apply only to integers. | |
739 | ||
740 | @defun lsh integer1 count | |
741 | @cindex logical shift | |
742 | @code{lsh}, which is an abbreviation for @dfn{logical shift}, shifts the | |
05fd2b65 RS |
743 | bits in @var{integer1} to the left @var{count} places, or to the right |
744 | if @var{count} is negative, bringing zeros into the vacated bits. If | |
745 | @var{count} is negative, @code{lsh} shifts zeros into the leftmost | |
746 | (most-significant) bit, producing a positive result even if | |
747 | @var{integer1} is negative. Contrast this with @code{ash}, below. | |
748 | ||
749 | Here are two examples of @code{lsh}, shifting a pattern of bits one | |
750 | place to the left. We show only the low-order eight bits of the binary | |
751 | pattern; the rest are all zero. | |
e6512bcf RS |
752 | |
753 | @example | |
754 | @group | |
755 | (lsh 5 1) | |
756 | @result{} 10 | |
757 | ;; @r{Decimal 5 becomes decimal 10.} | |
758 | 00000101 @result{} 00001010 | |
759 | ||
760 | (lsh 7 1) | |
761 | @result{} 14 | |
762 | ;; @r{Decimal 7 becomes decimal 14.} | |
763 | 00000111 @result{} 00001110 | |
764 | @end group | |
765 | @end example | |
766 | ||
767 | @noindent | |
768 | As the examples illustrate, shifting the pattern of bits one place to | |
769 | the left produces a number that is twice the value of the previous | |
770 | number. | |
771 | ||
bfe721d1 | 772 | Shifting a pattern of bits two places to the left produces results |
e6512bcf RS |
773 | like this (with 8-bit binary numbers): |
774 | ||
775 | @example | |
776 | @group | |
777 | (lsh 3 2) | |
778 | @result{} 12 | |
779 | ;; @r{Decimal 3 becomes decimal 12.} | |
177c0ea7 | 780 | 00000011 @result{} 00001100 |
e6512bcf RS |
781 | @end group |
782 | @end example | |
783 | ||
bfe721d1 | 784 | On the other hand, shifting one place to the right looks like this: |
e6512bcf RS |
785 | |
786 | @example | |
787 | @group | |
788 | (lsh 6 -1) | |
789 | @result{} 3 | |
790 | ;; @r{Decimal 6 becomes decimal 3.} | |
177c0ea7 | 791 | 00000110 @result{} 00000011 |
e6512bcf RS |
792 | @end group |
793 | ||
794 | @group | |
795 | (lsh 5 -1) | |
796 | @result{} 2 | |
797 | ;; @r{Decimal 5 becomes decimal 2.} | |
177c0ea7 | 798 | 00000101 @result{} 00000010 |
e6512bcf RS |
799 | @end group |
800 | @end example | |
801 | ||
802 | @noindent | |
bfe721d1 KH |
803 | As the example illustrates, shifting one place to the right divides the |
804 | value of a positive integer by two, rounding downward. | |
805 | ||
806 | The function @code{lsh}, like all Emacs Lisp arithmetic functions, does | |
807 | not check for overflow, so shifting left can discard significant bits | |
808 | and change the sign of the number. For example, left shifting | |
e1e44180 | 809 | 268,435,455 produces @minus{}2 on a 29-bit machine: |
bfe721d1 KH |
810 | |
811 | @example | |
e1e44180 | 812 | (lsh 268435455 1) ; @r{left shift} |
bfe721d1 KH |
813 | @result{} -2 |
814 | @end example | |
815 | ||
e1e44180 | 816 | In binary, in the 29-bit implementation, the argument looks like this: |
bfe721d1 KH |
817 | |
818 | @example | |
819 | @group | |
e1e44180 SM |
820 | ;; @r{Decimal 268,435,455} |
821 | 0 1111 1111 1111 1111 1111 1111 1111 | |
bfe721d1 KH |
822 | @end group |
823 | @end example | |
