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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. |
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 | |
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. |
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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 | |
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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 |
1ddd6622 | 39 | -2**29 |
b8d4c8d0 GM |
40 | @end ifnottex |
41 | @tex | |
1ddd6622 | 42 | @math{-2^{29}} |
b8d4c8d0 GM |
43 | @end tex |
44 | to | |
45 | @ifnottex | |
1ddd6622 | 46 | 2**29 - 1), |
b8d4c8d0 GM |
47 | @end ifnottex |
48 | @tex | |
1ddd6622 | 49 | @math{2^{29}-1}), |
b8d4c8d0 | 50 | @end tex |
001903b5 PE |
51 | but some machines provide a wider range. Many examples in this |
52 | chapter assume that an integer has 30 bits and that floating point | |
be14b9ab | 53 | numbers are IEEE double precision. |
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54 | @cindex overflow |
55 | ||
56 | The Lisp reader reads an integer as a sequence of digits with optional | |
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57 | initial sign and optional final period. An integer that is out of the |
58 | Emacs range is treated as a floating-point number. | |
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59 | |
60 | @example | |
61 | 1 ; @r{The integer 1.} | |
62 | 1. ; @r{The integer 1.} | |
63 | +1 ; @r{Also the integer 1.} | |
64 | -1 ; @r{The integer @minus{}1.} | |
fed14fd7 | 65 | 1073741825 ; @r{The floating point number 1073741825.0.} |
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66 | 0 ; @r{The integer 0.} |
67 | -0 ; @r{The integer 0.} | |
68 | @end example | |
69 | ||
70 | @cindex integers in specific radix | |
71 | @cindex radix for reading an integer | |
72 | @cindex base for reading an integer | |
73 | @cindex hex numbers | |
74 | @cindex octal numbers | |
75 | @cindex reading numbers in hex, octal, and binary | |
76 | The syntax for integers in bases other than 10 uses @samp{#} | |
77 | followed by a letter that specifies the radix: @samp{b} for binary, | |
78 | @samp{o} for octal, @samp{x} for hex, or @samp{@var{radix}r} to | |
79 | specify radix @var{radix}. Case is not significant for the letter | |
80 | that specifies the radix. Thus, @samp{#b@var{integer}} reads | |
81 | @var{integer} in binary, and @samp{#@var{radix}r@var{integer}} reads | |
82 | @var{integer} in radix @var{radix}. Allowed values of @var{radix} run | |
83 | from 2 to 36. For example: | |
84 | ||
85 | @example | |
86 | #b101100 @result{} 44 | |
87 | #o54 @result{} 44 | |
88 | #x2c @result{} 44 | |
89 | #24r1k @result{} 44 | |
90 | @end example | |
91 | ||
92 | To understand how various functions work on integers, especially the | |
93 | bitwise operators (@pxref{Bitwise Operations}), it is often helpful to | |
94 | view the numbers in their binary form. | |
95 | ||
1ddd6622 | 96 | In 30-bit binary, the decimal integer 5 looks like this: |
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97 | |
98 | @example | |
001903b5 | 99 | 0000...000101 (30 bits total) |
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100 | @end example |
101 | ||
102 | @noindent | |
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103 | (The @samp{...} stands for enough bits to fill out a 30-bit word; in |
104 | this case, @samp{...} stands for twenty 0 bits. Later examples also | |
105 | use the @samp{...} notation to make binary integers easier to read.) | |
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106 | |
107 | The integer @minus{}1 looks like this: | |
108 | ||
109 | @example | |
001903b5 | 110 | 1111...111111 (30 bits total) |
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111 | @end example |
112 | ||
113 | @noindent | |
114 | @cindex two's complement | |
1ddd6622 | 115 | @minus{}1 is represented as 30 ones. (This is called @dfn{two's |
b8d4c8d0 GM |
116 | complement} notation.) |
117 | ||
118 | The negative integer, @minus{}5, is creating by subtracting 4 from | |
119 | @minus{}1. In binary, the decimal integer 4 is 100. Consequently, | |
120 | @minus{}5 looks like this: | |
121 | ||
122 | @example | |
001903b5 | 123 | 1111...111011 (30 bits total) |
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124 | @end example |
125 | ||
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126 | In this implementation, the largest 30-bit binary integer value is |
127 | 536,870,911 in decimal. In binary, it looks like this: | |
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128 | |
129 | @example | |
001903b5 | 130 | 0111...111111 (30 bits total) |
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131 | @end example |
132 | ||
133 | Since the arithmetic functions do not check whether integers go | |
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134 | outside their range, when you add 1 to 536,870,911, the value is the |
135 | negative integer @minus{}536,870,912: | |
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136 | |
137 | @example | |
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138 | (+ 1 536870911) |
139 | @result{} -536870912 | |
001903b5 | 140 | @result{} 1000...000000 (30 bits total) |
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141 | @end example |
142 | ||
143 | Many of the functions described in this chapter accept markers for | |
