2 @c This is part of the GNU Emacs Lisp Reference Manual.
3 @c Copyright (C) 1990-1995, 1998-1999, 2001-2018 Free Software
5 @c See the file elisp.texi for copying conditions.
11 GNU Emacs supports two numeric data types: @dfn{integers} and
12 @dfn{floating-point numbers}. Integers are whole numbers such as
13 @minus{}3, 0, 7, 13, and 511. Floating-point numbers are numbers with
14 fractional parts, such as @minus{}4.5, 0.0, and 2.71828. They can
15 also be expressed in exponential notation: @samp{1.5e2} is the same as
16 @samp{150.0}; here, @samp{e2} stands for ten to the second power, and
17 that is multiplied by 1.5. Integer computations are exact, though
18 they may overflow. Floating-point computations often involve rounding
19 errors, as the numbers have a fixed amount of precision.
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.
35 @section Integer Basics
37 The range of values for an integer depends on the machine. The
38 minimum range is @minus{}536,870,912 to 536,870,911 (30 bits; i.e.,
52 but many machines provide a wider range. Many examples in this
53 chapter assume the minimum integer width of 30 bits.
56 The Lisp reader reads an integer as a nonempty sequence
57 of decimal digits with optional initial sign and optional
58 final period. A decimal integer that is out of the
59 Emacs range is treated as a floating-point number.
62 1 ; @r{The integer 1.}
63 1. ; @r{The integer 1.}
64 +1 ; @r{Also the integer 1.}
65 -1 ; @r{The integer @minus{}1.}
67 ; @r{The floating-point number 9e18.}
68 0 ; @r{The integer 0.}
69 -0 ; @r{The integer 0.}
72 @cindex integers in specific radix
73 @cindex radix for reading an integer
74 @cindex base for reading an integer
77 @cindex reading numbers in hex, octal, and binary
78 The syntax for integers in bases other than 10 consists of @samp{#}
79 followed by a radix indication followed by one or more digits. The
80 radix indications are @samp{b} for binary, @samp{o} for octal,
81 @samp{x} for hex, and @samp{@var{radix}r} for radix @var{radix}.
82 Thus, @samp{#b@var{integer}} reads
83 @var{integer} in binary, and @samp{#@var{radix}r@var{integer}} reads
84 @var{integer} in radix @var{radix}. Allowed values of @var{radix} run
85 from 2 to 36, and allowed digits are the first @var{radix} characters
86 taken from @samp{0}--@samp{9}, @samp{A}--@samp{Z}.
87 Letter case is ignored and there is no initial sign or final period.
97 To understand how various functions work on integers, especially the
98 bitwise operators (@pxref{Bitwise Operations}), it is often helpful to
99 view the numbers in their binary form.
101 In 30-bit binary, the decimal integer 5 looks like this:
104 0000...000101 (30 bits total)
108 (The @samp{...} stands for enough bits to fill out a 30-bit word; in
109 this case, @samp{...} stands for twenty 0 bits. Later examples also
110 use the @samp{...} notation to make binary integers easier to read.)
112 The integer @minus{}1 looks like this:
115 1111...111111 (30 bits total)
119 @cindex two's complement
120 @minus{}1 is represented as 30 ones. (This is called @dfn{two's
121 complement} notation.)
123 Subtracting 4 from @minus{}1 returns the negative integer @minus{}5.
124 In binary, the decimal integer 4 is 100. Consequently,
125 @minus{}5 looks like this:
128 1111...111011 (30 bits total)
131 In this implementation, the largest 30-bit binary integer is
132 536,870,911 in decimal. In binary, it looks like this:
135 0111...111111 (30 bits total)
138 Since the arithmetic functions do not check whether integers go
139 outside their range, when you add 1 to 536,870,911, the value is the
140 negative integer @minus{}536,870,912:
145 @result{} 1000...000000 (30 bits total)
148 Many of the functions described in this chapter accept markers for
149 arguments in place of numbers. (@xref{Markers}.) Since the actual
150 arguments to such functions may be either numbers or markers, we often
151 give these arguments the name @var{number-or-marker}. When the argument
152 value is a marker, its position value is used and its buffer is ignored.
