Arithmetic can be divided into some special purpose integer
predicates and a series of general predicates for integer, floating
point and rational arithmetic as appropriate. The general arithmetic
predicates all handle `expressions`. An expression is either a
simple number or a `function`. The arguments of a function are
expressions. The functions are described in section
4.27.2.6.

The predicates in this section provide more logical operations
between integers. They are not covered by the ISO standard, although
they are‘part of the community' and found as either library or
built-in in many other Prolog systems.

**between**(`+Low,
+High, ?Value`)`Low` and `High` are integers, `High` >=`Low`.
If
`Value` is an integer, `Low` =<`Value`
=<`High`. When `Value` is a variable it is
successively bound to all integers between `Low` and `High`.
If `High` is `inf`

or
`infinite`

^{117We prefer infinite,
but some other Prolog systems already use inf for infinity;
we accept both for the time being.}
between/3
is true iff `Value` >=`Low`, a feature
that is particularly interesting for generating integers from a certain
value.
**succ**(`?Int1,
?Int2`)- True if
`Int2` = `Int1` + 1 and `Int1`
>= 0. At least one of the arguments must be instantiated to a
natural number. This predicate raises the domain error `not_less_than_zero`

if called with a negative integer. E.g. `succ(X, 0)`

fails
silently and `succ(X, -1)`

raises a domain error.^{118The
behaviour to deal with natural numbers only was defined by Richard
O'Keefe to support the common count-down-to-zero in a natural way. Up to
5.1.8, succ/2
also accepted negative integers.}
**plus**(`?Int1,
?Int2, ?Int3`)- True if
`Int3` = `Int1` + `Int2`.
At least two of the three arguments must be instantiated to integers.
**divmod**(`+Dividend,
+Divisor, -Quotient, -Remainder`)- This predicate is a shorthand for computing both the
`Quotient`
and
`Remainder` of two integers in a single operation. This allows
for exploiting the fact that the low level implementation for computing
the quotient also produces the remainder. Timing confirms that this
predicate is almost twice as fast as performing the steps independently.
Semantically, divmod/4
is defined as below.
divmod(Dividend, Divisor, Quotient, Remainder) :-
Quotient is Dividend div Divisor,
Remainder is Dividend mod Divisor.

Note that this predicate is only available if SWI-Prolog is compiled
with unbounded integer support. This is the case for all packaged
versions.

**nth_integer_root_and_remainder**(`+N,
+I, -Root, -Remainder`)- True when
`Root ** N + Remainder = I`. `N` and `I`
must be integers.^{119This predicate
was suggested by Markus Triska. The final name and argument order is by
Richard O'Keefe. The decision to include the remainder is by Jan
Wielemaker. Including the remainder makes this predicate about twice as
slow if Root is not exact.}
`N` must be one or more. If `I` is negative and
`N` is *odd*, `Root` and `Remainder`
are negative, i.e., the following holds for `I` < 0:
% I < 0,
% N mod 2 =\= 0,
nth_integer_root_and_remainder(
N, I, Root, Remainder),
IPos is -I,
nth_integer_root_and_remainder(
N, IPos, RootPos, RemainderPos),
Root =:= -RootPos,
Remainder =:= -RemainderPos.

The general arithmetic predicates are optionally compiled (see
set_prolog_flag/2
and the **-O** command line option). Compiled arithmetic
reduces global stack requirements and improves performance.
Unfortunately compiled arithmetic cannot be traced, which is why it is
optional.

- [ISO]
`+Expr1` **>** `+Expr2` - True if expression
`Expr1` evaluates to a larger number than `Expr2`.
- [ISO]
`+Expr1` **<** `+Expr2` - True if expression
`Expr1` evaluates to a smaller number than `Expr2`.
- [ISO]
`+Expr1` **=<** `+Expr2` - True if expression
`Expr1` evaluates to a smaller or equal
number to `Expr2`.
- [ISO]
`+Expr1` **>=** `+Expr2` - True if expression
`Expr1` evaluates to a larger or equal
number to `Expr2`.
- [ISO]
`+Expr1` **=\=** `+Expr2` - True if expression
`Expr1` evaluates to a number non-equal to
`Expr2`.
- [ISO]
`+Expr1` **=:=** `+Expr2` - True if expression
`Expr1` evaluates to a number equal to `
Expr2`.
- [ISO]
`-Number` **is** `+Expr` - True when
`Number` is the value to which `Expr`
evaluates. Typically, is/2
should be used with unbound left operand. If equality is to be tested, =:=/2
should be used. For example:

