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1 | @c -*-texinfo-*- |
2 | @c This is part of the GNU Emacs Lisp Reference Manual. | |
e1e44180 | 3 | @c Copyright (C) 1990, 1991, 1992, 1993, 1994, 1995, 1998, 1999, 2003 |
177c0ea7 | 4 | @c Free Software Foundation, Inc. |
e6512bcf RS |
5 | @c See the file elisp.texi for copying conditions. |
6 | @setfilename ../info/numbers | |
05fd2b65 | 7 | @node Numbers, Strings and Characters, Lisp Data Types, Top |
e6512bcf RS |
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 | |
969fe9b5 RS |
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 | |
e6512bcf RS |
19 | exact; they have a fixed, limited amount of precision. |
20 | ||
e6512bcf RS |
21 | @menu |
22 | * Integer Basics:: Representation and range of integers. | |
23 | * Float Basics:: Representation and range of floating point. | |
24 | * Predicates on Numbers:: Testing for numbers. | |
25 | * Comparison of Numbers:: Equality and inequality predicates. | |
26 | * Numeric Conversions:: Converting float to integer and vice versa. | |
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. | |
4ac771a0 | 30 | * Math Functions:: Trig, exponential and logarithmic functions. |
e6512bcf RS |
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 | |
e1e44180 | 39 | minimum range is @minus{}268435456 to 268435455 (29 bits; i.e., |
37680279 | 40 | @ifnottex |
e1e44180 | 41 | -2**28 |
37680279 | 42 | @end ifnottex |
177c0ea7 | 43 | @tex |
e1e44180 | 44 | @math{-2^{28}} |
e6512bcf | 45 | @end tex |
177c0ea7 | 46 | to |
37680279 | 47 | @ifnottex |
e1e44180 | 48 | 2**28 - 1), |
37680279 | 49 | @end ifnottex |
177c0ea7 | 50 | @tex |
e1e44180 | 51 | @math{2^{28}-1}), |
e6512bcf | 52 | @end tex |
94b00316 | 53 | but some machines may provide a wider range. Many examples in this |
e1e44180 | 54 | chapter assume an integer has 29 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.} | |
b98ace27 | 65 | 536870913 ; @r{Also the integer 1, due to overflow.} |
e6512bcf RS |
66 | 0 ; @r{The integer 0.} |
67 | -0 ; @r{The integer 0.} | |
68 | @end example | |
69 | ||
d9940cbf EZ |
70 | @cindex integers in specific radix |
71 | @cindex radix for reading an integer | |
72 | @cindex base for reading an integer | |
75c65b20 EZ |
73 | @cindex hex numbers |
74 | @cindex octal numbers | |
75 | @cindex reading numbers in hex, octal, and binary | |
d9940cbf EZ |
76 | In addition, the Lisp reader recognizes a syntax for integers in |
77 | bases other than 10: @samp{#B@var{integer}} reads @var{integer} in | |
78 | binary (radix 2), @samp{#O@var{integer}} reads @var{integer} in octal | |
79 | (radix 8), @samp{#X@var{integer}} reads @var{integer} in hexadecimal | |
80 | (radix 16), and @samp{#@var{radix}r@var{integer}} reads @var{integer} | |
81 | in radix @var{radix} (where @var{radix} is between 2 and 36, | |
5fe3b9bc | 82 | inclusively). Case is not significant for the letter after @samp{#} |
d9940cbf EZ |
83 | (@samp{B}, @samp{O}, etc.) that denotes the radix. |
84 | ||
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85 | To understand how various functions work on integers, especially the |
86 | bitwise operators (@pxref{Bitwise Operations}), it is often helpful to | |
87 | view the numbers in their binary form. | |
88 | ||
e1e44180 | 89 | In 29-bit binary, the decimal integer 5 looks like this: |
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90 | |
91 | @example | |
e1e44180 | 92 | 0 0000 0000 0000 0000 0000 0000 0101 |
e6512bcf RS |
93 | @end example |
94 | ||
95 | @noindent | |
96 | (We have inserted spaces between groups of 4 bits, and two spaces | |
97 | between groups of 8 bits, to make the binary integer easier to read.) | |
98 | ||
99 | The integer @minus{}1 looks like this: | |
100 | ||
101 | @example | |
e1e44180 | 102 | 1 1111 1111 1111 1111 1111 1111 1111 |
e6512bcf RS |
103 | @end example |
104 | ||
105 | @noindent | |
106 | @cindex two's complement | |
e1e44180 | 107 | @minus{}1 is represented as 29 ones. (This is called @dfn{two's |
e6512bcf RS |
108 | complement} notation.) |
109 | ||
110 | The negative integer, @minus{}5, is creating by subtracting 4 from | |
111 | @minus{}1. In binary, the decimal integer 4 is 100. Consequently, | |
112 | @minus{}5 looks like this: | |
113 | ||
114 | @example | |
e1e44180 | 115 | 1 1111 1111 1111 1111 1111 1111 1011 |
e6512bcf RS |
116 | @end example |
117 | ||
e1e44180 SM |
118 | In this implementation, the largest 29-bit binary integer value is |
119 | 268,435,455 in decimal. In binary, it looks like this: | |
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120 | |
121 | @example | |
e1e44180 | 122 | 0 1111 1111 1111 1111 1111 1111 1111 |
e6512bcf RS |
123 | @end example |
124 | ||
125 | Since the arithmetic functions do not check whether integers go | |
e1e44180 SM |
126 | outside their range, when you add 1 to 268,435,455, the value is the |
127 | negative integer @minus{}268,435,456: | |
e6512bcf RS |
128 | |
129 | @example | |
e1e44180 SM |
130 | (+ 1 268435455) |
131 | @result{} -268435456 | |
132 | @result{} 1 0000 0000 0000 0000 0000 0000 0000 | |
e6512bcf RS |
133 | @end example |
134 | ||
f9f59935 RS |
135 | Many of the functions described in this chapter accept markers for |
136 | arguments in place of numbers. (@xref{Markers}.) Since the actual | |
137 | arguments to such functions may be either numbers or markers, we often | |
138 | give these arguments the name @var{number-or-marker}. When the argument | |
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139 | value is a marker, its position value is used and its buffer is ignored. |
140 | ||
24fc2020 RS |
141 | @defvar most-positive-fixnum |
142 | The value of this variable is the largest integer that Emacs Lisp | |
143 | can handle. | |
144 | @end defvar | |
145 | ||
146 | @defvar most-negative-fixnum | |
147 | The value of this variable is the smallest integer that Emacs Lisp can | |
148 | handle. It is negative. | |
149 | @end defvar | |
150 | ||
e6512bcf RS |
151 | @node Float Basics |
152 | @section Floating Point Basics | |
153 | ||
f9f59935 RS |
154 | Floating point numbers are useful for representing numbers that are |