144 | arguments in place of numbers. (@xref{Markers}.) Since the actual | |
145 | arguments to such functions may be either numbers or markers, we often | |
146 | give these arguments the name @var{number-or-marker}. When the argument | |
147 | value is a marker, its position value is used and its buffer is ignored. | |
148 | ||
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149 | @cindex largest Lisp integer number |
150 | @cindex maximum Lisp integer number | |
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151 | @defvar most-positive-fixnum |
152 | The value of this variable is the largest integer that Emacs Lisp | |
153 | can handle. | |
154 | @end defvar | |
155 | ||
eddf142c EZ |
156 | @cindex smallest Lisp integer number |
157 | @cindex minimum Lisp integer number | |
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158 | @defvar most-negative-fixnum |
159 | The value of this variable is the smallest integer that Emacs Lisp can | |
160 | handle. It is negative. | |
161 | @end defvar | |
162 | ||
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163 | @xref{Character Codes, max-char}, for the maximum value of a valid |
164 | character codepoint. | |
165 | ||
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166 | @node Float Basics |
167 | @section Floating Point Basics | |
168 | ||
fead402d | 169 | @cindex @acronym{IEEE} floating point |
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170 | Floating point numbers are useful for representing numbers that are |
171 | not integral. The precise range of floating point numbers is | |
172 | machine-specific; it is the same as the range of the C data type | |
fead402d CY |
173 | @code{double} on the machine you are using. Emacs uses the |
174 | @acronym{IEEE} floating point standard where possible (the standard is | |
175 | supported by most modern computers). | |
b8d4c8d0 | 176 | |
fead402d | 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 | |
180 | @samp{.15e4} are five ways of writing a floating point number whose | |
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181 | value is 1500. They are all equivalent. You can also use a minus |
182 | sign to write negative floating point numbers, as in @samp{-1.0}. | |
183 | ||
184 | Emacs Lisp treats @code{-0.0} as equal to ordinary zero (with | |
185 | respect to @code{equal} and @code{=}), even though the two are | |
186 | distinguishable in the @acronym{IEEE} floating point standard. | |
b8d4c8d0 | 187 | |
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188 | @cindex positive infinity |
189 | @cindex negative infinity | |
190 | @cindex infinity | |
191 | @cindex NaN | |
fead402d CY |
192 | The @acronym{IEEE} floating point standard supports positive |
193 | infinity and negative infinity as floating point values. It also | |
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 | |
196 | correct answer. For example, @code{(/ 0.0 0.0)} returns a NaN. (NaN | |
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 | ||
fead402d | 207 | Here are the read syntaxes for these special floating point values: |
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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 |
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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 | |
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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 |
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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 | |
275 | This predicate tests whether its argument is a floating point | |
276 | number and returns @code{t} if so, @code{nil} otherwise. | |
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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}. | |
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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 | |
313 | @code{=}, not @code{eq}. There can be many distinct floating point | |
314 | number objects with the same numeric value. If you use @code{eq} to | |
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 | ||
319 | At present, each integer value has a unique Lisp object in Emacs Lisp. | |
320 | Therefore, @code{eq} is equivalent to @code{=} where integers are | |
321 | concerned. It is sometimes convenient to use @code{eq} for comparing an | |
322 | 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 any | |
324 | type. By contrast, @code{=} signals an error if the arguments are not | |
325 | numbers or markers. However, it is a good idea to use @code{=} if you | |
326 | can, even for comparing integers, just in case we change the | |
327 | representation of integers in a future Emacs version. | |
328 | ||
329 | Sometimes it is useful to compare numbers with @code{equal}; it | |
330 | treats two numbers as equal if they have the same data type (both | |
331 | integers, or both floating point) and the same value. By contrast, | |
332 | @code{=} can treat an integer and a floating point number as equal. | |
333 | @xref{Equality Predicates}. | |
334 | ||
335 | There is another wrinkle: because floating point arithmetic is not | |
336 | exact, it is often a bad idea to check for equality of two floating | |
337 | point values. Usually it is better to test for approximate equality. | |
338 | Here's a function to do this: | |
339 | ||
340 | @example | |
341 | (defvar fuzz-factor 1.0e-6) | |
342 | (defun approx-equal (x y) | |
