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Arithmetic Functions |

- Documentation
- Reference manual
- Built-in Predicates
- Arithmetic
- General purpose arithmetic
- >/2
- </2
- =</2
- >=/2
- =\=/2
- =:=/2
- is/2
- Arithmetic types
- Rational number examples
- Rational numbers or floats
- IEEE 754 floating point arithmetic
- Floating point arithmetic precision
- Arithmetic Functions
- -/1
- +/1
- +/2
- -/2
- */2
- //2
- mod/2
- rem/2
- ///2
- div/2
- rdiv/2
- gcd/2
- lcm/2
- abs/1
- sign/1
- copysign/2
- nexttoward/2
- roundtoward/2
- max/2
- min/2
- ./2
- random/1
- random_float/0
- round/1
- integer/1
- float/1
- rational/1
- rationalize/1
- numerator/1
- denominator/1
- float_fractional_part/1
- float_integer_part/1
- truncate/1
- floor/1
- ceiling/1
- ceil/1
- >>/2
- <</2
- \//2
- /\/2
- xor/2
- \/1
- sqrt/1
- sin/1
- cos/1
- tan/1
- asin/1
- acos/1
- atan/1
- atan2/2
- atan/2
- sinh/1
- cosh/1
- tanh/1
- asinh/1
- acosh/1
- atanh/1
- log/1
- log10/1
- exp/1
- **/2
- ^/2
- powm/3
- lgamma/1
- erf/1
- erfc/1
- pi/0
- e/0
- epsilon/0
- inf/0
- nan/0
- cputime/0
- eval/1
- msb/1
- lsb/1
- popcount/1
- getbit/2

- General purpose arithmetic

- Arithmetic

- Built-in Predicates
- Packages

- Reference manual

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` . 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`

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`

.^{122Future 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
, 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. **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.
^{123The 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 <`. 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`

.^{124On 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.^{125Richard 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.^{126The names rational/1 and rationalize/1 as well as their semantics are inspired by Common Lisp.}^{127The 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. Ifthis is the same as`Expr`>=`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. 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`). 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`).^{128The 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.
^{129Some 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.^{130The 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.^{131This 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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