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Allegro CL

ANSI Common Lisp   12 Numbers   12.2 Dictionary of Numbers

 12.2.38 sqrt, isqrt Function

Syntax:
sqrt number    root

isqrt natural    natural-root

Arguments and Values:
number, root - a number.

natural, natural-root - a non-negative integer.

Description:
sqrt and isqrt compute square roots.

sqrt returns the principal square root of number. If the number is not a complex but is negative, then the result is a complex.

isqrt returns the greatest integer less than or equal to the exact positive square root of natural.

If number is a positive rational, it is implementation-dependent whether root is a rational or a float. If number is a negative rational, it is implementation-dependent whether root is a complex rational or a complex float.

The mathematical definition of complex square root (whether or not minus zero is supported) follows:

(sqrt x) = (exp (/ (log x) 2))

The branch cut for square root lies along the negative real axis, continuous with quadrant II. The range consists of the right half-plane, including the non-negative imaginary axis and excluding the negative imaginary axis.

Examples:
``` (sqrt 9.0)  3.0
(sqrt -9.0)  #C(0.0 3.0)
(isqrt 9)  3
(sqrt 12)  3.4641016
(isqrt 12)  3
(isqrt 300)  17
(isqrt 325)  18
(sqrt 25)
5
OR5.0
(isqrt 25)  5
(sqrt -1)  #C(0.0 1.0)
(sqrt #c(0 2))  #C(1.0 1.0)
```

Exceptional Situations:
The function sqrt should signal type-error if its argument is not a number.

The function isqrt should signal type-error if its argument is not a non-negative integer.

The functions sqrt and isqrt might signal arithmetic-error.

``` (isqrt x) ==(values (floor (sqrt x)))