4.27.2.6 Arithmetic Functions

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.^{124Future 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.

cmpr(+Expr1, +Expr2)

Exactly compares the values Expr1 and Expr2 and returns -1 if Expr1 < Expr2, 0 if they are equal, and 1 if Expr1 > Expr2. Evaluates to NaN if either or both Expr1 and Expr2 are NaN and the Prolog flag float_undefined is set to nan. See also minr/2 amd maxr/2.

This function relates to the Prolog numerical comparison predicates >/2, =:=/2, etc. The Prolog numerical comparison converts the rational in a mixed rational/float comparison to a float, possibly rounding the value. This function converts the float to a rational, comparing the exact values.

[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. If the Prolog flag float_undefined is set to nan and one of the arguments evaluates to NaN, the result is NaN.

The function maxr/2 is similar, but uses exact (rational) comparision if Expr1 and Expr2 have a different type, propagate the rational (integer) rather and the float if the two compare equal and propagate the non-NaN value in case one is NaN.

maxr(+Expr1, +Expr2)

Evaluate to the larger of Expr1 and Expr2 using exact comparison (see cmpr/2). If the two values are exactly equal, and one of the values is rational, the result will be that value; the objective being to avoid "pollution" of any precise calculation with a potentially imprecise float. So max(1,1.0) evaluates to 1.0 while maxr(1,1.0) evaluates to 1. This also means that 0 is preferred over 0.0 or -0.0; -0.0 is still considered smaller than 0.0.

maxr/2 also treats NaN's as missing values so maxr(1,nan) evaluates to 1.

[ISO]min(+Expr1, +Expr2)

Evaluate to the smaller of Expr1 and Expr2. See max/2 for a description of type handling.

minr(+Expr1, +Expr2)

Evaluate to the smaller of Expr1 and Expr2 using exact comparison (see cmpr/2). See maxr/2 for a description of type handling.

[deprecated].(+Char,[])

A list of one element evaluates to the character code of this element.^{125The 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' 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.^{126On 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.^{127Richard 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 1r4
?- A is rational(0.1).
A = 3602879701896397r36028797018963968

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.^{128The names rational/1 and rationalize/1 as well as their semantics are inspired by Common Lisp.129The 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 = 1r4
?- A is rationalize(0.1).

A = 1r10

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 ≥ 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 = &#8730;(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 [-π...π. 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).^{130The 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.^{131Some 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 π (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 2.15.1.7 and section 4.27.2.4. This value can be negated using -/1.

nan

Evaluate to Not a Number. See section 2.15.1.7 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.^{132The 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.^{133This 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.}