Commit | Line | Data |
---|---|---|
b8d4c8d0 GM |
1 | @c -*-texinfo-*- |
2 | @c This is part of the GNU Emacs Lisp Reference Manual. | |
acaf905b | 3 | @c Copyright (C) 1990-1995, 1998-1999, 2001-2012 |
1ddd6622 | 4 | @c Free Software Foundation, Inc. |
b8d4c8d0 | 5 | @c See the file elisp.texi for copying conditions. |
b8d4c8d0 GM |
6 | @node Numbers, Strings and Characters, Lisp Data Types, Top |
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 | |
15 | 2.71828. They can also be expressed in exponential notation: 1.5e2 | |
16 | equals 150; in this example, @samp{e2} stands for ten to the second | |
17 | power, and that is multiplied by 1.5. Floating point values are not | |
18 | exact; they have a fixed, limited amount of precision. | |
19 | ||
20 | @menu | |
21 | * Integer Basics:: Representation and range of integers. | |
d24880de | 22 | * Float Basics:: Representation and range of floating point. |
b8d4c8d0 GM |
23 | * Predicates on Numbers:: Testing for numbers. |
24 | * Comparison of Numbers:: Equality and inequality predicates. | |
d24880de | 25 | * Numeric Conversions:: Converting float to integer and vice versa. |
b8d4c8d0 GM |
26 | * Arithmetic Operations:: How to add, subtract, multiply and divide. |
27 | * Rounding Operations:: Explicitly rounding floating point numbers. | |
28 | * Bitwise Operations:: Logical and, or, not, shifting. | |
29 | * Math Functions:: Trig, exponential and logarithmic functions. | |
30 | * Random Numbers:: Obtaining random integers, predictable or not. | |
31 | @end menu | |
32 | ||
33 | @node Integer Basics | |
34 | @comment node-name, next, previous, up | |
35 | @section Integer Basics | |
36 | ||
37 | The range of values for an integer depends on the machine. The | |
1ddd6622 | 38 | minimum range is @minus{}536870912 to 536870911 (30 bits; i.e., |
b8d4c8d0 | 39 | @ifnottex |
1ddd6622 | 40 | -2**29 |
b8d4c8d0 GM |
41 | @end ifnottex |
42 | @tex | |
1ddd6622 | 43 | @math{-2^{29}} |
b8d4c8d0 GM |
44 | @end tex |
45 | to | |
46 | @ifnottex | |
1ddd6622 | 47 | 2**29 - 1), |
b8d4c8d0 GM |
48 | @end ifnottex |
49 | @tex | |
1ddd6622 | 50 | @math{2^{29}-1}), |
b8d4c8d0 | 51 | @end tex |
001903b5 PE |
52 | but some machines provide a wider range. Many examples in this |
53 | chapter assume that an integer has 30 bits and that floating point | |
be14b9ab | 54 | numbers are IEEE double precision. |
b8d4c8d0 GM |
55 | @cindex overflow |
56 | ||
57 | The Lisp reader reads an integer as a sequence of digits with optional | |
fed14fd7 PE |
58 | initial sign and optional final period. An integer that is out of the |
59 | Emacs range is treated as a floating-point number. | |
b8d4c8d0 GM |
60 | |
61 | @example | |
62 | 1 ; @r{The integer 1.} | |
63 | 1. ; @r{The integer 1.} | |
64 | +1 ; @r{Also the integer 1.} | |
65 | -1 ; @r{The integer @minus{}1.} | |
fed14fd7 | 66 | 1073741825 ; @r{The floating point number 1073741825.0.} |
b8d4c8d0 GM |
67 | 0 ; @r{The integer 0.} |
68 | -0 ; @r{The integer 0.} | |
69 | @end example | |
70 | ||
71 | @cindex integers in specific radix | |
72 | @cindex radix for reading an integer | |
73 | @cindex base for reading an integer | |
74 | @cindex hex numbers | |
75 | @cindex octal numbers | |
76 | @cindex reading numbers in hex, octal, and binary | |
77 | The syntax for integers in bases other than 10 uses @samp{#} | |
78 | followed by a letter that specifies the radix: @samp{b} for binary, | |
79 | @samp{o} for octal, @samp{x} for hex, or @samp{@var{radix}r} to | |
80 | specify radix @var{radix}. Case is not significant for the letter | |
81 | that specifies the radix. Thus, @samp{#b@var{integer}} reads | |
82 | @var{integer} in binary, and @samp{#@var{radix}r@var{integer}} reads | |
83 | @var{integer} in radix @var{radix}. Allowed values of @var{radix} run | |
84 | from 2 to 36. For example: | |
85 | ||
86 | @example | |
87 | #b101100 @result{} 44 | |
88 | #o54 @result{} 44 | |
89 | #x2c @result{} 44 | |
90 | #24r1k @result{} 44 | |
91 | @end example | |
92 | ||
93 | To understand how various functions work on integers, especially the | |
94 | bitwise operators (@pxref{Bitwise Operations}), it is often helpful to | |
95 | view the numbers in their binary form. | |
96 | ||
1ddd6622 | 97 | In 30-bit binary, the decimal integer 5 looks like this: |
b8d4c8d0 GM |
98 | |
99 | @example | |
001903b5 | 100 | 0000...000101 (30 bits total) |
b8d4c8d0 GM |
101 | @end example |
102 | ||
103 | @noindent | |
001903b5 PE |
104 | (The @samp{...} stands for enough bits to fill out a 30-bit word; in |
105 | this case, @samp{...} stands for twenty 0 bits. Later examples also | |
106 | use the @samp{...} notation to make binary integers easier to read.) | |
b8d4c8d0 GM |
107 | |
108 | The integer @minus{}1 looks like this: | |
109 | ||
110 | @example | |
001903b5 | 111 | 1111...111111 (30 bits total) |
b8d4c8d0 GM |
112 | @end example |
113 | ||
114 | @noindent | |
115 | @cindex two's complement | |
1ddd6622 | 116 | @minus{}1 is represented as 30 ones. (This is called @dfn{two's |
b8d4c8d0 GM |
117 | complement} notation.) |
118 | ||
119 | The negative integer, @minus{}5, is creating by subtracting 4 from | |
120 | @minus{}1. In binary, the decimal integer 4 is 100. Consequently, | |
121 | @minus{}5 looks like this: | |
122 | ||
123 | @example | |
001903b5 | 124 | 1111...111011 (30 bits total) |
b8d4c8d0 GM |
125 | @end example |
126 | ||
1ddd6622 GM |
127 | In this implementation, the largest 30-bit binary integer value is |
128 | 536,870,911 in decimal. In binary, it looks like this: | |
b8d4c8d0 GM |
129 | |
130 | @example | |
001903b5 | 131 | 0111...111111 (30 bits total) |
b8d4c8d0 GM |
132 | @end example |
133 | ||
134 | Since the arithmetic functions do not check whether integers go | |
1ddd6622 GM |
135 | outside their range, when you add 1 to 536,870,911, the value is the |
136 | negative integer @minus{}536,870,912: | |
b8d4c8d0 GM |
137 | |
138 | @example | |
1ddd6622 GM |
139 | (+ 1 536870911) |
140 | @result{} -536870912 | |
001903b5 | 141 | @result{} 1000...000000 (30 bits total) |
b8d4c8d0 GM |
142 | @end example |
143 | ||
144 | Many of the functions described in this chapter accept markers for | |
