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sqrt, isqrt

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 
<i><sup>or</sup>→</i> 5.0 
(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.

See Also:

exp, log, Section 12.1.3.3 (Rule of Float Substitutability)

Notes:

(isqrt x) ≡ (values (floor (sqrt x)))

but it is potentially more efficient.

Expanded Reference: sqrt, isqrt​

sqrt -- principal square root​

sqrt returns the principal square root of a number. For non-negative reals, the result is a non-negative real.

(sqrt 9.0)
=> 3.0
(sqrt 2.0)
=> 1.4142135
(sqrt 0)
=> 0.0
(sqrt 25)
=> 5.0

sqrt of negative numbers​

When the argument is a negative real, the result is a complex number.

(sqrt -1)
=> #C(0.0 1.0)
(sqrt -9.0)
=> #C(0.0 3.0)
(sqrt -4)
=> #C(0.0 2.0)

sqrt with complex arguments​

sqrt works with complex numbers directly.

(sqrt #c(0 2))
=> #C(1.0 1.0)
(sqrt #c(3 4))
=> #C(2.0 1.0)

isqrt -- integer square root​

isqrt returns the greatest integer less than or equal to the exact square root of a non-negative integer.

(isqrt 9)
=> 3
(isqrt 12)
=> 3
(isqrt 25)
=> 5
(isqrt 300)
=> 17
(isqrt 325)
=> 18
(isqrt 0)
=> 0

Practical use: checking for perfect squares​

isqrt is useful for testing whether a number is a perfect square.

(defun perfect-square-p (n)
  (let ((root (isqrt n)))
    (= (* root root) n)))

(perfect-square-p 16)
=> T
(perfect-square-p 15)
=> NIL
(perfect-square-p 144)
=> T