155 | not integral. The precise range of floating point numbers is | |
156 | machine-specific; it is the same as the range of the C data type | |
157 | @code{double} on the machine you are using. | |
e6512bcf | 158 | |
f9f59935 RS |
159 | The read-syntax for floating point numbers requires either a decimal |
160 | point (with at least one digit following), an exponent, or both. For | |
161 | example, @samp{1500.0}, @samp{15e2}, @samp{15.0e2}, @samp{1.5e3}, and | |
162 | @samp{.15e4} are five ways of writing a floating point number whose | |
163 | value is 1500. They are all equivalent. You can also use a minus sign | |
164 | to write negative floating point numbers, as in @samp{-1.0}. | |
e6512bcf | 165 | |
ad800164 | 166 | @cindex @acronym{IEEE} floating point |
e6512bcf RS |
167 | @cindex positive infinity |
168 | @cindex negative infinity | |
169 | @cindex infinity | |
170 | @cindex NaN | |
fa091c04 LT |
171 | Most modern computers support the @acronym{IEEE} floating point standard, |
172 | which provides for positive infinity and negative infinity as floating point | |
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173 | values. It also provides for a class of values called NaN or |
174 | ``not-a-number''; numerical functions return such values in cases where | |
903a1a91 | 175 | there is no correct answer. For example, @code{(/ 0.0 0.0)} returns a |
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176 | NaN. For practical purposes, there's no significant difference between |
177 | different NaN values in Emacs Lisp, and there's no rule for precisely | |
969fe9b5 | 178 | which NaN value should be used in a particular case, so Emacs Lisp |
f9f59935 | 179 | doesn't try to distinguish them. Here are the read syntaxes for |
969fe9b5 | 180 | these special floating point values: |
f9f59935 RS |
181 | |
182 | @table @asis | |
183 | @item positive infinity | |
184 | @samp{1.0e+INF} | |
185 | @item negative infinity | |
186 | @samp{-1.0e+INF} | |
187 | @item Not-a-number | |
188 | @samp{0.0e+NaN}. | |
189 | @end table | |
e6512bcf | 190 | |
969fe9b5 | 191 | In addition, the value @code{-0.0} is distinguishable from ordinary |
fa091c04 LT |
192 | zero in @acronym{IEEE} floating point (although @code{equal} and |
193 | @code{=} consider them equal values). | |
969fe9b5 | 194 | |
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195 | You can use @code{logb} to extract the binary exponent of a floating |
196 | point number (or estimate the logarithm of an integer): | |
197 | ||
198 | @defun logb number | |
199 | This function returns the binary exponent of @var{number}. More | |
200 | precisely, the value is the logarithm of @var{number} base 2, rounded | |
201 | down to an integer. | |
f9f59935 RS |
202 | |
203 | @example | |
204 | (logb 10) | |
205 | @result{} 3 | |
206 | (logb 10.0e20) | |
207 | @result{} 69 | |
208 | @end example | |
e6512bcf RS |
209 | @end defun |
210 | ||
211 | @node Predicates on Numbers | |
212 | @section Type Predicates for Numbers | |
213 | ||
214 | The functions in this section test whether the argument is a number or | |
215 | whether it is a certain sort of number. The functions @code{integerp} | |
216 | and @code{floatp} can take any type of Lisp object as argument (the | |
217 | predicates would not be of much use otherwise); but the @code{zerop} | |
218 | predicate requires a number as its argument. See also | |
219 | @code{integer-or-marker-p} and @code{number-or-marker-p}, in | |
220 | @ref{Predicates on Markers}. | |
221 | ||
222 | @defun floatp object | |
223 | This predicate tests whether its argument is a floating point | |
224 | number and returns @code{t} if so, @code{nil} otherwise. | |
225 | ||
226 | @code{floatp} does not exist in Emacs versions 18 and earlier. | |
227 | @end defun | |
228 | ||
229 | @defun integerp object | |
230 | This predicate tests whether its argument is an integer, and returns | |
231 | @code{t} if so, @code{nil} otherwise. | |
232 | @end defun | |
233 | ||
234 | @defun numberp object | |
235 | This predicate tests whether its argument is a number (either integer or | |
236 | floating point), and returns @code{t} if so, @code{nil} otherwise. | |
237 | @end defun | |
238 | ||
05fd2b65 | 239 | @defun wholenump object |
e6512bcf | 240 | @cindex natural numbers |
05fd2b65 RS |
241 | The @code{wholenump} predicate (whose name comes from the phrase |
242 | ``whole-number-p'') tests to see whether its argument is a nonnegative | |
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243 | integer, and returns @code{t} if so, @code{nil} otherwise. 0 is |
244 | considered non-negative. | |
245 | ||
05fd2b65 RS |
246 | @findex natnump |
247 | @code{natnump} is an obsolete synonym for @code{wholenump}. | |
e6512bcf RS |
248 | @end defun |
249 | ||
250 | @defun zerop number | |
251 | This predicate tests whether its argument is zero, and returns @code{t} | |
252 | if so, @code{nil} otherwise. The argument must be a number. | |
253 | ||
254 | These two forms are equivalent: @code{(zerop x)} @equiv{} @code{(= x 0)}. | |
255 | @end defun | |
256 | ||
257 | @node Comparison of Numbers | |
258 | @section Comparison of Numbers | |
259 | @cindex number equality | |
260 | ||
05fd2b65 RS |
261 | To test numbers for numerical equality, you should normally use |
262 | @code{=}, not @code{eq}. There can be many distinct floating point | |
263 | number objects with the same numeric value. If you use @code{eq} to | |
264 | compare them, then you test whether two values are the same | |
265 | @emph{object}. By contrast, @code{=} compares only the numeric values | |
266 | of the objects. | |
267 | ||
268 | At present, each integer value has a unique Lisp object in Emacs Lisp. | |
f9f59935 | 269 | Therefore, @code{eq} is equivalent to @code{=} where integers are |
05fd2b65 RS |
270 | concerned. It is sometimes convenient to use @code{eq} for comparing an |
271 | unknown value with an integer, because @code{eq} does not report an | |
272 | error if the unknown value is not a number---it accepts arguments of any | |
273 | type. By contrast, @code{=} signals an error if the arguments are not | |
274 | numbers or markers. However, it is a good idea to use @code{=} if you | |
