sqrt number => root
isqrt natural => natural-root
number, root---a number.
natural, natural-root---a non-negative integer.
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.
(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 OR=> 5.0 (isqrt 25) => 5 (sqrt -1) => #C(0.0 1.0) (sqrt #c(0 2)) => #C(1.0 1.0)
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.
@xref{exp; expt} , section log [Function] , section Rule of Float Substitutability
(isqrt x) == (values (floor (sqrt x)))
but it is potentially more efficient.
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