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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}), |
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52 | @end tex |
53 | but some machines may provide a wider range. Many examples in this | |
1ddd6622 | 54 | chapter assume an integer has 30 bits. |
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55 | @cindex overflow |
56 | ||
57 | The Lisp reader reads an integer as a sequence of digits with optional | |
58 | initial sign and optional final period. | |
59 | ||
60 | @example | |
61 | 1 ; @r{The integer 1.} | |
62 | 1. ; @r{The integer 1.} | |
63 | +1 ; @r{Also the integer 1.} | |
64 | -1 ; @r{The integer @minus{}1.} | |
1ddd6622 | 65 | 1073741825 ; @r{Also the integer 1, due to overflow.} |
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66 | 0 ; @r{The integer 0.} |
67 | -0 ; @r{The integer 0.} | |
68 | @end example | |
69 | ||
70 | @cindex integers in specific radix | |
71 | @cindex radix for reading an integer | |
72 | @cindex base for reading an integer | |
73 | @cindex hex numbers | |
74 | @cindex octal numbers | |
75 | @cindex reading numbers in hex, octal, and binary | |
76 | The syntax for integers in bases other than 10 uses @samp{#} | |
77 | followed by a letter that specifies the radix: @samp{b} for binary, | |
78 | @samp{o} for octal, @samp{x} for hex, or @samp{@var{radix}r} to | |
79 | specify radix @var{radix}. Case is not significant for the letter | |
80 | that specifies the radix. Thus, @samp{#b@var{integer}} reads | |
81 | @var{integer} in binary, and @samp{#@var{radix}r@var{integer}} reads | |
82 | @var{integer} in radix @var{radix}. Allowed values of @var{radix} run | |
83 | from 2 to 36. For example: | |
84 | ||
85 | @example | |
86 | #b101100 @result{} 44 | |
87 | #o54 @result{} 44 | |
88 | #x2c @result{} 44 | |
89 | #24r1k @result{} 44 | |
90 | @end example | |
91 | ||
92 | To understand how various functions work on integers, especially the | |
93 | bitwise operators (@pxref{Bitwise Operations}), it is often helpful to | |
94 | view the numbers in their binary form. | |
95 | ||
1ddd6622 | 96 | In 30-bit binary, the decimal integer 5 looks like this: |
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97 | |
98 | @example | |
1ddd6622 | 99 | 00 0000 0000 0000 0000 0000 0000 0101 |
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100 | @end example |
101 | ||
102 | @noindent | |
103 | (We have inserted spaces between groups of 4 bits, and two spaces | |
104 | between groups of 8 bits, to make the binary integer easier to read.) | |
105 | ||
106 | The integer @minus{}1 looks like this: | |
107 | ||
108 | @example | |
1ddd6622 | 109 | 11 1111 1111 1111 1111 1111 1111 1111 |
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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 | |
1ddd6622 | 122 | 11 1111 1111 1111 1111 1111 1111 1011 |
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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 | |
1ddd6622 | 129 | 01 1111 1111 1111 1111 1111 1111 1111 |
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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 | |
139 | @result{} 10 0000 0000 0000 0000 0000 0000 0000 | |
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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 | ||
148 | @defvar most-positive-fixnum | |
149 | The value of this variable is the largest integer that Emacs Lisp | |
150 | can handle. | |
151 | @end defvar | |
152 | ||
153 | @defvar most-negative-fixnum | |
154 | The value of this variable is the smallest integer that Emacs Lisp can | |
155 | handle. It is negative. | |
156 | @end defvar | |
157 | ||
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158 | @xref{Character Codes, max-char}, for the maximum value of a valid |
159 | character codepoint. | |
160 | ||
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161 | @node Float Basics |
162 | @section Floating Point Basics | |
163 | ||
164 | Floating point numbers are useful for representing numbers that are | |
165 | not integral. The precise range of floating point numbers is | |
166 | machine-specific; it is the same as the range of the C data type | |
167 | @code{double} on the machine you are using. | |
168 | ||
169 | The read-syntax for floating point numbers requires either a decimal | |
170 | point (with at least one digit following), an exponent, or both. For | |
171 | example, @samp{1500.0}, @samp{15e2}, @samp{15.0e2}, @samp{1.5e3}, and | |
172 | @samp{.15e4} are five ways of writing a floating point number whose | |
173 | value is 1500. They are all equivalent. You can also use a minus sign | |
174 | to write negative floating point numbers, as in @samp{-1.0}. | |
175 | ||
176 | @cindex @acronym{IEEE} floating point | |
177 | @cindex positive infinity | |
178 | @cindex negative infinity | |
179 | @cindex infinity | |
180 | @cindex NaN | |
181 | Most modern computers support the @acronym{IEEE} floating point standard, | |
182 | which provides for positive infinity and negative infinity as floating point | |
183 | values. It also provides for a class of values called NaN or | |
184 | ``not-a-number''; numerical functions return such values in cases where | |
185 | there is no correct answer. For example, @code{(/ 0.0 0.0)} returns a | |
186 | NaN. For practical purposes, there's no significant difference between | |
187 | different NaN values in Emacs Lisp, and there's no rule for precisely | |
188 | which NaN value should be used in a particular case, so Emacs Lisp | |
189 | doesn't try to distinguish them (but it does report the sign, if you | |
190 | print it). Here are the read syntaxes for these special floating | |
