Simplify Horner's Method code and add the link to it in main READMe.

Author: trekhleb

README.md

@@ -73,6 +73,7 @@ a set of rules that precisely define a sequence of operations.
   * `B` [Complex Number](src/algorithms/math/complex-number) - complex numbers and basic operations with them
   * `B` [Radian & Degree](src/algorithms/math/radian) - radians to degree and backwards conversion
   * `B` [Fast Powering](src/algorithms/math/fast-powering)
+  * `B` [Horner's method](src/algorithms/math/horner-method) - polynomial evaluation
   * `A` [Integer Partition](src/algorithms/math/integer-partition)
   * `A` [Square Root](src/algorithms/math/square-root) - Newton's method
   * `A` [Liu Hui π Algorithm](src/algorithms/math/liu-hui) - approximate π calculations based on N-gons

src/algorithms/math/horner-method/README.md

@@ -1,21 +1,20 @@
 # Horner's Method
 
-In mathematics, Horner's method (or Horner's scheme) is an algorithm for polynomial evaluation.
-With this method, it is possible to evaluate a polynomial with only n additions and n multiplications.
-Hence, its storage requirements are n times the number of bits of x.
+In mathematics, Horner's method (or Horner's scheme) is an algorithm for polynomial evaluation. With this method, it is possible to evaluate a polynomial with only `n` additions and `n` multiplications. Hence, its storage requirements are `n` times the number of bits of `x`.
 
 Horner's method can be based on the following identity:
-![](https://wikimedia.org/api/rest_v1/media/math/render/svg/2a576e42d875496f8b0f0dda5ebff7c2415532e4)
-, which is called Horner's rule.
 
-To solve the right part of the identity above, for a given x, we start by iterating through the polynomial from the inside out,
-accumulating each iteration result. After n iterations, with n being the order of the polynomial, the accumulated result gives
-us the polynomial evaluation. 
+![Horner's rule](https://wikimedia.org/api/rest_v1/media/math/render/svg/2a576e42d875496f8b0f0dda5ebff7c2415532e4)
 
-Using the polynomial:
-![](http://www.sciweavers.org/tex2img.php?eq=%244x%5E4%20%2B%202x%5E3%20%2B%203x%5E2%2B%20x%5E1%20%2B%203%24&bc=White&fc=Black&im=jpg&fs=12&ff=arev&edit=0), a traditional approach to evaluate it at x = 2, could be representing it as an array [3,1,3,2,4] and iterate over it saving each iteration value at an accumulator, such as acc += pow(x=2,index) * array[index]. In essence, each power of a number (pow) operation is n-1 multiplications. So, in this scenario, a total of 15 operations would have happened, composed of 5 additions, 5 multiplications, and 5 pows.
+This identity is called _Horner's rule_.
+
+To solve the right part of the identity above, for a given `x`, we start by iterating through the polynomial from the inside out, accumulating each iteration result. After `n` iterations, with `n` being the order of the polynomial, the accumulated result gives us the polynomial evaluation. 
+
+**Using the polynomial:**
+![Traditional approach](http://www.sciweavers.org/tex2img.php?eq=%244x%5E4%20%2B%202x%5E3%20%2B%203x%5E2%2B%20x%5E1%20%2B%203%24&bc=White&fc=Black&im=jpg&fs=12&ff=arev&edit=0), a traditional approach to evaluate it at `x = 2`, could be representing it as an array `[3, 1, 3, 2, 4]` and iterate over it saving each iteration value at an accumulator, such as `acc += pow(x=2, index) * array[index]`. In essence, each power of a number (`pow`) operation is `n-1` multiplications. So, in this scenario, a total of `14` operations would have happened, composed of `4` additions, `5` multiplications, and `5` pows (we're assuming that each power is calculated by repeated multiplication).
+
+Now, **using the same scenario but with Horner's rule**, the polynomial can be re-written as ![Horner's rule approach](http://www.sciweavers.org/tex2img.php?eq=%24x%28x%28x%284x%2B2%29%2B3%29%2B1%29%2B3%24&bc=White&fc=Black&im=jpg&fs=12&ff=arev&edit=0), representing it as `[4, 2, 3, 1, 3]` it is possible to save the first iteration as `acc = arr[0] * (x=2) + arr[1]`, and then finish iterations for `acc *= (x=2) + arr[index]`. In the same scenario but using Horner's rule, a total of `10` operations would have happened, composed of only `4` additions and `4` multiplications.
 
-Now, using the same scenario but with Horner's rule, the polynomial can be re-written as ![](http://www.sciweavers.org/tex2img.php?eq=%24x%28x%28x%284x%2B2%29%2B3%29%2B1%29%2B3%24&bc=White&fc=Black&im=jpg&fs=12&ff=arev&edit=0), representing it as [4,2,3,1,3] it is possible to save the first iteration as acc = arr[0]*(x=2) + arr[1], and then finish iterations for acc *= (x=2) + arr[index]. In the same scenario but using Horner's rule, a total of 10 operations would have happened, composed of only 5 additions and 5 multiplications.
 ## References
 