154 @cindex largest Lisp integer
155 @cindex maximum Lisp integer
156 @defvar most-positive-fixnum
157 The value of this variable is the largest integer that Emacs Lisp can
158 handle. Typical values are
175 @cindex smallest Lisp integer
176 @cindex minimum Lisp integer
177 @defvar most-negative-fixnum
178 The value of this variable is the smallest integer that Emacs Lisp can
179 handle. It is negative. Typical values are
196 In Emacs Lisp, text characters are represented by integers. Any
197 integer between zero and the value of @code{(max-char)}, inclusive, is
198 considered to be valid as a character. @xref{Character Codes}.
201 @section Floating-Point Basics
203 @cindex @acronym{IEEE} floating point
204 Floating-point numbers are useful for representing numbers that are
205 not integral. The range of floating-point numbers is
206 the same as the range of the C data type @code{double} on the machine
207 you are using. On all computers currently supported by Emacs, this is
208 double-precision @acronym{IEEE} floating point.
210 The read syntax for floating-point numbers requires either a decimal
211 point, an exponent, or both. Optional signs (@samp{+} or @samp{-})
212 precede the number and its exponent. For example, @samp{1500.0},
213 @samp{+15e2}, @samp{15.0e+2}, @samp{+1500000e-3}, and @samp{.15e4} are
214 five ways of writing a floating-point number whose value is 1500.
215 They are all equivalent. Like Common Lisp, Emacs Lisp requires at
216 least one digit after any decimal point in a floating-point number;
217 @samp{1500.} is an integer, not a floating-point number.
219 Emacs Lisp treats @code{-0.0} as numerically equal to ordinary zero
220 with respect to @code{equal} and @code{=}. This follows the
221 @acronym{IEEE} floating-point standard, which says @code{-0.0} and
222 @code{0.0} are numerically equal even though other operations can
225 @cindex positive infinity
226 @cindex negative infinity
229 The @acronym{IEEE} floating-point standard supports positive
230 infinity and negative infinity as floating-point values. It also
231 provides for a class of values called NaN, or ``not a number'';
232 numerical functions return such values in cases where there is no
233 correct answer. For example, @code{(/ 0.0 0.0)} returns a NaN@.
234 Although NaN values carry a sign, for practical purposes there is no other
235 significant difference between different NaN values in Emacs Lisp.
237 Here are read syntaxes for these special floating-point values:
241 @samp{1.0e+INF} and @samp{-1.0e+INF}
243 @samp{0.0e+NaN} and @samp{-0.0e+NaN}
246 The following functions are specialized for handling floating-point
250 This predicate returns @code{t} if its floating-point argument is a NaN,
251 @code{nil} otherwise.
255 This function returns a cons cell @code{(@var{s} . @var{e})},
256 where @var{s} and @var{e} are respectively the significand and
257 exponent of the floating-point number @var{x}.
259 If @var{x} is finite, then @var{s} is a floating-point number between 0.5
260 (inclusive) and 1.0 (exclusive), @var{e} is an integer, and
262 @var{x} = @var{s} * 2**@var{e}.
267 If @var{x} is zero or infinity, then @var{s} is the same as @var{x}.
268 If @var{x} is a NaN, then @var{s} is also a NaN@.
269 If @var{x} is zero, then @var{e} is 0.
273 Given a numeric significand @var{s} and an integer exponent @var{e},
274 this function returns the floating point number
276 @var{s} * 2**@var{e}.
283 @defun copysign x1 x2
284 This function copies the sign of @var{x2} to the value of @var{x1},
285 and returns the result. @var{x1} and @var{x2} must be floating point.
