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 sequence of digits with optional
57 initial sign and optional final period. An integer that is out of the
58 Emacs range is treated as a floating-point number.
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.}
66 ; @r{The floating-point number 9e18.}
67 0 ; @r{The integer 0.}
68 -0 ; @r{The integer 0.}
71 @cindex integers in specific radix
72 @cindex radix for reading an integer
73 @cindex base for reading an integer
76 @cindex reading numbers in hex, octal, and binary
77 The syntax for integers in bases other than 10 uses @samp{#}
78 followed by a letter that specifies the radix: @samp{b} for binary,
79 @samp{o} for octal, @samp{x} for hex, or @samp{@var{radix}r} to
80 specify radix @var{radix}. Case is not significant for the letter
81 that specifies the radix. Thus, @samp{#b@var{integer}} reads
82 @var{integer} in binary, and @samp{#@var{radix}r@var{integer}} reads
83 @var{integer} in radix @var{radix}. Allowed values of @var{radix} run
84 from 2 to 36. For example:
93 To understand how various functions work on integers, especially the
94 bitwise operators (@pxref{Bitwise Operations}), it is often helpful to
95 view the numbers in their binary form.
97 In 30-bit binary, the decimal integer 5 looks like this:
100 0000...000101 (30 bits total)
104 (The @samp{...} stands for enough bits to fill out a 30-bit word; in
105 this case, @samp{...} stands for twenty 0 bits. Later examples also
106 use the @samp{...} notation to make binary integers easier to read.)
108 The integer @minus{}1 looks like this:
111 1111...111111 (30 bits total)
115 @cindex two's complement
116 @minus{}1 is represented as 30 ones. (This is called @dfn{two's
117 complement} notation.)
119 Subtracting 4 from @minus{}1 returns the negative integer @minus{}5.
120 In binary, the decimal integer 4 is 100. Consequently,
121 @minus{}5 looks like this:
124 1111...111011 (30 bits total)
127 In this implementation, the largest 30-bit binary integer is
128 536,870,911 in decimal. In binary, it looks like this:
131 0111...111111 (30 bits total)
134 Since the arithmetic functions do not check whether integers go
135 outside their range, when you add 1 to 536,870,911, the value is the
136 negative integer @minus{}536,870,912:
141 @result{} 1000...000000 (30 bits total)
144 Many of the functions described in this chapter accept markers for
145 arguments in place of numbers. (@xref{Markers}.) Since the actual
146 arguments to such functions may be either numbers or markers, we often
147 give these arguments the name @var{number-or-marker}. When the argument
148 value is a marker, its position value is used and its buffer is ignored.
150 @cindex largest Lisp integer
151 @cindex maximum Lisp integer
152 @defvar most-positive-fixnum
153 The value of this variable is the largest integer that Emacs Lisp can
154 handle. Typical values are
171 @cindex smallest Lisp integer
172 @cindex minimum Lisp integer
173 @defvar most-negative-fixnum
174 The value of this variable is the smallest integer that Emacs Lisp can
175 handle. It is negative. Typical values are
192 In Emacs Lisp, text characters are represented by integers. Any
193 integer between zero and the value of @code{(max-char)}, inclusive, is
194 considered to be valid as a character. @xref{Character Codes}.
197 @section Floating-Point Basics
199 @cindex @acronym{IEEE} floating point
200 Floating-point numbers are useful for representing numbers that are
201 not integral. The range of floating-point numbers is
202 the same as the range of the C data type @code{double} on the machine
203 you are using. On all computers currently supported by Emacs, this is
204 double-precision @acronym{IEEE} floating point.
206 The read syntax for floating-point numbers requires either a decimal
207 point, an exponent, or both. Optional signs (@samp{+} or @samp{-})
208 precede the number and its exponent. For example, @samp{1500.0},
209 @samp{+15e2}, @samp{15.0e+2}, @samp{+1500000e-3}, and @samp{.15e4} are
210 five ways of writing a floating-point number whose value is 1500.
211 They are all equivalent. Like Common Lisp, Emacs Lisp requires at
212 least one digit after any decimal point in a floating-point number;
213 @samp{1500.} is an integer, not a floating-point number.
