Cubic Bezier Curves in After Effects

I won't get into detail of the function itself, as I don't understand the math behind it either. Generally speaking the function expects values between 0 and 1 and returns new values between 0 and 1 according to the cubic bezier curve. These values can be used in a linear function i.e. to modify your desired parameter. As the curve doesn't know anything about time yet, the In- and Out-Points of the animation need to be mapped to 0 and 1 first. We can use these values then in the cubic bezier function and run the return values through a linear function to control the animation.

Let's try it out.
In my example it's the position parameter and the function moves the layer on the x-Value.

1. Create a curve i.e. with cubic-bezier.com
2. Use the four values in my After Effects expression down below and paste them in the function BezierEasing
3. In my example it is (.65,1,.98,.16)
4. In-Point of the animation is the layers In-Point
5. Duration is 3 seconds

////////////// CUBIC-BEZIER FUNCTION BEGIN
// These values are established by empiricism with tests (tradeoff: performance VS precision)
var NEWTON_ITERATIONS = 4;
var NEWTON_MIN_SLOPE = 0.001;
var SUBDIVISION_PRECISION = 0.0000001;
var SUBDIVISION_MAX_ITERATIONS = 10;

var kSplineTableSize = 11;
var kSampleStepSize = 1.0 / (kSplineTableSize - 1.0);

var float32ArraySupported = false;

function A (aA1, aA2) { return 1.0 - 3.0 * aA2 + 3.0 * aA1; }
function B (aA1, aA2) { return 3.0 * aA2 - 6.0 * aA1; }
function C (aA1)      { return 3.0 * aA1; }

// Returns x(t) given t, x1, and x2, or y(t) given t, y1, and y2.
function calcBezier (aT, aA1, aA2) {
  return ((A(aA1, aA2)*aT + B(aA1, aA2))*aT + C(aA1))*aT;
}

// Returns dx/dt given t, x1, and x2, or dy/dt given t, y1, and y2.
function getSlope (aT, aA1, aA2) {
  return 3.0 * A(aA1, aA2)*aT*aT + 2.0 * B(aA1, aA2) * aT + C(aA1);
}

function binarySubdivide (aX, aA, aB) {
  var currentX, currentT, i = 0;
  do {
    currentT = aA + (aB - aA) / 2.0;
    currentX = calcBezier(currentT, mX1, mX2) - aX;
    if (currentX > 0.0) {
      aB = currentT;
    } else {
      aA = currentT;
    }
  } while (Math.abs(currentX) > SUBDIVISION_PRECISION && ++i < SUBDIVISION_MAX_ITERATIONS);
  return currentT;
}

function BezierEasing (mX1, mY1, mX2, mY2) {
  // Validate arguments
  if (arguments.length !== 4) {
    throw new Error("BezierEasing requires 4 arguments.");
  }
  for (var i=0; i<4; ++i) {
    if (typeof arguments[i] !== "number" || isNaN(arguments[i]) || !isFinite(arguments[i])) {
      throw new Error("BezierEasing arguments should be integers.");
    }
  }
  if (mX1 < 0 || mX1 > 1 || mX2 < 0 || mX2 > 1) {
    throw new Error("BezierEasing x values must be in [0, 1] range.");
  }

  var mSampleValues = float32ArraySupported ? new Float32Array(kSplineTableSize) : new Array(kSplineTableSize);

  function newtonRaphsonIterate (aX, aGuessT) {
    for (var i = 0; i < NEWTON_ITERATIONS; ++i) {
      var currentSlope = getSlope(aGuessT, mX1, mX2);
      if (currentSlope === 0.0) return aGuessT;
      var currentX = calcBezier(aGuessT, mX1, mX2) - aX;
      aGuessT -= currentX / currentSlope;
    }
    return aGuessT;
  }

  function calcSampleValues () {
    for (var i = 0; i < kSplineTableSize; ++i) {
      mSampleValues[i] = calcBezier(i * kSampleStepSize, mX1, mX2);
    }
  }

  function getTForX (aX) {
    var intervalStart = 0.0;
    var currentSample = 1;
    var lastSample = kSplineTableSize - 1;

    for (; currentSample != lastSample && mSampleValues[currentSample] <= aX; ++currentSample) {
      intervalStart += kSampleStepSize;
    }
    --currentSample;

    // Interpolate to provide an initial guess for t
    var dist = (aX - mSampleValues[currentSample]) / (mSampleValues[currentSample+1] - mSampleValues[currentSample]);
    var guessForT = intervalStart + dist * kSampleStepSize;

    var initialSlope = getSlope(guessForT, mX1, mX2);
    if (initialSlope >= NEWTON_MIN_SLOPE) {
      return newtonRaphsonIterate(aX, guessForT);
    } else if (initialSlope === 0.0) {
      return guessForT;
    } else {
      return binarySubdivide(aX, intervalStart, intervalStart + kSampleStepSize);
    }
  }

  var _precomputed = false;
  function precompute() {
    _precomputed = true;
    if (mX1 != mY1 || mX2 != mY2)
      calcSampleValues();
  }

  var f = function (aX) {
    if (!_precomputed) precompute();
    if (mX1 === mY1 && mX2 === mY2) return aX; // linear
    // Because JavaScript number are imprecise, we should guarantee the extremes are right.
    if (aX === 0) return 0;
    if (aX === 1) return 1;
    return calcBezier(getTForX(aX), mY1, mY2);
  };

  f.getControlPoints = function() { return [{ x: mX1, y: mY1 }, { x: mX2, y: mY2 }]; };

  var args = [mX1, mY1, mX2, mY2];
  var str = "BezierEasing("+args.join()+")";
  f.toString = function () { return str; };
  var css = "cubic-bezier("+args.join()+")";
  f.toCSS = function () { return css; };

  f.toString = function () { return args; };

  return f;
}
////////////// CUBIC-BEZIER FUNCTION END

var AniTimeIn = 3; // Seconds
var AniStart = inPoint; 
var AniStop = inPoint + AniTimeIn;
var OutX;
var AniTimerange;


var newValues = BezierEasing(.65,1,.98,.16); // Returns Y values between 0-1
var AniTimerange = linear(time, AniStart, AniStop, 0, 1); // Mapping AniStart and AniStop to 0-1 because BezierEasing expects a range between 0-1

OutX = linear(newValues(AniTimerange),0,1, 0, 1000);

[OutX,value[1]]