145 | arguments in place of numbers. (@xref{Markers}.) Since the actual | |
146 | arguments to such functions may be either numbers or markers, we often | |
147 | give these arguments the name @var{number-or-marker}. When the argument | |
148 | value is a marker, its position value is used and its buffer is ignored. | |
149 | ||
eddf142c EZ |
150 | @cindex largest Lisp integer number |
151 | @cindex maximum Lisp integer number | |
b8d4c8d0 GM |
152 | @defvar most-positive-fixnum |
153 | The value of this variable is the largest integer that Emacs Lisp | |
154 | can handle. | |
155 | @end defvar | |
156 | ||
eddf142c EZ |
157 | @cindex smallest Lisp integer number |
158 | @cindex minimum Lisp integer number | |
b8d4c8d0 GM |
159 | @defvar most-negative-fixnum |
160 | The value of this variable is the smallest integer that Emacs Lisp can | |
161 | handle. It is negative. | |
162 | @end defvar | |
163 | ||
57e2db6d EZ |
164 | @xref{Character Codes, max-char}, for the maximum value of a valid |
165 | character codepoint. | |
166 | ||
b8d4c8d0 GM |
167 | @node Float Basics |
168 | @section Floating Point Basics | |
169 | ||
fead402d | 170 | @cindex @acronym{IEEE} floating point |
b8d4c8d0 GM |
171 | Floating point numbers are useful for representing numbers that are |
172 | not integral. The precise range of floating point numbers is | |
173 | machine-specific; it is the same as the range of the C data type | |
fead402d CY |
174 | @code{double} on the machine you are using. Emacs uses the |
175 | @acronym{IEEE} floating point standard where possible (the standard is | |
176 | supported by most modern computers). | |
b8d4c8d0 | 177 | |
fead402d | 178 | The read syntax for floating point numbers requires either a decimal |
b8d4c8d0 GM |
179 | point (with at least one digit following), an exponent, or both. For |
180 | example, @samp{1500.0}, @samp{15e2}, @samp{15.0e2}, @samp{1.5e3}, and | |
181 | @samp{.15e4} are five ways of writing a floating point number whose | |
fead402d CY |
182 | value is 1500. They are all equivalent. You can also use a minus |
183 | sign to write negative floating point numbers, as in @samp{-1.0}. | |
184 | ||
185 | Emacs Lisp treats @code{-0.0} as equal to ordinary zero (with | |
186 | respect to @code{equal} and @code{=}), even though the two are | |
187 | distinguishable in the @acronym{IEEE} floating point standard. | |
b8d4c8d0 | 188 | |
b8d4c8d0 GM |
189 | @cindex positive infinity |
190 | @cindex negative infinity | |
191 | @cindex infinity | |
192 | @cindex NaN | |
fead402d CY |
193 | The @acronym{IEEE} floating point standard supports positive |
194 | infinity and negative infinity as floating point values. It also | |
195 | provides for a class of values called NaN or ``not-a-number''; | |
196 | numerical functions return such values in cases where there is no | |
197 | correct answer. For example, @code{(/ 0.0 0.0)} returns a NaN. (NaN | |
198 | values can also carry a sign, but for practical purposes there's no | |
199 | significant difference between different NaN values in Emacs Lisp.) | |
200 | Here are the read syntaxes for these special floating point values: | |
b8d4c8d0 GM |
201 | |
202 | @table @asis | |
203 | @item positive infinity | |
204 | @samp{1.0e+INF} | |
205 | @item negative infinity | |
206 | @samp{-1.0e+INF} | |
fed14fd7 | 207 | @item Not-a-number |
b8d4c8d0 GM |
208 | @samp{0.0e+NaN} or @samp{-0.0e+NaN}. |
209 | @end table | |
210 | ||
fead402d CY |
211 | @defun isnan number |
212 | This predicate tests whether its argument is NaN, and returns @code{t} | |
213 | if so, @code{nil} otherwise. The argument must be a number. | |
214 | @end defun | |
215 | ||
216 | The following functions are specialized for handling floating point | |
217 | numbers: | |
218 | ||
219 | @defun frexp x | |
220 | This function returns a cons cell @code{(@var{sig} . @var{exp})}, | |
221 | where @var{sig} and @var{exp} are respectively the significand and | |
222 | exponent of the floating point number @var{x}: | |
223 | ||
224 | @smallexample | |
225 | @var{x} = @var{sig} * 2^@var{exp} | |
226 | @end smallexample | |
227 | ||
228 | @var{sig} is a floating point number between 0.5 (inclusive) and 1.0 | |
229 | (exclusive). If @var{x} is zero, the return value is @code{(0 . 0)}. | |
230 | @end defun | |
b8d4c8d0 | 231 | |
fead402d CY |
232 | @defun ldexp sig &optional exp |
233 | This function returns a floating point number corresponding to the | |
234 | significand @var{sig} and exponent @var{exp}. | |
235 | @end defun | |
b8d4c8d0 | 236 | |
fead402d CY |
237 | @defun copysign x1 x2 |
238 | This function copies the sign of @var{x2} to the value of @var{x1}, | |
239 | and returns the result. @var{x1} and @var{x2} must be floating point | |
240 | numbers. | |
241 | @end defun | |
b8d4c8d0 GM |
242 | |
243 | @defun logb number | |
244 | This function returns the binary exponent of @var{number}. More | |
245 | precisely, the value is the logarithm of @var{number} base 2, rounded | |
246 | down to an integer. | |
247 | ||
248 | @example | |
249 | (logb 10) | |
250 | @result{} 3 | |
251 | (logb 10.0e20) | |
252 | @result{} 69 | |
253 | @end example | |
254 | @end defun | |
255 | ||
256 | @node Predicates on Numbers | |
257 | @section Type Predicates for Numbers | |
258 | @cindex predicates for numbers | |
259 | ||
260 | The functions in this section test for numbers, or for a specific | |
261 | type of number. The functions @code{integerp} and @code{floatp} can | |
262 | take any type of Lisp object as argument (they would not be of much | |
263 | use otherwise), but the @code{zerop} predicate requires a number as | |
264 | its argument. See also @code{integer-or-marker-p} and | |
265 | @code{number-or-marker-p}, in @ref{Predicates on Markers}. | |
266 | ||
267 | @defun floatp object | |
268 | This predicate tests whether its argument is a floating point | |
269 | number and returns @code{t} if so, @code{nil} otherwise. | |
270 | ||
271 | @code{floatp} does not exist in Emacs versions 18 and earlier. | |
272 | @end defun | |
273 | ||
274 | @defun integerp object | |