215 Emacs Lisp treats @code{-0.0} as numerically equal to ordinary zero
216 with respect to @code{equal} and @code{=}. This follows the
217 @acronym{IEEE} floating-point standard, which says @code{-0.0} and
218 @code{0.0} are numerically equal even though other operations can
221 @cindex positive infinity
222 @cindex negative infinity
225 The @acronym{IEEE} floating-point standard supports positive
226 infinity and negative infinity as floating-point values. It also
227 provides for a class of values called NaN, or ``not a number'';
228 numerical functions return such values in cases where there is no
229 correct answer. For example, @code{(/ 0.0 0.0)} returns a NaN@.
230 Although NaN values carry a sign, for practical purposes there is no other
231 significant difference between different NaN values in Emacs Lisp.
233 Here are read syntaxes for these special floating-point values:
237 @samp{1.0e+INF} and @samp{-1.0e+INF}
239 @samp{0.0e+NaN} and @samp{-0.0e+NaN}
242 The following functions are specialized for handling floating-point
246 This predicate returns @code{t} if its floating-point argument is a NaN,
247 @code{nil} otherwise.
251 This function returns a cons cell @code{(@var{s} . @var{e})},
252 where @var{s} and @var{e} are respectively the significand and
253 exponent of the floating-point number @var{x}.
255 If @var{x} is finite, then @var{s} is a floating-point number between 0.5
256 (inclusive) and 1.0 (exclusive), @var{e} is an integer, and
258 @var{x} = @var{s} * 2**@var{e}.
263 If @var{x} is zero or infinity, then @var{s} is the same as @var{x}.
264 If @var{x} is a NaN, then @var{s} is also a NaN@.
265 If @var{x} is zero, then @var{e} is 0.
269 Given a numeric significand @var{s} and an integer exponent @var{e},
270 this function returns the floating point number
272 @var{s} * 2**@var{e}.
279 @defun copysign x1 x2
280 This function copies the sign of @var{x2} to the value of @var{x1},
281 and returns the result. @var{x1} and @var{x2} must be floating point.
285 This function returns the binary exponent of @var{x}. More
286 precisely, the value is the logarithm base 2 of @math{|x|}, rounded
297 @node Predicates on Numbers
298 @section Type Predicates for Numbers
299 @cindex predicates for numbers
301 The functions in this section test for numbers, or for a specific
302 type of number. The functions @code{integerp} and @code{floatp} can
303 take any type of Lisp object as argument (they would not be of much
304 use otherwise), but the @code{zerop} predicate requires a number as
305 its argument. See also @code{integer-or-marker-p} and
306 @code{number-or-marker-p}, in @ref{Predicates on Markers}.
309 This predicate tests whether its argument is floating point
310 and returns @code{t} if so, @code{nil} otherwise.
313 @defun integerp object
314 This predicate tests whether its argument is an integer, and returns
315 @code{t} if so, @code{nil} otherwise.
318 @defun numberp object
319 This predicate tests whether its argument is a number (either integer or
320 floating point), and returns @code{t} if so, @code{nil} otherwise.
323 @defun natnump object
324 @cindex natural numbers
325 This predicate (whose name comes from the phrase ``natural number'')
326 tests to see whether its argument is a nonnegative integer, and
327 returns @code{t} if so, @code{nil} otherwise. 0 is considered
331 @code{wholenump} is a synonym for @code{natnump}.
335 This predicate tests whether its argument is zero, and returns @code{t}
336 if so, @code{nil} otherwise. The argument must be a number.
338 @code{(zerop x)} is equivalent to @code{(= x 0)}.
341 @node Comparison of Numbers
342 @section Comparison of Numbers
343 @cindex number comparison
344 @cindex comparing numbers
346 To test numbers for numerical equality, you should normally use
347 @code{=}, not @code{eq}. There can be many distinct floating-point
348 objects with the same numeric value. If you use @code{eq} to
349 compare them, then you test whether two values are the same
350 @emph{object}. By contrast, @code{=} compares only the numeric values
353 In Emacs Lisp, each integer is a unique Lisp object.
354 Therefore, @code{eq} is equivalent to @code{=} where integers are
355 concerned. It is sometimes convenient to use @code{eq} for comparing
356 an unknown value with an integer, because @code{eq} does not report an
357 error if the unknown value is not a number---it accepts arguments of
358 any type. By contrast, @code{=} signals an error if the arguments are
359 not numbers or markers. However, it is better programming practice to
360 use @code{=} if you can, even for comparing integers.
