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1 | @c -*-texinfo-*- |
2 | @c This is part of the GNU Emacs Lisp Reference Manual. | |
73b0cd50 | 3 | @c Copyright (C) 1990-1995, 1998-1999, 2001-2011 |
1ddd6622 | 4 | @c Free Software Foundation, Inc. |
b8d4c8d0 | 5 | @c See the file elisp.texi for copying conditions. |
6336d8c3 | 6 | @setfilename ../../info/numbers |
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7 | @node Numbers, Strings and Characters, Lisp Data Types, Top |
8 | @chapter Numbers | |
9 | @cindex integers | |
10 | @cindex numbers | |
11 | ||
12 | GNU Emacs supports two numeric data types: @dfn{integers} and | |
13 | @dfn{floating point numbers}. Integers are whole numbers such as | |
14 | @minus{}3, 0, 7, 13, and 511. Their values are exact. Floating point | |
15 | numbers are numbers with fractional parts, such as @minus{}4.5, 0.0, or | |
16 | 2.71828. They can also be expressed in exponential notation: 1.5e2 | |
17 | equals 150; in this example, @samp{e2} stands for ten to the second | |
18 | power, and that is multiplied by 1.5. Floating point values are not | |
19 | exact; they have a fixed, limited amount of precision. | |
20 | ||
21 | @menu | |
22 | * Integer Basics:: Representation and range of integers. | |
d24880de | 23 | * Float Basics:: Representation and range of floating point. |
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24 | * Predicates on Numbers:: Testing for numbers. |
25 | * Comparison of Numbers:: Equality and inequality predicates. | |
d24880de | 26 | * Numeric Conversions:: Converting float to integer and vice versa. |
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27 | * Arithmetic Operations:: How to add, subtract, multiply and divide. |
28 | * Rounding Operations:: Explicitly rounding floating point numbers. | |
29 | * Bitwise Operations:: Logical and, or, not, shifting. | |
30 | * Math Functions:: Trig, exponential and logarithmic functions. | |
31 | * Random Numbers:: Obtaining random integers, predictable or not. | |
32 | @end menu | |
33 | ||
34 | @node Integer Basics | |
35 | @comment node-name, next, previous, up | |
36 | @section Integer Basics | |
37 | ||
38 | The range of values for an integer depends on the machine. The | |
1ddd6622 | 39 | minimum range is @minus{}536870912 to 536870911 (30 bits; i.e., |
b8d4c8d0 | 40 | @ifnottex |
1ddd6622 | 41 | -2**29 |
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42 | @end ifnottex |
43 | @tex | |
1ddd6622 | 44 | @math{-2^{29}} |
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45 | @end tex |
46 | to | |
47 | @ifnottex | |
1ddd6622 | 48 | 2**29 - 1), |
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49 | @end ifnottex |
50 | @tex | |
1ddd6622 | 51 | @math{2^{29}-1}), |
b8d4c8d0 | 52 | @end tex |
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53 | but some machines provide a wider range. Many examples in this |
54 | chapter assume that an integer has 30 bits and that floating point | |
be14b9ab | 55 | numbers are IEEE double precision. |
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56 | @cindex overflow |
57 | ||
58 | The Lisp reader reads an integer as a sequence of digits with optional | |
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59 | initial sign and optional final period. An integer that is out of the |
60 | Emacs range is treated as a floating-point number. | |
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61 | |
62 | @example | |
63 | 1 ; @r{The integer 1.} | |
64 | 1. ; @r{The integer 1.} | |
65 | +1 ; @r{Also the integer 1.} | |
66 | -1 ; @r{The integer @minus{}1.} | |
fed14fd7 | 67 | 1073741825 ; @r{The floating point number 1073741825.0.} |
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68 | 0 ; @r{The integer 0.} |
69 | -0 ; @r{The integer 0.} | |
70 | @end example | |
71 | ||
72 | @cindex integers in specific radix | |
73 | @cindex radix for reading an integer | |
74 | @cindex base for reading an integer | |
75 | @cindex hex numbers | |
76 | @cindex octal numbers | |
77 | @cindex reading numbers in hex, octal, and binary | |
78 | The syntax for integers in bases other than 10 uses @samp{#} | |
79 | followed by a letter that specifies the radix: @samp{b} for binary, | |
80 | @samp{o} for octal, @samp{x} for hex, or @samp{@var{radix}r} to | |
81 | specify radix @var{radix}. Case is not significant for the letter | |
82 | that specifies the radix. Thus, @samp{#b@var{integer}} reads | |
83 | @var{integer} in binary, and @samp{#@var{radix}r@var{integer}} reads | |
84 | @var{integer} in radix @var{radix}. Allowed values of @var{radix} run | |
85 | from 2 to 36. For example: | |
86 | ||
87 | @example | |
88 | #b101100 @result{} 44 | |
89 | #o54 @result{} 44 | |
90 | #x2c @result{} 44 | |
91 | #24r1k @result{} 44 | |
92 | @end example | |
93 | ||
94 | To understand how various functions work on integers, especially the | |
95 | bitwise operators (@pxref{Bitwise Operations}), it is often helpful to | |
96 | view the numbers in their binary form. | |
97 | ||
1ddd6622 | 98 | In 30-bit binary, the decimal integer 5 looks like this: |
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99 | |
100 | @example | |
001903b5 | 101 | 0000...000101 (30 bits total) |
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102 | @end example |
103 | ||
104 | @noindent | |
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105 | (The @samp{...} stands for enough bits to fill out a 30-bit word; in |
106 | this case, @samp{...} stands for twenty 0 bits. Later examples also | |
107 | use the @samp{...} notation to make binary integers easier to read.) | |
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108 | |
109 | The integer @minus{}1 looks like this: | |
110 | ||
111 | @example | |
001903b5 | 112 | 1111...111111 (30 bits total) |
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113 | @end example |
114 | ||
115 | @noindent | |
116 | @cindex two's complement | |
1ddd6622 | 117 | @minus{}1 is represented as 30 ones. (This is called @dfn{two's |
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118 | complement} notation.) |
119 | ||
120 | The negative integer, @minus{}5, is creating by subtracting 4 from | |