`?- 1 is sin(pi/2).` | Fails! sin(pi/2)
evaluates to the float 1.0, which does not unify with the integer 1. |

`?- 1 =:= sin(pi/2).` | Succeeds as expected. |

SWI-Prolog
defines the following numeric types:

*integer*

If SWI-Prolog is built using the *GNU multiple precision arithmetic
library* (GMP), integer arithmetic is *unbounded*,
which means that the size of integers is limited by available memory
only. Without GMP, SWI-Prolog integers are 64-bits, regardless of the
native integer size of the platform. The type of integer support can be
detected using the Prolog flags bounded, min_integer
and
max_integer. As
the use of GMP is default, most of the following descriptions assume
unbounded integer arithmetic.
Internally, SWI-Prolog has three integer representations. Small
integers (defined by the Prolog flag max_tagged_integer)
are encoded directly. Larger integers are represented as 64-bit values
on the global stack. Integers that do not fit in 64 bits are represented
as serialised GNU MPZ structures on the global stack.

*rational number*

Rational numbers (`Q`) are quotients of two integers (`N/M`).
Rational arithmetic is only provided if GMP is used (see above).
Rational numbers satisfy the type tests rational/1, number/1
and atomic/1
and may satisfy the type test integer/1,
i.e., integers are considered rational numbers. Rational numbers are
always kept in *canonical representation*, which means `M`
is positive and `N` and `M` have no common divisors.
Rational numbers are introduced into the computation using the functions
rational/1, rationalize/1
or the rdiv/2
(rational division) function. If the Prolog flag
prefer_rationals
is `true`

(default), division (//2)
and integer power (^/2)
also produce a rational number.

*float*

Floating point numbers are represented using the C type `double`

.
On most of today's platforms these are 64-bit IEEE floating point
numbers.

Arithmetic functions that require integer arguments accept, in
addition to integers, rational numbers with (canonical) denominator‘1'.
If the required argument is a float the argument is converted to float.
Note that conversion of integers to floating point numbers may raise an
overflow exception. In all other cases, arguments are converted to the
same type using the order below.

integer `->` rational number `->`
floating point number

The use of rational numbers with unbounded integers allows for exact
integer or *fixed point* arithmetic under addition, subtraction,
multiplication, division and exponentiation (^/2).
Support for rational numbers depends on the Prolog flag
prefer_rationals.
If this is `true`

(default), the number division function (//2)
and exponentiation function (^/2)
generate a rational number on integer and rational arguments and read/1
and friends read `[-+][0-9_ ]+/[0-9_ ]+`

into a rational
number. See also section
2.16.1.6. Here are some examples.

A is 2/6 | A = 1/3 |

A is 4/3 + 1 | A = 7/3 |

A is 4/3 + 1.5 | A = 2.83333 |

A is 4/3 + rationalize(1.5) | A = 17/6 |

Note that floats cannot represent all decimal numbers exactly. The
function rational/1
creates an *exact* equivalent of the float, while rationalize/1
creates a rational number that is within the float rounding error from
the original float. Please check the documentation of these functions
for details and examples.

Rational numbers can be printed as decimal numbers with arbitrary
precision using the format/3
floating point conversion:

?- A is 4/3 + rational(1.5),
format('~50f~n', [A]).
2.83333333333333333333333333333333333333333333333333
A = 17/6

SWI-Prolog uses rational number arithmetic if the Prolog flag
prefer_rationals
is `true`

and if this is defined for a function on the given
operants. This results in perfectly precise answers. Unfortunately
rational numbers can get really large and, if a precise answer is not
needed, a big waste of memory and CPU time. In such cases one should use
floating point arithmetic. The Prolog flag
max_rational_size
provides a *tripwire* to detect cases where rational numbers get
big and react on these events.

Floating point arithmetic can be forced by forcing a float into an
argument at any point, i.e., the result of a function with at least one
float is always float except for the float-to-integer rounding and
truncating functions such as round/1, rational/1
or float_integer_part/1.

Float arithmetic is typically forced by using a floating point
constant as initial value or operant. Alternatively, the float/1
function forces conversion of the argument.

The Prolog ISO standard defines that floating point arithmetic
returns a valid floating point number or raises an exception. IEEE
floating point arithmetic defines two modes: raising exceptions and
propagating the special float values `NaN`

, `Inf`

, `-Inf`

and
`-0.0`

. SWI-Prolog implements a part of the
ECLiPSe
proposal to support non-exception based processing of floating point
numbers. There are four flags that define handling the four exceptional
events in floating point arithmetic, providing the choice between
`error`

and returning the IEEE special value. Note that these
flags *only* apply for floating point arithmetic. For example
rational division by zero always raises an exception.

