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1 | @c -*-texinfo-*- |
2 | @c This is part of the GNU Emacs Lisp Reference Manual. | |
fd897522 | 3 | @c Copyright (C) 1990, 1991, 1992, 1993, 1994, 1995, 1998, 1999 |
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 | |
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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 | |
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19 | exact; they have a fixed, limited amount of precision. |
20 | ||
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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. |
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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 | |
94b00316 | 39 | minimum range is @minus{}134217728 to 134217727 (28 bits; i.e., |
37680279 | 40 | @ifnottex |
8be5e0fc | 41 | -2**27 |
37680279 | 42 | @end ifnottex |
177c0ea7 | 43 | @tex |
8241495d | 44 | @math{-2^{27}} |
e6512bcf | 45 | @end tex |
177c0ea7 | 46 | to |
37680279 | 47 | @ifnottex |
94b00316 | 48 | 2**27 - 1), |
37680279 | 49 | @end ifnottex |
177c0ea7 | 50 | @tex |
8241495d | 51 | @math{2^{27}-1}), |
e6512bcf | 52 | @end tex |
94b00316 RS |
53 | but some machines may provide a wider range. Many examples in this |
54 | chapter assume an integer has 28 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.} | |
8be5e0fc | 65 | 268435457 ; @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 | ||
d9940cbf EZ |
70 | @cindex integers in specific radix |
71 | @cindex radix for reading an integer | |
72 | @cindex base for reading an integer | |
73 | In addition, the Lisp reader recognizes a syntax for integers in | |
74 | bases other than 10: @samp{#B@var{integer}} reads @var{integer} in | |
75 | binary (radix 2), @samp{#O@var{integer}} reads @var{integer} in octal | |
76 | (radix 8), @samp{#X@var{integer}} reads @var{integer} in hexadecimal | |
77 | (radix 16), and @samp{#@var{radix}r@var{integer}} reads @var{integer} | |
78 | in radix @var{radix} (where @var{radix} is between 2 and 36, | |
5fe3b9bc | 79 | inclusively). Case is not significant for the letter after @samp{#} |
d9940cbf EZ |
80 | (@samp{B}, @samp{O}, etc.) that denotes the radix. |
81 | ||
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82 | To understand how various functions work on integers, especially the |
83 | bitwise operators (@pxref{Bitwise Operations}), it is often helpful to | |
84 | view the numbers in their binary form. | |
85 | ||
8be5e0fc | 86 | In 28-bit binary, the decimal integer 5 looks like this: |
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87 | |
88 | @example | |
8be5e0fc | 89 | 0000 0000 0000 0000 0000 0000 0101 |
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90 | @end example |
91 | ||
92 | @noindent | |
93 | (We have inserted spaces between groups of 4 bits, and two spaces | |
94 | between groups of 8 bits, to make the binary integer easier to read.) | |
95 | ||
96 | The integer @minus{}1 looks like this: | |
97 | ||
98 | @example | |
8be5e0fc | 99 | 1111 1111 1111 1111 1111 1111 1111 |
e6512bcf RS |
100 | @end example |
101 | ||
102 | @noindent | |
103 | @cindex two's complement | |
8be5e0fc | 104 | @minus{}1 is represented as 28 ones. (This is called @dfn{two's |
e6512bcf RS |
105 | complement} notation.) |
106 | ||
107 | The negative integer, @minus{}5, is creating by subtracting 4 from | |
108 | @minus{}1. In binary, the decimal integer 4 is 100. Consequently, | |
109 | @minus{}5 looks like this: | |
110 | ||
111 | @example | |
8be5e0fc | 112 | 1111 1111 1111 1111 1111 1111 1011 |
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113 | @end example |
114 | ||
9e2b495b RS |
115 | In this implementation, the largest 28-bit binary integer value is |
116 | 134,217,727 in decimal. In binary, it looks like this: | |
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117 | |
118 | @example | |
8be5e0fc | 119 | 0111 1111 1111 1111 1111 1111 1111 |
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120 | @end example |
121 | ||
122 | Since the arithmetic functions do not check whether integers go | |
8be5e0fc RS |
123 | outside their range, when you add 1 to 134,217,727, the value is the |
124 | negative integer @minus{}134,217,728: | |
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125 | |
126 | @example | |
8be5e0fc RS |
127 | (+ 1 134217727) |
128 | @result{} -134217728 | |
129 | @result{} 1000 0000 0000 0000 0000 0000 0000 | |
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130 | @end example |
131 | ||
f9f59935 RS |
132 | Many of the functions described in this chapter accept markers for |
133 | arguments in place of numbers. (@xref{Markers}.) Since the actual | |
134 | arguments to such functions may be either numbers or markers, we often | |
135 | give these arguments the name @var{number-or-marker}. When the argument | |
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136 | value is a marker, its position value is used and its buffer is ignored. |
137 | ||
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138 | @node Float Basics |
139 | @section Floating Point Basics | |
140 | ||
f9f59935 RS |
141 | Floating point numbers are useful for representing numbers that are |
142 | not integral. The precise range of floating point numbers is | |
143 | machine-specific; it is the same as the range of the C data type | |
144 | @code{double} on the machine you are using. | |
e6512bcf | 145 | |
f9f59935 RS |
146 | The read-syntax for floating point numbers requires either a decimal |
147 | point (with at least one digit following), an exponent, or both. For | |
148 | example, @samp{1500.0}, @samp{15e2}, @samp{15.0e2}, @samp{1.5e3}, and | |
149 | @samp{.15e4} are five ways of writing a floating point number whose | |
150 | value is 1500. They are all equivalent. You can also use a minus sign | |
151 | to write negative floating point numbers, as in @samp{-1.0}. | |