275 | can, even for comparing integers, just in case we change the | |
276 | representation of integers in a future Emacs version. | |
e6512bcf | 277 | |
a9f0a989 RS |
278 | Sometimes it is useful to compare numbers with @code{equal}; it treats |
279 | two numbers as equal if they have the same data type (both integers, or | |
280 | both floating point) and the same value. By contrast, @code{=} can | |
281 | treat an integer and a floating point number as equal. | |
282 | ||
e6512bcf RS |
283 | There is another wrinkle: because floating point arithmetic is not |
284 | exact, it is often a bad idea to check for equality of two floating | |
285 | point values. Usually it is better to test for approximate equality. | |
286 | Here's a function to do this: | |
287 | ||
288 | @example | |
289 | (defvar fuzz-factor 1.0e-6) | |
290 | (defun approx-equal (x y) | |
bfe721d1 KH |
291 | (or (and (= x 0) (= y 0)) |
292 | (< (/ (abs (- x y)) | |
293 | (max (abs x) (abs y))) | |
294 | fuzz-factor))) | |
e6512bcf RS |
295 | @end example |
296 | ||
297 | @cindex CL note---integers vrs @code{eq} | |
298 | @quotation | |
05fd2b65 | 299 | @b{Common Lisp note:} Comparing numbers in Common Lisp always requires |
e6512bcf RS |
300 | @code{=} because Common Lisp implements multi-word integers, and two |
301 | distinct integer objects can have the same numeric value. Emacs Lisp | |
302 | can have just one integer object for any given value because it has a | |
303 | limited range of integer values. | |
304 | @end quotation | |
305 | ||
306 | @defun = number-or-marker1 number-or-marker2 | |
307 | This function tests whether its arguments are numerically equal, and | |
308 | returns @code{t} if so, @code{nil} otherwise. | |
309 | @end defun | |
310 | ||
311 | @defun /= number-or-marker1 number-or-marker2 | |
312 | This function tests whether its arguments are numerically equal, and | |
313 | returns @code{t} if they are not, and @code{nil} if they are. | |
314 | @end defun | |
315 | ||
316 | @defun < number-or-marker1 number-or-marker2 | |
317 | This function tests whether its first argument is strictly less than | |
318 | its second argument. It returns @code{t} if so, @code{nil} otherwise. | |
319 | @end defun | |
320 | ||
321 | @defun <= number-or-marker1 number-or-marker2 | |
322 | This function tests whether its first argument is less than or equal | |
323 | to its second argument. It returns @code{t} if so, @code{nil} | |
324 | otherwise. | |
325 | @end defun | |
326 | ||
327 | @defun > number-or-marker1 number-or-marker2 | |
328 | This function tests whether its first argument is strictly greater | |
329 | than its second argument. It returns @code{t} if so, @code{nil} | |
330 | otherwise. | |
331 | @end defun | |
332 | ||
333 | @defun >= number-or-marker1 number-or-marker2 | |
334 | This function tests whether its first argument is greater than or | |
335 | equal to its second argument. It returns @code{t} if so, @code{nil} | |
336 | otherwise. | |
337 | @end defun | |
338 | ||
339 | @defun max number-or-marker &rest numbers-or-markers | |
340 | This function returns the largest of its arguments. | |
8241495d RS |
341 | If any of the argument is floating-point, the value is returned |
342 | as floating point, even if it was given as an integer. | |
e6512bcf RS |
343 | |
344 | @example | |
345 | (max 20) | |
346 | @result{} 20 | |
347 | (max 1 2.5) | |
348 | @result{} 2.5 | |
349 | (max 1 3 2.5) | |
8241495d | 350 | @result{} 3.0 |
e6512bcf RS |
351 | @end example |
352 | @end defun | |
353 | ||
354 | @defun min number-or-marker &rest numbers-or-markers | |
355 | This function returns the smallest of its arguments. | |
8241495d RS |
356 | If any of the argument is floating-point, the value is returned |
357 | as floating point, even if it was given as an integer. | |
e6512bcf RS |
358 | |
359 | @example | |
360 | (min -4 1) | |
361 | @result{} -4 | |
362 | @end example | |
363 | @end defun | |
364 | ||
969fe9b5 | 365 | @defun abs number |
a9f0a989 | 366 | This function returns the absolute value of @var{number}. |
969fe9b5 RS |
367 | @end defun |
368 | ||
e6512bcf RS |
369 | @node Numeric Conversions |
370 | @section Numeric Conversions | |
371 | @cindex rounding in conversions | |
372 | ||
373 | To convert an integer to floating point, use the function @code{float}. | |
374 | ||
375 | @defun float number | |
376 | This returns @var{number} converted to floating point. | |
377 | If @var{number} is already a floating point number, @code{float} returns | |
378 | it unchanged. | |
379 | @end defun | |
380 | ||
381 | There are four functions to convert floating point numbers to integers; | |
fa091c04 LT |
382 | they differ in how they round. All accept an argument @var{number} |
383 | and an optional argument @var{divisor}. Both arguments may be | |
384 | integers or floating point numbers. @var{divisor} may also be | |
385 | @code{nil}. If @var{divisor} is @code{nil} or omitted, these | |
386 | functions convert @var{number} to an integer, or return it unchanged | |
387 | if it already is an integer. If @var{divisor} is non-@code{nil}, they | |
388 | divide @var{number} by @var{divisor} and convert the result to an | |
389 | integer. An @code{arith-error} results if @var{divisor} is 0. | |
390 | ||
391 | @defun truncate number &optional divisor | |
e6512bcf RS |
392 | This returns @var{number}, converted to an integer by rounding towards |
393 | zero. | |
b7a2fc9b RS |
394 | |
395 | @example | |
396 | (truncate 1.2) | |
397 | @result{} 1 | |
398 | (truncate 1.7) | |
399 | @result{} 1 | |
400 | (truncate -1.2) | |
401 | @result{} -1 | |
402 | (truncate -1.7) | |
403 | @result{} -1 | |
404 | @end example | |
e6512bcf RS |
405 | @end defun |
406 | ||
407 | @defun floor number &optional divisor | |
408 | This returns @var{number}, converted to an integer by rounding downward | |
409 | (towards negative infinity). | |
410 | ||
fa091c04 LT |
411 | If @var{divisor} is specified, this uses the kind of division |
412 | operation that corresponds to @code{mod}, rounding downward. | |
b7a2fc9b RS |
413 | |
414 | @example | |
415 | (floor 1.2) | |
416 | @result{} 1 | |
417 | (floor 1.7) | |
