Add Full Adder algorithm (math/bits) (trekhleb#334)

* Add Full Adder algorithm (math/bits)

* Full adder: minor spelling fixes

* Full adder: even better comments

Author: sergiitk

src/algorithms/math/bits/README.md

@@ -226,6 +226,42 @@ Number: 9 = (10 - 1) = 0b01001
 
 > See [isPowerOfTwo.js](isPowerOfTwo.js) for further details.
 
+
+#### Full Adder
+
+This method adds up two integer numbers using bitwise operators.
+
+It implements [full adder](https://en.wikipedia.org/wiki/Adder_(electronics))
+electronics circut logic to sum two 32-bit integers in two's complement format.
+It's using the boolean logic to cover all possible cases of adding two input bits:
+with and without a "carry bit" from adding the previous less-significant stage.
+
+Legend:
+- `A`: Number `A`
+- `B`: Number `B`
+- `ai`: ith bit of number `A`
+- `bi`: ith bit of number `B`
+- `carryIn`: a bit carried in from the previous less-significant stage
+- `carryOut`: a bit to carry to the next most-significant stage
+- `bitSum`: The sum of `ai`, `bi`, and `carryIn`
+- `resultBin`: The full result of adding current stage with all less-significant stages (in binary)
+- `resultBin`: The full result of adding current stage with all less-significant stages (in decimal)
+```
+A = 3: 011
+B = 6: 110
+┌──────┬────┬────┬─────────┬──────────┬─────────┬───────────┬───────────┐
+│  bit │ ai │ bi │ carryIn │ carryOut │  bitSum │ resultBin │ resultDec │
+├──────┼────┼────┼─────────┼──────────┼─────────┼───────────┼───────────┤
+│   0  │ 1  │ 0  │    0    │    0     │     1   │       1   │     1     │
+│   1  │ 1  │ 1  │    0    │    1     │     0   │      01   │     1     │
+│   2  │ 0  │ 1  │    1    │    1     │     0   │     001   │     1     │
+│   3  │ 0  │ 0  │    1    │    0     │     1   │    1001   │     9     │
+└──────┴────┴────┴─────────┴──────────┴─────────┴───────────┴───────────┘
+```
+
+> See [fullAdder.js](fullAdder.js) for further details.  
+> See [Full Adder on YouTube](https://www.youtube.com/watch?v=wvJc9CZcvBc).
+
 ## References
 
 - [Bit Manipulation on YouTube](https://www.youtube.com/watch?v=NLKQEOgBAnw&t=0s&index=28&list=PLLXdhg_r2hKA7DPDsunoDZ-Z769jWn4R8)

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

@@ -0,0 +1,18 @@
+import fullAdder from '../fullAdder';
+
+describe('Full adder', () => {
+  it('should add up two numbers', () => {
+    expect(fullAdder(0, 0)).toBe(0);
+    expect(fullAdder(2, 0)).toBe(2);
+    expect(fullAdder(0, 2)).toBe(2);
+    expect(fullAdder(1, 2)).toBe(3);
+    expect(fullAdder(2, 1)).toBe(3);
+    expect(fullAdder(6, 6)).toBe(12);
+    expect(fullAdder(-2, 4)).toBe(2);
+    expect(fullAdder(4, -2)).toBe(2);
+    expect(fullAdder(-4, -4)).toBe(-8);
+    expect(fullAdder(4, -5)).toBe(-1);
+    expect(fullAdder(2, 121)).toBe(123);
+    expect(fullAdder(121, 2)).toBe(123);
+  });
+});

src/algorithms/math/bits/fullAdder.js

@@ -0,0 +1,69 @@
+import getBit from './getBit';
+
+/**
+ * Add two numbers using only binary operators.
+ *
+ * This is an implementation of full adders logic circut.
+ * https://en.wikipedia.org/wiki/Adder_(electronics)
+ * Inspired by: https://www.youtube.com/watch?v=wvJc9CZcvBc
+ *
+ * Table(1)
+ *  INPUT  | OUT
+ *  C Ai Bi | C Si | Row
+ * -------- | -----| ---
+ *  0  0  0 | 0  0 | 1
+ *  0  0  1 | 0  1 | 2
+ *  0  1  0 | 0  1 | 3
+ *  0  1  1 | 1  0 | 4
+ * -------- | ---- | --
+ *  1  0  0 | 0  1 | 5
+ *  1  0  1 | 1  0 | 6
+ *  1  1  0 | 1  0 | 7
+ *  1  1  1 | 1  1 | 8
+ * ---------------------
+ *
+ * Legend:
+ * INPUT C = Carry in, from the previous less-significant stage
+ * INPUT Ai = ith bit of Number A
+ * INPUT Bi = ith bit of Number B
+ * OUT C = Carry out to the next most-significant stage
+ * OUT Si = Bit Sum, ith least significant bit of the result
+ *
+ *
+ * @param {number} a
+ * @param {number} b
+ * @return {number}
+ */
+export default function fullAdder(a, b) {
+  let result = 0;
+  let carry = 0;
+
+  // The operands of all bitwise operators are converted to signed
+  // 32-bit integers in two's complement format.
+  // https://developer.mozilla.org/en-US/docs/Web/JavaScript/Reference/Operators/Bitwise_Operators#Signed_32-bit_integers
+  for (let i = 0; i < 32; i += 1) {
+    const ai = getBit(a, i);
+    const bi = getBit(b, i);
+    const carryIn = carry;
+
+    // Calculate binary Ai + Bi without carry (half adder)
+    // See Table(1) rows 1 - 4: Si = Ai ^ Bi
+    const aiPlusBi = ai ^ bi;
+
+    // Calculate ith bit of the result by adding the carry bit to Ai + Bi
+    // For Table(1) rows 5 - 8 carryIn = 1: Si = Ai ^ Bi ^ 1, flip the bit
+    // Fpr Table(1) rows 1 - 4 carryIn = 0: Si = Ai ^ Bi ^ 0, a no-op.
+    const bitSum = aiPlusBi ^ carryIn;
+
+    // Carry out one to the next most-significant stage
+    // when at least one of these is true:
+    // 1) Table(1) rows 6, 7: one of Ai OR Bi is 1 AND carryIn = 1
+    // 2) Table(1) rows 4, 8: Both Ai AND Bi are 1
+    const carryOut = (aiPlusBi & carryIn) | (ai & bi);
+    carry = carryOut;
+
+    // Set ith least significant bit of the result to bitSum.
+    result |= bitSum << i;
+  }
+  return result;
+}