191 | point values: | |
192 | ||
193 | @table @asis | |
194 | @item positive infinity | |
195 | @samp{1.0e+INF} | |
196 | @item negative infinity | |
197 | @samp{-1.0e+INF} | |
198 | @item Not-a-number | |
199 | @samp{0.0e+NaN} or @samp{-0.0e+NaN}. | |
200 | @end table | |
201 | ||
202 | To test whether a floating point value is a NaN, compare it with | |
203 | itself using @code{=}. That returns @code{nil} for a NaN, and | |
204 | @code{t} for any other floating point value. | |
205 | ||
206 | The value @code{-0.0} is distinguishable from ordinary zero in | |
207 | @acronym{IEEE} floating point, but Emacs Lisp @code{equal} and | |
208 | @code{=} consider them equal values. | |
209 | ||
210 | You can use @code{logb} to extract the binary exponent of a floating | |
211 | point number (or estimate the logarithm of an integer): | |
212 | ||
213 | @defun logb number | |
214 | This function returns the binary exponent of @var{number}. More | |
215 | precisely, the value is the logarithm of @var{number} base 2, rounded | |
216 | down to an integer. | |
217 | ||
218 | @example | |
219 | (logb 10) | |
220 | @result{} 3 | |
221 | (logb 10.0e20) | |
222 | @result{} 69 | |
223 | @end example | |
224 | @end defun | |
225 | ||
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226 | @defvar float-e |
227 | The mathematical constant @math{e} (2.71828@dots{}). | |
228 | @end defvar | |
229 | ||
230 | @defvar float-pi | |
231 | The mathematical constant @math{pi} (3.14159@dots{}). | |
232 | @end defvar | |
233 | ||
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234 | @node Predicates on Numbers |
235 | @section Type Predicates for Numbers | |
236 | @cindex predicates for numbers | |
237 | ||
238 | The functions in this section test for numbers, or for a specific | |
239 | type of number. The functions @code{integerp} and @code{floatp} can | |
240 | take any type of Lisp object as argument (they would not be of much | |
241 | use otherwise), but the @code{zerop} predicate requires a number as | |
242 | its argument. See also @code{integer-or-marker-p} and | |
243 | @code{number-or-marker-p}, in @ref{Predicates on Markers}. | |
244 | ||
245 | @defun floatp object | |
246 | This predicate tests whether its argument is a floating point | |
247 | number and returns @code{t} if so, @code{nil} otherwise. | |
248 | ||
249 | @code{floatp} does not exist in Emacs versions 18 and earlier. | |
250 | @end defun | |
251 | ||
252 | @defun integerp object | |
253 | This predicate tests whether its argument is an integer, and returns | |
254 | @code{t} if so, @code{nil} otherwise. | |
255 | @end defun | |
256 | ||
257 | @defun numberp object | |
258 | This predicate tests whether its argument is a number (either integer or | |
259 | floating point), and returns @code{t} if so, @code{nil} otherwise. | |
260 | @end defun | |
261 | ||
262 | @defun wholenump object | |
263 | @cindex natural numbers | |
264 | The @code{wholenump} predicate (whose name comes from the phrase | |
265 | ``whole-number-p'') tests to see whether its argument is a nonnegative | |
266 | integer, and returns @code{t} if so, @code{nil} otherwise. 0 is | |
267 | considered non-negative. | |
268 | ||
269 | @findex natnump | |
270 | @code{natnump} is an obsolete synonym for @code{wholenump}. | |
271 | @end defun | |
272 | ||
273 | @defun zerop number | |
274 | This predicate tests whether its argument is zero, and returns @code{t} | |
275 | if so, @code{nil} otherwise. The argument must be a number. | |
276 | ||
277 | @code{(zerop x)} is equivalent to @code{(= x 0)}. | |
278 | @end defun | |
279 | ||
280 | @node Comparison of Numbers | |
281 | @section Comparison of Numbers | |
282 | @cindex number comparison | |
283 | @cindex comparing numbers | |
284 | ||
285 | To test numbers for numerical equality, you should normally use | |
286 | @code{=}, not @code{eq}. There can be many distinct floating point | |
287 | number objects with the same numeric value. If you use @code{eq} to | |
288 | compare them, then you test whether two values are the same | |
289 | @emph{object}. By contrast, @code{=} compares only the numeric values | |
290 | of the objects. | |
291 | ||
292 | At present, each integer value has a unique Lisp object in Emacs Lisp. | |
293 | Therefore, @code{eq} is equivalent to @code{=} where integers are | |
294 | concerned. It is sometimes convenient to use @code{eq} for comparing an | |
295 | unknown value with an integer, because @code{eq} does not report an | |
296 | error if the unknown value is not a number---it accepts arguments of any | |
297 | type. By contrast, @code{=} signals an error if the arguments are not | |
298 | numbers or markers. However, it is a good idea to use @code{=} if you | |
299 | can, even for comparing integers, just in case we change the | |
300 | representation of integers in a future Emacs version. | |
301 | ||
302 | Sometimes it is useful to compare numbers with @code{equal}; it | |
303 | treats two numbers as equal if they have the same data type (both | |
304 | integers, or both floating point) and the same value. By contrast, | |
305 | @code{=} can treat an integer and a floating point number as equal. | |
306 | @xref{Equality Predicates}. | |
307 | ||
308 | There is another wrinkle: because floating point arithmetic is not | |
309 | exact, it is often a bad idea to check for equality of two floating | |
310 | point values. Usually it is better to test for approximate equality. | |
311 | Here's a function to do this: | |
312 | ||
313 | @example | |
314 | (defvar fuzz-factor 1.0e-6) | |
315 | (defun approx-equal (x y) | |
316 | (or (and (= x 0) (= y 0)) | |
317 | (< (/ (abs (- x y)) | |