289 This function returns the binary exponent of @var{x}. More
290 precisely, the value is the logarithm base 2 of @math{|x|}, rounded
301 @node Predicates on Numbers
302 @section Type Predicates for Numbers
303 @cindex predicates for numbers
305 The functions in this section test for numbers, or for a specific
306 type of number. The functions @code{integerp} and @code{floatp} can
307 take any type of Lisp object as argument (they would not be of much
308 use otherwise), but the @code{zerop} predicate requires a number as
309 its argument. See also @code{integer-or-marker-p} and
310 @code{number-or-marker-p}, in @ref{Predicates on Markers}.
313 This predicate tests whether its argument is floating point
314 and returns @code{t} if so, @code{nil} otherwise.
317 @defun integerp object
318 This predicate tests whether its argument is an integer, and returns
319 @code{t} if so, @code{nil} otherwise.
322 @defun numberp object
323 This predicate tests whether its argument is a number (either integer or
324 floating point), and returns @code{t} if so, @code{nil} otherwise.
327 @defun natnump object
328 @cindex natural numbers
329 This predicate (whose name comes from the phrase ``natural number'')
330 tests to see whether its argument is a nonnegative integer, and
331 returns @code{t} if so, @code{nil} otherwise. 0 is considered
335 @code{wholenump} is a synonym for @code{natnump}.
339 This predicate tests whether its argument is zero, and returns @code{t}
340 if so, @code{nil} otherwise. The argument must be a number.
342 @code{(zerop x)} is equivalent to @code{(= x 0)}.
345 @node Comparison of Numbers
346 @section Comparison of Numbers
347 @cindex number comparison
348 @cindex comparing numbers
350 To test numbers for numerical equality, you should normally use
351 @code{=}, not @code{eq}. There can be many distinct floating-point
352 objects with the same numeric value. If you use @code{eq} to
353 compare them, then you test whether two values are the same
354 @emph{object}. By contrast, @code{=} compares only the numeric values
357 In Emacs Lisp, each integer is a unique Lisp object.
358 Therefore, @code{eq} is equivalent to @code{=} where integers are
359 concerned. It is sometimes convenient to use @code{eq} for comparing
360 an unknown value with an integer, because @code{eq} does not report an
361 error if the unknown value is not a number---it accepts arguments of
362 any type. By contrast, @code{=} signals an error if the arguments are
363 not numbers or markers. However, it is better programming practice to
364 use @code{=} if you can, even for comparing integers.
366 Sometimes it is useful to compare numbers with @code{equal}, which
367 treats two numbers as equal if they have the same data type (both
368 integers, or both floating point) and the same value. By contrast,
369 @code{=} can treat an integer and a floating-point number as equal.
370 @xref{Equality Predicates}.
372 There is another wrinkle: because floating-point arithmetic is not
373 exact, it is often a bad idea to check for equality of floating-point
374 values. Usually it is better to test for approximate equality.
375 Here's a function to do this:
378 (defvar fuzz-factor 1.0e-6)
379 (defun approx-equal (x y)
382 (max (abs x) (abs y)))
386 @cindex CL note---integers vrs @code{eq}
388 @b{Common Lisp note:} Comparing numbers in Common Lisp always requires
389 @code{=} because Common Lisp implements multi-word integers, and two
390 distinct integer objects can have the same numeric value. Emacs Lisp
391 can have just one integer object for any given value because it has a
392 limited range of integers.
395 @defun = number-or-marker &rest number-or-markers
396 This function tests whether all its arguments are numerically equal,
397 and returns @code{t} if so, @code{nil} otherwise.
400 @defun eql value1 value2
401 This function acts like @code{eq} except when both arguments are
402 numbers. It compares numbers by type and numeric value, so that
403 @code{(eql 1.0 1)} returns @code{nil}, but @code{(eql 1.0 1.0)} and
404 @code{(eql 1 1)} both return @code{t}.
407 @defun /= number-or-marker1 number-or-marker2
408 This function tests whether its arguments are numerically equal, and
409 returns @code{t} if they are not, and @code{nil} if they are.
