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GNU Emacs supports two numeric data types: integers and floating point numbers. Integers are whole numbers such as -3, 0, 7, 13, and 511. Their values are exact. Floating point numbers are numbers with fractional parts, such as -4.5, 0.0, or 2.71828. They can also be expressed in exponential notation: 1.5e2 equals 150; in this example, ‘e2’ stands for ten to the second power, and that is multiplied by 1.5. Floating point values are not exact; they have a fixed, limited amount of precision.
3.1 Integer Basics | Representation and range of integers. | |
3.2 Floating Point Basics | Representation and range of floating point. | |
3.3 Type Predicates for Numbers | Testing for numbers. | |
3.4 Comparison of Numbers | Equality and inequality predicates. | |
3.5 Numeric Conversions | Converting float to integer and vice versa. | |
3.6 Arithmetic Operations | How to add, subtract, multiply and divide. | |
3.7 Rounding Operations | Explicitly rounding floating point numbers. | |
3.8 Bitwise Operations on Integers | Logical and, or, not, shifting. | |
3.9 Standard Mathematical Functions | Trig, exponential and logarithmic functions. | |
3.10 Random Numbers | Obtaining random integers, predictable or not. |
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The range of values for an integer depends on the machine. The minimum range is -134217728 to 134217727 (28 bits; i.e., -2**27 to 2**27 - 1), but some machines may provide a wider range. Many examples in this chapter assume an integer has 28 bits.
The Lisp reader reads an integer as a sequence of digits with optional initial sign and optional final period.
1 ; The integer 1. 1. ; The integer 1. +1 ; Also the integer 1. -1 ; The integer -1. 268435457 ; Also the integer 1, due to overflow. 0 ; The integer 0. -0 ; The integer 0. |
In addition, the Lisp reader recognizes a syntax for integers in bases other than 10: ‘#Binteger’ reads integer in binary (radix 2), ‘#Ointeger’ reads integer in octal (radix 8), ‘#Xinteger’ reads integer in hexadecimal (radix 16), and ‘#radixrinteger’ reads integer in radix radix (where radix is between 2 and 36, inclusivley). Case is not significant for the letter after ‘#’ (‘B’, ‘O’, etc.) that denotes the radix.
To understand how various functions work on integers, especially the bitwise operators (see section Bitwise Operations on Integers), it is often helpful to view the numbers in their binary form.
In 28-bit binary, the decimal integer 5 looks like this:
0000 0000 0000 0000 0000 0000 0101 |
(We have inserted spaces between groups of 4 bits, and two spaces between groups of 8 bits, to make the binary integer easier to read.)
The integer -1 looks like this:
1111 1111 1111 1111 1111 1111 1111 |
-1 is represented as 28 ones. (This is called two’s complement notation.)
The negative integer, -5, is creating by subtracting 4 from -1. In binary, the decimal integer 4 is 100. Consequently, -5 looks like this:
1111 1111 1111 1111 1111 1111 1011 |
In this implementation, the largest 28-bit binary integer value is 134,217,727 in decimal. In binary, it looks like this:
0111 1111 1111 1111 1111 1111 1111 |
Since the arithmetic functions do not check whether integers go outside their range, when you add 1 to 134,217,727, the value is the negative integer -134,217,728:
(+ 1 134217727) ⇒ -134217728 ⇒ 1000 0000 0000 0000 0000 0000 0000 |
Many of the functions described in this chapter accept markers for arguments in place of numbers. (See section Markers.) Since the actual arguments to such functions may be either numbers or markers, we often give these arguments the name number-or-marker. When the argument value is a marker, its position value is used and its buffer is ignored.
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Floating point numbers are useful for representing numbers that are
not integral. The precise range of floating point numbers is
machine-specific; it is the same as the range of the C data type
double
on the machine you are using.
The read-syntax for floating point numbers requires either a decimal point (with at least one digit following), an exponent, or both. For example, ‘1500.0’, ‘15e2’, ‘15.0e2’, ‘1.5e3’, and ‘.15e4’ are five ways of writing a floating point number whose value is 1500. They are all equivalent. You can also use a minus sign to write negative floating point numbers, as in ‘-1.0’.