318 | (max (abs x) (abs y))) | |
319 | fuzz-factor))) | |
320 | @end example | |
321 | ||
322 | @cindex CL note---integers vrs @code{eq} | |
323 | @quotation | |
324 | @b{Common Lisp note:} Comparing numbers in Common Lisp always requires | |
325 | @code{=} because Common Lisp implements multi-word integers, and two | |
326 | distinct integer objects can have the same numeric value. Emacs Lisp | |
327 | can have just one integer object for any given value because it has a | |
328 | limited range of integer values. | |
329 | @end quotation | |
330 | ||
331 | @defun = number-or-marker1 number-or-marker2 | |
332 | This function tests whether its arguments are numerically equal, and | |
333 | returns @code{t} if so, @code{nil} otherwise. | |
334 | @end defun | |
335 | ||
336 | @defun eql value1 value2 | |
337 | This function acts like @code{eq} except when both arguments are | |
338 | numbers. It compares numbers by type and numeric value, so that | |
339 | @code{(eql 1.0 1)} returns @code{nil}, but @code{(eql 1.0 1.0)} and | |
340 | @code{(eql 1 1)} both return @code{t}. | |
341 | @end defun | |
342 | ||
343 | @defun /= number-or-marker1 number-or-marker2 | |
344 | This function tests whether its arguments are numerically equal, and | |
345 | returns @code{t} if they are not, and @code{nil} if they are. | |
346 | @end defun | |
347 | ||
348 | @defun < number-or-marker1 number-or-marker2 | |
349 | This function tests whether its first argument is strictly less than | |
350 | its second argument. It returns @code{t} if so, @code{nil} otherwise. | |
351 | @end defun | |
352 | ||
353 | @defun <= number-or-marker1 number-or-marker2 | |
354 | This function tests whether its first argument is less than or equal | |
355 | to its second argument. It returns @code{t} if so, @code{nil} | |
356 | otherwise. | |
357 | @end defun | |
358 | ||
359 | @defun > number-or-marker1 number-or-marker2 | |
360 | This function tests whether its first argument is strictly greater | |
361 | than its second argument. It returns @code{t} if so, @code{nil} | |
362 | otherwise. | |
363 | @end defun | |
364 | ||
365 | @defun >= number-or-marker1 number-or-marker2 | |
366 | This function tests whether its first argument is greater than or | |
367 | equal to its second argument. It returns @code{t} if so, @code{nil} | |
368 | otherwise. | |
369 | @end defun | |
370 | ||
371 | @defun max number-or-marker &rest numbers-or-markers | |
372 | This function returns the largest of its arguments. | |
373 | If any of the arguments is floating-point, the value is returned | |
374 | as floating point, even if it was given as an integer. | |
375 | ||
376 | @example | |
377 | (max 20) | |
378 | @result{} 20 | |
379 | (max 1 2.5) | |
380 | @result{} 2.5 | |
381 | (max 1 3 2.5) | |
382 | @result{} 3.0 | |
383 | @end example | |
384 | @end defun | |
385 | ||
386 | @defun min number-or-marker &rest numbers-or-markers | |
387 | This function returns the smallest of its arguments. | |
388 | If any of the arguments is floating-point, the value is returned | |
389 | as floating point, even if it was given as an integer. | |
390 | ||
391 | @example | |
392 | (min -4 1) | |
393 | @result{} -4 | |
394 | @end example | |
395 | @end defun | |
396 | ||
397 | @defun abs number | |
398 | This function returns the absolute value of @var{number}. | |
399 | @end defun | |
400 | ||
401 | @node Numeric Conversions | |
402 | @section Numeric Conversions | |
403 | @cindex rounding in conversions | |
404 | @cindex number conversions | |
405 | @cindex converting numbers | |
406 | ||
407 | To convert an integer to floating point, use the function @code{float}. | |
408 | ||
409 | @defun float number | |
410 | This returns @var{number} converted to floating point. | |
411 | If @var{number} is already a floating point number, @code{float} returns | |
412 | it unchanged. | |
413 | @end defun | |
414 | ||
415 | There are four functions to convert floating point numbers to integers; | |
416 | they differ in how they round. All accept an argument @var{number} | |
417 | and an optional argument @var{divisor}. Both arguments may be | |
418 | integers or floating point numbers. @var{divisor} may also be | |
419 | @code{nil}. If @var{divisor} is @code{nil} or omitted, these | |
420 | functions convert @var{number} to an integer, or return it unchanged | |
421 | if it already is an integer. If @var{divisor} is non-@code{nil}, they | |
422 | divide @var{number} by @var{divisor} and convert the result to an | |
423 | integer. An @code{arith-error} results if @var{divisor} is 0. | |
424 | ||
425 | @defun truncate number &optional divisor | |
426 | This returns @var{number}, converted to an integer by rounding towards | |
427 | zero. | |
428 | ||
429 | @example | |
430 | (truncate 1.2) | |
431 | @result{} 1 | |
432 | (truncate 1.7) | |
433 | @result{} 1 | |
434 | (truncate -1.2) | |
435 | @result{} -1 | |
436 | (truncate -1.7) | |
437 | @result{} -1 | |
438 | @end example | |
439 | @end defun | |
440 | ||
441 | @defun floor number &optional divisor | |
442 | This returns @var{number}, converted to an integer by rounding downward | |
443 | (towards negative infinity). | |
444 | ||
445 | If @var{divisor} is specified, this uses the kind of division | |
446 | operation that corresponds to @code{mod}, rounding downward. | |
447 | ||
448 | @example | |
449 | (floor 1.2) | |
450 | @result{} 1 | |
451 | (floor 1.7) | |
452 | @result{} 1 | |
453 | (floor -1.2) | |
454 | @result{} -2 | |
455 | (floor -1.7) | |
456 | @result{} -2 | |
457 | (floor 5.99 3) | |
458 | @result{} 1 | |
459 | @end example | |
460 | @end defun | |
461 | ||
462 | @defun ceiling number &optional divisor | |
