fixedPoint.scala

import math.abs

val tolerance = 0.0001
def isCloseEnough(x: Double, y: Double) =
  abs((x - y) / x) / x < tolerance

def fixedPoint(f: Double => Double)(firstGuess: Double) = {
  def iterate(guess: Double): Double = {
    val next = f(guess)
    if (isCloseEnough(guess, next)) next
    else iterate(next)
  }
  iterate(firstGuess)
}
fixedPoint(x => 1 + x / 2)(1)

// avaraging successive values
def avarageDamp(f: Double => Double)(x: Double) = (x + f(x)) / 2

def sqrt(x: Double) = fixedPoint(avarageDamp(y => x / y))(1.0)
sqrt(2)

you can express sqrt as fixedpoint function
y*y =>x -> y => x/y
but we need smaller steps so we y => (y + x/y) / 2

avaraging successive values will be extracted to a separate function avarageDamp which modifies input function