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152 | |
153 | @cindex IEEE floating point | |
154 | @cindex positive infinity | |
155 | @cindex negative infinity | |
156 | @cindex infinity | |
157 | @cindex NaN | |
158 | Most modern computers support the IEEE floating point standard, which | |
159 | provides for positive infinity and negative infinity as floating point | |
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160 | values. It also provides for a class of values called NaN or |
161 | ``not-a-number''; numerical functions return such values in cases where | |
162 | there is no correct answer. For example, @code{(sqrt -1.0)} returns a | |
163 | NaN. For practical purposes, there's no significant difference between | |
164 | different NaN values in Emacs Lisp, and there's no rule for precisely | |
969fe9b5 | 165 | which NaN value should be used in a particular case, so Emacs Lisp |
f9f59935 | 166 | doesn't try to distinguish them. Here are the read syntaxes for |
969fe9b5 | 167 | these special floating point values: |
f9f59935 RS |
168 | |
169 | @table @asis | |
170 | @item positive infinity | |
171 | @samp{1.0e+INF} | |
172 | @item negative infinity | |
173 | @samp{-1.0e+INF} | |
174 | @item Not-a-number | |
175 | @samp{0.0e+NaN}. | |
176 | @end table | |
e6512bcf | 177 | |
969fe9b5 RS |
178 | In addition, the value @code{-0.0} is distinguishable from ordinary |
179 | zero in IEEE floating point (although @code{equal} and @code{=} consider | |
180 | them equal values). | |
181 | ||
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182 | You can use @code{logb} to extract the binary exponent of a floating |
183 | point number (or estimate the logarithm of an integer): | |
184 | ||
185 | @defun logb number | |
186 | This function returns the binary exponent of @var{number}. More | |
187 | precisely, the value is the logarithm of @var{number} base 2, rounded | |
188 | down to an integer. | |
f9f59935 RS |
189 | |
190 | @example | |
191 | (logb 10) | |
192 | @result{} 3 | |
193 | (logb 10.0e20) | |
194 | @result{} 69 | |
195 | @end example | |
e6512bcf RS |
196 | @end defun |
197 | ||
198 | @node Predicates on Numbers | |
199 | @section Type Predicates for Numbers | |
200 | ||
201 | The functions in this section test whether the argument is a number or | |
202 | whether it is a certain sort of number. The functions @code{integerp} | |
203 | and @code{floatp} can take any type of Lisp object as argument (the | |
204 | predicates would not be of much use otherwise); but the @code{zerop} | |
205 | predicate requires a number as its argument. See also | |
206 | @code{integer-or-marker-p} and @code{number-or-marker-p}, in | |
207 | @ref{Predicates on Markers}. | |
208 | ||
209 | @defun floatp object | |
210 | This predicate tests whether its argument is a floating point | |
211 | number and returns @code{t} if so, @code{nil} otherwise. | |
212 | ||
213 | @code{floatp} does not exist in Emacs versions 18 and earlier. | |
214 | @end defun | |
215 | ||
216 | @defun integerp object | |
217 | This predicate tests whether its argument is an integer, and returns | |
218 | @code{t} if so, @code{nil} otherwise. | |
219 | @end defun | |
220 | ||
221 | @defun numberp object | |
222 | This predicate tests whether its argument is a number (either integer or | |
223 | floating point), and returns @code{t} if so, @code{nil} otherwise. | |
224 | @end defun | |
225 | ||
05fd2b65 | 226 | @defun wholenump object |
e6512bcf | 227 | @cindex natural numbers |
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228 | The @code{wholenump} predicate (whose name comes from the phrase |
229 | ``whole-number-p'') tests to see whether its argument is a nonnegative | |
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230 | integer, and returns @code{t} if so, @code{nil} otherwise. 0 is |
231 | considered non-negative. | |
232 | ||
05fd2b65 RS |
233 | @findex natnump |
234 | @code{natnump} is an obsolete synonym for @code{wholenump}. | |
e6512bcf RS |
235 | @end defun |
236 | ||
237 | @defun zerop number | |
238 | This predicate tests whether its argument is zero, and returns @code{t} | |
239 | if so, @code{nil} otherwise. The argument must be a number. | |
240 | ||
241 | These two forms are equivalent: @code{(zerop x)} @equiv{} @code{(= x 0)}. | |
242 | @end defun | |
243 | ||
244 | @node Comparison of Numbers | |
245 | @section Comparison of Numbers | |
246 | @cindex number equality | |
247 | ||
05fd2b65 RS |
248 | To test numbers for numerical equality, you should normally use |
249 | @code{=}, not @code{eq}. There can be many distinct floating point | |
250 | number objects with the same numeric value. If you use @code{eq} to | |
251 | compare them, then you test whether two values are the same | |
252 | @emph{object}. By contrast, @code{=} compares only the numeric values | |
253 | of the objects. | |
254 | ||
255 | At present, each integer value has a unique Lisp object in Emacs Lisp. | |
f9f59935 | 256 | Therefore, @code{eq} is equivalent to @code{=} where integers are |
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257 | concerned. It is sometimes convenient to use @code{eq} for comparing an |
258 | unknown value with an integer, because @code{eq} does not report an | |
259 | error if the unknown value is not a number---it accepts arguments of any | |
260 | type. By contrast, @code{=} signals an error if the arguments are not | |
261 | numbers or markers. However, it is a good idea to use @code{=} if you | |
262 | can, even for comparing integers, just in case we change the | |
263 | representation of integers in a future Emacs version. | |
e6512bcf | 264 | |
a9f0a989 RS |
265 | Sometimes it is useful to compare numbers with @code{equal}; it treats |
266 | two numbers as equal if they have the same data type (both integers, or | |
267 | both floating point) and the same value. By contrast, @code{=} can | |
268 | treat an integer and a floating point number as equal. | |
269 | ||
