cookie clicker grimoire estimation

calc.js

var fps = 30;

function calcMagicPerSecond(current, maxMagic) {
  return Math.max(0.002,Math.pow(current/Math.max(maxMagic,100),0.5))*0.002*fps;
}

/*

let z = Math.max(maxMagic, 100)

m' = k * sqrt(m / z) = k * sqrt(m) / sqrt(z) = k * sqrt(m) ; constant k absorbs other constant sqrt(z)
dm/dt = k * sqrt(m)
dm/sqrt(m) = k * dt ; separate variables
2 * sqrt(m) = kt + c ; integrate. now we have two unknown constants

solve m:
m(t) = ((kt + c) / 2)**2 = (kt + c)**2 / 4

solve t:
t(m) = (2 * sqrt(m) - c) / k

find k: let m = 1, maxMagic = 100
m'(m) = 0.006 = k * sqrt(1)
k = 0.006
OR
k = 0.06 / sqrt(z)

find c:
run simulation for how long it takes to go from m=0 to m=max

for maxMagic = 100, c is -0.004

.:. for maxMagic = 100, t(m) = (2 * sqrt(m) + 0.004) / 0.006

.:. general solution, t(m) = (2 * sqrt(m) - c) / (0.06 / sqrt(z))
                           = 16.6667 * (2 * sqrt(m) - c) * sqrt(z)

*/


var C_CACHE = {};
function findC(maxMagic) {
  if (C_CACHE[maxMagic] !== undefined) {
    return C_CACHE[maxMagic];
  }

  var m = 0;
  var t = 0;

  while (m < maxMagic) {
    m += calcMagicPerSecond(m, maxMagic);
    t += 1;
  }

  var z = Math.max(maxMagic, 100);
  var k = 0.06 / Math.pow(z, 0.5);
  var c = 2 * Math.pow(maxMagic, 0.5) - k * t;

  C_CACHE[maxMagic] = c;

  return c;
}

function timeFromZeroToM(m, maxMagic) {
  var z = Math.max(100, maxMagic);
  var c = findC(z);
  return 16.6667 * (2 * Math.pow(m, 0.5) - c) * Math.pow(z, 0.5);
}

// this will be inaccurate if m0 = 0.
function timeElapsed(m0, m1, maxMagic) {
  return timeFromZeroToM(m1, maxMagic) - timeFromZeroToM(m0, maxMagic);
}

var MAX = 72;
for (var i = 1; i < MAX; i++) {
  console.log(i - 1 + " to " + i + ":", timeElapsed(i - 1, i, MAX) + " seconds");
}