412 @defun < number-or-marker &rest number-or-markers
413 This function tests whether each argument is strictly less than the
414 following argument. It returns @code{t} if so, @code{nil} otherwise.
417 @defun <= number-or-marker &rest number-or-markers
418 This function tests whether each argument is less than or equal to
419 the following argument. It returns @code{t} if so, @code{nil} otherwise.
422 @defun > number-or-marker &rest number-or-markers
423 This function tests whether each argument is strictly greater than
424 the following argument. It returns @code{t} if so, @code{nil} otherwise.
427 @defun >= number-or-marker &rest number-or-markers
428 This function tests whether each argument is greater than or equal to
429 the following argument. It returns @code{t} if so, @code{nil} otherwise.
432 @defun max number-or-marker &rest numbers-or-markers
433 This function returns the largest of its arguments.
445 @defun min number-or-marker &rest numbers-or-markers
446 This function returns the smallest of its arguments.
455 This function returns the absolute value of @var{number}.
458 @node Numeric Conversions
459 @section Numeric Conversions
460 @cindex rounding in conversions
461 @cindex number conversions
462 @cindex converting numbers
464 To convert an integer to floating point, use the function @code{float}.
467 This returns @var{number} converted to floating point.
468 If @var{number} is already floating point, @code{float} returns
472 There are four functions to convert floating-point numbers to
473 integers; they differ in how they round. All accept an argument
474 @var{number} and an optional argument @var{divisor}. Both arguments
475 may be integers or floating-point numbers. @var{divisor} may also be
476 @code{nil}. If @var{divisor} is @code{nil} or omitted, these
477 functions convert @var{number} to an integer, or return it unchanged
478 if it already is an integer. If @var{divisor} is non-@code{nil}, they
479 divide @var{number} by @var{divisor} and convert the result to an
480 integer. If @var{divisor} is zero (whether integer or
481 floating point), Emacs signals an @code{arith-error} error.
483 @defun truncate number &optional divisor
484 This returns @var{number}, converted to an integer by rounding towards
499 @defun floor number &optional divisor
500 This returns @var{number}, converted to an integer by rounding downward
501 (towards negative infinity).
503 If @var{divisor} is specified, this uses the kind of division
504 operation that corresponds to @code{mod}, rounding downward.
520 @defun ceiling number &optional divisor
521 This returns @var{number}, converted to an integer by rounding upward
522 (towards positive infinity).
536 @defun round number &optional divisor
537 This returns @var{number}, converted to an integer by rounding towards the
538 nearest integer. Rounding a value equidistant between two integers
539 returns the even integer.
553 @node Arithmetic Operations
554 @section Arithmetic Operations
555 @cindex arithmetic operations
557 Emacs Lisp provides the traditional four arithmetic operations
558 (addition, subtraction, multiplication, and division), as well as
559 remainder and modulus functions, and functions to add or subtract 1.
560 Except for @code{%}, each of these functions accepts both integer and
561 floating-point arguments, and returns a floating-point number if any
562 argument is floating point.
564 Emacs Lisp arithmetic functions do not check for integer overflow.
565 Thus @code{(1+ 536870911)} may evaluate to
566 @minus{}536870912, depending on your hardware.
568 @defun 1+ number-or-marker
569 This function returns @var{number-or-marker} plus 1.
579 This function is not analogous to the C operator @code{++}---it does not
580 increment a variable. It just computes a sum. Thus, if we continue,
587 If you want to increment the variable, you must use @code{setq},
596 @defun 1- number-or-marker
597 This function returns @var{number-or-marker} minus 1.
600 @defun + &rest numbers-or-markers
601 This function adds its arguments together. When given no arguments,
614 @defun - &optional number-or-marker &rest more-numbers-or-markers
615 The @code{-} function serves two purposes: negation and subtraction.
616 When @code{-} has a single argument, the value is the negative of the
617 argument. When there are multiple arguments, @code{-} subtracts each of
618 the @var{more-numbers-or-markers} from @var{number-or-marker},
619 cumulatively. If there are no arguments, the result is 0.