463 | This returns @var{number}, converted to an integer by rounding upward | |
464 | (towards positive infinity). | |
465 | ||
466 | @example | |
467 | (ceiling 1.2) | |
468 | @result{} 2 | |
469 | (ceiling 1.7) | |
470 | @result{} 2 | |
471 | (ceiling -1.2) | |
472 | @result{} -1 | |
473 | (ceiling -1.7) | |
474 | @result{} -1 | |
475 | @end example | |
476 | @end defun | |
477 | ||
478 | @defun round number &optional divisor | |
479 | This returns @var{number}, converted to an integer by rounding towards the | |
480 | nearest integer. Rounding a value equidistant between two integers | |
481 | may choose the integer closer to zero, or it may prefer an even integer, | |
482 | depending on your machine. | |
483 | ||
484 | @example | |
485 | (round 1.2) | |
486 | @result{} 1 | |
487 | (round 1.7) | |
488 | @result{} 2 | |
489 | (round -1.2) | |
490 | @result{} -1 | |
491 | (round -1.7) | |
492 | @result{} -2 | |
493 | @end example | |
494 | @end defun | |
495 | ||
496 | @node Arithmetic Operations | |
497 | @section Arithmetic Operations | |
498 | @cindex arithmetic operations | |
499 | ||
500 | Emacs Lisp provides the traditional four arithmetic operations: | |
501 | addition, subtraction, multiplication, and division. Remainder and modulus | |
502 | functions supplement the division functions. The functions to | |
503 | add or subtract 1 are provided because they are traditional in Lisp and | |
504 | commonly used. | |
505 | ||
506 | All of these functions except @code{%} return a floating point value | |
507 | if any argument is floating. | |
508 | ||
509 | It is important to note that in Emacs Lisp, arithmetic functions | |
510 | do not check for overflow. Thus @code{(1+ 268435455)} may evaluate to | |
511 | @minus{}268435456, depending on your hardware. | |
512 | ||
513 | @defun 1+ number-or-marker | |
514 | This function returns @var{number-or-marker} plus 1. | |
515 | For example, | |
516 | ||
517 | @example | |
518 | (setq foo 4) | |
519 | @result{} 4 | |
520 | (1+ foo) | |
521 | @result{} 5 | |
522 | @end example | |
523 | ||
524 | This function is not analogous to the C operator @code{++}---it does not | |
525 | increment a variable. It just computes a sum. Thus, if we continue, | |
526 | ||
527 | @example | |
528 | foo | |
529 | @result{} 4 | |
530 | @end example | |
531 | ||
532 | If you want to increment the variable, you must use @code{setq}, | |
533 | like this: | |
534 | ||
535 | @example | |
536 | (setq foo (1+ foo)) | |
537 | @result{} 5 | |
538 | @end example | |
539 | @end defun | |
540 | ||
541 | @defun 1- number-or-marker | |
542 | This function returns @var{number-or-marker} minus 1. | |
543 | @end defun | |
544 | ||
545 | @defun + &rest numbers-or-markers | |
546 | This function adds its arguments together. When given no arguments, | |
547 | @code{+} returns 0. | |
548 | ||
549 | @example | |
550 | (+) | |
551 | @result{} 0 | |
552 | (+ 1) | |
553 | @result{} 1 | |
554 | (+ 1 2 3 4) | |
555 | @result{} 10 | |
556 | @end example | |
557 | @end defun | |
558 | ||
559 | @defun - &optional number-or-marker &rest more-numbers-or-markers | |
560 | The @code{-} function serves two purposes: negation and subtraction. | |
561 | When @code{-} has a single argument, the value is the negative of the | |
562 | argument. When there are multiple arguments, @code{-} subtracts each of | |
563 | the @var{more-numbers-or-markers} from @var{number-or-marker}, | |
564 | cumulatively. If there are no arguments, the result is 0. | |
565 | ||
566 | @example | |
567 | (- 10 1 2 3 4) | |
568 | @result{} 0 | |
569 | (- 10) | |
570 | @result{} -10 | |
571 | (-) | |
572 | @result{} 0 | |
573 | @end example | |
574 | @end defun | |
575 | ||
576 | @defun * &rest numbers-or-markers | |
577 | This function multiplies its arguments together, and returns the | |
578 | product. When given no arguments, @code{*} returns 1. | |
579 | ||
580 | @example | |
581 | (*) | |
582 | @result{} 1 | |
583 | (* 1) | |
584 | @result{} 1 | |
585 | (* 1 2 3 4) | |
586 | @result{} 24 | |
587 | @end example | |
588 | @end defun | |
589 | ||
590 | @defun / dividend divisor &rest divisors | |
591 | This function divides @var{dividend} by @var{divisor} and returns the | |
592 | quotient. If there are additional arguments @var{divisors}, then it | |
593 | divides @var{dividend} by each divisor in turn. Each argument may be a | |
594 | number or a marker. | |
595 | ||
596 | If all the arguments are integers, then the result is an integer too. | |
597 | This means the result has to be rounded. On most machines, the result | |
598 | is rounded towards zero after each division, but some machines may round | |
599 | differently with negative arguments. This is because the Lisp function | |
600 | @code{/} is implemented using the C division operator, which also | |
601 | permits machine-dependent rounding. As a practical matter, all known | |
602 | machines round in the standard fashion. | |
603 | ||
604 | @cindex @code{arith-error} in division | |
605 | If you divide an integer by 0, an @code{arith-error} error is signaled. | |
606 | (@xref{Errors}.) Floating point division by zero returns either | |
607 | infinity or a NaN if your machine supports @acronym{IEEE} floating point; | |
608 | otherwise, it signals an @code{arith-error} error. | |
609 | ||
610 | @example | |
611 | @group | |
612 | (/ 6 2) | |
613 | @result{} 3 | |
614 | @end group | |
615 | (/ 5 2) | |
616 | @result{} 2 | |
617 | (/ 5.0 2) | |
618 | @result{} 2.5 | |
619 | (/ 5 2.0) | |
620 | @result{} 2.5 | |
621 | (/ 5.0 2.0) | |
622 | @result{} 2.5 | |
623 | (/ 25 3 2) | |
624 | @result{} 4 | |
625 | @group | |
626 | (/ -17 6) | |
627 | @result{} -2 @r{(could in theory be @minus{}3 on some machines)} | |