 - [Wikipedia](https://en.wikipedia.org/wiki/Horner%27s_method)

src/algorithms/math/horner-method/__test__/classicPolynome.test.js

@@ -0,0 +1,14 @@
+import classicPolynome from '../classicPolynome';
+
+describe('classicPolynome', () => {
+  it('should evaluate the polynomial for the specified value of x correctly', () => {
+    expect(classicPolynome([8], 0.1)).toBe(8);
+    expect(classicPolynome([2, 4, 2, 5], 0.555)).toBe(7.68400775);
+    expect(classicPolynome([2, 4, 2, 5], 0.75)).toBe(9.59375);
+    expect(classicPolynome([1, 1, 1, 1, 1], 1.75)).toBe(20.55078125);
+    expect(classicPolynome([15, 3.5, 0, 2, 1.42, 0.41], 0.315)).toBe(1.1367300651406251);
+    expect(classicPolynome([0, 0, 2.77, 1.42, 0.41], 1.35)).toBe(7.375325000000001);
+    expect(classicPolynome([0, 0, 2.77, 1.42, 2.3311], 1.35)).toBe(9.296425000000001);
+    expect(classicPolynome([2, 0, 0, 5.757, 5.31412, 12.3213], 3.141)).toBe(697.2731167035034);
+  });
+});

src/algorithms/math/horner-method/__test__/hornerMethod.test.js

@@ -1,14 +1,21 @@
 import hornerMethod from '../hornerMethod';
+import classicPolynome from '../classicPolynome';
 
 describe('hornerMethod', () => {
-  it('should evaluate the polynomial on the specified point correctly', () => {
-    expect(hornerMethod([8],0.1)).toBe(8);
-    expect(hornerMethod([2,4,2,5],0.555)).toBe(7.68400775);
-    expect(hornerMethod([2,4,2,5],0.75)).toBe(9.59375);
-    expect(hornerMethod([1,1,1,1,1],1.75)).toBe(20.55078125);
-    expect(hornerMethod([15,3.5,0,2,1.42,0.41],0.315)).toBe(1.136730065140625);
-    expect(hornerMethod([0,0,2.77,1.42,0.41],1.35)).toBe(7.375325000000001);
-    expect(hornerMethod([0,0,2.77,1.42,2.3311],1.35)).toBe(9.296425000000001);
-    expect(hornerMethod([2,0,0,5.757,5.31412,12.3213],3.141)).toBe(697.2731167035034);
+  it('should evaluate the polynomial for the specified value of x correctly', () => {
+    expect(hornerMethod([8], 0.1)).toBe(8);
+    expect(hornerMethod([2, 4, 2, 5], 0.555)).toBe(7.68400775);
+    expect(hornerMethod([2, 4, 2, 5], 0.75)).toBe(9.59375);
+    expect(hornerMethod([1, 1, 1, 1, 1], 1.75)).toBe(20.55078125);
+    expect(hornerMethod([15, 3.5, 0, 2, 1.42, 0.41], 0.315)).toBe(1.136730065140625);
+    expect(hornerMethod([0, 0, 2.77, 1.42, 0.41], 1.35)).toBe(7.375325000000001);
+    expect(hornerMethod([0, 0, 2.77, 1.42, 2.3311], 1.35)).toBe(9.296425000000001);
+    expect(hornerMethod([2, 0, 0, 5.757, 5.31412, 12.3213], 3.141)).toBe(697.2731167035034);
   });
-});
+
+  it('should evaluate the same polynomial value as classical approach', () => {
+    expect(hornerMethod([8], 0.1)).toBe(classicPolynome([8], 0.1));
+    expect(hornerMethod([2, 4, 2, 5], 0.555)).toBe(classicPolynome([2, 4, 2, 5], 0.555));
+    expect(hornerMethod([2, 4, 2, 5], 0.75)).toBe(classicPolynome([2, 4, 2, 5], 0.75));
+  });
+});

src/algorithms/math/horner-method/classicPolynome.js

@@ -0,0 +1,16 @@
+/**
+ * Returns the evaluation of a polynomial function at a certain point.
+ * Uses straightforward approach with powers.
+ *
+ * @param {number[]} coefficients - i.e. [4, 3, 2] for (4 * x^2 + 3 * x + 2)
+ * @param {number} xVal
+ * @return {number}
+ */
+export default function classicPolynome(coefficients, xVal) {
+  return coefficients.reverse().reduce(
+    (accumulator, currentCoefficient, index) => {
+      return accumulator + currentCoefficient * (xVal ** index);
+    },
+    0,
+  );
+}

src/algorithms/math/horner-method/hornerMethod.js

@@ -1,17 +1,16 @@
 /**
  * Returns the evaluation of a polynomial function at a certain point.
  * Uses Horner's rule.
- * @param {number[]} numbers
+ *
+ * @param {number[]} coefficients - i.e. [4, 3, 2] for (4 * x^2 + 3 * x + 2)
+ * @param {number} xVal
  * @return {number}
  */
-export default function hornerMethod(numbers, point) {
-  // polynomial function is just a constant.
-  if (numbers.length === 1) {
-    return numbers[0];
-  }
-  return numbers.reduce((accumulator, currentValue, index) => {
-    return index === 1
-      ? numbers[0] * point + currentValue
-      : accumulator * point + currentValue;
-  });
+export default function hornerMethod(coefficients, xVal) {
+  return coefficients.reduce(
+    (accumulator, currentCoefficient) => {
+      return accumulator * xVal + currentCoefficient;
+    },
+    0,
+  );
 }