343 | (or (and (= x 0) (= y 0)) | |
344 | (< (/ (abs (- x y)) | |
345 | (max (abs x) (abs y))) | |
346 | fuzz-factor))) | |
347 | @end example | |
348 | ||
349 | @cindex CL note---integers vrs @code{eq} | |
350 | @quotation | |
351 | @b{Common Lisp note:} Comparing numbers in Common Lisp always requires | |
352 | @code{=} because Common Lisp implements multi-word integers, and two | |
353 | distinct integer objects can have the same numeric value. Emacs Lisp | |
354 | can have just one integer object for any given value because it has a | |
355 | limited range of integer values. | |
356 | @end quotation | |
357 | ||
358 | @defun = number-or-marker1 number-or-marker2 | |
359 | This function tests whether its arguments are numerically equal, and | |
360 | returns @code{t} if so, @code{nil} otherwise. | |
361 | @end defun | |
362 | ||
363 | @defun eql value1 value2 | |
364 | This function acts like @code{eq} except when both arguments are | |
365 | numbers. It compares numbers by type and numeric value, so that | |
366 | @code{(eql 1.0 1)} returns @code{nil}, but @code{(eql 1.0 1.0)} and | |
367 | @code{(eql 1 1)} both return @code{t}. | |
368 | @end defun | |
369 | ||
370 | @defun /= number-or-marker1 number-or-marker2 | |
371 | This function tests whether its arguments are numerically equal, and | |
372 | returns @code{t} if they are not, and @code{nil} if they are. | |
373 | @end defun | |
374 | ||
375 | @defun < number-or-marker1 number-or-marker2 | |
376 | This function tests whether its first argument is strictly less than | |
377 | its second argument. It returns @code{t} if so, @code{nil} otherwise. | |
378 | @end defun | |
379 | ||
380 | @defun <= number-or-marker1 number-or-marker2 | |
381 | This function tests whether its first argument is less than or equal | |
382 | to its second argument. It returns @code{t} if so, @code{nil} | |
383 | otherwise. | |
384 | @end defun | |
385 | ||
386 | @defun > number-or-marker1 number-or-marker2 | |
387 | This function tests whether its first argument is strictly greater | |
388 | than its second argument. It returns @code{t} if so, @code{nil} | |
389 | otherwise. | |
390 | @end defun | |
391 | ||
392 | @defun >= number-or-marker1 number-or-marker2 | |
393 | This function tests whether its first argument is greater than or | |
394 | equal to its second argument. It returns @code{t} if so, @code{nil} | |
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. | |
400 | If any of the arguments is floating-point, the value is returned | |
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. | |
415 | If any of the arguments is floating-point, the value is returned | |
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. | |
438 | If @var{number} is already a floating point number, @code{float} returns | |
439 | it unchanged. | |
440 | @end defun | |
441 | ||
442 | There are four functions to convert floating point numbers to integers; | |
443 | they differ in how they round. All accept an argument @var{number} | |
444 | and an optional argument @var{divisor}. Both arguments may be | |
445 | integers or floating point numbers. @var{divisor} may also be | |
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 | |
450 | integer. An @code{arith-error} results if @var{divisor} is 0. | |
451 | ||
452 | @defun truncate number &optional divisor | |
453 | This returns @var{number}, converted to an integer by rounding towards | |
454 | zero. | |
455 | ||
456 | @example | |
457 | (truncate 1.2) | |
458 | @result{} 1 | |
459 | (truncate 1.7) | |
460 | @result{} 1 | |
461 | (truncate -1.2) | |
462 | @result{} -1 | |
463 | (truncate -1.7) | |
464 | @result{} -1 | |
465 | @end example | |
466 | @end defun | |
467 | ||
468 | @defun floor number &optional divisor | |
469 | This returns @var{number}, converted to an integer by rounding downward | |
470 | (towards negative infinity). | |
471 | ||
472 | If @var{divisor} is specified, this uses the kind of division | |
473 | operation that corresponds to @code{mod}, rounding downward. | |
474 | ||
475 | @example | |
476 | (floor 1.2) | |
477 | @result{} 1 | |
478 | (floor 1.7) | |
479 | @result{} 1 | |
480 | (floor -1.2) | |
481 | @result{} -2 | |
482 | (floor -1.7) | |
483 | @result{} -2 | |
484 | (floor 5.99 3) | |
485 | @result{} 1 | |
486 | @end example | |
487 | @end defun | |
488 | ||
489 | @defun ceiling number &optional divisor | |
490 | This returns @var{number}, converted to an integer by rounding upward | |
491 | (towards positive infinity). | |
492 | ||
493 | @example | |
494 | (ceiling 1.2) | |
495 | @result{} 2 | |
496 | (ceiling 1.7) | |
497 | @result{} 2 | |
498 | (ceiling -1.2) | |
499 | @result{} -1 | |
500 | (ceiling -1.7) | |
501 | @result{} -1 | |
502 | @end example | |
503 | @end defun | |
504 | ||
505 | @defun round number &optional divisor | |
506 | This returns @var{number}, converted to an integer by rounding towards the | |
507 | nearest integer. Rounding a value equidistant between two integers | |
508 | may choose the integer closer to zero, or it may prefer an even integer, | |
509 | depending on your machine. | |
510 | ||
511 | @example | |
512 | (round 1.2) | |
513 | @result{} 1 | |
514 | (round 1.7) | |
515 | @result{} 2 | |
516 | (round -1.2) | |
517 | @result{} -1 | |
518 | (round -1.7) | |
519 | @result{} -2 | |
520 | @end example | |
521 | @end defun | |
522 | ||
523 | @node Arithmetic Operations | |
524 | @section Arithmetic Operations | |