628 | @end group | |
629 | @end example | |
630 | @end defun | |
631 | ||
632 | @defun % dividend divisor | |
633 | @cindex remainder | |
634 | This function returns the integer remainder after division of @var{dividend} | |
635 | by @var{divisor}. The arguments must be integers or markers. | |
636 | ||
637 | For negative arguments, the remainder is in principle machine-dependent | |
638 | since the quotient is; but in practice, all known machines behave alike. | |
639 | ||
640 | An @code{arith-error} results if @var{divisor} is 0. | |
641 | ||
642 | @example | |
643 | (% 9 4) | |
644 | @result{} 1 | |
645 | (% -9 4) | |
646 | @result{} -1 | |
647 | (% 9 -4) | |
648 | @result{} 1 | |
649 | (% -9 -4) | |
650 | @result{} -1 | |
651 | @end example | |
652 | ||
653 | For any two integers @var{dividend} and @var{divisor}, | |
654 | ||
655 | @example | |
656 | @group | |
657 | (+ (% @var{dividend} @var{divisor}) | |
658 | (* (/ @var{dividend} @var{divisor}) @var{divisor})) | |
659 | @end group | |
660 | @end example | |
661 | ||
662 | @noindent | |
663 | always equals @var{dividend}. | |
664 | @end defun | |
665 | ||
666 | @defun mod dividend divisor | |
667 | @cindex modulus | |
668 | This function returns the value of @var{dividend} modulo @var{divisor}; | |
669 | in other words, the remainder after division of @var{dividend} | |
670 | by @var{divisor}, but with the same sign as @var{divisor}. | |
671 | The arguments must be numbers or markers. | |
672 | ||
673 | Unlike @code{%}, @code{mod} returns a well-defined result for negative | |
674 | arguments. It also permits floating point arguments; it rounds the | |
675 | quotient downward (towards minus infinity) to an integer, and uses that | |
676 | quotient to compute the remainder. | |
677 | ||
678 | An @code{arith-error} results if @var{divisor} is 0. | |
679 | ||
680 | @example | |
681 | @group | |
682 | (mod 9 4) | |
683 | @result{} 1 | |
684 | @end group | |
685 | @group | |
686 | (mod -9 4) | |
687 | @result{} 3 | |
688 | @end group | |
689 | @group | |
690 | (mod 9 -4) | |
691 | @result{} -3 | |
692 | @end group | |
693 | @group | |
694 | (mod -9 -4) | |
695 | @result{} -1 | |
696 | @end group | |
697 | @group | |
698 | (mod 5.5 2.5) | |
699 | @result{} .5 | |
700 | @end group | |
701 | @end example | |
702 | ||
703 | For any two numbers @var{dividend} and @var{divisor}, | |
704 | ||
705 | @example | |
706 | @group | |
707 | (+ (mod @var{dividend} @var{divisor}) | |
708 | (* (floor @var{dividend} @var{divisor}) @var{divisor})) | |
709 | @end group | |
710 | @end example | |
711 | ||
712 | @noindent | |
713 | always equals @var{dividend}, subject to rounding error if either | |
714 | argument is floating point. For @code{floor}, see @ref{Numeric | |
715 | Conversions}. | |
716 | @end defun | |
717 | ||
718 | @node Rounding Operations | |
719 | @section Rounding Operations | |
720 | @cindex rounding without conversion | |
721 | ||
722 | The functions @code{ffloor}, @code{fceiling}, @code{fround}, and | |
723 | @code{ftruncate} take a floating point argument and return a floating | |
724 | point result whose value is a nearby integer. @code{ffloor} returns the | |
725 | nearest integer below; @code{fceiling}, the nearest integer above; | |
726 | @code{ftruncate}, the nearest integer in the direction towards zero; | |
727 | @code{fround}, the nearest integer. | |
728 | ||
729 | @defun ffloor float | |
730 | This function rounds @var{float} to the next lower integral value, and | |
731 | returns that value as a floating point number. | |
732 | @end defun | |
733 | ||
734 | @defun fceiling float | |
735 | This function rounds @var{float} to the next higher integral value, and | |
736 | returns that value as a floating point number. | |
737 | @end defun | |
738 | ||
739 | @defun ftruncate float | |
740 | This function rounds @var{float} towards zero to an integral value, and | |
741 | returns that value as a floating point number. | |
742 | @end defun | |
743 | ||
744 | @defun fround float | |
745 | This function rounds @var{float} to the nearest integral value, | |
746 | and returns that value as a floating point number. | |
747 | @end defun | |
748 | ||
749 | @node Bitwise Operations | |
750 | @section Bitwise Operations on Integers | |
751 | @cindex bitwise arithmetic | |
752 | @cindex logical arithmetic | |
753 | ||
754 | In a computer, an integer is represented as a binary number, a | |
755 | sequence of @dfn{bits} (digits which are either zero or one). A bitwise | |
756 | operation acts on the individual bits of such a sequence. For example, | |
757 | @dfn{shifting} moves the whole sequence left or right one or more places, | |
758 | reproducing the same pattern ``moved over.'' | |
759 | ||
760 | The bitwise operations in Emacs Lisp apply only to integers. | |
761 | ||
762 | @defun lsh integer1 count | |
763 | @cindex logical shift | |
764 | @code{lsh}, which is an abbreviation for @dfn{logical shift}, shifts the | |
765 | bits in @var{integer1} to the left @var{count} places, or to the right | |
766 | if @var{count} is negative, bringing zeros into the vacated bits. If | |
767 | @var{count} is negative, @code{lsh} shifts zeros into the leftmost | |
768 | (most-significant) bit, producing a positive result even if | |
769 | @var{integer1} is negative. Contrast this with @code{ash}, below. | |
770 | ||
771 | Here are two examples of @code{lsh}, shifting a pattern of bits one | |
772 | place to the left. We show only the low-order eight bits of the binary | |
773 | pattern; the rest are all zero. | |
774 | ||
775 | @example | |
776 | @group | |
777 | (lsh 5 1) | |
778 | @result{} 10 | |
779 | ;; @r{Decimal 5 becomes decimal 10.} | |
