| 1 | @c -*-texinfo-*- |
| 2 | @c This is part of the GNU Emacs Lisp Reference Manual. |
| 3 | @c Copyright (C) 1990, 1991, 1992, 1993, 1994, 1995, 1998, 1999, 2003 |
| 4 | @c Free Software Foundation, Inc. |
| 5 | @c See the file elisp.texi for copying conditions. |
| 6 | @setfilename ../info/numbers |
| 7 | @node Numbers, Strings and Characters, Lisp Data Types, Top |
| 8 | @chapter Numbers |
| 9 | @cindex integers |
| 10 | @cindex numbers |
| 11 | |
| 12 | GNU Emacs supports two numeric data types: @dfn{integers} and |
| 13 | @dfn{floating point numbers}. Integers are whole numbers such as |
| 14 | @minus{}3, 0, 7, 13, and 511. Their values are exact. Floating point |
| 15 | numbers are numbers with fractional parts, such as @minus{}4.5, 0.0, or |
| 16 | 2.71828. They can also be expressed in exponential notation: 1.5e2 |
| 17 | equals 150; in this example, @samp{e2} stands for ten to the second |
| 18 | power, and that is multiplied by 1.5. Floating point values are not |
| 19 | exact; they have a fixed, limited amount of precision. |
| 20 | |
| 21 | @menu |
| 22 | * Integer Basics:: Representation and range of integers. |
| 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. |
| 30 | * Math Functions:: Trig, exponential and logarithmic functions. |
| 31 | * Random Numbers:: Obtaining random integers, predictable or not. |
| 32 | @end menu |
| 33 | |
| 34 | @node Integer Basics |
| 35 | @comment node-name, next, previous, up |
| 36 | @section Integer Basics |
| 37 | |
| 38 | The range of values for an integer depends on the machine. The |
| 39 | minimum range is @minus{}268435456 to 268435455 (29 bits; i.e., |
| 40 | @ifnottex |
| 41 | -2**28 |
| 42 | @end ifnottex |
| 43 | @tex |
| 44 | @math{-2^{28}} |
| 45 | @end tex |
| 46 | to |
| 47 | @ifnottex |
| 48 | 2**28 - 1), |
| 49 | @end ifnottex |
| 50 | @tex |
| 51 | @math{2^{28}-1}), |
| 52 | @end tex |
| 53 | but some machines may provide a wider range. Many examples in this |
| 54 | chapter assume an integer has 29 bits. |
| 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.} |
| 65 | 536870913 ; @r{Also the integer 1, due to overflow.} |
| 66 | 0 ; @r{The integer 0.} |
| 67 | -0 ; @r{The integer 0.} |
| 68 | @end example |
| 69 | |
| 70 | @cindex integers in specific radix |
| 71 | @cindex radix for reading an integer |
| 72 | @cindex base for reading an integer |
| 73 | @cindex hex numbers |
| 74 | @cindex octal numbers |
| 75 | @cindex reading numbers in hex, octal, and binary |
| 76 | 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, |
| 82 | inclusively). Case is not significant for the letter after @samp{#} |
| 83 | (@samp{B}, @samp{O}, etc.) that denotes the radix. |
| 84 | |
| 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 | |
| 89 | In 29-bit binary, the decimal integer 5 looks like this: |
| 90 | |
| 91 | @example |
| 92 | 0 0000 0000 0000 0000 0000 0000 0101 |
| 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 |
| 102 | 1 1111 1111 1111 1111 1111 1111 1111 |
| 103 | @end example |
| 104 | |
| 105 | @noindent |
| 106 | @cindex two's complement |
| 107 | @minus{}1 is represented as 29 ones. (This is called @dfn{two's |
| 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 |
| 115 | 1 1111 1111 1111 1111 1111 1111 1011 |
| 116 | @end example |
| 117 | |
| 118 | In this implementation, the largest 29-bit binary integer value is |
| 119 | 268,435,455 in decimal. In binary, it looks like this: |
| 120 | |
| 121 | @example |
| 122 | 0 1111 1111 1111 1111 1111 1111 1111 |
| 123 | @end example |
| 124 | |
| 125 | Since the arithmetic functions do not check whether integers go |
| 126 | outside their range, when you add 1 to 268,435,455, the value is the |
| 127 | negative integer @minus{}268,435,456: |
| 128 | |
| 129 | @example |
| 130 | (+ 1 268435455) |
| 131 | @result{} -268435456 |
| 132 | @result{} 1 0000 0000 0000 0000 0000 0000 0000 |
