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1 | @c -*-texinfo-*- |
2 | @c This is part of the GNU Emacs Lisp Reference Manual. | |
ba318903 | 3 | @c Copyright (C) 1990-1995, 1998-1999, 2001-2014 Free Software |
ab422c4d | 4 | @c Foundation, Inc. |
b8d4c8d0 | 5 | @c See the file elisp.texi for copying conditions. |
ecc6530d | 6 | @node Numbers |
b8d4c8d0 GM |
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 | |
09b73f08 | 13 | @minus{}3, 0, 7, 13, and 511. Their values are exact. Floating-point |
b8d4c8d0 GM |
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 | 26 | * Arithmetic Operations:: How to add, subtract, multiply and divide. |
09b73f08 | 27 | * Rounding Operations:: Explicitly rounding floating-point numbers. |
b8d4c8d0 GM |
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 | |
b8d4c8d0 GM |
34 | @section Integer Basics |
35 | ||
36 | The range of values for an integer depends on the machine. The | |
1ddd6622 | 37 | minimum range is @minus{}536870912 to 536870911 (30 bits; i.e., |
b8d4c8d0 | 38 | @ifnottex |
09b73f08 | 39 | @minus{}2**29 |
b8d4c8d0 GM |
40 | @end ifnottex |
41 | @tex | |
1ddd6622 | 42 | @math{-2^{29}} |
b8d4c8d0 GM |
43 | @end tex |
44 | to | |
45 | @ifnottex | |
f99f1641 | 46 | 2**29 @minus{} 1), |
b8d4c8d0 GM |
47 | @end ifnottex |
48 | @tex | |
1ddd6622 | 49 | @math{2^{29}-1}), |
b8d4c8d0 | 50 | @end tex |
48de8b12 CY |
51 | but many machines provide a wider range. Many examples in this |
52 | chapter assume the minimum integer width of 30 bits. | |
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53 | @cindex overflow |
54 | ||
55 | The Lisp reader reads an integer as a sequence of digits with optional | |
fed14fd7 PE |
56 | initial sign and optional final period. An integer that is out of the |
57 | Emacs range is treated as a floating-point number. | |
b8d4c8d0 GM |
58 | |
59 | @example | |
60 | 1 ; @r{The integer 1.} | |
61 | 1. ; @r{The integer 1.} | |
62 | +1 ; @r{Also the integer 1.} | |
63 | -1 ; @r{The integer @minus{}1.} | |
fed14fd7 | 64 | 1073741825 ; @r{The floating point number 1073741825.0.} |
b8d4c8d0 GM |
65 | 0 ; @r{The integer 0.} |
66 | -0 ; @r{The integer 0.} | |
67 | @end example | |
68 | ||
69 | @cindex integers in specific radix | |
70 | @cindex radix for reading an integer | |
71 | @cindex base for reading an integer | |
72 | @cindex hex numbers | |
73 | @cindex octal numbers | |
74 | @cindex reading numbers in hex, octal, and binary | |
75 | The syntax for integers in bases other than 10 uses @samp{#} | |
76 | followed by a letter that specifies the radix: @samp{b} for binary, | |
77 | @samp{o} for octal, @samp{x} for hex, or @samp{@var{radix}r} to | |
78 | specify radix @var{radix}. Case is not significant for the letter | |
79 | that specifies the radix. Thus, @samp{#b@var{integer}} reads | |
80 | @var{integer} in binary, and @samp{#@var{radix}r@var{integer}} reads | |
81 | @var{integer} in radix @var{radix}. Allowed values of @var{radix} run | |
82 | from 2 to 36. For example: | |
83 | ||
84 | @example | |
85 | #b101100 @result{} 44 | |
86 | #o54 @result{} 44 | |
87 | #x2c @result{} 44 | |
88 | #24r1k @result{} 44 | |
89 | @end example | |
90 | ||
91 | To understand how various functions work on integers, especially the | |
92 | bitwise operators (@pxref{Bitwise Operations}), it is often helpful to | |
93 | view the numbers in their binary form. | |
94 | ||
1ddd6622 | 95 | In 30-bit binary, the decimal integer 5 looks like this: |
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96 | |
97 | @example | |
001903b5 | 98 | 0000...000101 (30 bits total) |
b8d4c8d0 GM |
99 | @end example |
100 | ||
101 | @noindent | |
001903b5 PE |
102 | (The @samp{...} stands for enough bits to fill out a 30-bit word; in |
103 | this case, @samp{...} stands for twenty 0 bits. Later examples also | |
104 | use the @samp{...} notation to make binary integers easier to read.) | |
b8d4c8d0 GM |
105 | |
106 | The integer @minus{}1 looks like this: | |
107 | ||
108 | @example | |
001903b5 | 109 | 1111...111111 (30 bits total) |
b8d4c8d0 GM |
110 | @end example |
111 | ||
112 | @noindent | |
113 | @cindex two's complement | |
1ddd6622 | 114 | @minus{}1 is represented as 30 ones. (This is called @dfn{two's |
b8d4c8d0 GM |
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 | |
001903b5 | 122 | 1111...111011 (30 bits total) |
b8d4c8d0 GM |
123 | @end example |
124 | ||
09b73f08 | 125 | In this implementation, the largest 30-bit binary integer is |
1ddd6622 | 126 | 536,870,911 in decimal. In binary, it looks like this: |
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127 | |
128 | @example | |
001903b5 | 129 | 0111...111111 (30 bits total) |
b8d4c8d0 GM |
130 | @end example |
131 | ||
132 | Since the arithmetic functions do not check whether integers go | |
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133 | outside their range, when you add 1 to 536,870,911, the value is the |
134 | negative integer @minus{}536,870,912: | |
b8d4c8d0 GM |
135 | |
136 | @example | |
1ddd6622 GM |
137 | (+ 1 536870911) |
138 | @result{} -536870912 | |
001903b5 | 139 | @result{} 1000...000000 (30 bits total) |
b8d4c8d0 GM |
140 | @end example |
141 | ||
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 | |
146 | value is a marker, its position value is used and its buffer is ignored. | |
147 | ||
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148 | @cindex largest Lisp integer |
149 | @cindex maximum Lisp integer | |
b8d4c8d0 GM |
150 | @defvar most-positive-fixnum |
151 | The value of this variable is the largest integer that Emacs Lisp | |
152 | can handle. | |
153 | @end defvar | |
154 | ||
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155 | @cindex smallest Lisp integer |
156 | @cindex minimum Lisp integer | |
b8d4c8d0 GM |
157 | @defvar most-negative-fixnum |
158 | The value of this variable is the smallest integer that Emacs Lisp can | |
159 | handle. It is negative. | |
160 | @end defvar | |
161 | ||
48de8b12 CY |
162 | In Emacs Lisp, text characters are represented by integers. Any |
163 | integer between zero and the value of @code{max-char}, inclusive, is | |
