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+// Copyright 2006 The Closure Library Authors. All Rights Reserved.
+//
+// Licensed under the Apache License, Version 2.0 (the "License");
+// you may not use this file except in compliance with the License.
+// You may obtain a copy of the License at
+//
+//      http://www.apache.org/licenses/LICENSE-2.0
+//
+// Unless required by applicable law or agreed to in writing, software
+// distributed under the License is distributed on an "AS-IS" BASIS,
+// WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
+// See the License for the specific language governing permissions and
+// limitations under the License.
+
+/**
+ * @fileoverview Additional mathematical functions.
+ */
+
+goog.provide('goog.math');
+
+goog.require('goog.array');
+goog.require('goog.asserts');
+
+
+/**
+ * Returns a random integer greater than or equal to 0 and less than {@code a}.
+ * @param {number} a  The upper bound for the random integer (exclusive).
+ * @return {number} A random integer N such that 0 <= N < a.
+ */
+goog.math.randomInt = function(a) {
+  return Math.floor(Math.random() * a);
+};
+
+
+/**
+ * Returns a random number greater than or equal to {@code a} and less than
+ * {@code b}.
+ * @param {number} a  The lower bound for the random number (inclusive).
+ * @param {number} b  The upper bound for the random number (exclusive).
+ * @return {number} A random number N such that a <= N < b.
+ */
+goog.math.uniformRandom = function(a, b) {
+  return a + Math.random() * (b - a);
+};
+
+
+/**
+ * Takes a number and clamps it to within the provided bounds.
+ * @param {number} value The input number.
+ * @param {number} min The minimum value to return.
+ * @param {number} max The maximum value to return.
+ * @return {number} The input number if it is within bounds, or the nearest
+ *     number within the bounds.
+ */
+goog.math.clamp = function(value, min, max) {
+  return Math.min(Math.max(value, min), max);
+};
+
+
+/**
+ * The % operator in JavaScript returns the remainder of a / b, but differs from
+ * some other languages in that the result will have the same sign as the
+ * dividend. For example, -1 % 8 == -1, whereas in some other languages
+ * (such as Python) the result would be 7. This function emulates the more
+ * correct modulo behavior, which is useful for certain applications such as
+ * calculating an offset index in a circular list.
+ *
+ * @param {number} a The dividend.
+ * @param {number} b The divisor.
+ * @return {number} a % b where the result is between 0 and b (either 0 <= x < b
+ *     or b < x <= 0, depending on the sign of b).
+ */
+goog.math.modulo = function(a, b) {
+  var r = a % b;
+  // If r and b differ in sign, add b to wrap the result to the correct sign.
+  return (r * b < 0) ? r + b : r;
+};
+
+
+/**
+ * Performs linear interpolation between values a and b. Returns the value
+ * between a and b proportional to x (when x is between 0 and 1. When x is
+ * outside this range, the return value is a linear extrapolation).
+ * @param {number} a A number.
+ * @param {number} b A number.
+ * @param {number} x The proportion between a and b.
+ * @return {number} The interpolated value between a and b.
+ */
+goog.math.lerp = function(a, b, x) {
+  return a + x * (b - a);
+};
+
+
+/**
+ * Tests whether the two values are equal to each other, within a certain
+ * tolerance to adjust for floating point errors.
+ * @param {number} a A number.
+ * @param {number} b A number.
+ * @param {number=} opt_tolerance Optional tolerance range. Defaults
+ *     to 0.000001. If specified, should be greater than 0.
+ * @return {boolean} Whether {@code a} and {@code b} are nearly equal.
+ */
+goog.math.nearlyEquals = function(a, b, opt_tolerance) {
+  return Math.abs(a - b) <= (opt_tolerance || 0.000001);
+};
+
+
+// TODO(user): Rename to normalizeAngle, retaining old name as deprecated
+// alias.
+/**
+ * Normalizes an angle to be in range [0-360). Angles outside this range will
+ * be normalized to be the equivalent angle with that range.
+ * @param {number} angle Angle in degrees.
+ * @return {number} Standardized angle.