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270 | There is another wrinkle: because floating point arithmetic is not |
271 | exact, it is often a bad idea to check for equality of two floating | |
272 | point values. Usually it is better to test for approximate equality. | |
273 | Here's a function to do this: | |
274 | ||
275 | @example | |
276 | (defvar fuzz-factor 1.0e-6) | |
277 | (defun approx-equal (x y) | |
bfe721d1 KH |
278 | (or (and (= x 0) (= y 0)) |
279 | (< (/ (abs (- x y)) | |
280 | (max (abs x) (abs y))) | |
281 | fuzz-factor))) | |
e6512bcf RS |
282 | @end example |
283 | ||
284 | @cindex CL note---integers vrs @code{eq} | |
285 | @quotation | |
05fd2b65 | 286 | @b{Common Lisp note:} Comparing numbers in Common Lisp always requires |
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287 | @code{=} because Common Lisp implements multi-word integers, and two |
288 | distinct integer objects can have the same numeric value. Emacs Lisp | |
289 | can have just one integer object for any given value because it has a | |
290 | limited range of integer values. | |
291 | @end quotation | |
292 | ||
293 | @defun = number-or-marker1 number-or-marker2 | |
294 | This function tests whether its arguments are numerically equal, and | |
295 | returns @code{t} if so, @code{nil} otherwise. | |
296 | @end defun | |
297 | ||
298 | @defun /= number-or-marker1 number-or-marker2 | |
299 | This function tests whether its arguments are numerically equal, and | |
300 | returns @code{t} if they are not, and @code{nil} if they are. | |
301 | @end defun | |
302 | ||
303 | @defun < number-or-marker1 number-or-marker2 | |
304 | This function tests whether its first argument is strictly less than | |
305 | its second argument. It returns @code{t} if so, @code{nil} otherwise. | |
306 | @end defun | |
307 | ||
308 | @defun <= number-or-marker1 number-or-marker2 | |
309 | This function tests whether its first argument is less than or equal | |
310 | to its second argument. It returns @code{t} if so, @code{nil} | |
311 | otherwise. | |
312 | @end defun | |
313 | ||
314 | @defun > number-or-marker1 number-or-marker2 | |
315 | This function tests whether its first argument is strictly greater | |
316 | than its second argument. It returns @code{t} if so, @code{nil} | |
317 | otherwise. | |
318 | @end defun | |
319 | ||
320 | @defun >= number-or-marker1 number-or-marker2 | |
321 | This function tests whether its first argument is greater than or | |
322 | equal to its second argument. It returns @code{t} if so, @code{nil} | |
323 | otherwise. | |
324 | @end defun | |
325 | ||
326 | @defun max number-or-marker &rest numbers-or-markers | |
327 | This function returns the largest of its arguments. | |
8241495d RS |
328 | If any of the argument is floating-point, the value is returned |
329 | as floating point, even if it was given as an integer. | |
e6512bcf RS |
330 | |
331 | @example | |
332 | (max 20) | |
333 | @result{} 20 | |
334 | (max 1 2.5) | |
335 | @result{} 2.5 | |
336 | (max 1 3 2.5) | |
8241495d | 337 | @result{} 3.0 |
e6512bcf RS |
338 | @end example |
339 | @end defun | |
340 | ||
341 | @defun min number-or-marker &rest numbers-or-markers | |
342 | This function returns the smallest of its arguments. | |
8241495d RS |
343 | If any of the argument is floating-point, the value is returned |
344 | as floating point, even if it was given as an integer. | |
e6512bcf RS |
345 | |
346 | @example | |
347 | (min -4 1) | |
348 | @result{} -4 | |
349 | @end example | |
350 | @end defun | |
351 | ||
969fe9b5 | 352 | @defun abs number |
a9f0a989 | 353 | This function returns the absolute value of @var{number}. |
969fe9b5 RS |
354 | @end defun |
355 | ||
e6512bcf RS |
356 | @node Numeric Conversions |
357 | @section Numeric Conversions | |
358 | @cindex rounding in conversions | |
359 | ||
360 | To convert an integer to floating point, use the function @code{float}. | |
361 | ||
362 | @defun float number | |
363 | This returns @var{number} converted to floating point. | |
364 | If @var{number} is already a floating point number, @code{float} returns | |
365 | it unchanged. | |
366 | @end defun | |
367 | ||
368 | There are four functions to convert floating point numbers to integers; | |
369 | they differ in how they round. These functions accept integer arguments | |
370 | also, and return such arguments unchanged. | |
371 | ||
372 | @defun truncate number | |
373 | This returns @var{number}, converted to an integer by rounding towards | |
374 | zero. | |
b7a2fc9b RS |
375 | |
376 | @example | |
377 | (truncate 1.2) | |
378 | @result{} 1 | |
379 | (truncate 1.7) | |
380 | @result{} 1 | |
381 | (truncate -1.2) | |
382 | @result{} -1 | |
383 | (truncate -1.7) | |
384 | @result{} -1 | |
385 | @end example | |
e6512bcf RS |
386 | @end defun |
387 | ||
388 | @defun floor number &optional divisor | |
389 | This returns @var{number}, converted to an integer by rounding downward | |
390 | (towards negative infinity). | |
391 | ||
b7a2fc9b RS |
392 | If @var{divisor} is specified, @code{floor} divides @var{number} by |
393 | @var{divisor} and then converts to an integer; this uses the kind of | |
394 | division operation that corresponds to @code{mod}, rounding downward. | |
395 | An @code{arith-error} results if @var{divisor} is 0. | |
396 | ||
397 | @example | |
398 | (floor 1.2) | |
399 | @result{} 1 | |
400 | (floor 1.7) | |
401 | @result{} 1 | |
402 | (floor -1.2) | |
403 | @result{} -2 | |
404 | (floor -1.7) | |
405 | @result{} -2 | |
406 | (floor 5.99 3) | |
407 | @result{} 1 | |
408 | @end example | |
e6512bcf RS |
409 | @end defun |
410 | ||
411 | @defun ceiling number | |
412 | This returns @var{number}, converted to an integer by rounding upward | |
413 | (towards positive infinity). | |
b7a2fc9b RS |
414 | |
415 | @example | |
416 | (ceiling 1.2) | |
417 | @result{} 2 | |
418 | (ceiling 1.7) | |
419 | @result{} 2 | |
420 | (ceiling -1.2) | |
421 | @result{} -1 | |
