Refactor bezier implementation from bezier-easing library

Summary:fast & accurate implementation
See https://github.com/gre/bezier-easing
the library is embedded in React Native

fixes facebook#6207 & to follow facebook#6340 (or to replace it)
cc vjeux

tests
 ---

[the lib tests](https://github.com/gre/bezier-easing/blob/master/test/test.js) ensure the library is accurate.
It is tested that the library have a precision better than ±0.000001 .

performance
 ---

On my macbook pro, [the lib benchmark](https://github.com/gre/bezier-easing/blob/master/benchmark.js) have:

```
BezierEasing: instanciation x 1,043,725 ops/sec ±1.46% (82 runs sampled)
BezierEasing: call x 7,866,642 ops/sec ±0.93% (85 runs sampled)
BezierEasing: instanciation + call x 803,051 ops/sec ±1.58% (74 runs sampled)
```
Closes facebook#6433

Differential Revision: D3045854

Pulled By: vjeux

fb-gh-sync-id: b3c5dba19195a6719967b4fdc8ef940cc067b1f4
shipit-source-id: b3c5dba19195a6719967b4fdc8ef940cc067b1f4

Libraries/Animated/src/Easing.js

@@ -104,17 +104,9 @@ class Easing {
     x1: number,
     y1: number,
     x2: number,
-    y2: number,
-    epsilon?: ?number,
+    y2: number
   ): (t: number) => number {
-    if (epsilon === undefined) {
-      // epsilon determines the precision of the solved values
-      // a good approximation is:
-      var duration = 500; // duration of animation in milliseconds.
-      epsilon = (1000 / 60 / duration) / 4;
-    }
-
-    return _bezier(x1, y1, x2, y2, epsilon);
+    return _bezier(x1, y1, x2, y2);
   }
 
   static in(

Libraries/Animated/src/__tests__/bezier-test.js

@@ -0,0 +1,96 @@
+/* eslint-disable */
+
+jest.dontMock('bezier');
+var bezier = require('bezier');
+
+var identity = function (x) { return x; };
+
+function assertClose (a, b, precision) {
+  expect(a).toBeCloseTo(b, 3);
+}
+
+function makeAssertCloseWithPrecision (precision) {
+  return function (a, b, message) {
+    assertClose(a, b, message, precision);
+  };
+}
+
+function allEquals (be1, be2, samples, assertion) {
+  if (!assertion) assertion = assertClose;
+  for (var i=0; i<=samples; ++i) {
+    var x = i / samples;
+    assertion(be1(x), be2(x), 'comparing '+be1+' and '+be2+' for value '+x);
+  }
+}
+
+function repeat (n) {
+  return function (f) {
+    for (var i=0; i<n; ++i) f(i);
+  };
+}
+
+describe('bezier', function(){
+  it('should be a function', function(){
+    expect(typeof bezier === 'function').toBe(true);
+  });
+  it('should creates an object', function(){
+    expect(typeof bezier(0, 0, 1, 1) === 'function').toBe(true);
+  });
+  it('should fail with wrong arguments', function () {
+    expect(function () { bezier(0.5, 0.5, -5, 0.5); }).toThrow();
+    expect(function () { bezier(0.5, 0.5, 5, 0.5); }).toThrow();
+    expect(function () { bezier(-2, 0.5, 0.5, 0.5); }).toThrow();
+    expect(function () { bezier(2, 0.5, 0.5, 0.5); }).toThrow();
+  });
+  describe('linear curves', function () {
+    it('should be linear', function () {
+      allEquals(bezier(0, 0, 1, 1), bezier(1, 1, 0, 0), 100);
+      allEquals(bezier(0, 0, 1, 1), identity, 100);
+    });
+  });
+  describe('common properties', function () {
+    it('should be the right value at extremes', function () {
+      repeat(10)(function () {
+        var a = Math.random(), b = 2*Math.random()-0.5, c = Math.random(), d = 2*Math.random()-0.5;
+        var easing = bezier(a, b, c, d);
+        expect(easing(0)).toBe(0);
+        expect(easing(1)).toBe(1);
+      });
+    });
+
+    it('should approach the projected value of its x=y projected curve', function () {
+      repeat(10)(function () {
+        var a = Math.random(), b = Math.random(), c = Math.random(), d = Math.random();
+        var easing = bezier(a, b, c, d);
+        var projected = bezier(b, a, d, c);
+        var composed = function (x) { return projected(easing(x)); };
+        allEquals(identity, composed, 100, makeAssertCloseWithPrecision(0.05));
+      });
+    });
+  });
+  describe('two same instances', function () {
+    it('should be strictly equals', function () {
+      repeat(10)(function () {
+        var a = Math.random(), b = 2*Math.random()-0.5, c = Math.random(), d = 2*Math.random()-0.5;
+        allEquals(bezier(a, b, c, d), bezier(a, b, c, d), 100, 0);
+      });
+    });
+  });
+  describe('symetric curves', function () {
+    it('should have a central value y~=0.5 at x=0.5', function () {
+      repeat(10)(function () {
+        var a = Math.random(), b = 2*Math.random()-0.5, c = 1-a, d = 1-b;
+        var easing = bezier(a, b, c, d);
+        assertClose(easing(0.5), 0.5, easing+'(0.5) should be 0.5');
+      });
+    });
+    it('should be symetrical', function () {
+      repeat(10)(function () {
+        var a = Math.random(), b = 2*Math.random()-0.5, c = 1-a, d = 1-b;
+        var easing = bezier(a, b, c, d);
+        var sym = function (x) { return 1 - easing(1-x); };
+        allEquals(easing, sym, 100);
+      });
+    });
+  });
+});

