sqrt, isqrt (Function)

Syntax:

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)

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.