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Derivation

1. Split input into 2 regimes

2. if beta < 0.4187459134614581

   1. Initial program 19.4

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}\]

   2. Initial simplification19.4

      \[\leadsto \frac{1.0 + \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0}}{2.0}\]
   3. Using strategy rm

   4. Applied div-sub19.4

      \[\leadsto \frac{1.0 + \color{blue}{\left(\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \frac{\alpha}{\left(\alpha + \beta\right) + 2.0}\right)}}{2.0}\]

   5. Applied associate-+r-19.4

      \[\leadsto \frac{\color{blue}{\left(1.0 + \frac{\beta}{\left(\alpha + \beta\right) + 2.0}\right) - \frac{\alpha}{\left(\alpha + \beta\right) + 2.0}}}{2.0}\]
   6. Using strategy rm

   7. Applied flip3-+19.4

      \[\leadsto \frac{\color{blue}{\frac{{1.0}^{3} + {\left(\frac{\beta}{\left(\alpha + \beta\right) + 2.0}\right)}^{3}}{1.0 \cdot 1.0 + \left(\frac{\beta}{\left(\alpha + \beta\right) + 2.0} \cdot \frac{\beta}{\left(\alpha + \beta\right) + 2.0} - 1.0 \cdot \frac{\beta}{\left(\alpha + \beta\right) + 2.0}\right)}} - \frac{\alpha}{\left(\alpha + \beta\right) + 2.0}}{2.0}\]

   8. Applied frac-sub19.0

      \[\leadsto \frac{\color{blue}{\frac{\left({1.0}^{3} + {\left(\frac{\beta}{\left(\alpha + \beta\right) + 2.0}\right)}^{3}\right) \cdot \left(\left(\alpha + \beta\right) + 2.0\right) - \left(1.0 \cdot 1.0 + \left(\frac{\beta}{\left(\alpha + \beta\right) + 2.0} \cdot \frac{\beta}{\left(\alpha + \beta\right) + 2.0} - 1.0 \cdot \frac{\beta}{\left(\alpha + \beta\right) + 2.0}\right)\right) \cdot \alpha}{\left(1.0 \cdot 1.0 + \left(\frac{\beta}{\left(\alpha + \beta\right) + 2.0} \cdot \frac{\beta}{\left(\alpha + \beta\right) + 2.0} - 1.0 \cdot \frac{\beta}{\left(\alpha + \beta\right) + 2.0}\right)\right) \cdot \left(\left(\alpha + \beta\right) + 2.0\right)}}}{2.0}\]

   9. Taylor expanded around 0 10.9

      \[\leadsto \frac{\frac{\color{blue}{2.0 + \left(1.0 \cdot \beta + 0.5 \cdot \left(\beta \cdot \alpha\right)\right)}}{\left(1.0 \cdot 1.0 + \left(\frac{\beta}{\left(\alpha + \beta\right) + 2.0} \cdot \frac{\beta}{\left(\alpha + \beta\right) + 2.0} - 1.0 \cdot \frac{\beta}{\left(\alpha + \beta\right) + 2.0}\right)\right) \cdot \left(\left(\alpha + \beta\right) + 2.0\right)}}{2.0}\]

   if 0.4187459134614581 < beta

   1. Initial program 10.0

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}\]

   2. Initial simplification10.0

      \[\leadsto \frac{1.0 + \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0}}{2.0}\]
   3. Using strategy rm

   4. Applied div-sub10.0

      \[\leadsto \frac{1.0 + \color{blue}{\left(\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \frac{\alpha}{\left(\alpha + \beta\right) + 2.0}\right)}}{2.0}\]

   5. Applied associate-+r-10.0

      \[\leadsto \frac{\color{blue}{\left(1.0 + \frac{\beta}{\left(\alpha + \beta\right) + 2.0}\right) - \frac{\alpha}{\left(\alpha + \beta\right) + 2.0}}}{2.0}\]
   6. Using strategy rm