164 | considered to be valid as a character. @xref{String Basics}. | |
57e2db6d | 165 | |
b8d4c8d0 | 166 | @node Float Basics |
09b73f08 | 167 | @section Floating-Point Basics |
b8d4c8d0 | 168 | |
fead402d | 169 | @cindex @acronym{IEEE} floating point |
09b73f08 PE |
170 | Floating-point numbers are useful for representing numbers that are |
171 | not integral. The precise range of floating-point numbers is | |
b8d4c8d0 | 172 | machine-specific; it is the same as the range of the C data type |
fead402d | 173 | @code{double} on the machine you are using. Emacs uses the |
09b73f08 | 174 | @acronym{IEEE} floating-point standard, which is supported by all |
48de8b12 | 175 | modern computers. |
b8d4c8d0 | 176 | |
09b73f08 | 177 | The read syntax for floating-point numbers requires either a decimal |
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178 | point (with at least one digit following), an exponent, or both. For |
179 | example, @samp{1500.0}, @samp{15e2}, @samp{15.0e2}, @samp{1.5e3}, and | |
09b73f08 | 180 | @samp{.15e4} are five ways of writing a floating-point number whose |
fead402d | 181 | value is 1500. They are all equivalent. You can also use a minus |
09b73f08 | 182 | sign to write negative floating-point numbers, as in @samp{-1.0}. |
fead402d | 183 | |
09b73f08 | 184 | Emacs Lisp treats @code{-0.0} as numerically equal to ordinary zero (with |
fead402d | 185 | respect to @code{equal} and @code{=}), even though the two are |
09b73f08 | 186 | distinguishable in the @acronym{IEEE} floating-point standard. |
b8d4c8d0 | 187 | |
b8d4c8d0 GM |
188 | @cindex positive infinity |
189 | @cindex negative infinity | |
190 | @cindex infinity | |
191 | @cindex NaN | |
09b73f08 PE |
192 | The @acronym{IEEE} floating-point standard supports positive |
193 | infinity and negative infinity as floating-point values. It also | |
fead402d CY |
194 | provides for a class of values called NaN or ``not-a-number''; |
195 | numerical functions return such values in cases where there is no | |
1df7defd | 196 | correct answer. For example, @code{(/ 0.0 0.0)} returns a NaN@. (NaN |
fead402d CY |
197 | values can also carry a sign, but for practical purposes there's no |
198 | significant difference between different NaN values in Emacs Lisp.) | |
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199 | |
200 | When a function is documented to return a NaN, it returns an | |
201 | implementation-defined value when Emacs is running on one of the | |
202 | now-rare platforms that do not use @acronym{IEEE} floating point. For | |
203 | example, @code{(log -1.0)} typically returns a NaN, but on | |
204 | non-@acronym{IEEE} platforms it returns an implementation-defined | |
205 | value. | |
206 | ||
09b73f08 | 207 | Here are the read syntaxes for these special floating-point values: |
b8d4c8d0 GM |
208 | |
209 | @table @asis | |
210 | @item positive infinity | |
211 | @samp{1.0e+INF} | |
212 | @item negative infinity | |
213 | @samp{-1.0e+INF} | |
fed14fd7 | 214 | @item Not-a-number |
b8d4c8d0 GM |
215 | @samp{0.0e+NaN} or @samp{-0.0e+NaN}. |
216 | @end table | |
217 | ||
fead402d CY |
218 | @defun isnan number |
219 | This predicate tests whether its argument is NaN, and returns @code{t} | |
220 | if so, @code{nil} otherwise. The argument must be a number. | |
221 | @end defun | |
222 | ||
223 | The following functions are specialized for handling floating point | |
224 | numbers: | |
225 | ||
226 | @defun frexp x | |
227 | This function returns a cons cell @code{(@var{sig} . @var{exp})}, | |
228 | where @var{sig} and @var{exp} are respectively the significand and | |
229 | exponent of the floating point number @var{x}: | |
230 | ||
231 | @smallexample | |
232 | @var{x} = @var{sig} * 2^@var{exp} | |
233 | @end smallexample | |
234 | ||
235 | @var{sig} is a floating point number between 0.5 (inclusive) and 1.0 | |
236 | (exclusive). If @var{x} is zero, the return value is @code{(0 . 0)}. | |
237 | @end defun | |
b8d4c8d0 | 238 | |
fead402d CY |
239 | @defun ldexp sig &optional exp |
240 | This function returns a floating point number corresponding to the | |
241 | significand @var{sig} and exponent @var{exp}. | |
242 | @end defun | |
b8d4c8d0 | 243 | |
fead402d CY |
244 | @defun copysign x1 x2 |
245 | This function copies the sign of @var{x2} to the value of @var{x1}, | |
246 | and returns the result. @var{x1} and @var{x2} must be floating point | |
247 | numbers. | |
248 | @end defun | |
b8d4c8d0 GM |
249 | |
250 | @defun logb number | |
251 | This function returns the binary exponent of @var{number}. More | |
c990426a | 252 | precisely, the value is the logarithm of |@var{number}| base 2, rounded |
b8d4c8d0 GM |
253 | down to an integer. |
254 | ||
255 | @example | |
256 | (logb 10) | |
257 | @result{} 3 | |
258 | (logb 10.0e20) | |
259 | @result{} 69 | |
260 | @end example | |
261 | @end defun | |
262 | ||
263 | @node Predicates on Numbers | |
264 | @section Type Predicates for Numbers | |
265 | @cindex predicates for numbers | |
266 | ||
267 | The functions in this section test for numbers, or for a specific | |
268 | type of number. The functions @code{integerp} and @code{floatp} can | |
269 | take any type of Lisp object as argument (they would not be of much | |
270 | use otherwise), but the @code{zerop} predicate requires a number as | |
271 | its argument. See also @code{integer-or-marker-p} and | |
272 | @code{number-or-marker-p}, in @ref{Predicates on Markers}. | |
273 | ||
274 | @defun floatp object | |
09b73f08 PE |
275 | This predicate tests whether its argument is floating point |
276 | and returns @code{t} if so, @code{nil} otherwise. | |
b8d4c8d0 GM |
277 | @end defun |
278 | ||
279 | @defun integerp object | |
280 | This predicate tests whether its argument is an integer, and returns | |
281 | @code{t} if so, @code{nil} otherwise. | |
282 | @end defun | |
283 | ||
284 | @defun numberp object | |
285 | This predicate tests whether its argument is a number (either integer or | |
286 | floating point), and returns @code{t} if so, @code{nil} otherwise. | |
287 | @end defun | |
288 | ||
0f29fa41 | 289 | @defun natnump object |