422 | (ceiling -1.7) | |
423 | @result{} -1 | |
424 | @end example | |
e6512bcf RS |
425 | @end defun |
426 | ||
427 | @defun round number | |
428 | This returns @var{number}, converted to an integer by rounding towards the | |
bfe721d1 KH |
429 | nearest integer. Rounding a value equidistant between two integers |
430 | may choose the integer closer to zero, or it may prefer an even integer, | |
431 | depending on your machine. | |
b7a2fc9b RS |
432 | |
433 | @example | |
434 | (round 1.2) | |
435 | @result{} 1 | |
436 | (round 1.7) | |
437 | @result{} 2 | |
438 | (round -1.2) | |
439 | @result{} -1 | |
440 | (round -1.7) | |
441 | @result{} -2 | |
442 | @end example | |
e6512bcf RS |
443 | @end defun |
444 | ||
445 | @node Arithmetic Operations | |
446 | @section Arithmetic Operations | |
447 | ||
448 | Emacs Lisp provides the traditional four arithmetic operations: | |
449 | addition, subtraction, multiplication, and division. Remainder and modulus | |
450 | functions supplement the division functions. The functions to | |
451 | add or subtract 1 are provided because they are traditional in Lisp and | |
452 | commonly used. | |
453 | ||
454 | All of these functions except @code{%} return a floating point value | |
455 | if any argument is floating. | |
456 | ||
969fe9b5 | 457 | It is important to note that in Emacs Lisp, arithmetic functions |
22697dac KH |
458 | do not check for overflow. Thus @code{(1+ 134217727)} may evaluate to |
459 | @minus{}134217728, depending on your hardware. | |
e6512bcf RS |
460 | |
461 | @defun 1+ number-or-marker | |
462 | This function returns @var{number-or-marker} plus 1. | |
463 | For example, | |
464 | ||
465 | @example | |
466 | (setq foo 4) | |
467 | @result{} 4 | |
468 | (1+ foo) | |
469 | @result{} 5 | |
470 | @end example | |
471 | ||
bfe721d1 KH |
472 | This function is not analogous to the C operator @code{++}---it does not |
473 | increment a variable. It just computes a sum. Thus, if we continue, | |
e6512bcf RS |
474 | |
475 | @example | |
476 | foo | |
477 | @result{} 4 | |
478 | @end example | |
479 | ||
480 | If you want to increment the variable, you must use @code{setq}, | |
481 | like this: | |
482 | ||
483 | @example | |
484 | (setq foo (1+ foo)) | |
485 | @result{} 5 | |
486 | @end example | |
487 | @end defun | |
488 | ||
489 | @defun 1- number-or-marker | |
490 | This function returns @var{number-or-marker} minus 1. | |
491 | @end defun | |
492 | ||
e6512bcf RS |
493 | @defun + &rest numbers-or-markers |
494 | This function adds its arguments together. When given no arguments, | |
bfe721d1 | 495 | @code{+} returns 0. |
e6512bcf RS |
496 | |
497 | @example | |
498 | (+) | |
499 | @result{} 0 | |
500 | (+ 1) | |
501 | @result{} 1 | |
502 | (+ 1 2 3 4) | |
503 | @result{} 10 | |
504 | @end example | |
505 | @end defun | |
506 | ||
f9f59935 | 507 | @defun - &optional number-or-marker &rest more-numbers-or-markers |
e6512bcf RS |
508 | The @code{-} function serves two purposes: negation and subtraction. |
509 | When @code{-} has a single argument, the value is the negative of the | |
510 | argument. When there are multiple arguments, @code{-} subtracts each of | |
f9f59935 | 511 | the @var{more-numbers-or-markers} from @var{number-or-marker}, |
bfe721d1 | 512 | cumulatively. If there are no arguments, the result is 0. |
e6512bcf RS |
513 | |
514 | @example | |
515 | (- 10 1 2 3 4) | |
516 | @result{} 0 | |
517 | (- 10) | |
518 | @result{} -10 | |
519 | (-) | |
520 | @result{} 0 | |
521 | @end example | |
522 | @end defun | |
523 | ||
524 | @defun * &rest numbers-or-markers | |
525 | This function multiplies its arguments together, and returns the | |
bfe721d1 | 526 | product. When given no arguments, @code{*} returns 1. |
e6512bcf RS |
527 | |
528 | @example | |
529 | (*) | |
530 | @result{} 1 | |
531 | (* 1) | |
532 | @result{} 1 | |
533 | (* 1 2 3 4) | |
534 | @result{} 24 | |
535 | @end example | |
536 | @end defun | |
537 | ||
538 | @defun / dividend divisor &rest divisors | |
05fd2b65 | 539 | This function divides @var{dividend} by @var{divisor} and returns the |
e6512bcf RS |
540 | quotient. If there are additional arguments @var{divisors}, then it |
541 | divides @var{dividend} by each divisor in turn. Each argument may be a | |
542 | number or a marker. | |
543 | ||
544 | If all the arguments are integers, then the result is an integer too. | |
545 | This means the result has to be rounded. On most machines, the result | |
546 | is rounded towards zero after each division, but some machines may round | |
547 | differently with negative arguments. This is because the Lisp function | |
548 | @code{/} is implemented using the C division operator, which also | |
549 | permits machine-dependent rounding. As a practical matter, all known | |
550 | machines round in the standard fashion. | |
551 | ||
552 | @cindex @code{arith-error} in division | |
969fe9b5 RS |
553 | If you divide an integer by 0, an @code{arith-error} error is signaled. |
554 | (@xref{Errors}.) Floating point division by zero returns either | |
555 | infinity or a NaN if your machine supports IEEE floating point; | |
556 | otherwise, it signals an @code{arith-error} error. | |
e6512bcf RS |
557 | |
558 | @example | |
bda144f4 | 559 | @group |
e6512bcf RS |
560 | (/ 6 2) |
561 | @result{} 3 | |
bda144f4 | 562 | @end group |
e6512bcf RS |
563 | (/ 5 2) |
564 | @result{} 2 | |
969fe9b5 RS |
565 | (/ 5.0 2) |
566 | @result{} 2.5 | |
567 | (/ 5 2.0) | |
568 | @result{} 2.5 | |
569 | (/ 5.0 2.0) | |
570 | @result{} 2.5 | |
e6512bcf RS |
571 | (/ 25 3 2) |
572 | @result{} 4 | |
573 | (/ -17 6) | |
574 | @result{} -2 | |
575 | @end example | |
576 | ||
577 | The result of @code{(/ -17 6)} could in principle be -3 on some | |
578 | machines. | |
579 | @end defun | |
580 | ||
581 | @defun % dividend divisor | |
582 | @cindex remainder | |
583 | This function returns the integer remainder after division of @var{dividend} | |