121 | @minus{}1. In binary, the decimal integer 4 is 100. Consequently, | |
122 | @minus{}5 looks like this: | |
123 | ||
124 | @example | |
001903b5 | 125 | 1111...111011 (30 bits total) |
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126 | @end example |
127 | ||
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128 | In this implementation, the largest 30-bit binary integer value is |
129 | 536,870,911 in decimal. In binary, it looks like this: | |
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130 | |
131 | @example | |
001903b5 | 132 | 0111...111111 (30 bits total) |
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133 | @end example |
134 | ||
135 | Since the arithmetic functions do not check whether integers go | |
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136 | outside their range, when you add 1 to 536,870,911, the value is the |
137 | negative integer @minus{}536,870,912: | |
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138 | |
139 | @example | |
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140 | (+ 1 536870911) |
141 | @result{} -536870912 | |
001903b5 | 142 | @result{} 1000...000000 (30 bits total) |
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143 | @end example |
144 | ||
145 | Many of the functions described in this chapter accept markers for | |
146 | arguments in place of numbers. (@xref{Markers}.) Since the actual | |
147 | arguments to such functions may be either numbers or markers, we often | |
148 | give these arguments the name @var{number-or-marker}. When the argument | |
149 | value is a marker, its position value is used and its buffer is ignored. | |
150 | ||
151 | @defvar most-positive-fixnum | |
152 | The value of this variable is the largest integer that Emacs Lisp | |
153 | can handle. | |
154 | @end defvar | |
155 | ||
156 | @defvar most-negative-fixnum | |
157 | The value of this variable is the smallest integer that Emacs Lisp can | |
158 | handle. It is negative. | |
159 | @end defvar | |
160 | ||
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161 | @xref{Character Codes, max-char}, for the maximum value of a valid |
162 | character codepoint. | |
163 | ||
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164 | @node Float Basics |
165 | @section Floating Point Basics | |
166 | ||
167 | Floating point numbers are useful for representing numbers that are | |
168 | not integral. The precise range of floating point numbers is | |
169 | machine-specific; it is the same as the range of the C data type | |
170 | @code{double} on the machine you are using. | |
171 | ||
172 | The read-syntax for floating point numbers requires either a decimal | |
173 | point (with at least one digit following), an exponent, or both. For | |
174 | example, @samp{1500.0}, @samp{15e2}, @samp{15.0e2}, @samp{1.5e3}, and | |
175 | @samp{.15e4} are five ways of writing a floating point number whose | |
176 | value is 1500. They are all equivalent. You can also use a minus sign | |
177 | to write negative floating point numbers, as in @samp{-1.0}. | |
178 | ||
179 | @cindex @acronym{IEEE} floating point | |
180 | @cindex positive infinity | |
181 | @cindex negative infinity | |
182 | @cindex infinity | |
183 | @cindex NaN | |
184 | Most modern computers support the @acronym{IEEE} floating point standard, | |
185 | which provides for positive infinity and negative infinity as floating point | |
186 | values. It also provides for a class of values called NaN or | |
187 | ``not-a-number''; numerical functions return such values in cases where | |
188 | there is no correct answer. For example, @code{(/ 0.0 0.0)} returns a | |
189 | NaN. For practical purposes, there's no significant difference between | |
190 | different NaN values in Emacs Lisp, and there's no rule for precisely | |
191 | which NaN value should be used in a particular case, so Emacs Lisp | |
192 | doesn't try to distinguish them (but it does report the sign, if you | |
193 | print it). Here are the read syntaxes for these special floating | |
194 | point values: | |
195 | ||
196 | @table @asis | |
197 | @item positive infinity | |
198 | @samp{1.0e+INF} | |
199 | @item negative infinity | |
200 | @samp{-1.0e+INF} | |
fed14fd7 | 201 | @item Not-a-number |
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202 | @samp{0.0e+NaN} or @samp{-0.0e+NaN}. |
203 | @end table | |
204 | ||
205 | To test whether a floating point value is a NaN, compare it with | |
206 | itself using @code{=}. That returns @code{nil} for a NaN, and | |
207 | @code{t} for any other floating point value. | |
208 | ||
209 | The value @code{-0.0} is distinguishable from ordinary zero in | |
210 | @acronym{IEEE} floating point, but Emacs Lisp @code{equal} and | |
211 | @code{=} consider them equal values. | |
212 | ||
213 | You can use @code{logb} to extract the binary exponent of a floating | |
214 | point number (or estimate the logarithm of an integer): | |
215 | ||
216 | @defun logb number | |
217 | This function returns the binary exponent of @var{number}. More | |
218 | precisely, the value is the logarithm of @var{number} base 2, rounded | |
219 | down to an integer. | |
220 | ||
221 | @example | |
222 | (logb 10) | |
223 | @result{} 3 | |
224 | (logb 10.0e20) | |
225 | @result{} 69 | |
226 | @end example | |
227 | @end defun | |
228 | ||
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229 | @defvar float-e |
230 | The mathematical constant @math{e} (2.71828@dots{}). | |
231 | @end defvar | |
232 | ||
233 | @defvar float-pi | |
234 | The mathematical constant @math{pi} (3.14159@dots{}). | |
235 | @end defvar | |
236 | ||
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237 | @node Predicates on Numbers |
238 | @section Type Predicates for Numbers | |
239 | @cindex predicates for numbers | |
240 | ||
241 | The functions in this section test for numbers, or for a specific | |
242 | type of number. The functions @code{integerp} and @code{floatp} can | |
243 | take any type of Lisp object as argument (they would not be of much | |
244 | use otherwise), but the @code{zerop} predicate requires a number as | |
245 | its argument. See also @code{integer-or-marker-p} and | |
246 | @code{number-or-marker-p}, in @ref{Predicates on Markers}. | |