   7. Applied flip3-+10.0

      \[\leadsto \frac{\color{blue}{\frac{{1.0}^{3} + {\left(\frac{\beta}{\left(\alpha + \beta\right) + 2.0}\right)}^{3}}{1.0 \cdot 1.0 + \left(\frac{\beta}{\left(\alpha + \beta\right) + 2.0} \cdot \frac{\beta}{\left(\alpha + \beta\right) + 2.0} - 1.0 \cdot \frac{\beta}{\left(\alpha + \beta\right) + 2.0}\right)}} - \frac{\alpha}{\left(\alpha + \beta\right) + 2.0}}{2.0}\]

   8. Applied frac-sub10.1

      \[\leadsto \frac{\color{blue}{\frac{\left({1.0}^{3} + {\left(\frac{\beta}{\left(\alpha + \beta\right) + 2.0}\right)}^{3}\right) \cdot \left(\left(\alpha + \beta\right) + 2.0\right) - \left(1.0 \cdot 1.0 + \left(\frac{\beta}{\left(\alpha + \beta\right) + 2.0} \cdot \frac{\beta}{\left(\alpha + \beta\right) + 2.0} - 1.0 \cdot \frac{\beta}{\left(\alpha + \beta\right) + 2.0}\right)\right) \cdot \alpha}{\left(1.0 \cdot 1.0 + \left(\frac{\beta}{\left(\alpha + \beta\right) + 2.0} \cdot \frac{\beta}{\left(\alpha + \beta\right) + 2.0} - 1.0 \cdot \frac{\beta}{\left(\alpha + \beta\right) + 2.0}\right)\right) \cdot \left(\left(\alpha + \beta\right) + 2.0\right)}}}{2.0}\]

   9. Taylor expanded around -inf 2.0

      \[\leadsto \frac{\frac{\color{blue}{\left(12.0 \cdot \frac{1}{\beta} + 2.0 \cdot \beta\right) - 2.0}}{\left(1.0 \cdot 1.0 + \left(\frac{\beta}{\left(\alpha + \beta\right) + 2.0} \cdot \frac{\beta}{\left(\alpha + \beta\right) + 2.0} - 1.0 \cdot \frac{\beta}{\left(\alpha + \beta\right) + 2.0}\right)\right) \cdot \left(\left(\alpha + \beta\right) + 2.0\right)}}{2.0}\]

   10. Simplified2.0

       \[\leadsto \frac{\frac{\color{blue}{\left(\frac{12.0}{\beta} - 2.0\right) + \beta \cdot 2.0}}{\left(1.0 \cdot 1.0 + \left(\frac{\beta}{\left(\alpha + \beta\right) + 2.0} \cdot \frac{\beta}{\left(\alpha + \beta\right) + 2.0} - 1.0 \cdot \frac{\beta}{\left(\alpha + \beta\right) + 2.0}\right)\right) \cdot \left(\left(\alpha + \beta\right) + 2.0\right)}}{2.0}\]
3. Recombined 2 regimes into one program.

4. Final simplification8.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \le 0.4187459134614581:\\ \;\;\;\;\frac{\frac{\left(1.0 \cdot \beta + 0.5 \cdot \left(\alpha \cdot \beta\right)\right) + 2.0}{\left(2.0 + \left(\beta + \alpha\right)\right) \cdot \left(\left(\frac{\beta}{2.0 + \left(\beta + \alpha\right)} \cdot \frac{\beta}{2.0 + \left(\beta + \alpha\right)} - 1.0 \cdot \frac{\beta}{2.0 + \left(\beta + \alpha\right)}\right) + 1.0 \cdot 1.0\right)}}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\frac{12.0}{\beta} - 2.0\right) + 2.0 \cdot \beta}{\left(2.0 + \left(\beta + \alpha\right)\right) \cdot \left(\left(\frac{\beta}{2.0 + \left(\beta + \alpha\right)} \cdot \frac{\beta}{2.0 + \left(\beta + \alpha\right)} - 1.0 \cdot \frac{\beta}{2.0 + \left(\beta + \alpha\right)}\right) + 1.0 \cdot 1.0\right)}}{2.0}\\ \end{array}\]

Runtime

Time bar (total: 39.7s)Debug logProfile

herbie shell --seed 2018296 
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/1"
  :pre (and (> alpha -1) (> beta -1))
  (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0))