584 | by @var{divisor}. The arguments must be integers or markers. | |
585 | ||
586 | For negative arguments, the remainder is in principle machine-dependent | |
587 | since the quotient is; but in practice, all known machines behave alike. | |
588 | ||
589 | An @code{arith-error} results if @var{divisor} is 0. | |
590 | ||
591 | @example | |
592 | (% 9 4) | |
593 | @result{} 1 | |
594 | (% -9 4) | |
595 | @result{} -1 | |
596 | (% 9 -4) | |
597 | @result{} 1 | |
598 | (% -9 -4) | |
599 | @result{} -1 | |
600 | @end example | |
601 | ||
602 | For any two integers @var{dividend} and @var{divisor}, | |
603 | ||
604 | @example | |
605 | @group | |
606 | (+ (% @var{dividend} @var{divisor}) | |
607 | (* (/ @var{dividend} @var{divisor}) @var{divisor})) | |
608 | @end group | |
609 | @end example | |
610 | ||
611 | @noindent | |
612 | always equals @var{dividend}. | |
613 | @end defun | |
614 | ||
615 | @defun mod dividend divisor | |
616 | @cindex modulus | |
617 | This function returns the value of @var{dividend} modulo @var{divisor}; | |
618 | in other words, the remainder after division of @var{dividend} | |
619 | by @var{divisor}, but with the same sign as @var{divisor}. | |
620 | The arguments must be numbers or markers. | |
621 | ||
622 | Unlike @code{%}, @code{mod} returns a well-defined result for negative | |
623 | arguments. It also permits floating point arguments; it rounds the | |
624 | quotient downward (towards minus infinity) to an integer, and uses that | |
625 | quotient to compute the remainder. | |
626 | ||
627 | An @code{arith-error} results if @var{divisor} is 0. | |
628 | ||
629 | @example | |
bda144f4 | 630 | @group |
e6512bcf RS |
631 | (mod 9 4) |
632 | @result{} 1 | |
bda144f4 MW |
633 | @end group |
634 | @group | |
e6512bcf RS |
635 | (mod -9 4) |
636 | @result{} 3 | |
bda144f4 MW |
637 | @end group |
638 | @group | |
e6512bcf RS |
639 | (mod 9 -4) |
640 | @result{} -3 | |
bda144f4 MW |
641 | @end group |
642 | @group | |
e6512bcf RS |
643 | (mod -9 -4) |
644 | @result{} -1 | |
bda144f4 MW |
645 | @end group |
646 | @group | |
e6512bcf RS |
647 | (mod 5.5 2.5) |
648 | @result{} .5 | |
bda144f4 | 649 | @end group |
e6512bcf RS |
650 | @end example |
651 | ||
652 | For any two numbers @var{dividend} and @var{divisor}, | |
653 | ||
654 | @example | |
655 | @group | |
656 | (+ (mod @var{dividend} @var{divisor}) | |
657 | (* (floor @var{dividend} @var{divisor}) @var{divisor})) | |
658 | @end group | |
659 | @end example | |
660 | ||
661 | @noindent | |
bfe721d1 KH |
662 | always equals @var{dividend}, subject to rounding error if either |
663 | argument is floating point. For @code{floor}, see @ref{Numeric | |
664 | Conversions}. | |
e6512bcf RS |
665 | @end defun |
666 | ||
667 | @node Rounding Operations | |
668 | @section Rounding Operations | |
669 | @cindex rounding without conversion | |
670 | ||
a9f0a989 | 671 | The functions @code{ffloor}, @code{fceiling}, @code{fround}, and |
e6512bcf RS |
672 | @code{ftruncate} take a floating point argument and return a floating |
673 | point result whose value is a nearby integer. @code{ffloor} returns the | |
bb61c6c6 | 674 | nearest integer below; @code{fceiling}, the nearest integer above; |
05fd2b65 | 675 | @code{ftruncate}, the nearest integer in the direction towards zero; |
e6512bcf RS |
676 | @code{fround}, the nearest integer. |
677 | ||
678 | @defun ffloor float | |
679 | This function rounds @var{float} to the next lower integral value, and | |
680 | returns that value as a floating point number. | |
681 | @end defun | |
682 | ||
bb61c6c6 | 683 | @defun fceiling float |
e6512bcf RS |
684 | This function rounds @var{float} to the next higher integral value, and |
685 | returns that value as a floating point number. | |
686 | @end defun | |
687 | ||
05fd2b65 | 688 | @defun ftruncate float |
e6512bcf RS |
689 | This function rounds @var{float} towards zero to an integral value, and |
690 | returns that value as a floating point number. | |
691 | @end defun | |
692 | ||
693 | @defun fround float | |
694 | This function rounds @var{float} to the nearest integral value, | |
695 | and returns that value as a floating point number. | |
696 | @end defun | |
697 | ||
698 | @node Bitwise Operations | |
699 | @section Bitwise Operations on Integers | |
700 | ||
701 | In a computer, an integer is represented as a binary number, a | |
702 | sequence of @dfn{bits} (digits which are either zero or one). A bitwise | |
703 | operation acts on the individual bits of such a sequence. For example, | |
704 | @dfn{shifting} moves the whole sequence left or right one or more places, | |
705 | reproducing the same pattern ``moved over''. | |
706 | ||
707 | The bitwise operations in Emacs Lisp apply only to integers. | |
708 | ||
709 | @defun lsh integer1 count | |
710 | @cindex logical shift | |
711 | @code{lsh}, which is an abbreviation for @dfn{logical shift}, shifts the | |
05fd2b65 RS |
712 | bits in @var{integer1} to the left @var{count} places, or to the right |
713 | if @var{count} is negative, bringing zeros into the vacated bits. If | |
714 | @var{count} is negative, @code{lsh} shifts zeros into the leftmost | |
715 | (most-significant) bit, producing a positive result even if | |
716 | @var{integer1} is negative. Contrast this with @code{ash}, below. | |
717 | ||
718 | Here are two examples of @code{lsh}, shifting a pattern of bits one | |
719 | place to the left. We show only the low-order eight bits of the binary | |
720 | pattern; the rest are all zero. | |
e6512bcf RS |
721 | |
722 | @example | |
723 | @group | |
724 | (lsh 5 1) | |
725 | @result{} 10 | |
726 | ;; @r{Decimal 5 becomes decimal 10.} | |
727 | 00000101 @result{} 00001010 | |
728 | ||
729 | (lsh 7 1) | |
730 | @result{} 14 | |
731 | ;; @r{Decimal 7 becomes decimal 14.} | |
732 | 00000111 @result{} 00001110 | |
733 | @end group | |
734 | @end example | |
735 | ||
736 | @noindent | |
737 | As the examples illustrate, shifting the pattern of bits one place to | |