b8d4c8d0 | 290 | @cindex natural numbers |
bccc0e40 | 291 | This predicate (whose name comes from the phrase ``natural number'') |
0f29fa41 CY |
292 | tests to see whether its argument is a nonnegative integer, and |
293 | returns @code{t} if so, @code{nil} otherwise. 0 is considered | |
294 | non-negative. | |
b8d4c8d0 | 295 | |
0f29fa41 CY |
296 | @findex wholenump number |
297 | This is a synonym for @code{natnump}. | |
b8d4c8d0 GM |
298 | @end defun |
299 | ||
300 | @defun zerop number | |
301 | This predicate tests whether its argument is zero, and returns @code{t} | |
302 | if so, @code{nil} otherwise. The argument must be a number. | |
303 | ||
304 | @code{(zerop x)} is equivalent to @code{(= x 0)}. | |
305 | @end defun | |
306 | ||
307 | @node Comparison of Numbers | |
308 | @section Comparison of Numbers | |
309 | @cindex number comparison | |
310 | @cindex comparing numbers | |
311 | ||
312 | To test numbers for numerical equality, you should normally use | |
09b73f08 PE |
313 | @code{=}, not @code{eq}. There can be many distinct floating-point |
314 | objects with the same numeric value. If you use @code{eq} to | |
b8d4c8d0 GM |
315 | compare them, then you test whether two values are the same |
316 | @emph{object}. By contrast, @code{=} compares only the numeric values | |
317 | of the objects. | |
318 | ||
09b73f08 | 319 | In Emacs Lisp, each integer is a unique Lisp object. |
b8d4c8d0 | 320 | Therefore, @code{eq} is equivalent to @code{=} where integers are |
48de8b12 CY |
321 | concerned. It is sometimes convenient to use @code{eq} for comparing |
322 | an unknown value with an integer, because @code{eq} does not report an | |
323 | error if the unknown value is not a number---it accepts arguments of | |
324 | any type. By contrast, @code{=} signals an error if the arguments are | |
325 | not numbers or markers. However, it is better programming practice to | |
326 | use @code{=} if you can, even for comparing integers. | |
327 | ||
328 | Sometimes it is useful to compare numbers with @code{equal}, which | |
b8d4c8d0 GM |
329 | treats two numbers as equal if they have the same data type (both |
330 | integers, or both floating point) and the same value. By contrast, | |
09b73f08 | 331 | @code{=} can treat an integer and a floating-point number as equal. |
b8d4c8d0 GM |
332 | @xref{Equality Predicates}. |
333 | ||
09b73f08 PE |
334 | There is another wrinkle: because floating-point arithmetic is not |
335 | exact, it is often a bad idea to check for equality of floating-point | |
336 | values. Usually it is better to test for approximate equality. | |
b8d4c8d0 GM |
337 | Here's a function to do this: |
338 | ||
339 | @example | |
340 | (defvar fuzz-factor 1.0e-6) | |
341 | (defun approx-equal (x y) | |
342 | (or (and (= x 0) (= y 0)) | |
343 | (< (/ (abs (- x y)) | |
344 | (max (abs x) (abs y))) | |
345 | fuzz-factor))) | |
346 | @end example | |
347 | ||
348 | @cindex CL note---integers vrs @code{eq} | |
349 | @quotation | |
350 | @b{Common Lisp note:} Comparing numbers in Common Lisp always requires | |
351 | @code{=} because Common Lisp implements multi-word integers, and two | |
352 | distinct integer objects can have the same numeric value. Emacs Lisp | |
353 | can have just one integer object for any given value because it has a | |
09b73f08 | 354 | limited range of integers. |
b8d4c8d0 GM |
355 | @end quotation |
356 | ||
3fbba716 TH |
357 | @defun = number-or-marker &rest number-or-markers |
358 | This function tests whether all its arguments are numerically equal, | |
359 | and returns @code{t} if so, @code{nil} otherwise. | |
b8d4c8d0 GM |
360 | @end defun |
361 | ||
362 | @defun eql value1 value2 | |
363 | This function acts like @code{eq} except when both arguments are | |
364 | numbers. It compares numbers by type and numeric value, so that | |
365 | @code{(eql 1.0 1)} returns @code{nil}, but @code{(eql 1.0 1.0)} and | |
366 | @code{(eql 1 1)} both return @code{t}. | |
367 | @end defun | |
368 | ||
369 | @defun /= number-or-marker1 number-or-marker2 | |
370 | This function tests whether its arguments are numerically equal, and | |
371 | returns @code{t} if they are not, and @code{nil} if they are. | |
372 | @end defun | |
373 | ||
3fbba716 TH |
374 | @defun < number-or-marker &rest number-or-markers |
375 | This function tests whether every argument is strictly less than the | |
376 | respective next argument. It returns @code{t} if so, @code{nil} | |
377 | otherwise. | |
b8d4c8d0 GM |
378 | @end defun |
379 | ||
3fbba716 TH |
380 | @defun <= number-or-marker &rest number-or-markers |
381 | This function tests whether every argument is less than or equal to | |
382 | the respective next argument. It returns @code{t} if so, @code{nil} | |
b8d4c8d0 GM |
383 | otherwise. |
384 | @end defun | |
385 | ||
3fbba716 TH |
386 | @defun > number-or-marker &rest number-or-markers |
387 | This function tests whether every argument is strictly greater than | |
388 | the respective next argument. It returns @code{t} if so, @code{nil} | |
b8d4c8d0 GM |
389 | otherwise. |
390 | @end defun | |
391 | ||
3fbba716 TH |
392 | @defun >= number-or-marker &rest number-or-markers |
393 | This function tests whether every argument is greater than or equal to | |
394 | the respective next argument. It returns @code{t} if so, @code{nil} | |
b8d4c8d0 GM |
395 | otherwise. |
396 | @end defun | |
397 | ||
398 | @defun max number-or-marker &rest numbers-or-markers | |
399 | This function returns the largest of its arguments. | |
09b73f08 | 400 | If any of the arguments is floating point, the value is returned |
b8d4c8d0 GM |
401 | as floating point, even if it was given as an integer. |
402 | ||
403 | @example | |
404 | (max 20) | |
405 | @result{} 20 | |
406 | (max 1 2.5) | |
407 | @result{} 2.5 | |
408 | (max 1 3 2.5) | |
409 | @result{} 3.0 | |
410 | @end example | |
411 | @end defun | |
412 | ||
413 | @defun min number-or-marker &rest numbers-or-markers | |
414 | This function returns the smallest of its arguments. | |
09b73f08 | 415 | If any of the arguments is floating point, the value is returned |
b8d4c8d0 GM |
416 | as floating point, even if it was given as an integer. |
