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Derivation

1. Split input into 2 regimes

2. if (/ (/ (/ (+ (+ (+ alpha beta) (* beta alpha)) 1.0) (+ (+ alpha beta) (* 2 1))) (+ (+ alpha beta) (* 2 1))) (+ (+ (+ alpha beta) (* 2 1)) 1.0)) < 1.2789894458039883e-87

   1. Initial program 0.1

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]

   2. Taylor expanded around inf 0.1

      \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{\beta + \left(3.0 + \alpha\right)}}\]
   3. Using strategy rm

   4. Applied add-sqr-sqrt0.3

      \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{\sqrt{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \sqrt{\left(\alpha + \beta\right) + 2 \cdot 1}}}}{\beta + \left(3.0 + \alpha\right)}\]

   5. Applied add-sqr-sqrt0.4

      \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\color{blue}{\sqrt{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \sqrt{\left(\alpha + \beta\right) + 2 \cdot 1}}}}{\sqrt{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \sqrt{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\beta + \left(3.0 + \alpha\right)}\]

   6. Applied add-sqr-sqrt0.3

      \[\leadsto \frac{\frac{\frac{\color{blue}{\sqrt{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0} \cdot \sqrt{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}}}{\sqrt{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \sqrt{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\sqrt{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \sqrt{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\beta + \left(3.0 + \alpha\right)}\]

   7. Applied times-frac0.3

      \[\leadsto \frac{\frac{\color{blue}{\frac{\sqrt{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}}{\sqrt{\left(\alpha + \beta\right) + 2 \cdot 1}} \cdot \frac{\sqrt{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}}{\sqrt{\left(\alpha + \beta\right) + 2 \cdot 1}}}}{\sqrt{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \sqrt{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\beta + \left(3.0 + \alpha\right)}\]

   8. Applied times-frac0.3

      \[\leadsto \frac{\color{blue}{\frac{\frac{\sqrt{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}}{\sqrt{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\sqrt{\left(\alpha + \beta\right) + 2 \cdot 1}} \cdot \frac{\frac{\sqrt{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}}{\sqrt{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\sqrt{\left(\alpha + \beta\right) + 2 \cdot 1}}}}{\beta + \left(3.0 + \alpha\right)}\]

   9. Simplified0.2

      \[\leadsto \frac{\color{blue}{\frac{\sqrt{\alpha \cdot \beta + \left(\beta + \left(1.0 + \alpha\right)\right)}}{2 + \left(\beta + \alpha\right)}} \cdot \frac{\frac{\sqrt{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}}{\sqrt{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\sqrt{\left(\alpha + \beta\right) + 2 \cdot 1}}}{\beta + \left(3.0 + \alpha\right)}\]

   10. Simplified0.3

       \[\leadsto \frac{\frac{\sqrt{\alpha \cdot \beta + \left(\beta + \left(1.0 + \alpha\right)\right)}}{2 + \left(\beta + \alpha\right)} \cdot \color{blue}{\frac{\sqrt{\alpha \cdot \beta + \left(\beta + \left(1.0 + \alpha\right)\right)}}{2 + \left(\beta + \alpha\right)}}}{\beta + \left(3.0 + \alpha\right)}\]

   if 1.2789894458039883e-87 < (/ (/ (/ (+ (+ (+ alpha beta) (* beta alpha)) 1.0) (+ (+ alpha beta) (* 2 1))) (+ (+ alpha beta) (* 2 1))) (+ (+ (+ alpha beta) (* 2 1)) 1.0))

   1. Initial program 6.4

      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]

   2. Taylor expanded around inf 6.4

      \[\leadsto \frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\color{blue}{\beta + \left(3.0 + \alpha\right)}}\]

   3. Taylor expanded around 0 2.7

      \[\leadsto \frac{\frac{\frac{\color{blue}{\alpha + \left(\beta + 1.0\right)}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\beta + \left(3.0 + \alpha\right)}\]
3. Recombined 2 regimes into one program.

4. Final simplification1.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\frac{1.0 + \left(\alpha \cdot \beta + \left(\beta + \alpha\right)\right)}{2 + \left(\beta + \alpha\right)}}{2 + \left(\beta + \alpha\right)}}{\left(2 + \left(\beta + \alpha\right)\right) + 1.0} \le 1.2789894458039883 \cdot 10^{-87}:\\ \;\;\;\;\frac{\frac{\sqrt{\alpha \cdot \beta + \left(\left(1.0 + \alpha\right) + \beta\right)}}{\left(\beta + \alpha\right) + 2} \cdot \frac{\sqrt{\alpha \cdot \beta + \left(\left(1.0 + \alpha\right) + \beta\right)}}{\left(\beta + \alpha\right) + 2}}{\beta + \left(\alpha + 3.0\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\alpha + \left(\beta + 1.0\right)}{2 + \left(\beta + \alpha\right)}}{2 + \left(\beta + \alpha\right)}}{\beta + \left(\alpha + 3.0\right)}\\ \end{array}\]

Runtime

Time bar (total: 6.3m)Debug logProfile

herbie shell --seed 2018225 
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/3"
  :pre (and (> alpha -1) (> beta -1))
  (/ (/ (/ (+ (+ (+ alpha beta) (* beta alpha)) 1.0) (+ (+ alpha beta) (* 2 1))) (+ (+ alpha beta) (* 2 1))) (+ (+ (+ alpha beta) (* 2 1)) 1.0)))