+ */
+goog.math.standardAngle = function(angle) {
+  return goog.math.modulo(angle, 360);
+};
+
+
+/**
+ * Normalizes an angle to be in range [0-2*PI). Angles outside this range will
+ * be normalized to be the equivalent angle with that range.
+ * @param {number} angle Angle in radians.
+ * @return {number} Standardized angle.
+ */
+goog.math.standardAngleInRadians = function(angle) {
+  return goog.math.modulo(angle, 2 * Math.PI);
+};
+
+
+/**
+ * Converts degrees to radians.
+ * @param {number} angleDegrees Angle in degrees.
+ * @return {number} Angle in radians.
+ */
+goog.math.toRadians = function(angleDegrees) {
+  return angleDegrees * Math.PI / 180;
+};
+
+
+/**
+ * Converts radians to degrees.
+ * @param {number} angleRadians Angle in radians.
+ * @return {number} Angle in degrees.
+ */
+goog.math.toDegrees = function(angleRadians) {
+  return angleRadians * 180 / Math.PI;
+};
+
+
+/**
+ * For a given angle and radius, finds the X portion of the offset.
+ * @param {number} degrees Angle in degrees (zero points in +X direction).
+ * @param {number} radius Radius.
+ * @return {number} The x-distance for the angle and radius.
+ */
+goog.math.angleDx = function(degrees, radius) {
+  return radius * Math.cos(goog.math.toRadians(degrees));
+};
+
+
+/**
+ * For a given angle and radius, finds the Y portion of the offset.
+ * @param {number} degrees Angle in degrees (zero points in +X direction).
+ * @param {number} radius Radius.
+ * @return {number} The y-distance for the angle and radius.
+ */
+goog.math.angleDy = function(degrees, radius) {
+  return radius * Math.sin(goog.math.toRadians(degrees));
+};
+
+
+/**
+ * Computes the angle between two points (x1,y1) and (x2,y2).
+ * Angle zero points in the +X direction, 90 degrees points in the +Y
+ * direction (down) and from there we grow clockwise towards 360 degrees.
+ * @param {number} x1 x of first point.
+ * @param {number} y1 y of first point.
+ * @param {number} x2 x of second point.
+ * @param {number} y2 y of second point.
+ * @return {number} Standardized angle in degrees of the vector from
+ *     x1,y1 to x2,y2.
+ */
+goog.math.angle = function(x1, y1, x2, y2) {
+  return goog.math.standardAngle(
+      goog.math.toDegrees(Math.atan2(y2 - y1, x2 - x1)));
+};
+
+
+/**
+ * Computes the difference between startAngle and endAngle (angles in degrees).
+ * @param {number} startAngle  Start angle in degrees.
+ * @param {number} endAngle  End angle in degrees.
+ * @return {number} The number of degrees that when added to
+ *     startAngle will result in endAngle. Positive numbers mean that the
+ *     direction is clockwise. Negative numbers indicate a counter-clockwise
+ *     direction.
+ *     The shortest route (clockwise vs counter-clockwise) between the angles
+ *     is used.
+ *     When the difference is 180 degrees, the function returns 180 (not -180)
+ *     angleDifference(30, 40) is 10, and angleDifference(40, 30) is -10.
+ *     angleDifference(350, 10) is 20, and angleDifference(10, 350) is -20.
+ */
+goog.math.angleDifference = function(startAngle, endAngle) {
+  var d =
+      goog.math.standardAngle(endAngle) - goog.math.standardAngle(startAngle);
+  if (d > 180) {
+    d = d - 360;
+  } else if (d <= -180) {
+    d = 360 + d;
+  }
+  return d;
+};
+
+
+/**
+ * Returns the sign of a number as per the "sign" or "signum" function.
+ * @param {number} x The number to take the sign of.
+ * @return {number} -1 when negative, 1 when positive, 0 when 0. Preserves
+ *     signed zeros and NaN.
+ */
+goog.math.sign = Math.sign || function(x) {
+  if (x > 0) {
+    return 1;
+  }
+  if (x < 0) {
+    return -1;
+  }
+  return x;  // Preserves signed zeros and NaN.
