Very good illustration indeed.
1. simplify the sqrt by log of each number, i.e. log(sqrt(x)) = 1/2 * log(x)
2. since sum of logs is the log of the products, i.e., log(a) + log(b) = log(ab)
you can simplify the whole expression by multiplying all the numbers, taking the log of the product once (which i presume is much faster), then multiplying by 1/2
since logs are strictly increasing, the resulting number is still going to be bigger if it originally was going to be bigger, and now you don't need to have performed all those sqrts.
Now you've reduced the problem to computing 1 log to an arbitrary precision...not sure how one does that actually...
1. Take the log of all terms (allowed, because log keeps the sum monotonic):
log(sqrt(a)) + log(sqrt(b)) + log(sqrt(c))
2. pull out the sqrt from the log:
1/2 * log(a) + 1/2 * log(b) +1/2 * log(c)
3. factor out the 1/2:
1/2 * (log(a) + log(b) + log(c))
4. sum of logs can be rewritten as a log of product:
1/2 * log (a * b * c)
5. compute log of (a * b * c), and halve it. Ditto with log of (d * e * f). This should give a number which is proportional to the original sum of sqrt.
It seems you're employing a + b < c <=> log(a) + log(b) < log(c), which doesn't hold (consider 10, 10, and 20).
(the real rule is a * b < c <=> log(a) + log(b) < log(c)), because log(a) + log(b) <=> log(a * b)
> sqrt(1) + sqrt(100)
11.0
> sqrt(25) + sqrt(25)
10.0
> log(sqrt(1)) + log(sqrt(100))
2.302585092994046
> log(sqrt(25)) + log(sqrt(25))
3.2188758248682006