The Prolog flag float_rounding
and the function
roundtoward/2
control the rounding mode for floating point arithmetic. The default
rounding is `to_nearest`

and the following alternatives are
provided: `to_positive`

, `to_negative`

and
`to_zero`

.

- [det]
**float_class**(`+Float,
-Class`) - Wraps C99 fpclassify() to access the class of a floating point number.
Raises a type error if
`Float` is not a float. Defined classes
are below.
**nan**`Float` is “Not a number''. See nan/0.
May be produced if the Prolog flag float_undefined
is set to `nan`

. Although IEEE 754 allows NaN to carry a *payload*
and have a sign, SWI-Prolog has only a single NaN values. Note that two
NaN
*terms* compare equal in the standard order of terms (==/2,
etc.), they compare non-equal for arithmetic (=:=/2,
etc.).
**infinite**`Float` is positive or negative infinity. See inf/0.
May be produced if the Prolog flag float_overflow
or the flag float_zero_div
is set to `infinity`

.
**zero**`Float` is zero (0.0 or -0.0)
**subnormal**`Float` is too small to be represented in normalized format.
May **not** be produced if the Prolog flag
float_underflow
is set to `error`

.
**normal**`Float` is a normal floating point number.

- [det]
**float_parts**(`+Float,
-Mantissa, -Base, -Exponent`) - True when
`Mantissa` is the normalized fraction of `Float`,
`Base` is the *radix* and `Exponent` is the
exponent. This uses the C function frexp(). If `Float` is NaN
or `±`Inf
`Mantissa` has the same value and `Exponent` is 0
(zero). In the current implementation `Base` is always 2. The
following relation is always true:
`Float =:= Mantissa × Base^Exponent`

- [det]
**bounded_number**(`?Low,
?High, +Num`) - True if
`Low` < `Num` < `High`. Raises
a type error if `Num` is not a number. This predicate can be
used both to check and generate bounds across the various numeric types.
Note that a number cannot be bounded by itself and `NaN`

, `Inf`

,
and `-Inf`

are not bounded numbers.
If `Low` and/or `High` are variables they will be
unified with *tightest* values that still meet the bounds
criteria. The generated bounds will be integers if `Num` is an
integer; otherwise they will be floats (also see nexttoward/2
for generating float bounds). Some examples:

?- bounded_number(0,10,1).
true.
?- bounded_number(0.0,1.0,1r2).
true.
?- bounded_number(L,H,1.0).
L = 0.9999999999999999,
H = 1.0000000000000002.
?- bounded_number(L,H,-1).
L = -2,
H = 0.
?- bounded_number(0,1r2,1).
false.
?- bounded_number(L,H,1.0Inf).
false.

SWI-Prolog represents floats using the C `double`

type. On
virtually all modern hardware this implies it uses 64-bit IEEE 754
floating point numbers. See also section
4.27.2.4. All floating point arithmetic is performed using C.
Different C compilers, different C math libraries and different hardware
floating point support may yield different results for the same
expression on different instances of SWI-Prolog.

Arithmetic functions are terms which are evaluated by the arithmetic
predicates described in section
4.27.2. There are four types of arguments to functions:

`Expr` | Arbitrary expression, returning either
a floating point value or an integer. |

`IntExpr` | Arbitrary expression that must
evaluate to an integer. |

`RatExpr` | Arbitrary expression that must
evaluate to a rational number. |

`FloatExpr` | Arbitrary expression that must
evaluate to a floating point. |

For systems using bounded integer arithmetic (default is unbounded,
see section 4.27.2.1
for details), integer operations that would cause overflow automatically
convert to floating point arithmetic.

SWI-Prolog provides many extensions to the set of floating point
functions defined by the ISO standard. The current policy is to provide
such functions on‘as-needed' basis if the function is widely
supported elsewhere and notably if it is part of the
C99
mathematical library. In addition, we try to maintain compatibility with
other Prolog implementations.

