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1 | @c -*-texinfo-*- |
2 | @c This is part of the GNU Emacs Lisp Reference Manual. | |
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3 | @c Copyright (C) 1990-1995, 1998-1999, 2001-2013 Free Software |
4 | @c Foundation, Inc. | |
b8d4c8d0 | 5 | @c See the file elisp.texi for copying conditions. |
ecc6530d | 6 | @node Numbers |
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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 | |
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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 |
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40 | @end ifnottex |
41 | @tex | |
1ddd6622 | 42 | @math{-2^{29}} |
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43 | @end tex |
44 | to | |
45 | @ifnottex | |
f99f1641 | 46 | 2**29 @minus{} 1), |
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47 | @end ifnottex |
48 | @tex | |
1ddd6622 | 49 | @math{2^{29}-1}), |
b8d4c8d0 | 50 | @end tex |
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51 | but many machines provide a wider range. Many examples in this |
52 | chapter assume the minimum integer width of 30 bits. | |
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53 | @cindex overflow |
54 | ||
55 | The Lisp reader reads an integer as a sequence of digits with optional | |
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56 | initial sign and optional final period. An integer that is out of the |
57 | Emacs range is treated as a floating-point number. | |
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58 | |
59 | @example | |
60 | 1 ; @r{The integer 1.} | |
61 | 1. ; @r{The integer 1.} | |
62 | +1 ; @r{Also the integer 1.} | |
63 | -1 ; @r{The integer @minus{}1.} | |
fed14fd7 | 64 | 1073741825 ; @r{The floating point number 1073741825.0.} |
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65 | 0 ; @r{The integer 0.} |
66 | -0 ; @r{The integer 0.} | |
67 | @end example | |
68 | ||
69 | @cindex integers in specific radix | |
70 | @cindex radix for reading an integer | |
71 | @cindex base for reading an integer | |
72 | @cindex hex numbers | |
73 | @cindex octal numbers | |
74 | @cindex reading numbers in hex, octal, and binary | |
75 | The syntax for integers in bases other than 10 uses @samp{#} | |
76 | followed by a letter that specifies the radix: @samp{b} for binary, | |
77 | @samp{o} for octal, @samp{x} for hex, or @samp{@var{radix}r} to | |
78 | specify radix @var{radix}. Case is not significant for the letter | |
79 | that specifies the radix. Thus, @samp{#b@var{integer}} reads | |
80 | @var{integer} in binary, and @samp{#@var{radix}r@var{integer}} reads | |
81 | @var{integer} in radix @var{radix}. Allowed values of @var{radix} run | |
82 | from 2 to 36. For example: | |
83 | ||
84 | @example | |
85 | #b101100 @result{} 44 | |
86 | #o54 @result{} 44 | |
87 | #x2c @result{} 44 | |
88 | #24r1k @result{} 44 | |
89 | @end example | |
90 | ||
91 | To understand how various functions work on integers, especially the | |
92 | bitwise operators (@pxref{Bitwise Operations}), it is often helpful to | |
93 | view the numbers in their binary form. | |
94 | ||
1ddd6622 | 95 | In 30-bit binary, the decimal integer 5 looks like this: |
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96 | |
97 | @example | |
001903b5 | 98 | 0000...000101 (30 bits total) |
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99 | @end example |
100 | ||
101 | @noindent | |
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102 | (The @samp{...} stands for enough bits to fill out a 30-bit word; in |
103 | this case, @samp{...} stands for twenty 0 bits. Later examples also | |
104 | use the @samp{...} notation to make binary integers easier to read.) | |
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105 | |
106 | The integer @minus{}1 looks like this: | |
107 | ||
108 | @example | |
001903b5 | 109 | 1111...111111 (30 bits total) |
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110 | @end example |
111 | ||
112 | @noindent | |
113 | @cindex two's complement | |
1ddd6622 | 114 | @minus{}1 is represented as 30 ones. (This is called @dfn{two's |
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115 | complement} notation.) |
116 | ||
117 | The negative integer, @minus{}5, is creating by subtracting 4 from | |
118 | @minus{}1. In binary, the decimal integer 4 is 100. Consequently, | |
119 | @minus{}5 looks like this: | |
120 | ||
121 | @example | |
001903b5 | 122 | 1111...111011 (30 bits total) |
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123 | @end example |
124 | ||
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125 | In this implementation, the largest 30-bit binary integer value is |
126 | 536,870,911 in decimal. In binary, it looks like this: | |
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127 | |
128 | @example | |
001903b5 | 129 | 0111...111111 (30 bits total) |
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130 | @end example |
131 | ||
132 | Since the arithmetic functions do not check whether integers go | |
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133 | outside their range, when you add 1 to 536,870,911, the value is the |
