Average Error: 52.7 → 11.4
Time: 5.9m
Precision: 64
Internal Precision: 128
\[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le 4.743023560006392 \cdot 10^{+155}:\\ \;\;\;\;\frac{\sqrt{\alpha + \left(\beta + i\right)}}{\frac{(\left(\beta + (2 \cdot i + \alpha)_*\right) \cdot \left(\beta + (2 \cdot i + \alpha)_*\right) + \left(-1.0\right))_*}{i}} \cdot \sqrt[3]{{\left(\frac{\frac{\sqrt{\left(\alpha + i\right) + \beta}}{\sqrt{\frac{\alpha + (2 \cdot i + \beta)_*}{\beta + i} \cdot \frac{\alpha + (2 \cdot i + \beta)_*}{\alpha + i}}}}{\sqrt{\frac{\alpha + (2 \cdot i + \beta)_*}{\beta + i} \cdot \frac{\alpha + (2 \cdot i + \beta)_*}{\alpha + i}}}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(\frac{\frac{\left(\alpha + i\right) + \beta}{\sqrt{\frac{\alpha + (2 \cdot i + \beta)_*}{\beta + i} \cdot \frac{\alpha + (2 \cdot i + \beta)_*}{\alpha + i}}}}{\left(4 \cdot \left(\beta + i\right) - \frac{1.0}{i}\right) \cdot \sqrt{\frac{\alpha + (2 \cdot i + \beta)_*}{\beta + i} \cdot \frac{\alpha + (2 \cdot i + \beta)_*}{\alpha + i}}}\right)}^{3}}\\ \end{array}\]

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

  1. Split input into 2 regimes

  2. if i < 4.743023560006392e+155

    1. Initial program 43.5

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

    3. Applied associate-/l*16.2

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

    5. Applied add-sqr-sqrt16.2

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

    6. Applied times-frac16.2

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

    7. Simplified16.2

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

    8. Simplified16.2

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

    10. Applied add-cbrt-cube47.6

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

    11. Applied add-cbrt-cube47.6

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

    12. Applied cbrt-undiv47.7

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

    13. Simplified14.9

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

    15. Applied *-un-lft-identity14.9

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

    16. Applied add-sqr-sqrt14.8

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

    17. Applied times-frac14.8

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

    18. Applied times-frac14.8

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

    19. Applied cube-prod15.2

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

    20. Applied cbrt-prod15.1

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

    21. Simplified11.5

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

    if 4.743023560006392e+155 < i

    1. Initial program 62.1

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

    3. Applied associate-/l*62.1

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

    5. Applied add-sqr-sqrt62.1

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

    6. Applied times-frac62.1

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

    7. Simplified62.1

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

    8. Simplified62.1

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

    10. Applied add-cbrt-cube62.1

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

    11. Applied add-cbrt-cube62.1

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

    12. Applied cbrt-undiv62.1

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

    13. Simplified61.9

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

    14. Taylor expanded around 0 11.3

      \[\leadsto \sqrt[3]{{\left(\frac{\frac{\beta + \left(i + \alpha\right)}{\sqrt{\frac{(2 \cdot i + \beta)_* + \alpha}{i + \alpha} \cdot \frac{(2 \cdot i + \beta)_* + \alpha}{\beta + i}}}}{\color{blue}{\left(\left(4 \cdot i + 4 \cdot \beta\right) - 1.0 \cdot \frac{1}{i}\right)} \cdot \sqrt{\frac{(2 \cdot i + \beta)_* + \alpha}{i + \alpha} \cdot \frac{(2 \cdot i + \beta)_* + \alpha}{\beta + i}}}\right)}^{3}}\]

    15. Simplified11.3

      \[\leadsto \sqrt[3]{{\left(\frac{\frac{\beta + \left(i + \alpha\right)}{\sqrt{\frac{(2 \cdot i + \beta)_* + \alpha}{i + \alpha} \cdot \frac{(2 \cdot i + \beta)_* + \alpha}{\beta + i}}}}{\color{blue}{\left(4 \cdot \left(i + \beta\right) - \frac{1.0}{i}\right)} \cdot \sqrt{\frac{(2 \cdot i + \beta)_* + \alpha}{i + \alpha} \cdot \frac{(2 \cdot i + \beta)_* + \alpha}{\beta + i}}}\right)}^{3}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification11.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \le 4.743023560006392 \cdot 10^{+155}:\\ \;\;\;\;\frac{\sqrt{\alpha + \left(\beta + i\right)}}{\frac{(\left(\beta + (2 \cdot i + \alpha)_*\right) \cdot \left(\beta + (2 \cdot i + \alpha)_*\right) + \left(-1.0\right))_*}{i}} \cdot \sqrt[3]{{\left(\frac{\frac{\sqrt{\left(\alpha + i\right) + \beta}}{\sqrt{\frac{\alpha + (2 \cdot i + \beta)_*}{\beta + i} \cdot \frac{\alpha + (2 \cdot i + \beta)_*}{\alpha + i}}}}{\sqrt{\frac{\alpha + (2 \cdot i + \beta)_*}{\beta + i} \cdot \frac{\alpha + (2 \cdot i + \beta)_*}{\alpha + i}}}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(\frac{\frac{\left(\alpha + i\right) + \beta}{\sqrt{\frac{\alpha + (2 \cdot i + \beta)_*}{\beta + i} \cdot \frac{\alpha + (2 \cdot i + \beta)_*}{\alpha + i}}}}{\left(4 \cdot \left(\beta + i\right) - \frac{1.0}{i}\right) \cdot \sqrt{\frac{\alpha + (2 \cdot i + \beta)_*}{\beta + i} \cdot \frac{\alpha + (2 \cdot i + \beta)_*}{\alpha + i}}}\right)}^{3}}\\ \end{array}\]

Runtime

Time bar (total: 5.9m)Debug logProfile

herbie shell --seed 2018346 +o rules:numerics
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/4"
  :pre (and (> alpha -1) (> beta -1) (> i 1))
  (/ (/ (* (* i (+ (+ alpha beta) i)) (+ (* beta alpha) (* i (+ (+ alpha beta) i)))) (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i)))) (- (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i))) 1.0)))