| 133 | @end example |
| 134 | |
| 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 |
| 139 | value is a marker, its position value is used and its buffer is ignored. |
| 140 | |
| 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 | |
| 151 | @node Float Basics |
| 152 | @section Floating Point Basics |
| 153 | |
| 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. |
| 158 | |
| 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}. |
| 165 | |
| 166 | @cindex @acronym{IEEE} floating point |
| 167 | @cindex positive infinity |
| 168 | @cindex negative infinity |
| 169 | @cindex infinity |
| 170 | @cindex NaN |
| 171 | Most modern computers support the @acronym{IEEE} floating point standard, |
| 172 | which provides for positive infinity and negative infinity as floating point |
| 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 |
| 175 | there is no correct answer. For example, @code{(/ 0.0 0.0)} returns a |
| 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 |
| 178 | which NaN value should be used in a particular case, so Emacs Lisp |
| 179 | doesn't try to distinguish them. Here are the read syntaxes for |
| 180 | these special floating point values: |
| 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 |
| 190 | |
| 191 | In addition, the value @code{-0.0} is distinguishable from ordinary |
| 192 | zero in @acronym{IEEE} floating point (although @code{equal} and |
| 193 | @code{=} consider them equal values). |
| 194 | |
| 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. |
| 202 | |
| 203 | @example |
| 204 | (logb 10) |
| 205 | @result{} 3 |
| 206 | (logb 10.0e20) |
| 207 | @result{} 69 |
| 208 | @end example |
| 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 | |
| 239 | @defun wholenump object |
| 240 | @cindex natural numbers |
| 241 | The @code{wholenump} predicate (whose name comes from the phrase |
| 242 | ``whole-number-p'') tests to see whether its argument is a nonnegative |
| 243 | integer, and returns @code{t} if so, @code{nil} otherwise. 0 is |
| 244 | considered non-negative. |
| 245 | |
| 246 | @findex natnump |
| 247 | @code{natnump} is an obsolete synonym for @code{wholenump}. |
| 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 | |
| 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. |
| 269 | Therefore, @code{eq} is equivalent to @code{=} where integers are |
| 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. |
| 277 | |
| 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 | |
| 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) |
| 291 | (or (and (= x 0) (= y 0)) |
| 292 | (< (/ (abs (- x y)) |
| 293 | (max (abs x) (abs y))) |
| 294 | fuzz-factor))) |
| 295 | @end example |
| 296 | |
| 297 | @cindex CL note---integers vrs @code{eq} |
| 298 | @quotation |
| 299 | @b{Common Lisp note:} Comparing numbers in Common Lisp always requires |
| 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. |
| 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. |
| 343 | |
| 344 | @example |
| 345 | (max 20) |
| 346 | @result{} 20 |
| 347 | (max 1 2.5) |
| 348 | @result{} 2.5 |
| 349 | (max 1 3 2.5) |
| 350 | @result{} 3.0 |
| 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. |
| 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. |
| 358 | |
| 359 | @example |
| 360 | (min -4 1) |
| 361 | @result{} -4 |
| 362 | @end example |
| 363 | @end defun |
| 364 | |
| 365 | @defun abs number |
| 366 | This function returns the absolute value of @var{number}. |
| 367 | @end defun |
| 368 | |
| 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; |
| 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 |
| 392 | This returns @var{number}, converted to an integer by rounding towards |
| 393 | zero. |
| 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 |
| 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 | |
