Square root functions intended for use without <math.h> dependencies.

sqrtX.c

/*
 *  Square Root X
 *  
 *  Programatic attempt to solve square roots without <math.h> with 
 *  brute force approach of either incrementing the square of 0.001
 *  until square value is reached (toggling iterators for accuracy) 
 *  or translate Newton's method into working C code.
 * 
 *  02.05.17 | Lucas Barbosa | Open source software
 */

#include <stdio.h>
#include <stdlib.h>
#include <assert.h>

#define CONVERGENCE_CHECK 2.25e-308
#define NON_REAL 0
#define DEFAULT_ITERATIONS 32

double sqrtB(double square);
double sqrtX(double square);
void tests(void);

int main (void) {
	tests();
	double mySquare = 0;
	scanf("%lf", &mySquare);
	double rootResult = sqrtX(mySquare);
	printf("%lf\n", rootResult);
	return EXIT_SUCCESS;
}

double sqrtB(double square) {
	if (square <= NON_REAL) return 0;
	double root = square / 3;
	int i = 0;
	/*
	*  @re-iterate arithmetic enough times for accurate successive 
	*  approximations, it doesn't take many iterations.
	*/
	while (i < DEFAULT_ITERATIONS) {
		root = (root + square / root) / 2;
		i++;
	}
	return root;
}

double sqrtX(double square) {
	if (square <= NON_REAL) return 0;
	double root = square / 3;
	double last, diff = 1;
   /*
	*  @perform the arithmetic over sufficient iterations, to provide 
	*  successive approximations, convergence check is performed on the
	*  while loop restrictions between the difference of the re-iterated
	*  roots and their previous values (as last).
	*/
	do {
		last = root;
		root = (root + square / root) / 2;
		diff = root - last;
	} while (diff > CONVERGENCE_CHECK || diff < -CONVERGENCE_CHECK);
	
	return root;
}

void tests(void) {
	assert((int)sqrtX(0) == 0);
	assert((int)sqrtX(49) == 7);
	assert((int)sqrtX(81) == 9);
	assert((int)sqrtX(-2) == 0);
	/// All tests were passed if code is compiled
	
}

Optimisation includes using long doubles etc.