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Derivation

1. Initial program 23.4

   \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
2. Using strategy rm

3. Applied add-sqr-sqrt23.4

   \[\leadsto \frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\color{blue}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} \cdot \sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}}} + 1.0}{2.0}\]

4. Applied *-un-lft-identity23.4

   \[\leadsto \frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\color{blue}{1 \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} \cdot \sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}} + 1.0}{2.0}\]

5. Applied times-frac12.3

   \[\leadsto \frac{\frac{\color{blue}{\frac{\alpha + \beta}{1} \cdot \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} \cdot \sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}} + 1.0}{2.0}\]

6. Applied times-frac12.3

   \[\leadsto \frac{\color{blue}{\frac{\frac{\alpha + \beta}{1}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}} \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}}} + 1.0}{2.0}\]

7. Simplified12.3

   \[\leadsto \frac{\color{blue}{\frac{\beta + \alpha}{\sqrt{\left(2 \cdot i + \left(2.0 + \beta\right)\right) + \alpha}}} \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}} + 1.0}{2.0}\]
8. Using strategy rm

9. Applied add-cbrt-cube12.3

   \[\leadsto \frac{\color{blue}{\sqrt[3]{\left(\left(\frac{\beta + \alpha}{\sqrt{\left(2 \cdot i + \left(2.0 + \beta\right)\right) + \alpha}} \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}} + 1.0\right) \cdot \left(\frac{\beta + \alpha}{\sqrt{\left(2 \cdot i + \left(2.0 + \beta\right)\right) + \alpha}} \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}} + 1.0\right)\right) \cdot \left(\frac{\beta + \alpha}{\sqrt{\left(2 \cdot i + \left(2.0 + \beta\right)\right) + \alpha}} \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}} + 1.0\right)}}}{2.0}\]
10. Using strategy rm

11. Applied add-log-exp12.3

    \[\leadsto \frac{\sqrt[3]{\left(\color{blue}{\log \left(e^{\frac{\beta + \alpha}{\sqrt{\left(2 \cdot i + \left(2.0 + \beta\right)\right) + \alpha}} \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}} + 1.0}\right)} \cdot \left(\frac{\beta + \alpha}{\sqrt{\left(2 \cdot i + \left(2.0 + \beta\right)\right) + \alpha}} \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}} + 1.0\right)\right) \cdot \left(\frac{\beta + \alpha}{\sqrt{\left(2 \cdot i + \left(2.0 + \beta\right)\right) + \alpha}} \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}} + 1.0\right)}}{2.0}\]
12. Using strategy rm

13. Applied flip3-+12.3

    \[\leadsto \frac{\sqrt[3]{\left(\log \left(e^{\frac{\beta + \alpha}{\sqrt{\left(2 \cdot i + \left(2.0 + \beta\right)\right) + \alpha}} \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}} + 1.0}\right) \cdot \color{blue}{\frac{{\left(\frac{\beta + \alpha}{\sqrt{\left(2 \cdot i + \left(2.0 + \beta\right)\right) + \alpha}} \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}}\right)}^{3} + {1.0}^{3}}{\left(\frac{\beta + \alpha}{\sqrt{\left(2 \cdot i + \left(2.0 + \beta\right)\right) + \alpha}} \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}}\right) \cdot \left(\frac{\beta + \alpha}{\sqrt{\left(2 \cdot i + \left(2.0 + \beta\right)\right) + \alpha}} \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}}\right) + \left(1.0 \cdot 1.0 - \left(\frac{\beta + \alpha}{\sqrt{\left(2 \cdot i + \left(2.0 + \beta\right)\right) + \alpha}} \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}}\right) \cdot 1.0\right)}}\right) \cdot \left(\frac{\beta + \alpha}{\sqrt{\left(2 \cdot i + \left(2.0 + \beta\right)\right) + \alpha}} \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}} + 1.0\right)}}{2.0}\]

14. Final simplification12.3

    \[\leadsto \frac{\sqrt[3]{\left(\log \left(e^{1.0 + \frac{\alpha + \beta}{\sqrt{\alpha + \left(2 \cdot i + \left(\beta + 2.0\right)\right)}} \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt{2.0 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}}\right) \cdot \frac{{\left(\frac{\alpha + \beta}{\sqrt{\alpha + \left(2 \cdot i + \left(\beta + 2.0\right)\right)}} \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt{2.0 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}\right)}^{3} + {1.0}^{3}}{\left(\frac{\alpha + \beta}{\sqrt{\alpha + \left(2 \cdot i + \left(\beta + 2.0\right)\right)}} \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt{2.0 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}\right) \cdot \left(\frac{\alpha + \beta}{\sqrt{\alpha + \left(2 \cdot i + \left(\beta + 2.0\right)\right)}} \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt{2.0 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}\right) + \left(1.0 \cdot 1.0 - \left(\frac{\alpha + \beta}{\sqrt{\alpha + \left(2 \cdot i + \left(\beta + 2.0\right)\right)}} \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt{2.0 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}\right) \cdot 1.0\right)}\right) \cdot \left(1.0 + \frac{\alpha + \beta}{\sqrt{\alpha + \left(2 \cdot i + \left(\beta + 2.0\right)\right)}} \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt{2.0 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}\right)}}{2.0}\]

Runtime

Time bar (total: 1.2m)Debug logProfile

herbie shell --seed 2018340 
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/2"
  :pre (and (> alpha -1) (> beta -1) (> i 0))
  (/ (+ (/ (/ (* (+ alpha beta) (- beta alpha)) (+ (+ alpha beta) (* 2 i))) (+ (+ (+ alpha beta) (* 2 i)) 2.0)) 1.0) 2.0))