275 | This predicate tests whether its argument is an integer, and returns | |
276 | @code{t} if so, @code{nil} otherwise. | |
277 | @end defun | |
278 | ||
279 | @defun numberp object | |
280 | This predicate tests whether its argument is a number (either integer or | |
281 | floating point), and returns @code{t} if so, @code{nil} otherwise. | |
282 | @end defun | |
283 | ||
0f29fa41 | 284 | @defun natnump object |
b8d4c8d0 | 285 | @cindex natural numbers |
bccc0e40 | 286 | This predicate (whose name comes from the phrase ``natural number'') |
0f29fa41 CY |
287 | tests to see whether its argument is a nonnegative integer, and |
288 | returns @code{t} if so, @code{nil} otherwise. 0 is considered | |
289 | non-negative. | |
b8d4c8d0 | 290 | |
0f29fa41 CY |
291 | @findex wholenump number |
292 | This is a synonym for @code{natnump}. | |
b8d4c8d0 GM |
293 | @end defun |
294 | ||
295 | @defun zerop number | |
296 | This predicate tests whether its argument is zero, and returns @code{t} | |
297 | if so, @code{nil} otherwise. The argument must be a number. | |
298 | ||
299 | @code{(zerop x)} is equivalent to @code{(= x 0)}. | |
300 | @end defun | |
301 | ||
302 | @node Comparison of Numbers | |
303 | @section Comparison of Numbers | |
304 | @cindex number comparison | |
305 | @cindex comparing numbers | |
306 | ||
307 | To test numbers for numerical equality, you should normally use | |
308 | @code{=}, not @code{eq}. There can be many distinct floating point | |
309 | number objects with the same numeric value. If you use @code{eq} to | |
310 | compare them, then you test whether two values are the same | |
311 | @emph{object}. By contrast, @code{=} compares only the numeric values | |
312 | of the objects. | |
313 | ||
314 | At present, each integer value has a unique Lisp object in Emacs Lisp. | |
315 | Therefore, @code{eq} is equivalent to @code{=} where integers are | |
316 | concerned. It is sometimes convenient to use @code{eq} for comparing an | |
317 | unknown value with an integer, because @code{eq} does not report an | |
318 | error if the unknown value is not a number---it accepts arguments of any | |
319 | type. By contrast, @code{=} signals an error if the arguments are not | |
320 | numbers or markers. However, it is a good idea to use @code{=} if you | |
321 | can, even for comparing integers, just in case we change the | |
322 | representation of integers in a future Emacs version. | |
323 | ||
324 | Sometimes it is useful to compare numbers with @code{equal}; it | |
325 | treats two numbers as equal if they have the same data type (both | |
326 | integers, or both floating point) and the same value. By contrast, | |
327 | @code{=} can treat an integer and a floating point number as equal. | |
328 | @xref{Equality Predicates}. | |
329 | ||
330 | There is another wrinkle: because floating point arithmetic is not | |
331 | exact, it is often a bad idea to check for equality of two floating | |
332 | point values. Usually it is better to test for approximate equality. | |
333 | Here's a function to do this: | |
334 | ||
335 | @example | |
336 | (defvar fuzz-factor 1.0e-6) | |
337 | (defun approx-equal (x y) | |
338 | (or (and (= x 0) (= y 0)) | |
339 | (< (/ (abs (- x y)) | |
340 | (max (abs x) (abs y))) | |
341 | fuzz-factor))) | |
342 | @end example | |
343 | ||
344 | @cindex CL note---integers vrs @code{eq} | |
345 | @quotation | |
346 | @b{Common Lisp note:} Comparing numbers in Common Lisp always requires | |
347 | @code{=} because Common Lisp implements multi-word integers, and two | |
348 | distinct integer objects can have the same numeric value. Emacs Lisp | |
349 | can have just one integer object for any given value because it has a | |
350 | limited range of integer values. | |
351 | @end quotation | |
352 | ||
353 | @defun = number-or-marker1 number-or-marker2 | |
354 | This function tests whether its arguments are numerically equal, and | |
355 | returns @code{t} if so, @code{nil} otherwise. | |
356 | @end defun | |
357 | ||
358 | @defun eql value1 value2 | |
359 | This function acts like @code{eq} except when both arguments are | |
360 | numbers. It compares numbers by type and numeric value, so that | |
361 | @code{(eql 1.0 1)} returns @code{nil}, but @code{(eql 1.0 1.0)} and | |
362 | @code{(eql 1 1)} both return @code{t}. | |
363 | @end defun | |
364 | ||
365 | @defun /= number-or-marker1 number-or-marker2 | |
366 | This function tests whether its arguments are numerically equal, and | |
367 | returns @code{t} if they are not, and @code{nil} if they are. | |
368 | @end defun | |
369 | ||
370 | @defun < number-or-marker1 number-or-marker2 | |
371 | This function tests whether its first argument is strictly less than | |
372 | its second argument. It returns @code{t} if so, @code{nil} otherwise. | |
373 | @end defun | |
374 | ||
375 | @defun <= number-or-marker1 number-or-marker2 | |
376 | This function tests whether its first argument is less than or equal | |
377 | to its second argument. It returns @code{t} if so, @code{nil} | |
378 | otherwise. | |
379 | @end defun | |
380 | ||
381 | @defun > number-or-marker1 number-or-marker2 | |
382 | This function tests whether its first argument is strictly greater | |
383 | than its second argument. It returns @code{t} if so, @code{nil} | |
384 | otherwise. | |
385 | @end defun | |
386 | ||
387 | @defun >= number-or-marker1 number-or-marker2 | |
388 | This function tests whether its first argument is greater than or | |
389 | equal to its second argument. It returns @code{t} if so, @code{nil} | |
390 | otherwise. | |
391 | @end defun | |
392 | ||
393 | @defun max number-or-marker &rest numbers-or-markers | |
394 | This function returns the largest of its arguments. | |
395 | If any of the arguments is floating-point, the value is returned | |
396 | as floating point, even if it was given as an integer. | |
397 | ||
398 | @example | |
399 | (max 20) | |
400 | @result{} 20 | |
401 | (max 1 2.5) | |
402 | @result{} 2.5 | |
403 | (max 1 3 2.5) | |
404 | @result{} 3.0 | |
405 | @end example | |
406 | @end defun | |
407 | ||
408 | @defun min number-or-marker &rest numbers-or-markers | |