631 @defun * &rest numbers-or-markers
632 This function multiplies its arguments together, and returns the
633 product. When given no arguments, @code{*} returns 1.
645 @defun / number &rest divisors
646 With one or more @var{divisors}, this function divides @var{number}
647 by each divisor in @var{divisors} in turn, and returns the quotient.
648 With no @var{divisors}, this function returns 1/@var{number}, i.e.,
649 the multiplicative inverse of @var{number}. Each argument may be a
652 If all the arguments are integers, the result is an integer, obtained
653 by rounding the quotient towards zero after each division.
694 @cindex @code{arith-error} in division
695 If you divide an integer by the integer 0, Emacs signals an
696 @code{arith-error} error (@pxref{Errors}). Floating-point division of
697 a nonzero number by zero yields either positive or negative infinity
698 (@pxref{Float Basics}).
701 @defun % dividend divisor
703 This function returns the integer remainder after division of @var{dividend}
704 by @var{divisor}. The arguments must be integers or markers.
706 For any two integers @var{dividend} and @var{divisor},
710 (+ (% @var{dividend} @var{divisor})
711 (* (/ @var{dividend} @var{divisor}) @var{divisor}))
716 always equals @var{dividend} if @var{divisor} is nonzero.
730 @defun mod dividend divisor
732 This function returns the value of @var{dividend} modulo @var{divisor};
733 in other words, the remainder after division of @var{dividend}
734 by @var{divisor}, but with the same sign as @var{divisor}.
735 The arguments must be numbers or markers.
737 Unlike @code{%}, @code{mod} permits floating-point arguments; it
738 rounds the quotient downward (towards minus infinity) to an integer,
739 and uses that quotient to compute the remainder.
741 If @var{divisor} is zero, @code{mod} signals an @code{arith-error}
742 error if both arguments are integers, and returns a NaN otherwise.
767 For any two numbers @var{dividend} and @var{divisor},
771 (+ (mod @var{dividend} @var{divisor})
772 (* (floor @var{dividend} @var{divisor}) @var{divisor}))
777 always equals @var{dividend}, subject to rounding error if either
778 argument is floating point and to an @code{arith-error} if @var{dividend} is an
779 integer and @var{divisor} is 0. For @code{floor}, see @ref{Numeric
783 @node Rounding Operations
784 @section Rounding Operations
785 @cindex rounding without conversion
787 The functions @code{ffloor}, @code{fceiling}, @code{fround}, and
788 @code{ftruncate} take a floating-point argument and return a floating-point
789 result whose value is a nearby integer. @code{ffloor} returns the
790 nearest integer below; @code{fceiling}, the nearest integer above;
791 @code{ftruncate}, the nearest integer in the direction towards zero;
792 @code{fround}, the nearest integer.
795 This function rounds @var{float} to the next lower integral value, and
796 returns that value as a floating-point number.
799 @defun fceiling float
800 This function rounds @var{float} to the next higher integral value, and
801 returns that value as a floating-point number.
804 @defun ftruncate float
805 This function rounds @var{float} towards zero to an integral value, and
806 returns that value as a floating-point number.
810 This function rounds @var{float} to the nearest integral value,
811 and returns that value as a floating-point number.
812 Rounding a value equidistant between two integers returns the even integer.
815 @node Bitwise Operations
816 @section Bitwise Operations on Integers
817 @cindex bitwise arithmetic
818 @cindex logical arithmetic
820 In a computer, an integer is represented as a binary number, a
821 sequence of @dfn{bits} (digits which are either zero or one). A bitwise
822 operation acts on the individual bits of such a sequence. For example,
823 @dfn{shifting} moves the whole sequence left or right one or more places,
824 reproducing the same pattern moved over.