Most modern computers support the IEEE floating point standard, which
provides for positive infinity and negative infinity as floating point
values. It also provides for a class of values called NaN or
“not-a-number”; numerical functions return such values in cases where
there is no correct answer. For example, (sqrt -1.0)
returns a
NaN. For practical purposes, there’s no significant difference between
different NaN values in Emacs Lisp, and there’s no rule for precisely
which NaN value should be used in a particular case, so Emacs Lisp
doesn’t try to distinguish them. Here are the read syntaxes for
these special floating point values:
‘1.0e+INF’
‘-1.0e+INF’
‘0.0e+NaN’.
In addition, the value -0.0
is distinguishable from ordinary
zero in IEEE floating point (although equal
and =
consider
them equal values).
You can use logb
to extract the binary exponent of a floating
point number (or estimate the logarithm of an integer):
This function returns the binary exponent of number. More precisely, the value is the logarithm of number base 2, rounded down to an integer.
(logb 10) ⇒ 3 (logb 10.0e20) ⇒ 69 |
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The functions in this section test whether the argument is a number or
whether it is a certain sort of number. The functions integerp
and floatp
can take any type of Lisp object as argument (the
predicates would not be of much use otherwise); but the zerop
predicate requires a number as its argument. See also
integer-or-marker-p
and number-or-marker-p
, in
Predicates on Markers.
This predicate tests whether its argument is a floating point
number and returns t
if so, nil
otherwise.
floatp
does not exist in Emacs versions 18 and earlier.
This predicate tests whether its argument is an integer, and returns
t
if so, nil
otherwise.
This predicate tests whether its argument is a number (either integer or
floating point), and returns t
if so, nil
otherwise.
The wholenump
predicate (whose name comes from the phrase
“whole-number-p”) tests to see whether its argument is a nonnegative
integer, and returns t
if so, nil
otherwise. 0 is
considered non-negative.
natnump
is an obsolete synonym for wholenump
.
This predicate tests whether its argument is zero, and returns t
if so, nil
otherwise. The argument must be a number.
These two forms are equivalent: (zerop x)
≡ (= x 0)
.
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To test numbers for numerical equality, you should normally use
=
, not eq
. There can be many distinct floating point
number objects with the same numeric value. If you use eq
to
compare them, then you test whether two values are the same
object. By contrast, =
compares only the numeric values
of the objects.
At present, each integer value has a unique Lisp object in Emacs Lisp.
Therefore, eq
is equivalent to =
where integers are
concerned. It is sometimes convenient to use eq
for comparing an
unknown value with an integer, because eq
does not report an
error if the unknown value is not a number—it accepts arguments of any
type. By contrast, =
signals an error if the arguments are not
numbers or markers. However, it is a good idea to use =
if you
can, even for comparing integers, just in case we change the
representation of integers in a future Emacs version.
Sometimes it is useful to compare numbers with equal
; it treats
two numbers as equal if they have the same data type (both integers, or
both floating point) and the same value. By contrast, =
can
treat an integer and a floating point number as equal.
There is another wrinkle: because floating point arithmetic is not exact, it is often a bad idea to check for equality of two floating point values. Usually it is better to test for approximate equality. Here’s a function to do this:
(defvar fuzz-factor 1.0e-6) (defun approx-equal (x y) (or (and (= x 0) (= y 0)) (< (/ (abs (- x y)) (max (abs x) (abs y))) fuzz-factor))) |
Common Lisp note: Comparing numbers in Common Lisp always requires
=
because Common Lisp implements multi-word integers, and two distinct integer objects can have the same numeric value. Emacs Lisp can have just one integer object for any given value because it has a limited range of integer values.
This function tests whether its arguments are numerically equal, and
returns t
if so, nil
otherwise.
This function tests whether its arguments are numerically equal, and
returns t
if they are not, and nil
if they are.