| 411 | If @var{divisor} is specified, this uses the kind of division |
| 412 | operation that corresponds to @code{mod}, rounding downward. |
| 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 |
| 426 | @end defun |
| 427 | |
| 428 | @defun ceiling number &optional divisor |
| 429 | This returns @var{number}, converted to an integer by rounding upward |
| 430 | (towards positive infinity). |
| 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 |
| 442 | @end defun |
| 443 | |
| 444 | @defun round number &optional divisor |
| 445 | This returns @var{number}, converted to an integer by rounding towards the |
| 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. |
| 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 |
| 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 | |
| 474 | It is important to note that in Emacs Lisp, arithmetic functions |
| 475 | do not check for overflow. Thus @code{(1+ 268435455)} may evaluate to |
| 476 | @minus{}268435456, depending on your hardware. |
| 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 | |
| 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, |
| 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 | |
| 510 | @defun + &rest numbers-or-markers |
| 511 | This function adds its arguments together. When given no arguments, |
| 512 | @code{+} returns 0. |
| 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 | |
| 524 | @defun - &optional number-or-marker &rest more-numbers-or-markers |
| 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 |
| 528 | the @var{more-numbers-or-markers} from @var{number-or-marker}, |
| 529 | cumulatively. If there are no arguments, the result is 0. |
| 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 |
| 543 | product. When given no arguments, @code{*} returns 1. |
| 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 |
| 556 | This function divides @var{dividend} by @var{divisor} and returns the |
| 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 |
| 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 |
| 572 | infinity or a NaN if your machine supports @acronym{IEEE} floating point; |
| 573 | otherwise, it signals an @code{arith-error} error. |
| 574 | |
| 575 | @example |
| 576 | @group |
| 577 | (/ 6 2) |
| 578 | @result{} 3 |
| 579 | @end group |
| 580 | (/ 5 2) |
| 581 | @result{} 2 |
| 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 |
| 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 |
| 647 | @group |
| 648 | (mod 9 4) |
| 649 | @result{} 1 |
| 650 | @end group |
| 651 | @group |
| 652 | (mod -9 4) |
| 653 | @result{} 3 |
| 654 | @end group |
| 655 | @group |
| 656 | (mod 9 -4) |
| 657 | @result{} -3 |
| 658 | @end group |
| 659 | @group |
| 660 | (mod -9 -4) |
| 661 | @result{} -1 |
| 662 | @end group |
| 663 | @group |
| 664 | (mod 5.5 2.5) |
| 665 | @result{} .5 |
| 666 | @end group |
| 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 |
| 679 | always equals @var{dividend}, subject to rounding error if either |
| 680 | argument is floating point. For @code{floor}, see @ref{Numeric |
| 681 | Conversions}. |
| 682 | @end defun |
| 683 | |
| 684 | @node Rounding Operations |
| 685 | @section Rounding Operations |
| 686 | @cindex rounding without conversion |
| 687 | |
| 688 | The functions @code{ffloor}, @code{fceiling}, @code{fround}, and |
| 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 |
| 691 | nearest integer below; @code{fceiling}, the nearest integer above; |
| 692 | @code{ftruncate}, the nearest integer in the direction towards zero; |
| 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 | |
| 700 | @defun fceiling float |
| 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 | |
| 705 | @defun ftruncate float |
| 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 |
| 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. |
| 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 | |
