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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. |
b8d4c8d0 GM |
6 | @node Numbers, Strings and Characters, Lisp Data Types, Top |
7 | @chapter Numbers | |
8 | @cindex integers | |
9 | @cindex numbers | |
10 | ||
11 | GNU Emacs supports two numeric data types: @dfn{integers} and | |
12 | @dfn{floating point numbers}. Integers are whole numbers such as | |
13 | @minus{}3, 0, 7, 13, and 511. Their values are exact. Floating point | |
14 | numbers are numbers with fractional parts, such as @minus{}4.5, 0.0, or | |
15 | 2.71828. They can also be expressed in exponential notation: 1.5e2 | |
16 | equals 150; in this example, @samp{e2} stands for ten to the second | |
17 | power, and that is multiplied by 1.5. Floating point values are not | |
18 | exact; they have a fixed, limited amount of precision. | |
19 | ||
20 | @menu | |
21 | * Integer Basics:: Representation and range of integers. | |
d24880de | 22 | * Float Basics:: Representation and range of floating point. |
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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 | |
34 | @comment node-name, next, previous, up | |
35 | @section Integer Basics | |
36 | ||
37 | The range of values for an integer depends on the machine. The | |
1ddd6622 | 38 | minimum range is @minus{}536870912 to 536870911 (30 bits; i.e., |
b8d4c8d0 | 39 | @ifnottex |
1ddd6622 | 40 | -2**29 |
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41 | @end ifnottex |
42 | @tex | |
1ddd6622 | 43 | @math{-2^{29}} |
b8d4c8d0 GM |
44 | @end tex |
45 | to | |
46 | @ifnottex | |
1ddd6622 | 47 | 2**29 - 1), |
b8d4c8d0 GM |
48 | @end ifnottex |
49 | @tex | |
1ddd6622 | 50 | @math{2^{29}-1}), |
b8d4c8d0 | 51 | @end tex |
001903b5 PE |
52 | but some machines provide a wider range. Many examples in this |
53 | chapter assume that an integer has 30 bits and that floating point | |
be14b9ab | 54 | numbers are IEEE double precision. |
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55 | @cindex overflow |
56 | ||
57 | The Lisp reader reads an integer as a sequence of digits with optional | |
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58 | initial sign and optional final period. An integer that is out of the |
59 | Emacs range is treated as a floating-point number. | |
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60 | |
61 | @example | |
62 | 1 ; @r{The integer 1.} | |
63 | 1. ; @r{The integer 1.} | |
64 | +1 ; @r{Also the integer 1.} | |
65 | -1 ; @r{The integer @minus{}1.} | |
fed14fd7 | 66 | 1073741825 ; @r{The floating point number 1073741825.0.} |
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67 | 0 ; @r{The integer 0.} |
68 | -0 ; @r{The integer 0.} | |
69 | @end example | |
70 | ||
71 | @cindex integers in specific radix | |
72 | @cindex radix for reading an integer | |
73 | @cindex base for reading an integer | |
74 | @cindex hex numbers | |
75 | @cindex octal numbers | |
76 | @cindex reading numbers in hex, octal, and binary | |
77 | The syntax for integers in bases other than 10 uses @samp{#} | |
78 | followed by a letter that specifies the radix: @samp{b} for binary, | |
79 | @samp{o} for octal, @samp{x} for hex, or @samp{@var{radix}r} to | |
80 | specify radix @var{radix}. Case is not significant for the letter | |
81 | that specifies the radix. Thus, @samp{#b@var{integer}} reads | |
82 | @var{integer} in binary, and @samp{#@var{radix}r@var{integer}} reads | |
83 | @var{integer} in radix @var{radix}. Allowed values of @var{radix} run | |
84 | from 2 to 36. For example: | |
85 | ||
86 | @example | |
87 | #b101100 @result{} 44 | |
88 | #o54 @result{} 44 | |
89 | #x2c @result{} 44 | |
90 | #24r1k @result{} 44 | |
91 | @end example | |
92 | ||
93 | To understand how various functions work on integers, especially the | |
94 | bitwise operators (@pxref{Bitwise Operations}), it is often helpful to | |
95 | view the numbers in their binary form. | |
96 | ||
1ddd6622 | 97 | In 30-bit binary, the decimal integer 5 looks like this: |
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98 | |
99 | @example | |
001903b5 | 100 | 0000...000101 (30 bits total) |
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101 | @end example |
102 | ||
103 | @noindent | |
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104 | (The @samp{...} stands for enough bits to fill out a 30-bit word; in |
105 | this case, @samp{...} stands for twenty 0 bits. Later examples also | |
106 | use the @samp{...} notation to make binary integers easier to read.) | |
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107 | |
108 | The integer @minus{}1 looks like this: | |
109 | ||
110 | @example | |
001903b5 | 111 | 1111...111111 (30 bits total) |
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112 | @end example |
113 | ||
114 | @noindent | |
115 | @cindex two's complement | |
1ddd6622 | 116 | @minus{}1 is represented as 30 ones. (This is called @dfn{two's |
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117 | complement} notation.) |
118 | ||
119 | The negative integer, @minus{}5, is creating by subtracting 4 from | |
120 | @minus{}1. In binary, the decimal integer 4 is 100. Consequently, | |
121 | @minus{}5 looks like this: | |
122 | ||
123 | @example | |
001903b5 | 124 | 1111...111011 (30 bits total) |
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125 | @end example |
126 | ||
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127 | In this implementation, the largest 30-bit binary integer value is |
128 | 536,870,911 in decimal. In binary, it looks like this: | |
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129 | |
130 | @example | |
001903b5 | 131 | 0111...111111 (30 bits total) |
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132 | @end example |
133 | ||
134 | Since the arithmetic functions do not check whether integers go | |
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135 | outside their range, when you add 1 to 536,870,911, the value is the |
136 | negative integer @minus{}536,870,912: | |
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137 | |
138 | @example | |
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139 | (+ 1 536870911) |
140 | @result{} -536870912 | |