824 | ||
825 | @noindent | |
826 | which becomes the following when left shifted: | |
827 | ||
828 | @example | |
829 | @group | |
830 | ;; @r{Decimal @minus{}2} | |
e1e44180 | 831 | 1 1111 1111 1111 1111 1111 1111 1110 |
bfe721d1 KH |
832 | @end group |
833 | @end example | |
e6512bcf RS |
834 | @end defun |
835 | ||
836 | @defun ash integer1 count | |
837 | @cindex arithmetic shift | |
838 | @code{ash} (@dfn{arithmetic shift}) shifts the bits in @var{integer1} | |
839 | to the left @var{count} places, or to the right if @var{count} | |
840 | is negative. | |
841 | ||
842 | @code{ash} gives the same results as @code{lsh} except when | |
843 | @var{integer1} and @var{count} are both negative. In that case, | |
bfe721d1 KH |
844 | @code{ash} puts ones in the empty bit positions on the left, while |
845 | @code{lsh} puts zeros in those bit positions. | |
e6512bcf RS |
846 | |
847 | Thus, with @code{ash}, shifting the pattern of bits one place to the right | |
848 | looks like this: | |
849 | ||
850 | @example | |
851 | @group | |
177c0ea7 | 852 | (ash -6 -1) @result{} -3 |
e6512bcf | 853 | ;; @r{Decimal @minus{}6 becomes decimal @minus{}3.} |
e1e44180 | 854 | 1 1111 1111 1111 1111 1111 1111 1010 |
177c0ea7 | 855 | @result{} |
e1e44180 | 856 | 1 1111 1111 1111 1111 1111 1111 1101 |
e6512bcf RS |
857 | @end group |
858 | @end example | |
859 | ||
860 | In contrast, shifting the pattern of bits one place to the right with | |
861 | @code{lsh} looks like this: | |
862 | ||
863 | @example | |
864 | @group | |
e1e44180 SM |
865 | (lsh -6 -1) @result{} 268435453 |
866 | ;; @r{Decimal @minus{}6 becomes decimal 268,435,453.} | |
867 | 1 1111 1111 1111 1111 1111 1111 1010 | |
177c0ea7 | 868 | @result{} |
e1e44180 | 869 | 0 1111 1111 1111 1111 1111 1111 1101 |
e6512bcf RS |
870 | @end group |
871 | @end example | |
872 | ||
e6512bcf RS |
873 | Here are other examples: |
874 | ||
875 | @c !!! Check if lined up in smallbook format! XDVI shows problem | |
876 | @c with smallbook but not with regular book! --rjc 16mar92 | |
877 | @smallexample | |
878 | @group | |
e1e44180 | 879 | ; @r{ 29-bit binary values} |
e6512bcf | 880 | |
e1e44180 SM |
881 | (lsh 5 2) ; 5 = @r{0 0000 0000 0000 0000 0000 0000 0101} |
882 | @result{} 20 ; = @r{0 0000 0000 0000 0000 0000 0001 0100} | |
e6512bcf RS |
883 | @end group |
884 | @group | |
885 | (ash 5 2) | |
886 | @result{} 20 | |
e1e44180 SM |
887 | (lsh -5 2) ; -5 = @r{1 1111 1111 1111 1111 1111 1111 1011} |
888 | @result{} -20 ; = @r{1 1111 1111 1111 1111 1111 1110 1100} | |
e6512bcf RS |
889 | (ash -5 2) |
890 | @result{} -20 | |
891 | @end group | |
892 | @group | |
e1e44180 SM |
893 | (lsh 5 -2) ; 5 = @r{0 0000 0000 0000 0000 0000 0000 0101} |
894 | @result{} 1 ; = @r{0 0000 0000 0000 0000 0000 0000 0001} | |
e6512bcf RS |
895 | @end group |
896 | @group | |
897 | (ash 5 -2) | |
898 | @result{} 1 | |
899 | @end group | |
900 | @group | |
e1e44180 SM |
901 | (lsh -5 -2) ; -5 = @r{1 1111 1111 1111 1111 1111 1111 1011} |
902 | @result{} 134217726 ; = @r{0 0111 1111 1111 1111 1111 1111 1110} | |
e6512bcf RS |
903 | @end group |
904 | @group | |
e1e44180 SM |
905 | (ash -5 -2) ; -5 = @r{1 1111 1111 1111 1111 1111 1111 1011} |
906 | @result{} -2 ; = @r{1 1111 1111 1111 1111 1111 1111 1110} | |