417 | ||
418 | @example | |
419 | (min -4 1) | |
420 | @result{} -4 | |
421 | @end example | |
422 | @end defun | |
423 | ||
424 | @defun abs number | |
425 | This function returns the absolute value of @var{number}. | |
426 | @end defun | |
427 | ||
428 | @node Numeric Conversions | |
429 | @section Numeric Conversions | |
430 | @cindex rounding in conversions | |
431 | @cindex number conversions | |
432 | @cindex converting numbers | |
433 | ||
434 | To convert an integer to floating point, use the function @code{float}. | |
435 | ||
436 | @defun float number | |
437 | This returns @var{number} converted to floating point. | |
09b73f08 | 438 | If @var{number} is already floating point, @code{float} returns |
b8d4c8d0 GM |
439 | it unchanged. |
440 | @end defun | |
441 | ||
09b73f08 | 442 | There are four functions to convert floating-point numbers to |
48de8b12 CY |
443 | integers; they differ in how they round. All accept an argument |
444 | @var{number} and an optional argument @var{divisor}. Both arguments | |
09b73f08 | 445 | may be integers or floating-point numbers. @var{divisor} may also be |
b8d4c8d0 GM |
446 | @code{nil}. If @var{divisor} is @code{nil} or omitted, these |
447 | functions convert @var{number} to an integer, or return it unchanged | |
448 | if it already is an integer. If @var{divisor} is non-@code{nil}, they | |
449 | divide @var{number} by @var{divisor} and convert the result to an | |
33f846fb | 450 | integer. If @var{divisor} is zero (whether integer or |
09b73f08 | 451 | floating point), Emacs signals an @code{arith-error} error. |
b8d4c8d0 GM |
452 | |
453 | @defun truncate number &optional divisor | |
454 | This returns @var{number}, converted to an integer by rounding towards | |
455 | zero. | |
456 | ||
457 | @example | |
458 | (truncate 1.2) | |
459 | @result{} 1 | |
460 | (truncate 1.7) | |
461 | @result{} 1 | |
462 | (truncate -1.2) | |
463 | @result{} -1 | |
464 | (truncate -1.7) | |
465 | @result{} -1 | |
466 | @end example | |
467 | @end defun | |
468 | ||
469 | @defun floor number &optional divisor | |
470 | This returns @var{number}, converted to an integer by rounding downward | |
471 | (towards negative infinity). | |
472 | ||
473 | If @var{divisor} is specified, this uses the kind of division | |
474 | operation that corresponds to @code{mod}, rounding downward. | |
475 | ||
476 | @example | |
477 | (floor 1.2) | |
478 | @result{} 1 | |
479 | (floor 1.7) | |
480 | @result{} 1 | |
481 | (floor -1.2) | |
482 | @result{} -2 | |
483 | (floor -1.7) | |
484 | @result{} -2 | |
485 | (floor 5.99 3) | |
486 | @result{} 1 | |
487 | @end example | |
488 | @end defun | |
489 | ||
490 | @defun ceiling number &optional divisor | |
491 | This returns @var{number}, converted to an integer by rounding upward | |
492 | (towards positive infinity). | |
493 | ||
494 | @example | |
495 | (ceiling 1.2) | |
496 | @result{} 2 | |
497 | (ceiling 1.7) | |
498 | @result{} 2 | |
499 | (ceiling -1.2) | |
500 | @result{} -1 | |
501 | (ceiling -1.7) | |
502 | @result{} -1 | |
503 | @end example | |
504 | @end defun | |
505 | ||
506 | @defun round number &optional divisor | |
507 | This returns @var{number}, converted to an integer by rounding towards the | |
508 | nearest integer. Rounding a value equidistant between two integers | |
509 | may choose the integer closer to zero, or it may prefer an even integer, | |
510 | depending on your machine. | |
511 | ||
512 | @example | |
513 | (round 1.2) | |
514 | @result{} 1 | |
515 | (round 1.7) | |
516 | @result{} 2 | |
517 | (round -1.2) | |
518 | @result{} -1 | |
519 | (round -1.7) | |
520 | @result{} -2 | |
521 | @end example | |
522 | @end defun | |
523 | ||
524 | @node Arithmetic Operations | |
525 | @section Arithmetic Operations | |
526 | @cindex arithmetic operations | |
527 | ||
48de8b12 CY |
528 | Emacs Lisp provides the traditional four arithmetic operations |
529 | (addition, subtraction, multiplication, and division), as well as | |
530 | remainder and modulus functions, and functions to add or subtract 1. | |
531 | Except for @code{%}, each of these functions accepts both integer and | |
09b73f08 PE |
532 | floating-point arguments, and returns a floating-point number if any |
533 | argument is floating point. | |
b8d4c8d0 | 534 | |
c717b326 | 535 | It is important to note that in Emacs Lisp, arithmetic functions |
001903b5 PE |
536 | do not check for overflow. Thus @code{(1+ 536870911)} may evaluate to |
537 | @minus{}536870912, depending on your hardware. | |
b8d4c8d0 GM |
538 | |
539 | @defun 1+ number-or-marker | |
540 | This function returns @var{number-or-marker} plus 1. | |
541 | For example, | |
542 | ||
543 | @example | |
544 | (setq foo 4) | |
545 | @result{} 4 | |
546 | (1+ foo) | |
547 | @result{} 5 | |
548 | @end example | |
549 | ||
550 | This function is not analogous to the C operator @code{++}---it does not | |
551 | increment a variable. It just computes a sum. Thus, if we continue, | |
552 | ||
553 | @example | |
554 | foo | |
555 | @result{} 4 | |
556 | @end example | |
557 | ||
558 | If you want to increment the variable, you must use @code{setq}, | |
559 | like this: | |
560 | ||
561 | @example | |
562 | (setq foo (1+ foo)) | |
563 | @result{} 5 | |
564 | @end example | |
565 | @end defun | |
566 | ||
567 | @defun 1- number-or-marker | |
568 | This function returns @var{number-or-marker} minus 1. | |
569 | @end defun | |
570 | ||
571 | @defun + &rest numbers-or-markers | |
572 | This function adds its arguments together. When given no arguments, | |
573 | @code{+} returns 0. | |
574 | ||
575 | @example | |
576 | (+) | |
577 | @result{} 0 | |
578 | (+ 1) | |
579 | @result{} 1 | |
580 | (+ 1 2 3 4) | |
581 | @result{} 10 | |
582 | @end example | |
583 | @end defun | |
584 | ||
585 | @defun - &optional number-or-marker &rest more-numbers-or-markers | |
586 | The @code{-} function serves two purposes: negation and subtraction. | |
587 | When @code{-} has a single argument, the value is the negative of the | |
588 | argument. When there are multiple arguments, @code{-} subtracts each of | |