247 | ||
248 | @defun floatp object | |
249 | This predicate tests whether its argument is a floating point | |
250 | number and returns @code{t} if so, @code{nil} otherwise. | |
251 | ||
252 | @code{floatp} does not exist in Emacs versions 18 and earlier. | |
253 | @end defun | |
254 | ||
255 | @defun integerp object | |
256 | This predicate tests whether its argument is an integer, and returns | |
257 | @code{t} if so, @code{nil} otherwise. | |
258 | @end defun | |
259 | ||
260 | @defun numberp object | |
261 | This predicate tests whether its argument is a number (either integer or | |
262 | floating point), and returns @code{t} if so, @code{nil} otherwise. | |
263 | @end defun | |
264 | ||
265 | @defun wholenump object | |
266 | @cindex natural numbers | |
267 | The @code{wholenump} predicate (whose name comes from the phrase | |
268 | ``whole-number-p'') tests to see whether its argument is a nonnegative | |
269 | integer, and returns @code{t} if so, @code{nil} otherwise. 0 is | |
270 | considered non-negative. | |
271 | ||
272 | @findex natnump | |
273 | @code{natnump} is an obsolete synonym for @code{wholenump}. | |
274 | @end defun | |
275 | ||
276 | @defun zerop number | |
277 | This predicate tests whether its argument is zero, and returns @code{t} | |
278 | if so, @code{nil} otherwise. The argument must be a number. | |
279 | ||
280 | @code{(zerop x)} is equivalent to @code{(= x 0)}. | |
281 | @end defun | |
282 | ||
283 | @node Comparison of Numbers | |
284 | @section Comparison of Numbers | |
285 | @cindex number comparison | |
286 | @cindex comparing numbers | |
287 | ||
288 | To test numbers for numerical equality, you should normally use | |
289 | @code{=}, not @code{eq}. There can be many distinct floating point | |
290 | number objects with the same numeric value. If you use @code{eq} to | |
291 | compare them, then you test whether two values are the same | |
292 | @emph{object}. By contrast, @code{=} compares only the numeric values | |
293 | of the objects. | |
294 | ||
295 | At present, each integer value has a unique Lisp object in Emacs Lisp. | |
296 | Therefore, @code{eq} is equivalent to @code{=} where integers are | |
297 | concerned. It is sometimes convenient to use @code{eq} for comparing an | |
298 | unknown value with an integer, because @code{eq} does not report an | |
299 | error if the unknown value is not a number---it accepts arguments of any | |
300 | type. By contrast, @code{=} signals an error if the arguments are not | |
301 | numbers or markers. However, it is a good idea to use @code{=} if you | |
302 | can, even for comparing integers, just in case we change the | |
303 | representation of integers in a future Emacs version. | |
304 | ||
305 | Sometimes it is useful to compare numbers with @code{equal}; it | |
306 | treats two numbers as equal if they have the same data type (both | |
307 | integers, or both floating point) and the same value. By contrast, | |
308 | @code{=} can treat an integer and a floating point number as equal. | |
309 | @xref{Equality Predicates}. | |
310 | ||
311 | There is another wrinkle: because floating point arithmetic is not | |
312 | exact, it is often a bad idea to check for equality of two floating | |
313 | point values. Usually it is better to test for approximate equality. | |
314 | Here's a function to do this: | |
315 | ||
316 | @example | |
317 | (defvar fuzz-factor 1.0e-6) | |
318 | (defun approx-equal (x y) | |
319 | (or (and (= x 0) (= y 0)) | |
320 | (< (/ (abs (- x y)) | |
321 | (max (abs x) (abs y))) | |
322 | fuzz-factor))) | |
323 | @end example | |
324 | ||
325 | @cindex CL note---integers vrs @code{eq} | |
326 | @quotation | |
327 | @b{Common Lisp note:} Comparing numbers in Common Lisp always requires | |
328 | @code{=} because Common Lisp implements multi-word integers, and two | |
329 | distinct integer objects can have the same numeric value. Emacs Lisp | |
330 | can have just one integer object for any given value because it has a | |
331 | limited range of integer values. | |
332 | @end quotation | |
333 | ||
334 | @defun = number-or-marker1 number-or-marker2 | |
335 | This function tests whether its arguments are numerically equal, and | |
336 | returns @code{t} if so, @code{nil} otherwise. | |
337 | @end defun | |
338 | ||
339 | @defun eql value1 value2 | |
340 | This function acts like @code{eq} except when both arguments are | |
341 | numbers. It compares numbers by type and numeric value, so that | |
342 | @code{(eql 1.0 1)} returns @code{nil}, but @code{(eql 1.0 1.0)} and | |
343 | @code{(eql 1 1)} both return @code{t}. | |
344 | @end defun | |
345 | ||
346 | @defun /= number-or-marker1 number-or-marker2 | |
347 | This function tests whether its arguments are numerically equal, and | |
348 | returns @code{t} if they are not, and @code{nil} if they are. | |
349 | @end defun | |
350 | ||
351 | @defun < number-or-marker1 number-or-marker2 | |
352 | This function tests whether its first argument is strictly less than | |
353 | its second argument. It returns @code{t} if so, @code{nil} otherwise. | |
354 | @end defun | |
355 | ||
356 | @defun <= number-or-marker1 number-or-marker2 | |
357 | This function tests whether its first argument is less than or equal | |
358 | to its second argument. It returns @code{t} if so, @code{nil} | |
359 | otherwise. | |
360 | @end defun | |
361 | ||
362 | @defun > number-or-marker1 number-or-marker2 | |
363 | This function tests whether its first argument is strictly greater | |
364 | than its second argument. It returns @code{t} if so, @code{nil} | |
365 | otherwise. | |
366 | @end defun | |
367 | ||
368 | @defun >= number-or-marker1 number-or-marker2 | |
369 | This function tests whether its first argument is greater than or | |
370 | equal to its second argument. It returns @code{t} if so, @code{nil} | |
371 | otherwise. | |
372 | @end defun | |
373 | ||
374 | @defun max number-or-marker &rest numbers-or-markers | |
375 | This function returns the largest of its arguments. | |
376 | If any of the arguments is floating-point, the value is returned | |
377 | as floating point, even if it was given as an integer. | |
378 | ||
379 | @example | |
380 | (max 20) | |