This function tests whether its first argument is strictly less than
its second argument. It returns t
if so, nil
otherwise.
This function tests whether its first argument is less than or equal
to its second argument. It returns t
if so, nil
otherwise.
This function tests whether its first argument is strictly greater
than its second argument. It returns t
if so, nil
otherwise.
This function tests whether its first argument is greater than or
equal to its second argument. It returns t
if so, nil
otherwise.
This function returns the largest of its arguments. If any of the argument is floating-point, the value is returned as floating point, even if it was given as an integer.
(max 20) ⇒ 20 (max 1 2.5) ⇒ 2.5 (max 1 3 2.5) ⇒ 3.0 |
This function returns the smallest of its arguments. If any of the argument is floating-point, the value is returned as floating point, even if it was given as an integer.
(min -4 1) ⇒ -4 |
This function returns the absolute value of number.
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To convert an integer to floating point, use the function float
.
This returns number converted to floating point.
If number is already a floating point number, float
returns
it unchanged.
There are four functions to convert floating point numbers to integers; they differ in how they round. These functions accept integer arguments also, and return such arguments unchanged.
This returns number, converted to an integer by rounding towards zero.
(truncate 1.2) ⇒ 1 (truncate 1.7) ⇒ 1 (truncate -1.2) ⇒ -1 (truncate -1.7) ⇒ -1 |
This returns number, converted to an integer by rounding downward (towards negative infinity).
If divisor is specified, floor
divides number by
divisor and then converts to an integer; this uses the kind of
division operation that corresponds to mod
, rounding downward.
An arith-error
results if divisor is 0.
(floor 1.2) ⇒ 1 (floor 1.7) ⇒ 1 (floor -1.2) ⇒ -2 (floor -1.7) ⇒ -2 (floor 5.99 3) ⇒ 1 |
This returns number, converted to an integer by rounding upward (towards positive infinity).
(ceiling 1.2) ⇒ 2 (ceiling 1.7) ⇒ 2 (ceiling -1.2) ⇒ -1 (ceiling -1.7) ⇒ -1 |
This returns number, converted to an integer by rounding towards the nearest integer. Rounding a value equidistant between two integers may choose the integer closer to zero, or it may prefer an even integer, depending on your machine.
(round 1.2) ⇒ 1 (round 1.7) ⇒ 2 (round -1.2) ⇒ -1 (round -1.7) ⇒ -2 |
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Emacs Lisp provides the traditional four arithmetic operations: addition, subtraction, multiplication, and division. Remainder and modulus functions supplement the division functions. The functions to add or subtract 1 are provided because they are traditional in Lisp and commonly used.
All of these functions except %
return a floating point value
if any argument is floating.
It is important to note that in Emacs Lisp, arithmetic functions
do not check for overflow. Thus (1+ 134217727)
may evaluate to
-134217728, depending on your hardware.
This function returns number-or-marker plus 1. For example,
(setq foo 4) ⇒ 4 (1+ foo) ⇒ 5 |
This function is not analogous to the C operator ++
—it does not
increment a variable. It just computes a sum. Thus, if we continue,
foo ⇒ 4 |
If you want to increment the variable, you must use setq
,
like this:
(setq foo (1+ foo)) ⇒ 5 |
This function returns number-or-marker minus 1.
This function adds its arguments together. When given no arguments,
+
returns 0.
(+) ⇒ 0 (+ 1) ⇒ 1 (+ 1 2 3 4) ⇒ 10 |
The -
function serves two purposes: negation and subtraction.
When -
has a single argument, the value is the negative of the
argument. When there are multiple arguments, -
subtracts each of
the more-numbers-or-markers from number-or-marker,
cumulatively. If there are no arguments, the result is 0.
(- 10 1 2 3 4) ⇒ 0 (- 10) ⇒ -10 (-) ⇒ 0 |
This function multiplies its arguments together, and returns the
product. When given no arguments, *
returns 1.
