Apparently this algorithm is well known.

The algorithm will build the square root answer bit by bit starting from the left-most bit to the last.

Let's say we will support squaring up to a 8 bit unsigned integer for simplicity.

let res be result

let x be 8 bit unsigned integer

let n be the nth bit in the result where the 0th bit is the last bit.

set res to 0
for nth bit from 3 down to 0
    change the nth bit of the res to 1
    if res * res > x then: <- The Comparison
         restore the nth bit back to 0
return res

The return result will be a 4 bit unsigned integer

For Example: Finding the square root of 4

(see image) enter image description here

Is this perhaps the Digit by Digit Calculation: Binary Numeral System (base 2)? Wiki Link here

Thank you for your time!

  • 1
    I'm not sure what is the question here.
    – Paul92
    Sep 29, 2016 at 18:46
  • 1
    I would describe it as optimized binary search. Sep 30, 2016 at 0:41
  • 1
    What is shown in the question is a digit-by-digit calculation of the square root (I would claim not a particular fast one, and therefore of little practical use). What is shown at the Wiki link is the classical longhand digit-by-digit square rooting method based on the binomial theorem, which goes back to the Middle Ages (possibly Fibonacci's "Liber Abaci"?) and until a few decades ago, was taught to high-school students.
    – njuffa
    Nov 29, 2016 at 1:40

3 Answers 3


i might call it successive approximation or maybe binary search to invert a monotonic function.

if you are able to perform a function f(x), and if you know that f(x) is strictly monotonic (strictly increasing or strictly decreasing) in the domain of x that is salient, then you can always invert f(x) to whatever finite degree of precision you want with this binary search. one iteration for every bit you need in your result.


Yes, that's essentially the same algorithm described at the article that you linked.

The main difference is your pseudo-code's use of multiplication, which costs more than what can be done, as the same article points out, "... the operations needed to compute ... e * e can be replaced with faster bit shift operations."



Put simply, if it is given that X == Y^2 (a reformulation of sqrt(X) == Y), and you would like to solve (N*N*X + m) == (N*Y + k)^2 for k where the value of N is fixed for the entire algorithm, you can simply substitute values into k from k = 0, 1, ..., N-1 and compare the magnitude between the left hand side (which is fixed) and the right hand side (which depend on the value substituted into k).

For N = 2, the possible value of k is simply 0, 1 so that it reduces to a binary search.

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