409 | This function returns the smallest of its arguments. | |
410 | If any of the arguments is floating-point, the value is returned | |
411 | as floating point, even if it was given as an integer. | |
412 | ||
413 | @example | |
414 | (min -4 1) | |
415 | @result{} -4 | |
416 | @end example | |
417 | @end defun | |
418 | ||
419 | @defun abs number | |
420 | This function returns the absolute value of @var{number}. | |
421 | @end defun | |
422 | ||
423 | @node Numeric Conversions | |
424 | @section Numeric Conversions | |
425 | @cindex rounding in conversions | |
426 | @cindex number conversions | |
427 | @cindex converting numbers | |
428 | ||
429 | To convert an integer to floating point, use the function @code{float}. | |
430 | ||
431 | @defun float number | |
432 | This returns @var{number} converted to floating point. | |
433 | If @var{number} is already a floating point number, @code{float} returns | |
434 | it unchanged. | |
435 | @end defun | |
436 | ||
437 | There are four functions to convert floating point numbers to integers; | |
438 | they differ in how they round. All accept an argument @var{number} | |
439 | and an optional argument @var{divisor}. Both arguments may be | |
440 | integers or floating point numbers. @var{divisor} may also be | |
441 | @code{nil}. If @var{divisor} is @code{nil} or omitted, these | |
442 | functions convert @var{number} to an integer, or return it unchanged | |
443 | if it already is an integer. If @var{divisor} is non-@code{nil}, they | |
444 | divide @var{number} by @var{divisor} and convert the result to an | |
445 | integer. An @code{arith-error} results if @var{divisor} is 0. | |
446 | ||
447 | @defun truncate number &optional divisor | |
448 | This returns @var{number}, converted to an integer by rounding towards | |
449 | zero. | |
450 | ||
451 | @example | |
452 | (truncate 1.2) | |
453 | @result{} 1 | |
454 | (truncate 1.7) | |
455 | @result{} 1 | |
456 | (truncate -1.2) | |
457 | @result{} -1 | |
458 | (truncate -1.7) | |
459 | @result{} -1 | |
460 | @end example | |
461 | @end defun | |
462 | ||
463 | @defun floor number &optional divisor | |
464 | This returns @var{number}, converted to an integer by rounding downward | |
465 | (towards negative infinity). | |
466 | ||
467 | If @var{divisor} is specified, this uses the kind of division | |
468 | operation that corresponds to @code{mod}, rounding downward. | |
469 | ||
470 | @example | |
471 | (floor 1.2) | |
472 | @result{} 1 | |
473 | (floor 1.7) | |
474 | @result{} 1 | |
475 | (floor -1.2) | |
476 | @result{} -2 | |
477 | (floor -1.7) | |
478 | @result{} -2 | |
479 | (floor 5.99 3) | |
480 | @result{} 1 | |
481 | @end example | |
482 | @end defun | |
483 | ||
484 | @defun ceiling number &optional divisor | |
485 | This returns @var{number}, converted to an integer by rounding upward | |
486 | (towards positive infinity). | |
487 | ||
488 | @example | |
489 | (ceiling 1.2) | |
490 | @result{} 2 | |
491 | (ceiling 1.7) | |
492 | @result{} 2 | |
493 | (ceiling -1.2) | |
494 | @result{} -1 | |
495 | (ceiling -1.7) | |
496 | @result{} -1 | |
497 | @end example | |
498 | @end defun | |
499 | ||
500 | @defun round number &optional divisor | |
501 | This returns @var{number}, converted to an integer by rounding towards the | |
502 | nearest integer. Rounding a value equidistant between two integers | |
503 | may choose the integer closer to zero, or it may prefer an even integer, | |
504 | depending on your machine. | |
505 | ||
506 | @example | |
507 | (round 1.2) | |
508 | @result{} 1 | |
509 | (round 1.7) | |
510 | @result{} 2 | |
511 | (round -1.2) | |
512 | @result{} -1 | |
513 | (round -1.7) | |
514 | @result{} -2 | |
515 | @end example | |
516 | @end defun | |
517 | ||
518 | @node Arithmetic Operations | |
519 | @section Arithmetic Operations | |
520 | @cindex arithmetic operations | |
521 | ||
522 | Emacs Lisp provides the traditional four arithmetic operations: | |
523 | addition, subtraction, multiplication, and division. Remainder and modulus | |
524 | functions supplement the division functions. The functions to | |
525 | add or subtract 1 are provided because they are traditional in Lisp and | |
526 | commonly used. | |
527 | ||
528 | All of these functions except @code{%} return a floating point value | |
529 | if any argument is floating. | |
530 | ||
c717b326 | 531 | It is important to note that in Emacs Lisp, arithmetic functions |
001903b5 PE |
532 | do not check for overflow. Thus @code{(1+ 536870911)} may evaluate to |
533 | @minus{}536870912, depending on your hardware. | |
b8d4c8d0 GM |
534 | |
535 | @defun 1+ number-or-marker | |
536 | This function returns @var{number-or-marker} plus 1. | |
537 | For example, | |
538 | ||
539 | @example | |
540 | (setq foo 4) | |
541 | @result{} 4 | |
542 | (1+ foo) | |
543 | @result{} 5 | |
544 | @end example | |
545 | ||
546 | This function is not analogous to the C operator @code{++}---it does not | |
547 | increment a variable. It just computes a sum. Thus, if we continue, | |
548 | ||
549 | @example | |
550 | foo | |
551 | @result{} 4 | |
552 | @end example | |
553 | ||
554 | If you want to increment the variable, you must use @code{setq}, | |
555 | like this: | |
556 | ||
557 | @example | |
558 | (setq foo (1+ foo)) | |
559 | @result{} 5 | |
560 | @end example | |
561 | @end defun | |
562 | ||
563 | @defun 1- number-or-marker | |
564 | This function returns @var{number-or-marker} minus 1. | |
565 | @end defun | |
566 | ||
567 | @defun + &rest numbers-or-markers | |
568 | This function adds its arguments together. When given no arguments, | |
569 | @code{+} returns 0. | |
570 | ||
571 | @example | |
572 | (+) | |
573 | @result{} 0 | |
574 | (+ 1) | |
575 | @result{} 1 | |
576 | (+ 1 2 3 4) | |
577 | @result{} 10 | |
578 | @end example | |
579 | @end defun | |
580 | ||
581 | @defun - &optional number-or-marker &rest more-numbers-or-markers | |
582 | The @code{-} function serves two purposes: negation and subtraction. | |
583 | When @code{-} has a single argument, the value is the negative of the | |
584 | argument. When there are multiple arguments, @code{-} subtracts each of | |