362 Sometimes it is useful to compare numbers with @code{equal}, which
363 treats two numbers as equal if they have the same data type (both
364 integers, or both floating point) and the same value. By contrast,
365 @code{=} can treat an integer and a floating-point number as equal.
366 @xref{Equality Predicates}.
368 There is another wrinkle: because floating-point arithmetic is not
369 exact, it is often a bad idea to check for equality of floating-point
370 values. Usually it is better to test for approximate equality.
371 Here's a function to do this:
374 (defvar fuzz-factor 1.0e-6)
375 (defun approx-equal (x y)
378 (max (abs x) (abs y)))
382 @cindex CL note---integers vrs @code{eq}
384 @b{Common Lisp note:} Comparing numbers in Common Lisp always requires
385 @code{=} because Common Lisp implements multi-word integers, and two
386 distinct integer objects can have the same numeric value. Emacs Lisp
387 can have just one integer object for any given value because it has a
388 limited range of integers.
391 @defun = number-or-marker &rest number-or-markers
392 This function tests whether all its arguments are numerically equal,
393 and returns @code{t} if so, @code{nil} otherwise.
396 @defun eql value1 value2
397 This function acts like @code{eq} except when both arguments are
398 numbers. It compares numbers by type and numeric value, so that
399 @code{(eql 1.0 1)} returns @code{nil}, but @code{(eql 1.0 1.0)} and
400 @code{(eql 1 1)} both return @code{t}.
403 @defun /= number-or-marker1 number-or-marker2
404 This function tests whether its arguments are numerically equal, and
405 returns @code{t} if they are not, and @code{nil} if they are.
408 @defun < number-or-marker &rest number-or-markers
409 This function tests whether each argument is strictly less than the
410 following argument. It returns @code{t} if so, @code{nil} otherwise.
413 @defun <= number-or-marker &rest number-or-markers
414 This function tests whether each argument is less than or equal to
415 the following argument. It returns @code{t} if so, @code{nil} otherwise.
418 @defun > number-or-marker &rest number-or-markers
419 This function tests whether each argument is strictly greater than
420 the following argument. It returns @code{t} if so, @code{nil} otherwise.
423 @defun >= number-or-marker &rest number-or-markers
424 This function tests whether each argument is greater than or equal to
425 the following argument. It returns @code{t} if so, @code{nil} otherwise.
428 @defun max number-or-marker &rest numbers-or-markers
429 This function returns the largest of its arguments.
441 @defun min number-or-marker &rest numbers-or-markers
442 This function returns the smallest of its arguments.
451 This function returns the absolute value of @var{number}.
454 @node Numeric Conversions
455 @section Numeric Conversions
456 @cindex rounding in conversions
457 @cindex number conversions
458 @cindex converting numbers
460 To convert an integer to floating point, use the function @code{float}.
463 This returns @var{number} converted to floating point.
464 If @var{number} is already floating point, @code{float} returns
468 There are four functions to convert floating-point numbers to
469 integers; they differ in how they round. All accept an argument
470 @var{number} and an optional argument @var{divisor}. Both arguments
471 may be integers or floating-point numbers. @var{divisor} may also be
472 @code{nil}. If @var{divisor} is @code{nil} or omitted, these
473 functions convert @var{number} to an integer, or return it unchanged
474 if it already is an integer. If @var{divisor} is non-@code{nil}, they
475 divide @var{number} by @var{divisor} and convert the result to an
476 integer. If @var{divisor} is zero (whether integer or
477 floating point), Emacs signals an @code{arith-error} error.
479 @defun truncate number &optional divisor
480 This returns @var{number}, converted to an integer by rounding towards
495 @defun floor number &optional divisor
496 This returns @var{number}, converted to an integer by rounding downward
497 (towards negative infinity).
499 If @var{divisor} is specified, this uses the kind of division
500 operation that corresponds to @code{mod}, rounding downward.
516 @defun ceiling number &optional divisor
517 This returns @var{number}, converted to an integer by rounding upward
518 (towards positive infinity).