738 | the left produces a number that is twice the value of the previous | |
739 | number. | |
740 | ||
bfe721d1 | 741 | Shifting a pattern of bits two places to the left produces results |
e6512bcf RS |
742 | like this (with 8-bit binary numbers): |
743 | ||
744 | @example | |
745 | @group | |
746 | (lsh 3 2) | |
747 | @result{} 12 | |
748 | ;; @r{Decimal 3 becomes decimal 12.} | |
177c0ea7 | 749 | 00000011 @result{} 00001100 |
e6512bcf RS |
750 | @end group |
751 | @end example | |
752 | ||
bfe721d1 | 753 | On the other hand, shifting one place to the right looks like this: |
e6512bcf RS |
754 | |
755 | @example | |
756 | @group | |
757 | (lsh 6 -1) | |
758 | @result{} 3 | |
759 | ;; @r{Decimal 6 becomes decimal 3.} | |
177c0ea7 | 760 | 00000110 @result{} 00000011 |
e6512bcf RS |
761 | @end group |
762 | ||
763 | @group | |
764 | (lsh 5 -1) | |
765 | @result{} 2 | |
766 | ;; @r{Decimal 5 becomes decimal 2.} | |
177c0ea7 | 767 | 00000101 @result{} 00000010 |
e6512bcf RS |
768 | @end group |
769 | @end example | |
770 | ||
771 | @noindent | |
bfe721d1 KH |
772 | As the example illustrates, shifting one place to the right divides the |
773 | value of a positive integer by two, rounding downward. | |
774 | ||
775 | The function @code{lsh}, like all Emacs Lisp arithmetic functions, does | |
776 | not check for overflow, so shifting left can discard significant bits | |
777 | and change the sign of the number. For example, left shifting | |
778 | 134,217,727 produces @minus{}2 on a 28-bit machine: | |
779 | ||
780 | @example | |
781 | (lsh 134217727 1) ; @r{left shift} | |
782 | @result{} -2 | |
783 | @end example | |
784 | ||
785 | In binary, in the 28-bit implementation, the argument looks like this: | |
786 | ||
787 | @example | |
788 | @group | |
cc8c51f1 | 789 | ;; @r{Decimal 134,217,727} |
177c0ea7 | 790 | 0111 1111 1111 1111 1111 1111 1111 |
bfe721d1 KH |
791 | @end group |
792 | @end example | |
793 | ||
794 | @noindent | |
795 | which becomes the following when left shifted: | |
796 | ||
797 | @example | |
798 | @group | |
799 | ;; @r{Decimal @minus{}2} | |
177c0ea7 | 800 | 1111 1111 1111 1111 1111 1111 1110 |
bfe721d1 KH |
801 | @end group |
802 | @end example | |
e6512bcf RS |
803 | @end defun |
804 | ||
805 | @defun ash integer1 count | |
806 | @cindex arithmetic shift | |
807 | @code{ash} (@dfn{arithmetic shift}) shifts the bits in @var{integer1} | |
808 | to the left @var{count} places, or to the right if @var{count} | |
809 | is negative. | |
810 | ||
811 | @code{ash} gives the same results as @code{lsh} except when | |
812 | @var{integer1} and @var{count} are both negative. In that case, | |
bfe721d1 KH |
813 | @code{ash} puts ones in the empty bit positions on the left, while |
814 | @code{lsh} puts zeros in those bit positions. | |
e6512bcf RS |
815 | |
816 | Thus, with @code{ash}, shifting the pattern of bits one place to the right | |
817 | looks like this: | |
818 | ||
819 | @example | |
820 | @group | |
177c0ea7 | 821 | (ash -6 -1) @result{} -3 |
e6512bcf | 822 | ;; @r{Decimal @minus{}6 becomes decimal @minus{}3.} |
8be5e0fc | 823 | 1111 1111 1111 1111 1111 1111 1010 |
177c0ea7 | 824 | @result{} |
8be5e0fc | 825 | 1111 1111 1111 1111 1111 1111 1101 |
e6512bcf RS |
826 | @end group |
827 | @end example | |
828 | ||
829 | In contrast, shifting the pattern of bits one place to the right with | |
830 | @code{lsh} looks like this: | |
831 | ||
832 | @example | |
833 | @group | |
8be5e0fc RS |
834 | (lsh -6 -1) @result{} 134217725 |
835 | ;; @r{Decimal @minus{}6 becomes decimal 134,217,725.} | |
836 | 1111 1111 1111 1111 1111 1111 1010 | |
177c0ea7 | 837 | @result{} |
8be5e0fc | 838 | 0111 1111 1111 1111 1111 1111 1101 |
e6512bcf RS |
839 | @end group |
840 | @end example | |
841 | ||
e6512bcf RS |
842 | Here are other examples: |
843 | ||
844 | @c !!! Check if lined up in smallbook format! XDVI shows problem | |
845 | @c with smallbook but not with regular book! --rjc 16mar92 | |
846 | @smallexample | |
847 | @group | |
8be5e0fc | 848 | ; @r{ 28-bit binary values} |
e6512bcf | 849 | |
8be5e0fc RS |
850 | (lsh 5 2) ; 5 = @r{0000 0000 0000 0000 0000 0000 0101} |
851 | @result{} 20 ; = @r{0000 0000 0000 0000 0000 0001 0100} | |
e6512bcf RS |
852 | @end group |
853 | @group | |
854 | (ash 5 2) | |
855 | @result{} 20 | |
8be5e0fc RS |
856 | (lsh -5 2) ; -5 = @r{1111 1111 1111 1111 1111 1111 1011} |
857 | @result{} -20 ; = @r{1111 1111 1111 1111 1111 1110 1100} | |
e6512bcf RS |
858 | (ash -5 2) |
859 | @result{} -20 | |
860 | @end group | |
861 | @group | |
8be5e0fc RS |
862 | (lsh 5 -2) ; 5 = @r{0000 0000 0000 0000 0000 0000 0101} |
863 | @result{} 1 ; = @r{0000 0000 0000 0000 0000 0000 0001} | |
e6512bcf RS |
864 | @end group |
865 | @group | |
866 | (ash 5 -2) | |
867 | @result{} 1 | |
868 | @end group | |
869 | @group | |
8be5e0fc RS |
870 | (lsh -5 -2) ; -5 = @r{1111 1111 1111 1111 1111 1111 1011} |
871 | @result{} 4194302 ; = @r{0011 1111 1111 1111 1111 1111 1110} | |
e6512bcf RS |
872 | @end group |
873 | @group | |
8be5e0fc RS |
874 | (ash -5 -2) ; -5 = @r{1111 1111 1111 1111 1111 1111 1011} |
875 | @result{} -2 ; = @r{1111 1111 1111 1111 1111 1111 1110} | |
e6512bcf RS |
876 | @end group |
877 | @end smallexample | |
878 | @end defun | |
879 | ||
880 | @defun logand &rest ints-or-markers | |
881 | @cindex logical and | |
882 | @cindex bitwise and | |
883 | This function returns the ``logical and'' of the arguments: the | |
884 | @var{n}th bit is set in the result if, and only if, the @var{n}th bit is | |
885 | set in all the arguments. (``Set'' means that the value of the bit is 1 | |
886 | rather than 0.) | |
887 | ||
888 | For example, using 4-bit binary numbers, the ``logical and'' of 13 and | |
889 | 12 is 12: 1101 combined with 1100 produces 1100. | |
e6512bcf RS |
890 | In both the binary numbers, the leftmost two bits are set (i.e., they |
891 | are 1's), so the leftmost two bits of the returned value are set. | |