585 | the @var{more-numbers-or-markers} from @var{number-or-marker}, | |
586 | cumulatively. If there are no arguments, the result is 0. | |
587 | ||
588 | @example | |
589 | (- 10 1 2 3 4) | |
590 | @result{} 0 | |
591 | (- 10) | |
592 | @result{} -10 | |
593 | (-) | |
594 | @result{} 0 | |
595 | @end example | |
596 | @end defun | |
597 | ||
598 | @defun * &rest numbers-or-markers | |
599 | This function multiplies its arguments together, and returns the | |
600 | product. When given no arguments, @code{*} returns 1. | |
601 | ||
602 | @example | |
603 | (*) | |
604 | @result{} 1 | |
605 | (* 1) | |
606 | @result{} 1 | |
607 | (* 1 2 3 4) | |
608 | @result{} 24 | |
609 | @end example | |
610 | @end defun | |
611 | ||
612 | @defun / dividend divisor &rest divisors | |
613 | This function divides @var{dividend} by @var{divisor} and returns the | |
614 | quotient. If there are additional arguments @var{divisors}, then it | |
615 | divides @var{dividend} by each divisor in turn. Each argument may be a | |
616 | number or a marker. | |
617 | ||
618 | If all the arguments are integers, then the result is an integer too. | |
619 | This means the result has to be rounded. On most machines, the result | |
620 | is rounded towards zero after each division, but some machines may round | |
621 | differently with negative arguments. This is because the Lisp function | |
622 | @code{/} is implemented using the C division operator, which also | |
623 | permits machine-dependent rounding. As a practical matter, all known | |
624 | machines round in the standard fashion. | |
625 | ||
626 | @cindex @code{arith-error} in division | |
627 | If you divide an integer by 0, an @code{arith-error} error is signaled. | |
628 | (@xref{Errors}.) Floating point division by zero returns either | |
629 | infinity or a NaN if your machine supports @acronym{IEEE} floating point; | |
630 | otherwise, it signals an @code{arith-error} error. | |
631 | ||
632 | @example | |
633 | @group | |
634 | (/ 6 2) | |
635 | @result{} 3 | |
636 | @end group | |
637 | (/ 5 2) | |
638 | @result{} 2 | |
639 | (/ 5.0 2) | |
640 | @result{} 2.5 | |
641 | (/ 5 2.0) | |
642 | @result{} 2.5 | |
643 | (/ 5.0 2.0) | |
644 | @result{} 2.5 | |
645 | (/ 25 3 2) | |
646 | @result{} 4 | |
647 | @group | |
648 | (/ -17 6) | |
649 | @result{} -2 @r{(could in theory be @minus{}3 on some machines)} | |
650 | @end group | |
651 | @end example | |
652 | @end defun | |
653 | ||
654 | @defun % dividend divisor | |
655 | @cindex remainder | |
656 | This function returns the integer remainder after division of @var{dividend} | |
657 | by @var{divisor}. The arguments must be integers or markers. | |
658 | ||
659 | For negative arguments, the remainder is in principle machine-dependent | |
660 | since the quotient is; but in practice, all known machines behave alike. | |
661 | ||
662 | An @code{arith-error} results if @var{divisor} is 0. | |
663 | ||
664 | @example | |
665 | (% 9 4) | |
666 | @result{} 1 | |
667 | (% -9 4) | |
668 | @result{} -1 | |
669 | (% 9 -4) | |
670 | @result{} 1 | |
671 | (% -9 -4) | |
672 | @result{} -1 | |
673 | @end example | |
674 | ||
675 | For any two integers @var{dividend} and @var{divisor}, | |
676 | ||
677 | @example | |
678 | @group | |
679 | (+ (% @var{dividend} @var{divisor}) | |
680 | (* (/ @var{dividend} @var{divisor}) @var{divisor})) | |
681 | @end group | |
682 | @end example | |
683 | ||
684 | @noindent | |
685 | always equals @var{dividend}. | |
686 | @end defun | |
687 | ||
688 | @defun mod dividend divisor | |
689 | @cindex modulus | |
690 | This function returns the value of @var{dividend} modulo @var{divisor}; | |
691 | in other words, the remainder after division of @var{dividend} | |
692 | by @var{divisor}, but with the same sign as @var{divisor}. | |
693 | The arguments must be numbers or markers. | |
694 | ||
695 | Unlike @code{%}, @code{mod} returns a well-defined result for negative | |
696 | arguments. It also permits floating point arguments; it rounds the | |
697 | quotient downward (towards minus infinity) to an integer, and uses that | |
698 | quotient to compute the remainder. | |
699 | ||
700 | An @code{arith-error} results if @var{divisor} is 0. | |
701 | ||
702 | @example | |
703 | @group | |
704 | (mod 9 4) | |
705 | @result{} 1 | |
706 | @end group | |
707 | @group | |
708 | (mod -9 4) | |
709 | @result{} 3 | |
710 | @end group | |
711 | @group | |
712 | (mod 9 -4) | |
713 | @result{} -3 | |
714 | @end group | |
715 | @group | |
716 | (mod -9 -4) | |
717 | @result{} -1 | |
718 | @end group | |
719 | @group | |
720 | (mod 5.5 2.5) | |
721 | @result{} .5 | |
722 | @end group | |
723 | @end example | |
724 | ||
725 | For any two numbers @var{dividend} and @var{divisor}, | |
726 | ||
727 | @example | |
728 | @group | |
729 | (+ (mod @var{dividend} @var{divisor}) | |
730 | (* (floor @var{dividend} @var{divisor}) @var{divisor})) | |
731 | @end group | |
732 | @end example | |
733 | ||
734 | @noindent | |
735 | always equals @var{dividend}, subject to rounding error if either | |
736 | argument is floating point. For @code{floor}, see @ref{Numeric | |
737 | Conversions}. | |
738 | @end defun | |
739 | ||
740 | @node Rounding Operations | |
741 | @section Rounding Operations | |
742 | @cindex rounding without conversion | |
743 | ||
744 | The functions @code{ffloor}, @code{fceiling}, @code{fround}, and | |
745 | @code{ftruncate} take a floating point argument and return a floating | |
746 | point result whose value is a nearby integer. @code{ffloor} returns the | |
747 | nearest integer below; @code{fceiling}, the nearest integer above; | |
748 | @code{ftruncate}, the nearest integer in the direction towards zero; | |
749 | @code{fround}, the nearest integer. | |
750 | ||
751 | @defun ffloor float | |
752 | This function rounds @var{float} to the next lower integral value, and | |
753 | returns that value as a floating point number. | |
754 | @end defun | |
755 | ||
756 | @defun fceiling float | |
757 | This function rounds @var{float} to the next higher integral value, and | |
758 | returns that value as a floating point number. | |
759 | @end defun | |
760 | ||
