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:
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.
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
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
126.96.36.199. 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
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
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
-0.0. SWI-Prolog implements a part of the
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
control the rounding mode for floating point arithmetic. The default
to_nearest and the following alternatives are
- [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
- 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.).
- 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
- Float is zero (0.0 or -0.0)
- Float is too small to be represented in normalized format.
May not be produced if the Prolog flag
is set to
- 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
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
-Infare 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.
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 188.8.131.52 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 YAP.
- [ISO]- +Expr
- Result = -Expr
- [ISO]+ +Expr
- Result = Expr. Note that if
is followed by a number, the parser discards the
- [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
trueor 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
falsethe 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 184.108.40.206, ///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
10or 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
divis 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
In the C99 standard, C-rounding is defined as
towards_zero.114Future 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
mod/2, i.e., the
following holds for every pair of integers
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 220.127.116.11 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.
- Evaluate Expr and return the absolute value of it.
- 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
This function can be nested. The supported values for RoundMode
are the same as the flag values:
- [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.
- [ISO]min(+Expr1, +Expr2)
- Evaluate to the smaller of Expr1 and Expr2. See max/2 for a description of type handling.
- A list of one element evaluates to the element. 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. 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"also works of the Prolog flag double_quotes is set to
string. The recommended way to specify the character code of the letter‘a' is
- 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.115On 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
- 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 0.0.116Richard 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.''
- 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) =:= 2and
round(-1.5) =:= -2.
- Same as round/1 (backward compatibility).
- 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.
- 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
rational(0.1). The function rationalize/1 remedies this. See section 18.104.22.168 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
This function raises an
evaluation_error(undefined)if Expr is NaN and
evaluation_error(rational_overflow)if Expr is Inf.
- 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.117The
as well as their semantics are inspired by Common Lisp.118The
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
This function raises the same exceptions as rational/1 on non-normal floating point numbers.
- 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
The following is true for any rational
X =:= numerator(X)/denominator(X).
- 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
following is true for any rational X.
X =:= numerator(X)/denominator(X).
- 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).
- Integer part of floating point number. Negative if Expr is negative, Expr if Expr is integer.
- 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.
- Evaluate Expr and return the largest integer smaller or equal to the result of the evaluation.
- Evaluate Expr and return the smallest integer larger or equal to the result of the evaluation.
- 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.
- Result = sqrt(Expr).
- Result = sin(Expr). Expr is the angle in radians.
- Result = cos(Expr). Expr is the angle in radians.
- Result = tan(Expr). Expr is the angle in radians.
- Result = arcsin(Expr). Result is the angle in radians.
- Result = arccos(Expr). Result is the angle in radians.
- 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
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).
- Result = sinh(Expr). The hyperbolic sine of X is defined as e ** X - e ** -X / 2.
- Result = cosh(Expr). The hyperbolic cosine of X is defined as e ** X + e ** -X / 2.
- Result = tanh(Expr). The hyperbolic tangent of X is defined as sinh( X ) / cosh( X ).
- Result = arcsinh(Expr) (inverse hyperbolic sine).
- Result = arccosh(Expr) (inverse hyperbolic cosine).
- Result = arctanh(Expr). (inverse hyperbolic tangent).
- Natural logarithm. Result = ln(Expr)
- Base-10 logarithm. Result = log10(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
resourceerror 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).119The 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.
- Return the natural logarithm of the absolute value of the Gamma function.120Some 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.
- 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.''
- Wikipedia: “The complementary error function.''
- Evaluate to the mathematical constant pi (3.14159 ... ).
- Evaluate to the mathematical constant e (2.71828 ... ).
- 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.
- Evaluate to positive infinity. See section 22.214.171.124 and section 126.96.36.199. This value can be negated using -/1.
- Evaluate to Not a Number. See section 188.8.131.52 and section 184.108.40.206.
- 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.
- 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.121The eval/1 function was first introduced by ECLiPSe and is under consideration for YAP.
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
clrbit(Vector,Index). The others are SWI-Prolog
extensions that improve handling of ---unbounded--- integers as
- 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.
- 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.
- 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)/\1to a call to getbit/2, providing both portability and performance.122This 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.