381 | @result{} 20 | |
382 | (max 1 2.5) | |
383 | @result{} 2.5 | |
384 | (max 1 3 2.5) | |
385 | @result{} 3.0 | |
386 | @end example | |
387 | @end defun | |
388 | ||
389 | @defun min number-or-marker &rest numbers-or-markers | |
390 | This function returns the smallest of its arguments. | |
391 | If any of the arguments is floating-point, the value is returned | |
392 | as floating point, even if it was given as an integer. | |
393 | ||
394 | @example | |
395 | (min -4 1) | |
396 | @result{} -4 | |
397 | @end example | |
398 | @end defun | |
399 | ||
400 | @defun abs number | |
401 | This function returns the absolute value of @var{number}. | |
402 | @end defun | |
403 | ||
404 | @node Numeric Conversions | |
405 | @section Numeric Conversions | |
406 | @cindex rounding in conversions | |
407 | @cindex number conversions | |
408 | @cindex converting numbers | |
409 | ||
410 | To convert an integer to floating point, use the function @code{float}. | |
411 | ||
412 | @defun float number | |
413 | This returns @var{number} converted to floating point. | |
414 | If @var{number} is already a floating point number, @code{float} returns | |
415 | it unchanged. | |
416 | @end defun | |
417 | ||
418 | There are four functions to convert floating point numbers to integers; | |
419 | they differ in how they round. All accept an argument @var{number} | |
420 | and an optional argument @var{divisor}. Both arguments may be | |
421 | integers or floating point numbers. @var{divisor} may also be | |
422 | @code{nil}. If @var{divisor} is @code{nil} or omitted, these | |
423 | functions convert @var{number} to an integer, or return it unchanged | |
424 | if it already is an integer. If @var{divisor} is non-@code{nil}, they | |
425 | divide @var{number} by @var{divisor} and convert the result to an | |
426 | integer. An @code{arith-error} results if @var{divisor} is 0. | |
427 | ||
428 | @defun truncate number &optional divisor | |
429 | This returns @var{number}, converted to an integer by rounding towards | |
430 | zero. | |
431 | ||
432 | @example | |
433 | (truncate 1.2) | |
434 | @result{} 1 | |
435 | (truncate 1.7) | |
436 | @result{} 1 | |
437 | (truncate -1.2) | |
438 | @result{} -1 | |
439 | (truncate -1.7) | |
440 | @result{} -1 | |
441 | @end example | |
442 | @end defun | |
443 | ||
444 | @defun floor number &optional divisor | |
445 | This returns @var{number}, converted to an integer by rounding downward | |
446 | (towards negative infinity). | |
447 | ||
448 | If @var{divisor} is specified, this uses the kind of division | |
449 | operation that corresponds to @code{mod}, rounding downward. | |
450 | ||
451 | @example | |
452 | (floor 1.2) | |
453 | @result{} 1 | |
454 | (floor 1.7) | |
455 | @result{} 1 | |
456 | (floor -1.2) | |
457 | @result{} -2 | |
458 | (floor -1.7) | |
459 | @result{} -2 | |
460 | (floor 5.99 3) | |
461 | @result{} 1 | |
462 | @end example | |
463 | @end defun | |
464 | ||
465 | @defun ceiling number &optional divisor | |
466 | This returns @var{number}, converted to an integer by rounding upward | |
467 | (towards positive infinity). | |
468 | ||
469 | @example | |
470 | (ceiling 1.2) | |
471 | @result{} 2 | |
472 | (ceiling 1.7) | |
473 | @result{} 2 | |
474 | (ceiling -1.2) | |
475 | @result{} -1 | |
476 | (ceiling -1.7) | |
477 | @result{} -1 | |
478 | @end example | |
479 | @end defun | |
480 | ||
481 | @defun round number &optional divisor | |
482 | This returns @var{number}, converted to an integer by rounding towards the | |
483 | nearest integer. Rounding a value equidistant between two integers | |
484 | may choose the integer closer to zero, or it may prefer an even integer, | |
485 | depending on your machine. | |
486 | ||
487 | @example | |
488 | (round 1.2) | |
489 | @result{} 1 | |
490 | (round 1.7) | |
491 | @result{} 2 | |
492 | (round -1.2) | |
493 | @result{} -1 | |
494 | (round -1.7) | |
495 | @result{} -2 | |
496 | @end example | |
497 | @end defun | |
498 | ||
499 | @node Arithmetic Operations | |
500 | @section Arithmetic Operations | |
501 | @cindex arithmetic operations | |
502 | ||
503 | Emacs Lisp provides the traditional four arithmetic operations: | |
504 | addition, subtraction, multiplication, and division. Remainder and modulus | |
505 | functions supplement the division functions. The functions to | |
506 | add or subtract 1 are provided because they are traditional in Lisp and | |
507 | commonly used. | |
508 | ||
509 | All of these functions except @code{%} return a floating point value | |
510 | if any argument is floating. | |
511 | ||
c717b326 | 512 | It is important to note that in Emacs Lisp, arithmetic functions |
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513 | do not check for overflow. Thus @code{(1+ 536870911)} may evaluate to |
514 | @minus{}536870912, depending on your hardware. | |
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515 | |
516 | @defun 1+ number-or-marker | |
517 | This function returns @var{number-or-marker} plus 1. | |
518 | For example, | |
519 | ||
520 | @example | |
521 | (setq foo 4) | |
522 | @result{} 4 | |
523 | (1+ foo) | |
524 | @result{} 5 | |
525 | @end example | |
526 | ||
527 | This function is not analogous to the C operator @code{++}---it does not | |
528 | increment a variable. It just computes a sum. Thus, if we continue, | |
529 | ||
530 | @example | |
531 | foo | |
532 | @result{} 4 | |
533 | @end example | |
534 | ||
535 | If you want to increment the variable, you must use @code{setq}, | |
536 | like this: | |
537 | ||
538 | @example | |
539 | (setq foo (1+ foo)) | |
540 | @result{} 5 | |
541 | @end example | |
542 | @end defun | |
543 | ||
544 | @defun 1- number-or-marker | |
545 | This function returns @var{number-or-marker} minus 1. | |
546 | @end defun | |
547 | ||
548 | @defun + &rest numbers-or-markers | |
549 | This function adds its arguments together. When given no arguments, | |
550 | @code{+} returns 0. | |
551 | ||
552 | @example | |
553 | (+) | |
554 | @result{} 0 | |
555 | (+ 1) | |
556 | @result{} 1 | |
557 | (+ 1 2 3 4) | |
558 | @result{} 10 | |
559 | @end example | |
560 | @end defun | |
561 | ||
562 | @defun - &optional number-or-marker &rest more-numbers-or-markers | |