761 | @defun ftruncate float | |
762 | This function rounds @var{float} towards zero to an integral value, and | |
763 | returns that value as a floating point number. | |
764 | @end defun | |
765 | ||
766 | @defun fround float | |
767 | This function rounds @var{float} to the nearest integral value, | |
768 | and returns that value as a floating point number. | |
769 | @end defun | |
770 | ||
771 | @node Bitwise Operations | |
772 | @section Bitwise Operations on Integers | |
773 | @cindex bitwise arithmetic | |
774 | @cindex logical arithmetic | |
775 | ||
776 | In a computer, an integer is represented as a binary number, a | |
777 | sequence of @dfn{bits} (digits which are either zero or one). A bitwise | |
778 | operation acts on the individual bits of such a sequence. For example, | |
779 | @dfn{shifting} moves the whole sequence left or right one or more places, | |
16152b76 | 780 | reproducing the same pattern ``moved over''. |
b8d4c8d0 GM |
781 | |
782 | The bitwise operations in Emacs Lisp apply only to integers. | |
783 | ||
784 | @defun lsh integer1 count | |
785 | @cindex logical shift | |
786 | @code{lsh}, which is an abbreviation for @dfn{logical shift}, shifts the | |
787 | bits in @var{integer1} to the left @var{count} places, or to the right | |
788 | if @var{count} is negative, bringing zeros into the vacated bits. If | |
789 | @var{count} is negative, @code{lsh} shifts zeros into the leftmost | |
790 | (most-significant) bit, producing a positive result even if | |
791 | @var{integer1} is negative. Contrast this with @code{ash}, below. | |
792 | ||
793 | Here are two examples of @code{lsh}, shifting a pattern of bits one | |
794 | place to the left. We show only the low-order eight bits of the binary | |
795 | pattern; the rest are all zero. | |
796 | ||
797 | @example | |
798 | @group | |
799 | (lsh 5 1) | |
800 | @result{} 10 | |
801 | ;; @r{Decimal 5 becomes decimal 10.} | |
802 | 00000101 @result{} 00001010 | |
803 | ||
804 | (lsh 7 1) | |
805 | @result{} 14 | |
806 | ;; @r{Decimal 7 becomes decimal 14.} | |
807 | 00000111 @result{} 00001110 | |
808 | @end group | |
809 | @end example | |
810 | ||
811 | @noindent | |
812 | As the examples illustrate, shifting the pattern of bits one place to | |
813 | the left produces a number that is twice the value of the previous | |
814 | number. | |
815 | ||
816 | Shifting a pattern of bits two places to the left produces results | |
817 | like this (with 8-bit binary numbers): | |
818 | ||
819 | @example | |
820 | @group | |
821 | (lsh 3 2) | |
822 | @result{} 12 | |
823 | ;; @r{Decimal 3 becomes decimal 12.} | |
824 | 00000011 @result{} 00001100 | |
825 | @end group | |
826 | @end example | |
827 | ||
828 | On the other hand, shifting one place to the right looks like this: | |
829 | ||
830 | @example | |
831 | @group | |
832 | (lsh 6 -1) | |
833 | @result{} 3 | |
834 | ;; @r{Decimal 6 becomes decimal 3.} | |
835 | 00000110 @result{} 00000011 | |
836 | @end group | |
837 | ||
838 | @group | |
839 | (lsh 5 -1) | |
840 | @result{} 2 | |
841 | ;; @r{Decimal 5 becomes decimal 2.} | |
842 | 00000101 @result{} 00000010 | |
843 | @end group | |
844 | @end example | |
845 | ||
846 | @noindent | |
847 | As the example illustrates, shifting one place to the right divides the | |
848 | value of a positive integer by two, rounding downward. | |
849 | ||
c717b326 | 850 | The function @code{lsh}, like all Emacs Lisp arithmetic functions, does |
b8d4c8d0 GM |
851 | not check for overflow, so shifting left can discard significant bits |
852 | and change the sign of the number. For example, left shifting | |
001903b5 | 853 | 536,870,911 produces @minus{}2 in the 30-bit implementation: |
b8d4c8d0 GM |
854 | |
855 | @example | |
1ddd6622 | 856 | (lsh 536870911 1) ; @r{left shift} |
b8d4c8d0 GM |
857 | @result{} -2 |
858 | @end example | |
859 | ||
001903b5 | 860 | In binary, the argument looks like this: |
b8d4c8d0 GM |
861 | |
862 | @example | |
863 | @group | |
1ddd6622 | 864 | ;; @r{Decimal 536,870,911} |
001903b5 | 865 | 0111...111111 (30 bits total) |
b8d4c8d0 GM |
866 | @end group |
867 | @end example | |
868 | ||
869 | @noindent | |
870 | which becomes the following when left shifted: | |
871 | ||
872 | @example | |
873 | @group | |
874 | ;; @r{Decimal @minus{}2} | |
001903b5 | 875 | 1111...111110 (30 bits total) |
b8d4c8d0 GM |
876 | @end group |
877 | @end example | |
878 | @end defun | |
879 | ||
880 | @defun ash integer1 count | |
881 | @cindex arithmetic shift | |
882 | @code{ash} (@dfn{arithmetic shift}) shifts the bits in @var{integer1} | |
883 | to the left @var{count} places, or to the right if @var{count} | |
884 | is negative. | |
885 | ||
886 | @code{ash} gives the same results as @code{lsh} except when | |
887 | @var{integer1} and @var{count} are both negative. In that case, | |
888 | @code{ash} puts ones in the empty bit positions on the left, while | |
889 | @code{lsh} puts zeros in those bit positions. | |
890 | ||
891 | Thus, with @code{ash}, shifting the pattern of bits one place to the right | |
892 | looks like this: | |
893 | ||
894 | @example | |
895 | @group | |
896 | (ash -6 -1) @result{} -3 | |
897 | ;; @r{Decimal @minus{}6 becomes decimal @minus{}3.} | |
001903b5 | 898 | 1111...111010 (30 bits total) |
b8d4c8d0 | 899 | @result{} |
001903b5 | 900 | 1111...111101 (30 bits total) |
b8d4c8d0 GM |
901 | @end group |
902 | @end example | |
903 | ||
904 | In contrast, shifting the pattern of bits one place to the right with | |
905 | @code{lsh} looks like this: | |
906 | ||
907 | @example | |
908 | @group | |
1ddd6622 GM |
909 | (lsh -6 -1) @result{} 536870909 |
910 | ;; @r{Decimal @minus{}6 becomes decimal 536,870,909.} | |
001903b5 | 911 | 1111...111010 (30 bits total) |
b8d4c8d0 | 912 | @result{} |
001903b5 | 913 | 0111...111101 (30 bits total) |
b8d4c8d0 GM |
914 | @end group |
915 | @end example | |
916 | ||
917 | Here are other examples: | |
918 | ||
919 | @c !!! Check if lined up in smallbook format! XDVI shows problem | |
920 | @c with smallbook but not with regular book! --rjc 16mar92 | |
921 | @smallexample | |
922 | @group | |
001903b5 | 923 | ; @r{ 30-bit binary values} |