- [ISO]
**-** `+Expr` `Result` = -`Expr`
- [ISO]
**+** `+Expr` `Result` = `Expr`. Note that if `+`

is followed by a number, the parser discards the `+`

.
I.e. `?- integer(+1)`

succeeds.
- [ISO]
`+Expr1` **+** `+Expr2` `Result` = `Expr1` + `Expr2`
- [ISO]
`+Expr1` **-** `+Expr2` `Result` = `Expr1` - `Expr2`
- [ISO]
`+Expr1` ***** `+Expr2` `Result` = `Expr1` × `Expr2`
- [ISO]
`+Expr1` **/** `+Expr2` `Result` = `Expr1`/`Expr2`. If the
flag iso is `true`

or one of the arguments is a float, both arguments are converted to
float and the return value is a float. Otherwise the result type depends
on the Prolog flag
prefer_rationals.
If `true`

, the result is always a rational number. If `false`

the result is rational if at least one of the arguments is rational.
Otherwise (both arguments are integer) the result is integer if the
division is exact and float otherwise. See also section
4.27.2.2, ///2, and rdiv/2.
The current default for the Prolog flag prefer_rationals
is
`false`

. Future version may switch this to `true`

,
providing precise results when possible. The pitfall is that in general
rational arithmetic is slower and can become very slow and produce huge
numbers that require a lot of (global stack) memory. Code for which the
exact results provided by rational numbers is not needed should force
float results by making one of the operants float, for example by
dividing by
`10.0`

rather than `10`

or by using float/1.
Note that when one of the arguments is forced to a float the division is
a float operation while if the result is forced to the float the
division is done using rational arithmetic.

- [ISO]
`+IntExpr1` **mod** `+IntExpr2` - Modulo, defined as
`Result` = `IntExpr1` - (`IntExpr1`
div `IntExpr2`) ` × ` `IntExpr2`, where `div`

is
*floored* division.
- [ISO]
`+IntExpr1` **rem** `+IntExpr2` - Remainder of integer division. Behaves as if defined by
`Result` is `IntExpr1` - (`IntExpr1` // `IntExpr2`) ` × ` `IntExpr2`
- [ISO]
`+IntExpr1` **//** `+IntExpr2` - Integer division, defined as
`Result` is `rnd_I`(`Expr1`/`Expr2`)
. The function `rnd_I` is the default rounding used by the C
compiler and available through the Prolog flag
integer_rounding_function.
In the C99 standard, C-rounding is defined as `towards_zero`

.^{120Future
versions might guarantee rounding towards zero.}
- [ISO]
**div**(`+IntExpr1,
+IntExpr2`) - Integer division, defined as
`Result` is (`IntExpr1` - `IntExpr1` `mod` `IntExpr2`)
// `IntExpr2`. In other words, this is integer division that
rounds towards -infinity. This function guarantees behaviour that is
consistent with
mod/2, i.e., the
following holds for every pair of integers
`X,Y` where `Y =\= 0`

.
Q is div(X, Y),
M is mod(X, Y),
X =:= Y*Q+M.

`+RatExpr` **rdiv** `+RatExpr`- Rational number division. This function is only available if SWI-Prolog
has been compiled with rational number support. See
section 4.27.2.2 for
details.
`+IntExpr1` **gcd** `+IntExpr2`- Result is the greatest common divisor of
`IntExpr1` and
`IntExpr2`. The GCD is always a positive integer. If either
expression evaluates to zero the GCD is the result of the other
expression.
`+IntExpr1` **lcm** `+IntExpr2`- Result is the least common multiple of
`IntExpr1`,
`IntExpr2`.^{bugIf the
system is compiled for bounded integers only lcm/2
produces an integer overflow if the product of the two expressions does
not fit in a 64 bit signed integer. The default build with unbounded
integer support has no such limit.} If either expression
evaluates to zero the LCM is zero.
- [ISO]
**abs**(`+Expr`) - Evaluate
`Expr` and return the absolute value of it.
- [ISO]
**sign**(`+Expr`) - Evaluate to -1 if
`Expr` < 0, 1 if `Expr`
> 0 and 0 if
`Expr` = 0. If `Expr` evaluates to a float,
the return value is a float (e.g., -1.0, 0.0 or 1.0). In particular,
note that sign(-0.0) evaluates to 0.0. See also copysign/2.
- [ISO]
**copysign**(`+Expr1,
+Expr2`) - Evaluate to
`X`, where the absolute value of `X`
equals the absolute value of `Expr1` and the sign of `X`
matches the sign of `Expr2`. This function is based on
copysign() from C99, which works on double precision floats and deals
with handling the sign of special floating point values such as -0.0.
Our implementation follows C99 if both arguments are floats. Otherwise, copysign/2
evaluates to `Expr1` if the sign of both expressions matches or
-`Expr1` if the signs do not match. Here, we use the extended
notion of signs for floating point numbers, where the sign of -0.0 and
other special floats is negative.
**nexttoward**(`+Expr1,
+Expr2`)- Evaluates to floating point number following
`Expr1` in the
direction of `Expr2`. This relates to epsilon/0
in the following way:
?- epsilon =:= nexttoward(1,2)-1.
true.

