Average Error: 23.4 → 6.7
Time: 9.6m
Precision: 64
Internal Precision: 1344
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\sqrt[3]{{\left(\frac{\frac{\sqrt[3]{\alpha + \beta} \cdot \sqrt[3]{\alpha + \beta}}{\sqrt[3]{\left(2 \cdot i + \left(\alpha + 2.0\right)\right) + \beta}}}{\left(\frac{\sqrt[3]{\left(2 \cdot i + \left(\alpha + 2.0\right)\right) + \beta}}{\frac{\sqrt[3]{\alpha + \beta}}{2 \cdot i + \left(\alpha + \beta\right)}} \cdot \frac{\sqrt[3]{\sqrt[3]{\left(2 \cdot i + \left(\alpha + 2.0\right)\right) + \beta} \cdot \sqrt[3]{\left(2 \cdot i + \left(\alpha + 2.0\right)\right) + \beta}}}{\sqrt[3]{\beta - \alpha} \cdot \sqrt[3]{\beta - \alpha}}\right) \cdot \frac{\sqrt[3]{\sqrt[3]{\left(2 \cdot i + \left(\alpha + 2.0\right)\right) + \beta}}}{\sqrt[3]{\beta - \alpha}}} + 1.0\right)}^{3}}}{2.0} \le 0.0:\\ \;\;\;\;\frac{\left(\beta \cdot \frac{2}{\alpha}\right) \cdot \left(\frac{1.0}{\beta} - \frac{3.0}{\alpha}\right) + \beta \cdot \frac{2}{\alpha}}{2.0}\\ \mathbf{if}\;\frac{\sqrt[3]{{\left(\frac{\frac{\sqrt[3]{\alpha + \beta} \cdot \sqrt[3]{\alpha + \beta}}{\sqrt[3]{\left(2 \cdot i + \left(\alpha + 2.0\right)\right) + \beta}}}{\left(\frac{\sqrt[3]{\left(2 \cdot i + \left(\alpha + 2.0\right)\right) + \beta}}{\frac{\sqrt[3]{\alpha + \beta}}{2 \cdot i + \left(\alpha + \beta\right)}} \cdot \frac{\sqrt[3]{\sqrt[3]{\left(2 \cdot i + \left(\alpha + 2.0\right)\right) + \beta} \cdot \sqrt[3]{\left(2 \cdot i + \left(\alpha + 2.0\right)\right) + \beta}}}{\sqrt[3]{\beta - \alpha} \cdot \sqrt[3]{\beta - \alpha}}\right) \cdot \frac{\sqrt[3]{\sqrt[3]{\left(2 \cdot i + \left(\alpha + 2.0\right)\right) + \beta}}}{\sqrt[3]{\beta - \alpha}}} + 1.0\right)}^{3}}}{2.0} \le 3.4416913763379853 \cdot 10^{-15}:\\ \;\;\;\;\frac{\frac{2.0}{\alpha} + \frac{\frac{8.0}{\alpha} - 4.0}{\alpha \cdot \alpha}}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{{\left(\frac{\frac{\sqrt[3]{\beta + \alpha}}{\sqrt[3]{\left(\beta + i \cdot 2\right) + \left(2.0 + \alpha\right)}}}{\frac{\sqrt[3]{\left(\beta + i \cdot 2\right) + \left(2.0 + \alpha\right)}}{\frac{\sqrt[3]{\beta + \alpha}}{i \cdot 2 + \left(\beta + \alpha\right)}}} \cdot \frac{\sqrt[3]{\beta + \alpha}}{\frac{\sqrt[3]{\left(\beta + i \cdot 2\right) + \left(2.0 + \alpha\right)}}{\beta - \alpha}} + 1.0\right)}^{3}}}{2.0}\\ \end{array}\]

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

  1. Split input into 3 regimes

  2. if (/ (cbrt (pow (+ (/ (/ (* (cbrt (+ alpha beta)) (cbrt (+ alpha beta))) (cbrt (+ (+ (* 2 i) (+ alpha 2.0)) beta))) (* (* (/ (cbrt (+ (+ (* 2 i) (+ alpha 2.0)) beta)) (/ (cbrt (+ alpha beta)) (+ (* 2 i) (+ alpha beta)))) (/ (cbrt (* (cbrt (+ (+ (* 2 i) (+ alpha 2.0)) beta)) (cbrt (+ (+ (* 2 i) (+ alpha 2.0)) beta)))) (* (cbrt (- beta alpha)) (cbrt (- beta alpha))))) (/ (cbrt (cbrt (+ (+ (* 2 i) (+ alpha 2.0)) beta))) (cbrt (- beta alpha))))) 1.0) 3)) 2.0) < 0.0