001903b5 | 141 | @result{} 1000...000000 (30 bits total) |
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142 | @end example |
143 | ||
144 | Many of the functions described in this chapter accept markers for | |
145 | arguments in place of numbers. (@xref{Markers}.) Since the actual | |
146 | arguments to such functions may be either numbers or markers, we often | |
147 | give these arguments the name @var{number-or-marker}. When the argument | |
148 | value is a marker, its position value is used and its buffer is ignored. | |
149 | ||
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150 | @cindex largest Lisp integer number |
151 | @cindex maximum Lisp integer number | |
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152 | @defvar most-positive-fixnum |
153 | The value of this variable is the largest integer that Emacs Lisp | |
154 | can handle. | |
155 | @end defvar | |
156 | ||
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157 | @cindex smallest Lisp integer number |
158 | @cindex minimum Lisp integer number | |
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159 | @defvar most-negative-fixnum |
160 | The value of this variable is the smallest integer that Emacs Lisp can | |
161 | handle. It is negative. | |
162 | @end defvar | |
163 | ||
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164 | @xref{Character Codes, max-char}, for the maximum value of a valid |
165 | character codepoint. | |
166 | ||
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167 | @node Float Basics |
168 | @section Floating Point Basics | |
169 | ||
fead402d | 170 | @cindex @acronym{IEEE} floating point |
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171 | Floating point numbers are useful for representing numbers that are |
172 | not integral. The precise range of floating point numbers is | |
173 | machine-specific; it is the same as the range of the C data type | |
fead402d CY |
174 | @code{double} on the machine you are using. Emacs uses the |
175 | @acronym{IEEE} floating point standard where possible (the standard is | |
176 | supported by most modern computers). | |
b8d4c8d0 | 177 | |
fead402d | 178 | The read syntax for floating point numbers requires either a decimal |
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179 | point (with at least one digit following), an exponent, or both. For |
180 | example, @samp{1500.0}, @samp{15e2}, @samp{15.0e2}, @samp{1.5e3}, and | |
181 | @samp{.15e4} are five ways of writing a floating point number whose | |
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182 | value is 1500. They are all equivalent. You can also use a minus |
183 | sign to write negative floating point numbers, as in @samp{-1.0}. | |
184 | ||
185 | Emacs Lisp treats @code{-0.0} as equal to ordinary zero (with | |
186 | respect to @code{equal} and @code{=}), even though the two are | |
187 | distinguishable in the @acronym{IEEE} floating point standard. | |
b8d4c8d0 | 188 | |
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189 | @cindex positive infinity |
190 | @cindex negative infinity | |
191 | @cindex infinity | |
192 | @cindex NaN | |
fead402d CY |
193 | The @acronym{IEEE} floating point standard supports positive |
194 | infinity and negative infinity as floating point values. It also | |
195 | provides for a class of values called NaN or ``not-a-number''; | |
196 | numerical functions return such values in cases where there is no | |
197 | correct answer. For example, @code{(/ 0.0 0.0)} returns a NaN. (NaN | |
198 | values can also carry a sign, but for practical purposes there's no | |
199 | significant difference between different NaN values in Emacs Lisp.) | |
200 | Here are the read syntaxes for these special floating point values: | |
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201 | |
202 | @table @asis | |
203 | @item positive infinity | |
204 | @samp{1.0e+INF} | |
205 | @item negative infinity | |
206 | @samp{-1.0e+INF} | |
fed14fd7 | 207 | @item Not-a-number |
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208 | @samp{0.0e+NaN} or @samp{-0.0e+NaN}. |
209 | @end table | |
210 | ||
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211 | @defun isnan number |
212 | This predicate tests whether its argument is NaN, and returns @code{t} | |
213 | if so, @code{nil} otherwise. The argument must be a number. | |
214 | @end defun | |
215 | ||
216 | The following functions are specialized for handling floating point | |
217 | numbers: | |
218 | ||
219 | @defun frexp x | |
220 | This function returns a cons cell @code{(@var{sig} . @var{exp})}, | |
221 | where @var{sig} and @var{exp} are respectively the significand and | |
222 | exponent of the floating point number @var{x}: | |
223 | ||
224 | @smallexample | |
225 | @var{x} = @var{sig} * 2^@var{exp} | |
226 | @end smallexample | |
227 | ||
228 | @var{sig} is a floating point number between 0.5 (inclusive) and 1.0 | |
229 | (exclusive). If @var{x} is zero, the return value is @code{(0 . 0)}. | |
230 | @end defun | |
b8d4c8d0 | 231 | |
fead402d CY |
232 | @defun ldexp sig &optional exp |
233 | This function returns a floating point number corresponding to the | |
234 | significand @var{sig} and exponent @var{exp}. | |
235 | @end defun | |
b8d4c8d0 | 236 | |
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237 | @defun copysign x1 x2 |
238 | This function copies the sign of @var{x2} to the value of @var{x1}, | |
239 | and returns the result. @var{x1} and @var{x2} must be floating point | |
240 | numbers. | |
241 | @end defun | |
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242 | |
243 | @defun logb number | |
244 | This function returns the binary exponent of @var{number}. More | |
245 | precisely, the value is the logarithm of @var{number} base 2, rounded | |
246 | down to an integer. | |
247 | ||
248 | @example | |
249 | (logb 10) | |
250 | @result{} 3 | |
251 | (logb 10.0e20) | |
252 | @result{} 69 | |
253 | @end example | |
254 | @end defun | |
255 | ||
256 | @node Predicates on Numbers | |
257 | @section Type Predicates for Numbers | |
258 | @cindex predicates for numbers | |
259 | ||
260 | The functions in this section test for numbers, or for a specific | |
261 | type of number. The functions @code{integerp} and @code{floatp} can | |