780 | 00000101 @result{} 00001010 | |
781 | ||
782 | (lsh 7 1) | |
783 | @result{} 14 | |
784 | ;; @r{Decimal 7 becomes decimal 14.} | |
785 | 00000111 @result{} 00001110 | |
786 | @end group | |
787 | @end example | |
788 | ||
789 | @noindent | |
790 | As the examples illustrate, shifting the pattern of bits one place to | |
791 | the left produces a number that is twice the value of the previous | |
792 | number. | |
793 | ||
794 | Shifting a pattern of bits two places to the left produces results | |
795 | like this (with 8-bit binary numbers): | |
796 | ||
797 | @example | |
798 | @group | |
799 | (lsh 3 2) | |
800 | @result{} 12 | |
801 | ;; @r{Decimal 3 becomes decimal 12.} | |
802 | 00000011 @result{} 00001100 | |
803 | @end group | |
804 | @end example | |
805 | ||
806 | On the other hand, shifting one place to the right looks like this: | |
807 | ||
808 | @example | |
809 | @group | |
810 | (lsh 6 -1) | |
811 | @result{} 3 | |
812 | ;; @r{Decimal 6 becomes decimal 3.} | |
813 | 00000110 @result{} 00000011 | |
814 | @end group | |
815 | ||
816 | @group | |
817 | (lsh 5 -1) | |
818 | @result{} 2 | |
819 | ;; @r{Decimal 5 becomes decimal 2.} | |
820 | 00000101 @result{} 00000010 | |
821 | @end group | |
822 | @end example | |
823 | ||
824 | @noindent | |
825 | As the example illustrates, shifting one place to the right divides the | |
826 | value of a positive integer by two, rounding downward. | |
827 | ||
828 | The function @code{lsh}, like all Emacs Lisp arithmetic functions, does | |
829 | not check for overflow, so shifting left can discard significant bits | |
830 | and change the sign of the number. For example, left shifting | |
1ddd6622 | 831 | 536,870,911 produces @minus{}2 on a 30-bit machine: |
b8d4c8d0 GM |
832 | |
833 | @example | |
1ddd6622 | 834 | (lsh 536870911 1) ; @r{left shift} |
b8d4c8d0 GM |
835 | @result{} -2 |
836 | @end example | |
837 | ||
1ddd6622 | 838 | In binary, in the 30-bit implementation, the argument looks like this: |
b8d4c8d0 GM |
839 | |
840 | @example | |
841 | @group | |
1ddd6622 GM |
842 | ;; @r{Decimal 536,870,911} |
843 | 01 1111 1111 1111 1111 1111 1111 1111 | |
b8d4c8d0 GM |
844 | @end group |
845 | @end example | |
846 | ||
847 | @noindent | |
848 | which becomes the following when left shifted: | |
849 | ||
850 | @example | |
851 | @group | |
852 | ;; @r{Decimal @minus{}2} | |
1ddd6622 | 853 | 11 1111 1111 1111 1111 1111 1111 1110 |
b8d4c8d0 GM |
854 | @end group |
855 | @end example | |
856 | @end defun | |
857 | ||
858 | @defun ash integer1 count | |
859 | @cindex arithmetic shift | |
860 | @code{ash} (@dfn{arithmetic shift}) shifts the bits in @var{integer1} | |
861 | to the left @var{count} places, or to the right if @var{count} | |
862 | is negative. | |
863 | ||
864 | @code{ash} gives the same results as @code{lsh} except when | |
865 | @var{integer1} and @var{count} are both negative. In that case, | |
866 | @code{ash} puts ones in the empty bit positions on the left, while | |
867 | @code{lsh} puts zeros in those bit positions. | |
868 | ||
869 | Thus, with @code{ash}, shifting the pattern of bits one place to the right | |
870 | looks like this: | |
871 | ||
872 | @example | |
873 | @group | |
874 | (ash -6 -1) @result{} -3 | |
875 | ;; @r{Decimal @minus{}6 becomes decimal @minus{}3.} | |
1ddd6622 | 876 | 11 1111 1111 1111 1111 1111 1111 1010 |
b8d4c8d0 | 877 | @result{} |
1ddd6622 | 878 | 11 1111 1111 1111 1111 1111 1111 1101 |
b8d4c8d0 GM |
879 | @end group |
880 | @end example | |
881 | ||
882 | In contrast, shifting the pattern of bits one place to the right with | |
883 | @code{lsh} looks like this: | |
884 | ||
885 | @example | |
886 | @group | |
1ddd6622 GM |
887 | (lsh -6 -1) @result{} 536870909 |
888 | ;; @r{Decimal @minus{}6 becomes decimal 536,870,909.} | |
889 | 11 1111 1111 1111 1111 1111 1111 1010 | |
b8d4c8d0 | 890 | @result{} |
1ddd6622 | 891 | 01 1111 1111 1111 1111 1111 1111 1101 |
b8d4c8d0 GM |
892 | @end group |
893 | @end example | |
894 | ||
895 | Here are other examples: | |
896 | ||
897 | @c !!! Check if lined up in smallbook format! XDVI shows problem | |
898 | @c with smallbook but not with regular book! --rjc 16mar92 | |
899 | @smallexample | |
900 | @group | |
1ddd6622 | 901 | ; @r{ 30-bit binary values} |
b8d4c8d0 | 902 | |
1ddd6622 GM |
903 | (lsh 5 2) ; 5 = @r{00 0000 0000 0000 0000 0000 0000 0101} |
904 | @result{} 20 ; = @r{00 0000 0000 0000 0000 0000 0001 0100} | |
b8d4c8d0 GM |
905 | @end group |
906 | @group | |
907 | (ash 5 2) | |
908 | @result{} 20 | |
1ddd6622 GM |
909 | (lsh -5 2) ; -5 = @r{11 1111 1111 1111 1111 1111 1111 1011} |
910 | @result{} -20 ; = @r{11 1111 1111 1111 1111 1111 1110 1100} | |
b8d4c8d0 GM |
911 | (ash -5 2) |
912 | @result{} -20 | |
913 | @end group | |
914 | @group | |
1ddd6622 GM |
915 | (lsh 5 -2) ; 5 = @r{00 0000 0000 0000 0000 0000 0000 0101} |
916 | @result{} 1 ; = @r{00 0000 0000 0000 0000 0000 0000 0001} | |
b8d4c8d0 GM |
917 | @end group |
918 | @group | |
919 | (ash 5 -2) | |
920 | @result{} 1 | |
921 | @end group | |
922 | @group | |
1ddd6622 GM |
923 | (lsh -5 -2) ; -5 = @r{11 1111 1111 1111 1111 1111 1111 1011} |
924 | @result{} 268435454 ; = @r{00 0111 1111 1111 1111 1111 1111 1110} | |
b8d4c8d0 GM |
925 | @end group |
926 | @group | |
1ddd6622 GM |
927 | (ash -5 -2) ; -5 = @r{11 1111 1111 1111 1111 1111 1111 1011} |
928 | @result{} -2 ; = @r{11 1111 1111 1111 1111 1111 1111 1110} | |
b8d4c8d0 GM |
929 | @end group |
930 | @end smallexample | |
931 | @end defun | |
932 | ||
933 | @defun logand &rest ints-or-markers | |
934 | This function returns the ``logical and'' of the arguments: the | |
935 | @var{n}th bit is set in the result if, and only if, the @var{n}th bit is | |