563 | The @code{-} function serves two purposes: negation and subtraction. | |
564 | When @code{-} has a single argument, the value is the negative of the | |
565 | argument. When there are multiple arguments, @code{-} subtracts each of | |
566 | the @var{more-numbers-or-markers} from @var{number-or-marker}, | |
567 | cumulatively. If there are no arguments, the result is 0. | |
568 | ||
569 | @example | |
570 | (- 10 1 2 3 4) | |
571 | @result{} 0 | |
572 | (- 10) | |
573 | @result{} -10 | |
574 | (-) | |
575 | @result{} 0 | |
576 | @end example | |
577 | @end defun | |
578 | ||
579 | @defun * &rest numbers-or-markers | |
580 | This function multiplies its arguments together, and returns the | |
581 | product. When given no arguments, @code{*} returns 1. | |
582 | ||
583 | @example | |
584 | (*) | |
585 | @result{} 1 | |
586 | (* 1) | |
587 | @result{} 1 | |
588 | (* 1 2 3 4) | |
589 | @result{} 24 | |
590 | @end example | |
591 | @end defun | |
592 | ||
593 | @defun / dividend divisor &rest divisors | |
594 | This function divides @var{dividend} by @var{divisor} and returns the | |
595 | quotient. If there are additional arguments @var{divisors}, then it | |
596 | divides @var{dividend} by each divisor in turn. Each argument may be a | |
597 | number or a marker. | |
598 | ||
599 | If all the arguments are integers, then the result is an integer too. | |
600 | This means the result has to be rounded. On most machines, the result | |
601 | is rounded towards zero after each division, but some machines may round | |
602 | differently with negative arguments. This is because the Lisp function | |
603 | @code{/} is implemented using the C division operator, which also | |
604 | permits machine-dependent rounding. As a practical matter, all known | |
605 | machines round in the standard fashion. | |
606 | ||
607 | @cindex @code{arith-error} in division | |
608 | If you divide an integer by 0, an @code{arith-error} error is signaled. | |
609 | (@xref{Errors}.) Floating point division by zero returns either | |
610 | infinity or a NaN if your machine supports @acronym{IEEE} floating point; | |
611 | otherwise, it signals an @code{arith-error} error. | |
612 | ||
613 | @example | |
614 | @group | |
615 | (/ 6 2) | |
616 | @result{} 3 | |
617 | @end group | |
618 | (/ 5 2) | |
619 | @result{} 2 | |
620 | (/ 5.0 2) | |
621 | @result{} 2.5 | |
622 | (/ 5 2.0) | |
623 | @result{} 2.5 | |
624 | (/ 5.0 2.0) | |
625 | @result{} 2.5 | |
626 | (/ 25 3 2) | |
627 | @result{} 4 | |
628 | @group | |
629 | (/ -17 6) | |
630 | @result{} -2 @r{(could in theory be @minus{}3 on some machines)} | |
631 | @end group | |
632 | @end example | |
633 | @end defun | |
634 | ||
635 | @defun % dividend divisor | |
636 | @cindex remainder | |
637 | This function returns the integer remainder after division of @var{dividend} | |
638 | by @var{divisor}. The arguments must be integers or markers. | |
639 | ||
640 | For negative arguments, the remainder is in principle machine-dependent | |
641 | since the quotient is; but in practice, all known machines behave alike. | |
642 | ||
643 | An @code{arith-error} results if @var{divisor} is 0. | |
644 | ||
645 | @example | |
646 | (% 9 4) | |
647 | @result{} 1 | |
648 | (% -9 4) | |
649 | @result{} -1 | |
650 | (% 9 -4) | |
651 | @result{} 1 | |
652 | (% -9 -4) | |
653 | @result{} -1 | |
654 | @end example | |
655 | ||
656 | For any two integers @var{dividend} and @var{divisor}, | |
657 | ||
658 | @example | |
659 | @group | |
660 | (+ (% @var{dividend} @var{divisor}) | |
661 | (* (/ @var{dividend} @var{divisor}) @var{divisor})) | |
662 | @end group | |
663 | @end example | |
664 | ||
665 | @noindent | |
666 | always equals @var{dividend}. | |
667 | @end defun | |
668 | ||
669 | @defun mod dividend divisor | |
670 | @cindex modulus | |
671 | This function returns the value of @var{dividend} modulo @var{divisor}; | |
672 | in other words, the remainder after division of @var{dividend} | |
673 | by @var{divisor}, but with the same sign as @var{divisor}. | |
674 | The arguments must be numbers or markers. | |
675 | ||
676 | Unlike @code{%}, @code{mod} returns a well-defined result for negative | |
677 | arguments. It also permits floating point arguments; it rounds the | |
678 | quotient downward (towards minus infinity) to an integer, and uses that | |
679 | quotient to compute the remainder. | |
680 | ||
681 | An @code{arith-error} results if @var{divisor} is 0. | |
682 | ||
683 | @example | |
684 | @group | |
685 | (mod 9 4) | |
686 | @result{} 1 | |
687 | @end group | |
688 | @group | |
689 | (mod -9 4) | |
690 | @result{} 3 | |
691 | @end group | |
692 | @group | |
693 | (mod 9 -4) | |
694 | @result{} -3 | |
695 | @end group | |
696 | @group | |
697 | (mod -9 -4) | |
698 | @result{} -1 | |
699 | @end group | |
700 | @group | |
701 | (mod 5.5 2.5) | |
702 | @result{} .5 | |
703 | @end group | |
704 | @end example | |
705 | ||
706 | For any two numbers @var{dividend} and @var{divisor}, | |
707 | ||
708 | @example | |
709 | @group | |
710 | (+ (mod @var{dividend} @var{divisor}) | |
711 | (* (floor @var{dividend} @var{divisor}) @var{divisor})) | |
712 | @end group | |
713 | @end example | |
714 | ||
715 | @noindent | |
716 | always equals @var{dividend}, subject to rounding error if either | |
717 | argument is floating point. For @code{floor}, see @ref{Numeric | |
718 | Conversions}. | |
719 | @end defun | |
720 | ||
721 | @node Rounding Operations | |
722 | @section Rounding Operations | |
723 | @cindex rounding without conversion | |
724 | ||
725 | The functions @code{ffloor}, @code{fceiling}, @code{fround}, and | |
726 | @code{ftruncate} take a floating point argument and return a floating | |
727 | point result whose value is a nearby integer. @code{ffloor} returns the | |
728 | nearest integer below; @code{fceiling}, the nearest integer above; | |
729 | @code{ftruncate}, the nearest integer in the direction towards zero; | |
730 | @code{fround}, the nearest integer. | |
731 | ||
732 | @defun ffloor float | |
733 | This function rounds @var{float} to the next lower integral value, and | |
734 | returns that value as a floating point number. | |
735 | @end defun | |
736 | ||