134 | negative integer @minus{}536,870,912: | |
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135 | |
136 | @example | |
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137 | (+ 1 536870911) |
138 | @result{} -536870912 | |
001903b5 | 139 | @result{} 1000...000000 (30 bits total) |
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140 | @end example |
141 | ||
142 | Many of the functions described in this chapter accept markers for | |
143 | arguments in place of numbers. (@xref{Markers}.) Since the actual | |
144 | arguments to such functions may be either numbers or markers, we often | |
145 | give these arguments the name @var{number-or-marker}. When the argument | |
146 | value is a marker, its position value is used and its buffer is ignored. | |
147 | ||
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148 | @cindex largest Lisp integer number |
149 | @cindex maximum Lisp integer number | |
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150 | @defvar most-positive-fixnum |
151 | The value of this variable is the largest integer that Emacs Lisp | |
152 | can handle. | |
153 | @end defvar | |
154 | ||
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155 | @cindex smallest Lisp integer number |
156 | @cindex minimum Lisp integer number | |
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157 | @defvar most-negative-fixnum |
158 | The value of this variable is the smallest integer that Emacs Lisp can | |
159 | handle. It is negative. | |
160 | @end defvar | |
161 | ||
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162 | In Emacs Lisp, text characters are represented by integers. Any |
163 | integer between zero and the value of @code{max-char}, inclusive, is | |
164 | considered to be valid as a character. @xref{String Basics}. | |
57e2db6d | 165 | |
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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 | 173 | @code{double} on the machine you are using. Emacs uses the |
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174 | @acronym{IEEE} floating point standard, which is supported by all |
175 | 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 | |
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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 | |
1df7defd | 196 | correct answer. For example, @code{(/ 0.0 0.0)} returns a NaN@. (NaN |
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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 | ||
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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 | |
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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 | |
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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'') |
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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 | |
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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 | ||
48de8b12 | 319 | In Emacs Lisp, each integer value is a unique Lisp object. |
b8d4c8d0 | 320 | Therefore, @code{eq} is equivalent to @code{=} where integers are |
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321 | concerned. It is sometimes convenient to use @code{eq} for comparing |
322 | an unknown value with an integer, because @code{eq} does not report an | |
323 | error if the unknown value is not a number---it accepts arguments of | |
324 | any type. By contrast, @code{=} signals an error if the arguments are | |
325 | not numbers or markers. However, it is better programming practice to | |
326 | use @code{=} if you can, even for comparing integers. | |
327 | ||
328 | Sometimes it is useful to compare numbers with @code{equal}, which | |
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329 | treats two numbers as equal if they have the same data type (both |
330 | integers, or both floating point) and the same value. By contrast, | |
331 | @code{=} can treat an integer and a floating point number as equal. | |
332 | @xref{Equality Predicates}. | |
333 | ||
334 | There is another wrinkle: because floating point arithmetic is not | |
335 | exact, it is often a bad idea to check for equality of two floating | |
336 | point values. Usually it is better to test for approximate equality. | |
337 | Here's a function to do this: | |
338 | ||
339 | @example | |
340 | (defvar fuzz-factor 1.0e-6) | |
341 | (defun approx-equal (x y) | |
342 | (or (and (= x 0) (= y 0)) | |
343 | (< (/ (abs (- x y)) | |
344 | (max (abs x) (abs y))) | |
345 | fuzz-factor))) | |
346 | @end example | |
347 | ||
348 | @cindex CL note---integers vrs @code{eq} | |
349 | @quotation | |
350 | @b{Common Lisp note:} Comparing numbers in Common Lisp always requires | |
351 | @code{=} because Common Lisp implements multi-word integers, and two | |
352 | distinct integer objects can have the same numeric value. Emacs Lisp | |
353 | can have just one integer object for any given value because it has a | |
354 | limited range of integer values. | |