892 | However, for the rightmost two bits, each is zero in at least one of | |
893 | the arguments, so the rightmost two bits of the returned value are 0's. | |
894 | ||
895 | @noindent | |
896 | Therefore, | |
897 | ||
898 | @example | |
899 | @group | |
900 | (logand 13 12) | |
901 | @result{} 12 | |
902 | @end group | |
903 | @end example | |
904 | ||
905 | If @code{logand} is not passed any argument, it returns a value of | |
906 | @minus{}1. This number is an identity element for @code{logand} | |
907 | because its binary representation consists entirely of ones. If | |
908 | @code{logand} is passed just one argument, it returns that argument. | |
909 | ||
910 | @smallexample | |
911 | @group | |
8be5e0fc | 912 | ; @r{ 28-bit binary values} |
e6512bcf | 913 | |
8be5e0fc RS |
914 | (logand 14 13) ; 14 = @r{0000 0000 0000 0000 0000 0000 1110} |
915 | ; 13 = @r{0000 0000 0000 0000 0000 0000 1101} | |
916 | @result{} 12 ; 12 = @r{0000 0000 0000 0000 0000 0000 1100} | |
e6512bcf RS |
917 | @end group |
918 | ||
919 | @group | |
8be5e0fc RS |
920 | (logand 14 13 4) ; 14 = @r{0000 0000 0000 0000 0000 0000 1110} |
921 | ; 13 = @r{0000 0000 0000 0000 0000 0000 1101} | |
922 | ; 4 = @r{0000 0000 0000 0000 0000 0000 0100} | |
923 | @result{} 4 ; 4 = @r{0000 0000 0000 0000 0000 0000 0100} | |
e6512bcf RS |
924 | @end group |
925 | ||
926 | @group | |
927 | (logand) | |
8be5e0fc | 928 | @result{} -1 ; -1 = @r{1111 1111 1111 1111 1111 1111 1111} |
e6512bcf RS |
929 | @end group |
930 | @end smallexample | |
931 | @end defun | |
932 | ||
933 | @defun logior &rest ints-or-markers | |
934 | @cindex logical inclusive or | |
935 | @cindex bitwise or | |
936 | This function returns the ``inclusive or'' of its arguments: the @var{n}th bit | |
937 | is set in the result if, and only if, the @var{n}th bit is set in at least | |
938 | one of the arguments. If there are no arguments, the result is zero, | |
939 | which is an identity element for this operation. If @code{logior} is | |
940 | passed just one argument, it returns that argument. | |
941 | ||
942 | @smallexample | |
943 | @group | |
8be5e0fc | 944 | ; @r{ 28-bit binary values} |
e6512bcf | 945 | |
8be5e0fc RS |
946 | (logior 12 5) ; 12 = @r{0000 0000 0000 0000 0000 0000 1100} |
947 | ; 5 = @r{0000 0000 0000 0000 0000 0000 0101} | |
948 | @result{} 13 ; 13 = @r{0000 0000 0000 0000 0000 0000 1101} | |
e6512bcf RS |
949 | @end group |
950 | ||
951 | @group | |
8be5e0fc RS |
952 | (logior 12 5 7) ; 12 = @r{0000 0000 0000 0000 0000 0000 1100} |
953 | ; 5 = @r{0000 0000 0000 0000 0000 0000 0101} | |
954 | ; 7 = @r{0000 0000 0000 0000 0000 0000 0111} | |
955 | @result{} 15 ; 15 = @r{0000 0000 0000 0000 0000 0000 1111} | |
e6512bcf RS |
956 | @end group |
957 | @end smallexample | |
958 | @end defun | |
959 | ||
960 | @defun logxor &rest ints-or-markers | |
961 | @cindex bitwise exclusive or | |
962 | @cindex logical exclusive or | |
963 | This function returns the ``exclusive or'' of its arguments: the | |
05fd2b65 RS |
964 | @var{n}th bit is set in the result if, and only if, the @var{n}th bit is |
965 | set in an odd number of the arguments. If there are no arguments, the | |
966 | result is 0, which is an identity element for this operation. If | |
967 | @code{logxor} is passed just one argument, it returns that argument. | |
e6512bcf RS |
968 | |
969 | @smallexample | |
970 | @group | |
8be5e0fc | 971 | ; @r{ 28-bit binary values} |
e6512bcf | 972 | |
8be5e0fc RS |
973 | (logxor 12 5) ; 12 = @r{0000 0000 0000 0000 0000 0000 1100} |
974 | ; 5 = @r{0000 0000 0000 0000 0000 0000 0101} | |
975 | @result{} 9 ; 9 = @r{0000 0000 0000 0000 0000 0000 1001} | |
e6512bcf RS |
976 | @end group |
977 | ||
978 | @group | |
8be5e0fc RS |
979 | (logxor 12 5 7) ; 12 = @r{0000 0000 0000 0000 0000 0000 1100} |
980 | ; 5 = @r{0000 0000 0000 0000 0000 0000 0101} | |
981 | ; 7 = @r{0000 0000 0000 0000 0000 0000 0111} | |
982 | @result{} 14 ; 14 = @r{0000 0000 0000 0000 0000 0000 1110} | |
e6512bcf RS |
983 | @end group |
984 | @end smallexample | |
985 | @end defun | |
986 | ||
987 | @defun lognot integer | |
988 | @cindex logical not | |
989 | @cindex bitwise not | |
990 | This function returns the logical complement of its argument: the @var{n}th | |
991 | bit is one in the result if, and only if, the @var{n}th bit is zero in | |
992 | @var{integer}, and vice-versa. | |
993 | ||
994 | @example | |
177c0ea7 | 995 | (lognot 5) |
e6512bcf | 996 | @result{} -6 |
8be5e0fc | 997 | ;; 5 = @r{0000 0000 0000 0000 0000 0000 0101} |
e6512bcf | 998 | ;; @r{becomes} |
8be5e0fc | 999 | ;; -6 = @r{1111 1111 1111 1111 1111 1111 1010} |
e6512bcf RS |
1000 | @end example |
1001 | @end defun | |
1002 | ||
4ac771a0 RS |
1003 | @node Math Functions |
1004 | @section Standard Mathematical Functions | |
e6512bcf RS |
1005 | @cindex transcendental functions |
1006 | @cindex mathematical functions | |
1007 | ||
969fe9b5 RS |
1008 | These mathematical functions allow integers as well as floating point |
1009 | numbers as arguments. | |
e6512bcf RS |
1010 | |
1011 | @defun sin arg | |
1012 | @defunx cos arg | |
1013 | @defunx tan arg | |
1014 | These are the ordinary trigonometric functions, with argument measured | |
1015 | in radians. | |
1016 | @end defun | |
1017 | ||
1018 | @defun asin arg | |
15da7853 | 1019 | The value of @code{(asin @var{arg})} is a number between |
37680279 | 1020 | @ifnottex |
15da7853 | 1021 | @minus{}pi/2 |
37680279 | 1022 | @end ifnottex |
15da7853 | 1023 | @tex |
8241495d | 1024 | @math{-\pi/2} |
15da7853 KH |
1025 | @end tex |
1026 | and | |
37680279 | 1027 | @ifnottex |
15da7853 | 1028 | pi/2 |
37680279 | 1029 | @end ifnottex |
15da7853 | 1030 | @tex |
8241495d | 1031 | @math{\pi/2} |
15da7853 KH |
1032 | @end tex |
1033 | (inclusive) whose sine is @var{arg}; if, however, @var{arg} | |
e6512bcf RS |
1034 | is out of range (outside [-1, 1]), then the result is a NaN. |
1035 | @end defun | |
1036 | ||