(*) ⇒ 1 (* 1) ⇒ 1 (* 1 2 3 4) ⇒ 24 |
This function divides dividend by divisor and returns the quotient. If there are additional arguments divisors, then it divides dividend by each divisor in turn. Each argument may be a number or a marker.
If all the arguments are integers, then the result is an integer too.
This means the result has to be rounded. On most machines, the result
is rounded towards zero after each division, but some machines may round
differently with negative arguments. This is because the Lisp function
/
is implemented using the C division operator, which also
permits machine-dependent rounding. As a practical matter, all known
machines round in the standard fashion.
If you divide an integer by 0, an arith-error
error is signaled.
(See section Errors.) Floating point division by zero returns either
infinity or a NaN if your machine supports IEEE floating point;
otherwise, it signals an arith-error
error.
(/ 6 2) ⇒ 3 (/ 5 2) ⇒ 2 (/ 5.0 2) ⇒ 2.5 (/ 5 2.0) ⇒ 2.5 (/ 5.0 2.0) ⇒ 2.5 (/ 25 3 2) ⇒ 4 (/ -17 6) ⇒ -2 |
The result of (/ -17 6)
could in principle be -3 on some
machines.
This function returns the integer remainder after division of dividend by divisor. The arguments must be integers or markers.
For negative arguments, the remainder is in principle machine-dependent since the quotient is; but in practice, all known machines behave alike.
An arith-error
results if divisor is 0.
(% 9 4) ⇒ 1 (% -9 4) ⇒ -1 (% 9 -4) ⇒ 1 (% -9 -4) ⇒ -1 |
For any two integers dividend and divisor,
(+ (% dividend divisor) (* (/ dividend divisor) divisor)) |
always equals dividend.
This function returns the value of dividend modulo divisor; in other words, the remainder after division of dividend by divisor, but with the same sign as divisor. The arguments must be numbers or markers.
Unlike %
, mod
returns a well-defined result for negative
arguments. It also permits floating point arguments; it rounds the
quotient downward (towards minus infinity) to an integer, and uses that
quotient to compute the remainder.
An arith-error
results if divisor is 0.
(mod 9 4) ⇒ 1 (mod -9 4) ⇒ 3 (mod 9 -4) ⇒ -3 (mod -9 -4) ⇒ -1 (mod 5.5 2.5) ⇒ .5 |
For any two numbers dividend and divisor,
(+ (mod dividend divisor) (* (floor dividend divisor) divisor)) |
always equals dividend, subject to rounding error if either
argument is floating point. For floor
, see Numeric Conversions.
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The functions ffloor
, fceiling
, fround
, and
ftruncate
take a floating point argument and return a floating
point result whose value is a nearby integer. ffloor
returns the
nearest integer below; fceiling
, the nearest integer above;
ftruncate
, the nearest integer in the direction towards zero;
fround
, the nearest integer.
This function rounds float to the next lower integral value, and returns that value as a floating point number.
This function rounds float to the next higher integral value, and returns that value as a floating point number.
This function rounds float towards zero to an integral value, and returns that value as a floating point number.
This function rounds float to the nearest integral value, and returns that value as a floating point number.
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In a computer, an integer is represented as a binary number, a sequence of bits (digits which are either zero or one). A bitwise operation acts on the individual bits of such a sequence. For example, shifting moves the whole sequence left or right one or more places, reproducing the same pattern “moved over”.
The bitwise operations in Emacs Lisp apply only to integers.
lsh
, which is an abbreviation for logical shift, shifts the
bits in integer1 to the left count places, or to the right
if count is negative, bringing zeros into the vacated bits. If
count is negative, lsh
shifts zeros into the leftmost
(most-significant) bit, producing a positive result even if
integer1 is negative. Contrast this with ash
, below.
Here are two examples of lsh
, shifting a pattern of bits one
place to the left. We show only the low-order eight bits of the binary
pattern; the rest are all zero.
(lsh 5 1) ⇒ 10 ;; Decimal 5 becomes decimal 10. 00000101 ⇒ 00001010 (lsh 7 1) ⇒ 14 ;; Decimal 7 becomes decimal 14. 00000111 ⇒ 00001110 |
As the examples illustrate, shifting the pattern of bits one place to the left produces a number that is twice the value of the previous number.