525 | @cindex arithmetic operations | |
526 | ||
527 | Emacs Lisp provides the traditional four arithmetic operations: | |
528 | addition, subtraction, multiplication, and division. Remainder and modulus | |
529 | functions supplement the division functions. The functions to | |
530 | add or subtract 1 are provided because they are traditional in Lisp and | |
531 | commonly used. | |
532 | ||
533 | All of these functions except @code{%} return a floating point value | |
534 | if any argument is floating. | |
535 | ||
c717b326 | 536 | It is important to note that in Emacs Lisp, arithmetic functions |
001903b5 PE |
537 | do not check for overflow. Thus @code{(1+ 536870911)} may evaluate to |
538 | @minus{}536870912, depending on your hardware. | |
b8d4c8d0 GM |
539 | |
540 | @defun 1+ number-or-marker | |
541 | This function returns @var{number-or-marker} plus 1. | |
542 | For example, | |
543 | ||
544 | @example | |
545 | (setq foo 4) | |
546 | @result{} 4 | |
547 | (1+ foo) | |
548 | @result{} 5 | |
549 | @end example | |
550 | ||
551 | This function is not analogous to the C operator @code{++}---it does not | |
552 | increment a variable. It just computes a sum. Thus, if we continue, | |
553 | ||
554 | @example | |
555 | foo | |
556 | @result{} 4 | |
557 | @end example | |
558 | ||
559 | If you want to increment the variable, you must use @code{setq}, | |
560 | like this: | |
561 | ||
562 | @example | |
563 | (setq foo (1+ foo)) | |
564 | @result{} 5 | |
565 | @end example | |
566 | @end defun | |
567 | ||
568 | @defun 1- number-or-marker | |
569 | This function returns @var{number-or-marker} minus 1. | |
570 | @end defun | |
571 | ||
572 | @defun + &rest numbers-or-markers | |
573 | This function adds its arguments together. When given no arguments, | |
574 | @code{+} returns 0. | |
575 | ||
576 | @example | |
577 | (+) | |
578 | @result{} 0 | |
579 | (+ 1) | |
580 | @result{} 1 | |
581 | (+ 1 2 3 4) | |
582 | @result{} 10 | |
583 | @end example | |
584 | @end defun | |
585 | ||
586 | @defun - &optional number-or-marker &rest more-numbers-or-markers | |
587 | The @code{-} function serves two purposes: negation and subtraction. | |
588 | When @code{-} has a single argument, the value is the negative of the | |
589 | argument. When there are multiple arguments, @code{-} subtracts each of | |
590 | the @var{more-numbers-or-markers} from @var{number-or-marker}, | |
591 | cumulatively. If there are no arguments, the result is 0. | |
592 | ||
593 | @example | |
594 | (- 10 1 2 3 4) | |
595 | @result{} 0 | |
596 | (- 10) | |
597 | @result{} -10 | |
598 | (-) | |
599 | @result{} 0 | |
600 | @end example | |
601 | @end defun | |
602 | ||
603 | @defun * &rest numbers-or-markers | |
604 | This function multiplies its arguments together, and returns the | |
605 | product. When given no arguments, @code{*} returns 1. | |
606 | ||
607 | @example | |
608 | (*) | |
609 | @result{} 1 | |
610 | (* 1) | |
611 | @result{} 1 | |
612 | (* 1 2 3 4) | |
613 | @result{} 24 | |
614 | @end example | |
615 | @end defun | |
616 | ||
617 | @defun / dividend divisor &rest divisors | |
618 | This function divides @var{dividend} by @var{divisor} and returns the | |
619 | quotient. If there are additional arguments @var{divisors}, then it | |
620 | divides @var{dividend} by each divisor in turn. Each argument may be a | |
621 | number or a marker. | |
622 | ||
623 | If all the arguments are integers, then the result is an integer too. | |
624 | This means the result has to be rounded. On most machines, the result | |
625 | is rounded towards zero after each division, but some machines may round | |
626 | differently with negative arguments. This is because the Lisp function | |
627 | @code{/} is implemented using the C division operator, which also | |
628 | permits machine-dependent rounding. As a practical matter, all known | |
629 | machines round in the standard fashion. | |
630 | ||
631 | @cindex @code{arith-error} in division | |
632 | If you divide an integer by 0, an @code{arith-error} error is signaled. | |
633 | (@xref{Errors}.) Floating point division by zero returns either | |
634 | infinity or a NaN if your machine supports @acronym{IEEE} floating point; | |
635 | otherwise, it signals an @code{arith-error} error. | |
636 | ||
637 | @example | |
638 | @group | |
639 | (/ 6 2) | |
640 | @result{} 3 | |
641 | @end group | |
642 | (/ 5 2) | |
643 | @result{} 2 | |
644 | (/ 5.0 2) | |
645 | @result{} 2.5 | |
646 | (/ 5 2.0) | |
647 | @result{} 2.5 | |
648 | (/ 5.0 2.0) | |
649 | @result{} 2.5 | |
650 | (/ 25 3 2) | |
651 | @result{} 4 | |
652 | @group | |
653 | (/ -17 6) | |
654 | @result{} -2 @r{(could in theory be @minus{}3 on some machines)} | |
655 | @end group | |
656 | @end example | |
657 | @end defun | |
658 | ||
659 | @defun % dividend divisor | |
660 | @cindex remainder | |
661 | This function returns the integer remainder after division of @var{dividend} | |
662 | by @var{divisor}. The arguments must be integers or markers. | |
663 | ||
664 | For negative arguments, the remainder is in principle machine-dependent | |
665 | since the quotient is; but in practice, all known machines behave alike. | |
666 | ||
667 | An @code{arith-error} results if @var{divisor} is 0. | |
668 | ||
669 | @example | |
670 | (% 9 4) | |
671 | @result{} 1 | |
672 | (% -9 4) | |
673 | @result{} -1 | |
674 | (% 9 -4) | |
675 | @result{} 1 | |
676 | (% -9 -4) | |
677 | @result{} -1 | |
678 | @end example | |
