Error

Derivation

1. Split input into 2 regimes

2. if (/ (+ (* (/ (/ (+ beta alpha) (cbrt (+ (+ beta alpha) (* 2 i)))) (cbrt (+ (+ beta alpha) (* 2 i)))) (/ (/ (- beta alpha) (cbrt (+ (+ alpha beta) (* 2 i)))) (+ (+ (+ alpha beta) (* 2 i)) 2.0))) 1.0) 2.0) < 4.029418662203658e-10

   1. Initial program 62.2

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

   2. Taylor expanded around inf 29.4

      \[\leadsto \frac{\color{blue}{\left(8.0 \cdot \frac{1}{{\alpha}^{3}} + 2.0 \cdot \frac{1}{\alpha}\right) - 4.0 \cdot \frac{1}{{\alpha}^{2}}}}{2.0}\]

   3. Applied simplify29.4

      \[\leadsto \color{blue}{\frac{\frac{2.0}{\alpha} + \frac{\frac{8.0}{\alpha} - 4.0}{\alpha \cdot \alpha}}{2.0}}\]

   if 4.029418662203658e-10 < (/ (+ (* (/ (/ (+ beta alpha) (cbrt (+ (+ beta alpha) (* 2 i)))) (cbrt (+ (+ beta alpha) (* 2 i)))) (/ (/ (- beta alpha) (cbrt (+ (+ alpha beta) (* 2 i)))) (+ (+ (+ alpha beta) (* 2 i)) 2.0))) 1.0) 2.0)

   1. Initial program 13.5

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

   3. Applied *-un-lft-identity13.5

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

   4. Applied *-un-lft-identity13.5

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

   5. Applied times-frac0.2

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

   6. Applied times-frac0.2

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

   7. Applied simplify0.2

      \[\leadsto \frac{\color{blue}{\frac{\beta + \alpha}{1}} \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
   8. Using strategy rm

   9. Applied div-sub0.2

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

   10. Applied div-sub0.2

       \[\leadsto \frac{\frac{\beta + \alpha}{1} \cdot \color{blue}{\left(\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} - \frac{\frac{\alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}\right)} + 1.0}{2.0}\]
   11. Using strategy rm

   12. Applied add-cbrt-cube0.2

       \[\leadsto \frac{\color{blue}{\sqrt[3]{\left(\left(\frac{\beta + \alpha}{1} \cdot \left(\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} - \frac{\frac{\alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}\right) + 1.0\right) \cdot \left(\frac{\beta + \alpha}{1} \cdot \left(\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} - \frac{\frac{\alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}\right) + 1.0\right)\right) \cdot \left(\frac{\beta + \alpha}{1} \cdot \left(\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} - \frac{\frac{\alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}\right) + 1.0\right)}}}{2.0}\]

   13. Applied simplify0.2

       \[\leadsto \frac{\sqrt[3]{\color{blue}{{\left(1.0 + \left(\frac{\frac{\beta}{i \cdot 2 + \left(\alpha + \beta\right)}}{2.0 + \left(i \cdot 2 + \left(\alpha + \beta\right)\right)} - \frac{\frac{\alpha}{i \cdot 2 + \left(\alpha + \beta\right)}}{2.0 + \left(i \cdot 2 + \left(\alpha + \beta\right)\right)}\right) \cdot \frac{\alpha + \beta}{1}\right)}^{3}}}}{2.0}\]
   14. Using strategy rm

   15. Applied add-sqr-sqrt0.2

       \[\leadsto \frac{\sqrt[3]{{\left(1.0 + \left(\frac{\frac{\beta}{i \cdot 2 + \left(\alpha + \beta\right)}}{2.0 + \left(i \cdot 2 + \left(\alpha + \beta\right)\right)} - \frac{\frac{\alpha}{i \cdot 2 + \left(\alpha + \beta\right)}}{\color{blue}{\sqrt{2.0 + \left(i \cdot 2 + \left(\alpha + \beta\right)\right)} \cdot \sqrt{2.0 + \left(i \cdot 2 + \left(\alpha + \beta\right)\right)}}}\right) \cdot \frac{\alpha + \beta}{1}\right)}^{3}}}{2.0}\]

   16. Applied *-un-lft-identity0.2

       \[\leadsto \frac{\sqrt[3]{{\left(1.0 + \left(\frac{\frac{\beta}{i \cdot 2 + \left(\alpha + \beta\right)}}{2.0 + \left(i \cdot 2 + \left(\alpha + \beta\right)\right)} - \frac{\color{blue}{1 \cdot \frac{\alpha}{i \cdot 2 + \left(\alpha + \beta\right)}}}{\sqrt{2.0 + \left(i \cdot 2 + \left(\alpha + \beta\right)\right)} \cdot \sqrt{2.0 + \left(i \cdot 2 + \left(\alpha + \beta\right)\right)}}\right) \cdot \frac{\alpha + \beta}{1}\right)}^{3}}}{2.0}\]

   17. Applied times-frac0.2

       \[\leadsto \frac{\sqrt[3]{{\left(1.0 + \left(\frac{\frac{\beta}{i \cdot 2 + \left(\alpha + \beta\right)}}{2.0 + \left(i \cdot 2 + \left(\alpha + \beta\right)\right)} - \color{blue}{\frac{1}{\sqrt{2.0 + \left(i \cdot 2 + \left(\alpha + \beta\right)\right)}} \cdot \frac{\frac{\alpha}{i \cdot 2 + \left(\alpha + \beta\right)}}{\sqrt{2.0 + \left(i \cdot 2 + \left(\alpha + \beta\right)\right)}}}\right) \cdot \frac{\alpha + \beta}{1}\right)}^{3}}}{2.0}\]

