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Task/Fibonacci-sequence/Potion/fibonacci-sequence-3.potion

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+sqr = (x): x * x.
+
+# Based on the fact that
+# F2n = Fn(2Fn+1 - Fn)
+# F2n+1 = Fn ^2 + Fn+1 ^2
+matrix = (n) :
+   algorithm = (n) :
+      "computes (Fn, Fn+1)"
+      if (n < 2): return ((0, 1), (1, 1)) at(n).
+
+      # n = e + {0, 1}
+      q = algorithm(n / 2)  # q = (Fe/2, Fe/2+1)
+      q = (q(0) * (2 * q(1) - q(0)), sqr(q(0)) + sqr(q(1)))  # q => (Fe, Fe+1)
+      if (n % 2 == 1) :  # q => (Fe+{0, 1}, Fe+1+{0,1}) = (Fn, Fn+1)
+         q = (q(1), q(1) + q(0))
+      .
+      q
+   .
+   algorithm(n)(0)
+.