262 | take any type of Lisp object as argument (they would not be of much | |
263 | use otherwise), but the @code{zerop} predicate requires a number as | |
264 | its argument. See also @code{integer-or-marker-p} and | |
265 | @code{number-or-marker-p}, in @ref{Predicates on Markers}. | |
266 | ||
267 | @defun floatp object | |
268 | This predicate tests whether its argument is a floating point | |
269 | number and returns @code{t} if so, @code{nil} otherwise. | |
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270 | @end defun |
271 | ||
272 | @defun integerp object | |
273 | This predicate tests whether its argument is an integer, and returns | |
274 | @code{t} if so, @code{nil} otherwise. | |
275 | @end defun | |
276 | ||
277 | @defun numberp object | |
278 | This predicate tests whether its argument is a number (either integer or | |
279 | floating point), and returns @code{t} if so, @code{nil} otherwise. | |
280 | @end defun | |
281 | ||
0f29fa41 | 282 | @defun natnump object |
b8d4c8d0 | 283 | @cindex natural numbers |
bccc0e40 | 284 | This predicate (whose name comes from the phrase ``natural number'') |
0f29fa41 CY |
285 | tests to see whether its argument is a nonnegative integer, and |
286 | returns @code{t} if so, @code{nil} otherwise. 0 is considered | |
287 | non-negative. | |
b8d4c8d0 | 288 | |
0f29fa41 CY |
289 | @findex wholenump number |
290 | This is a synonym for @code{natnump}. | |
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291 | @end defun |
292 | ||
293 | @defun zerop number | |
294 | This predicate tests whether its argument is zero, and returns @code{t} | |
295 | if so, @code{nil} otherwise. The argument must be a number. | |
296 | ||
297 | @code{(zerop x)} is equivalent to @code{(= x 0)}. | |
298 | @end defun | |
299 | ||
300 | @node Comparison of Numbers | |
301 | @section Comparison of Numbers | |
302 | @cindex number comparison | |
303 | @cindex comparing numbers | |
304 | ||
305 | To test numbers for numerical equality, you should normally use | |
306 | @code{=}, not @code{eq}. There can be many distinct floating point | |
307 | number objects with the same numeric value. If you use @code{eq} to | |
308 | compare them, then you test whether two values are the same | |
309 | @emph{object}. By contrast, @code{=} compares only the numeric values | |
310 | of the objects. | |
311 | ||
312 | At present, each integer value has a unique Lisp object in Emacs Lisp. | |
313 | Therefore, @code{eq} is equivalent to @code{=} where integers are | |
314 | concerned. It is sometimes convenient to use @code{eq} for comparing an | |
315 | unknown value with an integer, because @code{eq} does not report an | |
316 | error if the unknown value is not a number---it accepts arguments of any | |
317 | type. By contrast, @code{=} signals an error if the arguments are not | |
318 | numbers or markers. However, it is a good idea to use @code{=} if you | |
319 | can, even for comparing integers, just in case we change the | |
320 | representation of integers in a future Emacs version. | |
321 | ||
322 | Sometimes it is useful to compare numbers with @code{equal}; it | |
323 | treats two numbers as equal if they have the same data type (both | |
324 | integers, or both floating point) and the same value. By contrast, | |
325 | @code{=} can treat an integer and a floating point number as equal. | |
326 | @xref{Equality Predicates}. | |
327 | ||
328 | There is another wrinkle: because floating point arithmetic is not | |
329 | exact, it is often a bad idea to check for equality of two floating | |
330 | point values. Usually it is better to test for approximate equality. | |
331 | Here's a function to do this: | |
332 | ||
333 | @example | |
334 | (defvar fuzz-factor 1.0e-6) | |
335 | (defun approx-equal (x y) | |
336 | (or (and (= x 0) (= y 0)) | |
337 | (< (/ (abs (- x y)) | |
338 | (max (abs x) (abs y))) | |
339 | fuzz-factor))) | |
340 | @end example | |
341 | ||
342 | @cindex CL note---integers vrs @code{eq} | |
343 | @quotation | |
344 | @b{Common Lisp note:} Comparing numbers in Common Lisp always requires | |
345 | @code{=} because Common Lisp implements multi-word integers, and two | |
346 | distinct integer objects can have the same numeric value. Emacs Lisp | |
347 | can have just one integer object for any given value because it has a | |
348 | limited range of integer values. | |
349 | @end quotation | |
350 | ||
351 | @defun = number-or-marker1 number-or-marker2 | |
352 | This function tests whether its arguments are numerically equal, and | |
353 | returns @code{t} if so, @code{nil} otherwise. | |
354 | @end defun | |
355 | ||
356 | @defun eql value1 value2 | |
357 | This function acts like @code{eq} except when both arguments are | |
358 | numbers. It compares numbers by type and numeric value, so that | |
359 | @code{(eql 1.0 1)} returns @code{nil}, but @code{(eql 1.0 1.0)} and | |
360 | @code{(eql 1 1)} both return @code{t}. | |
361 | @end defun | |
362 | ||
363 | @defun /= number-or-marker1 number-or-marker2 | |
364 | This function tests whether its arguments are numerically equal, and | |
365 | returns @code{t} if they are not, and @code{nil} if they are. | |
366 | @end defun | |
367 | ||
368 | @defun < number-or-marker1 number-or-marker2 | |
369 | This function tests whether its first argument is strictly less than | |
370 | its second argument. It returns @code{t} if so, @code{nil} otherwise. | |
371 | @end defun | |
372 | ||
373 | @defun <= number-or-marker1 number-or-marker2 | |
374 | This function tests whether its first argument is less than or equal | |
375 | to its second argument. It returns @code{t} if so, @code{nil} | |
376 | otherwise. | |
377 | @end defun | |
378 | ||
379 | @defun > number-or-marker1 number-or-marker2 | |
380 | This function tests whether its first argument is strictly greater | |
381 | than its second argument. It returns @code{t} if so, @code{nil} | |
382 | otherwise. | |
383 | @end defun | |
384 | ||
385 | @defun >= number-or-marker1 number-or-marker2 | |
386 | This function tests whether its first argument is greater than or | |
387 | equal to its second argument. It returns @code{t} if so, @code{nil} | |
388 | otherwise. | |
389 | @end defun | |
390 | ||