589 | the @var{more-numbers-or-markers} from @var{number-or-marker}, | |
590 | cumulatively. If there are no arguments, the result is 0. | |
591 | ||
592 | @example | |
593 | (- 10 1 2 3 4) | |
594 | @result{} 0 | |
595 | (- 10) | |
596 | @result{} -10 | |
597 | (-) | |
598 | @result{} 0 | |
599 | @end example | |
600 | @end defun | |
601 | ||
602 | @defun * &rest numbers-or-markers | |
603 | This function multiplies its arguments together, and returns the | |
604 | product. When given no arguments, @code{*} returns 1. | |
605 | ||
606 | @example | |
607 | (*) | |
608 | @result{} 1 | |
609 | (* 1) | |
610 | @result{} 1 | |
611 | (* 1 2 3 4) | |
612 | @result{} 24 | |
613 | @end example | |
614 | @end defun | |
615 | ||
616 | @defun / dividend divisor &rest divisors | |
617 | This function divides @var{dividend} by @var{divisor} and returns the | |
618 | quotient. If there are additional arguments @var{divisors}, then it | |
619 | divides @var{dividend} by each divisor in turn. Each argument may be a | |
620 | number or a marker. | |
621 | ||
48de8b12 CY |
622 | If all the arguments are integers, the result is an integer, obtained |
623 | by rounding the quotient towards zero after each division. | |
624 | (Hypothetically, some machines may have different rounding behavior | |
625 | for negative arguments, because @code{/} is implemented using the C | |
626 | division operator, which permits machine-dependent rounding; but this | |
627 | does not happen in practice.) | |
b8d4c8d0 GM |
628 | |
629 | @example | |
630 | @group | |
631 | (/ 6 2) | |
632 | @result{} 3 | |
633 | @end group | |
48de8b12 | 634 | @group |
b8d4c8d0 GM |
635 | (/ 5 2) |
636 | @result{} 2 | |
48de8b12 CY |
637 | @end group |
638 | @group | |
b8d4c8d0 GM |
639 | (/ 5.0 2) |
640 | @result{} 2.5 | |
48de8b12 CY |
641 | @end group |
642 | @group | |
b8d4c8d0 GM |
643 | (/ 5 2.0) |
644 | @result{} 2.5 | |
48de8b12 CY |
645 | @end group |
646 | @group | |
b8d4c8d0 GM |
647 | (/ 5.0 2.0) |
648 | @result{} 2.5 | |
48de8b12 CY |
649 | @end group |
650 | @group | |
b8d4c8d0 GM |
651 | (/ 25 3 2) |
652 | @result{} 4 | |
48de8b12 | 653 | @end group |
b8d4c8d0 GM |
654 | @group |
655 | (/ -17 6) | |
48de8b12 | 656 | @result{} -2 |
b8d4c8d0 GM |
657 | @end group |
658 | @end example | |
48de8b12 CY |
659 | |
660 | @cindex @code{arith-error} in division | |
661 | If you divide an integer by the integer 0, Emacs signals an | |
09b73f08 PE |
662 | @code{arith-error} error (@pxref{Errors}). Floating-point division of |
663 | a nonzero number by zero yields either positive or negative infinity | |
664 | (@pxref{Float Basics}). | |
b8d4c8d0 GM |
665 | @end defun |
666 | ||
667 | @defun % dividend divisor | |
668 | @cindex remainder | |
669 | This function returns the integer remainder after division of @var{dividend} | |
670 | by @var{divisor}. The arguments must be integers or markers. | |
671 | ||
48de8b12 CY |
672 | For any two integers @var{dividend} and @var{divisor}, |
673 | ||
674 | @example | |
675 | @group | |
676 | (+ (% @var{dividend} @var{divisor}) | |
677 | (* (/ @var{dividend} @var{divisor}) @var{divisor})) | |
678 | @end group | |
679 | @end example | |
b8d4c8d0 | 680 | |
48de8b12 CY |
681 | @noindent |
682 | always equals @var{dividend}. If @var{divisor} is zero, Emacs signals | |
683 | an @code{arith-error} error. | |
b8d4c8d0 GM |
684 | |
685 | @example | |
686 | (% 9 4) | |
687 | @result{} 1 | |
688 | (% -9 4) | |
689 | @result{} -1 | |
690 | (% 9 -4) | |
691 | @result{} 1 | |
692 | (% -9 -4) | |
693 | @result{} -1 | |
694 | @end example | |
b8d4c8d0 GM |
695 | @end defun |
696 | ||
697 | @defun mod dividend divisor | |
698 | @cindex modulus | |
699 | This function returns the value of @var{dividend} modulo @var{divisor}; | |
700 | in other words, the remainder after division of @var{dividend} | |
701 | by @var{divisor}, but with the same sign as @var{divisor}. | |
702 | The arguments must be numbers or markers. | |
703 | ||
09b73f08 | 704 | Unlike @code{%}, @code{mod} permits floating-point arguments; it |
48de8b12 CY |
705 | rounds the quotient downward (towards minus infinity) to an integer, |
706 | and uses that quotient to compute the remainder. | |
b8d4c8d0 | 707 | |
c990426a PE |
708 | If @var{divisor} is zero, @code{mod} signals an @code{arith-error} |
709 | error if both arguments are integers, and returns a NaN otherwise. | |
b8d4c8d0 GM |
710 | |
711 | @example | |
712 | @group | |
713 | (mod 9 4) | |
714 | @result{} 1 | |
715 | @end group | |
716 | @group | |
717 | (mod -9 4) | |
718 | @result{} 3 | |
719 | @end group | |
720 | @group | |
721 | (mod 9 -4) | |
722 | @result{} -3 | |
723 | @end group | |
724 | @group | |
725 | (mod -9 -4) | |
726 | @result{} -1 | |
727 | @end group | |
728 | @group | |
729 | (mod 5.5 2.5) | |
730 | @result{} .5 | |
731 | @end group | |
732 | @end example | |
733 | ||
734 | For any two numbers @var{dividend} and @var{divisor}, | |
735 | ||
736 | @example | |
737 | @group | |
738 | (+ (mod @var{dividend} @var{divisor}) | |
739 | (* (floor @var{dividend} @var{divisor}) @var{divisor})) | |
740 | @end group | |
741 | @end example | |
742 | ||
743 | @noindent | |
744 | always equals @var{dividend}, subject to rounding error if either | |
745 | argument is floating point. For @code{floor}, see @ref{Numeric | |
746 | Conversions}. | |
747 | @end defun | |
748 | ||
749 | @node Rounding Operations | |
750 | @section Rounding Operations | |
751 | @cindex rounding without conversion | |
752 | ||
753 | The functions @code{ffloor}, @code{fceiling}, @code{fround}, and | |
09b73f08 PE |
754 | @code{ftruncate} take a floating-point argument and return a floating-point |
755 | result whose value is a nearby integer. @code{ffloor} returns the | |
b8d4c8d0 GM |
756 | nearest integer below; @code{fceiling}, the nearest integer above; |
757 | @code{ftruncate}, the nearest integer in the direction towards zero; | |
758 | @code{fround}, the nearest integer. | |
759 | ||
760 | @defun ffloor float | |
761 | This function rounds @var{float} to the next lower integral value, and | |
09b73f08 | 762 | returns that value as a floating-point number. |
b8d4c8d0 GM |
763 | @end defun |
