Newton's Method

main.go

package main

import (
	"fmt"
	"math"
)

func Sqrt(x float64) float64 {
	z := 3.0
	threshold := 1e-10 // Define a small threshold value
	var prevZ float64  // Variable to store the previous estimate

	for {
		prevZ = z
		z -= (z*z - x) / (2*z)
		fmt.Println(z)

		// Check if the change in z is smaller than the threshold
		if math.Abs(z-prevZ) < threshold {
			break
		}
	}

	return z
}

func main() {
	x := 2.0
	fmt.Println(Sqrt(x), math.Sqrt(x))
}

Details of the algorithm: the z² − x above is how far away z² is from where it needs to be (x), and the division by 2z is the derivative of z², to scale how much we adjust z by how quickly z² is changing. This general approach is called Newton's method. It works well for many functions but especially well for square root.

z in the case of line 9, defined in sqrt function is the first guess and can be changed to amend the first result set.