418 | @result{} 1 | |
419 | (floor -1.2) | |
420 | @result{} -2 | |
421 | (floor -1.7) | |
422 | @result{} -2 | |
423 | (floor 5.99 3) | |
424 | @result{} 1 | |
425 | @end example | |
e6512bcf RS |
426 | @end defun |
427 | ||
fa091c04 | 428 | @defun ceiling number &optional divisor |
e6512bcf RS |
429 | This returns @var{number}, converted to an integer by rounding upward |
430 | (towards positive infinity). | |
b7a2fc9b RS |
431 | |
432 | @example | |
433 | (ceiling 1.2) | |
434 | @result{} 2 | |
435 | (ceiling 1.7) | |
436 | @result{} 2 | |
437 | (ceiling -1.2) | |
438 | @result{} -1 | |
439 | (ceiling -1.7) | |
440 | @result{} -1 | |
441 | @end example | |
e6512bcf RS |
442 | @end defun |
443 | ||
fa091c04 | 444 | @defun round number &optional divisor |
e6512bcf | 445 | This returns @var{number}, converted to an integer by rounding towards the |
bfe721d1 KH |
446 | nearest integer. Rounding a value equidistant between two integers |
447 | may choose the integer closer to zero, or it may prefer an even integer, | |
448 | depending on your machine. | |
b7a2fc9b RS |
449 | |
450 | @example | |
451 | (round 1.2) | |
452 | @result{} 1 | |
453 | (round 1.7) | |
454 | @result{} 2 | |
455 | (round -1.2) | |
456 | @result{} -1 | |
457 | (round -1.7) | |
458 | @result{} -2 | |
459 | @end example | |
e6512bcf RS |
460 | @end defun |
461 | ||
462 | @node Arithmetic Operations | |
463 | @section Arithmetic Operations | |
464 | ||
465 | Emacs Lisp provides the traditional four arithmetic operations: | |
466 | addition, subtraction, multiplication, and division. Remainder and modulus | |
467 | functions supplement the division functions. The functions to | |
468 | add or subtract 1 are provided because they are traditional in Lisp and | |
469 | commonly used. | |
470 | ||
471 | All of these functions except @code{%} return a floating point value | |
472 | if any argument is floating. | |
473 | ||
969fe9b5 | 474 | It is important to note that in Emacs Lisp, arithmetic functions |
e1e44180 SM |
475 | do not check for overflow. Thus @code{(1+ 268435455)} may evaluate to |
476 | @minus{}268435456, depending on your hardware. | |
e6512bcf RS |
477 | |
478 | @defun 1+ number-or-marker | |
479 | This function returns @var{number-or-marker} plus 1. | |
480 | For example, | |
481 | ||
482 | @example | |
483 | (setq foo 4) | |
484 | @result{} 4 | |
485 | (1+ foo) | |
486 | @result{} 5 | |
487 | @end example | |
488 | ||
bfe721d1 KH |
489 | This function is not analogous to the C operator @code{++}---it does not |
490 | increment a variable. It just computes a sum. Thus, if we continue, | |
e6512bcf RS |
491 | |
492 | @example | |
493 | foo | |
494 | @result{} 4 | |
495 | @end example | |
496 | ||
497 | If you want to increment the variable, you must use @code{setq}, | |
498 | like this: | |
499 | ||
500 | @example | |
501 | (setq foo (1+ foo)) | |
502 | @result{} 5 | |
503 | @end example | |
504 | @end defun | |
505 | ||
506 | @defun 1- number-or-marker | |
507 | This function returns @var{number-or-marker} minus 1. | |
508 | @end defun | |
509 | ||
e6512bcf RS |
510 | @defun + &rest numbers-or-markers |
511 | This function adds its arguments together. When given no arguments, | |
bfe721d1 | 512 | @code{+} returns 0. |
e6512bcf RS |
513 | |
514 | @example | |
515 | (+) | |
516 | @result{} 0 | |
517 | (+ 1) | |
518 | @result{} 1 | |
519 | (+ 1 2 3 4) | |
520 | @result{} 10 | |
521 | @end example | |
522 | @end defun | |
523 | ||
f9f59935 | 524 | @defun - &optional number-or-marker &rest more-numbers-or-markers |
e6512bcf RS |
525 | The @code{-} function serves two purposes: negation and subtraction. |
526 | When @code{-} has a single argument, the value is the negative of the | |
527 | argument. When there are multiple arguments, @code{-} subtracts each of | |
f9f59935 | 528 | the @var{more-numbers-or-markers} from @var{number-or-marker}, |
bfe721d1 | 529 | cumulatively. If there are no arguments, the result is 0. |
e6512bcf RS |
530 | |
531 | @example | |
532 | (- 10 1 2 3 4) | |
533 | @result{} 0 | |
534 | (- 10) | |
535 | @result{} -10 | |
536 | (-) | |
537 | @result{} 0 | |
538 | @end example | |
539 | @end defun | |
540 | ||
541 | @defun * &rest numbers-or-markers | |
542 | This function multiplies its arguments together, and returns the | |
bfe721d1 | 543 | product. When given no arguments, @code{*} returns 1. |
e6512bcf RS |
544 | |
545 | @example | |
546 | (*) | |
547 | @result{} 1 | |
548 | (* 1) | |
549 | @result{} 1 | |
550 | (* 1 2 3 4) | |
551 | @result{} 24 | |
552 | @end example | |
553 | @end defun | |
554 | ||
555 | @defun / dividend divisor &rest divisors | |
05fd2b65 | 556 | This function divides @var{dividend} by @var{divisor} and returns the |
e6512bcf RS |
557 | quotient. If there are additional arguments @var{divisors}, then it |
558 | divides @var{dividend} by each divisor in turn. Each argument may be a | |
559 | number or a marker. | |
560 | ||
561 | If all the arguments are integers, then the result is an integer too. | |
562 | This means the result has to be rounded. On most machines, the result | |
563 | is rounded towards zero after each division, but some machines may round | |
564 | differently with negative arguments. This is because the Lisp function | |
565 | @code{/} is implemented using the C division operator, which also | |
566 | permits machine-dependent rounding. As a practical matter, all known | |
567 | machines round in the standard fashion. | |
568 | ||
569 | @cindex @code{arith-error} in division | |
969fe9b5 RS |
570 | If you divide an integer by 0, an @code{arith-error} error is signaled. |
571 | (@xref{Errors}.) Floating point division by zero returns either | |
ad800164 | 572 | infinity or a NaN if your machine supports @acronym{IEEE} floating point; |
969fe9b5 | 573 | otherwise, it signals an @code{arith-error} error. |
e6512bcf RS |
574 | |
575 | @example | |
bda144f4 | 576 | @group |
e6512bcf RS |
577 | (/ 6 2) |
578 | @result{} 3 | |
bda144f4 | 579 | @end group |
e6512bcf RS |
580 | (/ 5 2) |
581 | @result{} 2 | |
969fe9b5 RS |
582 | (/ 5.0 2) |
583 | @result{} 2.5 | |
584 | (/ 5 2.0) | |
585 | @result{} 2.5 | |
586 | (/ 5.0 2.0) | |