391 | @defun max number-or-marker &rest numbers-or-markers | |
392 | This function returns the largest of its arguments. | |
393 | If any of the arguments is floating-point, the value is returned | |
394 | as floating point, even if it was given as an integer. | |
395 | ||
396 | @example | |
397 | (max 20) | |
398 | @result{} 20 | |
399 | (max 1 2.5) | |
400 | @result{} 2.5 | |
401 | (max 1 3 2.5) | |
402 | @result{} 3.0 | |
403 | @end example | |
404 | @end defun | |
405 | ||
406 | @defun min number-or-marker &rest numbers-or-markers | |
407 | This function returns the smallest of its arguments. | |
408 | If any of the arguments is floating-point, the value is returned | |
409 | as floating point, even if it was given as an integer. | |
410 | ||
411 | @example | |
412 | (min -4 1) | |
413 | @result{} -4 | |
414 | @end example | |
415 | @end defun | |
416 | ||
417 | @defun abs number | |
418 | This function returns the absolute value of @var{number}. | |
419 | @end defun | |
420 | ||
421 | @node Numeric Conversions | |
422 | @section Numeric Conversions | |
423 | @cindex rounding in conversions | |
424 | @cindex number conversions | |
425 | @cindex converting numbers | |
426 | ||
427 | To convert an integer to floating point, use the function @code{float}. | |
428 | ||
429 | @defun float number | |
430 | This returns @var{number} converted to floating point. | |
431 | If @var{number} is already a floating point number, @code{float} returns | |
432 | it unchanged. | |
433 | @end defun | |
434 | ||
435 | There are four functions to convert floating point numbers to integers; | |
436 | they differ in how they round. All accept an argument @var{number} | |
437 | and an optional argument @var{divisor}. Both arguments may be | |
438 | integers or floating point numbers. @var{divisor} may also be | |
439 | @code{nil}. If @var{divisor} is @code{nil} or omitted, these | |
440 | functions convert @var{number} to an integer, or return it unchanged | |
441 | if it already is an integer. If @var{divisor} is non-@code{nil}, they | |
442 | divide @var{number} by @var{divisor} and convert the result to an | |
443 | integer. An @code{arith-error} results if @var{divisor} is 0. | |
444 | ||
445 | @defun truncate number &optional divisor | |
446 | This returns @var{number}, converted to an integer by rounding towards | |
447 | zero. | |
448 | ||
449 | @example | |
450 | (truncate 1.2) | |
451 | @result{} 1 | |
452 | (truncate 1.7) | |
453 | @result{} 1 | |
454 | (truncate -1.2) | |
455 | @result{} -1 | |
456 | (truncate -1.7) | |
457 | @result{} -1 | |
458 | @end example | |
459 | @end defun | |
460 | ||
461 | @defun floor number &optional divisor | |
462 | This returns @var{number}, converted to an integer by rounding downward | |
463 | (towards negative infinity). | |
464 | ||
465 | If @var{divisor} is specified, this uses the kind of division | |
466 | operation that corresponds to @code{mod}, rounding downward. | |
467 | ||
468 | @example | |
469 | (floor 1.2) | |
470 | @result{} 1 | |
471 | (floor 1.7) | |
472 | @result{} 1 | |
473 | (floor -1.2) | |
474 | @result{} -2 | |
475 | (floor -1.7) | |
476 | @result{} -2 | |
477 | (floor 5.99 3) | |
478 | @result{} 1 | |
479 | @end example | |
480 | @end defun | |
481 | ||
482 | @defun ceiling number &optional divisor | |
483 | This returns @var{number}, converted to an integer by rounding upward | |
484 | (towards positive infinity). | |
485 | ||
486 | @example | |
487 | (ceiling 1.2) | |
488 | @result{} 2 | |
489 | (ceiling 1.7) | |
490 | @result{} 2 | |
491 | (ceiling -1.2) | |
492 | @result{} -1 | |
493 | (ceiling -1.7) | |
494 | @result{} -1 | |
495 | @end example | |
496 | @end defun | |
497 | ||
498 | @defun round number &optional divisor | |
499 | This returns @var{number}, converted to an integer by rounding towards the | |
500 | nearest integer. Rounding a value equidistant between two integers | |
501 | may choose the integer closer to zero, or it may prefer an even integer, | |
502 | depending on your machine. | |
503 | ||
504 | @example | |
505 | (round 1.2) | |
506 | @result{} 1 | |
507 | (round 1.7) | |
508 | @result{} 2 | |
509 | (round -1.2) | |
510 | @result{} -1 | |
511 | (round -1.7) | |
512 | @result{} -2 | |
513 | @end example | |
514 | @end defun | |
515 | ||
516 | @node Arithmetic Operations | |
517 | @section Arithmetic Operations | |
518 | @cindex arithmetic operations | |
519 | ||
520 | Emacs Lisp provides the traditional four arithmetic operations: | |
521 | addition, subtraction, multiplication, and division. Remainder and modulus | |
522 | functions supplement the division functions. The functions to | |
523 | add or subtract 1 are provided because they are traditional in Lisp and | |
524 | commonly used. | |
525 | ||
526 | All of these functions except @code{%} return a floating point value | |
527 | if any argument is floating. | |
528 | ||
c717b326 | 529 | It is important to note that in Emacs Lisp, arithmetic functions |
001903b5 PE |
530 | do not check for overflow. Thus @code{(1+ 536870911)} may evaluate to |
531 | @minus{}536870912, depending on your hardware. | |
b8d4c8d0 GM |
532 | |
533 | @defun 1+ number-or-marker | |
534 | This function returns @var{number-or-marker} plus 1. | |
535 | For example, | |
536 | ||
537 | @example | |
538 | (setq foo 4) | |
539 | @result{} 4 | |
540 | (1+ foo) | |
541 | @result{} 5 | |
542 | @end example | |
543 | ||
544 | This function is not analogous to the C operator @code{++}---it does not | |
545 | increment a variable. It just computes a sum. Thus, if we continue, | |
546 | ||
547 | @example | |
548 | foo | |
549 | @result{} 4 | |
550 | @end example | |
551 | ||
552 | If you want to increment the variable, you must use @code{setq}, | |
553 | like this: | |
554 | ||
555 | @example | |
556 | (setq foo (1+ foo)) | |
557 | @result{} 5 | |
558 | @end example | |
559 | @end defun | |
560 | ||
561 | @defun 1- number-or-marker | |
562 | This function returns @var{number-or-marker} minus 1. | |
563 | @end defun | |
564 | ||
565 | @defun + &rest numbers-or-markers | |
566 | This function adds its arguments together. When given no arguments, | |
567 | @code{+} returns 0. | |
568 | ||
569 | @example | |
570 | (+) | |
571 | @result{} 0 | |
572 | (+ 1) | |