679 | ||
680 | For any two integers @var{dividend} and @var{divisor}, | |
681 | ||
682 | @example | |
683 | @group | |
684 | (+ (% @var{dividend} @var{divisor}) | |
685 | (* (/ @var{dividend} @var{divisor}) @var{divisor})) | |
686 | @end group | |
687 | @end example | |
688 | ||
689 | @noindent | |
690 | always equals @var{dividend}. | |
691 | @end defun | |
692 | ||
693 | @defun mod dividend divisor | |
694 | @cindex modulus | |
695 | This function returns the value of @var{dividend} modulo @var{divisor}; | |
696 | in other words, the remainder after division of @var{dividend} | |
697 | by @var{divisor}, but with the same sign as @var{divisor}. | |
698 | The arguments must be numbers or markers. | |
699 | ||
700 | Unlike @code{%}, @code{mod} returns a well-defined result for negative | |
701 | arguments. It also permits floating point arguments; it rounds the | |
702 | quotient downward (towards minus infinity) to an integer, and uses that | |
703 | quotient to compute the remainder. | |
704 | ||
c990426a PE |
705 | If @var{divisor} is zero, @code{mod} signals an @code{arith-error} |
706 | error if both arguments are integers, and returns a NaN otherwise. | |
b8d4c8d0 GM |
707 | |
708 | @example | |
709 | @group | |
710 | (mod 9 4) | |
711 | @result{} 1 | |
712 | @end group | |
713 | @group | |
714 | (mod -9 4) | |
715 | @result{} 3 | |
716 | @end group | |
717 | @group | |
718 | (mod 9 -4) | |
719 | @result{} -3 | |
720 | @end group | |
721 | @group | |
722 | (mod -9 -4) | |
723 | @result{} -1 | |
724 | @end group | |
725 | @group | |
726 | (mod 5.5 2.5) | |
727 | @result{} .5 | |
728 | @end group | |
729 | @end example | |
730 | ||
731 | For any two numbers @var{dividend} and @var{divisor}, | |
732 | ||
733 | @example | |
734 | @group | |
735 | (+ (mod @var{dividend} @var{divisor}) | |
736 | (* (floor @var{dividend} @var{divisor}) @var{divisor})) | |
737 | @end group | |
738 | @end example | |
739 | ||
740 | @noindent | |
741 | always equals @var{dividend}, subject to rounding error if either | |
742 | argument is floating point. For @code{floor}, see @ref{Numeric | |
743 | Conversions}. | |
744 | @end defun | |
745 | ||
746 | @node Rounding Operations | |
747 | @section Rounding Operations | |
748 | @cindex rounding without conversion | |
749 | ||
750 | The functions @code{ffloor}, @code{fceiling}, @code{fround}, and | |
751 | @code{ftruncate} take a floating point argument and return a floating | |
752 | point result whose value is a nearby integer. @code{ffloor} returns the | |
753 | nearest integer below; @code{fceiling}, the nearest integer above; | |
754 | @code{ftruncate}, the nearest integer in the direction towards zero; | |
755 | @code{fround}, the nearest integer. | |
756 | ||
757 | @defun ffloor float | |
758 | This function rounds @var{float} to the next lower integral value, and | |
759 | returns that value as a floating point number. | |
760 | @end defun | |
761 | ||
762 | @defun fceiling float | |
763 | This function rounds @var{float} to the next higher integral value, and | |
764 | returns that value as a floating point number. | |
765 | @end defun | |
766 | ||
767 | @defun ftruncate float | |
768 | This function rounds @var{float} towards zero to an integral value, and | |
769 | returns that value as a floating point number. | |
770 | @end defun | |
771 | ||
772 | @defun fround float | |
773 | This function rounds @var{float} to the nearest integral value, | |
774 | and returns that value as a floating point number. | |
775 | @end defun | |
776 | ||
777 | @node Bitwise Operations | |
778 | @section Bitwise Operations on Integers | |
779 | @cindex bitwise arithmetic | |
780 | @cindex logical arithmetic | |
781 | ||
782 | In a computer, an integer is represented as a binary number, a | |
783 | sequence of @dfn{bits} (digits which are either zero or one). A bitwise | |
784 | operation acts on the individual bits of such a sequence. For example, | |
785 | @dfn{shifting} moves the whole sequence left or right one or more places, | |
16152b76 | 786 | reproducing the same pattern ``moved over''. |
b8d4c8d0 GM |
787 | |
788 | The bitwise operations in Emacs Lisp apply only to integers. | |
789 | ||
790 | @defun lsh integer1 count | |
791 | @cindex logical shift | |
792 | @code{lsh}, which is an abbreviation for @dfn{logical shift}, shifts the | |
793 | bits in @var{integer1} to the left @var{count} places, or to the right | |
794 | if @var{count} is negative, bringing zeros into the vacated bits. If | |
795 | @var{count} is negative, @code{lsh} shifts zeros into the leftmost | |
796 | (most-significant) bit, producing a positive result even if | |
797 | @var{integer1} is negative. Contrast this with @code{ash}, below. | |
798 | ||
799 | Here are two examples of @code{lsh}, shifting a pattern of bits one | |
800 | place to the left. We show only the low-order eight bits of the binary | |
801 | pattern; the rest are all zero. | |
802 | ||
803 | @example | |
804 | @group | |
805 | (lsh 5 1) | |
806 | @result{} 10 | |
807 | ;; @r{Decimal 5 becomes decimal 10.} | |
808 | 00000101 @result{} 00001010 | |
809 | ||
810 | (lsh 7 1) | |
811 | @result{} 14 | |
812 | ;; @r{Decimal 7 becomes decimal 14.} | |
813 | 00000111 @result{} 00001110 | |
814 | @end group | |
815 | @end example | |
816 | ||
817 | @noindent | |
818 | As the examples illustrate, shifting the pattern of bits one place to | |
819 | the left produces a number that is twice the value of the previous | |