587 | @result{} 2.5 | |
e6512bcf RS |
588 | (/ 25 3 2) |
589 | @result{} 4 | |
590 | (/ -17 6) | |
591 | @result{} -2 | |
592 | @end example | |
593 | ||
594 | The result of @code{(/ -17 6)} could in principle be -3 on some | |
595 | machines. | |
596 | @end defun | |
597 | ||
598 | @defun % dividend divisor | |
599 | @cindex remainder | |
600 | This function returns the integer remainder after division of @var{dividend} | |
601 | by @var{divisor}. The arguments must be integers or markers. | |
602 | ||
603 | For negative arguments, the remainder is in principle machine-dependent | |
604 | since the quotient is; but in practice, all known machines behave alike. | |
605 | ||
606 | An @code{arith-error} results if @var{divisor} is 0. | |
607 | ||
608 | @example | |
609 | (% 9 4) | |
610 | @result{} 1 | |
611 | (% -9 4) | |
612 | @result{} -1 | |
613 | (% 9 -4) | |
614 | @result{} 1 | |
615 | (% -9 -4) | |
616 | @result{} -1 | |
617 | @end example | |
618 | ||
619 | For any two integers @var{dividend} and @var{divisor}, | |
620 | ||
621 | @example | |
622 | @group | |
623 | (+ (% @var{dividend} @var{divisor}) | |
624 | (* (/ @var{dividend} @var{divisor}) @var{divisor})) | |
625 | @end group | |
626 | @end example | |
627 | ||
628 | @noindent | |
629 | always equals @var{dividend}. | |
630 | @end defun | |
631 | ||
632 | @defun mod dividend divisor | |
633 | @cindex modulus | |
634 | This function returns the value of @var{dividend} modulo @var{divisor}; | |
635 | in other words, the remainder after division of @var{dividend} | |
636 | by @var{divisor}, but with the same sign as @var{divisor}. | |
637 | The arguments must be numbers or markers. | |
638 | ||
639 | Unlike @code{%}, @code{mod} returns a well-defined result for negative | |
640 | arguments. It also permits floating point arguments; it rounds the | |
641 | quotient downward (towards minus infinity) to an integer, and uses that | |
642 | quotient to compute the remainder. | |
643 | ||
644 | An @code{arith-error} results if @var{divisor} is 0. | |
645 | ||
646 | @example | |
bda144f4 | 647 | @group |
e6512bcf RS |
648 | (mod 9 4) |
649 | @result{} 1 | |
bda144f4 MW |
650 | @end group |
651 | @group | |
e6512bcf RS |
652 | (mod -9 4) |
653 | @result{} 3 | |
bda144f4 MW |
654 | @end group |
655 | @group | |
e6512bcf RS |
656 | (mod 9 -4) |
657 | @result{} -3 | |
bda144f4 MW |
658 | @end group |
659 | @group | |
e6512bcf RS |
660 | (mod -9 -4) |
661 | @result{} -1 | |
bda144f4 MW |
662 | @end group |
663 | @group | |
e6512bcf RS |
664 | (mod 5.5 2.5) |
665 | @result{} .5 | |
bda144f4 | 666 | @end group |
e6512bcf RS |
667 | @end example |
668 | ||
669 | For any two numbers @var{dividend} and @var{divisor}, | |
670 | ||
671 | @example | |
672 | @group | |
673 | (+ (mod @var{dividend} @var{divisor}) | |
674 | (* (floor @var{dividend} @var{divisor}) @var{divisor})) | |
675 | @end group | |
676 | @end example | |
677 | ||
678 | @noindent | |
bfe721d1 KH |
679 | always equals @var{dividend}, subject to rounding error if either |
680 | argument is floating point. For @code{floor}, see @ref{Numeric | |
681 | Conversions}. | |
e6512bcf RS |
682 | @end defun |
683 | ||
684 | @node Rounding Operations | |
685 | @section Rounding Operations | |
686 | @cindex rounding without conversion | |
687 | ||
a9f0a989 | 688 | The functions @code{ffloor}, @code{fceiling}, @code{fround}, and |
e6512bcf RS |
689 | @code{ftruncate} take a floating point argument and return a floating |
690 | point result whose value is a nearby integer. @code{ffloor} returns the | |
bb61c6c6 | 691 | nearest integer below; @code{fceiling}, the nearest integer above; |
05fd2b65 | 692 | @code{ftruncate}, the nearest integer in the direction towards zero; |
e6512bcf RS |
693 | @code{fround}, the nearest integer. |
694 | ||
695 | @defun ffloor float | |
696 | This function rounds @var{float} to the next lower integral value, and | |
697 | returns that value as a floating point number. | |
698 | @end defun | |
699 | ||
bb61c6c6 | 700 | @defun fceiling float |
e6512bcf RS |
701 | This function rounds @var{float} to the next higher integral value, and |
702 | returns that value as a floating point number. | |
703 | @end defun | |
704 | ||
05fd2b65 | 705 | @defun ftruncate float |
e6512bcf RS |
706 | This function rounds @var{float} towards zero to an integral value, and |
707 | returns that value as a floating point number. | |
708 | @end defun | |
709 | ||
710 | @defun fround float | |
711 | This function rounds @var{float} to the nearest integral value, | |
712 | and returns that value as a floating point number. | |
713 | @end defun | |
714 | ||
715 | @node Bitwise Operations | |
716 | @section Bitwise Operations on Integers | |
717 | ||
718 | In a computer, an integer is represented as a binary number, a | |
719 | sequence of @dfn{bits} (digits which are either zero or one). A bitwise | |
720 | operation acts on the individual bits of such a sequence. For example, | |
721 | @dfn{shifting} moves the whole sequence left or right one or more places, | |
722 | reproducing the same pattern ``moved over''. | |
723 | ||
724 | The bitwise operations in Emacs Lisp apply only to integers. | |
725 | ||
726 | @defun lsh integer1 count | |
727 | @cindex logical shift | |
728 | @code{lsh}, which is an abbreviation for @dfn{logical shift}, shifts the | |
05fd2b65 RS |
729 | bits in @var{integer1} to the left @var{count} places, or to the right |
730 | if @var{count} is negative, bringing zeros into the vacated bits. If | |
731 | @var{count} is negative, @code{lsh} shifts zeros into the leftmost | |
732 | (most-significant) bit, producing a positive result even if | |
733 | @var{integer1} is negative. Contrast this with @code{ash}, below. | |
734 | ||
735 | Here are two examples of @code{lsh}, shifting a pattern of bits one | |
736 | place to the left. We show only the low-order eight bits of the binary | |
737 | pattern; the rest are all zero. | |
e6512bcf RS |
738 | |
739 | @example | |
740 | @group | |
741 | (lsh 5 1) | |
742 | @result{} 10 | |
743 | ;; @r{Decimal 5 becomes decimal 10.} | |
744 | 00000101 @result{} 00001010 | |
745 | ||
746 | (lsh 7 1) | |
747 | @result{} 14 | |
