Perform multiplication of any two integers positive or negative through bit manipulations (#201)

src/algorithms/math/bits/README.md

@@ -35,6 +35,20 @@ This method is a combination of "Clear Bit" and "Set Bit" methods.
 
 > See [updateBit.js](updateBit.js) for further details.
 
+#### isEven
+
+This method determines if the number provided is even.
+
+```
+Number: 5
+isEven: false
+
+Number: 4
+isEven: true
+```
+
+> See [isEven.js](isEven.js) for further details.
+
 #### Multiply By Two
 
 This method shifts original number by one bit to the left.
@@ -91,7 +105,25 @@ inverting all of the bits of the number and adding 1 to it.
 
 > See [switchSign.js](switchSign.js) for further details.
 
-#### Multiply Two Numbers
+#### Multiply Two Signed Numbers
+
+This method multiplies two signed integer numbers using bitwise operators.
+This method is based on the following :
+
+```text
+a * b can be written in the below formats
+  0                     if a is zero or b is zero or both a and b are zeroes
+  2a * (b/2)            if b is even
+  2a * (b - 1)/2 + a    if b is odd and positive
+  2a * (b + 1)/2 - a    if b is odd and negative
+```
+
+The advantage of this approach is that in each recursive step one of the operands reduces to half its original value.
+Hence, the run time complexity is O(log b) where b is the operand that reduces to half on each recursive step.
+
+> See [multiply.js](multiply.js) for further details.
+
+#### Multiply Two Unsigned Numbers
 
 This method multiplies two integer numbers using bitwise operators.
 This method is based on that "Every number can be denoted as the sum of powers of 2".

src/algorithms/math/bits/__test__/isEven.test.js

@@ -0,0 +1,11 @@
+import isEven from '../isEven';
+
+describe('isEven', () => {
+  it('should detect if a number is even', () => {
+    expect(isEven(0)).toBe(true);
+    expect(isEven(2)).toBe(true);
+    expect(isEven(-2)).toBe(true);
+    expect(isEven(1)).toBe(false);
+    expect(isEven(-1)).toBe(false);
+  });
+});

src/algorithms/math/bits/__test__/multiply.test.js

@@ -0,0 +1,16 @@
+import multiply from '../multiply';
+
+describe('multiply', () => {
+  it('should multiply two numbers', () => {
+    expect(multiply(0, 0)).toBe(0);
+    expect(multiply(2, 0)).toBe(0);
+    expect(multiply(0, 2)).toBe(0);
+    expect(multiply(1, 2)).toBe(2);
+    expect(multiply(2, 1)).toBe(2);
+    expect(multiply(6, 6)).toBe(36);
+    expect(multiply(-2, 4)).toBe(-8);
+    expect(multiply(4, -2)).toBe(-8);
+    expect(multiply(-4, -4)).toBe(16);
+    expect(multiply(4, -5)).toBe(-20);
+  });
+});

src/algorithms/math/bits/isEven.js

@@ -0,0 +1,7 @@
+/**
+ * @param {number} number
+ * @return bool
+ */
+export default function isEven(number) {
+  return (number & 1) === 0;
+}

src/algorithms/math/bits/multiply.js

@@ -0,0 +1,29 @@
+import divideByTwo from './divideByTwo';
+import isEven from './isEven';
+import multiplyByTwo from './multiplyByTwo';
+
+/**
+ * FUNCTION DEFINITION
+ * multiply(a, b) = 0 if a is zero or b is zero or if both a and b are zeros
+ * multiply(a, b) = multiply(2a, b/2) if b is even
+ * multiply(a, b) = multiply(2a, (b-1)/2) + a if b is odd and b is positive
+ * multiply(a, b) = multiply(2a, (b+1)/2) - a if b is odd and b is negative
+ *
+ * COMPLEXITY
+ * O(log b)
+ * @param {number} a
+ * @param {number} b
+ * @return {number} a * b
+ */
+export default function multiply(a, b) {
+  if (b === 0 || a === 0) {
+    return 0;
+  }
+
+  const multiplyByOddPositive = () => multiply(multiplyByTwo(a), divideByTwo(b - 1)) + a;
+  const multiplyByOddNegative = () => multiply(multiplyByTwo(a), divideByTwo(b + 1)) - a;
+  const multiplyByEven = () => multiply(multiplyByTwo(a), divideByTwo(b));
+  const multiplyByOdd = () => (b > 0 ? multiplyByOddPositive() : multiplyByOddNegative());
+
+  return isEven(b) ? multiplyByEven() : multiplyByOdd();
+}