936 | set in all the arguments. (``Set'' means that the value of the bit is 1 | |
937 | rather than 0.) | |
938 | ||
939 | For example, using 4-bit binary numbers, the ``logical and'' of 13 and | |
940 | 12 is 12: 1101 combined with 1100 produces 1100. | |
941 | In both the binary numbers, the leftmost two bits are set (i.e., they | |
942 | are 1's), so the leftmost two bits of the returned value are set. | |
943 | However, for the rightmost two bits, each is zero in at least one of | |
944 | the arguments, so the rightmost two bits of the returned value are 0's. | |
945 | ||
946 | @noindent | |
947 | Therefore, | |
948 | ||
949 | @example | |
950 | @group | |
951 | (logand 13 12) | |
952 | @result{} 12 | |
953 | @end group | |
954 | @end example | |
955 | ||
956 | If @code{logand} is not passed any argument, it returns a value of | |
957 | @minus{}1. This number is an identity element for @code{logand} | |
958 | because its binary representation consists entirely of ones. If | |
959 | @code{logand} is passed just one argument, it returns that argument. | |
960 | ||
961 | @smallexample | |
962 | @group | |
1ddd6622 | 963 | ; @r{ 30-bit binary values} |
b8d4c8d0 | 964 | |
1ddd6622 GM |
965 | (logand 14 13) ; 14 = @r{00 0000 0000 0000 0000 0000 0000 1110} |
966 | ; 13 = @r{00 0000 0000 0000 0000 0000 0000 1101} | |
967 | @result{} 12 ; 12 = @r{00 0000 0000 0000 0000 0000 0000 1100} | |
b8d4c8d0 GM |
968 | @end group |
969 | ||
970 | @group | |
1ddd6622 GM |
971 | (logand 14 13 4) ; 14 = @r{00 0000 0000 0000 0000 0000 0000 1110} |
972 | ; 13 = @r{00 0000 0000 0000 0000 0000 0000 1101} | |
973 | ; 4 = @r{00 0000 0000 0000 0000 0000 0000 0100} | |
974 | @result{} 4 ; 4 = @r{00 0000 0000 0000 0000 0000 0000 0100} | |
b8d4c8d0 GM |
975 | @end group |
976 | ||
977 | @group | |
978 | (logand) | |
1ddd6622 | 979 | @result{} -1 ; -1 = @r{11 1111 1111 1111 1111 1111 1111 1111} |
b8d4c8d0 GM |
980 | @end group |
981 | @end smallexample | |
982 | @end defun | |
983 | ||
984 | @defun logior &rest ints-or-markers | |
985 | This function returns the ``inclusive or'' of its arguments: the @var{n}th bit | |
986 | is set in the result if, and only if, the @var{n}th bit is set in at least | |
987 | one of the arguments. If there are no arguments, the result is zero, | |
988 | which is an identity element for this operation. If @code{logior} is | |
989 | passed just one argument, it returns that argument. | |
990 | ||
991 | @smallexample | |
992 | @group | |
1ddd6622 | 993 | ; @r{ 30-bit binary values} |
b8d4c8d0 | 994 | |
1ddd6622 GM |
995 | (logior 12 5) ; 12 = @r{00 0000 0000 0000 0000 0000 0000 1100} |
996 | ; 5 = @r{00 0000 0000 0000 0000 0000 0000 0101} | |
997 | @result{} 13 ; 13 = @r{00 0000 0000 0000 0000 0000 0000 1101} | |
b8d4c8d0 GM |
998 | @end group |
999 | ||
1000 | @group | |
1ddd6622 GM |
1001 | (logior 12 5 7) ; 12 = @r{00 0000 0000 0000 0000 0000 0000 1100} |
1002 | ; 5 = @r{00 0000 0000 0000 0000 0000 0000 0101} | |
1003 | ; 7 = @r{00 0000 0000 0000 0000 0000 0000 0111} | |
1004 | @result{} 15 ; 15 = @r{00 0000 0000 0000 0000 0000 0000 1111} | |
b8d4c8d0 GM |
1005 | @end group |
1006 | @end smallexample | |
1007 | @end defun | |
1008 | ||
1009 | @defun logxor &rest ints-or-markers | |
1010 | This function returns the ``exclusive or'' of its arguments: the | |
1011 | @var{n}th bit is set in the result if, and only if, the @var{n}th bit is | |
1012 | set in an odd number of the arguments. If there are no arguments, the | |
1013 | result is 0, which is an identity element for this operation. If | |
1014 | @code{logxor} is passed just one argument, it returns that argument. | |
1015 | ||
1016 | @smallexample | |
1017 | @group | |
1ddd6622 | 1018 | ; @r{ 30-bit binary values} |
b8d4c8d0 | 1019 | |
1ddd6622 GM |
1020 | (logxor 12 5) ; 12 = @r{00 0000 0000 0000 0000 0000 0000 1100} |
1021 | ; 5 = @r{00 0000 0000 0000 0000 0000 0000 0101} | |
1022 | @result{} 9 ; 9 = @r{00 0000 0000 0000 0000 0000 0000 1001} | |
b8d4c8d0 GM |
1023 | @end group |
1024 | ||
1025 | @group | |
1ddd6622 GM |
1026 | (logxor 12 5 7) ; 12 = @r{00 0000 0000 0000 0000 0000 0000 1100} |
1027 | ; 5 = @r{00 0000 0000 0000 0000 0000 0000 0101} | |
1028 | ; 7 = @r{00 0000 0000 0000 0000 0000 0000 0111} | |
1029 | @result{} 14 ; 14 = @r{00 0000 0000 0000 0000 0000 0000 1110} | |
b8d4c8d0 GM |
1030 | @end group |
1031 | @end smallexample | |
1032 | @end defun | |
1033 | ||
1034 | @defun lognot integer | |
1035 | This function returns the logical complement of its argument: the @var{n}th | |
1036 | bit is one in the result if, and only if, the @var{n}th bit is zero in | |
1037 | @var{integer}, and vice-versa. | |
1038 | ||
1039 | @example | |
1040 | (lognot 5) | |
1041 | @result{} -6 | |
1ddd6622 | 1042 | ;; 5 = @r{00 0000 0000 0000 0000 0000 0000 0101} |
b8d4c8d0 | 1043 | ;; @r{becomes} |
1ddd6622 | 1044 | ;; -6 = @r{11 1111 1111 1111 1111 1111 1111 1010} |
b8d4c8d0 GM |
1045 | @end example |
1046 | @end defun | |
1047 | ||
1048 | @node Math Functions | |
1049 | @section Standard Mathematical Functions | |
1050 | @cindex transcendental functions | |
1051 | @cindex mathematical functions | |
1052 | @cindex floating-point functions | |
1053 | ||
1054 | These mathematical functions allow integers as well as floating point | |
1055 | numbers as arguments. | |
1056 | ||
1057 | @defun sin arg | |
1058 | @defunx cos arg | |
1059 | @defunx tan arg | |
1060 | These are the ordinary trigonometric functions, with argument measured | |
1061 | in radians. | |
1062 | @end defun | |
1063 | ||
1064 | @defun asin arg | |
1065 | The value of @code{(asin @var{arg})} is a number between | |
1066 | @ifnottex | |
1067 | @minus{}pi/2 | |