Libraries/Animated/src/bezier.js

@@ -1,82 +1,106 @@
 /**
- * https://github.com/arian/cubic-bezier
- *
- * MIT License
- *
- * Copyright (c) 2013 Arian Stolwijk
- *
- * Permission is hereby granted, free of charge, to any person obtaining
- * a copy of this software and associated documentation files (the
- * "Software"), to deal in the Software without restriction, including
- * without limitation the rights to use, copy, modify, merge, publish,
- * distribute, sublicense, and/or sell copies of the Software, and to
- * permit persons to whom the Software is furnished to do so, subject to
- * the following conditions:
- *
- * The above copyright notice and this permission notice shall be
- * included in all copies or substantial portions of the Software.
- *
- * THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND,
- * EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF
- * MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT.
- * IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY
- * CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT,
- * TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE
- * SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.
+ * https://github.com/gre/bezier-easing
+ * BezierEasing - use bezier curve for transition easing function
+ * by Gaëtan Renaudeau 2014 - 2015 – MIT License
  *
  * @providesModule bezier
- * @nolint
  */
 
-module.exports = function(x1, y1, x2, y2, epsilon){
+ // 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 curveX = function(t){
-    var v = 1 - t;
-    return 3 * v * v * t * x1 + 3 * v * t * t * x2 + t * t * t;
-  };
+ var float32ArraySupported = typeof Float32Array === 'function';
 
-  var curveY = function(t){
-    var v = 1 - t;
-    return 3 * v * v * t * y1 + 3 * v * t * t * y2 + t * t * t;
-  };
+ 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; }
 
-  var derivativeCurveX = function(t){
-    var v = 1 - t;
-    return 3 * (2 * (t - 1) * t + v * v) * x1 + 3 * (-t * t * t + 2 * v * t) * x2;
-  };
+ // 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; }
 
-  return function(t){
+ // 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); }
 
-    var x = t, t0, t1, t2, x2, d2, i;
+ function binarySubdivide (aX, aA, aB, mX1, mX2) {
+   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;
+ }
 
-    // First try a few iterations of Newton's method -- normally very fast.
-    for (t2 = x, i = 0; i < 8; i++){
-      x2 = curveX(t2) - x;
-      if (Math.abs(x2) < epsilon) { return curveY(t2); }
-      d2 = derivativeCurveX(t2);
-      if (Math.abs(d2) < 1e-6) { break; }
-      t2 = t2 - x2 / d2;
+ function newtonRaphsonIterate (aX, aGuessT, mX1, mX2) {
+  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;
+ }
 
-    t0 = 0;
-    t1 = 1;
-    t2 = x;
+ module.exports = function bezier (mX1, mY1, mX2, mY2) {
+   if (!(0 <= mX1 && mX1 <= 1 && 0 <= mX2 && mX2 <= 1)) { // eslint-disable-line yoda
+     throw new Error('bezier x values must be in [0, 1] range');
+   }
 
-    if (t2 < t0) { return curveY(t0); }
-    if (t2 > t1) { return curveY(t1); }
+   // Precompute samples table
+   var sampleValues = float32ArraySupported ? new Float32Array(kSplineTableSize) : new Array(kSplineTableSize);
+   if (mX1 !== mY1 || mX2 !== mY2) {
+     for (var i = 0; i < kSplineTableSize; ++i) {
+       sampleValues[i] = calcBezier(i * kSampleStepSize, mX1, mX2);
+     }
+   }
 
-    // Fallback to the bisection method for reliability.
-    while (t0 < t1){
-      x2 = curveX(t2);
-      if (Math.abs(x2 - x) < epsilon) { return curveY(t2); }
-      if (x > x2) { t0 = t2; }
-      else { t1 = t2; }
-      t2 = (t1 - t0) * 0.5 + t0;
-    }
+   function getTForX (aX) {
+     var intervalStart = 0.0;
+     var currentSample = 1;
+     var lastSample = kSplineTableSize - 1;
+
+     for (; currentSample !== lastSample && sampleValues[currentSample] <= aX; ++currentSample) {
+       intervalStart += kSampleStepSize;
+     }
+     --currentSample;
 
-    // Failure
-    return curveY(t2);
+     // Interpolate to provide an initial guess for t
+     var dist = (aX - sampleValues[currentSample]) / (sampleValues[currentSample + 1] - sampleValues[currentSample]);
+     var guessForT = intervalStart + dist * kSampleStepSize;
 
-  };
+     var initialSlope = getSlope(guessForT, mX1, mX2);
+     if (initialSlope >= NEWTON_MIN_SLOPE) {
+       return newtonRaphsonIterate(aX, guessForT, mX1, mX2);
+     } else if (initialSlope === 0.0) {
+       return guessForT;
+     } else {
+       return binarySubdivide(aX, intervalStart, intervalStart + kSampleStepSize, mX1, mX2);
+     }
+   }
 
-};
+   return function BezierEasing (x) {
+     if (mX1 === mY1 && mX2 === mY2) {
+       return x; // linear
+     }
+     // Because JavaScript number are imprecise, we should guarantee the extremes are right.
+     if (x === 0) {
+       return 0;
+     }
+     if (x === 1) {
+       return 1;
+     }
+     return calcBezier(getTForX(x), mY1, mY2);
+   };
+ };