748 | ;; @r{Decimal 7 becomes decimal 14.} | |
749 | 00000111 @result{} 00001110 | |
750 | @end group | |
751 | @end example | |
752 | ||
753 | @noindent | |
754 | As the examples illustrate, shifting the pattern of bits one place to | |
755 | the left produces a number that is twice the value of the previous | |
756 | number. | |
757 | ||
bfe721d1 | 758 | Shifting a pattern of bits two places to the left produces results |
e6512bcf RS |
759 | like this (with 8-bit binary numbers): |
760 | ||
761 | @example | |
762 | @group | |
763 | (lsh 3 2) | |
764 | @result{} 12 | |
765 | ;; @r{Decimal 3 becomes decimal 12.} | |
177c0ea7 | 766 | 00000011 @result{} 00001100 |
e6512bcf RS |
767 | @end group |
768 | @end example | |
769 | ||
bfe721d1 | 770 | On the other hand, shifting one place to the right looks like this: |
e6512bcf RS |
771 | |
772 | @example | |
773 | @group | |
774 | (lsh 6 -1) | |
775 | @result{} 3 | |
776 | ;; @r{Decimal 6 becomes decimal 3.} | |
177c0ea7 | 777 | 00000110 @result{} 00000011 |
e6512bcf RS |
778 | @end group |
779 | ||
780 | @group | |
781 | (lsh 5 -1) | |
782 | @result{} 2 | |
783 | ;; @r{Decimal 5 becomes decimal 2.} | |
177c0ea7 | 784 | 00000101 @result{} 00000010 |
e6512bcf RS |
785 | @end group |
786 | @end example | |
787 | ||
788 | @noindent | |
bfe721d1 KH |
789 | As the example illustrates, shifting one place to the right divides the |
790 | value of a positive integer by two, rounding downward. | |
791 | ||
792 | The function @code{lsh}, like all Emacs Lisp arithmetic functions, does | |
793 | not check for overflow, so shifting left can discard significant bits | |
794 | and change the sign of the number. For example, left shifting | |
e1e44180 | 795 | 268,435,455 produces @minus{}2 on a 29-bit machine: |
bfe721d1 KH |
796 | |
797 | @example | |
e1e44180 | 798 | (lsh 268435455 1) ; @r{left shift} |
bfe721d1 KH |
799 | @result{} -2 |
800 | @end example | |
801 | ||
e1e44180 | 802 | In binary, in the 29-bit implementation, the argument looks like this: |
bfe721d1 KH |
803 | |
804 | @example | |
805 | @group | |
e1e44180 SM |
806 | ;; @r{Decimal 268,435,455} |
807 | 0 1111 1111 1111 1111 1111 1111 1111 | |
bfe721d1 KH |
808 | @end group |
809 | @end example | |
810 | ||
811 | @noindent | |
812 | which becomes the following when left shifted: | |
813 | ||
814 | @example | |
815 | @group | |
816 | ;; @r{Decimal @minus{}2} | |
e1e44180 | 817 | 1 1111 1111 1111 1111 1111 1111 1110 |
bfe721d1 KH |
818 | @end group |
819 | @end example | |
e6512bcf RS |
820 | @end defun |
821 | ||
822 | @defun ash integer1 count | |
823 | @cindex arithmetic shift | |
824 | @code{ash} (@dfn{arithmetic shift}) shifts the bits in @var{integer1} | |
825 | to the left @var{count} places, or to the right if @var{count} | |
826 | is negative. | |
827 | ||
828 | @code{ash} gives the same results as @code{lsh} except when | |
829 | @var{integer1} and @var{count} are both negative. In that case, | |
bfe721d1 KH |
830 | @code{ash} puts ones in the empty bit positions on the left, while |
831 | @code{lsh} puts zeros in those bit positions. | |
e6512bcf RS |
832 | |
833 | Thus, with @code{ash}, shifting the pattern of bits one place to the right | |
834 | looks like this: | |
835 | ||
836 | @example | |
837 | @group | |
177c0ea7 | 838 | (ash -6 -1) @result{} -3 |
e6512bcf | 839 | ;; @r{Decimal @minus{}6 becomes decimal @minus{}3.} |
e1e44180 | 840 | 1 1111 1111 1111 1111 1111 1111 1010 |
177c0ea7 | 841 | @result{} |
e1e44180 | 842 | 1 1111 1111 1111 1111 1111 1111 1101 |
e6512bcf RS |
843 | @end group |
844 | @end example | |
845 | ||
846 | In contrast, shifting the pattern of bits one place to the right with | |
847 | @code{lsh} looks like this: | |
848 | ||
849 | @example | |
850 | @group | |
e1e44180 SM |
851 | (lsh -6 -1) @result{} 268435453 |
852 | ;; @r{Decimal @minus{}6 becomes decimal 268,435,453.} | |
853 | 1 1111 1111 1111 1111 1111 1111 1010 | |
177c0ea7 | 854 | @result{} |
e1e44180 | 855 | 0 1111 1111 1111 1111 1111 1111 1101 |
e6512bcf RS |
856 | @end group |
857 | @end example | |
858 | ||
e6512bcf RS |
859 | Here are other examples: |
860 | ||
861 | @c !!! Check if lined up in smallbook format! XDVI shows problem | |
862 | @c with smallbook but not with regular book! --rjc 16mar92 | |
863 | @smallexample | |
864 | @group | |
e1e44180 | 865 | ; @r{ 29-bit binary values} |
e6512bcf | 866 | |
e1e44180 SM |
867 | (lsh 5 2) ; 5 = @r{0 0000 0000 0000 0000 0000 0000 0101} |
868 | @result{} 20 ; = @r{0 0000 0000 0000 0000 0000 0001 0100} | |
e6512bcf RS |
869 | @end group |
870 | @group | |
871 | (ash 5 2) | |
872 | @result{} 20 | |
e1e44180 SM |
873 | (lsh -5 2) ; -5 = @r{1 1111 1111 1111 1111 1111 1111 1011} |
874 | @result{} -20 ; = @r{1 1111 1111 1111 1111 1111 1110 1100} | |
e6512bcf RS |
875 | (ash -5 2) |
876 | @result{} -20 | |
877 | @end group | |
878 | @group | |
e1e44180 SM |
879 | (lsh 5 -2) ; 5 = @r{0 0000 0000 0000 0000 0000 0000 0101} |
880 | @result{} 1 ; = @r{0 0000 0000 0000 0000 0000 0000 0001} | |
e6512bcf RS |
881 | @end group |
882 | @group | |
883 | (ash 5 -2) | |
884 | @result{} 1 | |
885 | @end group | |
886 | @group | |
e1e44180 SM |
887 | (lsh -5 -2) ; -5 = @r{1 1111 1111 1111 1111 1111 1111 1011} |
888 | @result{} 134217726 ; = @r{0 0111 1111 1111 1111 1111 1111 1110} | |
e6512bcf RS |
889 | @end group |
890 | @group | |
e1e44180 SM |
891 | (ash -5 -2) ; -5 = @r{1 1111 1111 1111 1111 1111 1111 1011} |
892 | @result{} -2 ; = @r{1 1111 1111 1111 1111 1111 1111 1110} | |
e6512bcf RS |
893 | @end group |
894 | @end smallexample | |
895 | @end defun | |
896 | ||
897 | @defun logand &rest ints-or-markers | |
898 | @cindex logical and | |
899 | @cindex bitwise and | |
900 | This function returns the ``logical and'' of the arguments: the | |
901 | @var{n}th bit is set in the result if, and only if, the @var{n}th bit is | |
902 | set in all the arguments. (``Set'' means that the value of the bit is 1 | |
903 | rather than 0.) | |
904 | ||
905 | For example, using 4-bit binary numbers, the ``logical and'' of 13 and | |
906 | 12 is 12: 1101 combined with 1100 produces 1100. | |
e6512bcf RS |
907 | In both the binary numbers, the leftmost two bits are set (i.e., they |
