Numbers

Datatypes and constructors

The numeric datatypes of ABS are Int (arbitrary-length integers), Rat (arbitrary-precision rational numbers) and Float (64-bit floating point). See Built-in types for their syntax.

Note

Support for floating-point calculations is under development; calculations resulting in Inf or NaN currently have unspecified runtime behavior.

Operators

Syntax

Expression	Meaning	Associativity	Result type
e1 == e2	equality	left	Bool
e1 != e2	inequality	left	Bool
e1 < e2	less than	left	Bool
e1 \<= e2	less or equal	left	Bool
e1 > e2	greater than	left	Bool
e1 >= e2	greater or equal	left	Bool
e1 + e2	addition	left	number
e1 - e2	subtraction	left	number
e1 * e2	multiplication	left	number
e1 / e2	division	left	Rat or Float
e1 % e2	modulo	left	number

Functions

min, max

These functions calculate the maximum and minimum of their two arguments. Since ABS datatypes are ordered, they can be applied to arguments of all types.

max(5, 3)
// => 5

min(2.5, 1.5)
// => 1.5

abs

This function calculates the absolute (positive) value of its argument.

max(-2)
// => 2

truncate

Converts a rational number to an integer by truncating towards zero.

truncate(7/5)
// => 1

float

Converts an integer or rational number into a floating-point number.

float(1)
// => 1.0
float(1/2)
// => 0.5

Note

Very large integers and some rational numbers cannot be converted exactly into floating-point numbers. More in general, rat(float(x)) == x might not be true.

rat

Converts a floating-point number into a rational number.

rat(0.5)
// => 1/2

Note

Conversion from floating-point to rational numbers is inexact and backend-specific. In general, float(rat(x)) == x might not be true.

floor

Returns the largest integer smaller or equal to the floating-point argument.

floor(5.3)
// => 5

ceil

Returns the smallest integer larger or equal to the floating-point argument.

ceil(5.3)
// => 6

numerator

Returns the numerator of a rational number, or the number itself for an integer.

numerator(1/2)
// => 1

denominator

Returns the denominator of a rational number, or 1 for an integer.

denominator(1/2)
// => 2

pow

This function calculates \(b^n\).

pow(1/2, 4)
// => 1/16

sqrt_newton

This function approximates \(\sqrt{x}\); it stops when two subsequent estimates (as per Newton’s algorithm) differ by less than epsilon, its third argument. The second argument is an initial estimate of the square root.

sqrt_newton(4, 2, 1/100)
// => 2

exp_newton

This function approximates \(e^x\); it stops when two subsequent estimates (as per Newton’s algorithm) differ by less than its second argument epsilon.

exp_newton(1, 1/100)
// => 163/60

sqrt

This function returns \(\sqrt{x}\). It is an error if its argument is negative.

sqrt(4.0)
// => 2.0

log

This function returns \(ln(x)\), the natural logarithm of its argument.

log(1.0)
// => 0.0

exp

This function returns \(e^x\), Euler’s number \(e\) raised to the power of x.

exp(1.0)
// => 2.7182818284590455

random

Returns an integer between 0 (inclusive) and its argument (exclusive).

random(5)
// => 4