573 | @result{} 1 | |
574 | (+ 1 2 3 4) | |
575 | @result{} 10 | |
576 | @end example | |
577 | @end defun | |
578 | ||
579 | @defun - &optional number-or-marker &rest more-numbers-or-markers | |
580 | The @code{-} function serves two purposes: negation and subtraction. | |
581 | When @code{-} has a single argument, the value is the negative of the | |
582 | argument. When there are multiple arguments, @code{-} subtracts each of | |
583 | the @var{more-numbers-or-markers} from @var{number-or-marker}, | |
584 | cumulatively. If there are no arguments, the result is 0. | |
585 | ||
586 | @example | |
587 | (- 10 1 2 3 4) | |
588 | @result{} 0 | |
589 | (- 10) | |
590 | @result{} -10 | |
591 | (-) | |
592 | @result{} 0 | |
593 | @end example | |
594 | @end defun | |
595 | ||
596 | @defun * &rest numbers-or-markers | |
597 | This function multiplies its arguments together, and returns the | |
598 | product. When given no arguments, @code{*} returns 1. | |
599 | ||
600 | @example | |
601 | (*) | |
602 | @result{} 1 | |
603 | (* 1) | |
604 | @result{} 1 | |
605 | (* 1 2 3 4) | |
606 | @result{} 24 | |
607 | @end example | |
608 | @end defun | |
609 | ||
610 | @defun / dividend divisor &rest divisors | |
611 | This function divides @var{dividend} by @var{divisor} and returns the | |
612 | quotient. If there are additional arguments @var{divisors}, then it | |
613 | divides @var{dividend} by each divisor in turn. Each argument may be a | |
614 | number or a marker. | |
615 | ||
616 | If all the arguments are integers, then the result is an integer too. | |
617 | This means the result has to be rounded. On most machines, the result | |
618 | is rounded towards zero after each division, but some machines may round | |
619 | differently with negative arguments. This is because the Lisp function | |
620 | @code{/} is implemented using the C division operator, which also | |
621 | permits machine-dependent rounding. As a practical matter, all known | |
622 | machines round in the standard fashion. | |
623 | ||
624 | @cindex @code{arith-error} in division | |
625 | If you divide an integer by 0, an @code{arith-error} error is signaled. | |
626 | (@xref{Errors}.) Floating point division by zero returns either | |
627 | infinity or a NaN if your machine supports @acronym{IEEE} floating point; | |
628 | otherwise, it signals an @code{arith-error} error. | |
629 | ||
630 | @example | |
631 | @group | |
632 | (/ 6 2) | |
633 | @result{} 3 | |
634 | @end group | |
635 | (/ 5 2) | |
636 | @result{} 2 | |
637 | (/ 5.0 2) | |
638 | @result{} 2.5 | |
639 | (/ 5 2.0) | |
640 | @result{} 2.5 | |
641 | (/ 5.0 2.0) | |
642 | @result{} 2.5 | |
643 | (/ 25 3 2) | |
644 | @result{} 4 | |
645 | @group | |
646 | (/ -17 6) | |
647 | @result{} -2 @r{(could in theory be @minus{}3 on some machines)} | |
648 | @end group | |
649 | @end example | |
650 | @end defun | |
651 | ||
652 | @defun % dividend divisor | |
653 | @cindex remainder | |
654 | This function returns the integer remainder after division of @var{dividend} | |
655 | by @var{divisor}. The arguments must be integers or markers. | |
656 | ||
657 | For negative arguments, the remainder is in principle machine-dependent | |
658 | since the quotient is; but in practice, all known machines behave alike. | |
659 | ||
660 | An @code{arith-error} results if @var{divisor} is 0. | |
661 | ||
662 | @example | |
663 | (% 9 4) | |
664 | @result{} 1 | |
665 | (% -9 4) | |
666 | @result{} -1 | |
667 | (% 9 -4) | |
668 | @result{} 1 | |
669 | (% -9 -4) | |
670 | @result{} -1 | |
671 | @end example | |
672 | ||
673 | For any two integers @var{dividend} and @var{divisor}, | |
674 | ||
675 | @example | |
676 | @group | |
677 | (+ (% @var{dividend} @var{divisor}) | |
678 | (* (/ @var{dividend} @var{divisor}) @var{divisor})) | |
679 | @end group | |
680 | @end example | |
681 | ||
682 | @noindent | |
683 | always equals @var{dividend}. | |
684 | @end defun | |
685 | ||
686 | @defun mod dividend divisor | |
687 | @cindex modulus | |
688 | This function returns the value of @var{dividend} modulo @var{divisor}; | |
689 | in other words, the remainder after division of @var{dividend} | |
690 | by @var{divisor}, but with the same sign as @var{divisor}. | |
691 | The arguments must be numbers or markers. | |
692 | ||
693 | Unlike @code{%}, @code{mod} returns a well-defined result for negative | |
694 | arguments. It also permits floating point arguments; it rounds the | |
695 | quotient downward (towards minus infinity) to an integer, and uses that | |
696 | quotient to compute the remainder. | |
697 | ||
698 | An @code{arith-error} results if @var{divisor} is 0. | |
699 | ||
700 | @example | |
701 | @group | |
702 | (mod 9 4) | |
703 | @result{} 1 | |
704 | @end group | |
705 | @group | |
706 | (mod -9 4) | |
707 | @result{} 3 | |
708 | @end group | |
709 | @group | |
710 | (mod 9 -4) | |
711 | @result{} -3 | |
712 | @end group | |
713 | @group | |
714 | (mod -9 -4) | |
715 | @result{} -1 | |
716 | @end group | |
717 | @group | |
718 | (mod 5.5 2.5) | |
719 | @result{} .5 | |
720 | @end group | |
721 | @end example | |
722 | ||
723 | For any two numbers @var{dividend} and @var{divisor}, | |
724 | ||
725 | @example | |
726 | @group | |
727 | (+ (mod @var{dividend} @var{divisor}) | |
728 | (* (floor @var{dividend} @var{divisor}) @var{divisor})) | |
729 | @end group | |
730 | @end example | |
731 | ||
732 | @noindent | |
733 | always equals @var{dividend}, subject to rounding error if either | |
734 | argument is floating point. For @code{floor}, see @ref{Numeric | |
735 | Conversions}. | |
736 | @end defun | |
737 | ||
738 | @node Rounding Operations | |
739 | @section Rounding Operations | |
740 | @cindex rounding without conversion | |
741 | ||
742 | The functions @code{ffloor}, @code{fceiling}, @code{fround}, and | |
743 | @code{ftruncate} take a floating point argument and return a floating | |
744 | point result whose value is a nearby integer. @code{ffloor} returns the | |
745 | nearest integer below; @code{fceiling}, the nearest integer above; | |
746 | @code{ftruncate}, the nearest integer in the direction towards zero; | |
747 | @code{fround}, the nearest integer. | |
748 | ||
749 | @defun ffloor float | |
750 | This function rounds @var{float} to the next lower integral value, and | |
