Average Error: 23.0 → 12.4
Time: 8.5m
Precision: 64
Internal Precision: 128
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 3.9922356107388413 \cdot 10^{+36}:\\ \;\;\;\;\frac{1.0 + \frac{\beta + \alpha}{\sqrt{2.0 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)} \cdot \frac{\sqrt{2.0 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)}}{\frac{\beta - \alpha}{i \cdot 2 + \left(\beta + \alpha\right)}}}}{2.0}\\ \mathbf{elif}\;\alpha \le 7.121571446054871 \cdot 10^{+128}:\\ \;\;\;\;\frac{\frac{2.0}{\alpha} + \frac{\frac{\frac{8.0}{\alpha} - 4.0}{\alpha}}{\alpha}}{2.0}\\ \mathbf{elif}\;\alpha \le 7.617578652767246 \cdot 10^{+149}:\\ \;\;\;\;\frac{1.0 + \frac{\frac{\beta + \alpha}{\frac{1}{\frac{\sqrt{\beta} + \sqrt{\alpha}}{\sqrt[3]{i \cdot 2 + \left(\beta + \alpha\right)} \cdot \sqrt[3]{i \cdot 2 + \left(\beta + \alpha\right)}}}}}{\frac{2.0 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)}{\frac{\sqrt{\beta} - \sqrt{\alpha}}{\sqrt[3]{i \cdot 2 + \left(\beta + \alpha\right)}}}}}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2.0}{\alpha} + \frac{\frac{\frac{8.0}{\alpha} - 4.0}{\alpha}}{\alpha}}{2.0}\\ \end{array}\]

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

  1. Split input into 3 regimes

  2. if alpha < 3.9922356107388413e+36

    1. Initial program 11.2

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

    3. Applied *-un-lft-identity11.2

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

    4. Applied times-frac0.9

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

    5. Applied associate-/l*0.9

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

    6. Simplified0.9

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

    8. Applied *-un-lft-identity0.9

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

    9. Applied add-sqr-sqrt1.0

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

    10. Applied times-frac1.0

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

    if 3.9922356107388413e+36 < alpha < 7.121571446054871e+128 or 7.617578652767246e+149 < alpha

    1. Initial program 53.4

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

    2. Taylor expanded around inf 41.1

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

    3. Simplified41.1

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

    if 7.121571446054871e+128 < alpha < 7.617578652767246e+149

    1. Initial program 44.7

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

    3. Applied *-un-lft-identity44.7

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

    4. Applied times-frac32.4

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

    5. Applied associate-/l*32.5

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

    6. Simplified32.5

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

    8. Applied add-cube-cbrt32.4

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

    9. Applied add-sqr-sqrt32.4

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

    10. Applied add-sqr-sqrt41.6

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

    11. Applied difference-of-squares41.6

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

    12. Applied times-frac41.6

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

    13. Applied *-un-lft-identity41.6

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

    14. Applied times-frac41.6

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

    15. Applied associate-/r*41.6

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

  4. Final simplification12.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 3.9922356107388413 \cdot 10^{+36}:\\ \;\;\;\;\frac{1.0 + \frac{\beta + \alpha}{\sqrt{2.0 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)} \cdot \frac{\sqrt{2.0 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)}}{\frac{\beta - \alpha}{i \cdot 2 + \left(\beta + \alpha\right)}}}}{2.0}\\ \mathbf{elif}\;\alpha \le 7.121571446054871 \cdot 10^{+128}:\\ \;\;\;\;\frac{\frac{2.0}{\alpha} + \frac{\frac{\frac{8.0}{\alpha} - 4.0}{\alpha}}{\alpha}}{2.0}\\ \mathbf{elif}\;\alpha \le 7.617578652767246 \cdot 10^{+149}:\\ \;\;\;\;\frac{1.0 + \frac{\frac{\beta + \alpha}{\frac{1}{\frac{\sqrt{\beta} + \sqrt{\alpha}}{\sqrt[3]{i \cdot 2 + \left(\beta + \alpha\right)} \cdot \sqrt[3]{i \cdot 2 + \left(\beta + \alpha\right)}}}}}{\frac{2.0 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)}{\frac{\sqrt{\beta} - \sqrt{\alpha}}{\sqrt[3]{i \cdot 2 + \left(\beta + \alpha\right)}}}}}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2.0}{\alpha} + \frac{\frac{\frac{8.0}{\alpha} - 4.0}{\alpha}}{\alpha}}{2.0}\\ \end{array}\]

Reproduce

herbie shell --seed 2019091 
(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))