1068 | @end ifnottex | |
1069 | @tex | |
1070 | @math{-\pi/2} | |
1071 | @end tex | |
1072 | and | |
1073 | @ifnottex | |
1074 | pi/2 | |
1075 | @end ifnottex | |
1076 | @tex | |
1077 | @math{\pi/2} | |
1078 | @end tex | |
1079 | (inclusive) whose sine is @var{arg}; if, however, @var{arg} is out of | |
1080 | range (outside [@minus{}1, 1]), it signals a @code{domain-error} error. | |
1081 | @end defun | |
1082 | ||
1083 | @defun acos arg | |
1084 | The value of @code{(acos @var{arg})} is a number between 0 and | |
1085 | @ifnottex | |
1086 | pi | |
1087 | @end ifnottex | |
1088 | @tex | |
1089 | @math{\pi} | |
1090 | @end tex | |
1091 | (inclusive) whose cosine is @var{arg}; if, however, @var{arg} is out | |
1092 | of range (outside [@minus{}1, 1]), it signals a @code{domain-error} error. | |
1093 | @end defun | |
1094 | ||
1095 | @defun atan y &optional x | |
1096 | The value of @code{(atan @var{y})} 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 | |
1110 | (exclusive) whose tangent is @var{y}. If the optional second | |
1111 | argument @var{x} is given, the value of @code{(atan y x)} is the | |
1112 | angle in radians between the vector @code{[@var{x}, @var{y}]} and the | |
1113 | @code{X} axis. | |
1114 | @end defun | |
1115 | ||
1116 | @defun exp arg | |
1117 | This is the exponential function; it returns | |
1118 | @tex | |
1119 | @math{e} | |
1120 | @end tex | |
1121 | @ifnottex | |
1122 | @i{e} | |
1123 | @end ifnottex | |
1124 | to the power @var{arg}. | |
1125 | @tex | |
1126 | @math{e} | |
1127 | @end tex | |
1128 | @ifnottex | |
1129 | @i{e} | |
1130 | @end ifnottex | |
1131 | is a fundamental mathematical constant also called the base of natural | |
1132 | logarithms. | |
1133 | @end defun | |
1134 | ||
1135 | @defun log arg &optional base | |
1136 | This function returns the logarithm of @var{arg}, with base @var{base}. | |
1137 | If you don't specify @var{base}, the base | |
1138 | @tex | |
1139 | @math{e} | |
1140 | @end tex | |
1141 | @ifnottex | |
1142 | @i{e} | |
1143 | @end ifnottex | |
1144 | is used. If @var{arg} is negative, it signals a @code{domain-error} | |
1145 | error. | |
1146 | @end defun | |
1147 | ||
1148 | @ignore | |
1149 | @defun expm1 arg | |
1150 | This function returns @code{(1- (exp @var{arg}))}, but it is more | |
1151 | accurate than that when @var{arg} is negative and @code{(exp @var{arg})} | |
1152 | is close to 1. | |
1153 | @end defun | |
1154 | ||
1155 | @defun log1p arg | |
1156 | This function returns @code{(log (1+ @var{arg}))}, but it is more | |
1157 | accurate than that when @var{arg} is so small that adding 1 to it would | |
1158 | lose accuracy. | |
1159 | @end defun | |
1160 | @end ignore | |
1161 | ||
1162 | @defun log10 arg | |
1163 | This function returns the logarithm of @var{arg}, with base 10. If | |
1164 | @var{arg} is negative, it signals a @code{domain-error} error. | |
1165 | @code{(log10 @var{x})} @equiv{} @code{(log @var{x} 10)}, at least | |
1166 | approximately. | |
1167 | @end defun | |
1168 | ||
1169 | @defun expt x y | |
1170 | This function returns @var{x} raised to power @var{y}. If both | |
1171 | arguments are integers and @var{y} is positive, the result is an | |
1172 | integer; in this case, overflow causes truncation, so watch out. | |
1173 | @end defun | |
1174 | ||
1175 | @defun sqrt arg | |
1176 | This returns the square root of @var{arg}. If @var{arg} is negative, | |
1177 | it signals a @code{domain-error} error. | |
1178 | @end defun | |
1179 | ||
1180 | @node Random Numbers | |
1181 | @section Random Numbers | |
1182 | @cindex random numbers | |
1183 | ||
1184 | A deterministic computer program cannot generate true random numbers. | |
1185 | For most purposes, @dfn{pseudo-random numbers} suffice. A series of | |
1186 | pseudo-random numbers is generated in a deterministic fashion. The | |
1187 | numbers are not truly random, but they have certain properties that | |
1188 | mimic a random series. For example, all possible values occur equally | |
1189 | often in a pseudo-random series. | |
1190 | ||
1191 | In Emacs, pseudo-random numbers are generated from a ``seed'' number. | |
1192 | Starting from any given seed, the @code{random} function always | |
1193 | generates the same sequence of numbers. Emacs always starts with the | |
1194 | same seed value, so the sequence of values of @code{random} is actually | |
1195 | the same in each Emacs run! For example, in one operating system, the | |
1196 | first call to @code{(random)} after you start Emacs always returns | |
1197 | @minus{}1457731, and the second one always returns @minus{}7692030. This | |
1198 | repeatability is helpful for debugging. | |
1199 | ||
1200 | If you want random numbers that don't always come out the same, execute | |
1201 | @code{(random t)}. This chooses a new seed based on the current time of | |
1202 | day and on Emacs's process @acronym{ID} number. | |
1203 | ||
1204 | @defun random &optional limit | |
1205 | This function returns a pseudo-random integer. Repeated calls return a | |
1206 | series of pseudo-random integers. | |
1207 | ||
1208 | If @var{limit} is a positive integer, the value is chosen to be | |
1209 | nonnegative and less than @var{limit}. | |
1210 | ||
1211 | If @var{limit} is @code{t}, it means to choose a new seed based on the | |
1212 | current time of day and on Emacs's process @acronym{ID} number. | |
1213 | @c "Emacs'" is incorrect usage! | |
1214 | ||
1215 | On some machines, any integer representable in Lisp may be the result | |
1216 | of @code{random}. On other machines, the result can never be larger | |
1217 | than a certain maximum or less than a certain (negative) minimum. | |
1218 | @end defun |