b8d4c8d0 | 924 | |
be14b9ab PE |
925 | (lsh 5 2) ; 5 = @r{0000...000101} |
926 | @result{} 20 ; = @r{0000...010100} | |
b8d4c8d0 GM |
927 | @end group |
928 | @group | |
929 | (ash 5 2) | |
930 | @result{} 20 | |
be14b9ab PE |
931 | (lsh -5 2) ; -5 = @r{1111...111011} |
932 | @result{} -20 ; = @r{1111...101100} | |
b8d4c8d0 GM |
933 | (ash -5 2) |
934 | @result{} -20 | |
935 | @end group | |
936 | @group | |
be14b9ab PE |
937 | (lsh 5 -2) ; 5 = @r{0000...000101} |
938 | @result{} 1 ; = @r{0000...000001} | |
b8d4c8d0 GM |
939 | @end group |
940 | @group | |
941 | (ash 5 -2) | |
942 | @result{} 1 | |
943 | @end group | |
944 | @group | |
be14b9ab | 945 | (lsh -5 -2) ; -5 = @r{1111...111011} |
001903b5 | 946 | @result{} 268435454 |
be14b9ab | 947 | ; = @r{0011...111110} |
b8d4c8d0 GM |
948 | @end group |
949 | @group | |
be14b9ab PE |
950 | (ash -5 -2) ; -5 = @r{1111...111011} |
951 | @result{} -2 ; = @r{1111...111110} | |
b8d4c8d0 GM |
952 | @end group |
953 | @end smallexample | |
954 | @end defun | |
955 | ||
956 | @defun logand &rest ints-or-markers | |
957 | This function returns the ``logical and'' of the arguments: the | |
958 | @var{n}th bit is set in the result if, and only if, the @var{n}th bit is | |
959 | set in all the arguments. (``Set'' means that the value of the bit is 1 | |
960 | rather than 0.) | |
961 | ||
962 | For example, using 4-bit binary numbers, the ``logical and'' of 13 and | |
963 | 12 is 12: 1101 combined with 1100 produces 1100. | |
964 | In both the binary numbers, the leftmost two bits are set (i.e., they | |
965 | are 1's), so the leftmost two bits of the returned value are set. | |
966 | However, for the rightmost two bits, each is zero in at least one of | |
967 | the arguments, so the rightmost two bits of the returned value are 0's. | |
968 | ||
969 | @noindent | |
970 | Therefore, | |
971 | ||
972 | @example | |
973 | @group | |
974 | (logand 13 12) | |
975 | @result{} 12 | |
976 | @end group | |
977 | @end example | |
978 | ||
979 | If @code{logand} is not passed any argument, it returns a value of | |
980 | @minus{}1. This number is an identity element for @code{logand} | |
981 | because its binary representation consists entirely of ones. If | |
982 | @code{logand} is passed just one argument, it returns that argument. | |
983 | ||
984 | @smallexample | |
985 | @group | |
001903b5 | 986 | ; @r{ 30-bit binary values} |
b8d4c8d0 | 987 | |
be14b9ab PE |
988 | (logand 14 13) ; 14 = @r{0000...001110} |
989 | ; 13 = @r{0000...001101} | |
990 | @result{} 12 ; 12 = @r{0000...001100} | |
b8d4c8d0 GM |
991 | @end group |
992 | ||
993 | @group | |
be14b9ab PE |
994 | (logand 14 13 4) ; 14 = @r{0000...001110} |
995 | ; 13 = @r{0000...001101} | |
996 | ; 4 = @r{0000...000100} | |
997 | @result{} 4 ; 4 = @r{0000...000100} | |
b8d4c8d0 GM |
998 | @end group |
999 | ||
1000 | @group | |
1001 | (logand) | |
be14b9ab | 1002 | @result{} -1 ; -1 = @r{1111...111111} |
b8d4c8d0 GM |
1003 | @end group |
1004 | @end smallexample | |
1005 | @end defun | |
1006 | ||
1007 | @defun logior &rest ints-or-markers | |
1008 | This function returns the ``inclusive or'' of its arguments: the @var{n}th bit | |
1009 | is set in the result if, and only if, the @var{n}th bit is set in at least | |
1010 | one of the arguments. If there are no arguments, the result is zero, | |
1011 | which is an identity element for this operation. If @code{logior} is | |
1012 | passed just one argument, it returns that argument. | |
1013 | ||
1014 | @smallexample | |
1015 | @group | |
001903b5 | 1016 | ; @r{ 30-bit binary values} |
b8d4c8d0 | 1017 | |
be14b9ab PE |
1018 | (logior 12 5) ; 12 = @r{0000...001100} |
1019 | ; 5 = @r{0000...000101} | |
1020 | @result{} 13 ; 13 = @r{0000...001101} | |
b8d4c8d0 GM |
1021 | @end group |
1022 | ||
1023 | @group | |
be14b9ab PE |
1024 | (logior 12 5 7) ; 12 = @r{0000...001100} |
1025 | ; 5 = @r{0000...000101} | |
1026 | ; 7 = @r{0000...000111} | |
1027 | @result{} 15 ; 15 = @r{0000...001111} | |
b8d4c8d0 GM |
1028 | @end group |
1029 | @end smallexample | |
1030 | @end defun | |
1031 | ||
1032 | @defun logxor &rest ints-or-markers | |
1033 | This function returns the ``exclusive or'' of its arguments: the | |
1034 | @var{n}th bit is set in the result if, and only if, the @var{n}th bit is | |
1035 | set in an odd number of the arguments. If there are no arguments, the | |
1036 | result is 0, which is an identity element for this operation. If | |
1037 | @code{logxor} is passed just one argument, it returns that argument. | |
1038 | ||
1039 | @smallexample | |
1040 | @group | |
001903b5 | 1041 | ; @r{ 30-bit binary values} |
b8d4c8d0 | 1042 | |
be14b9ab PE |
1043 | (logxor 12 5) ; 12 = @r{0000...001100} |
1044 | ; 5 = @r{0000...000101} | |
1045 | @result{} 9 ; 9 = @r{0000...001001} | |
b8d4c8d0 GM |
1046 | @end group |
1047 | ||
1048 | @group | |
be14b9ab PE |
1049 | (logxor 12 5 7) ; 12 = @r{0000...001100} |
1050 | ; 5 = @r{0000...000101} | |
1051 | ; 7 = @r{0000...000111} | |
1052 | @result{} 14 ; 14 = @r{0000...001110} | |
b8d4c8d0 GM |
1053 | @end group |
1054 | @end smallexample | |
1055 | @end defun | |
1056 | ||
1057 | @defun lognot integer | |
1058 | This function returns the logical complement of its argument: the @var{n}th | |
1059 | bit is one in the result if, and only if, the @var{n}th bit is zero in | |
1060 | @var{integer}, and vice-versa. | |
1061 | ||
1062 | @example | |
1063 | (lognot 5) | |
1064 | @result{} -6 | |
001903b5 | 1065 | ;; 5 = @r{0000...000101} (30 bits total) |
b8d4c8d0 | 1066 | ;; @r{becomes} |
001903b5 | 1067 | ;; -6 = @r{1111...111010} (30 bits total) |
b8d4c8d0 GM |
1068 | @end example |
1069 | @end defun | |
1070 | ||
1071 | @node Math Functions | |
1072 | @section Standard Mathematical Functions | |
1073 | @cindex transcendental functions | |
1074 | @cindex mathematical functions | |
1075 | @cindex floating-point functions | |
1076 | ||
1077 | These mathematical functions allow integers as well as floating point | |
1078 | numbers as arguments. | |
1079 | ||
1080 | @defun sin arg | |
1081 | @defunx cos arg | |
1082 | @defunx tan arg | |
1083 | These are the ordinary trigonometric functions, with argument measured | |