Shifting a pattern of bits two places to the left produces results like this (with 8-bit binary numbers):
(lsh 3 2)
⇒ 12
;; Decimal 3 becomes decimal 12.
00000011 ⇒ 00001100
|
On the other hand, shifting one place to the right looks like this:
(lsh 6 -1)
⇒ 3
;; Decimal 6 becomes decimal 3.
00000110 ⇒ 00000011
(lsh 5 -1)
⇒ 2
;; Decimal 5 becomes decimal 2.
00000101 ⇒ 00000010
|
As the example illustrates, shifting one place to the right divides the value of a positive integer by two, rounding downward.
The function lsh
, like all Emacs Lisp arithmetic functions, does
not check for overflow, so shifting left can discard significant bits
and change the sign of the number. For example, left shifting
134,217,727 produces -2 on a 28-bit machine:
(lsh 134217727 1) ; left shift
⇒ -2
|
In binary, in the 28-bit implementation, the argument looks like this:
;; Decimal 134,217,727
0111 1111 1111 1111 1111 1111 1111
|
which becomes the following when left shifted:
;; Decimal -2
1111 1111 1111 1111 1111 1111 1110
|
ash
(arithmetic shift) shifts the bits in integer1
to the left count places, or to the right if count
is negative.
ash
gives the same results as lsh
except when
integer1 and count are both negative. In that case,
ash
puts ones in the empty bit positions on the left, while
lsh
puts zeros in those bit positions.
Thus, with ash
, shifting the pattern of bits one place to the right
looks like this:
(ash -6 -1) ⇒ -3
;; Decimal -6 becomes decimal -3.
1111 1111 1111 1111 1111 1111 1010
⇒
1111 1111 1111 1111 1111 1111 1101
|
In contrast, shifting the pattern of bits one place to the right with
lsh
looks like this:
(lsh -6 -1) ⇒ 134217725
;; Decimal -6 becomes decimal 134,217,725.
1111 1111 1111 1111 1111 1111 1010
⇒
0111 1111 1111 1111 1111 1111 1101
|
Here are other examples:
; 28-bit binary values (lsh 5 2) ; 5 = 0000 0000 0000 0000 0000 0000 0101 ⇒ 20 ; = 0000 0000 0000 0000 0000 0001 0100 (ash 5 2) ⇒ 20 (lsh -5 2) ; -5 = 1111 1111 1111 1111 1111 1111 1011 ⇒ -20 ; = 1111 1111 1111 1111 1111 1110 1100 (ash -5 2) ⇒ -20 (lsh 5 -2) ; 5 = 0000 0000 0000 0000 0000 0000 0101 ⇒ 1 ; = 0000 0000 0000 0000 0000 0000 0001 (ash 5 -2) ⇒ 1 (lsh -5 -2) ; -5 = 1111 1111 1111 1111 1111 1111 1011 ⇒ 4194302 ; = 0011 1111 1111 1111 1111 1111 1110 (ash -5 -2) ; -5 = 1111 1111 1111 1111 1111 1111 1011 ⇒ -2 ; = 1111 1111 1111 1111 1111 1111 1110 |
This function returns the “logical and” of the arguments: the nth bit is set in the result if, and only if, the nth bit is set in all the arguments. (“Set” means that the value of the bit is 1 rather than 0.)
For example, using 4-bit binary numbers, the “logical and” of 13 and 12 is 12: 1101 combined with 1100 produces 1100. In both the binary numbers, the leftmost two bits are set (i.e., they are 1’s), so the leftmost two bits of the returned value are set. However, for the rightmost two bits, each is zero in at least one of the arguments, so the rightmost two bits of the returned value are 0’s.
Therefore,
(logand 13 12) ⇒ 12 |
If logand
is not passed any argument, it returns a value of
-1. This number is an identity element for logand
because its binary representation consists entirely of ones. If
logand
is passed just one argument, it returns that argument.