**roundtoward**(`+Expr1,
+RoundMode`)- Evaluate
`Expr1` using the floating point rounding mode
`RoundMode`. This provides a local alternative to the Prolog
flag
float_rounding.
This function can be nested. The supported values for `RoundMode`
are the same as the flag values:
`to_nearest`

, `to_positive`

, `to_negative`

or
`to_zero`

.
Note that floating point arithmetic is provided by the C compiler
and C runtime library. Unfortunately most C libraries do not
correctly implement the rounding modes for notably the trigonometry and
exponential functions. There exist correct libraries such as
crlibm, but
these libraries are large, most of them are poorly maintained or have an
incompatible license. C runtime libraries do a better job using the
default
*to nearest* rounding mode. SWI-Prolog now assumes this mode is
correct and translates upward rounding to be the nexttoward/2
infinity and downward rounding nexttoward/2
-infinity. If the “to nearest'' rounding mode is correct, this
ensures that the true value is between the downward and upward rounded
values, although the generated interval is larger than needed.
Unfortunately this is not the case as shown in Accuracy
of Mathematical Functions in Single, Double, Extended Double and
Quadruple Precision by *Vincenzo Innocente and Paul Zimmermann*.

- [ISO]
**max**(`+Expr1,
+Expr2`) - Evaluate to the larger of
`Expr1` and `Expr2`. Both
arguments are compared after converting to the same type, but the return
value is in the original type. For example, max(2.5, 3) compares the two
values after converting to float, but returns the integer 3. If both
values are numerical equal the returned max is of the type used for the
comparison. For example, the max of 1 and 1.0 is 1.0 because both
numbers are converted to float for the comparison. However, the special
float -0.0 is smaller than 0.0 as well as the integer 0.
- [ISO]
**min**(`+Expr1,
+Expr2`) - Evaluate to the smaller of
`Expr1` and `Expr2`. See
max/2 for a
description of type handling.
- [deprecated]
**.**(`+Char,[]`) - A list of one element evaluates to the character code of this element.
^{121The
function is documented as ./2. Using
SWI-Prolog v7 and later the actual functor is [|]/2.}
This implies `"a"`

evaluates to the character code of the
letter‘a' (97) using the traditional mapping of double quoted
string to a list of character codes. `Char` is either a valid
code point (non-negative integer up to the Prolog flag max_char_code)
or a one-character atom. Arithmetic evaluation also translates a string
object (see section 5.2)
of one character length into the character code for that character. This
implies that expression `"a"`

works if the Prolog flag double_quotes
is set to one of
`codes`

, `chars`

or `string`

.
Getting access to character codes this way originates from DEC10
Prolog. ISO has the `0'a`

syntax and the predicate char_code/2.
Future versions may drop support for `X is "a"`

.

**random**(`+IntExpr`)- Evaluate to a random integer
`i` for which `0 =< i < ``IntExpr`.
The system has two implementations. If it is compiled with support for
unbounded arithmetic (default) it uses the GMP library random functions.
In this case, each thread keeps its own random state. The default
algorithm is the *Mersenne Twister* algorithm. The seed is set
when the first random number in a thread is generated. If available, it
is set from `/dev/random`

.^{122On
Windows the state is initialised from CryptGenRandom().}
Otherwise it is set from the system clock. If unbounded arithmetic is
not supported, random numbers are shared between threads and the seed is
initialised from the clock when SWI-Prolog was started. The predicate set_random/1
can be used to control the random number generator.
**Warning!** Although properly seeded (if supported on the OS),
the Mersenne Twister algorithm does *not* produce
cryptographically secure random numbers. To generate cryptographically
secure random numbers, use crypto_n_random_bytes/2
from library `library(crypto)`

provided by the `ssl`

package.