532 @defun round number &optional divisor
533 This returns @var{number}, converted to an integer by rounding towards the
534 nearest integer. Rounding a value equidistant between two integers
535 returns the even integer.
549 @node Arithmetic Operations
550 @section Arithmetic Operations
551 @cindex arithmetic operations
553 Emacs Lisp provides the traditional four arithmetic operations
554 (addition, subtraction, multiplication, and division), as well as
555 remainder and modulus functions, and functions to add or subtract 1.
556 Except for @code{%}, each of these functions accepts both integer and
557 floating-point arguments, and returns a floating-point number if any
558 argument is floating point.
560 Emacs Lisp arithmetic functions do not check for integer overflow.
561 Thus @code{(1+ 536870911)} may evaluate to
562 @minus{}536870912, depending on your hardware.
564 @defun 1+ number-or-marker
565 This function returns @var{number-or-marker} plus 1.
575 This function is not analogous to the C operator @code{++}---it does not
576 increment a variable. It just computes a sum. Thus, if we continue,
583 If you want to increment the variable, you must use @code{setq},
592 @defun 1- number-or-marker
593 This function returns @var{number-or-marker} minus 1.
596 @defun + &rest numbers-or-markers
597 This function adds its arguments together. When given no arguments,
610 @defun - &optional number-or-marker &rest more-numbers-or-markers
611 The @code{-} function serves two purposes: negation and subtraction.
612 When @code{-} has a single argument, the value is the negative of the
613 argument. When there are multiple arguments, @code{-} subtracts each of
614 the @var{more-numbers-or-markers} from @var{number-or-marker},
615 cumulatively. If there are no arguments, the result is 0.
627 @defun * &rest numbers-or-markers
628 This function multiplies its arguments together, and returns the
629 product. When given no arguments, @code{*} returns 1.
641 @defun / number &rest divisors
642 With one or more @var{divisors}, this function divides @var{number}
643 by each divisor in @var{divisors} in turn, and returns the quotient.
644 With no @var{divisors}, this function returns 1/@var{number}, i.e.,
645 the multiplicative inverse of @var{number}. Each argument may be a
648 If all the arguments are integers, the result is an integer, obtained
649 by rounding the quotient towards zero after each division.
690 @cindex @code{arith-error} in division
691 If you divide an integer by the integer 0, Emacs signals an
692 @code{arith-error} error (@pxref{Errors}). Floating-point division of
693 a nonzero number by zero yields either positive or negative infinity
694 (@pxref{Float Basics}).
697 @defun % dividend divisor
699 This function returns the integer remainder after division of @var{dividend}
700 by @var{divisor}. The arguments must be integers or markers.
702 For any two integers @var{dividend} and @var{divisor},
706 (+ (% @var{dividend} @var{divisor})
707 (* (/ @var{dividend} @var{divisor}) @var{divisor}))
712 always equals @var{dividend} if @var{divisor} is nonzero.
726 @defun mod dividend divisor
728 This function returns the value of @var{dividend} modulo @var{divisor};
729 in other words, the remainder after division of @var{dividend}
730 by @var{divisor}, but with the same sign as @var{divisor}.
731 The arguments must be numbers or markers.
733 Unlike @code{%}, @code{mod} permits floating-point arguments; it
734 rounds the quotient downward (towards minus infinity) to an integer,
735 and uses that quotient to compute the remainder.
737 If @var{divisor} is zero, @code{mod} signals an @code{arith-error}
738 error if both arguments are integers, and returns a NaN otherwise.
763 For any two numbers @var{dividend} and @var{divisor},
767 (+ (mod @var{dividend} @var{divisor})
768 (* (floor @var{dividend} @var{divisor}) @var{divisor}))
773 always equals @var{dividend}, subject to rounding error if either
774 argument is floating point and to an @code{arith-error} if @var{dividend} is an
775 integer and @var{divisor} is 0. For @code{floor}, see @ref{Numeric
779 @node Rounding Operations
780 @section Rounding Operations
781 @cindex rounding without conversion
783 The functions @code{ffloor}, @code{fceiling}, @code{fround}, and
784 @code{ftruncate} take a floating-point argument and return a floating-point
785 result whose value is a nearby integer. @code{ffloor} returns the
786 nearest integer below; @code{fceiling}, the nearest integer above;
787 @code{ftruncate}, the nearest integer in the direction towards zero;
788 @code{fround}, the nearest integer.