e6512bcf RS |
907 | @end group |
908 | @end smallexample | |
909 | @end defun | |
910 | ||
911 | @defun logand &rest ints-or-markers | |
912 | @cindex logical and | |
913 | @cindex bitwise and | |
914 | This function returns the ``logical and'' of the arguments: the | |
915 | @var{n}th bit is set in the result if, and only if, the @var{n}th bit is | |
916 | set in all the arguments. (``Set'' means that the value of the bit is 1 | |
917 | rather than 0.) | |
918 | ||
919 | For example, using 4-bit binary numbers, the ``logical and'' of 13 and | |
920 | 12 is 12: 1101 combined with 1100 produces 1100. | |
e6512bcf RS |
921 | In both the binary numbers, the leftmost two bits are set (i.e., they |
922 | are 1's), so the leftmost two bits of the returned value are set. | |
923 | However, for the rightmost two bits, each is zero in at least one of | |
924 | the arguments, so the rightmost two bits of the returned value are 0's. | |
925 | ||
926 | @noindent | |
927 | Therefore, | |
928 | ||
929 | @example | |
930 | @group | |
931 | (logand 13 12) | |
932 | @result{} 12 | |
933 | @end group | |
934 | @end example | |
935 | ||
936 | If @code{logand} is not passed any argument, it returns a value of | |
937 | @minus{}1. This number is an identity element for @code{logand} | |
938 | because its binary representation consists entirely of ones. If | |
939 | @code{logand} is passed just one argument, it returns that argument. | |
940 | ||
941 | @smallexample | |
942 | @group | |
e1e44180 | 943 | ; @r{ 29-bit binary values} |
e6512bcf | 944 | |
e1e44180 SM |
945 | (logand 14 13) ; 14 = @r{0 0000 0000 0000 0000 0000 0000 1110} |
946 | ; 13 = @r{0 0000 0000 0000 0000 0000 0000 1101} | |
947 | @result{} 12 ; 12 = @r{0 0000 0000 0000 0000 0000 0000 1100} | |
e6512bcf RS |
948 | @end group |
949 | ||
950 | @group | |
e1e44180 SM |
951 | (logand 14 13 4) ; 14 = @r{0 0000 0000 0000 0000 0000 0000 1110} |
952 | ; 13 = @r{0 0000 0000 0000 0000 0000 0000 1101} | |
953 | ; 4 = @r{0 0000 0000 0000 0000 0000 0000 0100} | |
954 | @result{} 4 ; 4 = @r{0 0000 0000 0000 0000 0000 0000 0100} | |
e6512bcf RS |
955 | @end group |
956 | ||
957 | @group | |
958 | (logand) | |
e1e44180 | 959 | @result{} -1 ; -1 = @r{1 1111 1111 1111 1111 1111 1111 1111} |
e6512bcf RS |
960 | @end group |
961 | @end smallexample | |
962 | @end defun | |
963 | ||
964 | @defun logior &rest ints-or-markers | |
965 | @cindex logical inclusive or | |
966 | @cindex bitwise or | |
967 | This function returns the ``inclusive or'' of its arguments: the @var{n}th bit | |
968 | is set in the result if, and only if, the @var{n}th bit is set in at least | |
969 | one of the arguments. If there are no arguments, the result is zero, | |
970 | which is an identity element for this operation. If @code{logior} is | |
971 | passed just one argument, it returns that argument. | |
972 | ||
973 | @smallexample | |
974 | @group | |
e1e44180 | 975 | ; @r{ 29-bit binary values} |
e6512bcf | 976 | |
e1e44180 SM |
977 | (logior 12 5) ; 12 = @r{0 0000 0000 0000 0000 0000 0000 1100} |
978 | ; 5 = @r{0 0000 0000 0000 0000 0000 0000 0101} | |
979 | @result{} 13 ; 13 = @r{0 0000 0000 0000 0000 0000 0000 1101} | |
e6512bcf RS |
980 | @end group |
981 | ||
982 | @group | |
e1e44180 SM |