826 The bitwise operations in Emacs Lisp apply only to integers.
828 @defun lsh integer1 count
829 @cindex logical shift
830 @code{lsh}, which is an abbreviation for @dfn{logical shift}, shifts the
831 bits in @var{integer1} to the left @var{count} places, or to the right
832 if @var{count} is negative, bringing zeros into the vacated bits. If
833 @var{count} is negative, @code{lsh} shifts zeros into the leftmost
834 (most-significant) bit, producing a positive result even if
835 @var{integer1} is negative. Contrast this with @code{ash}, below.
837 Here are two examples of @code{lsh}, shifting a pattern of bits one
838 place to the left. We show only the low-order eight bits of the binary
839 pattern; the rest are all zero.
845 ;; @r{Decimal 5 becomes decimal 10.}
846 00000101 @result{} 00001010
850 ;; @r{Decimal 7 becomes decimal 14.}
851 00000111 @result{} 00001110
856 As the examples illustrate, shifting the pattern of bits one place to
857 the left produces a number that is twice the value of the previous
860 Shifting a pattern of bits two places to the left produces results
861 like this (with 8-bit binary numbers):
867 ;; @r{Decimal 3 becomes decimal 12.}
868 00000011 @result{} 00001100
872 On the other hand, shifting one place to the right looks like this:
878 ;; @r{Decimal 6 becomes decimal 3.}
879 00000110 @result{} 00000011
885 ;; @r{Decimal 5 becomes decimal 2.}
886 00000101 @result{} 00000010
891 As the example illustrates, shifting one place to the right divides the
892 value of a positive integer by two, rounding downward.
894 The function @code{lsh}, like all Emacs Lisp arithmetic functions, does
895 not check for overflow, so shifting left can discard significant bits
896 and change the sign of the number. For example, left shifting
897 536,870,911 produces @minus{}2 in the 30-bit implementation:
900 (lsh 536870911 1) ; @r{left shift}
904 In binary, the argument looks like this:
908 ;; @r{Decimal 536,870,911}
909 0111...111111 (30 bits total)
914 which becomes the following when left shifted:
918 ;; @r{Decimal @minus{}2}
919 1111...111110 (30 bits total)
924 @defun ash integer1 count
925 @cindex arithmetic shift
926 @code{ash} (@dfn{arithmetic shift}) shifts the bits in @var{integer1}
927 to the left @var{count} places, or to the right if @var{count}
930 @code{ash} gives the same results as @code{lsh} except when
931 @var{integer1} and @var{count} are both negative. In that case,
932 @code{ash} puts ones in the empty bit positions on the left, while
933 @code{lsh} puts zeros in those bit positions.
935 Thus, with @code{ash}, shifting the pattern of bits one place to the right
940 (ash -6 -1) @result{} -3
941 ;; @r{Decimal @minus{}6 becomes decimal @minus{}3.}
942 1111...111010 (30 bits total)
944 1111...111101 (30 bits total)
948 In contrast, shifting the pattern of bits one place to the right with
949 @code{lsh} looks like this:
953 (lsh -6 -1) @result{} 536870909
954 ;; @r{Decimal @minus{}6 becomes decimal 536,870,909.}
955 1111...111010 (30 bits total)
957 0111...111101 (30 bits total)
961 Here are other examples:
963 @c !!! Check if lined up in smallbook format! XDVI shows problem
964 @c with smallbook but not with regular book! --rjc 16mar92
967 ; @r{ 30-bit binary values}
969 (lsh 5 2) ; 5 = @r{0000...000101}
970 @result{} 20 ; = @r{0000...010100}
975 (lsh -5 2) ; -5 = @r{1111...111011}
976 @result{} -20 ; = @r{1111...101100}
981 (lsh 5 -2) ; 5 = @r{0000...000101}
982 @result{} 1 ; = @r{0000...000001}
989 (lsh -5 -2) ; -5 = @r{1111...111011}
991 ; = @r{0011...111110}
994 (ash -5 -2) ; -5 = @r{1111...111011}
995 @result{} -2 ; = @r{1111...111110}
1000 @defun logand &rest ints-or-markers
1001 This function returns the bitwise AND of the arguments: the @var{n}th
1002 bit is 1 in the result if, and only if, the @var{n}th bit is 1 in all
1005 For example, using 4-bit binary numbers, the bitwise AND of 13 and
1006 12 is 12: 1101 combined with 1100 produces 1100.