1084 | in radians. | |
1085 | @end defun | |
1086 | ||
1087 | @defun asin arg | |
1088 | The value of @code{(asin @var{arg})} is a number between | |
1089 | @ifnottex | |
1090 | @minus{}pi/2 | |
1091 | @end ifnottex | |
1092 | @tex | |
1093 | @math{-\pi/2} | |
1094 | @end tex | |
1095 | and | |
1096 | @ifnottex | |
1097 | pi/2 | |
1098 | @end ifnottex | |
1099 | @tex | |
1100 | @math{\pi/2} | |
1101 | @end tex | |
1102 | (inclusive) whose sine is @var{arg}; if, however, @var{arg} is out of | |
1103 | range (outside [@minus{}1, 1]), it signals a @code{domain-error} error. | |
1104 | @end defun | |
1105 | ||
1106 | @defun acos arg | |
1107 | The value of @code{(acos @var{arg})} is a number between 0 and | |
1108 | @ifnottex | |
1109 | pi | |
1110 | @end ifnottex | |
1111 | @tex | |
1112 | @math{\pi} | |
1113 | @end tex | |
1114 | (inclusive) whose cosine is @var{arg}; if, however, @var{arg} is out | |
1115 | of range (outside [@minus{}1, 1]), it signals a @code{domain-error} error. | |
1116 | @end defun | |
1117 | ||
1118 | @defun atan y &optional x | |
1119 | The value of @code{(atan @var{y})} is a number between | |
1120 | @ifnottex | |
1121 | @minus{}pi/2 | |
1122 | @end ifnottex | |
1123 | @tex | |
1124 | @math{-\pi/2} | |
1125 | @end tex | |
1126 | and | |
1127 | @ifnottex | |
1128 | pi/2 | |
1129 | @end ifnottex | |
1130 | @tex | |
1131 | @math{\pi/2} | |
1132 | @end tex | |
1133 | (exclusive) whose tangent is @var{y}. If the optional second | |
1134 | argument @var{x} is given, the value of @code{(atan y x)} is the | |
1135 | angle in radians between the vector @code{[@var{x}, @var{y}]} and the | |
1136 | @code{X} axis. | |
1137 | @end defun | |
1138 | ||
1139 | @defun exp arg | |
fead402d CY |
1140 | This is the exponential function; it returns @math{e} to the power |
1141 | @var{arg}. | |
b8d4c8d0 GM |
1142 | @end defun |
1143 | ||
1144 | @defun log arg &optional base | |
fead402d CY |
1145 | This function returns the logarithm of @var{arg}, with base |
1146 | @var{base}. If you don't specify @var{base}, the natural base | |
1147 | @math{e} is used. If @var{arg} is negative, it signals a | |
1148 | @code{domain-error} error. | |
b8d4c8d0 GM |
1149 | @end defun |
1150 | ||
1151 | @ignore | |
1152 | @defun expm1 arg | |
1153 | This function returns @code{(1- (exp @var{arg}))}, but it is more | |
1154 | accurate than that when @var{arg} is negative and @code{(exp @var{arg})} | |
1155 | is close to 1. | |
1156 | @end defun | |
1157 | ||
1158 | @defun log1p arg | |
1159 | This function returns @code{(log (1+ @var{arg}))}, but it is more | |
1160 | accurate than that when @var{arg} is so small that adding 1 to it would | |
1161 | lose accuracy. | |
1162 | @end defun | |
1163 | @end ignore | |
1164 | ||
1165 | @defun log10 arg | |
1166 | This function returns the logarithm of @var{arg}, with base 10. If | |
1167 | @var{arg} is negative, it signals a @code{domain-error} error. | |
1168 | @code{(log10 @var{x})} @equiv{} @code{(log @var{x} 10)}, at least | |
1169 | approximately. | |
1170 | @end defun | |
1171 | ||
1172 | @defun expt x y | |
1173 | This function returns @var{x} raised to power @var{y}. If both | |
c717b326 PE |
1174 | arguments are integers and @var{y} is positive, the result is an |
1175 | integer; in this case, overflow causes truncation, so watch out. | |
b8d4c8d0 GM |
1176 | @end defun |
1177 | ||
1178 | @defun sqrt arg | |
1179 | This returns the square root of @var{arg}. If @var{arg} is negative, | |
1180 | it signals a @code{domain-error} error. | |
1181 | @end defun | |
1182 | ||
fead402d CY |
1183 | In addition, Emacs defines the following common mathematical |
1184 | constants: | |
1185 | ||
1186 | @defvar float-e | |
1187 | The mathematical constant @math{e} (2.71828@dots{}). | |
1188 | @end defvar | |
1189 | ||
1190 | @defvar float-pi | |
1191 | The mathematical constant @math{pi} (3.14159@dots{}). | |
1192 | @end defvar | |
1193 | ||
b8d4c8d0 GM |
1194 | @node Random Numbers |
1195 | @section Random Numbers | |
1196 | @cindex random numbers | |
1197 | ||
1198 | A deterministic computer program cannot generate true random numbers. | |
1199 | For most purposes, @dfn{pseudo-random numbers} suffice. A series of | |
1200 | pseudo-random numbers is generated in a deterministic fashion. The | |
1201 | numbers are not truly random, but they have certain properties that | |
1202 | mimic a random series. For example, all possible values occur equally | |
1203 | often in a pseudo-random series. | |
1204 | ||
1205 | In Emacs, pseudo-random numbers are generated from a ``seed'' number. | |
1206 | Starting from any given seed, the @code{random} function always | |
1207 | generates the same sequence of numbers. Emacs always starts with the | |
1208 | same seed value, so the sequence of values of @code{random} is actually | |
1209 | the same in each Emacs run! For example, in one operating system, the | |
1210 | first call to @code{(random)} after you start Emacs always returns | |
1211 | @minus{}1457731, and the second one always returns @minus{}7692030. This | |
1212 | repeatability is helpful for debugging. | |
1213 | ||
1214 | If you want random numbers that don't always come out the same, execute | |
1215 | @code{(random t)}. This chooses a new seed based on the current time of | |
1216 | day and on Emacs's process @acronym{ID} number. | |
1217 | ||
1218 | @defun random &optional limit | |
1219 | This function returns a pseudo-random integer. Repeated calls return a | |
1220 | series of pseudo-random integers. | |
1221 | ||
1222 | If @var{limit} is a positive integer, the value is chosen to be | |
1223 | nonnegative and less than @var{limit}. | |
1224 | ||
1225 | If @var{limit} is @code{t}, it means to choose a new seed based on the | |
1226 | current time of day and on Emacs's process @acronym{ID} number. | |
b8d4c8d0 GM |
1227 | |
1228 | On some machines, any integer representable in Lisp may be the result | |
1229 | of @code{random}. On other machines, the result can never be larger | |
1230 | than a certain maximum or less than a certain (negative) minimum. | |
1231 | @end defun |