983 | (logior 12 5 7) ; 12 = @r{0 0000 0000 0000 0000 0000 0000 1100} |
984 | ; 5 = @r{0 0000 0000 0000 0000 0000 0000 0101} | |
985 | ; 7 = @r{0 0000 0000 0000 0000 0000 0000 0111} | |
986 | @result{} 15 ; 15 = @r{0 0000 0000 0000 0000 0000 0000 1111} | |
e6512bcf RS |
987 | @end group |
988 | @end smallexample | |
989 | @end defun | |
990 | ||
991 | @defun logxor &rest ints-or-markers | |
992 | @cindex bitwise exclusive or | |
993 | @cindex logical exclusive or | |
994 | This function returns the ``exclusive or'' of its arguments: the | |
05fd2b65 RS |
995 | @var{n}th bit is set in the result if, and only if, the @var{n}th bit is |
996 | set in an odd number of the arguments. If there are no arguments, the | |
997 | result is 0, which is an identity element for this operation. If | |
998 | @code{logxor} is passed just one argument, it returns that argument. | |
e6512bcf RS |
999 | |
1000 | @smallexample | |
1001 | @group | |
e1e44180 | 1002 | ; @r{ 29-bit binary values} |
e6512bcf | 1003 | |
e1e44180 SM |
1004 | (logxor 12 5) ; 12 = @r{0 0000 0000 0000 0000 0000 0000 1100} |
1005 | ; 5 = @r{0 0000 0000 0000 0000 0000 0000 0101} | |
1006 | @result{} 9 ; 9 = @r{0 0000 0000 0000 0000 0000 0000 1001} | |
e6512bcf RS |
1007 | @end group |
1008 | ||
1009 | @group | |
e1e44180 SM |
1010 | (logxor 12 5 7) ; 12 = @r{0 0000 0000 0000 0000 0000 0000 1100} |
1011 | ; 5 = @r{0 0000 0000 0000 0000 0000 0000 0101} | |
1012 | ; 7 = @r{0 0000 0000 0000 0000 0000 0000 0111} | |
1013 | @result{} 14 ; 14 = @r{0 0000 0000 0000 0000 0000 0000 1110} | |
e6512bcf RS |
1014 | @end group |
1015 | @end smallexample | |
1016 | @end defun | |
1017 | ||
1018 | @defun lognot integer | |
1019 | @cindex logical not | |
1020 | @cindex bitwise not | |
1021 | This function returns the logical complement of its argument: the @var{n}th | |
1022 | bit is one in the result if, and only if, the @var{n}th bit is zero in | |
1023 | @var{integer}, and vice-versa. | |
1024 | ||
1025 | @example | |
177c0ea7 | 1026 | (lognot 5) |
e6512bcf | 1027 | @result{} -6 |
e1e44180 | 1028 | ;; 5 = @r{0 0000 0000 0000 0000 0000 0000 0101} |
e6512bcf | 1029 | ;; @r{becomes} |
e1e44180 | 1030 | ;; -6 = @r{1 1111 1111 1111 1111 1111 1111 1010} |
e6512bcf RS |
1031 | @end example |
1032 | @end defun | |
1033 | ||
4ac771a0 RS |
1034 | @node Math Functions |
1035 | @section Standard Mathematical Functions | |
e6512bcf RS |
1036 | @cindex transcendental functions |
1037 | @cindex mathematical functions | |
1038 | ||
969fe9b5 RS |
1039 | These mathematical functions allow integers as well as floating point |
1040 | numbers as arguments. | |
e6512bcf RS |
1041 | |
1042 | @defun sin arg | |
1043 | @defunx cos arg | |
1044 | @defunx tan arg | |
1045 | These are the ordinary trigonometric functions, with argument measured | |
1046 | in radians. | |
1047 | @end defun | |
1048 | ||
1049 | @defun asin arg | |
15da7853 | 1050 | The value of @code{(asin @var{arg})} is a number between |
37680279 | 1051 | @ifnottex |
15da7853 | 1052 | @minus{}pi/2 |
37680279 | 1053 | @end ifnottex |
15da7853 | 1054 | @tex |
8241495d | 1055 | @math{-\pi/2} |
15da7853 KH |
1056 | @end tex |
1057 | and | |
37680279 | 1058 | @ifnottex |
15da7853 | 1059 | pi/2 |
37680279 | 1060 | @end ifnottex |
