I have a little bit of mathematical background and I would suggest using
a Newton algorithm to compute the sqrt of a:
Just find the positive root of the function x^2-a by iterating
x_{n+1} = 1/2(x_n + a/x_n)
This converges quadratically, i.e. the number of correct digits doubles
with every iteration.
As a starting value x_0 one can choose M*2^{E/2} if a=M*2^E, i.e. just
divide the exponent of a by two and leave its mantissa unchanged.
I don't know how fast the alpha is at divisions, if it is slow, one
could perhaps replace the division a/x_n by a some series expansion.
Greetings,
-Gerhard