355 | @end quotation | |
356 | ||
357 | @defun = number-or-marker1 number-or-marker2 | |
358 | This function tests whether its arguments are numerically equal, and | |
359 | returns @code{t} if so, @code{nil} otherwise. | |
360 | @end defun | |
361 | ||
362 | @defun eql value1 value2 | |
363 | This function acts like @code{eq} except when both arguments are | |
364 | numbers. It compares numbers by type and numeric value, so that | |
365 | @code{(eql 1.0 1)} returns @code{nil}, but @code{(eql 1.0 1.0)} and | |
366 | @code{(eql 1 1)} both return @code{t}. | |
367 | @end defun | |
368 | ||
369 | @defun /= number-or-marker1 number-or-marker2 | |
370 | This function tests whether its arguments are numerically equal, and | |
371 | returns @code{t} if they are not, and @code{nil} if they are. | |
372 | @end defun | |
373 | ||
374 | @defun < number-or-marker1 number-or-marker2 | |
375 | This function tests whether its first argument is strictly less than | |
376 | its second argument. It returns @code{t} if so, @code{nil} otherwise. | |
377 | @end defun | |
378 | ||
379 | @defun <= number-or-marker1 number-or-marker2 | |
380 | This function tests whether its first argument is less than or equal | |
381 | to 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 strictly greater | |
387 | than its second argument. It returns @code{t} if so, @code{nil} | |
388 | otherwise. | |
389 | @end defun | |
390 | ||
391 | @defun >= number-or-marker1 number-or-marker2 | |
392 | This function tests whether its first argument is greater than or | |
393 | equal to its second argument. It returns @code{t} if so, @code{nil} | |
394 | otherwise. | |
395 | @end defun | |
396 | ||
397 | @defun max number-or-marker &rest numbers-or-markers | |
398 | This function returns the largest of its arguments. | |
399 | If any of the arguments is floating-point, the value is returned | |
400 | as floating point, even if it was given as an integer. | |
401 | ||
402 | @example | |
403 | (max 20) | |
404 | @result{} 20 | |
405 | (max 1 2.5) | |
406 | @result{} 2.5 | |
407 | (max 1 3 2.5) | |
408 | @result{} 3.0 | |
409 | @end example | |
410 | @end defun | |
411 | ||
412 | @defun min number-or-marker &rest numbers-or-markers | |
413 | This function returns the smallest of its arguments. | |
414 | If any of the arguments is floating-point, the value is returned | |
415 | as floating point, even if it was given as an integer. | |
416 | ||
417 | @example | |
418 | (min -4 1) | |
419 | @result{} -4 | |
420 | @end example | |
421 | @end defun | |
422 | ||
423 | @defun abs number | |
424 | This function returns the absolute value of @var{number}. | |
425 | @end defun | |
426 | ||
427 | @node Numeric Conversions | |
428 | @section Numeric Conversions | |
429 | @cindex rounding in conversions | |
430 | @cindex number conversions | |
431 | @cindex converting numbers | |
432 | ||
433 | To convert an integer to floating point, use the function @code{float}. | |
434 | ||
435 | @defun float number | |
436 | This returns @var{number} converted to floating point. | |
437 | If @var{number} is already a floating point number, @code{float} returns | |
438 | it unchanged. | |
439 | @end defun | |
440 | ||
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441 | There are four functions to convert floating point numbers to |
442 | integers; they differ in how they round. All accept an argument | |
443 | @var{number} and an optional argument @var{divisor}. Both arguments | |
444 | may be integers or floating point numbers. @var{divisor} may also be | |
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445 | @code{nil}. If @var{divisor} is @code{nil} or omitted, these |
446 | functions convert @var{number} to an integer, or return it unchanged | |
447 | if it already is an integer. If @var{divisor} is non-@code{nil}, they | |
448 | divide @var{number} by @var{divisor} and convert the result to an | |
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449 | integer. integer. If @var{divisor} is zero (whether integer or |
450 | floating-point), Emacs signals an @code{arith-error} error. | |
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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 | ||
48de8b12 CY |
527 | Emacs Lisp provides the traditional four arithmetic operations |
528 | (addition, subtraction, multiplication, and division), as well as | |
529 | remainder and modulus functions, and functions to add or subtract 1. | |
530 | Except for @code{%}, each of these functions accepts both integer and | |
531 | floating point arguments, and returns a floating point number if any | |
532 | argument is a floating point number. | |
b8d4c8d0 | 533 | |
c717b326 | 534 | It is important to note that in Emacs Lisp, arithmetic functions |
001903b5 PE |
535 | do not check for overflow. Thus @code{(1+ 536870911)} may evaluate to |
536 | @minus{}536870912, depending on your hardware. | |
b8d4c8d0 GM |
537 | |
538 | @defun 1+ number-or-marker | |
539 | This function returns @var{number-or-marker} plus 1. | |
540 | For example, | |
541 | ||
542 | @example | |
543 | (setq foo 4) | |
544 | @result{} 4 | |
545 | (1+ foo) | |
546 | @result{} 5 | |
547 | @end example | |
