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Analysis

For any integer $k \geq 1$ , if $a > b \geq 0$ and b < F_k+1, then Euclid(a,b) makes fewer than k recursive calls.

Remember that $F_k \;=\; F_{k-1} \;+\; F_{k-2}$ , $F_k \approx \frac{\phi^k}{\sqrt{5}}$ , $\phi \;=\; \frac{1+\sqrt{5}}{2}$

$b \;<\; F_{k+1} \;=\; \frac{\phi^{k+1}}{\sqrt{5}}$ , $b \;<\; \frac{\phi \phi^k}{\sqrt{5}}$
$\phi^k \;>\; \frac{\sqrt{5}}{\phi}b$ , $k \;>\; log_{\phi} \frac{\sqrt{5}}{\phi}b$
k = O(lg b) recursive calls
O( $\beta$ ) arithmetic operations
O( $\beta^3$ ) bit operations

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