737 | @defun fceiling float | |
738 | This function rounds @var{float} to the next higher integral value, and | |
739 | returns that value as a floating point number. | |
740 | @end defun | |
741 | ||
742 | @defun ftruncate float | |
743 | This function rounds @var{float} towards zero to an integral value, and | |
744 | returns that value as a floating point number. | |
745 | @end defun | |
746 | ||
747 | @defun fround float | |
748 | This function rounds @var{float} to the nearest integral value, | |
749 | and returns that value as a floating point number. | |
750 | @end defun | |
751 | ||
752 | @node Bitwise Operations | |
753 | @section Bitwise Operations on Integers | |
754 | @cindex bitwise arithmetic | |
755 | @cindex logical arithmetic | |
756 | ||
757 | In a computer, an integer is represented as a binary number, a | |
758 | sequence of @dfn{bits} (digits which are either zero or one). A bitwise | |
759 | operation acts on the individual bits of such a sequence. For example, | |
760 | @dfn{shifting} moves the whole sequence left or right one or more places, | |
761 | reproducing the same pattern ``moved over.'' | |
762 | ||
763 | The bitwise operations in Emacs Lisp apply only to integers. | |
764 | ||
765 | @defun lsh integer1 count | |
766 | @cindex logical shift | |
767 | @code{lsh}, which is an abbreviation for @dfn{logical shift}, shifts the | |
768 | bits in @var{integer1} to the left @var{count} places, or to the right | |
769 | if @var{count} is negative, bringing zeros into the vacated bits. If | |
770 | @var{count} is negative, @code{lsh} shifts zeros into the leftmost | |
771 | (most-significant) bit, producing a positive result even if | |
772 | @var{integer1} is negative. Contrast this with @code{ash}, below. | |
773 | ||
774 | Here are two examples of @code{lsh}, shifting a pattern of bits one | |
775 | place to the left. We show only the low-order eight bits of the binary | |
776 | pattern; the rest are all zero. | |
777 | ||
778 | @example | |
779 | @group | |
780 | (lsh 5 1) | |
781 | @result{} 10 | |
782 | ;; @r{Decimal 5 becomes decimal 10.} | |
783 | 00000101 @result{} 00001010 | |
784 | ||
785 | (lsh 7 1) | |
786 | @result{} 14 | |
787 | ;; @r{Decimal 7 becomes decimal 14.} | |
788 | 00000111 @result{} 00001110 | |
789 | @end group | |
790 | @end example | |
791 | ||
792 | @noindent | |
793 | As the examples illustrate, shifting the pattern of bits one place to | |
794 | the left produces a number that is twice the value of the previous | |
795 | number. | |
796 | ||
797 | Shifting a pattern of bits two places to the left produces results | |
798 | like this (with 8-bit binary numbers): | |
799 | ||
800 | @example | |
801 | @group | |
802 | (lsh 3 2) | |
803 | @result{} 12 | |
804 | ;; @r{Decimal 3 becomes decimal 12.} | |
805 | 00000011 @result{} 00001100 | |
806 | @end group | |
807 | @end example | |
808 | ||
809 | On the other hand, shifting one place to the right looks like this: | |
810 | ||
811 | @example | |
812 | @group | |
813 | (lsh 6 -1) | |
814 | @result{} 3 | |
815 | ;; @r{Decimal 6 becomes decimal 3.} | |
816 | 00000110 @result{} 00000011 | |
817 | @end group | |
818 | ||
819 | @group | |
820 | (lsh 5 -1) | |
821 | @result{} 2 | |
822 | ;; @r{Decimal 5 becomes decimal 2.} | |
823 | 00000101 @result{} 00000010 | |
824 | @end group | |
825 | @end example | |
826 | ||
827 | @noindent | |
828 | As the example illustrates, shifting one place to the right divides the | |
829 | value of a positive integer by two, rounding downward. | |
830 | ||
c717b326 | 831 | The function @code{lsh}, like all Emacs Lisp arithmetic functions, does |
b8d4c8d0 GM |
832 | not check for overflow, so shifting left can discard significant bits |
833 | and change the sign of the number. For example, left shifting | |
001903b5 | 834 | 536,870,911 produces @minus{}2 in the 30-bit implementation: |
b8d4c8d0 GM |
835 | |
836 | @example | |
1ddd6622 | 837 | (lsh 536870911 1) ; @r{left shift} |
b8d4c8d0 GM |
838 | @result{} -2 |
839 | @end example | |
840 | ||
001903b5 | 841 | In binary, the argument looks like this: |
b8d4c8d0 GM |
842 | |
843 | @example | |
844 | @group | |
1ddd6622 | 845 | ;; @r{Decimal 536,870,911} |
001903b5 | 846 | 0111...111111 (30 bits total) |
b8d4c8d0 GM |
847 | @end group |
848 | @end example | |
849 | ||
850 | @noindent | |
851 | which becomes the following when left shifted: | |
852 | ||
853 | @example | |
854 | @group | |
855 | ;; @r{Decimal @minus{}2} | |
001903b5 | 856 | 1111...111110 (30 bits total) |
b8d4c8d0 GM |
857 | @end group |
858 | @end example | |
859 | @end defun | |
860 | ||
861 | @defun ash integer1 count | |
862 | @cindex arithmetic shift | |
863 | @code{ash} (@dfn{arithmetic shift}) shifts the bits in @var{integer1} | |
864 | to the left @var{count} places, or to the right if @var{count} | |
865 | is negative. | |
866 | ||
867 | @code{ash} gives the same results as @code{lsh} except when | |
868 | @var{integer1} and @var{count} are both negative. In that case, | |
869 | @code{ash} puts ones in the empty bit positions on the left, while | |
870 | @code{lsh} puts zeros in those bit positions. | |
871 | ||
872 | Thus, with @code{ash}, shifting the pattern of bits one place to the right | |
873 | looks like this: | |
874 | ||
875 | @example | |
876 | @group | |
877 | (ash -6 -1) @result{} -3 | |
878 | ;; @r{Decimal @minus{}6 becomes decimal @minus{}3.} | |
001903b5 | 879 | 1111...111010 (30 bits total) |
b8d4c8d0 | 880 | @result{} |
001903b5 | 881 | 1111...111101 (30 bits total) |
b8d4c8d0 GM |
882 | @end group |
883 | @end example | |
884 | ||
885 | In contrast, shifting the pattern of bits one place to the right with | |
886 | @code{lsh} looks like this: | |
887 | ||
888 | @example | |
889 | @group | |
1ddd6622 GM |
890 | (lsh -6 -1) @result{} 536870909 |
891 | ;; @r{Decimal @minus{}6 becomes decimal 536,870,909.} | |
001903b5 | 892 | 1111...111010 (30 bits total) |
b8d4c8d0 | 893 | @result{} |
001903b5 | 894 | 0111...111101 (30 bits total) |
b8d4c8d0 GM |
895 | @end group |
896 | @end example | |
897 | ||
898 | Here are other examples: | |
899 | ||
900 | @c !!! Check if lined up in smallbook format! XDVI shows problem | |
901 | @c with smallbook but not with regular book! --rjc 16mar92 | |