908 | are 1's), so the leftmost two bits of the returned value are set. | |
909 | However, for the rightmost two bits, each is zero in at least one of | |
910 | the arguments, so the rightmost two bits of the returned value are 0's. | |
911 | ||
912 | @noindent | |
913 | Therefore, | |
914 | ||
915 | @example | |
916 | @group | |
917 | (logand 13 12) | |
918 | @result{} 12 | |
919 | @end group | |
920 | @end example | |
921 | ||
922 | If @code{logand} is not passed any argument, it returns a value of | |
923 | @minus{}1. This number is an identity element for @code{logand} | |
924 | because its binary representation consists entirely of ones. If | |
925 | @code{logand} is passed just one argument, it returns that argument. | |
926 | ||
927 | @smallexample | |
928 | @group | |
e1e44180 | 929 | ; @r{ 29-bit binary values} |
e6512bcf | 930 | |
e1e44180 SM |
931 | (logand 14 13) ; 14 = @r{0 0000 0000 0000 0000 0000 0000 1110} |
932 | ; 13 = @r{0 0000 0000 0000 0000 0000 0000 1101} | |
933 | @result{} 12 ; 12 = @r{0 0000 0000 0000 0000 0000 0000 1100} | |
e6512bcf RS |
934 | @end group |
935 | ||
936 | @group | |
e1e44180 SM |
937 | (logand 14 13 4) ; 14 = @r{0 0000 0000 0000 0000 0000 0000 1110} |
938 | ; 13 = @r{0 0000 0000 0000 0000 0000 0000 1101} | |
939 | ; 4 = @r{0 0000 0000 0000 0000 0000 0000 0100} | |
940 | @result{} 4 ; 4 = @r{0 0000 0000 0000 0000 0000 0000 0100} | |
e6512bcf RS |
941 | @end group |
942 | ||
943 | @group | |
944 | (logand) | |
e1e44180 | 945 | @result{} -1 ; -1 = @r{1 1111 1111 1111 1111 1111 1111 1111} |
e6512bcf RS |
946 | @end group |
947 | @end smallexample | |
948 | @end defun | |
949 | ||
950 | @defun logior &rest ints-or-markers | |
951 | @cindex logical inclusive or | |
952 | @cindex bitwise or | |
953 | This function returns the ``inclusive or'' of its arguments: the @var{n}th bit | |
954 | is set in the result if, and only if, the @var{n}th bit is set in at least | |
955 | one of the arguments. If there are no arguments, the result is zero, | |
956 | which is an identity element for this operation. If @code{logior} is | |
957 | passed just one argument, it returns that argument. | |
958 | ||
959 | @smallexample | |
960 | @group | |
e1e44180 | 961 | ; @r{ 29-bit binary values} |
e6512bcf | 962 | |
e1e44180 SM |
963 | (logior 12 5) ; 12 = @r{0 0000 0000 0000 0000 0000 0000 1100} |
964 | ; 5 = @r{0 0000 0000 0000 0000 0000 0000 0101} | |
965 | @result{} 13 ; 13 = @r{0 0000 0000 0000 0000 0000 0000 1101} | |
e6512bcf RS |
966 | @end group |
967 | ||
968 | @group | |
e1e44180 SM |
969 | (logior 12 5 7) ; 12 = @r{0 0000 0000 0000 0000 0000 0000 1100} |
970 | ; 5 = @r{0 0000 0000 0000 0000 0000 0000 0101} | |
971 | ; 7 = @r{0 0000 0000 0000 0000 0000 0000 0111} | |
972 | @result{} 15 ; 15 = @r{0 0000 0000 0000 0000 0000 0000 1111} | |
e6512bcf RS |
973 | @end group |
974 | @end smallexample | |
975 | @end defun | |
976 | ||
977 | @defun logxor &rest ints-or-markers | |
978 | @cindex bitwise exclusive or | |
979 | @cindex logical exclusive or | |
980 | This function returns the ``exclusive or'' of its arguments: the | |
05fd2b65 RS |
981 | @var{n}th bit is set in the result if, and only if, the @var{n}th bit is |
982 | set in an odd number of the arguments. If there are no arguments, the | |
983 | result is 0, which is an identity element for this operation. If | |
984 | @code{logxor} is passed just one argument, it returns that argument. | |
e6512bcf RS |
985 | |
986 | @smallexample | |
987 | @group | |
e1e44180 | 988 | ; @r{ 29-bit binary values} |
e6512bcf | 989 | |
e1e44180 SM |
990 | (logxor 12 5) ; 12 = @r{0 0000 0000 0000 0000 0000 0000 1100} |
991 | ; 5 = @r{0 0000 0000 0000 0000 0000 0000 0101} | |
992 | @result{} 9 ; 9 = @r{0 0000 0000 0000 0000 0000 0000 1001} | |
e6512bcf RS |
993 | @end group |
994 | ||
995 | @group | |
e1e44180 SM |
996 | (logxor 12 5 7) ; 12 = @r{0 0000 0000 0000 0000 0000 0000 1100} |
997 | ; 5 = @r{0 0000 0000 0000 0000 0000 0000 0101} | |
998 | ; 7 = @r{0 0000 0000 0000 0000 0000 0000 0111} | |
999 | @result{} 14 ; 14 = @r{0 0000 0000 0000 0000 0000 0000 1110} | |
e6512bcf RS |
1000 | @end group |
1001 | @end smallexample | |
1002 | @end defun | |
1003 | ||
1004 | @defun lognot integer | |
1005 | @cindex logical not | |
1006 | @cindex bitwise not | |
1007 | This function returns the logical complement of its argument: the @var{n}th | |
1008 | bit is one in the result if, and only if, the @var{n}th bit is zero in | |
1009 | @var{integer}, and vice-versa. | |
1010 | ||
1011 | @example | |
177c0ea7 | 1012 | (lognot 5) |
e6512bcf | 1013 | @result{} -6 |
e1e44180 | 1014 | ;; 5 = @r{0 0000 0000 0000 0000 0000 0000 0101} |
e6512bcf | 1015 | ;; @r{becomes} |
e1e44180 | 1016 | ;; -6 = @r{1 1111 1111 1111 1111 1111 1111 1010} |
e6512bcf RS |
1017 | @end example |
1018 | @end defun | |
1019 | ||
4ac771a0 RS |
1020 | @node Math Functions |
1021 | @section Standard Mathematical Functions | |
e6512bcf RS |
1022 | @cindex transcendental functions |
1023 | @cindex mathematical functions | |
1024 | ||
969fe9b5 RS |
1025 | These mathematical functions allow integers as well as floating point |
1026 | numbers as arguments. | |
e6512bcf RS |
1027 | |
1028 | @defun sin arg | |
1029 | @defunx cos arg | |
1030 | @defunx tan arg | |
1031 | These are the ordinary trigonometric functions, with argument measured | |
1032 | in radians. | |
1033 | @end defun | |
1034 | ||
1035 | @defun asin arg | |
15da7853 | 1036 | The value of @code{(asin @var{arg})} is a number between |
37680279 | 1037 | @ifnottex |
15da7853 | 1038 | @minus{}pi/2 |
37680279 | 1039 | @end ifnottex |
15da7853 | 1040 | @tex |
8241495d | 1041 | @math{-\pi/2} |
15da7853 KH |
1042 | @end tex |
1043 | and | |
37680279 | 1044 | @ifnottex |
15da7853 | 1045 | pi/2 |
37680279 | 1046 | @end ifnottex |
15da7853 | 1047 | @tex |
8241495d | 1048 | @math{\pi/2} |
15da7853 | 1049 | @end tex |
f3d95494 RS |
1050 | (inclusive) whose sine is @var{arg}; if, however, @var{arg} is out of |
1051 | range (outside [-1, 1]), it signals a @code{domain-error} error. | |
e6512bcf RS |
1052 | @end defun |
1053 | ||
1054 | @defun acos arg | |
15da7853 | 1055 | The value of @code{(acos @var{arg})} is a number between 0 and |