548 | ||
549 | This function is not analogous to the C operator @code{++}---it does not | |
550 | increment a variable. It just computes a sum. Thus, if we continue, | |
551 | ||
552 | @example | |
553 | foo | |
554 | @result{} 4 | |
555 | @end example | |
556 | ||
557 | If you want to increment the variable, you must use @code{setq}, | |
558 | like this: | |
559 | ||
560 | @example | |
561 | (setq foo (1+ foo)) | |
562 | @result{} 5 | |
563 | @end example | |
564 | @end defun | |
565 | ||
566 | @defun 1- number-or-marker | |
567 | This function returns @var{number-or-marker} minus 1. | |
568 | @end defun | |
569 | ||
570 | @defun + &rest numbers-or-markers | |
571 | This function adds its arguments together. When given no arguments, | |
572 | @code{+} returns 0. | |
573 | ||
574 | @example | |
575 | (+) | |
576 | @result{} 0 | |
577 | (+ 1) | |
578 | @result{} 1 | |
579 | (+ 1 2 3 4) | |
580 | @result{} 10 | |
581 | @end example | |
582 | @end defun | |
583 | ||
584 | @defun - &optional number-or-marker &rest more-numbers-or-markers | |
585 | The @code{-} function serves two purposes: negation and subtraction. | |
586 | When @code{-} has a single argument, the value is the negative of the | |
587 | argument. When there are multiple arguments, @code{-} subtracts each of | |
588 | the @var{more-numbers-or-markers} from @var{number-or-marker}, | |
589 | cumulatively. If there are no arguments, the result is 0. | |
590 | ||
591 | @example | |
592 | (- 10 1 2 3 4) | |
593 | @result{} 0 | |
594 | (- 10) | |
595 | @result{} -10 | |
596 | (-) | |
597 | @result{} 0 | |
598 | @end example | |
599 | @end defun | |
600 | ||
601 | @defun * &rest numbers-or-markers | |
602 | This function multiplies its arguments together, and returns the | |
603 | product. When given no arguments, @code{*} returns 1. | |
604 | ||
605 | @example | |
606 | (*) | |
607 | @result{} 1 | |
608 | (* 1) | |
609 | @result{} 1 | |
610 | (* 1 2 3 4) | |
611 | @result{} 24 | |
612 | @end example | |
613 | @end defun | |
614 | ||
615 | @defun / dividend divisor &rest divisors | |
616 | This function divides @var{dividend} by @var{divisor} and returns the | |
617 | quotient. If there are additional arguments @var{divisors}, then it | |
618 | divides @var{dividend} by each divisor in turn. Each argument may be a | |
619 | number or a marker. | |
620 | ||
48de8b12 CY |
621 | If all the arguments are integers, the result is an integer, obtained |
622 | by rounding the quotient towards zero after each division. | |
623 | (Hypothetically, some machines may have different rounding behavior | |
624 | for negative arguments, because @code{/} is implemented using the C | |
625 | division operator, which permits machine-dependent rounding; but this | |
626 | does not happen in practice.) | |
b8d4c8d0 GM |
627 | |
628 | @example | |
629 | @group | |
630 | (/ 6 2) | |
631 | @result{} 3 | |
632 | @end group | |
48de8b12 | 633 | @group |
b8d4c8d0 GM |
634 | (/ 5 2) |
635 | @result{} 2 | |
48de8b12 CY |
636 | @end group |
637 | @group | |
b8d4c8d0 GM |
638 | (/ 5.0 2) |
639 | @result{} 2.5 | |
48de8b12 CY |
640 | @end group |
641 | @group | |
b8d4c8d0 GM |
642 | (/ 5 2.0) |
643 | @result{} 2.5 | |
48de8b12 CY |
644 | @end group |
645 | @group | |
b8d4c8d0 GM |
646 | (/ 5.0 2.0) |
647 | @result{} 2.5 | |
48de8b12 CY |
648 | @end group |
649 | @group | |
b8d4c8d0 GM |
650 | (/ 25 3 2) |
651 | @result{} 4 | |
48de8b12 | 652 | @end group |
b8d4c8d0 GM |
653 | @group |
654 | (/ -17 6) | |
48de8b12 | 655 | @result{} -2 |
b8d4c8d0 GM |
656 | @end group |
657 | @end example | |
48de8b12 CY |
658 | |
659 | @cindex @code{arith-error} in division | |
660 | If you divide an integer by the integer 0, Emacs signals an | |
661 | @code{arith-error} error (@pxref{Errors}). If you divide a floating | |
662 | point number by 0, or divide by the floating point number 0.0, the | |
663 | result is either positive or negative infinity (@pxref{Float Basics}). | |
b8d4c8d0 GM |
664 | @end defun |
665 | ||
666 | @defun % dividend divisor | |
667 | @cindex remainder | |
668 | This function returns the integer remainder after division of @var{dividend} | |
669 | by @var{divisor}. The arguments must be integers or markers. | |
670 | ||
48de8b12 CY |
671 | For any two integers @var{dividend} and @var{divisor}, |
672 | ||
673 | @example | |
674 | @group | |
675 | (+ (% @var{dividend} @var{divisor}) | |
676 | (* (/ @var{dividend} @var{divisor}) @var{divisor})) | |
677 | @end group | |
678 | @end example | |
b8d4c8d0 | 679 | |
48de8b12 CY |
680 | @noindent |
681 | always equals @var{dividend}. If @var{divisor} is zero, Emacs signals | |
682 | an @code{arith-error} error. | |
b8d4c8d0 GM |
683 | |
684 | @example | |
685 | (% 9 4) | |
686 | @result{} 1 | |
687 | (% -9 4) | |
688 | @result{} -1 | |
689 | (% 9 -4) | |
690 | @result{} 1 | |
691 | (% -9 -4) | |
692 | @result{} -1 | |
693 | @end example | |
b8d4c8d0 GM |
694 | @end defun |
695 | ||
696 | @defun mod dividend divisor | |
697 | @cindex modulus | |
698 | This function returns the value of @var{dividend} modulo @var{divisor}; | |
699 | in other words, the remainder after division of @var{dividend} | |
