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General purpose arithmetic |

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. Rational arithmetic is only provided if GMP is used (see above). Rational numbers are currently not supported by a Prolog type. They are represented by the compound term`rdiv(N,M)`

. Rational numbers that are returned from is/2 are*canonical*, which means`M`is positive and`N`and`M`have no common divisors. Rational numbers are introduced in the computation using the rational/1, rationalize/1 or the rdiv/2 (rational division) function. Using the same functor for rational division and for representing rational numbers allows for passing rational numbers between computations as well as for using format/3 for printing.In the long term, it is likely that rational numbers will become

*atomic*as well as a subtype of*number*. User code that creates or inspects the`rdiv(M,N)`

terms will not be portable to future versions. Rationals are created using one of the functions mentioned above and inspected using rational/3.*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 and division. To exploit rational arithmetic rdiv/2
should be used instead of `/' and floating point numbers must be
converted to rational using rational/1.
Omitting the rational/1
on floats will convert a rational operand to float and continue the
arithmetic using floating point numbers. Here are some examples.

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

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

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

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

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 YAP.

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

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` . If the flag iso is`Result`=`Expr1`/`Expr2``true`

, both arguments are converted to float and the return value is a float. Otherwise (default), if both arguments are integers the operation returns an integer if the division is exact. If at least one of the arguments is rational and the other argument is integer, the operation returns a rational number. In all other cases the return value is a float. See also ///2 and rdiv/2.- [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`

.^{116Future 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`,`IntExpr2`. - [ISO]
**abs**(`+Expr`) - Evaluate
`Expr`and return the absolute value of it. - [ISO]
**sign**(`+Expr`) - Evaluate to -1 if
, 1 if`Expr`< 0and 0 if`Expr`> 0. If`Expr`= 0`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. - [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. **.**(`+Int,[]`)- 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`0'a`

. **random**(`+IntExpr`)- Evaluate to a random integer
`i`for which`0 =< i <`. 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`IntExpr`*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`

.^{117On 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.^{118Richard 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

**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.^{119The names rational/1 and rationalize/1 as well as their semantics are inspired by Common Lisp.}?- A is rationalize(0.25). A = 1 rdiv 4 ?- A is rationalize(0.1). A = 1 rdiv 10

- [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. Ifthis is the same as`Expr`>= 0`floor(Expr)`

. Forthis is the same as`Expr`< 0`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`). The hyperbolic sine of`Result`= sinh(`Expr`)`X`is defined as`e ** X - e ** -X / 2`.**cosh**(`+Expr`). The hyperbolic cosine of`Result`= cosh(`Expr`)`X`is defined as`e ** X + e ** -X / 2`.**tanh**(`+Expr`). The hyperbolic tangent of`Result`= tanh(`Expr`)`X`is defined as`sinh( X ) / cosh( X )`.**asinh**(`+Expr`)(inverse hyperbolic sine).`Result`= arcsinh(`Expr`)**acosh**(`+Expr`)(inverse hyperbolic cosine).`Result`= arccosh(`Expr`)**atanh**(`+Expr`). (inverse hyperbolic tangent).`Result`= arctanh(`Expr`)- [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` . 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`Result`=`Expr1`**`Expr2``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.`Expr1``Expr2`Function SWI ISO Int Int **/2 Int or Float 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 Float 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`). 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.`Result`= (`IntExprBase`**`IntExprExp`) modulo`IntExprMod``IntExprBase`and`IntExprExp`must be non-negative (`>=0`),`IntExprMod`must be positive (`>0`).^{120The 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.
^{121Some interfaces also provide the sign of the Gamma function. We canot 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.
**inf**- Evaluate to positive infinity. See section 2.17.1.6. This value can be negated using -/1.
**nan**- Evaluate to
*Not a Number*. See section 2.17.1.6. **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.^{122The 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.^{123This 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.}

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