See http://www.phailed.me/2012/08/somewhat-fast-square-root/ for the full derivation, including the error compensation trick, except I later discovered 3/7 worked better than 2/5

qsqrt.c

/*
 * qsqrt.c
 *
 *  Created on: Jun 6, 2012
 *      Author: Lee
 */

#include <stdio.h>
#include <math.h>

#define SQRT2 1.4142135623730951

// we use integer arithmetic on the bit-representation of x to accelerate the
// newton ralphson iteration by computing an initial guess of what sqrt(x) should be
// to 2 to 3 significant digits.
float qsqrt(float x){
	// x = [0 eeeeeeeeeee mmmmmmmmmmmmmmmmmmmmmmm]
	int i = *(int*)&x;
	int j = (((i/2-0x1fc00000)+0x3f800000)&0x3ff800000)+((i&0x7fffff)*3/7);

	float y = (*(float*)&j) * ((i-0x3f800000)&(0x800000) ? SQRT2 : 1); // 2-3 sig figs of significance

	// two iterations of newton: y = y - (y^2 - x)/2y yields 2^-18 points of precision
	y = (y + x/y)/2; // this yields 2^-8 points of precision
	return (y + x/y)/2;
}

int main(){
	int i = 1;
	for (; i < 1000; i++){
		float f = (qsqrt(i)-sqrt(i))/(sqrt(i));
		printf("%3d %10f %10f \t rel err: %10f\n",i,qsqrt(i),sqrt(i),f);
	}
	return 0;
}