diff --git a/03_euler_gen_alg.py b/03_euler_gen_alg.py
index 04ccc08..8ab78e8 100644
--- a/03_euler_gen_alg.py
+++ b/03_euler_gen_alg.py
@@ -10,49 +10,20 @@ def generate_random_individuals():
 
     return val
 
-# Function to flip a bit
-# represented as character.
-def flip(c): 
-    return '1' if c == '0' else '0'
-
-# accepts string binary array
 def grey_to_bin(gray):
-    binary = ""
+    """Convert Gray code to binary, operating on the integer value directly"""
+    num = int(gray, 2)  # Convert string to integer
+    mask = num
+    while mask != 0:
+        mask >>= 1
+        num ^= mask
+    return format(num, f'0{len(gray)}b')  # Convert back to binary string with same length
 
-    # MSB of binary code is same as gray code
-    binary += gray[0]
-
-    # Compute remaining bits
-    for i in range(1, len(gray)):
-        
-        # If current bit is 0, concatenate
-        # previous bit
-        if gray[i] == '0':
-            binary += binary[i - 1]
-
-        # Else, concatenate invert of
-        # previous bit
-        else:
-            binary += flip(binary[i - 1])
-
-    return binary
-
-# accepts string binary array
 def bin_to_grey(binary):
-    gray = ""
-
-    # MSB of gray code is same as binary code
-    gray += binary[0]
-
-    # Compute remaining bits, next bit is computed by
-    # doing XOR of previous and current in Binary
-    for i in range(1, len(binary)):
-        
-        # Concatenate XOR of previous bit
-        # with current bit
-        gray += xorChar(binary[i - 1], binary[i])
-
-    return gray
+    """Convert binary to Gray code using XOR with right shift"""
+    num = int(binary, 2)  # Convert string to integer
+    gray = num ^ (num >> 1)  # Gray code formula: G = B ^ (B >> 1)
+    return format(gray, f'0{len(binary)}b')  # Convert back to binary string with same length
 
 def quadratic_error(original_fn, approx_fn, n):
     error = 0.0