15da7853 | 1061 | @tex |
8241495d | 1062 | @math{\pi/2} |
15da7853 | 1063 | @end tex |
f3d95494 RS |
1064 | (inclusive) whose sine is @var{arg}; if, however, @var{arg} is out of |
1065 | range (outside [-1, 1]), it signals a @code{domain-error} error. | |
e6512bcf RS |
1066 | @end defun |
1067 | ||
1068 | @defun acos arg | |
15da7853 | 1069 | The value of @code{(acos @var{arg})} is a number between 0 and |
37680279 | 1070 | @ifnottex |
15da7853 | 1071 | pi |
37680279 | 1072 | @end ifnottex |
15da7853 | 1073 | @tex |
8241495d | 1074 | @math{\pi} |
15da7853 | 1075 | @end tex |
f3d95494 RS |
1076 | (inclusive) whose cosine is @var{arg}; if, however, @var{arg} is out |
1077 | of range (outside [-1, 1]), it signals a @code{domain-error} error. | |
e6512bcf RS |
1078 | @end defun |
1079 | ||
f120431c EZ |
1080 | @defun atan y &optional x |
1081 | The value of @code{(atan @var{y})} is a number between | |
37680279 | 1082 | @ifnottex |
15da7853 | 1083 | @minus{}pi/2 |
37680279 | 1084 | @end ifnottex |
15da7853 | 1085 | @tex |
8241495d | 1086 | @math{-\pi/2} |
15da7853 KH |
1087 | @end tex |
1088 | and | |
37680279 | 1089 | @ifnottex |
15da7853 | 1090 | pi/2 |
37680279 | 1091 | @end ifnottex |
15da7853 | 1092 | @tex |
8241495d | 1093 | @math{\pi/2} |
15da7853 | 1094 | @end tex |
f120431c EZ |
1095 | (exclusive) whose tangent is @var{y}. If the optional second |
1096 | argument @var{x} is given, the value of @code{(atan y x)} is the | |
1097 | angle in radians between the vector @code{[@var{x}, @var{y}]} and the | |
1098 | @code{X} axis. | |
e6512bcf RS |
1099 | @end defun |
1100 | ||
1101 | @defun exp arg | |
a9f0a989 RS |
1102 | This is the exponential function; it returns |
1103 | @tex | |
8241495d | 1104 | @math{e} |
a9f0a989 | 1105 | @end tex |
37680279 | 1106 | @ifnottex |
a9f0a989 | 1107 | @i{e} |
37680279 | 1108 | @end ifnottex |
a9f0a989 RS |
1109 | to the power @var{arg}. |
1110 | @tex | |
8241495d | 1111 | @math{e} |
a9f0a989 | 1112 | @end tex |
37680279 | 1113 | @ifnottex |
a9f0a989 | 1114 | @i{e} |
37680279 | 1115 | @end ifnottex |
a9f0a989 RS |
1116 | is a fundamental mathematical constant also called the base of natural |
1117 | logarithms. | |
e6512bcf RS |
1118 | @end defun |
1119 | ||
1120 | @defun log arg &optional base | |
1121 | This function returns the logarithm of @var{arg}, with base @var{base}. | |
a9f0a989 RS |
1122 | If you don't specify @var{base}, the base |
1123 | @tex | |
8241495d | 1124 | @math{e} |
a9f0a989 | 1125 | @end tex |
37680279 | 1126 | @ifnottex |
a9f0a989 | 1127 | @i{e} |
37680279 | 1128 | @end ifnottex |
f3d95494 RS |
1129 | is used. If @var{arg} is negative, it signals a @code{domain-error} |
1130 | error. | |
e6512bcf RS |
1131 | @end defun |
1132 | ||
1133 | @ignore | |
1134 | @defun expm1 arg | |
1135 | This function returns @code{(1- (exp @var{arg}))}, but it is more | |
1136 | accurate than that when @var{arg} is negative and @code{(exp @var{arg})} | |
1137 | is close to 1. | |
1138 | @end defun | |
1139 | ||
1140 | @defun log1p arg | |
1141 | This function returns @code{(log (1+ @var{arg}))}, but it is more | |
1142 | accurate than that when @var{arg} is so small that adding 1 to it would | |
1143 | lose accuracy. | |
1144 | @end defun | |
1145 | @end ignore | |
1146 | ||