**random_float**- Evaluate to a random float
`I` for which `0.0 < i <
1.0`. This function shares the random state with random/1.
All remarks with the function random/1
also apply for random_float/0.
Note that both sides of the domain are *open*. This avoids
evaluation errors on, e.g., log/1
or //2 while no
practical application can expect 0.0.^{123Richard
O'Keefe said: “If you are generating IEEE doubles with
the claimed uniformity, then 0 has a 1 in 2^53 = 1 in
9,007,199,254,740,992 chance of turning up. No program that
expects [0.0,1.0) is going to be surprised when 0.0 fails to turn up in
a few millions of millions of trials, now is it? But a program that
expects (0.0,1.0) could be devastated if 0.0 did turn up.''}
- [ISO]
**round**(`+Expr`) - Evaluate
`Expr` and round the result to the nearest integer.
According to ISO, round/1
is defined as
`floor(Expr+1/2)`

, i.e., rounding *down*. This is an
unconventional choice under which the relation
`round(Expr) == -round(-Expr)`

does not hold. SWI-Prolog
rounds *outward*, e.g., `round(1.5) =:= 2`

and
`round(-1.5) =:= -2`

.
**integer**(`+Expr`)- Same as round/1
(backward compatibility).
- [ISO]
**float**(`+Expr`) - Translate the result to a floating point number. Normally, Prolog will
use integers whenever possible. When used around the 2nd argument of
is/2,
the result will be returned as a floating point number. In other
contexts, the operation has no effect.
**rational**(`+Expr`)- Convert the
`Expr` to a rational number or integer. The
function returns the input on integers and rational numbers. For
floating point numbers, the returned rational number *exactly*
represents the float. As floats cannot exactly represent all decimal
numbers the results may be surprising. In the examples below, doubles
can represent 0.25 and the result is as expected, in contrast to the
result of `rational(0.1)`

. The function rationalize/1
remedies this. See section
4.27.2.2 for more information on rational number support.
?- A is rational(0.25).
A is 1 rdiv 4
?- A is rational(0.1).
A = 3602879701896397 rdiv 36028797018963968

For every *normal* float `X` the relation
`X` `=:=`

rational(`X`) holds.

This function raises an `evaluation_error(undefined)`

if `Expr`
is NaN and `evaluation_error(rational_overflow)`

if `Expr`
is Inf.

**rationalize**(`+Expr`)- Convert the
`Expr` to a rational number or integer. The
function is similar to rational/1,
but the result is only accurate within the rounding error of floating
point numbers, generally producing a much smaller denominator.^{124The
names rational/1
and rationalize/1
as well as their semantics are inspired by Common Lisp.}^{125The
implementation of rationalize as well as converting a rational number
into a float is copied from ECLiPSe and covered by the Cisco-style
Mozilla Public License Version 1.1.}
?- A is rationalize(0.25).
A = 1 rdiv 4
?- A is rationalize(0.1).
A = 1 rdiv 10

For every *normal* float `X` the relation
`X` `=:=`

rationalize(`X`)
holds.

This function raises the same exceptions as rational/1
on non-normal floating point numbers.

**numerator**(`+RationalExpr`)- If
`RationalExpr` evaluates to a rational number or integer,
evaluate to the top/left value. Evaluates to itself if
`RationalExpr` evaluates to an integer. See also
denominator/1.
The following is true for any rational
`X`.
X =:= numerator(X)/denominator(X).

**denominator**(`+RationalExpr`)- If
`RationalExpr` evaluates to a rational number or integer,
evaluate to the bottom/right value. Evaluates to 1 (one) if
`RationalExpr` evaluates to an integer. See also
numerator/1. The
following is true for any rational `X`.
X =:= numerator(X)/denominator(X).

- [ISO]
**float_fractional_part**(`+Expr`) - Fractional part of a floating point number. Negative if
`Expr`
is negative, rational if `Expr` is rational and 0 if `Expr`
is integer. The following relation is always true:
`X is float_fractional_part(X) + float_integer_part(X)`.
- [ISO]
**float_integer_part**(`+Expr`) - Integer part of floating point number. Negative if
`Expr` is
negative, `Expr` if `Expr` is integer.
- [ISO]
**truncate**(`+Expr`) - Truncate
`Expr` to an integer. If `Expr` >= 0
this is the same as `floor(Expr)`