+};
+
+
+/**
+ * JavaScript implementation of Longest Common Subsequence problem.
+ * http://en.wikipedia.org/wiki/Longest_common_subsequence
+ *
+ * Returns the longest possible array that is subarray of both of given arrays.
+ *
+ * @param {IArrayLike<S>} array1 First array of objects.
+ * @param {IArrayLike<T>} array2 Second array of objects.
+ * @param {Function=} opt_compareFn Function that acts as a custom comparator
+ *     for the array ojects. Function should return true if objects are equal,
+ *     otherwise false.
+ * @param {Function=} opt_collectorFn Function used to decide what to return
+ *     as a result subsequence. It accepts 2 arguments: index of common element
+ *     in the first array and index in the second. The default function returns
+ *     element from the first array.
+ * @return {!Array<S|T>} A list of objects that are common to both arrays
+ *     such that there is no common subsequence with size greater than the
+ *     length of the list.
+ * @template S,T
+ */
+goog.math.longestCommonSubsequence = function(
+    array1, array2, opt_compareFn, opt_collectorFn) {
+
+  var compare = opt_compareFn || function(a, b) { return a == b; };
+
+  var collect = opt_collectorFn || function(i1, i2) { return array1[i1]; };
+
+  var length1 = array1.length;
+  var length2 = array2.length;
+
+  var arr = [];
+  for (var i = 0; i < length1 + 1; i++) {
+    arr[i] = [];
+    arr[i][0] = 0;
+  }
+
+  for (var j = 0; j < length2 + 1; j++) {
+    arr[0][j] = 0;
+  }
+
+  for (i = 1; i <= length1; i++) {
+    for (j = 1; j <= length2; j++) {
+      if (compare(array1[i - 1], array2[j - 1])) {
+        arr[i][j] = arr[i - 1][j - 1] + 1;
+      } else {
+        arr[i][j] = Math.max(arr[i - 1][j], arr[i][j - 1]);
+      }
+    }
+  }
+
+  // Backtracking
+  var result = [];
+  var i = length1, j = length2;
+  while (i > 0 && j > 0) {
+    if (compare(array1[i - 1], array2[j - 1])) {
+      result.unshift(collect(i - 1, j - 1));
+      i--;
+      j--;
+    } else {
+      if (arr[i - 1][j] > arr[i][j - 1]) {
+        i--;
+      } else {
+        j--;
+      }
+    }
+  }
+
+  return result;
+};
+
+
+/**
+ * Returns the sum of the arguments.
+ * @param {...number} var_args Numbers to add.
+ * @return {number} The sum of the arguments (0 if no arguments were provided,
+ *     {@code NaN} if any of the arguments is not a valid number).
+ */
+goog.math.sum = function(var_args) {
+  return /** @type {number} */ (
+      goog.array.reduce(
+          arguments, function(sum, value) { return sum + value; }, 0));
+};
+
+
+/**
+ * Returns the arithmetic mean of the arguments.
+ * @param {...number} var_args Numbers to average.
+ * @return {number} The average of the arguments ({@code NaN} if no arguments
+ *     were provided or any of the arguments is not a valid number).
+ */
+goog.math.average = function(var_args) {
+  return goog.math.sum.apply(null, arguments) / arguments.length;
+};
+
+
+/**
+ * Returns the unbiased sample variance of the arguments. For a definition,
+ * see e.g. http://en.wikipedia.org/wiki/Variance
+ * @param {...number} var_args Number samples to analyze.
+ * @return {number} The unbiased sample variance of the arguments (0 if fewer
+ *     than two samples were provided, or {@code NaN} if any of the samples is
+ *     not a valid number).
+ */
+goog.math.sampleVariance = function(var_args) {
+  var sampleSize = arguments.length;
+  if (sampleSize < 2) {
+    return 0;
+  }
+
+  var mean = goog.math.average.apply(null, arguments);
+  var variance =
+      goog.math.sum.apply(null, goog.array.map(arguments, function(val) {
+        return Math.pow(val - mean, 2);
+      })) / (sampleSize - 1);
+
+  return variance;
+};
+
+
+/**
+ * Returns the sample standard deviation of the arguments.  For a definition of
+ * sample standard deviation, see e.g.