1007 In both the binary numbers, the leftmost two bits are both 1
1008 so the leftmost two bits of the returned value are both 1.
1009 However, for the rightmost two bits, each is 0 in at least one of
1010 the arguments, so the rightmost two bits of the returned value are both 0.
1022 If @code{logand} is not passed any argument, it returns a value of
1023 @minus{}1. This number is an identity element for @code{logand}
1024 because its binary representation consists entirely of ones. If
1025 @code{logand} is passed just one argument, it returns that argument.
1029 ; @r{ 30-bit binary values}
1031 (logand 14 13) ; 14 = @r{0000...001110}
1032 ; 13 = @r{0000...001101}
1033 @result{} 12 ; 12 = @r{0000...001100}
1037 (logand 14 13 4) ; 14 = @r{0000...001110}
1038 ; 13 = @r{0000...001101}
1039 ; 4 = @r{0000...000100}
1040 @result{} 4 ; 4 = @r{0000...000100}
1045 @result{} -1 ; -1 = @r{1111...111111}
1050 @defun logior &rest ints-or-markers
1051 This function returns the bitwise inclusive OR of its arguments: the @var{n}th
1052 bit is 1 in the result if, and only if, the @var{n}th bit is 1 in at
1053 least one of the arguments. If there are no arguments, the result is 0,
1054 which is an identity element for this operation. If @code{logior} is
1055 passed just one argument, it returns that argument.
1059 ; @r{ 30-bit binary values}
1061 (logior 12 5) ; 12 = @r{0000...001100}
1062 ; 5 = @r{0000...000101}
1063 @result{} 13 ; 13 = @r{0000...001101}
1067 (logior 12 5 7) ; 12 = @r{0000...001100}
1068 ; 5 = @r{0000...000101}
1069 ; 7 = @r{0000...000111}
1070 @result{} 15 ; 15 = @r{0000...001111}
1075 @defun logxor &rest ints-or-markers
1076 This function returns the bitwise exclusive OR of its arguments: the
1077 @var{n}th bit is 1 in the result if, and only if, the @var{n}th bit is
1078 1 in an odd number of the arguments. If there are no arguments, the
1079 result is 0, which is an identity element for this operation. If
1080 @code{logxor} is passed just one argument, it returns that argument.
1084 ; @r{ 30-bit binary values}
1086 (logxor 12 5) ; 12 = @r{0000...001100}
1087 ; 5 = @r{0000...000101}
1088 @result{} 9 ; 9 = @r{0000...001001}
1092 (logxor 12 5 7) ; 12 = @r{0000...001100}
1093 ; 5 = @r{0000...000101}
1094 ; 7 = @r{0000...000111}
1095 @result{} 14 ; 14 = @r{0000...001110}
1100 @defun lognot integer
1101 This function returns the bitwise complement of its argument: the @var{n}th
1102 bit is one in the result if, and only if, the @var{n}th bit is zero in
1103 @var{integer}, and vice-versa.
1108 ;; 5 = @r{0000...000101} (30 bits total)
1110 ;; -6 = @r{1111...111010} (30 bits total)
1115 @cindex Hamming weight
1116 @cindex counting set bits
1117 @defun logcount integer
1118 This function returns the @dfn{Hamming weight} of @var{integer}: the
1119 number of ones in the binary representation of @var{integer}.
1120 If @var{integer} is negative, it returns the number of zero bits in
1121 its two's complement binary representation. The result is always
1125 (logcount 43) ; 43 = #b101011
1127 (logcount -43) ; -43 = #b111...1010101
1132 @node Math Functions
1133 @section Standard Mathematical Functions
1134 @cindex transcendental functions
1135 @cindex mathematical functions
1136 @cindex floating-point functions
1138 These mathematical functions allow integers as well as floating-point
1139 numbers as arguments.