751 | returns that value as a floating point number. | |
752 | @end defun | |
753 | ||
754 | @defun fceiling float | |
755 | This function rounds @var{float} to the next higher integral value, and | |
756 | returns that value as a floating point number. | |
757 | @end defun | |
758 | ||
759 | @defun ftruncate float | |
760 | This function rounds @var{float} towards zero to an integral value, and | |
761 | returns that value as a floating point number. | |
762 | @end defun | |
763 | ||
764 | @defun fround float | |
765 | This function rounds @var{float} to the nearest integral value, | |
766 | and returns that value as a floating point number. | |
767 | @end defun | |
768 | ||
769 | @node Bitwise Operations | |
770 | @section Bitwise Operations on Integers | |
771 | @cindex bitwise arithmetic | |
772 | @cindex logical arithmetic | |
773 | ||
774 | In a computer, an integer is represented as a binary number, a | |
775 | sequence of @dfn{bits} (digits which are either zero or one). A bitwise | |
776 | operation acts on the individual bits of such a sequence. For example, | |
777 | @dfn{shifting} moves the whole sequence left or right one or more places, | |
16152b76 | 778 | reproducing the same pattern ``moved over''. |
b8d4c8d0 GM |
779 | |
780 | The bitwise operations in Emacs Lisp apply only to integers. | |
781 | ||
782 | @defun lsh integer1 count | |
783 | @cindex logical shift | |
784 | @code{lsh}, which is an abbreviation for @dfn{logical shift}, shifts the | |
785 | bits in @var{integer1} to the left @var{count} places, or to the right | |
786 | if @var{count} is negative, bringing zeros into the vacated bits. If | |
787 | @var{count} is negative, @code{lsh} shifts zeros into the leftmost | |
788 | (most-significant) bit, producing a positive result even if | |
789 | @var{integer1} is negative. Contrast this with @code{ash}, below. | |
790 | ||
791 | Here are two examples of @code{lsh}, shifting a pattern of bits one | |
792 | place to the left. We show only the low-order eight bits of the binary | |
793 | pattern; the rest are all zero. | |
794 | ||
795 | @example | |
796 | @group | |
797 | (lsh 5 1) | |
798 | @result{} 10 | |
799 | ;; @r{Decimal 5 becomes decimal 10.} | |
800 | 00000101 @result{} 00001010 | |
801 | ||
802 | (lsh 7 1) | |
803 | @result{} 14 | |
804 | ;; @r{Decimal 7 becomes decimal 14.} | |
805 | 00000111 @result{} 00001110 | |
806 | @end group | |
807 | @end example | |
808 | ||
809 | @noindent | |
810 | As the examples illustrate, shifting the pattern of bits one place to | |
811 | the left produces a number that is twice the value of the previous | |
812 | number. | |
813 | ||
814 | Shifting a pattern of bits two places to the left produces results | |
815 | like this (with 8-bit binary numbers): | |
816 | ||
817 | @example | |
818 | @group | |
819 | (lsh 3 2) | |
820 | @result{} 12 | |
821 | ;; @r{Decimal 3 becomes decimal 12.} | |
822 | 00000011 @result{} 00001100 | |
823 | @end group | |
824 | @end example | |
825 | ||
826 | On the other hand, shifting one place to the right looks like this: | |
827 | ||
828 | @example | |
829 | @group | |
830 | (lsh 6 -1) | |
831 | @result{} 3 | |
832 | ;; @r{Decimal 6 becomes decimal 3.} | |
833 | 00000110 @result{} 00000011 | |
834 | @end group | |
835 | ||
836 | @group | |
837 | (lsh 5 -1) | |
838 | @result{} 2 | |
839 | ;; @r{Decimal 5 becomes decimal 2.} | |
840 | 00000101 @result{} 00000010 | |
841 | @end group | |
842 | @end example | |
843 | ||
844 | @noindent | |
845 | As the example illustrates, shifting one place to the right divides the | |
846 | value of a positive integer by two, rounding downward. | |
847 | ||
c717b326 | 848 | The function @code{lsh}, like all Emacs Lisp arithmetic functions, does |
b8d4c8d0 GM |
849 | not check for overflow, so shifting left can discard significant bits |
850 | and change the sign of the number. For example, left shifting | |
001903b5 | 851 | 536,870,911 produces @minus{}2 in the 30-bit implementation: |
b8d4c8d0 GM |
852 | |
853 | @example | |
1ddd6622 | 854 | (lsh 536870911 1) ; @r{left shift} |
b8d4c8d0 GM |
855 | @result{} -2 |
856 | @end example | |
857 | ||
001903b5 | 858 | In binary, the argument looks like this: |
b8d4c8d0 GM |
859 | |
860 | @example | |
861 | @group | |
1ddd6622 | 862 | ;; @r{Decimal 536,870,911} |
001903b5 | 863 | 0111...111111 (30 bits total) |
b8d4c8d0 GM |
864 | @end group |
865 | @end example | |
866 | ||
867 | @noindent | |
868 | which becomes the following when left shifted: | |
869 | ||
870 | @example | |
871 | @group | |
872 | ;; @r{Decimal @minus{}2} | |
001903b5 | 873 | 1111...111110 (30 bits total) |
b8d4c8d0 GM |
874 | @end group |
875 | @end example | |
876 | @end defun | |
877 | ||
878 | @defun ash integer1 count | |
879 | @cindex arithmetic shift | |
880 | @code{ash} (@dfn{arithmetic shift}) shifts the bits in @var{integer1} | |
881 | to the left @var{count} places, or to the right if @var{count} | |
882 | is negative. | |
883 | ||
884 | @code{ash} gives the same results as @code{lsh} except when | |
885 | @var{integer1} and @var{count} are both negative. In that case, | |
886 | @code{ash} puts ones in the empty bit positions on the left, while | |
887 | @code{lsh} puts zeros in those bit positions. | |
888 | ||
889 | Thus, with @code{ash}, shifting the pattern of bits one place to the right | |
890 | looks like this: | |
891 | ||
892 | @example | |
893 | @group | |
894 | (ash -6 -1) @result{} -3 | |
895 | ;; @r{Decimal @minus{}6 becomes decimal @minus{}3.} | |
001903b5 | 896 | 1111...111010 (30 bits total) |
b8d4c8d0 | 897 | @result{} |
001903b5 | 898 | 1111...111101 (30 bits total) |
b8d4c8d0 GM |
899 | @end group |
900 | @end example | |
901 | ||
902 | In contrast, shifting the pattern of bits one place to the right with | |
903 | @code{lsh} looks like this: | |
904 | ||
905 | @example | |
906 | @group | |
1ddd6622 GM |
907 | (lsh -6 -1) @result{} 536870909 |
908 | ;; @r{Decimal @minus{}6 becomes decimal 536,870,909.} | |
001903b5 | 909 | 1111...111010 (30 bits total) |
b8d4c8d0 | 910 | @result{} |
001903b5 | 911 | 0111...111101 (30 bits total) |
b8d4c8d0 GM |
912 | @end group |
913 | @end example | |
914 | ||
915 | Here are other examples: | |
916 | ||