37680279 | 1056 | @ifnottex |
15da7853 | 1057 | pi |
37680279 | 1058 | @end ifnottex |
15da7853 | 1059 | @tex |
8241495d | 1060 | @math{\pi} |
15da7853 | 1061 | @end tex |
f3d95494 RS |
1062 | (inclusive) whose cosine is @var{arg}; if, however, @var{arg} is out |
1063 | of range (outside [-1, 1]), it signals a @code{domain-error} error. | |
e6512bcf RS |
1064 | @end defun |
1065 | ||
f120431c EZ |
1066 | @defun atan y &optional x |
1067 | The value of @code{(atan @var{y})} is a number between | |
37680279 | 1068 | @ifnottex |
15da7853 | 1069 | @minus{}pi/2 |
37680279 | 1070 | @end ifnottex |
15da7853 | 1071 | @tex |
8241495d | 1072 | @math{-\pi/2} |
15da7853 KH |
1073 | @end tex |
1074 | and | |
37680279 | 1075 | @ifnottex |
15da7853 | 1076 | pi/2 |
37680279 | 1077 | @end ifnottex |
15da7853 | 1078 | @tex |
8241495d | 1079 | @math{\pi/2} |
15da7853 | 1080 | @end tex |
f120431c EZ |
1081 | (exclusive) whose tangent is @var{y}. If the optional second |
1082 | argument @var{x} is given, the value of @code{(atan y x)} is the | |
1083 | angle in radians between the vector @code{[@var{x}, @var{y}]} and the | |
1084 | @code{X} axis. | |
e6512bcf RS |
1085 | @end defun |
1086 | ||
1087 | @defun exp arg | |
a9f0a989 RS |
1088 | This is the exponential function; it returns |
1089 | @tex | |
8241495d | 1090 | @math{e} |
a9f0a989 | 1091 | @end tex |
37680279 | 1092 | @ifnottex |
a9f0a989 | 1093 | @i{e} |
37680279 | 1094 | @end ifnottex |
a9f0a989 RS |
1095 | to the power @var{arg}. |
1096 | @tex | |
8241495d | 1097 | @math{e} |
a9f0a989 | 1098 | @end tex |
37680279 | 1099 | @ifnottex |
a9f0a989 | 1100 | @i{e} |
37680279 | 1101 | @end ifnottex |
a9f0a989 RS |
1102 | is a fundamental mathematical constant also called the base of natural |
1103 | logarithms. | |
e6512bcf RS |
1104 | @end defun |
1105 | ||
1106 | @defun log arg &optional base | |
1107 | This function returns the logarithm of @var{arg}, with base @var{base}. | |
a9f0a989 RS |
1108 | If you don't specify @var{base}, the base |
1109 | @tex | |
8241495d | 1110 | @math{e} |
a9f0a989 | 1111 | @end tex |
37680279 | 1112 | @ifnottex |
a9f0a989 | 1113 | @i{e} |
37680279 | 1114 | @end ifnottex |
f3d95494 RS |
1115 | is used. If @var{arg} is negative, it signals a @code{domain-error} |
1116 | error. | |
e6512bcf RS |
1117 | @end defun |
1118 | ||
1119 | @ignore | |
1120 | @defun expm1 arg | |
1121 | This function returns @code{(1- (exp @var{arg}))}, but it is more | |
1122 | accurate than that when @var{arg} is negative and @code{(exp @var{arg})} | |
1123 | is close to 1. | |
1124 | @end defun | |
1125 | ||
1126 | @defun log1p arg | |
1127 | This function returns @code{(log (1+ @var{arg}))}, but it is more | |
1128 | accurate than that when @var{arg} is so small that adding 1 to it would | |
1129 | lose accuracy. | |
1130 | @end defun | |
1131 | @end ignore | |
1132 | ||
1133 | @defun log10 arg | |
1134 | This function returns the logarithm of @var{arg}, with base 10. If | |
f3d95494 RS |
1135 | @var{arg} is negative, it signals a @code{domain-error} error. |
1136 | @code{(log10 @var{x})} @equiv{} @code{(log @var{x} 10)}, at least | |
1137 | approximately. | |
e6512bcf RS |
1138 | @end defun |
1139 | ||
1140 | @defun expt x y | |
8be5e0fc RS |
1141 | This function returns @var{x} raised to power @var{y}. If both |
1142 | arguments are integers and @var{y} is positive, the result is an | |
1143 | integer; in this case, it is truncated to fit the range of possible | |
1144 | integer values. | |
e6512bcf RS |
1145 | @end defun |
1146 | ||
1147 | @defun sqrt arg | |
1148 | This returns the square root of @var{arg}. If @var{arg} is negative, | |
903a1a91 | 1149 | it signals a @code{domain-error} error. |
e6512bcf RS |
1150 | @end defun |
1151 | ||
1152 | @node Random Numbers | |
1153 | @section Random Numbers | |
1154 | @cindex random numbers | |
1155 | ||
1156 | A deterministic computer program cannot generate true random numbers. | |
1157 | For most purposes, @dfn{pseudo-random numbers} suffice. A series of | |
1158 | pseudo-random numbers is generated in a deterministic fashion. The | |
1159 | numbers are not truly random, but they have certain properties that | |
1160 | mimic a random series. For example, all possible values occur equally | |
1161 | often in a pseudo-random series. | |
1162 | ||
1163 | In Emacs, pseudo-random numbers are generated from a ``seed'' number. | |
1164 | Starting from any given seed, the @code{random} function always | |
1165 | generates the same sequence of numbers. Emacs always starts with the | |
1166 | same seed value, so the sequence of values of @code{random} is actually | |
1167 | the same in each Emacs run! For example, in one operating system, the | |
1168 | first call to @code{(random)} after you start Emacs always returns | |
1169 | -1457731, and the second one always returns -7692030. This | |
1170 | repeatability is helpful for debugging. | |
1171 | ||
8241495d RS |
1172 | If you want random numbers that don't always come out the same, execute |
1173 | @code{(random t)}. This chooses a new seed based on the current time of | |
ad800164 | 1174 | day and on Emacs's process @acronym{ID} number. |
e6512bcf RS |
1175 | |
1176 | @defun random &optional limit | |
1177 | This function returns a pseudo-random integer. Repeated calls return a | |
1178 | series of pseudo-random integers. | |
1179 | ||
22697dac | 1180 | If @var{limit} is a positive integer, the value is chosen to be |
bfe721d1 | 1181 | nonnegative and less than @var{limit}. |
e6512bcf RS |
1182 | |
1183 | If @var{limit} is @code{t}, it means to choose a new seed based on the | |
ad800164 | 1184 | current time of day and on Emacs's process @acronym{ID} number. |
e6512bcf RS |
1185 | @c "Emacs'" is incorrect usage! |
1186 | ||
1187 | On some machines, any integer representable in Lisp may be the result | |
1188 | of @code{random}. On other machines, the result can never be larger | |
1189 | than a certain maximum or less than a certain (negative) minimum. | |
1190 | @end defun | |
ab5796a9 MB |
1191 | |
1192 | @ignore | |
1193 | arch-tag: 574e8dd2-d513-4616-9844-c9a27869782e | |
1194 | @end ignore |