764 | ||
765 | @defun fceiling float | |
766 | This function rounds @var{float} to the next higher integral value, and | |
09b73f08 | 767 | returns that value as a floating-point number. |
b8d4c8d0 GM |
768 | @end defun |
769 | ||
770 | @defun ftruncate float | |
771 | This function rounds @var{float} towards zero to an integral value, and | |
09b73f08 | 772 | returns that value as a floating-point number. |
b8d4c8d0 GM |
773 | @end defun |
774 | ||
775 | @defun fround float | |
776 | This function rounds @var{float} to the nearest integral value, | |
09b73f08 | 777 | and returns that value as a floating-point number. |
b8d4c8d0 GM |
778 | @end defun |
779 | ||
780 | @node Bitwise Operations | |
781 | @section Bitwise Operations on Integers | |
782 | @cindex bitwise arithmetic | |
783 | @cindex logical arithmetic | |
784 | ||
785 | In a computer, an integer is represented as a binary number, a | |
786 | sequence of @dfn{bits} (digits which are either zero or one). A bitwise | |
787 | operation acts on the individual bits of such a sequence. For example, | |
788 | @dfn{shifting} moves the whole sequence left or right one or more places, | |
16152b76 | 789 | reproducing the same pattern ``moved over''. |
b8d4c8d0 GM |
790 | |
791 | The bitwise operations in Emacs Lisp apply only to integers. | |
792 | ||
793 | @defun lsh integer1 count | |
794 | @cindex logical shift | |
795 | @code{lsh}, which is an abbreviation for @dfn{logical shift}, shifts the | |
796 | bits in @var{integer1} to the left @var{count} places, or to the right | |
797 | if @var{count} is negative, bringing zeros into the vacated bits. If | |
798 | @var{count} is negative, @code{lsh} shifts zeros into the leftmost | |
799 | (most-significant) bit, producing a positive result even if | |
800 | @var{integer1} is negative. Contrast this with @code{ash}, below. | |
801 | ||
802 | Here are two examples of @code{lsh}, shifting a pattern of bits one | |
803 | place to the left. We show only the low-order eight bits of the binary | |
804 | pattern; the rest are all zero. | |
805 | ||
806 | @example | |
807 | @group | |
808 | (lsh 5 1) | |
809 | @result{} 10 | |
810 | ;; @r{Decimal 5 becomes decimal 10.} | |
811 | 00000101 @result{} 00001010 | |
812 | ||
813 | (lsh 7 1) | |
814 | @result{} 14 | |
815 | ;; @r{Decimal 7 becomes decimal 14.} | |
816 | 00000111 @result{} 00001110 | |
817 | @end group | |
818 | @end example | |
819 | ||
820 | @noindent | |
821 | As the examples illustrate, shifting the pattern of bits one place to | |
822 | the left produces a number that is twice the value of the previous | |
823 | number. | |
824 | ||
825 | Shifting a pattern of bits two places to the left produces results | |
826 | like this (with 8-bit binary numbers): | |
827 | ||
828 | @example | |
829 | @group | |
830 | (lsh 3 2) | |
831 | @result{} 12 | |
832 | ;; @r{Decimal 3 becomes decimal 12.} | |
833 | 00000011 @result{} 00001100 | |
834 | @end group | |
835 | @end example | |
836 | ||
837 | On the other hand, shifting one place to the right looks like this: | |
838 | ||
839 | @example | |
840 | @group | |
841 | (lsh 6 -1) | |
842 | @result{} 3 | |
843 | ;; @r{Decimal 6 becomes decimal 3.} | |
844 | 00000110 @result{} 00000011 | |
845 | @end group | |
846 | ||
847 | @group | |
848 | (lsh 5 -1) | |
849 | @result{} 2 | |
850 | ;; @r{Decimal 5 becomes decimal 2.} | |
851 | 00000101 @result{} 00000010 | |
852 | @end group | |
853 | @end example | |
854 | ||
855 | @noindent | |
856 | As the example illustrates, shifting one place to the right divides the | |
857 | value of a positive integer by two, rounding downward. | |
858 | ||
c717b326 | 859 | The function @code{lsh}, like all Emacs Lisp arithmetic functions, does |
b8d4c8d0 GM |
860 | not check for overflow, so shifting left can discard significant bits |
861 | and change the sign of the number. For example, left shifting | |
001903b5 | 862 | 536,870,911 produces @minus{}2 in the 30-bit implementation: |
b8d4c8d0 GM |
863 | |
864 | @example | |
1ddd6622 | 865 | (lsh 536870911 1) ; @r{left shift} |
b8d4c8d0 GM |
866 | @result{} -2 |
867 | @end example | |
868 | ||
001903b5 | 869 | In binary, the argument looks like this: |
b8d4c8d0 GM |
870 | |
871 | @example | |
872 | @group | |
1ddd6622 | 873 | ;; @r{Decimal 536,870,911} |
001903b5 | 874 | 0111...111111 (30 bits total) |
b8d4c8d0 GM |
875 | @end group |
876 | @end example | |
877 | ||
878 | @noindent | |
879 | which becomes the following when left shifted: | |
880 | ||
881 | @example | |
882 | @group | |
883 | ;; @r{Decimal @minus{}2} | |
001903b5 | 884 | 1111...111110 (30 bits total) |
b8d4c8d0 GM |
885 | @end group |
886 | @end example | |
887 | @end defun | |
888 | ||
889 | @defun ash integer1 count | |
890 | @cindex arithmetic shift | |
891 | @code{ash} (@dfn{arithmetic shift}) shifts the bits in @var{integer1} | |
892 | to the left @var{count} places, or to the right if @var{count} | |
893 | is negative. | |
894 | ||
895 | @code{ash} gives the same results as @code{lsh} except when | |
896 | @var{integer1} and @var{count} are both negative. In that case, | |
897 | @code{ash} puts ones in the empty bit positions on the left, while | |
898 | @code{lsh} puts zeros in those bit positions. | |
899 | ||
900 | Thus, with @code{ash}, shifting the pattern of bits one place to the right | |
901 | looks like this: | |
902 | ||
903 | @example | |
904 | @group | |
905 | (ash -6 -1) @result{} -3 | |
906 | ;; @r{Decimal @minus{}6 becomes decimal @minus{}3.} | |
001903b5 | 907 | 1111...111010 (30 bits total) |
b8d4c8d0 | 908 | @result{} |
001903b5 | 909 | 1111...111101 (30 bits total) |
b8d4c8d0 GM |
910 | @end group |
911 | @end example | |
912 | ||
913 | In contrast, shifting the pattern of bits one place to the right with | |
914 | @code{lsh} looks like this: | |
915 | ||
916 | @example | |
917 | @group | |
1ddd6622 GM |
918 | (lsh -6 -1) @result{} 536870909 |
919 | ;; @r{Decimal @minus{}6 becomes decimal 536,870,909.} | |
001903b5 | 920 | 1111...111010 (30 bits total) |
b8d4c8d0 | 921 | @result{} |
001903b5 | 922 | 0111...111101 (30 bits total) |
b8d4c8d0 GM |
923 | @end group |