820 | number. | |
821 | ||
822 | Shifting a pattern of bits two places to the left produces results | |
823 | like this (with 8-bit binary numbers): | |
824 | ||
825 | @example | |
826 | @group | |
827 | (lsh 3 2) | |
828 | @result{} 12 | |
829 | ;; @r{Decimal 3 becomes decimal 12.} | |
830 | 00000011 @result{} 00001100 | |
831 | @end group | |
832 | @end example | |
833 | ||
834 | On the other hand, shifting one place to the right looks like this: | |
835 | ||
836 | @example | |
837 | @group | |
838 | (lsh 6 -1) | |
839 | @result{} 3 | |
840 | ;; @r{Decimal 6 becomes decimal 3.} | |
841 | 00000110 @result{} 00000011 | |
842 | @end group | |
843 | ||
844 | @group | |
845 | (lsh 5 -1) | |
846 | @result{} 2 | |
847 | ;; @r{Decimal 5 becomes decimal 2.} | |
848 | 00000101 @result{} 00000010 | |
849 | @end group | |
850 | @end example | |
851 | ||
852 | @noindent | |
853 | As the example illustrates, shifting one place to the right divides the | |
854 | value of a positive integer by two, rounding downward. | |
855 | ||
c717b326 | 856 | The function @code{lsh}, like all Emacs Lisp arithmetic functions, does |
b8d4c8d0 GM |
857 | not check for overflow, so shifting left can discard significant bits |
858 | and change the sign of the number. For example, left shifting | |
001903b5 | 859 | 536,870,911 produces @minus{}2 in the 30-bit implementation: |
b8d4c8d0 GM |
860 | |
861 | @example | |
1ddd6622 | 862 | (lsh 536870911 1) ; @r{left shift} |
b8d4c8d0 GM |
863 | @result{} -2 |
864 | @end example | |
865 | ||
001903b5 | 866 | In binary, the argument looks like this: |
b8d4c8d0 GM |
867 | |
868 | @example | |
869 | @group | |
1ddd6622 | 870 | ;; @r{Decimal 536,870,911} |
001903b5 | 871 | 0111...111111 (30 bits total) |
b8d4c8d0 GM |
872 | @end group |
873 | @end example | |
874 | ||
875 | @noindent | |
876 | which becomes the following when left shifted: | |
877 | ||
878 | @example | |
879 | @group | |
880 | ;; @r{Decimal @minus{}2} | |
001903b5 | 881 | 1111...111110 (30 bits total) |
b8d4c8d0 GM |
882 | @end group |
883 | @end example | |
884 | @end defun | |
885 | ||
886 | @defun ash integer1 count | |
887 | @cindex arithmetic shift | |
888 | @code{ash} (@dfn{arithmetic shift}) shifts the bits in @var{integer1} | |
889 | to the left @var{count} places, or to the right if @var{count} | |
890 | is negative. | |
891 | ||
892 | @code{ash} gives the same results as @code{lsh} except when | |
893 | @var{integer1} and @var{count} are both negative. In that case, | |
894 | @code{ash} puts ones in the empty bit positions on the left, while | |
895 | @code{lsh} puts zeros in those bit positions. | |
896 | ||
897 | Thus, with @code{ash}, shifting the pattern of bits one place to the right | |
898 | looks like this: | |
899 | ||
900 | @example | |
901 | @group | |
902 | (ash -6 -1) @result{} -3 | |
903 | ;; @r{Decimal @minus{}6 becomes decimal @minus{}3.} | |
001903b5 | 904 | 1111...111010 (30 bits total) |
b8d4c8d0 | 905 | @result{} |
001903b5 | 906 | 1111...111101 (30 bits total) |
b8d4c8d0 GM |
907 | @end group |
908 | @end example | |
909 | ||
910 | In contrast, shifting the pattern of bits one place to the right with | |
911 | @code{lsh} looks like this: | |
912 | ||
913 | @example | |
914 | @group | |
1ddd6622 GM |
915 | (lsh -6 -1) @result{} 536870909 |
916 | ;; @r{Decimal @minus{}6 becomes decimal 536,870,909.} | |
001903b5 | 917 | 1111...111010 (30 bits total) |
b8d4c8d0 | 918 | @result{} |
001903b5 | 919 | 0111...111101 (30 bits total) |
b8d4c8d0 GM |
920 | @end group |
921 | @end example | |
922 | ||
923 | Here are other examples: | |
924 | ||
925 | @c !!! Check if lined up in smallbook format! XDVI shows problem | |
926 | @c with smallbook but not with regular book! --rjc 16mar92 | |
927 | @smallexample | |
928 | @group | |
001903b5 | 929 | ; @r{ 30-bit binary values} |
b8d4c8d0 | 930 | |
be14b9ab PE |
931 | (lsh 5 2) ; 5 = @r{0000...000101} |
932 | @result{} 20 ; = @r{0000...010100} | |
b8d4c8d0 GM |
933 | @end group |
934 | @group | |
935 | (ash 5 2) | |
936 | @result{} 20 | |
be14b9ab PE |
937 | (lsh -5 2) ; -5 = @r{1111...111011} |
938 | @result{} -20 ; = @r{1111...101100} | |
b8d4c8d0 GM |
939 | (ash -5 2) |
940 | @result{} -20 | |
941 | @end group | |
942 | @group | |
be14b9ab PE |
943 | (lsh 5 -2) ; 5 = @r{0000...000101} |
944 | @result{} 1 ; = @r{0000...000001} | |
b8d4c8d0 GM |
945 | @end group |
946 | @group | |
947 | (ash 5 -2) | |
948 | @result{} 1 | |
949 | @end group | |
950 | @group | |
be14b9ab | 951 | (lsh -5 -2) ; -5 = @r{1111...111011} |
001903b5 | 952 | @result{} 268435454 |
be14b9ab | 953 | ; = @r{0011...111110} |
b8d4c8d0 GM |
954 | @end group |
955 | @group | |
be14b9ab PE |
956 | (ash -5 -2) ; -5 = @r{1111...111011} |
957 | @result{} -2 ; = @r{1111...111110} | |
b8d4c8d0 GM |
958 | @end group |
959 | @end smallexample | |
960 | @end defun | |
961 | ||
962 | @defun logand &rest ints-or-markers | |
963 | This function returns the ``logical and'' of the arguments: the | |
964 | @var{n}th bit is set in the result if, and only if, the @var{n}th bit is | |
965 | set in all the arguments. (``Set'' means that the value of the bit is 1 | |
966 | rather than 0.) | |
967 | ||
968 | For example, using 4-bit binary numbers, the ``logical and'' of 13 and | |
969 | 12 is 12: 1101 combined with 1100 produces 1100. | |
970 | In both the binary numbers, the leftmost two bits are set (i.e., they | |
