gistfile1.scala

case class LoopState(i: Int = 0, acc: Double = 0.0, isCompleted: Boolean = false)

lazy val twoPowers = for (i <- 0 until 50) yield Math.pow(2.0, i)

val Epsilon: Double = 1e-17

val TaskSize = 2500000

def modPow16(exp: Double, m: Double): Double = {
  if (exp < 1) Math.pow(16.0, exp)
  else {

    @tailrec
    def step(i: Int, pow2: Double, pow1: Double, acc: Double): Double = {
      i match {
        case 0 => acc
        case _ if pow1 >= pow2 => step(i, pow2, pow1 - pow2, (16.0 * acc) % m)
        case _ if pow2 >= 2 => step(i - 1, pow2 / 2, pow1, (acc * acc) % m)
        case _ => step(i - 1, pow2 / 2, pow1, acc)
      }
    }

    val i = twoPowers indexWhere (_ > exp)
    step(i, twoPowers(i - 1), exp, 1.0)
  }
}
  
def task(m: Int, digitPos: Int, i: Int, acc: Double): LoopState = {

  @tailrec
  def loop(i: Int, acc: Double): LoopState = {

    digitPos - i match {
      case -100 => LoopState(i, acc, isCompleted = true) // task complete
      case pow => {
        val denom = 8 * i + m
        val term = modPow16(pow, denom) / denom
        if (i >= digitPos && term < Epsilon) LoopState(i, acc, isCompleted = true) // task complete
        else {
          if (i % TaskSize == 0) LoopState(i + 1, (acc + term) % 1.0, isCompleted = false) // reached limit
          else loop(i + 1, (acc + term) % 1.0) // continue
        }
      }
    }
  }

  loop(i, acc)
}

gistfile2.js

var TASK_SIZE = 2500000;
var EPSILON = 1e-17;

var twoPowersCache = {};
function twoPowers(i) {
    if (!twoPowersCache.hasOwnProperty(i)) twoPowersCache[i] = Math.pow(2.0, i);
    return twoPowersCache[i];
}

function modPow16(exp, m) {
    var i = 0;
    var pow2 = 0;
    while(pow2 < exp) {
        pow2 = twoPowers(i);
        i++;
    }
    var pow1 = exp;
    var acc = 1.0;

    if (exp < 1) return Math.pow(16.0, exp);
    else {
        while (true) {
            if (i == 0) return acc;
            else if (pow1 >= pow2) {
                pow1 = pow1 - pow2;
                acc = (16.0 * acc) % m;
            } else if (pow2 >= 2) {
                i -= 1;
                pow2 = pow2 / 2;
                acc = (acc * acc) % m;
            } else {
                i -= 1;
                pow2 = pow2 / 2;
            }

        }
    }
}

function task(m, digitPos, i, acc) {
    while (true) {
        if (digitPos - i == -100) return [i, acc, true]; // task complete
        else {
            var pow = digitPos - i;
            var denom = 8 * i + m;
            var term = modPow16(pow, denom) / denom;
            if (i >= digitPos && term < EPSILON) return [i, acc, true]; // task complete
            else {
                if (i % TASK_SIZE == 0) return [i + 1, (acc + term) % 1.0, false]; // reached limit
                else {
                    i++;
                    acc = (acc + term) % 1.0;
                }
            }
        }
    }
}