902 | @smallexample | |
903 | @group | |
001903b5 | 904 | ; @r{ 30-bit binary values} |
b8d4c8d0 | 905 | |
be14b9ab PE |
906 | (lsh 5 2) ; 5 = @r{0000...000101} |
907 | @result{} 20 ; = @r{0000...010100} | |
b8d4c8d0 GM |
908 | @end group |
909 | @group | |
910 | (ash 5 2) | |
911 | @result{} 20 | |
be14b9ab PE |
912 | (lsh -5 2) ; -5 = @r{1111...111011} |
913 | @result{} -20 ; = @r{1111...101100} | |
b8d4c8d0 GM |
914 | (ash -5 2) |
915 | @result{} -20 | |
916 | @end group | |
917 | @group | |
be14b9ab PE |
918 | (lsh 5 -2) ; 5 = @r{0000...000101} |
919 | @result{} 1 ; = @r{0000...000001} | |
b8d4c8d0 GM |
920 | @end group |
921 | @group | |
922 | (ash 5 -2) | |
923 | @result{} 1 | |
924 | @end group | |
925 | @group | |
be14b9ab | 926 | (lsh -5 -2) ; -5 = @r{1111...111011} |
001903b5 | 927 | @result{} 268435454 |
be14b9ab | 928 | ; = @r{0011...111110} |
b8d4c8d0 GM |
929 | @end group |
930 | @group | |
be14b9ab PE |
931 | (ash -5 -2) ; -5 = @r{1111...111011} |
932 | @result{} -2 ; = @r{1111...111110} | |
b8d4c8d0 GM |
933 | @end group |
934 | @end smallexample | |
935 | @end defun | |
936 | ||
937 | @defun logand &rest ints-or-markers | |
938 | This function returns the ``logical and'' of the arguments: the | |
939 | @var{n}th bit is set in the result if, and only if, the @var{n}th bit is | |
940 | set in all the arguments. (``Set'' means that the value of the bit is 1 | |
941 | rather than 0.) | |
942 | ||
943 | For example, using 4-bit binary numbers, the ``logical and'' of 13 and | |
944 | 12 is 12: 1101 combined with 1100 produces 1100. | |
945 | In both the binary numbers, the leftmost two bits are set (i.e., they | |
946 | are 1's), so the leftmost two bits of the returned value are set. | |
947 | However, for the rightmost two bits, each is zero in at least one of | |
948 | the arguments, so the rightmost two bits of the returned value are 0's. | |
949 | ||
950 | @noindent | |
951 | Therefore, | |
952 | ||
953 | @example | |
954 | @group | |
955 | (logand 13 12) | |
956 | @result{} 12 | |
957 | @end group | |
958 | @end example | |
959 | ||
960 | If @code{logand} is not passed any argument, it returns a value of | |
961 | @minus{}1. This number is an identity element for @code{logand} | |
962 | because its binary representation consists entirely of ones. If | |
963 | @code{logand} is passed just one argument, it returns that argument. | |
964 | ||
965 | @smallexample | |
966 | @group | |
001903b5 | 967 | ; @r{ 30-bit binary values} |
b8d4c8d0 | 968 | |
be14b9ab PE |
969 | (logand 14 13) ; 14 = @r{0000...001110} |
970 | ; 13 = @r{0000...001101} | |
971 | @result{} 12 ; 12 = @r{0000...001100} | |
b8d4c8d0 GM |
972 | @end group |
973 | ||
974 | @group | |
be14b9ab PE |
975 | (logand 14 13 4) ; 14 = @r{0000...001110} |
976 | ; 13 = @r{0000...001101} | |
977 | ; 4 = @r{0000...000100} | |
978 | @result{} 4 ; 4 = @r{0000...000100} | |
b8d4c8d0 GM |
979 | @end group |
980 | ||
981 | @group | |
982 | (logand) | |
be14b9ab | 983 | @result{} -1 ; -1 = @r{1111...111111} |
b8d4c8d0 GM |
984 | @end group |
985 | @end smallexample | |
986 | @end defun | |
987 | ||
988 | @defun logior &rest ints-or-markers | |
989 | This function returns the ``inclusive or'' of its arguments: the @var{n}th bit | |
990 | is set in the result if, and only if, the @var{n}th bit is set in at least | |
991 | one of the arguments. If there are no arguments, the result is zero, | |
992 | which is an identity element for this operation. If @code{logior} is | |
993 | passed just one argument, it returns that argument. | |
994 | ||
995 | @smallexample | |
996 | @group | |
001903b5 | 997 | ; @r{ 30-bit binary values} |
b8d4c8d0 | 998 | |
be14b9ab PE |
999 | (logior 12 5) ; 12 = @r{0000...001100} |
1000 | ; 5 = @r{0000...000101} | |
1001 | @result{} 13 ; 13 = @r{0000...001101} | |
b8d4c8d0 GM |
1002 | @end group |
1003 | ||
1004 | @group | |
be14b9ab PE |
1005 | (logior 12 5 7) ; 12 = @r{0000...001100} |
1006 | ; 5 = @r{0000...000101} | |
1007 | ; 7 = @r{0000...000111} | |
1008 | @result{} 15 ; 15 = @r{0000...001111} | |
b8d4c8d0 GM |
1009 | @end group |
1010 | @end smallexample | |
1011 | @end defun | |
1012 | ||
1013 | @defun logxor &rest ints-or-markers | |
1014 | This function returns the ``exclusive or'' of its arguments: the | |
1015 | @var{n}th bit is set in the result if, and only if, the @var{n}th bit is | |
1016 | set in an odd number of the arguments. If there are no arguments, the | |
1017 | result is 0, which is an identity element for this operation. If | |
1018 | @code{logxor} is passed just one argument, it returns that argument. | |
1019 | ||
1020 | @smallexample | |
1021 | @group | |
001903b5 | 1022 | ; @r{ 30-bit binary values} |
b8d4c8d0 | 1023 | |
be14b9ab PE |
1024 | (logxor 12 5) ; 12 = @r{0000...001100} |
1025 | ; 5 = @r{0000...000101} | |
1026 | @result{} 9 ; 9 = @r{0000...001001} | |
b8d4c8d0 GM |
1027 | @end group |
1028 | ||
1029 | @group | |
be14b9ab PE |
1030 | (logxor 12 5 7) ; 12 = @r{0000...001100} |
1031 | ; 5 = @r{0000...000101} | |
1032 | ; 7 = @r{0000...000111} | |
1033 | @result{} 14 ; 14 = @r{0000...001110} | |
b8d4c8d0 GM |
1034 | @end group |
1035 | @end smallexample | |
1036 | @end defun | |
1037 | ||
1038 | @defun lognot integer | |
1039 | This function returns the logical complement of its argument: the @var{n}th | |
1040 | bit is one in the result if, and only if, the @var{n}th bit is zero in | |
1041 | @var{integer}, and vice-versa. | |
1042 | ||
1043 | @example | |
1044 | (lognot 5) | |
1045 | @result{} -6 | |
001903b5 | 1046 | ;; 5 = @r{0000...000101} (30 bits total) |
b8d4c8d0 | 1047 | ;; @r{becomes} |
001903b5 | 1048 | ;; -6 = @r{1111...111010} (30 bits total) |
b8d4c8d0 GM |
1049 | @end example |
1050 | @end defun | |
1051 | ||
1052 | @node Math Functions | |
1053 | @section Standard Mathematical Functions | |
1054 | @cindex transcendental functions | |
1055 | @cindex mathematical functions | |
1056 | @cindex floating-point functions | |
1057 | ||
1058 | These mathematical functions allow integers as well as floating point | |
1059 | numbers as arguments. | |
1060 | ||
1061 | @defun sin arg | |
1062 | @defunx cos arg | |
1063 | @defunx tan arg | |