1147 | @defun log10 arg | |
1148 | This function returns the logarithm of @var{arg}, with base 10. If | |
f3d95494 RS |
1149 | @var{arg} is negative, it signals a @code{domain-error} error. |
1150 | @code{(log10 @var{x})} @equiv{} @code{(log @var{x} 10)}, at least | |
1151 | approximately. | |
e6512bcf RS |
1152 | @end defun |
1153 | ||
1154 | @defun expt x y | |
8be5e0fc RS |
1155 | This function returns @var{x} raised to power @var{y}. If both |
1156 | arguments are integers and @var{y} is positive, the result is an | |
4182ecfc | 1157 | integer; in this case, overflow causes truncation, so watch out. |
e6512bcf RS |
1158 | @end defun |
1159 | ||
1160 | @defun sqrt arg | |
1161 | This returns the square root of @var{arg}. If @var{arg} is negative, | |
903a1a91 | 1162 | it signals a @code{domain-error} error. |
e6512bcf RS |
1163 | @end defun |
1164 | ||
1165 | @node Random Numbers | |
1166 | @section Random Numbers | |
1167 | @cindex random numbers | |
1168 | ||
1169 | A deterministic computer program cannot generate true random numbers. | |
1170 | For most purposes, @dfn{pseudo-random numbers} suffice. A series of | |
1171 | pseudo-random numbers is generated in a deterministic fashion. The | |
1172 | numbers are not truly random, but they have certain properties that | |
1173 | mimic a random series. For example, all possible values occur equally | |
1174 | often in a pseudo-random series. | |
1175 | ||
1176 | In Emacs, pseudo-random numbers are generated from a ``seed'' number. | |
1177 | Starting from any given seed, the @code{random} function always | |
1178 | generates the same sequence of numbers. Emacs always starts with the | |
1179 | same seed value, so the sequence of values of @code{random} is actually | |
1180 | the same in each Emacs run! For example, in one operating system, the | |
1181 | first call to @code{(random)} after you start Emacs always returns | |
1182 | -1457731, and the second one always returns -7692030. This | |
1183 | repeatability is helpful for debugging. | |
1184 | ||
8241495d RS |
1185 | If you want random numbers that don't always come out the same, execute |
1186 | @code{(random t)}. This chooses a new seed based on the current time of | |
ad800164 | 1187 | day and on Emacs's process @acronym{ID} number. |
e6512bcf RS |
1188 | |
1189 | @defun random &optional limit | |
1190 | This function returns a pseudo-random integer. Repeated calls return a | |
1191 | series of pseudo-random integers. | |
1192 | ||
22697dac | 1193 | If @var{limit} is a positive integer, the value is chosen to be |
bfe721d1 | 1194 | nonnegative and less than @var{limit}. |
e6512bcf RS |
1195 | |
1196 | If @var{limit} is @code{t}, it means to choose a new seed based on the | |
ad800164 | 1197 | current time of day and on Emacs's process @acronym{ID} number. |
e6512bcf RS |
1198 | @c "Emacs'" is incorrect usage! |
1199 | ||
1200 | On some machines, any integer representable in Lisp may be the result | |
1201 | of @code{random}. On other machines, the result can never be larger | |
1202 | than a certain maximum or less than a certain (negative) minimum. | |
1203 | @end defun | |
ab5796a9 MB |
1204 | |
1205 | @ignore | |
1206 | arch-tag: 574e8dd2-d513-4616-9844-c9a27869782e | |
1207 | @end ignore |