| 758 | Shifting a pattern of bits two places to the left produces results |
| 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.} |
| 766 | 00000011 @result{} 00001100 |
| 767 | @end group |
| 768 | @end example |
| 769 | |
| 770 | On the other hand, shifting one place to the right looks like this: |
| 771 | |
| 772 | @example |
| 773 | @group |
| 774 | (lsh 6 -1) |
| 775 | @result{} 3 |
| 776 | ;; @r{Decimal 6 becomes decimal 3.} |
| 777 | 00000110 @result{} 00000011 |
| 778 | @end group |
| 779 | |
| 780 | @group |
| 781 | (lsh 5 -1) |
| 782 | @result{} 2 |
| 783 | ;; @r{Decimal 5 becomes decimal 2.} |
| 784 | 00000101 @result{} 00000010 |
| 785 | @end group |
| 786 | @end example |
| 787 | |
| 788 | @noindent |
| 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 |
| 795 | 268,435,455 produces @minus{}2 on a 29-bit machine: |
| 796 | |
| 797 | @example |
| 798 | (lsh 268435455 1) ; @r{left shift} |
| 799 | @result{} -2 |
| 800 | @end example |
| 801 | |
| 802 | In binary, in the 29-bit implementation, the argument looks like this: |
| 803 | |
| 804 | @example |
| 805 | @group |
| 806 | ;; @r{Decimal 268,435,455} |
| 807 | 0 1111 1111 1111 1111 1111 1111 1111 |
| 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} |
| 817 | 1 1111 1111 1111 1111 1111 1111 1110 |
| 818 | @end group |
| 819 | @end example |
| 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, |
| 830 | @code{ash} puts ones in the empty bit positions on the left, while |
| 831 | @code{lsh} puts zeros in those bit positions. |
| 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 |
| 838 | (ash -6 -1) @result{} -3 |
| 839 | ;; @r{Decimal @minus{}6 becomes decimal @minus{}3.} |
| 840 | 1 1111 1111 1111 1111 1111 1111 1010 |
| 841 | @result{} |
| 842 | 1 1111 1111 1111 1111 1111 1111 1101 |
| 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 |
| 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 |
| 854 | @result{} |
| 855 | 0 1111 1111 1111 1111 1111 1111 1101 |
| 856 | @end group |
| 857 | @end example |
| 858 | |
| 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 |
| 865 | ; @r{ 29-bit binary values} |
| 866 | |
| 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} |
| 869 | @end group |
| 870 | @group |
| 871 | (ash 5 2) |
| 872 | @result{} 20 |
| 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} |
| 875 | (ash -5 2) |
| 876 | @result{} -20 |
| 877 | @end group |
| 878 | @group |
| 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} |
| 881 | @end group |
| 882 | @group |
| 883 | (ash 5 -2) |
| 884 | @result{} 1 |
| 885 | @end group |
| 886 | @group |
| 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} |
| 889 | @end group |
| 890 | @group |
| 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} |
| 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. |
| 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 |
| 929 | ; @r{ 29-bit binary values} |
| 930 | |
| 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} |
| 934 | @end group |
| 935 | |
| 936 | @group |
| 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} |
| 941 | @end group |
| 942 | |
| 943 | @group |
| 944 | (logand) |
| 945 | @result{} -1 ; -1 = @r{1 1111 1111 1111 1111 1111 1111 1111} |
| 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 |
| 961 | ; @r{ 29-bit binary values} |
| 962 | |
| 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} |
| 966 | @end group |
| 967 | |
| 968 | @group |
| 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} |
| 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 |
| 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. |
| 985 | |
| 986 | @smallexample |
| 987 | @group |
| 988 | ; @r{ 29-bit binary values} |
| 989 | |
| 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} |
| 993 | @end group |
| 994 | |
| 995 | @group |
| 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} |