1037 | @defun acos arg | |
15da7853 | 1038 | The value of @code{(acos @var{arg})} is a number between 0 and |
37680279 | 1039 | @ifnottex |
15da7853 | 1040 | pi |
37680279 | 1041 | @end ifnottex |
15da7853 | 1042 | @tex |
8241495d | 1043 | @math{\pi} |
15da7853 | 1044 | @end tex |
e6512bcf RS |
1045 | (inclusive) whose cosine is @var{arg}; if, however, @var{arg} |
1046 | is out of range (outside [-1, 1]), then the result is a NaN. | |
1047 | @end defun | |
1048 | ||
f120431c EZ |
1049 | @defun atan y &optional x |
1050 | The value of @code{(atan @var{y})} is a number between | |
37680279 | 1051 | @ifnottex |
15da7853 | 1052 | @minus{}pi/2 |
37680279 | 1053 | @end ifnottex |
15da7853 | 1054 | @tex |
8241495d | 1055 | @math{-\pi/2} |
15da7853 KH |
1056 | @end tex |
1057 | and | |
37680279 | 1058 | @ifnottex |
15da7853 | 1059 | pi/2 |
37680279 | 1060 | @end ifnottex |
15da7853 | 1061 | @tex |
8241495d | 1062 | @math{\pi/2} |
15da7853 | 1063 | @end tex |
f120431c EZ |
1064 | (exclusive) whose tangent is @var{y}. If the optional second |
1065 | argument @var{x} is given, the value of @code{(atan y x)} is the | |
1066 | angle in radians between the vector @code{[@var{x}, @var{y}]} and the | |
1067 | @code{X} axis. | |
e6512bcf RS |
1068 | @end defun |
1069 | ||
1070 | @defun exp arg | |
a9f0a989 RS |
1071 | This is the exponential function; it returns |
1072 | @tex | |
8241495d | 1073 | @math{e} |
a9f0a989 | 1074 | @end tex |
37680279 | 1075 | @ifnottex |
a9f0a989 | 1076 | @i{e} |
37680279 | 1077 | @end ifnottex |
a9f0a989 RS |
1078 | to the power @var{arg}. |
1079 | @tex | |
8241495d | 1080 | @math{e} |
a9f0a989 | 1081 | @end tex |
37680279 | 1082 | @ifnottex |
a9f0a989 | 1083 | @i{e} |
37680279 | 1084 | @end ifnottex |
a9f0a989 RS |
1085 | is a fundamental mathematical constant also called the base of natural |
1086 | logarithms. | |
e6512bcf RS |
1087 | @end defun |
1088 | ||
1089 | @defun log arg &optional base | |
1090 | This function returns the logarithm of @var{arg}, with base @var{base}. | |
a9f0a989 RS |
1091 | If you don't specify @var{base}, the base |
1092 | @tex | |
8241495d | 1093 | @math{e} |
a9f0a989 | 1094 | @end tex |
37680279 | 1095 | @ifnottex |
a9f0a989 | 1096 | @i{e} |
37680279 | 1097 | @end ifnottex |
a9f0a989 | 1098 | is used. If @var{arg} |
e6512bcf RS |
1099 | is negative, the result is a NaN. |
1100 | @end defun | |
1101 | ||
1102 | @ignore | |
1103 | @defun expm1 arg | |
1104 | This function returns @code{(1- (exp @var{arg}))}, but it is more | |
1105 | accurate than that when @var{arg} is negative and @code{(exp @var{arg})} | |
1106 | is close to 1. | |
1107 | @end defun | |
1108 | ||
1109 | @defun log1p arg | |
1110 | This function returns @code{(log (1+ @var{arg}))}, but it is more | |
1111 | accurate than that when @var{arg} is so small that adding 1 to it would | |
1112 | lose accuracy. | |
1113 | @end defun | |
1114 | @end ignore | |
1115 | ||
1116 | @defun log10 arg | |
1117 | This function returns the logarithm of @var{arg}, with base 10. If | |
05fd2b65 RS |
1118 | @var{arg} is negative, the result is a NaN. @code{(log10 @var{x})} |
1119 | @equiv{} @code{(log @var{x} 10)}, at least approximately. | |
e6512bcf RS |
1120 | @end defun |
1121 | ||
1122 | @defun expt x y | |
8be5e0fc RS |
1123 | This function returns @var{x} raised to power @var{y}. If both |
1124 | arguments are integers and @var{y} is positive, the result is an | |
1125 | integer; in this case, it is truncated to fit the range of possible | |
1126 | integer values. | |
e6512bcf RS |
1127 | @end defun |
1128 | ||
1129 | @defun sqrt arg | |
1130 | This returns the square root of @var{arg}. If @var{arg} is negative, | |
1131 | the value is a NaN. | |
1132 | @end defun | |
1133 | ||
1134 | @node Random Numbers | |
1135 | @section Random Numbers | |
1136 | @cindex random numbers | |
1137 | ||
1138 | A deterministic computer program cannot generate true random numbers. | |
1139 | For most purposes, @dfn{pseudo-random numbers} suffice. A series of | |
1140 | pseudo-random numbers is generated in a deterministic fashion. The | |
1141 | numbers are not truly random, but they have certain properties that | |
1142 | mimic a random series. For example, all possible values occur equally | |
1143 | often in a pseudo-random series. | |
1144 | ||
1145 | In Emacs, pseudo-random numbers are generated from a ``seed'' number. | |
1146 | Starting from any given seed, the @code{random} function always | |
1147 | generates the same sequence of numbers. Emacs always starts with the | |
1148 | same seed value, so the sequence of values of @code{random} is actually | |
1149 | the same in each Emacs run! For example, in one operating system, the | |
1150 | first call to @code{(random)} after you start Emacs always returns | |
1151 | -1457731, and the second one always returns -7692030. This | |
1152 | repeatability is helpful for debugging. | |
1153 | ||
8241495d RS |
1154 | If you want random numbers that don't always come out the same, execute |
1155 | @code{(random t)}. This chooses a new seed based on the current time of | |
1156 | day and on Emacs's process @sc{id} number. | |
e6512bcf RS |
1157 | |
1158 | @defun random &optional limit | |
1159 | This function returns a pseudo-random integer. Repeated calls return a | |
1160 | series of pseudo-random integers. | |
1161 | ||
22697dac | 1162 | If @var{limit} is a positive integer, the value is chosen to be |
bfe721d1 | 1163 | nonnegative and less than @var{limit}. |
e6512bcf RS |
1164 | |
1165 | If @var{limit} is @code{t}, it means to choose a new seed based on the | |
1166 | current time of day and on Emacs's process @sc{id} number. | |
1167 | @c "Emacs'" is incorrect usage! | |
1168 | ||
1169 | On some machines, any integer representable in Lisp may be the result | |
1170 | of @code{random}. On other machines, the result can never be larger | |
1171 | than a certain maximum or less than a certain (negative) minimum. | |
1172 | @end defun |