. For `Expr` <
0 this is the same as
`ceil(Expr)`

. That is, truncate/1
rounds towards zero.
- [ISO]
**floor**(`+Expr`) - Evaluate
`Expr` and return the largest integer smaller or equal
to the result of the evaluation.
- [ISO]
**ceiling**(`+Expr`) - Evaluate
`Expr` and return the smallest integer larger or equal
to the result of the evaluation.
**ceil**(`+Expr`)- Same as ceiling/1
(backward compatibility).
- [ISO]
`+IntExpr1` **>>** `+IntExpr2` - Bitwise shift
`IntExpr1` by `IntExpr2` bits to the
right. The operation performs *arithmetic shift*, which implies
that the inserted most significant bits are copies of the original most
significant bits.
- [ISO]
`+IntExpr1` **<<** `+IntExpr2` - Bitwise shift
`IntExpr1` by `IntExpr2` bits to the
left.
- [ISO]
`+IntExpr1` **\/** `+IntExpr2` - Bitwise‘or'
`IntExpr1` and `IntExpr2`.
- [ISO]
`+IntExpr1` **/\** `+IntExpr2` - Bitwise‘and'
`IntExpr1` and `IntExpr2`.
- [ISO]
`+IntExpr1` **xor** `+IntExpr2` - Bitwise‘exclusive or'
`IntExpr1` and `IntExpr2`.
- [ISO]
**\** `+IntExpr` - Bitwise negation. The returned value is the one's complement of
`IntExpr`.
- [ISO]
**sqrt**(`+Expr`) `Result` = sqrt(`Expr`).
- [ISO]
**sin**(`+Expr`) `Result` = sin(`Expr`). `Expr` is
the angle in radians.
- [ISO]
**cos**(`+Expr`) `Result` = cos(`Expr`). `Expr` is
the angle in radians.
- [ISO]
**tan**(`+Expr`) `Result` = tan(`Expr`). `Expr` is
the angle in radians.
- [ISO]
**asin**(`+Expr`) `Result` = arcsin(`Expr`). `Result`
is the angle in radians.
- [ISO]
**acos**(`+Expr`) `Result` = arccos(`Expr`). `Result`
is the angle in radians.
- [ISO]
**atan**(`+Expr`) `Result` = arctan(`Expr`). `Result`
is the angle in radians.
- [ISO]
**atan2**(`+YExpr,
+XExpr`) `Result` = arctan(`YExpr`/`XExpr`). `Result`
is the angle in radians. The return value is in the range `[- pi ...
pi ]`. Used to convert between rectangular and polar coordinate
system.
Note that the ISO Prolog standard demands `atan2(0.0,0.0)`

to raise an evaluation error, whereas the C99 and POSIX standards demand
this to evaluate to 0.0. SWI-Prolog follows C99 and POSIX.

**atan**(`+YExpr,
+XExpr`)- Same as atan2/2
(backward compatibility).
**sinh**(`+Expr`)`Result` = sinh(`Expr`). The hyperbolic
sine of `X` is defined as `e ** X - e ** -X / 2`.
**cosh**(`+Expr`)`Result` = cosh(`Expr`). The hyperbolic
cosine of `X` is defined as `e ** X + e ** -X / 2`.
**tanh**(`+Expr`)`Result` = tanh(`Expr`). The hyperbolic
tangent of `X` is defined as `sinh( X ) / cosh( X )`.
**asinh**(`+Expr`)`Result` = arcsinh(`Expr`) (inverse
hyperbolic sine).
**acosh**(`+Expr`)`Result` = arccosh(`Expr`) (inverse
hyperbolic cosine).
**atanh**(`+Expr`)`Result` = arctanh(`Expr`). (inverse
hyperbolic tangent).
- [ISO]
**log**(`+Expr`) - Natural logarithm.
`Result` = ln(`Expr`)
**log10**(`+Expr`)- Base-10 logarithm.
`Result` = log10(`Expr`)
- [ISO]
**exp**(`+Expr`) `Result` = e **`Expr`
- [ISO]
`+Expr1` ****** `+Expr2` `Result` = `Expr1`**`Expr2`. The
result is a float, unless SWI-Prolog is compiled with unbounded integer
support and the inputs are integers and produce an integer result. The
integer expressions `0 ** I`, `1 ** I` and `-1 **
I` are guaranteed to work for any integer `I`. Other
integer base values generate a
`resource`

error if the result does not fit in memory.
The ISO standard demands a float result for all inputs and introduces
^/2 for integer
exponentiation. The function
float/1 can be used
on one or both arguments to force a floating point result. Note that
casting the *input* result in a floating point computation, while
casting the *output* performs integer exponentiation followed by
a conversion to float.

- [ISO]
`+Expr1` **^** `+Expr2` -
In SWI-Prolog, ^/2 is
equivalent to **/2. The
ISO version is similar, except that it produces a evaluation error if
both
`Expr1` and `Expr2` are integers and the result is not
an integer. The table below illustrates the behaviour of the
exponentiation functions in ISO and SWI. Note that if the exponent is
negative the behavior of `Int``^`

`Int`
depends on the flag
prefer_rationals,
producing either a rational number or a floating point number.