924 | @end example | |
925 | ||
926 | Here are other examples: | |
927 | ||
928 | @c !!! Check if lined up in smallbook format! XDVI shows problem | |
929 | @c with smallbook but not with regular book! --rjc 16mar92 | |
930 | @smallexample | |
931 | @group | |
001903b5 | 932 | ; @r{ 30-bit binary values} |
b8d4c8d0 | 933 | |
be14b9ab PE |
934 | (lsh 5 2) ; 5 = @r{0000...000101} |
935 | @result{} 20 ; = @r{0000...010100} | |
b8d4c8d0 GM |
936 | @end group |
937 | @group | |
938 | (ash 5 2) | |
939 | @result{} 20 | |
be14b9ab PE |
940 | (lsh -5 2) ; -5 = @r{1111...111011} |
941 | @result{} -20 ; = @r{1111...101100} | |
b8d4c8d0 GM |
942 | (ash -5 2) |
943 | @result{} -20 | |
944 | @end group | |
945 | @group | |
be14b9ab PE |
946 | (lsh 5 -2) ; 5 = @r{0000...000101} |
947 | @result{} 1 ; = @r{0000...000001} | |
b8d4c8d0 GM |
948 | @end group |
949 | @group | |
950 | (ash 5 -2) | |
951 | @result{} 1 | |
952 | @end group | |
953 | @group | |
be14b9ab | 954 | (lsh -5 -2) ; -5 = @r{1111...111011} |
001903b5 | 955 | @result{} 268435454 |
be14b9ab | 956 | ; = @r{0011...111110} |
b8d4c8d0 GM |
957 | @end group |
958 | @group | |
be14b9ab PE |
959 | (ash -5 -2) ; -5 = @r{1111...111011} |
960 | @result{} -2 ; = @r{1111...111110} | |
b8d4c8d0 GM |
961 | @end group |
962 | @end smallexample | |
963 | @end defun | |
964 | ||
965 | @defun logand &rest ints-or-markers | |
966 | This function returns the ``logical and'' of the arguments: the | |
967 | @var{n}th bit is set in the result if, and only if, the @var{n}th bit is | |
968 | set in all the arguments. (``Set'' means that the value of the bit is 1 | |
969 | rather than 0.) | |
970 | ||
971 | For example, using 4-bit binary numbers, the ``logical and'' of 13 and | |
972 | 12 is 12: 1101 combined with 1100 produces 1100. | |
973 | In both the binary numbers, the leftmost two bits are set (i.e., they | |
974 | are 1's), so the leftmost two bits of the returned value are set. | |
975 | However, for the rightmost two bits, each is zero in at least one of | |
976 | the arguments, so the rightmost two bits of the returned value are 0's. | |
977 | ||
978 | @noindent | |
979 | Therefore, | |
980 | ||
981 | @example | |
982 | @group | |
983 | (logand 13 12) | |
984 | @result{} 12 | |
985 | @end group | |
986 | @end example | |
987 | ||
988 | If @code{logand} is not passed any argument, it returns a value of | |
989 | @minus{}1. This number is an identity element for @code{logand} | |
990 | because its binary representation consists entirely of ones. If | |
991 | @code{logand} is passed just one argument, it returns that argument. | |
992 | ||
993 | @smallexample | |
994 | @group | |
001903b5 | 995 | ; @r{ 30-bit binary values} |
b8d4c8d0 | 996 | |
be14b9ab PE |
997 | (logand 14 13) ; 14 = @r{0000...001110} |
998 | ; 13 = @r{0000...001101} | |
999 | @result{} 12 ; 12 = @r{0000...001100} | |
b8d4c8d0 GM |
1000 | @end group |
1001 | ||
1002 | @group | |
be14b9ab PE |
1003 | (logand 14 13 4) ; 14 = @r{0000...001110} |
1004 | ; 13 = @r{0000...001101} | |
1005 | ; 4 = @r{0000...000100} | |
1006 | @result{} 4 ; 4 = @r{0000...000100} | |
b8d4c8d0 GM |
1007 | @end group |
1008 | ||
1009 | @group | |
1010 | (logand) | |
be14b9ab | 1011 | @result{} -1 ; -1 = @r{1111...111111} |
b8d4c8d0 GM |
1012 | @end group |
1013 | @end smallexample | |
1014 | @end defun | |
1015 | ||
1016 | @defun logior &rest ints-or-markers | |
1017 | This function returns the ``inclusive or'' of its arguments: the @var{n}th bit | |
1018 | is set in the result if, and only if, the @var{n}th bit is set in at least | |
1019 | one of the arguments. If there are no arguments, the result is zero, | |
1020 | which is an identity element for this operation. If @code{logior} is | |
1021 | passed just one argument, it returns that argument. | |
1022 | ||
1023 | @smallexample | |
1024 | @group | |
001903b5 | 1025 | ; @r{ 30-bit binary values} |
b8d4c8d0 | 1026 | |
be14b9ab PE |
1027 | (logior 12 5) ; 12 = @r{0000...001100} |
1028 | ; 5 = @r{0000...000101} | |
1029 | @result{} 13 ; 13 = @r{0000...001101} | |
b8d4c8d0 GM |
1030 | @end group |
1031 | ||
1032 | @group | |
be14b9ab PE |
1033 | (logior 12 5 7) ; 12 = @r{0000...001100} |
1034 | ; 5 = @r{0000...000101} | |
1035 | ; 7 = @r{0000...000111} | |
1036 | @result{} 15 ; 15 = @r{0000...001111} | |
b8d4c8d0 GM |
1037 | @end group |
1038 | @end smallexample | |
1039 | @end defun | |
1040 | ||
1041 | @defun logxor &rest ints-or-markers | |
1042 | This function returns the ``exclusive or'' of its arguments: the | |
1043 | @var{n}th bit is set in the result if, and only if, the @var{n}th bit is | |
1044 | set in an odd number of the arguments. If there are no arguments, the | |
1045 | result is 0, which is an identity element for this operation. If | |
1046 | @code{logxor} is passed just one argument, it returns that argument. | |
1047 | ||
1048 | @smallexample | |
1049 | @group | |
001903b5 | 1050 | ; @r{ 30-bit binary values} |
b8d4c8d0 | 1051 | |
be14b9ab PE |
1052 | (logxor 12 5) ; 12 = @r{0000...001100} |
1053 | ; 5 = @r{0000...000101} | |
1054 | @result{} 9 ; 9 = @r{0000...001001} | |
b8d4c8d0 GM |
1055 | @end group |
1056 | ||
1057 | @group | |
be14b9ab PE |
1058 | (logxor 12 5 7) ; 12 = @r{0000...001100} |
1059 | ; 5 = @r{0000...000101} | |
1060 | ; 7 = @r{0000...000111} | |
1061 | @result{} 14 ; 14 = @r{0000...001110} | |
b8d4c8d0 GM |
1062 | @end group |
1063 | @end smallexample | |
1064 | @end defun | |
1065 | ||
1066 | @defun lognot integer | |
1067 | This function returns the logical complement of its argument: the @var{n}th | |
1068 | bit is one in the result if, and only if, the @var{n}th bit is zero in | |
1069 | @var{integer}, and vice-versa. | |
1070 | ||
1071 | @example | |
1072 | (lognot 5) | |
1073 | @result{} -6 | |
001903b5 | 1074 | ;; 5 = @r{0000...000101} (30 bits total) |
b8d4c8d0 | 1075 | ;; @r{becomes} |
001903b5 | 1076 | ;; -6 = @r{1111...111010} (30 bits total) |
b8d4c8d0 GM |
1077 | @end example |
1078 | @end defun | |
1079 | ||
1080 | @node Math Functions | |
1081 | @section Standard Mathematical Functions | |
1082 | @cindex transcendental functions | |
1083 | @cindex mathematical functions | |
