less_retarded_wiki/sqrt.md

Square Root

Square root (sometimes shortened to sqrt) of number a is such a number b that b^2 = a, for example 3 is a square root of 9 because 3^2 = 9. Finding square root is one of the most basic and important operations in math and programming, e.g. for computing distances, solving quadratic equations etc. Square root is a special case of finding Nth root of a number for N = 2. Square root of a number doesn't have to be a whole number; in fact if the square isn't a whole number, it is always an irrational number (i.e. it can't be expressed as a fraction of two integers, for example square root of two is approximately 1.414...); and it doesn't even have to be a real number (e.g. square root of -1 is i). Strictly speaking there may exist multiple square roots of a number, for example both 5 and -5 are square roots of 25 -- the positive square root is called principal square root; principal square root of x is the same number we get when we raise x to 1/2, and this is what we are usually interested in -- from now on by square root we will implicitly mean principal square root. Programmers write square root of x as sqrt(x) (which should give the same result as raising to 1/2, i.e. pow(x,0.5)), mathematicians write it as:

  _    1/2
\/x = x

TODO

Programming

TODO

Within desired precision square root can be relatively quickly computed iteratively by binary search. Of course if we need extreme speed, we may use a look up table with precomputed values.

TODO: C code for binary search

The following is a non-iterative approximation of integer square root in C that has acceptable accuracy to about 1 million (maximum error from 1000 to 1000000 is about 7%): { Painstakingly made by me. ~drummyfish }

int32_t sqrtApprox(int32_t x)
{
  return
    (x < 1024) ?
    (-2400 / (x + 120) + x / 64 + 20) :
      ((x < 93580) ?
       (-1000000 / (x + 8000) + x / 512 + 142) :
       (-75000000 / (x + 160000) + x / 2048 + 565));
}