700 | by @var{divisor}, but with the same sign as @var{divisor}. | |
701 | The arguments must be numbers or markers. | |
702 | ||
48de8b12 CY |
703 | Unlike @code{%}, @code{mod} permits floating point arguments; it |
704 | rounds the quotient downward (towards minus infinity) to an integer, | |
705 | and uses that quotient to compute the remainder. | |
b8d4c8d0 | 706 | |
c990426a PE |
707 | If @var{divisor} is zero, @code{mod} signals an @code{arith-error} |
708 | error if both arguments are integers, and returns a NaN otherwise. | |
b8d4c8d0 GM |
709 | |
710 | @example | |
711 | @group | |
712 | (mod 9 4) | |
713 | @result{} 1 | |
714 | @end group | |
715 | @group | |
716 | (mod -9 4) | |
717 | @result{} 3 | |
718 | @end group | |
719 | @group | |
720 | (mod 9 -4) | |
721 | @result{} -3 | |
722 | @end group | |
723 | @group | |
724 | (mod -9 -4) | |
725 | @result{} -1 | |
726 | @end group | |
727 | @group | |
728 | (mod 5.5 2.5) | |
729 | @result{} .5 | |
730 | @end group | |
731 | @end example | |
732 | ||
733 | For any two numbers @var{dividend} and @var{divisor}, | |
734 | ||
735 | @example | |
736 | @group | |
737 | (+ (mod @var{dividend} @var{divisor}) | |
738 | (* (floor @var{dividend} @var{divisor}) @var{divisor})) | |
739 | @end group | |
740 | @end example | |
741 | ||
742 | @noindent | |
743 | always equals @var{dividend}, subject to rounding error if either | |
744 | argument is floating point. For @code{floor}, see @ref{Numeric | |
745 | Conversions}. | |
746 | @end defun | |
747 | ||
748 | @node Rounding Operations | |
749 | @section Rounding Operations | |
750 | @cindex rounding without conversion | |
751 | ||
752 | The functions @code{ffloor}, @code{fceiling}, @code{fround}, and | |
753 | @code{ftruncate} take a floating point argument and return a floating | |
754 | point result whose value is a nearby integer. @code{ffloor} returns the | |
755 | nearest integer below; @code{fceiling}, the nearest integer above; | |
756 | @code{ftruncate}, the nearest integer in the direction towards zero; | |
757 | @code{fround}, the nearest integer. | |
758 | ||
759 | @defun ffloor float | |
760 | This function rounds @var{float} to the next lower integral value, and | |
761 | returns that value as a floating point number. | |
762 | @end defun | |
763 | ||
764 | @defun fceiling float | |
765 | This function rounds @var{float} to the next higher integral value, and | |
766 | returns that value as a floating point number. | |
767 | @end defun | |
768 | ||
769 | @defun ftruncate float | |
770 | This function rounds @var{float} towards zero to an integral value, and | |
771 | returns that value as a floating point number. | |
772 | @end defun | |
773 | ||
774 | @defun fround float | |
775 | This function rounds @var{float} to the nearest integral value, | |
776 | and returns that value as a floating point number. | |
777 | @end defun | |
778 | ||
779 | @node Bitwise Operations | |
780 | @section Bitwise Operations on Integers | |
781 | @cindex bitwise arithmetic | |
782 | @cindex logical arithmetic | |
783 | ||
784 | In a computer, an integer is represented as a binary number, a | |
785 | sequence of @dfn{bits} (digits which are either zero or one). A bitwise | |
786 | operation acts on the individual bits of such a sequence. For example, | |
787 | @dfn{shifting} moves the whole sequence left or right one or more places, | |
16152b76 | 788 | reproducing the same pattern ``moved over''. |
b8d4c8d0 GM |
789 | |
790 | The bitwise operations in Emacs Lisp apply only to integers. | |
791 | ||
792 | @defun lsh integer1 count | |
793 | @cindex logical shift | |
794 | @code{lsh}, which is an abbreviation for @dfn{logical shift}, shifts the | |
795 | bits in @var{integer1} to the left @var{count} places, or to the right | |
796 | if @var{count} is negative, bringing zeros into the vacated bits. If | |
797 | @var{count} is negative, @code{lsh} shifts zeros into the leftmost | |
798 | (most-significant) bit, producing a positive result even if | |
799 | @var{integer1} is negative. Contrast this with @code{ash}, below. | |
800 | ||
801 | Here are two examples of @code{lsh}, shifting a pattern of bits one | |
802 | place to the left. We show only the low-order eight bits of the binary | |
803 | pattern; the rest are all zero. | |
804 | ||
805 | @example | |
806 | @group | |
807 | (lsh 5 1) | |
808 | @result{} 10 | |
809 | ;; @r{Decimal 5 becomes decimal 10.} | |
810 | 00000101 @result{} 00001010 | |
811 | ||
812 | (lsh 7 1) | |
813 | @result{} 14 | |
814 | ;; @r{Decimal 7 becomes decimal 14.} | |
815 | 00000111 @result{} 00001110 | |
816 | @end group | |
817 | @end example | |
818 | ||
819 | @noindent | |
820 | As the examples illustrate, shifting the pattern of bits one place to | |
821 | the left produces a number that is twice the value of the previous | |
822 | number. | |
823 | ||
824 | Shifting a pattern of bits two places to the left produces results | |
825 | like this (with 8-bit binary numbers): | |
826 | ||
827 | @example | |
828 | @group | |
829 | (lsh 3 2) | |
830 | @result{} 12 | |
831 | ;; @r{Decimal 3 becomes decimal 12.} | |
832 | 00000011 @result{} 00001100 | |
833 | @end group | |
834 | @end example | |