    1. Initial program 62.6

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

    3. Applied *-un-lft-identity62.6

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

    4. Applied times-frac60.8

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

    5. Applied associate-/l*60.8

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

    7. Applied div-inv60.7

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

    8. Applied add-cube-cbrt60.4

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

    9. Applied times-frac60.5

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

    10. Applied add-cube-cbrt61.1

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

    11. Applied times-frac61.0

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

    12. Applied simplify61.1

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

    13. Applied simplify61.0

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

    15. Applied add-cbrt-cube61.0

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

    16. Applied simplify60.9

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

    17. Taylor expanded around inf 43.8

      \[\leadsto \frac{\color{blue}{\left(e^{\left(\log \left(\frac{1}{\alpha}\right) + \log 2\right) - \log \left(\frac{1}{\beta}\right)} + 1.0 \cdot \frac{e^{\left(\log \left(\frac{1}{\alpha}\right) + \log 2\right) - \log \left(\frac{1}{\beta}\right)}}{\beta}\right) - 3.0 \cdot \frac{e^{\left(\log \left(\frac{1}{\alpha}\right) + \log 2\right) - \log \left(\frac{1}{\beta}\right)}}{\alpha}}}{2.0}\]

    18. Applied simplify32.8

      \[\leadsto \color{blue}{\frac{\left(\beta \cdot \frac{2}{\alpha}\right) \cdot \left(\frac{1.0}{\beta} - \frac{3.0}{\alpha}\right) + \beta \cdot \frac{2}{\alpha}}{2.0}}\]

    if 0.0 < (/ (cbrt (pow (+ (/ (/ (* (cbrt (+ alpha beta)) (cbrt (+ alpha beta))) (cbrt (+ (+ (* 2 i) (+ alpha 2.0)) beta))) (* (* (/ (cbrt (+ (+ (* 2 i) (+ alpha 2.0)) beta)) (/ (cbrt (+ alpha beta)) (+ (* 2 i) (+ alpha beta)))) (/ (cbrt (* (cbrt (+ (+ (* 2 i) (+ alpha 2.0)) beta)) (cbrt (+ (+ (* 2 i) (+ alpha 2.0)) beta)))) (* (cbrt (- beta alpha)) (cbrt (- beta alpha))))) (/ (cbrt (cbrt (+ (+ (* 2 i) (+ alpha 2.0)) beta))) (cbrt (- beta alpha))))) 1.0) 3)) 2.0) < 3.4416913763379853e-15

    1. Initial program 62.4

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

    2. Taylor expanded around inf 29.1

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

    3. Applied simplify29.1

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

    if 3.4416913763379853e-15 < (/ (cbrt (pow (+ (/ (/ (* (cbrt (+ alpha beta)) (cbrt (+ alpha beta))) (cbrt (+ (+ (* 2 i) (+ alpha 2.0)) beta))) (* (* (/ (cbrt (+ (+ (* 2 i) (+ alpha 2.0)) beta)) (/ (cbrt (+ alpha beta)) (+ (* 2 i) (+ alpha beta)))) (/ (cbrt (* (cbrt (+ (+ (* 2 i) (+ alpha 2.0)) beta)) (cbrt (+ (+ (* 2 i) (+ alpha 2.0)) beta)))) (* (cbrt (- beta alpha)) (cbrt (- beta alpha))))) (/ (cbrt (cbrt (+ (+ (* 2 i) (+ alpha 2.0)) beta))) (cbrt (- beta alpha))))) 1.0) 3)) 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

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

    4. Applied times-frac0.4

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

    5. Applied associate-/l*0.4

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

    7. Applied div-inv0.4

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

    8. Applied add-cube-cbrt0.7

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

    9. Applied times-frac5.6

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

    10. Applied add-cube-cbrt5.5

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

    11. Applied times-frac5.5

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

    12. Applied simplify5.5

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

    13. Applied simplify0.5

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

    15. Applied add-cbrt-cube0.5

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

    16. Applied simplify0.5

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

    18. Applied add-cbrt-cube0.5

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

    19. Applied simplify0.5

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

Runtime

Time bar (total: 9.6m)Debug logProfile

herbie shell --seed '#(1071373924 2949776965 1885069702 3247780810 90874544 2263903749)' 
(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))