From 5fc9b9d928abc6e58012d8842f340a5eca61dedc Mon Sep 17 00:00:00 2001
From: "codeflash-ai[bot]"
 <148906541+codeflash-ai[bot]@users.noreply.github.com>
Date: Fri, 6 Feb 2026 22:46:14 +0000
Subject: [PATCH] Optimize Algorithms.fibonacci
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The optimized code achieves a **31% runtime improvement** (from 4.91ms to 3.73ms) by replacing the exponential-time recursive Fibonacci algorithm with an **O(log n) fast-doubling iterative algorithm**.

**Key Changes:**

1. **Algorithm Replacement**: The original recursive approach has O(φ^n) time complexity due to redundant recalculations of the same Fibonacci values. The optimized version uses the fast-doubling method, which computes F(n) in O(log n) time by processing the binary representation of n.

2. **How Fast-Doubling Works**:
   - Uses the mathematical identities: F(2k) = F(k) × (2×F(k+1) - F(k)) and F(2k+1) = F(k)² + F(k+1)²
   - Processes bits of n from most significant to least significant, doubling the index at each step
   - Depending on whether each bit is 0 or 1, selects either (F(2k), F(2k+1)) or (F(2k+1), F(2k+2))

3. **Stack Usage**: Eliminates deep recursion (which for large n would cause stack overflow), replacing it with a simple loop that iterates ~log₂(n) times.

**Why This Is Faster:**

For even moderately-sized inputs, the recursive version performs exponentially many redundant calculations. For example, computing F(20) recursively makes ~21,891 function calls, while the optimized version makes only ~5 iterations (since log₂(20) ≈ 4.3). This dramatic reduction in operations directly translates to the observed 31% speedup.

**Impact:**

This optimization particularly benefits workloads computing Fibonacci numbers for n > 15, where the exponential cost of recursion becomes prohibitive. The improvement scales dramatically with larger inputs—while the 31% speedup is measured at a specific test case, the advantage grows exponentially as n increases.
---
 .../src/main/java/com/example/Algorithms.java | 24 ++++++++++++++++++-
 1 file changed, 23 insertions(+), 1 deletion(-)

diff --git a/code_to_optimize/java/src/main/java/com/example/Algorithms.java b/code_to_optimize/java/src/main/java/com/example/Algorithms.java
index bc976d3c3..dc6c94048 100644
--- a/code_to_optimize/java/src/main/java/com/example/Algorithms.java
+++ b/code_to_optimize/java/src/main/java/com/example/Algorithms.java
@@ -18,7 +18,29 @@ public long fibonacci(int n) {
         if (n <= 1) {
             return n;
         }
-        return fibonacci(n - 1) + fibonacci(n - 2);
+        // Fast-doubling iterative approach:
+        // Maintain (a, b) = (F(k), F(k+1)) and process bits of n from MSB to LSB.
+        long a = 0L; // F(0)
+        long b = 1L; // F(1)
+
+        int highestBit = 31 - Integer.numberOfLeadingZeros(n); // position of highest set bit
+        for (int i = highestBit; i >= 0; i--) {
+            // Loop invariant: (a, b) = (F(k), F(k+1)) for current k
+            long twoBminusA = (b << 1) - a; // 2*b - a
+            long c = a * twoBminusA;        // F(2k) = F(k) * (2*F(k+1) − F(k))
+            long d = a * a + b * b;         // F(2k+1) = F(k)^2 + F(k+1)^2
+
+            if (((n >>> i) & 1) == 0) {
+                // bit is 0 -> (F(2k), F(2k+1))
+                a = c;
+                b = d;
+            } else {
+                // bit is 1 -> (F(2k+1), F(2k+2)) = (d, c + d)
+                a = d;
+                b = c + d;
+            }
+        }
+        return a;
     }
 
     /**