1064 | These are the ordinary trigonometric functions, with argument measured | |
1065 | in radians. | |
1066 | @end defun | |
1067 | ||
1068 | @defun asin arg | |
1069 | The value of @code{(asin @var{arg})} is a number between | |
1070 | @ifnottex | |
1071 | @minus{}pi/2 | |
1072 | @end ifnottex | |
1073 | @tex | |
1074 | @math{-\pi/2} | |
1075 | @end tex | |
1076 | and | |
1077 | @ifnottex | |
1078 | pi/2 | |
1079 | @end ifnottex | |
1080 | @tex | |
1081 | @math{\pi/2} | |
1082 | @end tex | |
1083 | (inclusive) whose sine is @var{arg}; if, however, @var{arg} is out of | |
1084 | range (outside [@minus{}1, 1]), it signals a @code{domain-error} error. | |
1085 | @end defun | |
1086 | ||
1087 | @defun acos arg | |
1088 | The value of @code{(acos @var{arg})} is a number between 0 and | |
1089 | @ifnottex | |
1090 | pi | |
1091 | @end ifnottex | |
1092 | @tex | |
1093 | @math{\pi} | |
1094 | @end tex | |
1095 | (inclusive) whose cosine is @var{arg}; if, however, @var{arg} is out | |
1096 | of range (outside [@minus{}1, 1]), it signals a @code{domain-error} error. | |
1097 | @end defun | |
1098 | ||
1099 | @defun atan y &optional x | |
1100 | The value of @code{(atan @var{y})} is a number between | |
1101 | @ifnottex | |
1102 | @minus{}pi/2 | |
1103 | @end ifnottex | |
1104 | @tex | |
1105 | @math{-\pi/2} | |
1106 | @end tex | |
1107 | and | |
1108 | @ifnottex | |
1109 | pi/2 | |
1110 | @end ifnottex | |
1111 | @tex | |
1112 | @math{\pi/2} | |
1113 | @end tex | |
1114 | (exclusive) whose tangent is @var{y}. If the optional second | |
1115 | argument @var{x} is given, the value of @code{(atan y x)} is the | |
1116 | angle in radians between the vector @code{[@var{x}, @var{y}]} and the | |
1117 | @code{X} axis. | |
1118 | @end defun | |
1119 | ||
1120 | @defun exp arg | |
1121 | This is the exponential function; it returns | |
1122 | @tex | |
1123 | @math{e} | |
1124 | @end tex | |
1125 | @ifnottex | |
1126 | @i{e} | |
1127 | @end ifnottex | |
1128 | to the power @var{arg}. | |
1129 | @tex | |
1130 | @math{e} | |
1131 | @end tex | |
1132 | @ifnottex | |
1133 | @i{e} | |
1134 | @end ifnottex | |
1135 | is a fundamental mathematical constant also called the base of natural | |
1136 | logarithms. | |
1137 | @end defun | |
1138 | ||
1139 | @defun log arg &optional base | |
1140 | This function returns the logarithm of @var{arg}, with base @var{base}. | |
1141 | If you don't specify @var{base}, the base | |
1142 | @tex | |
1143 | @math{e} | |
1144 | @end tex | |
1145 | @ifnottex | |
1146 | @i{e} | |
1147 | @end ifnottex | |
1148 | is used. If @var{arg} is negative, it signals a @code{domain-error} | |
1149 | error. | |
1150 | @end defun | |
1151 | ||
1152 | @ignore | |
1153 | @defun expm1 arg | |
1154 | This function returns @code{(1- (exp @var{arg}))}, but it is more | |
1155 | accurate than that when @var{arg} is negative and @code{(exp @var{arg})} | |
1156 | is close to 1. | |
1157 | @end defun | |
1158 | ||
1159 | @defun log1p arg | |
1160 | This function returns @code{(log (1+ @var{arg}))}, but it is more | |
1161 | accurate than that when @var{arg} is so small that adding 1 to it would | |
1162 | lose accuracy. | |
1163 | @end defun | |
1164 | @end ignore | |
1165 | ||
1166 | @defun log10 arg | |
1167 | This function returns the logarithm of @var{arg}, with base 10. If | |
1168 | @var{arg} is negative, it signals a @code{domain-error} error. | |
1169 | @code{(log10 @var{x})} @equiv{} @code{(log @var{x} 10)}, at least | |
1170 | approximately. | |
1171 | @end defun | |
1172 | ||
1173 | @defun expt x y | |
1174 | This function returns @var{x} raised to power @var{y}. If both | |
c717b326 PE |
1175 | arguments are integers and @var{y} is positive, the result is an |
1176 | integer; in this case, overflow causes truncation, so watch out. | |
b8d4c8d0 GM |
1177 | @end defun |
1178 | ||
1179 | @defun sqrt arg | |
1180 | This returns the square root of @var{arg}. If @var{arg} is negative, | |
1181 | it signals a @code{domain-error} error. | |
1182 | @end defun | |
1183 | ||
1184 | @node Random Numbers | |
1185 | @section Random Numbers | |
1186 | @cindex random numbers | |
1187 | ||
1188 | A deterministic computer program cannot generate true random numbers. | |
1189 | For most purposes, @dfn{pseudo-random numbers} suffice. A series of | |
1190 | pseudo-random numbers is generated in a deterministic fashion. The | |
1191 | numbers are not truly random, but they have certain properties that | |
1192 | mimic a random series. For example, all possible values occur equally | |
1193 | often in a pseudo-random series. | |
1194 | ||
1195 | In Emacs, pseudo-random numbers are generated from a ``seed'' number. | |
1196 | Starting from any given seed, the @code{random} function always | |
1197 | generates the same sequence of numbers. Emacs always starts with the | |
1198 | same seed value, so the sequence of values of @code{random} is actually | |
1199 | the same in each Emacs run! For example, in one operating system, the | |
1200 | first call to @code{(random)} after you start Emacs always returns | |
1201 | @minus{}1457731, and the second one always returns @minus{}7692030. This | |
1202 | repeatability is helpful for debugging. | |
1203 | ||
1204 | If you want random numbers that don't always come out the same, execute | |
1205 | @code{(random t)}. This chooses a new seed based on the current time of | |
1206 | day and on Emacs's process @acronym{ID} number. | |
1207 | ||
1208 | @defun random &optional limit | |
1209 | This function returns a pseudo-random integer. Repeated calls return a | |
1210 | series of pseudo-random integers. | |
1211 | ||
1212 | If @var{limit} is a positive integer, the value is chosen to be | |
1213 | nonnegative and less than @var{limit}. | |
1214 | ||
1215 | If @var{limit} is @code{t}, it means to choose a new seed based on the | |
1216 | current time of day and on Emacs's process @acronym{ID} number. | |
1217 | @c "Emacs'" is incorrect usage! | |
1218 | ||
1219 | On some machines, any integer representable in Lisp may be the result | |
1220 | of @code{random}. On other machines, the result can never be larger | |
1221 | than a certain maximum or less than a certain (negative) minimum. | |
1222 | @end defun |