791 This function rounds @var{float} to the next lower integral value, and
792 returns that value as a floating-point number.
795 @defun fceiling float
796 This function rounds @var{float} to the next higher integral value, and
797 returns that value as a floating-point number.
800 @defun ftruncate float
801 This function rounds @var{float} towards zero to an integral value, and
802 returns that value as a floating-point number.
806 This function rounds @var{float} to the nearest integral value,
807 and returns that value as a floating-point number.
808 Rounding a value equidistant between two integers returns the even integer.
811 @node Bitwise Operations
812 @section Bitwise Operations on Integers
813 @cindex bitwise arithmetic
814 @cindex logical arithmetic
816 In a computer, an integer is represented as a binary number, a
817 sequence of @dfn{bits} (digits which are either zero or one). A bitwise
818 operation acts on the individual bits of such a sequence. For example,
819 @dfn{shifting} moves the whole sequence left or right one or more places,
820 reproducing the same pattern moved over.
822 The bitwise operations in Emacs Lisp apply only to integers.
824 @defun lsh integer1 count
825 @cindex logical shift
826 @code{lsh}, which is an abbreviation for @dfn{logical shift}, shifts the
827 bits in @var{integer1} to the left @var{count} places, or to the right
828 if @var{count} is negative, bringing zeros into the vacated bits. If
829 @var{count} is negative, @code{lsh} shifts zeros into the leftmost
830 (most-significant) bit, producing a positive result even if
831 @var{integer1} is negative. Contrast this with @code{ash}, below.
833 Here are two examples of @code{lsh}, shifting a pattern of bits one
834 place to the left. We show only the low-order eight bits of the binary
835 pattern; the rest are all zero.
841 ;; @r{Decimal 5 becomes decimal 10.}
842 00000101 @result{} 00001010
846 ;; @r{Decimal 7 becomes decimal 14.}
847 00000111 @result{} 00001110
852 As the examples illustrate, shifting the pattern of bits one place to
853 the left produces a number that is twice the value of the previous
856 Shifting a pattern of bits two places to the left produces results
857 like this (with 8-bit binary numbers):
863 ;; @r{Decimal 3 becomes decimal 12.}
864 00000011 @result{} 00001100
868 On the other hand, shifting one place to the right looks like this:
874 ;; @r{Decimal 6 becomes decimal 3.}
875 00000110 @result{} 00000011
881 ;; @r{Decimal 5 becomes decimal 2.}
882 00000101 @result{} 00000010
887 As the example illustrates, shifting one place to the right divides the
888 value of a positive integer by two, rounding downward.
890 The function @code{lsh}, like all Emacs Lisp arithmetic functions, does
891 not check for overflow, so shifting left can discard significant bits
892 and change the sign of the number. For example, left shifting
893 536,870,911 produces @minus{}2 in the 30-bit implementation:
896 (lsh 536870911 1) ; @r{left shift}
900 In binary, the argument looks like this:
904 ;; @r{Decimal 536,870,911}
905 0111...111111 (30 bits total)
910 which becomes the following when left shifted:
914 ;; @r{Decimal @minus{}2}
915 1111...111110 (30 bits total)
920 @defun ash integer1 count
921 @cindex arithmetic shift
922 @code{ash} (@dfn{arithmetic shift}) shifts the bits in @var{integer1}
923 to the left @var{count} places, or to the right if @var{count}
926 @code{ash} gives the same results as @code{lsh} except when
927 @var{integer1} and @var{count} are both negative. In that case,
928 @code{ash} puts ones in the empty bit positions on the left, while
929 @code{lsh} puts zeros in those bit positions.