835 | ||
836 | On the other hand, shifting one place to the right looks like this: | |
837 | ||
838 | @example | |
839 | @group | |
840 | (lsh 6 -1) | |
841 | @result{} 3 | |
842 | ;; @r{Decimal 6 becomes decimal 3.} | |
843 | 00000110 @result{} 00000011 | |
844 | @end group | |
845 | ||
846 | @group | |
847 | (lsh 5 -1) | |
848 | @result{} 2 | |
849 | ;; @r{Decimal 5 becomes decimal 2.} | |
850 | 00000101 @result{} 00000010 | |
851 | @end group | |
852 | @end example | |
853 | ||
854 | @noindent | |
855 | As the example illustrates, shifting one place to the right divides the | |
856 | value of a positive integer by two, rounding downward. | |
857 | ||
c717b326 | 858 | The function @code{lsh}, like all Emacs Lisp arithmetic functions, does |
b8d4c8d0 GM |
859 | not check for overflow, so shifting left can discard significant bits |
860 | and change the sign of the number. For example, left shifting | |
001903b5 | 861 | 536,870,911 produces @minus{}2 in the 30-bit implementation: |
b8d4c8d0 GM |
862 | |
863 | @example | |
1ddd6622 | 864 | (lsh 536870911 1) ; @r{left shift} |
b8d4c8d0 GM |
865 | @result{} -2 |
866 | @end example | |
867 | ||
001903b5 | 868 | In binary, the argument looks like this: |
b8d4c8d0 GM |
869 | |
870 | @example | |
871 | @group | |
1ddd6622 | 872 | ;; @r{Decimal 536,870,911} |
001903b5 | 873 | 0111...111111 (30 bits total) |
b8d4c8d0 GM |
874 | @end group |
875 | @end example | |
876 | ||
877 | @noindent | |
878 | which becomes the following when left shifted: | |
879 | ||
880 | @example | |
881 | @group | |
882 | ;; @r{Decimal @minus{}2} | |
001903b5 | 883 | 1111...111110 (30 bits total) |
b8d4c8d0 GM |
884 | @end group |
885 | @end example | |
886 | @end defun | |
887 | ||
888 | @defun ash integer1 count | |
889 | @cindex arithmetic shift | |
890 | @code{ash} (@dfn{arithmetic shift}) shifts the bits in @var{integer1} | |
891 | to the left @var{count} places, or to the right if @var{count} | |
892 | is negative. | |
893 | ||
894 | @code{ash} gives the same results as @code{lsh} except when | |
895 | @var{integer1} and @var{count} are both negative. In that case, | |
896 | @code{ash} puts ones in the empty bit positions on the left, while | |
897 | @code{lsh} puts zeros in those bit positions. | |
898 | ||
899 | Thus, with @code{ash}, shifting the pattern of bits one place to the right | |
900 | looks like this: | |
901 | ||
902 | @example | |
903 | @group | |
904 | (ash -6 -1) @result{} -3 | |
905 | ;; @r{Decimal @minus{}6 becomes decimal @minus{}3.} | |
001903b5 | 906 | 1111...111010 (30 bits total) |
b8d4c8d0 | 907 | @result{} |
001903b5 | 908 | 1111...111101 (30 bits total) |
b8d4c8d0 GM |
909 | @end group |
910 | @end example | |
911 | ||
912 | In contrast, shifting the pattern of bits one place to the right with | |
913 | @code{lsh} looks like this: | |
914 | ||
915 | @example | |
916 | @group | |
1ddd6622 GM |
917 | (lsh -6 -1) @result{} 536870909 |
918 | ;; @r{Decimal @minus{}6 becomes decimal 536,870,909.} | |
001903b5 | 919 | 1111...111010 (30 bits total) |
b8d4c8d0 | 920 | @result{} |
001903b5 | 921 | 0111...111101 (30 bits total) |
b8d4c8d0 GM |
922 | @end group |
923 | @end example | |
924 | ||
925 | Here are other examples: | |
926 | ||
927 | @c !!! Check if lined up in smallbook format! XDVI shows problem | |
928 | @c with smallbook but not with regular book! --rjc 16mar92 | |
929 | @smallexample | |
930 | @group | |
001903b5 | 931 | ; @r{ 30-bit binary values} |
b8d4c8d0 | 932 | |
be14b9ab PE |
933 | (lsh 5 2) ; 5 = @r{0000...000101} |
934 | @result{} 20 ; = @r{0000...010100} | |
b8d4c8d0 GM |
935 | @end group |
936 | @group | |
937 | (ash 5 2) | |
938 | @result{} 20 | |
be14b9ab PE |
939 | (lsh -5 2) ; -5 = @r{1111...111011} |
940 | @result{} -20 ; = @r{1111...101100} | |
b8d4c8d0 GM |
941 | (ash -5 2) |
942 | @result{} -20 | |
943 | @end group | |
944 | @group | |
be14b9ab PE |
945 | (lsh 5 -2) ; 5 = @r{0000...000101} |
946 | @result{} 1 ; = @r{0000...000001} | |
b8d4c8d0 GM |
947 | @end group |
948 | @group | |
949 | (ash 5 -2) | |
950 | @result{} 1 | |
951 | @end group | |
952 | @group | |
be14b9ab | 953 | (lsh -5 -2) ; -5 = @r{1111...111011} |
001903b5 | 954 | @result{} 268435454 |
be14b9ab | 955 | ; = @r{0011...111110} |
b8d4c8d0 GM |
956 | @end group |
957 | @group | |
be14b9ab PE |
958 | (ash -5 -2) ; -5 = @r{1111...111011} |
959 | @result{} -2 ; = @r{1111...111110} | |
b8d4c8d0 GM |
960 | @end group |
961 | @end smallexample | |
962 | @end defun | |
963 | ||
964 | @defun logand &rest ints-or-markers | |
965 | This function returns the ``logical and'' of the arguments: the | |
966 | @var{n}th bit is set in the result if, and only if, the @var{n}th bit is | |
967 | set in all the arguments. (``Set'' means that the value of the bit is 1 | |
968 | rather than 0.) | |
969 | ||
970 | For example, using 4-bit binary numbers, the ``logical and'' of 13 and | |
971 | 12 is 12: 1101 combined with 1100 produces 1100. | |
972 | In both the binary numbers, the leftmost two bits are set (i.e., they | |
973 | are 1's), so the leftmost two bits of the returned value are set. | |
974 | However, for the rightmost two bits, each is zero in at least one of | |