1084 | @cindex floating-point functions | |
1085 | ||
09b73f08 | 1086 | These mathematical functions allow integers as well as floating-point |
b8d4c8d0 GM |
1087 | numbers as arguments. |
1088 | ||
1089 | @defun sin arg | |
1090 | @defunx cos arg | |
1091 | @defunx tan arg | |
48de8b12 CY |
1092 | These are the basic trigonometric functions, with argument @var{arg} |
1093 | measured in radians. | |
b8d4c8d0 GM |
1094 | @end defun |
1095 | ||
1096 | @defun asin arg | |
1097 | The value of @code{(asin @var{arg})} is a number between | |
1098 | @ifnottex | |
1099 | @minus{}pi/2 | |
1100 | @end ifnottex | |
1101 | @tex | |
1102 | @math{-\pi/2} | |
1103 | @end tex | |
1104 | and | |
1105 | @ifnottex | |
1106 | pi/2 | |
1107 | @end ifnottex | |
1108 | @tex | |
1109 | @math{\pi/2} | |
1110 | @end tex | |
c990426a PE |
1111 | (inclusive) whose sine is @var{arg}. If @var{arg} is out of range |
1112 | (outside [@minus{}1, 1]), @code{asin} returns a NaN. | |
b8d4c8d0 GM |
1113 | @end defun |
1114 | ||
1115 | @defun acos arg | |
1116 | The value of @code{(acos @var{arg})} is a number between 0 and | |
1117 | @ifnottex | |
1118 | pi | |
1119 | @end ifnottex | |
1120 | @tex | |
1121 | @math{\pi} | |
1122 | @end tex | |
c990426a PE |
1123 | (inclusive) whose cosine is @var{arg}. If @var{arg} is out of range |
1124 | (outside [@minus{}1, 1]), @code{acos} returns a NaN. | |
b8d4c8d0 GM |
1125 | @end defun |
1126 | ||
1127 | @defun atan y &optional x | |
1128 | The value of @code{(atan @var{y})} is a number between | |
1129 | @ifnottex | |
1130 | @minus{}pi/2 | |
1131 | @end ifnottex | |
1132 | @tex | |
1133 | @math{-\pi/2} | |
1134 | @end tex | |
1135 | and | |
1136 | @ifnottex | |
1137 | pi/2 | |
1138 | @end ifnottex | |
1139 | @tex | |
1140 | @math{\pi/2} | |
1141 | @end tex | |
1142 | (exclusive) whose tangent is @var{y}. If the optional second | |
1143 | argument @var{x} is given, the value of @code{(atan y x)} is the | |
1144 | angle in radians between the vector @code{[@var{x}, @var{y}]} and the | |
1145 | @code{X} axis. | |
1146 | @end defun | |
1147 | ||
1148 | @defun exp arg | |
fead402d CY |
1149 | This is the exponential function; it returns @math{e} to the power |
1150 | @var{arg}. | |
b8d4c8d0 GM |
1151 | @end defun |
1152 | ||
1153 | @defun log arg &optional base | |
fead402d CY |
1154 | This function returns the logarithm of @var{arg}, with base |
1155 | @var{base}. If you don't specify @var{base}, the natural base | |
c990426a PE |
1156 | @math{e} is used. If @var{arg} or @var{base} is negative, @code{log} |
1157 | returns a NaN. | |
b8d4c8d0 GM |
1158 | @end defun |
1159 | ||
b8d4c8d0 GM |
1160 | @defun expt x y |
1161 | This function returns @var{x} raised to power @var{y}. If both | |
c717b326 PE |
1162 | arguments are integers and @var{y} is positive, the result is an |
1163 | integer; in this case, overflow causes truncation, so watch out. | |
c990426a PE |
1164 | If @var{x} is a finite negative number and @var{y} is a finite |
1165 | non-integer, @code{expt} returns a NaN. | |
b8d4c8d0 GM |
1166 | @end defun |
1167 | ||
1168 | @defun sqrt arg | |
1169 | This returns the square root of @var{arg}. If @var{arg} is negative, | |
c990426a | 1170 | @code{sqrt} returns a NaN. |
b8d4c8d0 GM |
1171 | @end defun |
1172 | ||
fead402d CY |
1173 | In addition, Emacs defines the following common mathematical |
1174 | constants: | |
1175 | ||
1176 | @defvar float-e | |
1177 | The mathematical constant @math{e} (2.71828@dots{}). | |
1178 | @end defvar | |
1179 | ||
1180 | @defvar float-pi | |
1181 | The mathematical constant @math{pi} (3.14159@dots{}). | |
1182 | @end defvar | |
1183 | ||
b8d4c8d0 GM |
1184 | @node Random Numbers |
1185 | @section Random Numbers | |
1186 | @cindex random numbers | |
1187 | ||
48de8b12 CY |
1188 | A deterministic computer program cannot generate true random |
1189 | numbers. For most purposes, @dfn{pseudo-random numbers} suffice. A | |
1190 | series of pseudo-random numbers is generated in a deterministic | |
1191 | fashion. The numbers are not truly random, but they have certain | |
1192 | properties that mimic a random series. For example, all possible | |
1193 | values occur equally often in a pseudo-random series. | |
b8d4c8d0 | 1194 | |
48de8b12 CY |
1195 | Pseudo-random numbers are generated from a ``seed''. Starting from |
1196 | any given seed, the @code{random} function always generates the same | |
1197 | sequence of numbers. By default, Emacs initializes the random seed at | |
1198 | startup, in such a way that the sequence of values of @code{random} | |
1199 | (with overwhelming likelihood) differs in each Emacs run. | |
0e23ef9d | 1200 | |
48de8b12 | 1201 | Sometimes you want the random number sequence to be repeatable. For |
0e23ef9d PE |
1202 | example, when debugging a program whose behavior depends on the random |
1203 | number sequence, it is helpful to get the same behavior in each | |
1204 | program run. To make the sequence repeat, execute @code{(random "")}. | |
1205 | This sets the seed to a constant value for your particular Emacs | |
1206 | executable (though it may differ for other Emacs builds). You can use | |
1207 | other strings to choose various seed values. | |
b8d4c8d0 GM |
1208 | |
1209 | @defun random &optional limit | |
1210 | This function returns a pseudo-random integer. Repeated calls return a | |
1211 | series of pseudo-random integers. | |
1212 | ||
1213 | If @var{limit} is a positive integer, the value is chosen to be | |
48de8b12 | 1214 | nonnegative and less than @var{limit}. Otherwise, the value might be |
1df7defd | 1215 | any integer representable in Lisp, i.e., an integer between |
48de8b12 CY |
1216 | @code{most-negative-fixnum} and @code{most-positive-fixnum} |
1217 | (@pxref{Integer Basics}). | |
b8d4c8d0 GM |
1218 | |
1219 | If @var{limit} is @code{t}, it means to choose a new seed based on the | |
1220 | current time of day and on Emacs's process @acronym{ID} number. | |
b8d4c8d0 | 1221 | |
0e23ef9d PE |
1222 | If @var{limit} is a string, it means to choose a new seed based on the |
1223 | string's contents. | |
1224 | ||
b8d4c8d0 | 1225 | @end defun |