| 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 |
| 1012 | (lognot 5) |
| 1013 | @result{} -6 |
| 1014 | ;; 5 = @r{0 0000 0000 0000 0000 0000 0000 0101} |
| 1015 | ;; @r{becomes} |
| 1016 | ;; -6 = @r{1 1111 1111 1111 1111 1111 1111 1010} |
| 1017 | @end example |
| 1018 | @end defun |
| 1019 | |
| 1020 | @node Math Functions |
| 1021 | @section Standard Mathematical Functions |
| 1022 | @cindex transcendental functions |
| 1023 | @cindex mathematical functions |
| 1024 | |
| 1025 | These mathematical functions allow integers as well as floating point |
| 1026 | numbers as arguments. |
| 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 |
| 1036 | The value of @code{(asin @var{arg})} is a number between |
| 1037 | @ifnottex |
| 1038 | @minus{}pi/2 |
| 1039 | @end ifnottex |
| 1040 | @tex |
| 1041 | @math{-\pi/2} |
| 1042 | @end tex |
| 1043 | and |
| 1044 | @ifnottex |
| 1045 | pi/2 |
| 1046 | @end ifnottex |
| 1047 | @tex |
| 1048 | @math{\pi/2} |
| 1049 | @end tex |
| 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. |
| 1052 | @end defun |
| 1053 | |
| 1054 | @defun acos arg |
| 1055 | The value of @code{(acos @var{arg})} is a number between 0 and |
| 1056 | @ifnottex |
| 1057 | pi |
| 1058 | @end ifnottex |
| 1059 | @tex |
| 1060 | @math{\pi} |
| 1061 | @end tex |
| 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. |
| 1064 | @end defun |
| 1065 | |
| 1066 | @defun atan y &optional x |
| 1067 | The value of @code{(atan @var{y})} is a number between |
| 1068 | @ifnottex |
| 1069 | @minus{}pi/2 |
| 1070 | @end ifnottex |
| 1071 | @tex |
| 1072 | @math{-\pi/2} |
| 1073 | @end tex |
| 1074 | and |
| 1075 | @ifnottex |
| 1076 | pi/2 |
| 1077 | @end ifnottex |
| 1078 | @tex |
| 1079 | @math{\pi/2} |
| 1080 | @end tex |
| 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. |
| 1085 | @end defun |
| 1086 | |
| 1087 | @defun exp arg |
| 1088 | This is the exponential function; it returns |
| 1089 | @tex |
| 1090 | @math{e} |
| 1091 | @end tex |
| 1092 | @ifnottex |
| 1093 | @i{e} |
| 1094 | @end ifnottex |
| 1095 | to the power @var{arg}. |
| 1096 | @tex |
| 1097 | @math{e} |
| 1098 | @end tex |
| 1099 | @ifnottex |
| 1100 | @i{e} |
| 1101 | @end ifnottex |
| 1102 | is a fundamental mathematical constant also called the base of natural |
| 1103 | logarithms. |
| 1104 | @end defun |
| 1105 | |
| 1106 | @defun log arg &optional base |
| 1107 | This function returns the logarithm of @var{arg}, with base @var{base}. |
| 1108 | If you don't specify @var{base}, the base |
| 1109 | @tex |
| 1110 | @math{e} |
| 1111 | @end tex |
| 1112 | @ifnottex |
| 1113 | @i{e} |
| 1114 | @end ifnottex |
| 1115 | is used. If @var{arg} is negative, it signals a @code{domain-error} |
| 1116 | error. |
| 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 |
| 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. |
| 1138 | @end defun |
| 1139 | |
| 1140 | @defun expt x y |
| 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. |
| 1145 | @end defun |
| 1146 | |
| 1147 | @defun sqrt arg |
| 1148 | This returns the square root of @var{arg}. If @var{arg} is negative, |
| 1149 | it signals a @code{domain-error} error. |
| 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 | |
| 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 |
| 1174 | day and on Emacs's process @acronym{ID} number. |
| 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 | |
| 1180 | If @var{limit} is a positive integer, the value is chosen to be |
| 1181 | nonnegative and less than @var{limit}. |
| 1182 | |
| 1183 | If @var{limit} is @code{t}, it means to choose a new seed based on the |
| 1184 | current time of day and on Emacs's process @acronym{ID} number. |
| 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 |
| 1191 | |
| 1192 | @ignore |
| 1193 | arch-tag: 574e8dd2-d513-4616-9844-c9a27869782e |
| 1194 | @end ignore |