975 | the arguments, so the rightmost two bits of the returned value are 0's. | |
976 | ||
977 | @noindent | |
978 | Therefore, | |
979 | ||
980 | @example | |
981 | @group | |
982 | (logand 13 12) | |
983 | @result{} 12 | |
984 | @end group | |
985 | @end example | |
986 | ||
987 | If @code{logand} is not passed any argument, it returns a value of | |
988 | @minus{}1. This number is an identity element for @code{logand} | |
989 | because its binary representation consists entirely of ones. If | |
990 | @code{logand} is passed just one argument, it returns that argument. | |
991 | ||
992 | @smallexample | |
993 | @group | |
001903b5 | 994 | ; @r{ 30-bit binary values} |
b8d4c8d0 | 995 | |
be14b9ab PE |
996 | (logand 14 13) ; 14 = @r{0000...001110} |
997 | ; 13 = @r{0000...001101} | |
998 | @result{} 12 ; 12 = @r{0000...001100} | |
b8d4c8d0 GM |
999 | @end group |
1000 | ||
1001 | @group | |
be14b9ab PE |
1002 | (logand 14 13 4) ; 14 = @r{0000...001110} |
1003 | ; 13 = @r{0000...001101} | |
1004 | ; 4 = @r{0000...000100} | |
1005 | @result{} 4 ; 4 = @r{0000...000100} | |
b8d4c8d0 GM |
1006 | @end group |
1007 | ||
1008 | @group | |
1009 | (logand) | |
be14b9ab | 1010 | @result{} -1 ; -1 = @r{1111...111111} |
b8d4c8d0 GM |
1011 | @end group |
1012 | @end smallexample | |
1013 | @end defun | |
1014 | ||
1015 | @defun logior &rest ints-or-markers | |
1016 | This function returns the ``inclusive or'' of its arguments: the @var{n}th bit | |
1017 | is set in the result if, and only if, the @var{n}th bit is set in at least | |
1018 | one of the arguments. If there are no arguments, the result is zero, | |
1019 | which is an identity element for this operation. If @code{logior} is | |
1020 | passed just one argument, it returns that argument. | |
1021 | ||
1022 | @smallexample | |
1023 | @group | |
001903b5 | 1024 | ; @r{ 30-bit binary values} |
b8d4c8d0 | 1025 | |
be14b9ab PE |
1026 | (logior 12 5) ; 12 = @r{0000...001100} |
1027 | ; 5 = @r{0000...000101} | |
1028 | @result{} 13 ; 13 = @r{0000...001101} | |
b8d4c8d0 GM |
1029 | @end group |
1030 | ||
1031 | @group | |
be14b9ab PE |
1032 | (logior 12 5 7) ; 12 = @r{0000...001100} |
1033 | ; 5 = @r{0000...000101} | |
1034 | ; 7 = @r{0000...000111} | |
1035 | @result{} 15 ; 15 = @r{0000...001111} | |
b8d4c8d0 GM |
1036 | @end group |
1037 | @end smallexample | |
1038 | @end defun | |
1039 | ||
1040 | @defun logxor &rest ints-or-markers | |
1041 | This function returns the ``exclusive or'' of its arguments: the | |
1042 | @var{n}th bit is set in the result if, and only if, the @var{n}th bit is | |
1043 | set in an odd number of the arguments. If there are no arguments, the | |
1044 | result is 0, which is an identity element for this operation. If | |
1045 | @code{logxor} is passed just one argument, it returns that argument. | |
1046 | ||
1047 | @smallexample | |
1048 | @group | |
001903b5 | 1049 | ; @r{ 30-bit binary values} |
b8d4c8d0 | 1050 | |
be14b9ab PE |
1051 | (logxor 12 5) ; 12 = @r{0000...001100} |
1052 | ; 5 = @r{0000...000101} | |
1053 | @result{} 9 ; 9 = @r{0000...001001} | |
b8d4c8d0 GM |
1054 | @end group |
1055 | ||
1056 | @group | |
be14b9ab PE |
1057 | (logxor 12 5 7) ; 12 = @r{0000...001100} |
1058 | ; 5 = @r{0000...000101} | |
1059 | ; 7 = @r{0000...000111} | |
1060 | @result{} 14 ; 14 = @r{0000...001110} | |
b8d4c8d0 GM |
1061 | @end group |
1062 | @end smallexample | |
1063 | @end defun | |
1064 | ||
1065 | @defun lognot integer | |
1066 | This function returns the logical complement of its argument: the @var{n}th | |
1067 | bit is one in the result if, and only if, the @var{n}th bit is zero in | |
1068 | @var{integer}, and vice-versa. | |
1069 | ||
1070 | @example | |
1071 | (lognot 5) | |
1072 | @result{} -6 | |
001903b5 | 1073 | ;; 5 = @r{0000...000101} (30 bits total) |
b8d4c8d0 | 1074 | ;; @r{becomes} |
001903b5 | 1075 | ;; -6 = @r{1111...111010} (30 bits total) |
b8d4c8d0 GM |
1076 | @end example |
1077 | @end defun | |
1078 | ||
1079 | @node Math Functions | |
1080 | @section Standard Mathematical Functions | |
1081 | @cindex transcendental functions | |
1082 | @cindex mathematical functions | |
1083 | @cindex floating-point functions | |
1084 | ||
1085 | These mathematical functions allow integers as well as floating point | |
1086 | numbers as arguments. | |
1087 | ||
1088 | @defun sin arg | |
1089 | @defunx cos arg | |
1090 | @defunx tan arg | |
48de8b12 CY |
1091 | These are the basic trigonometric functions, with argument @var{arg} |
1092 | measured in radians. | |
b8d4c8d0 GM |
1093 | @end defun |
1094 | ||
1095 | @defun asin arg | |
1096 | The value of @code{(asin @var{arg})} is a number between | |
1097 | @ifnottex | |
1098 | @minus{}pi/2 | |
1099 | @end ifnottex | |
1100 | @tex | |
1101 | @math{-\pi/2} | |
1102 | @end tex | |
1103 | and | |
1104 | @ifnottex | |
1105 | pi/2 | |
1106 | @end ifnottex | |
1107 | @tex | |
1108 | @math{\pi/2} | |
1109 | @end tex | |
c990426a PE |
1110 | (inclusive) whose sine is @var{arg}. If @var{arg} is out of range |
1111 | (outside [@minus{}1, 1]), @code{asin} returns a NaN. | |
b8d4c8d0 GM |
1112 | @end defun |
1113 | ||
1114 | @defun acos arg | |