917 | @c !!! Check if lined up in smallbook format! XDVI shows problem | |
918 | @c with smallbook but not with regular book! --rjc 16mar92 | |
919 | @smallexample | |
920 | @group | |
001903b5 | 921 | ; @r{ 30-bit binary values} |
b8d4c8d0 | 922 | |
be14b9ab PE |
923 | (lsh 5 2) ; 5 = @r{0000...000101} |
924 | @result{} 20 ; = @r{0000...010100} | |
b8d4c8d0 GM |
925 | @end group |
926 | @group | |
927 | (ash 5 2) | |
928 | @result{} 20 | |
be14b9ab PE |
929 | (lsh -5 2) ; -5 = @r{1111...111011} |
930 | @result{} -20 ; = @r{1111...101100} | |
b8d4c8d0 GM |
931 | (ash -5 2) |
932 | @result{} -20 | |
933 | @end group | |
934 | @group | |
be14b9ab PE |
935 | (lsh 5 -2) ; 5 = @r{0000...000101} |
936 | @result{} 1 ; = @r{0000...000001} | |
b8d4c8d0 GM |
937 | @end group |
938 | @group | |
939 | (ash 5 -2) | |
940 | @result{} 1 | |
941 | @end group | |
942 | @group | |
be14b9ab | 943 | (lsh -5 -2) ; -5 = @r{1111...111011} |
001903b5 | 944 | @result{} 268435454 |
be14b9ab | 945 | ; = @r{0011...111110} |
b8d4c8d0 GM |
946 | @end group |
947 | @group | |
be14b9ab PE |
948 | (ash -5 -2) ; -5 = @r{1111...111011} |
949 | @result{} -2 ; = @r{1111...111110} | |
b8d4c8d0 GM |
950 | @end group |
951 | @end smallexample | |
952 | @end defun | |
953 | ||
954 | @defun logand &rest ints-or-markers | |
955 | This function returns the ``logical and'' of the arguments: the | |
956 | @var{n}th bit is set in the result if, and only if, the @var{n}th bit is | |
957 | set in all the arguments. (``Set'' means that the value of the bit is 1 | |
958 | rather than 0.) | |
959 | ||
960 | For example, using 4-bit binary numbers, the ``logical and'' of 13 and | |
961 | 12 is 12: 1101 combined with 1100 produces 1100. | |
962 | In both the binary numbers, the leftmost two bits are set (i.e., they | |
963 | are 1's), so the leftmost two bits of the returned value are set. | |
964 | However, for the rightmost two bits, each is zero in at least one of | |
965 | the arguments, so the rightmost two bits of the returned value are 0's. | |
966 | ||
967 | @noindent | |
968 | Therefore, | |
969 | ||
970 | @example | |
971 | @group | |
972 | (logand 13 12) | |
973 | @result{} 12 | |
974 | @end group | |
975 | @end example | |
976 | ||
977 | If @code{logand} is not passed any argument, it returns a value of | |
978 | @minus{}1. This number is an identity element for @code{logand} | |
979 | because its binary representation consists entirely of ones. If | |
980 | @code{logand} is passed just one argument, it returns that argument. | |
981 | ||
982 | @smallexample | |
983 | @group | |
001903b5 | 984 | ; @r{ 30-bit binary values} |
b8d4c8d0 | 985 | |
be14b9ab PE |
986 | (logand 14 13) ; 14 = @r{0000...001110} |
987 | ; 13 = @r{0000...001101} | |
988 | @result{} 12 ; 12 = @r{0000...001100} | |
b8d4c8d0 GM |
989 | @end group |
990 | ||
991 | @group | |
be14b9ab PE |
992 | (logand 14 13 4) ; 14 = @r{0000...001110} |
993 | ; 13 = @r{0000...001101} | |
994 | ; 4 = @r{0000...000100} | |
995 | @result{} 4 ; 4 = @r{0000...000100} | |
b8d4c8d0 GM |
996 | @end group |
997 | ||
998 | @group | |
999 | (logand) | |
be14b9ab | 1000 | @result{} -1 ; -1 = @r{1111...111111} |
b8d4c8d0 GM |
1001 | @end group |
1002 | @end smallexample | |
1003 | @end defun | |
1004 | ||
1005 | @defun logior &rest ints-or-markers | |
1006 | This function returns the ``inclusive or'' of its arguments: the @var{n}th bit | |
1007 | is set in the result if, and only if, the @var{n}th bit is set in at least | |
1008 | one of the arguments. If there are no arguments, the result is zero, | |
1009 | which is an identity element for this operation. If @code{logior} is | |
1010 | passed just one argument, it returns that argument. | |
1011 | ||
1012 | @smallexample | |
1013 | @group | |
001903b5 | 1014 | ; @r{ 30-bit binary values} |
b8d4c8d0 | 1015 | |
be14b9ab PE |
1016 | (logior 12 5) ; 12 = @r{0000...001100} |
1017 | ; 5 = @r{0000...000101} | |
1018 | @result{} 13 ; 13 = @r{0000...001101} | |
b8d4c8d0 GM |
1019 | @end group |
1020 | ||
1021 | @group | |
be14b9ab PE |
1022 | (logior 12 5 7) ; 12 = @r{0000...001100} |
1023 | ; 5 = @r{0000...000101} | |
1024 | ; 7 = @r{0000...000111} | |
1025 | @result{} 15 ; 15 = @r{0000...001111} | |
b8d4c8d0 GM |
1026 | @end group |
1027 | @end smallexample | |
1028 | @end defun | |
1029 | ||
1030 | @defun logxor &rest ints-or-markers | |
1031 | This function returns the ``exclusive or'' of its arguments: the | |
1032 | @var{n}th bit is set in the result if, and only if, the @var{n}th bit is | |
1033 | set in an odd number of the arguments. If there are no arguments, the | |
1034 | result is 0, which is an identity element for this operation. If | |
1035 | @code{logxor} is passed just one argument, it returns that argument. | |
1036 | ||
1037 | @smallexample | |
1038 | @group | |
001903b5 | 1039 | ; @r{ 30-bit binary values} |
b8d4c8d0 | 1040 | |
be14b9ab PE |
1041 | (logxor 12 5) ; 12 = @r{0000...001100} |
1042 | ; 5 = @r{0000...000101} | |
1043 | @result{} 9 ; 9 = @r{0000...001001} | |
b8d4c8d0 GM |
1044 | @end group |
1045 | ||
1046 | @group | |
be14b9ab PE |
1047 | (logxor 12 5 7) ; 12 = @r{0000...001100} |
1048 | ; 5 = @r{0000...000101} | |
1049 | ; 7 = @r{0000...000111} | |
1050 | @result{} 14 ; 14 = @r{0000...001110} | |
b8d4c8d0 GM |
1051 | @end group |
1052 | @end smallexample | |
1053 | @end defun | |
1054 | ||
1055 | @defun lognot integer | |
1056 | This function returns the logical complement of its argument: the @var{n}th | |
1057 | bit is one in the result if, and only if, the @var{n}th bit is zero in | |
1058 | @var{integer}, and vice-versa. | |
1059 | ||
1060 | @example | |
1061 | (lognot 5) | |
1062 | @result{} -6 | |
001903b5 | 1063 | ;; 5 = @r{0000...000101} (30 bits total) |
b8d4c8d0 | 1064 | ;; @r{becomes} |
001903b5 | 1065 | ;; -6 = @r{1111...111010} (30 bits total) |
b8d4c8d0 GM |
1066 | @end example |
1067 | @end defun | |
1068 | ||
1069 | @node Math Functions | |
1070 | @section Standard Mathematical Functions | |
1071 | @cindex transcendental functions | |
1072 | @cindex mathematical functions | |
1073 | @cindex floating-point functions | |
1074 | ||
1075 | These mathematical functions allow integers as well as floating point | |
1076 | numbers as arguments. | |
1077 | ||
