add karatsuba multiplication algorithm

Author: appleJax

src/algorithms/math/karatsuba-multiplication/README.md

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+import karatsuba from '../karatsuba';
+
+describe('karatsuba multiplication', () => {
+  xit('should multiply numbers correctly', () => {
+    expect(karatsuba(1234, 5678)).toEqual(7006652);
+  });
+});

src/algorithms/math/karatsuba-multiplication/__test__/karatsuba.test.js

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+/**
+ *
+ * @param {number} x
+ * @param {number} y
+ * @return {number}
+ */
+export default function karatsuba(x, y) {
+  // BASE CASE:
+  // if numbers are sufficiently small,
+  // multiply them together in the traditional way
+  if (x < 10 || y < 10) {
+    return x * y;
+  }
+
+  const minDigits = Math.min(
+    String(x).length,
+    String(y).length,
+  );
+
+  // scaleFactor is used to split the numbers
+  // into smaller numbers for recursion.
+  // when combining the subcomputations back
+  // together, the scaleFactor is used to
+  // recreate the components of the original number
+  const scaleFactor = 10 ** Math.floor(minDigits / 2);
+
+  // a b are the two components of x
+  // c d are the two components of y
+  //
+  // e.g.
+  // x = 1234 -> a = 12, b = 34
+  // y = 5678 -> c = 56, d = 78
+  //
+  // example of component computations:
+  // x = 1234, y = 5678
+  // scaleFactor = 100
+  // a = floor(1234 / 100) = floor(12.34) = 12
+  const a = Math.floor(x / scaleFactor);
+
+  // b = 1234 - (12 * 100) = 1234 - 1200 = 34
+  const b = x - (a * scaleFactor);
+
+  // c = floor(5678 / 100) = floor(56.78) = 56
+  const c = Math.floor(y / scaleFactor);
+
+  // d = 5678 - (56 * 100) = 5678 - 5600 = 78
+  const d = y - (c * scaleFactor);
+
+  // compute sub-expressions:
+  // since a + b is less than x, and c + d is less than y
+  // the recursion is guaranteed to reach the base case
+  const ac = karatsuba(a, c);
+  const bd = karatsuba(b, d);
+  const abcd = karatsuba(a + b, c + d);
+
+  // combine sub-expressions:
+  // since the scaleFactor was used to
+  // artificially reduce the size of the components,
+  // the scaleFactor must be applied in reverse
+  // to reconstruct the original components
+  const A = ac * (scaleFactor ** 2);
+  const B = (abcd - ac - bd) * scaleFactor;
+  const C = bd;
+
+  return A + B + C;
+}