971 | are 1's), so the leftmost two bits of the returned value are set. | |
972 | However, for the rightmost two bits, each is zero in at least one of | |
973 | the arguments, so the rightmost two bits of the returned value are 0's. | |
974 | ||
975 | @noindent | |
976 | Therefore, | |
977 | ||
978 | @example | |
979 | @group | |
980 | (logand 13 12) | |
981 | @result{} 12 | |
982 | @end group | |
983 | @end example | |
984 | ||
985 | If @code{logand} is not passed any argument, it returns a value of | |
986 | @minus{}1. This number is an identity element for @code{logand} | |
987 | because its binary representation consists entirely of ones. If | |
988 | @code{logand} is passed just one argument, it returns that argument. | |
989 | ||
990 | @smallexample | |
991 | @group | |
001903b5 | 992 | ; @r{ 30-bit binary values} |
b8d4c8d0 | 993 | |
be14b9ab PE |
994 | (logand 14 13) ; 14 = @r{0000...001110} |
995 | ; 13 = @r{0000...001101} | |
996 | @result{} 12 ; 12 = @r{0000...001100} | |
b8d4c8d0 GM |
997 | @end group |
998 | ||
999 | @group | |
be14b9ab PE |
1000 | (logand 14 13 4) ; 14 = @r{0000...001110} |
1001 | ; 13 = @r{0000...001101} | |
1002 | ; 4 = @r{0000...000100} | |
1003 | @result{} 4 ; 4 = @r{0000...000100} | |
b8d4c8d0 GM |
1004 | @end group |
1005 | ||
1006 | @group | |
1007 | (logand) | |
be14b9ab | 1008 | @result{} -1 ; -1 = @r{1111...111111} |
b8d4c8d0 GM |
1009 | @end group |
1010 | @end smallexample | |
1011 | @end defun | |
1012 | ||
1013 | @defun logior &rest ints-or-markers | |
1014 | This function returns the ``inclusive or'' of its arguments: the @var{n}th bit | |
1015 | is set in the result if, and only if, the @var{n}th bit is set in at least | |
1016 | one of the arguments. If there are no arguments, the result is zero, | |
1017 | which is an identity element for this operation. If @code{logior} is | |
1018 | passed just one argument, it returns that argument. | |
1019 | ||
1020 | @smallexample | |
1021 | @group | |
001903b5 | 1022 | ; @r{ 30-bit binary values} |
b8d4c8d0 | 1023 | |
be14b9ab PE |
1024 | (logior 12 5) ; 12 = @r{0000...001100} |
1025 | ; 5 = @r{0000...000101} | |
1026 | @result{} 13 ; 13 = @r{0000...001101} | |
b8d4c8d0 GM |
1027 | @end group |
1028 | ||
1029 | @group | |
be14b9ab PE |
1030 | (logior 12 5 7) ; 12 = @r{0000...001100} |
1031 | ; 5 = @r{0000...000101} | |
1032 | ; 7 = @r{0000...000111} | |
1033 | @result{} 15 ; 15 = @r{0000...001111} | |
b8d4c8d0 GM |
1034 | @end group |
1035 | @end smallexample | |
1036 | @end defun | |
1037 | ||
1038 | @defun logxor &rest ints-or-markers | |
1039 | This function returns the ``exclusive or'' of its arguments: the | |
1040 | @var{n}th bit is set in the result if, and only if, the @var{n}th bit is | |
1041 | set in an odd number of the arguments. If there are no arguments, the | |
1042 | result is 0, which is an identity element for this operation. If | |
1043 | @code{logxor} is passed just one argument, it returns that argument. | |
1044 | ||
1045 | @smallexample | |
1046 | @group | |
001903b5 | 1047 | ; @r{ 30-bit binary values} |
b8d4c8d0 | 1048 | |
be14b9ab PE |
1049 | (logxor 12 5) ; 12 = @r{0000...001100} |
1050 | ; 5 = @r{0000...000101} | |
1051 | @result{} 9 ; 9 = @r{0000...001001} | |
b8d4c8d0 GM |
1052 | @end group |
1053 | ||
1054 | @group | |
be14b9ab PE |
1055 | (logxor 12 5 7) ; 12 = @r{0000...001100} |
1056 | ; 5 = @r{0000...000101} | |
1057 | ; 7 = @r{0000...000111} | |
1058 | @result{} 14 ; 14 = @r{0000...001110} | |
b8d4c8d0 GM |
1059 | @end group |
1060 | @end smallexample | |
1061 | @end defun | |
1062 | ||
1063 | @defun lognot integer | |
1064 | This function returns the logical complement of its argument: the @var{n}th | |
1065 | bit is one in the result if, and only if, the @var{n}th bit is zero in | |
1066 | @var{integer}, and vice-versa. | |
1067 | ||
1068 | @example | |
1069 | (lognot 5) | |
1070 | @result{} -6 | |
001903b5 | 1071 | ;; 5 = @r{0000...000101} (30 bits total) |
b8d4c8d0 | 1072 | ;; @r{becomes} |
001903b5 | 1073 | ;; -6 = @r{1111...111010} (30 bits total) |
b8d4c8d0 GM |
1074 | @end example |
1075 | @end defun | |
1076 | ||
1077 | @node Math Functions | |
1078 | @section Standard Mathematical Functions | |
1079 | @cindex transcendental functions | |
1080 | @cindex mathematical functions | |
1081 | @cindex floating-point functions | |
1082 | ||
1083 | These mathematical functions allow integers as well as floating point | |
1084 | numbers as arguments. | |
1085 | ||
1086 | @defun sin arg | |
1087 | @defunx cos arg | |
1088 | @defunx tan arg | |
1089 | These are the ordinary trigonometric functions, with argument measured | |
1090 | in radians. | |
1091 | @end defun | |
1092 | ||
1093 | @defun asin arg | |
1094 | The value of @code{(asin @var{arg})} is a number between | |
1095 | @ifnottex | |
1096 | @minus{}pi/2 | |
1097 | @end ifnottex | |
1098 | @tex | |
1099 | @math{-\pi/2} | |
1100 | @end tex | |
1101 | and | |
1102 | @ifnottex | |
1103 | pi/2 | |
1104 | @end ifnottex | |
1105 | @tex | |
1106 | @math{\pi/2} | |
1107 | @end tex | |
c990426a PE |
1108 | (inclusive) whose sine is @var{arg}. If @var{arg} is out of range |
1109 | (outside [@minus{}1, 1]), @code{asin} returns a NaN. | |
b8d4c8d0 GM |
1110 | @end defun |
1111 | ||
1112 | @defun acos arg | |
1113 | The value of @code{(acos @var{arg})} is a number between 0 and | |
1114 | @ifnottex | |
1115 | pi | |
1116 | @end ifnottex | |
1117 | @tex | |
1118 | @math{\pi} | |
1119 | @end tex | |