1115 | The value of @code{(acos @var{arg})} is a number between 0 and | |
1116 | @ifnottex | |
1117 | pi | |
1118 | @end ifnottex | |
1119 | @tex | |
1120 | @math{\pi} | |
1121 | @end tex | |
c990426a PE |
1122 | (inclusive) whose cosine is @var{arg}. If @var{arg} is out of range |
1123 | (outside [@minus{}1, 1]), @code{acos} returns a NaN. | |
b8d4c8d0 GM |
1124 | @end defun |
1125 | ||
1126 | @defun atan y &optional x | |
1127 | The value of @code{(atan @var{y})} is a number between | |
1128 | @ifnottex | |
1129 | @minus{}pi/2 | |
1130 | @end ifnottex | |
1131 | @tex | |
1132 | @math{-\pi/2} | |
1133 | @end tex | |
1134 | and | |
1135 | @ifnottex | |
1136 | pi/2 | |
1137 | @end ifnottex | |
1138 | @tex | |
1139 | @math{\pi/2} | |
1140 | @end tex | |
1141 | (exclusive) whose tangent is @var{y}. If the optional second | |
1142 | argument @var{x} is given, the value of @code{(atan y x)} is the | |
1143 | angle in radians between the vector @code{[@var{x}, @var{y}]} and the | |
1144 | @code{X} axis. | |
1145 | @end defun | |
1146 | ||
1147 | @defun exp arg | |
fead402d CY |
1148 | This is the exponential function; it returns @math{e} to the power |
1149 | @var{arg}. | |
b8d4c8d0 GM |
1150 | @end defun |
1151 | ||
1152 | @defun log arg &optional base | |
fead402d CY |
1153 | This function returns the logarithm of @var{arg}, with base |
1154 | @var{base}. If you don't specify @var{base}, the natural base | |
c990426a PE |
1155 | @math{e} is used. If @var{arg} or @var{base} is negative, @code{log} |
1156 | returns a NaN. | |
b8d4c8d0 GM |
1157 | @end defun |
1158 | ||
b8d4c8d0 | 1159 | @defun log10 arg |
c990426a PE |
1160 | This function returns the logarithm of @var{arg}, with base 10: |
1161 | @code{(log10 @var{x})} @equiv{} @code{(log @var{x} 10)}. | |
b8d4c8d0 GM |
1162 | @end defun |
1163 | ||
1164 | @defun expt x y | |
1165 | This function returns @var{x} raised to power @var{y}. If both | |
c717b326 PE |
1166 | arguments are integers and @var{y} is positive, the result is an |
1167 | integer; in this case, overflow causes truncation, so watch out. | |
c990426a PE |
1168 | If @var{x} is a finite negative number and @var{y} is a finite |
1169 | non-integer, @code{expt} returns a NaN. | |
b8d4c8d0 GM |
1170 | @end defun |
1171 | ||
1172 | @defun sqrt arg | |
1173 | This returns the square root of @var{arg}. If @var{arg} is negative, | |
c990426a | 1174 | @code{sqrt} returns a NaN. |
b8d4c8d0 GM |
1175 | @end defun |
1176 | ||
fead402d CY |
1177 | In addition, Emacs defines the following common mathematical |
1178 | constants: | |
1179 | ||
1180 | @defvar float-e | |
1181 | The mathematical constant @math{e} (2.71828@dots{}). | |
1182 | @end defvar | |
1183 | ||
1184 | @defvar float-pi | |
1185 | The mathematical constant @math{pi} (3.14159@dots{}). | |
1186 | @end defvar | |
1187 | ||
b8d4c8d0 GM |
1188 | @node Random Numbers |
1189 | @section Random Numbers | |
1190 | @cindex random numbers | |
1191 | ||
48de8b12 CY |
1192 | A deterministic computer program cannot generate true random |
1193 | numbers. For most purposes, @dfn{pseudo-random numbers} suffice. A | |
1194 | series of pseudo-random numbers is generated in a deterministic | |
1195 | fashion. The numbers are not truly random, but they have certain | |
1196 | properties that mimic a random series. For example, all possible | |
1197 | values occur equally often in a pseudo-random series. | |
b8d4c8d0 | 1198 | |
48de8b12 CY |
1199 | Pseudo-random numbers are generated from a ``seed''. Starting from |
1200 | any given seed, the @code{random} function always generates the same | |
1201 | sequence of numbers. By default, Emacs initializes the random seed at | |
1202 | startup, in such a way that the sequence of values of @code{random} | |
1203 | (with overwhelming likelihood) differs in each Emacs run. | |
0e23ef9d | 1204 | |
48de8b12 | 1205 | Sometimes you want the random number sequence to be repeatable. For |
0e23ef9d PE |
1206 | example, when debugging a program whose behavior depends on the random |
1207 | number sequence, it is helpful to get the same behavior in each | |
1208 | program run. To make the sequence repeat, execute @code{(random "")}. | |
1209 | This sets the seed to a constant value for your particular Emacs | |
1210 | executable (though it may differ for other Emacs builds). You can use | |
1211 | other strings to choose various seed values. | |
b8d4c8d0 GM |
1212 | |
1213 | @defun random &optional limit | |
1214 | This function returns a pseudo-random integer. Repeated calls return a | |
1215 | series of pseudo-random integers. | |
1216 | ||
1217 | If @var{limit} is a positive integer, the value is chosen to be | |
48de8b12 | 1218 | nonnegative and less than @var{limit}. Otherwise, the value might be |
1df7defd | 1219 | any integer representable in Lisp, i.e., an integer between |
48de8b12 CY |
1220 | @code{most-negative-fixnum} and @code{most-positive-fixnum} |
1221 | (@pxref{Integer Basics}). | |
b8d4c8d0 GM |
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 | 1225 | |
0e23ef9d PE |
1226 | If @var{limit} is a string, it means to choose a new seed based on the |
1227 | string's contents. | |
1228 | ||
b8d4c8d0 | 1229 | @end defun |