Let x + i y = sqrt(a + i b). Square both sides to yield a pair of simultaneous quadratic equations in x and y. Their solution substituted back into the original equation yields
sqrt(a + i b) = (sqrt(2) / 2) ( sqrt(sqrt(a^2 + b^2) + a) + i (sqrt(sqrt(a^2 + b^2) - a) sign(b))
where the sign function is defined as
sign(x) = 1 if x > 0, -1 if x < 0, and 0 otherwise (that is, if x = 0)
This is the Algebraic reduction of the complex square-root to real operations.. However, the logarithmic-exponential expression is more direct.
1 - x^2 = (1 - x) (1 + x)
The right-hand side provides less round-off errors for x near 1.
Substitution of 1 - x for x in the foregoing yields
1 - (1 - x)^2 = (1 - (1 - x)) (1 + (1 - x))
which simplifies to
1 - (1 - x)^2 = x ( 2 - x)
Again, the right-hand side provides less round-off errors for x near 0.
Copywrite (c) 1997 R. I. 'Sciibor-Marchocki last modified on Tuesday 15-th April 1997.