c990426a PE |
1120 | (inclusive) whose cosine is @var{arg}. If @var{arg} is out of range |
1121 | (outside [@minus{}1, 1]), @code{acos} returns a NaN. | |
b8d4c8d0 GM |
1122 | @end defun |
1123 | ||
1124 | @defun atan y &optional x | |
1125 | The value of @code{(atan @var{y})} is a number between | |
1126 | @ifnottex | |
1127 | @minus{}pi/2 | |
1128 | @end ifnottex | |
1129 | @tex | |
1130 | @math{-\pi/2} | |
1131 | @end tex | |
1132 | and | |
1133 | @ifnottex | |
1134 | pi/2 | |
1135 | @end ifnottex | |
1136 | @tex | |
1137 | @math{\pi/2} | |
1138 | @end tex | |
1139 | (exclusive) whose tangent is @var{y}. If the optional second | |
1140 | argument @var{x} is given, the value of @code{(atan y x)} is the | |
1141 | angle in radians between the vector @code{[@var{x}, @var{y}]} and the | |
1142 | @code{X} axis. | |
1143 | @end defun | |
1144 | ||
1145 | @defun exp arg | |
fead402d CY |
1146 | This is the exponential function; it returns @math{e} to the power |
1147 | @var{arg}. | |
b8d4c8d0 GM |
1148 | @end defun |
1149 | ||
1150 | @defun log arg &optional base | |
fead402d CY |
1151 | This function returns the logarithm of @var{arg}, with base |
1152 | @var{base}. If you don't specify @var{base}, the natural base | |
c990426a PE |
1153 | @math{e} is used. If @var{arg} or @var{base} is negative, @code{log} |
1154 | returns a NaN. | |
b8d4c8d0 GM |
1155 | @end defun |
1156 | ||
1157 | @ignore | |
1158 | @defun expm1 arg | |
1159 | This function returns @code{(1- (exp @var{arg}))}, but it is more | |
1160 | accurate than that when @var{arg} is negative and @code{(exp @var{arg})} | |
1161 | is close to 1. | |
1162 | @end defun | |
1163 | ||
1164 | @defun log1p arg | |
1165 | This function returns @code{(log (1+ @var{arg}))}, but it is more | |
1166 | accurate than that when @var{arg} is so small that adding 1 to it would | |
1167 | lose accuracy. | |
1168 | @end defun | |
1169 | @end ignore | |
1170 | ||
1171 | @defun log10 arg | |
c990426a PE |
1172 | This function returns the logarithm of @var{arg}, with base 10: |
1173 | @code{(log10 @var{x})} @equiv{} @code{(log @var{x} 10)}. | |
b8d4c8d0 GM |
1174 | @end defun |
1175 | ||
1176 | @defun expt x y | |
1177 | This function returns @var{x} raised to power @var{y}. If both | |
c717b326 PE |
1178 | arguments are integers and @var{y} is positive, the result is an |
1179 | integer; in this case, overflow causes truncation, so watch out. | |
c990426a PE |
1180 | If @var{x} is a finite negative number and @var{y} is a finite |
1181 | non-integer, @code{expt} returns a NaN. | |
b8d4c8d0 GM |
1182 | @end defun |
1183 | ||
1184 | @defun sqrt arg | |
1185 | This returns the square root of @var{arg}. If @var{arg} is negative, | |
c990426a | 1186 | @code{sqrt} returns a NaN. |
b8d4c8d0 GM |
1187 | @end defun |
1188 | ||
fead402d CY |
1189 | In addition, Emacs defines the following common mathematical |
1190 | constants: | |
1191 | ||
1192 | @defvar float-e | |
1193 | The mathematical constant @math{e} (2.71828@dots{}). | |
1194 | @end defvar | |
1195 | ||
1196 | @defvar float-pi | |
1197 | The mathematical constant @math{pi} (3.14159@dots{}). | |
1198 | @end defvar | |
1199 | ||
b8d4c8d0 GM |
1200 | @node Random Numbers |
1201 | @section Random Numbers | |
1202 | @cindex random numbers | |
1203 | ||
1204 | A deterministic computer program cannot generate true random numbers. | |
1205 | For most purposes, @dfn{pseudo-random numbers} suffice. A series of | |
1206 | pseudo-random numbers is generated in a deterministic fashion. The | |
1207 | numbers are not truly random, but they have certain properties that | |
1208 | mimic a random series. For example, all possible values occur equally | |
1209 | often in a pseudo-random series. | |
1210 | ||
0e23ef9d | 1211 | In Emacs, pseudo-random numbers are generated from a ``seed''. |
b8d4c8d0 | 1212 | Starting from any given seed, the @code{random} function always |
0e23ef9d PE |
1213 | generates the same sequence of numbers. Emacs typically starts with a |
1214 | different seed each time, so the sequence of values of @code{random} | |
1215 | typically differs in each Emacs run. | |
1216 | ||
1217 | Sometimes you want the random number sequence to be repeatable. For | |
1218 | example, when debugging a program whose behavior depends on the random | |
1219 | number sequence, it is helpful to get the same behavior in each | |
1220 | program run. To make the sequence repeat, execute @code{(random "")}. | |
1221 | This sets the seed to a constant value for your particular Emacs | |
1222 | executable (though it may differ for other Emacs builds). You can use | |
1223 | other strings to choose various seed values. | |
b8d4c8d0 GM |
1224 | |
1225 | @defun random &optional limit | |
1226 | This function returns a pseudo-random integer. Repeated calls return a | |
1227 | series of pseudo-random integers. | |
1228 | ||
1229 | If @var{limit} is a positive integer, the value is chosen to be | |
0e23ef9d PE |
1230 | nonnegative and less than @var{limit}. Otherwise, the value |
1231 | might be any integer representable in Lisp. | |
b8d4c8d0 GM |
1232 | |
1233 | If @var{limit} is @code{t}, it means to choose a new seed based on the | |
1234 | current time of day and on Emacs's process @acronym{ID} number. | |
b8d4c8d0 | 1235 | |
0e23ef9d PE |
1236 | If @var{limit} is a string, it means to choose a new seed based on the |
1237 | string's contents. | |
1238 | ||
b8d4c8d0 | 1239 | @end defun |