CLHS: Function SQRT, ISQRT

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Function SQRT, ISQRT

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
    OR=>  5.0
     (isqrt 25) =>  5
     (sqrt -1) =>  #C(0.0 1.0)
     (sqrt #c(0 2)) =>  #C(1.0 1.0)

Side Effects: None.

Affected By: None.

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.

The following X3J13 cleanup issue, not part of the specification, applies to this section:

- IEEE-ATAN-BRANCH-CUT:SPLIT

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