; 28-bit binary values (logand 14 13) ; 14 = 0000 0000 0000 0000 0000 0000 1110 ; 13 = 0000 0000 0000 0000 0000 0000 1101 ⇒ 12 ; 12 = 0000 0000 0000 0000 0000 0000 1100 (logand 14 13 4) ; 14 = 0000 0000 0000 0000 0000 0000 1110 ; 13 = 0000 0000 0000 0000 0000 0000 1101 ; 4 = 0000 0000 0000 0000 0000 0000 0100 ⇒ 4 ; 4 = 0000 0000 0000 0000 0000 0000 0100 (logand)
⇒ -1 ; -1 = 1111 1111 1111 1111 1111 1111 1111
|
This function returns the “inclusive or” of its arguments: the nth bit
is set in the result if, and only if, the nth bit is set in at least
one of the arguments. If there are no arguments, the result is zero,
which is an identity element for this operation. If logior
is
passed just one argument, it returns that argument.
; 28-bit binary values (logior 12 5) ; 12 = 0000 0000 0000 0000 0000 0000 1100 ; 5 = 0000 0000 0000 0000 0000 0000 0101 ⇒ 13 ; 13 = 0000 0000 0000 0000 0000 0000 1101 (logior 12 5 7) ; 12 = 0000 0000 0000 0000 0000 0000 1100 ; 5 = 0000 0000 0000 0000 0000 0000 0101 ; 7 = 0000 0000 0000 0000 0000 0000 0111 ⇒ 15 ; 15 = 0000 0000 0000 0000 0000 0000 1111 |
This function returns the “exclusive or” of its arguments: the
nth bit is set in the result if, and only if, the nth bit is
set in an odd number of the arguments. If there are no arguments, the
result is 0, which is an identity element for this operation. If
logxor
is passed just one argument, it returns that argument.
; 28-bit binary values (logxor 12 5) ; 12 = 0000 0000 0000 0000 0000 0000 1100 ; 5 = 0000 0000 0000 0000 0000 0000 0101 ⇒ 9 ; 9 = 0000 0000 0000 0000 0000 0000 1001 (logxor 12 5 7) ; 12 = 0000 0000 0000 0000 0000 0000 1100 ; 5 = 0000 0000 0000 0000 0000 0000 0101 ; 7 = 0000 0000 0000 0000 0000 0000 0111 ⇒ 14 ; 14 = 0000 0000 0000 0000 0000 0000 1110 |
This function returns the logical complement of its argument: the nth bit is one in the result if, and only if, the nth bit is zero in integer, and vice-versa.
(lognot 5) ⇒ -6 ;; 5 = 0000 0000 0000 0000 0000 0000 0101 ;; becomes ;; -6 = 1111 1111 1111 1111 1111 1111 1010 |
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These mathematical functions allow integers as well as floating point numbers as arguments.
These are the ordinary trigonometric functions, with argument measured in radians.
The value of (asin arg)
is a number between
-pi/2
and
pi/2
(inclusive) whose sine is arg; if, however, arg
is out of range (outside [-1, 1]), then the result is a NaN.
The value of (acos arg)
is a number between 0 and
pi
(inclusive) whose cosine is arg; if, however, arg
is out of range (outside [-1, 1]), then the result is a NaN.
The value of (atan arg)
is a number between
-pi/2
and
pi/2
(exclusive) whose tangent is arg.
This is the exponential function; it returns e to the power arg. e is a fundamental mathematical constant also called the base of natural logarithms.
This function returns the logarithm of arg, with base base. If you don’t specify base, the base e is used. If arg is negative, the result is a NaN.
This function returns the logarithm of arg, with base 10. If
arg is negative, the result is a NaN. (log10 x)
≡ (log x 10)
, at least approximately.
This function returns x raised to power y. If both arguments are integers and y is positive, the result is an integer; in this case, it is truncated to fit the range of possible integer values.
This returns the square root of arg. If arg is negative, the value is a NaN.
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