1144 These are the basic trigonometric functions, with argument @var{arg}
1145 measured in radians.
1149 The value of @code{(asin @var{arg})} is a number between
1163 (inclusive) whose sine is @var{arg}. If @var{arg} is out of range
1164 (outside [@minus{}1, 1]), @code{asin} returns a NaN.
1168 The value of @code{(acos @var{arg})} is a number between 0 and
1175 (inclusive) whose cosine is @var{arg}. If @var{arg} is out of range
1176 (outside [@minus{}1, 1]), @code{acos} returns a NaN.
1179 @defun atan y &optional x
1180 The value of @code{(atan @var{y})} is a number between
1194 (exclusive) whose tangent is @var{y}. If the optional second
1195 argument @var{x} is given, the value of @code{(atan y x)} is the
1196 angle in radians between the vector @code{[@var{x}, @var{y}]} and the
1201 This is the exponential function; it returns @math{e} to the power
1205 @defun log arg &optional base
1206 This function returns the logarithm of @var{arg}, with base
1207 @var{base}. If you don't specify @var{base}, the natural base
1208 @math{e} is used. If @var{arg} or @var{base} is negative, @code{log}
1213 This function returns @var{x} raised to power @var{y}. If both
1214 arguments are integers and @var{y} is positive, the result is an
1215 integer; in this case, overflow causes truncation, so watch out.
1216 If @var{x} is a finite negative number and @var{y} is a finite
1217 non-integer, @code{expt} returns a NaN.
1221 This returns the square root of @var{arg}. If @var{arg} is finite
1222 and less than zero, @code{sqrt} returns a NaN.
1225 In addition, Emacs defines the following common mathematical
1229 The mathematical constant @math{e} (2.71828@dots{}).
1233 The mathematical constant @math{pi} (3.14159@dots{}).
1236 @node Random Numbers
1237 @section Random Numbers
1238 @cindex random numbers
1240 A deterministic computer program cannot generate true random
1241 numbers. For most purposes, @dfn{pseudo-random numbers} suffice. A
1242 series of pseudo-random numbers is generated in a deterministic
1243 fashion. The numbers are not truly random, but they have certain
1244 properties that mimic a random series. For example, all possible
1245 values occur equally often in a pseudo-random series.
1247 @cindex seed, for random number generation
1248 Pseudo-random numbers are generated from a @dfn{seed value}. Starting from
1249 any given seed, the @code{random} function always generates the same
1250 sequence of numbers. By default, Emacs initializes the random seed at
1251 startup, in such a way that the sequence of values of @code{random}
1252 (with overwhelming likelihood) differs in each Emacs run.
1254 Sometimes you want the random number sequence to be repeatable. For
1255 example, when debugging a program whose behavior depends on the random
1256 number sequence, it is helpful to get the same behavior in each
1257 program run. To make the sequence repeat, execute @code{(random "")}.
1258 This sets the seed to a constant value for your particular Emacs
1259 executable (though it may differ for other Emacs builds). You can use
1260 other strings to choose various seed values.
1262 @defun random &optional limit
1263 This function returns a pseudo-random integer. Repeated calls return a
1264 series of pseudo-random integers.
1266 If @var{limit} is a positive integer, the value is chosen to be
1267 nonnegative and less than @var{limit}. Otherwise, the value might be
1268 any integer representable in Lisp, i.e., an integer between
1269 @code{most-negative-fixnum} and @code{most-positive-fixnum}
1270 (@pxref{Integer Basics}).
1272 If @var{limit} is @code{t}, it means to choose a new seed as if Emacs
1273 were restarting, typically from the system entropy. On systems
1274 lacking entropy pools, choose the seed from less-random volatile data
1275 such as the current time.
1277 If @var{limit} is a string, it means to choose a new seed based on the