+ * http://en.wikipedia.org/wiki/Standard_deviation
+ * @param {...number} var_args Number samples to analyze.
+ * @return {number} The sample standard deviation of the arguments (0 if fewer
+ *     than two samples were provided, or {@code NaN} if any of the samples is
+ *     not a valid number).
+ */
+goog.math.standardDeviation = function(var_args) {
+  return Math.sqrt(goog.math.sampleVariance.apply(null, arguments));
+};
+
+
+/**
+ * Returns whether the supplied number represents an integer, i.e. that is has
+ * no fractional component.  No range-checking is performed on the number.
+ * @param {number} num The number to test.
+ * @return {boolean} Whether {@code num} is an integer.
+ */
+goog.math.isInt = function(num) {
+  return isFinite(num) && num % 1 == 0;
+};
+
+
+/**
+ * Returns whether the supplied number is finite and not NaN.
+ * @param {number} num The number to test.
+ * @return {boolean} Whether {@code num} is a finite number.
+ */
+goog.math.isFiniteNumber = function(num) {
+  return isFinite(num) && !isNaN(num);
+};
+
+
+/**
+ * @param {number} num The number to test.
+ * @return {boolean} Whether it is negative zero.
+ */
+goog.math.isNegativeZero = function(num) {
+  return num == 0 && 1 / num < 0;
+};
+
+
+/**
+ * Returns the precise value of floor(log10(num)).
+ * Simpler implementations didn't work because of floating point rounding
+ * errors. For example
+ * <ul>
+ * <li>Math.floor(Math.log(num) / Math.LN10) is off by one for num == 1e+3.
+ * <li>Math.floor(Math.log(num) * Math.LOG10E) is off by one for num == 1e+15.
+ * <li>Math.floor(Math.log10(num)) is off by one for num == 1e+15 - 1.
+ * </ul>
+ * @param {number} num A floating point number.
+ * @return {number} Its logarithm to base 10 rounded down to the nearest
+ *     integer if num > 0. -Infinity if num == 0. NaN if num < 0.
+ */
+goog.math.log10Floor = function(num) {
+  if (num > 0) {
+    var x = Math.round(Math.log(num) * Math.LOG10E);
+    return x - (parseFloat('1e' + x) > num ? 1 : 0);
+  }
+  return num == 0 ? -Infinity : NaN;
+};
+
+
+/**
+ * A tweaked variant of {@code Math.floor} which tolerates if the passed number
+ * is infinitesimally smaller than the closest integer. It often happens with
+ * the results of floating point calculations because of the finite precision
+ * of the intermediate results. For example {@code Math.floor(Math.log(1000) /
+ * Math.LN10) == 2}, not 3 as one would expect.
+ * @param {number} num A number.
+ * @param {number=} opt_epsilon An infinitesimally small positive number, the
+ *     rounding error to tolerate.
+ * @return {number} The largest integer less than or equal to {@code num}.
+ */
+goog.math.safeFloor = function(num, opt_epsilon) {
+  goog.asserts.assert(!goog.isDef(opt_epsilon) || opt_epsilon > 0);
+  return Math.floor(num + (opt_epsilon || 2e-15));
+};
+
+
+/**
+ * A tweaked variant of {@code Math.ceil}. See {@code goog.math.safeFloor} for
+ * details.
+ * @param {number} num A number.
+ * @param {number=} opt_epsilon An infinitesimally small positive number, the
+ *     rounding error to tolerate.
+ * @return {number} The smallest integer greater than or equal to {@code num}.
+ */
+goog.math.safeCeil = function(num, opt_epsilon) {
+  goog.asserts.assert(!goog.isDef(opt_epsilon) || opt_epsilon > 0);
+  return Math.ceil(num - (opt_epsilon || 2e-15));
+};