`Expr1` | `Expr2` | Function | SWI | ISO |

Int | Int | **/2 | Int
or Rational | Float |

Int | Float | **/2 | Float | Float |

Rational | Int | **/2 | Rational | - |

Float | Int | **/2 | Float | Float |

Float | Float | **/2 | Float | Float |

Int | Int | ^/2 | Int
or Rational | Int or error |

Int | Float | ^/2 | Float | Float |

Rational | Int | ^/2 | Rational | - |

Float | Int | ^/2 | Float | Float |

Float | Float | ^/2 | Float | Float |

**powm**(`+IntExprBase,
+IntExprExp, +IntExprMod`)`Result` = (`IntExprBase`**`IntExprExp`)
modulo `IntExprMod`. Only available when compiled with
unbounded integer support. This formula is required for Diffie-Hellman
key-exchange, a technique where two parties can establish a secret key
over a public network.
`IntExprBase` and `IntExprExp` must be non-negative (`>=0`),
`IntExprMod` must be positive (`>0`).^{126The
underlying GMP mpz_powm() function allows negative values under some
conditions. As the conditions are expensive to pre-compute, error
handling from GMP is non-trivial and negative values are not needed for
Diffie-Hellman key-exchange we do not support these.}
**lgamma**(`+Expr`)- Return the natural logarithm of the absolute value of the Gamma
function.
^{127Some interfaces also
provide the sign of the Gamma function. We cannot do that in an
arithmetic function. Future versions may provide a predicate
lgamma/3 that returns both the value and the sign.}
**erf**(`+Expr`)- Wikipedia: “In
mathematics, the error function (also called the Gauss error function)
is a special function (non-elementary) of sigmoid shape which occurs in
probability, statistics and partial differential equations.''
**erfc**(`+Expr`)- Wikipedia: “The
complementary error function.''
- [ISO]
**pi** - Evaluate to the mathematical constant
`pi` (3.14159 ... ).
**e**- Evaluate to the mathematical constant
`e` (2.71828 ... ).
**epsilon**- Evaluate to the difference between the float 1.0 and the first larger
floating point number. Deprecated. The function nexttoward/2
provides a better alternative.
**inf**- Evaluate to positive infinity. See section
2.16.1.7 and
section 4.27.2.4. This
value can be negated using -/1.
**nan**- Evaluate to
*Not a Number*. See section
2.16.1.7 and
section 4.27.2.4.
**cputime**- Evaluate to a floating point number expressing the CPU
time (in seconds) used by Prolog up till now. See also statistics/2
and time/1.
**eval**(`+Expr`)- Evaluate
`Expr`. Although ISO standard dictates that‘`A`=1+2, `B`
is
`A`’works and unifies `B` to 3, it is widely
felt that source level variables in arithmetic expressions should have
been limited to numbers. In this view the eval function can be used to
evaluate arbitrary expressions.^{128The eval/1
function was first introduced by ECLiPSe and is under consideration for
YAP.}

**Bitvector functions**

The functions below are not covered by the standard. The
msb/1 function also
appears in hProlog and SICStus Prolog. The getbit/2
function also appears in ECLiPSe, which also provides `setbit(Vector,Index)`

and `clrbit(Vector,Index)`

. The others are SWI-Prolog
extensions that improve handling of ---unbounded--- integers as
bit-vectors.

**msb**(`+IntExpr`)- Return the largest integer
`N` such that `(IntExpr >> N) /\ 1 =:= 1`

.
This is the (zero-origin) index of the most significant 1 bit in the
value of `IntExpr`, which must evaluate to a positive integer.
Errors for 0, negative integers, and non-integers.
**lsb**(`+IntExpr`)- Return the smallest integer
`N` such that `(IntExpr >> N) /\ 1 =:= 1`

.
This is the (zero-origin) index of the least significant 1 bit in the
value of `IntExpr`, which must evaluate to a positive integer.
Errors for 0, negative integers, and non-integers.
**popcount**(`+IntExpr`)- Return the number of 1s in the binary representation of the non-negative
integer
`IntExpr`.
**getbit**(`+IntExprV,
+IntExprI`)- Evaluates to the bit value (0 or 1) of the
`IntExprI`-th bit of
`IntExprV`. Both arguments must evaluate to non-negative
integers. The result is equivalent to `(IntExprV >> IntExprI)/\1`

,
but more efficient because materialization of the shifted value is
avoided. Future versions will optimise `(IntExprV >> IntExprI)/\1`

to a call to getbit/2,
providing both portability and performance.^{129This
issue was fiercely debated at the ISO standard mailinglist. The name getbit
was selected for compatibility with ECLiPSe, the only system providing
this support. Richard O'Keefe disliked the name and argued that
efficient handling of the above implementation is the best choice for
this functionality.}