931 Thus, with @code{ash}, shifting the pattern of bits one place to the right
936 (ash -6 -1) @result{} -3
937 ;; @r{Decimal @minus{}6 becomes decimal @minus{}3.}
938 1111...111010 (30 bits total)
940 1111...111101 (30 bits total)
944 In contrast, shifting the pattern of bits one place to the right with
945 @code{lsh} looks like this:
949 (lsh -6 -1) @result{} 536870909
950 ;; @r{Decimal @minus{}6 becomes decimal 536,870,909.}
951 1111...111010 (30 bits total)
953 0111...111101 (30 bits total)
957 Here are other examples:
959 @c !!! Check if lined up in smallbook format! XDVI shows problem
960 @c with smallbook but not with regular book! --rjc 16mar92
963 ; @r{ 30-bit binary values}
965 (lsh 5 2) ; 5 = @r{0000...000101}
966 @result{} 20 ; = @r{0000...010100}
971 (lsh -5 2) ; -5 = @r{1111...111011}
972 @result{} -20 ; = @r{1111...101100}
977 (lsh 5 -2) ; 5 = @r{0000...000101}
978 @result{} 1 ; = @r{0000...000001}
985 (lsh -5 -2) ; -5 = @r{1111...111011}
987 ; = @r{0011...111110}
990 (ash -5 -2) ; -5 = @r{1111...111011}
991 @result{} -2 ; = @r{1111...111110}
996 @defun logand &rest ints-or-markers
997 This function returns the bitwise AND of the arguments: the @var{n}th
998 bit is 1 in the result if, and only if, the @var{n}th bit is 1 in all
1001 For example, using 4-bit binary numbers, the bitwise AND of 13 and
1002 12 is 12: 1101 combined with 1100 produces 1100.
1003 In both the binary numbers, the leftmost two bits are both 1
1004 so the leftmost two bits of the returned value are both 1.
1005 However, for the rightmost two bits, each is 0 in at least one of
1006 the arguments, so the rightmost two bits of the returned value are both 0.
1018 If @code{logand} is not passed any argument, it returns a value of
1019 @minus{}1. This number is an identity element for @code{logand}
1020 because its binary representation consists entirely of ones. If
1021 @code{logand} is passed just one argument, it returns that argument.
1025 ; @r{ 30-bit binary values}
1027 (logand 14 13) ; 14 = @r{0000...001110}
1028 ; 13 = @r{0000...001101}
1029 @result{} 12 ; 12 = @r{0000...001100}
1033 (logand 14 13 4) ; 14 = @r{0000...001110}
1034 ; 13 = @r{0000...001101}
1035 ; 4 = @r{0000...000100}
1036 @result{} 4 ; 4 = @r{0000...000100}
1041 @result{} -1 ; -1 = @r{1111...111111}
1046 @defun logior &rest ints-or-markers
1047 This function returns the bitwise inclusive OR of its arguments: the @var{n}th
1048 bit is 1 in the result if, and only if, the @var{n}th bit is 1 in at
1049 least one of the arguments. If there are no arguments, the result is 0,
1050 which is an identity element for this operation. If @code{logior} is
1051 passed just one argument, it returns that argument.
1055 ; @r{ 30-bit binary values}
1057 (logior 12 5) ; 12 = @r{0000...001100}
1058 ; 5 = @r{0000...000101}
1059 @result{} 13 ; 13 = @r{0000...001101}
1063 (logior 12 5 7) ; 12 = @r{0000...001100}
1064 ; 5 = @r{0000...000101}
1065 ; 7 = @r{0000...000111}
1066 @result{} 15 ; 15 = @r{0000...001111}
1071 @defun logxor &rest ints-or-markers
1072 This function returns the bitwise exclusive OR of its arguments: the
1073 @var{n}th bit is 1 in the result if, and only if, the @var{n}th bit is
1074 1 in an odd number of the arguments. If there are no arguments, the
1075 result is 0, which is an identity element for this operation. If
1076 @code{logxor} is passed just one argument, it returns that argument.
1080 ; @r{ 30-bit binary values}
1082 (logxor 12 5) ; 12 = @r{0000...001100}
1083 ; 5 = @r{0000...000101}
1084 @result{} 9 ; 9 = @r{0000...001001}
1088 (logxor 12 5 7) ; 12 = @r{0000...001100}
1089 ; 5 = @r{0000...000101}
1090 ; 7 = @r{0000...000111}
1091 @result{} 14 ; 14 = @r{0000...001110}
1096 @defun lognot integer
1097 This function returns the bitwise complement of its argument: the @var{n}th
1098 bit is one in the result if, and only if, the @var{n}th bit is zero in
1099 @var{integer}, and vice-versa.