   18. Applied add-sqr-sqrt0.2

       \[\leadsto \frac{\sqrt[3]{{\left(1.0 + \left(\frac{\frac{\beta}{i \cdot 2 + \left(\alpha + \beta\right)}}{\color{blue}{\sqrt{2.0 + \left(i \cdot 2 + \left(\alpha + \beta\right)\right)} \cdot \sqrt{2.0 + \left(i \cdot 2 + \left(\alpha + \beta\right)\right)}}} - \frac{1}{\sqrt{2.0 + \left(i \cdot 2 + \left(\alpha + \beta\right)\right)}} \cdot \frac{\frac{\alpha}{i \cdot 2 + \left(\alpha + \beta\right)}}{\sqrt{2.0 + \left(i \cdot 2 + \left(\alpha + \beta\right)\right)}}\right) \cdot \frac{\alpha + \beta}{1}\right)}^{3}}}{2.0}\]

   19. Applied *-un-lft-identity0.2

       \[\leadsto \frac{\sqrt[3]{{\left(1.0 + \left(\frac{\color{blue}{1 \cdot \frac{\beta}{i \cdot 2 + \left(\alpha + \beta\right)}}}{\sqrt{2.0 + \left(i \cdot 2 + \left(\alpha + \beta\right)\right)} \cdot \sqrt{2.0 + \left(i \cdot 2 + \left(\alpha + \beta\right)\right)}} - \frac{1}{\sqrt{2.0 + \left(i \cdot 2 + \left(\alpha + \beta\right)\right)}} \cdot \frac{\frac{\alpha}{i \cdot 2 + \left(\alpha + \beta\right)}}{\sqrt{2.0 + \left(i \cdot 2 + \left(\alpha + \beta\right)\right)}}\right) \cdot \frac{\alpha + \beta}{1}\right)}^{3}}}{2.0}\]

   20. Applied times-frac0.2

       \[\leadsto \frac{\sqrt[3]{{\left(1.0 + \left(\color{blue}{\frac{1}{\sqrt{2.0 + \left(i \cdot 2 + \left(\alpha + \beta\right)\right)}} \cdot \frac{\frac{\beta}{i \cdot 2 + \left(\alpha + \beta\right)}}{\sqrt{2.0 + \left(i \cdot 2 + \left(\alpha + \beta\right)\right)}}} - \frac{1}{\sqrt{2.0 + \left(i \cdot 2 + \left(\alpha + \beta\right)\right)}} \cdot \frac{\frac{\alpha}{i \cdot 2 + \left(\alpha + \beta\right)}}{\sqrt{2.0 + \left(i \cdot 2 + \left(\alpha + \beta\right)\right)}}\right) \cdot \frac{\alpha + \beta}{1}\right)}^{3}}}{2.0}\]

   21. Applied distribute-lft-out--0.2

       \[\leadsto \frac{\sqrt[3]{{\left(1.0 + \color{blue}{\left(\frac{1}{\sqrt{2.0 + \left(i \cdot 2 + \left(\alpha + \beta\right)\right)}} \cdot \left(\frac{\frac{\beta}{i \cdot 2 + \left(\alpha + \beta\right)}}{\sqrt{2.0 + \left(i \cdot 2 + \left(\alpha + \beta\right)\right)}} - \frac{\frac{\alpha}{i \cdot 2 + \left(\alpha + \beta\right)}}{\sqrt{2.0 + \left(i \cdot 2 + \left(\alpha + \beta\right)\right)}}\right)\right)} \cdot \frac{\alpha + \beta}{1}\right)}^{3}}}{2.0}\]

   22. Applied associate-*l*0.2

       \[\leadsto \frac{\sqrt[3]{{\left(1.0 + \color{blue}{\frac{1}{\sqrt{2.0 + \left(i \cdot 2 + \left(\alpha + \beta\right)\right)}} \cdot \left(\left(\frac{\frac{\beta}{i \cdot 2 + \left(\alpha + \beta\right)}}{\sqrt{2.0 + \left(i \cdot 2 + \left(\alpha + \beta\right)\right)}} - \frac{\frac{\alpha}{i \cdot 2 + \left(\alpha + \beta\right)}}{\sqrt{2.0 + \left(i \cdot 2 + \left(\alpha + \beta\right)\right)}}\right) \cdot \frac{\alpha + \beta}{1}\right)}\right)}^{3}}}{2.0}\]

   23. Applied simplify0.2

       \[\leadsto \frac{\sqrt[3]{{\left(1.0 + \frac{1}{\sqrt{2.0 + \left(i \cdot 2 + \left(\alpha + \beta\right)\right)}} \cdot \color{blue}{\left(\frac{\frac{\alpha + \beta}{1}}{\sqrt{\left(\beta + 2 \cdot i\right) + \left(\alpha + 2.0\right)}} \cdot \left(\frac{-\alpha}{\left(\alpha + \beta\right) + 2 \cdot i} + \frac{\beta}{\left(\alpha + \beta\right) + 2 \cdot i}\right)\right)}\right)}^{3}}}{2.0}\]
3. Recombined 2 regimes into one program.

Runtime

Time bar (total: 5.7m)Debug logProfile

herbie shell --seed '#(1070578969 3140398606 632207097 462683394 1189254563 964980650)' 
(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))