1078 | @defun sin arg | |
1079 | @defunx cos arg | |
1080 | @defunx tan arg | |
1081 | These are the ordinary trigonometric functions, with argument measured | |
1082 | in radians. | |
1083 | @end defun | |
1084 | ||
1085 | @defun asin arg | |
1086 | The value of @code{(asin @var{arg})} is a number between | |
1087 | @ifnottex | |
1088 | @minus{}pi/2 | |
1089 | @end ifnottex | |
1090 | @tex | |
1091 | @math{-\pi/2} | |
1092 | @end tex | |
1093 | and | |
1094 | @ifnottex | |
1095 | pi/2 | |
1096 | @end ifnottex | |
1097 | @tex | |
1098 | @math{\pi/2} | |
1099 | @end tex | |
1100 | (inclusive) whose sine is @var{arg}; if, however, @var{arg} is out of | |
1101 | range (outside [@minus{}1, 1]), it signals a @code{domain-error} error. | |
1102 | @end defun | |
1103 | ||
1104 | @defun acos arg | |
1105 | The value of @code{(acos @var{arg})} is a number between 0 and | |
1106 | @ifnottex | |
1107 | pi | |
1108 | @end ifnottex | |
1109 | @tex | |
1110 | @math{\pi} | |
1111 | @end tex | |
1112 | (inclusive) whose cosine is @var{arg}; if, however, @var{arg} is out | |
1113 | of range (outside [@minus{}1, 1]), it signals a @code{domain-error} error. | |
1114 | @end defun | |
1115 | ||
1116 | @defun atan y &optional x | |
1117 | The value of @code{(atan @var{y})} is a number between | |
1118 | @ifnottex | |
1119 | @minus{}pi/2 | |
1120 | @end ifnottex | |
1121 | @tex | |
1122 | @math{-\pi/2} | |
1123 | @end tex | |
1124 | and | |
1125 | @ifnottex | |
1126 | pi/2 | |
1127 | @end ifnottex | |
1128 | @tex | |
1129 | @math{\pi/2} | |
1130 | @end tex | |
1131 | (exclusive) whose tangent is @var{y}. If the optional second | |
1132 | argument @var{x} is given, the value of @code{(atan y x)} is the | |
1133 | angle in radians between the vector @code{[@var{x}, @var{y}]} and the | |
1134 | @code{X} axis. | |
1135 | @end defun | |
1136 | ||
1137 | @defun exp arg | |
fead402d CY |
1138 | This is the exponential function; it returns @math{e} to the power |
1139 | @var{arg}. | |
b8d4c8d0 GM |
1140 | @end defun |
1141 | ||
1142 | @defun log arg &optional base | |
fead402d CY |
1143 | This function returns the logarithm of @var{arg}, with base |
1144 | @var{base}. If you don't specify @var{base}, the natural base | |
1145 | @math{e} is used. If @var{arg} is negative, it signals a | |
1146 | @code{domain-error} error. | |
b8d4c8d0 GM |
1147 | @end defun |
1148 | ||
1149 | @ignore | |
1150 | @defun expm1 arg | |
1151 | This function returns @code{(1- (exp @var{arg}))}, but it is more | |
1152 | accurate than that when @var{arg} is negative and @code{(exp @var{arg})} | |
1153 | is close to 1. | |
1154 | @end defun | |
1155 | ||
1156 | @defun log1p arg | |
1157 | This function returns @code{(log (1+ @var{arg}))}, but it is more | |
1158 | accurate than that when @var{arg} is so small that adding 1 to it would | |
1159 | lose accuracy. | |
1160 | @end defun | |
1161 | @end ignore | |
1162 | ||
1163 | @defun log10 arg | |
1164 | This function returns the logarithm of @var{arg}, with base 10. If | |
1165 | @var{arg} is negative, it signals a @code{domain-error} error. | |
1166 | @code{(log10 @var{x})} @equiv{} @code{(log @var{x} 10)}, at least | |
1167 | approximately. | |
1168 | @end defun | |
1169 | ||
1170 | @defun expt x y | |
1171 | This function returns @var{x} raised to power @var{y}. If both | |
c717b326 PE |
1172 | arguments are integers and @var{y} is positive, the result is an |
1173 | integer; in this case, overflow causes truncation, so watch out. | |
b8d4c8d0 GM |
1174 | @end defun |
1175 | ||
1176 | @defun sqrt arg | |
1177 | This returns the square root of @var{arg}. If @var{arg} is negative, | |
1178 | it signals a @code{domain-error} error. | |
1179 | @end defun | |
1180 | ||
fead402d CY |
1181 | In addition, Emacs defines the following common mathematical |
1182 | constants: | |
1183 | ||
1184 | @defvar float-e | |
1185 | The mathematical constant @math{e} (2.71828@dots{}). | |
1186 | @end defvar | |
1187 | ||
1188 | @defvar float-pi | |
1189 | The mathematical constant @math{pi} (3.14159@dots{}). | |
1190 | @end defvar | |
1191 | ||
b8d4c8d0 GM |
1192 | @node Random Numbers |
1193 | @section Random Numbers | |
1194 | @cindex random numbers | |
1195 | ||
1196 | A deterministic computer program cannot generate true random numbers. | |
1197 | For most purposes, @dfn{pseudo-random numbers} suffice. A series of | |
1198 | pseudo-random numbers is generated in a deterministic fashion. The | |
1199 | numbers are not truly random, but they have certain properties that | |
1200 | mimic a random series. For example, all possible values occur equally | |
1201 | often in a pseudo-random series. | |
1202 | ||
1203 | In Emacs, pseudo-random numbers are generated from a ``seed'' number. | |
1204 | Starting from any given seed, the @code{random} function always | |
1205 | generates the same sequence of numbers. Emacs always starts with the | |
1206 | same seed value, so the sequence of values of @code{random} is actually | |
1207 | the same in each Emacs run! For example, in one operating system, the | |
1208 | first call to @code{(random)} after you start Emacs always returns | |
1209 | @minus{}1457731, and the second one always returns @minus{}7692030. This | |
1210 | repeatability is helpful for debugging. | |
1211 | ||
1212 | If you want random numbers that don't always come out the same, execute | |
1213 | @code{(random t)}. This chooses a new seed based on the current time of | |
1214 | day and on Emacs's process @acronym{ID} number. | |
1215 | ||
1216 | @defun random &optional limit | |
1217 | This function returns a pseudo-random integer. Repeated calls return a | |
1218 | series of pseudo-random integers. | |
1219 | ||
1220 | If @var{limit} is a positive integer, the value is chosen to be | |
1221 | nonnegative and less than @var{limit}. | |
1222 | ||
1223 | If @var{limit} is @code{t}, it means to choose a new seed based on the | |
1224 | current time of day and on Emacs's process @acronym{ID} number. | |
b8d4c8d0 GM |
1225 | |
1226 | On some machines, any integer representable in Lisp may be the result | |
1227 | of @code{random}. On other machines, the result can never be larger | |
1228 | than a certain maximum or less than a certain (negative) minimum. | |
1229 | @end defun |