1104 ;; 5 = @r{0000...000101} (30 bits total)
1106 ;; -6 = @r{1111...111010} (30 bits total)
1110 @node Math Functions
1111 @section Standard Mathematical Functions
1112 @cindex transcendental functions
1113 @cindex mathematical functions
1114 @cindex floating-point functions
1116 These mathematical functions allow integers as well as floating-point
1117 numbers as arguments.
1122 These are the basic trigonometric functions, with argument @var{arg}
1123 measured in radians.
1127 The value of @code{(asin @var{arg})} is a number between
1141 (inclusive) whose sine is @var{arg}. If @var{arg} is out of range
1142 (outside [@minus{}1, 1]), @code{asin} returns a NaN.
1146 The value of @code{(acos @var{arg})} is a number between 0 and
1153 (inclusive) whose cosine is @var{arg}. If @var{arg} is out of range
1154 (outside [@minus{}1, 1]), @code{acos} returns a NaN.
1157 @defun atan y &optional x
1158 The value of @code{(atan @var{y})} is a number between
1172 (exclusive) whose tangent is @var{y}. If the optional second
1173 argument @var{x} is given, the value of @code{(atan y x)} is the
1174 angle in radians between the vector @code{[@var{x}, @var{y}]} and the
1179 This is the exponential function; it returns @math{e} to the power
1183 @defun log arg &optional base
1184 This function returns the logarithm of @var{arg}, with base
1185 @var{base}. If you don't specify @var{base}, the natural base
1186 @math{e} is used. If @var{arg} or @var{base} is negative, @code{log}
1191 This function returns @var{x} raised to power @var{y}. If both
1192 arguments are integers and @var{y} is positive, the result is an
1193 integer; in this case, overflow causes truncation, so watch out.
1194 If @var{x} is a finite negative number and @var{y} is a finite
1195 non-integer, @code{expt} returns a NaN.
1199 This returns the square root of @var{arg}. If @var{arg} is finite
1200 and less than zero, @code{sqrt} returns a NaN.
1203 In addition, Emacs defines the following common mathematical
1207 The mathematical constant @math{e} (2.71828@dots{}).
1211 The mathematical constant @math{pi} (3.14159@dots{}).
1214 @node Random Numbers
1215 @section Random Numbers
1216 @cindex random numbers
1218 A deterministic computer program cannot generate true random
1219 numbers. For most purposes, @dfn{pseudo-random numbers} suffice. A
1220 series of pseudo-random numbers is generated in a deterministic
1221 fashion. The numbers are not truly random, but they have certain
1222 properties that mimic a random series. For example, all possible
1223 values occur equally often in a pseudo-random series.
1225 @cindex seed, for random number generation
1226 Pseudo-random numbers are generated from a @dfn{seed value}. Starting from
1227 any given seed, the @code{random} function always generates the same
1228 sequence of numbers. By default, Emacs initializes the random seed at
1229 startup, in such a way that the sequence of values of @code{random}
1230 (with overwhelming likelihood) differs in each Emacs run.
1232 Sometimes you want the random number sequence to be repeatable. For
1233 example, when debugging a program whose behavior depends on the random
1234 number sequence, it is helpful to get the same behavior in each
1235 program run. To make the sequence repeat, execute @code{(random "")}.
1236 This sets the seed to a constant value for your particular Emacs
1237 executable (though it may differ for other Emacs builds). You can use
1238 other strings to choose various seed values.
1240 @defun random &optional limit
1241 This function returns a pseudo-random integer. Repeated calls return a
1242 series of pseudo-random integers.
1244 If @var{limit} is a positive integer, the value is chosen to be
1245 nonnegative and less than @var{limit}. Otherwise, the value might be
1246 any integer representable in Lisp, i.e., an integer between
1247 @code{most-negative-fixnum} and @code{most-positive-fixnum}
1248 (@pxref{Integer Basics}).
1250 If @var{limit} is @code{t}, it means to choose a new seed as if Emacs
1251 were restarting, typically from the system entropy. On systems
1252 lacking entropy pools, choose the seed from less-random volatile data
1253 such as the current time.
1255 If @var{limit} is a string, it means to choose a new seed based on the