4.7.5 Arithmetic Expressions

Arithmetic evaluation and testing is performed by predicates that take arithmetic expressions as arguments. An arithmetic expression is a term built from numbers, variables, and functors that represent arithmetic functions. These expressions are evaluated to yield an arithmetic result, which may be either an integer or a float; the type is determined by the rules described below.

At the time of evaluation, each variable in an arithmetic expression must be bound to a number or another arithmetic expression. If the expression is not sufficiently bound or if it is bound to terms of the wrong type, then Prolog raises exceptions of the appropriate type (see ref-ere-hex). Some arithmetic operations can also detect overflows. They also raise exceptions, e.g. division by zero results in an evaluation error being raised.

Only certain functors are permitted in arithmetic expressions. These are listed below, together with a description of their arithmetic meanings. For the rest of the section, X and Y are considered to be arithmetic expressions. Unless stated otherwise, the arguments of an expression may be any numbers and its value is a float if any of its arguments is a float; otherwise, the value is an integer. Any implicit coercions are performed with the integer/1 and float/1 functions. All trigonometric and transcendental functions take float arguments and deliver float values. The trigonometric functions take arguments or deliver values in radians.

Integers can for all practical purposes be considered to be of infinite size. Negative integers can be considered to be infinitely sign extended.

The arithmetic functors are annotated with ISO, with the same meaning as for the built-in predicates; see ISO Compliance.

+X   ISO

The value is X.

-X   ISO

The value is the negative of X.

X+Y   ISO

The value is the sum of X and Y.

X-Y   ISO

The value is the difference between X and Y.

X*Y   ISO

The value is the product of X and Y.

X/Y   ISO

The value is the float quotient of X and Y.

X//Y   ISO

The value is the integer quotient of X and Y, truncated towards zero. X and Y have to be integers.

X div Y   since release 4.3, ISO

The value is the integer quotient of X and Y, rounded downwards to the nearest integer. X and Y have to be integers.

X rem Y   ISO

The value is the integer remainder after truncated division of X by Y, i.e. X-Y*(X//Y). The sign of a nonzero remainder will thus be the same as that of the dividend. X and Y have to be integers.

X mod Y   ISO

The value is the integer remainder after floored division of X by Y, i.e. X-Y*(X div Y). The sign of a nonzero remainder will thus be the same as that of the divisor. X and Y have to be integers.

integer(X)

The value is the closest integer between X and 0, if X is a float; otherwise, X itself.

float_integer_part(X)   ISO

The same as float(integer(X)). X has to be a float.

float_fractional_part(X)   ISO

The value is the fractional part of X, i.e. X - float_integer_part(X).

X has to be a float.

float(X)   ISO

The value is the float equivalent of X, if X is an integer; otherwise, X itself.

X/\Y   ISO

The value is the bitwise conjunction of the integers X and Y. X and Y have to be integers, treating negative integers as infinitely sign extended.

X\/Y   ISO

The value is the bitwise disjunction of the integers X and Y. X and Y have to be integers, treating negative integers as infinitely sign extended.

xor(X,Y)   since release 4.3, ISO

The value is the bitwise exclusive or of the integers X and Y. X and Y have to be integers, treating negative integers as infinitely sign extended.

X\Y

The same as xor(X,Y).

\(X)   ISO

The value is the bitwise negation of the integer X. X has to be an integer, treating negative integers as infinitely sign extended.

X<<Y   ISO

The value is the integer X shifted arithmetically left by Y places. i.e. filling with a copy of the sign bit. X and Y have to be integers, and Y can be negative, in which case the shift is right.

X>>Y   ISO

The value is the integer X shifted arithmetically right by Y places, i.e. filling with a copy of the sign bit. X and Y have to be integers, and Y can be negative, in which case the shift is left.

[X]

A list of just one number X evaluates to X. Since a quoted string is just a list of integers, this allows a quoted character to be used in place of its character code; e.g. "A" behaves within arithmetic expressions as the integer 65.

abs(X)   ISO

The value is the absolute value of X.

sign(X)   ISO

The value is the sign of X, i.e. -1, if X is negative, 0, if X is zero, and 1, if X is positive, coerced into the same type as X (i.e. the result is an integer, if and only if X is an integer).

gcd(X,Y)

The value is the greatest common divisor of the two integers X and Y. X and Y have to be integers.

min(X,Y)   ISO

The value is the lesser value of X and Y.

max(X,Y)   ISO

The value is the greater value of X and Y.

msb(X)

The value is the position of the most significant nonzero bit of the integer X, counting bit positions from zero. It is equivalent to, but more efficient than, integer(log(2,X)). X must be greater than zero, and X has to be an integer.

round(X)   ISO

The value is the closest integer to X. If X is exactly half-way between two integers, then it is rounded up (i.e. the value is the least integer greater than X).

truncate(X)   ISO

The value is the closest integer between X and 0.

floor(X)   ISO

The value is the greatest integer less or equal to X.

ceiling(X)   ISO

The value is the least integer greater or equal to X.

sin(X)   ISO

The value is the sine of X.

cos(X)   ISO

The value is the cosine of X.

tan(X)   ISO

The value is the tangent of X.

cot(X)

The value is the cotangent of X.

sinh(X)

The value is the hyperbolic sine of X.

cosh(X)

The value is the hyperbolic cosine of X.

tanh(X)

The value is the hyperbolic tangent of X.

coth(X)

The value is the hyperbolic cotangent of X.

asin(X)   ISO

The value is the arc sine of X.

acos(X)   ISO

The value is the arc cosine of X.

atan(X)   ISO

The value is the arc tangent of X.

atan2(X,Y)   ISO

The value is the four-quadrant arc tangent of X and Y.

acot(X)

The value is the arc cotangent of X.

acot2(X,Y)

The value is the four-quadrant arc cotangent of X and Y.

asinh(X)

The value is the hyperbolic arc sine of X.

acosh(X)

The value is the hyperbolic arc cosine of X.

atanh(X)

The value is the hyperbolic arc tangent of X.

acoth(X)

The value is the hyperbolic arc cotangent of X.

sqrt(X)   ISO

The value is the square root of X.

log(X)   ISO

The value is the natural logarithm of X.

log(Base,X)

The value is the logarithm of X in the base Base.

exp(X)   ISO

The value is the natural exponent of X.

X ** Y   ISO

The value is X raised to the power of Y, represented as a float. In particular, the value of 0.0 ** 0.0 is 1.0.

exp(X,Y)

The same as X ** Y.

X ^ Y   since release 4.3, ISO

The value is X raised to the power of Y, represented as a float if any of X and Y is a float; otherwise, as an integer. In particular, the value of 0^0 is 1.

pi   since release 4.3, ISO

The value is approximately 3.14159.

The following operation is included in order to allow integer arithmetic on character codes.

[X]

Evaluates to X for numeric X. This is relevant because character strings in Prolog are lists of character codes, that is, integers. Thus, for those integers that correspond to character codes, the user can write a string of one character in place of that integer in an arithmetic expression. For example, the expression (A) is equivalent to (B), which in turn becomes (C) in which case X is unified with 2:

X is "c" - "a"  (A)

X is [99] - [97]  (B)

X is 99 - 97  (C)

A cleaner way to do the same thing is

X is 0'c - 0'a