Average Error: 23.4 → 7.2
Time: 2.1m
Precision: 64
Internal Precision: 1344
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\beta + \alpha\right) \cdot \left(\beta - \alpha\right)}{2 \cdot i + \left(\beta + \alpha\right)}}{2.0 + \left(2 \cdot i + \left(\beta + \alpha\right)\right)} \le -1.0:\\ \;\;\;\;\frac{\frac{\frac{8.0}{\alpha} - 4.0}{\alpha \cdot \alpha} + \frac{2.0}{\alpha}}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{\left(\left(\left(\frac{\sqrt[3]{\beta + \alpha}}{\sqrt[3]{2 \cdot i + \left(\beta + \alpha\right)}} \cdot \frac{\sqrt[3]{\beta + \alpha}}{\sqrt[3]{2 \cdot i + \left(\beta + \alpha\right)}}\right) \cdot \frac{\sqrt[3]{\beta + \alpha}}{\frac{2.0 + \left(2 \cdot i + \left(\beta + \alpha\right)\right)}{\frac{\beta - \alpha}{\sqrt[3]{2 \cdot i + \left(\beta + \alpha\right)}}}} + 1.0\right) \cdot \log \left(e^{\left(\frac{\sqrt[3]{\beta + \alpha}}{\sqrt[3]{2 \cdot i + \left(\beta + \alpha\right)}} \cdot \frac{\sqrt[3]{\beta + \alpha}}{\sqrt[3]{2 \cdot i + \left(\beta + \alpha\right)}}\right) \cdot \frac{\sqrt[3]{\beta + \alpha}}{\frac{2.0 + \left(2 \cdot i + \left(\beta + \alpha\right)\right)}{\frac{\beta - \alpha}{\sqrt[3]{2 \cdot i + \left(\beta + \alpha\right)}}}} + 1.0}\right)\right) \cdot \left(\left(\frac{\sqrt[3]{\beta + \alpha}}{\sqrt[3]{2 \cdot i + \left(\beta + \alpha\right)}} \cdot \frac{\sqrt[3]{\beta + \alpha}}{\sqrt[3]{2 \cdot i + \left(\beta + \alpha\right)}}\right) \cdot \frac{\sqrt[3]{\beta + \alpha}}{\frac{2.0 + \left(2 \cdot i + \left(\beta + \alpha\right)\right)}{\frac{\beta - \alpha}{\sqrt[3]{2 \cdot i + \left(\beta + \alpha\right)}}}} + 1.0\right)}}{2.0}\\ \end{array}\]

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if (/ (/ (* (+ alpha beta) (- beta alpha)) (+ (+ alpha beta) (* 2 i))) (+ (+ (+ alpha beta) (* 2 i)) 2.0)) < -1.0

    1. Initial program 62.8

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

    2. Taylor expanded around inf 31.3

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

    3. Simplified31.3

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

    if -1.0 < (/ (/ (* (+ alpha beta) (- beta alpha)) (+ (+ alpha beta) (* 2 i))) (+ (+ (+ alpha beta) (* 2 i)) 2.0))

    1. Initial program 12.6

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

    3. Applied *-un-lft-identity12.6

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

    4. Applied times-frac0.5

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

    5. Applied associate-/l*0.5

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

    6. Simplified0.5

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

    8. Applied add-cube-cbrt0.8

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

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

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

    10. Applied times-frac0.8

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

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

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

    12. Applied times-frac0.8

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

    13. Applied add-cube-cbrt0.6

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

    14. Applied times-frac0.6

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

    15. Simplified0.6

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

    17. Applied add-cbrt-cube0.6

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

    19. Applied add-log-exp0.6

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

  4. Final simplification7.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\beta + \alpha\right) \cdot \left(\beta - \alpha\right)}{2 \cdot i + \left(\beta + \alpha\right)}}{2.0 + \left(2 \cdot i + \left(\beta + \alpha\right)\right)} \le -1.0:\\ \;\;\;\;\frac{\frac{\frac{8.0}{\alpha} - 4.0}{\alpha \cdot \alpha} + \frac{2.0}{\alpha}}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{\left(\left(\left(\frac{\sqrt[3]{\beta + \alpha}}{\sqrt[3]{2 \cdot i + \left(\beta + \alpha\right)}} \cdot \frac{\sqrt[3]{\beta + \alpha}}{\sqrt[3]{2 \cdot i + \left(\beta + \alpha\right)}}\right) \cdot \frac{\sqrt[3]{\beta + \alpha}}{\frac{2.0 + \left(2 \cdot i + \left(\beta + \alpha\right)\right)}{\frac{\beta - \alpha}{\sqrt[3]{2 \cdot i + \left(\beta + \alpha\right)}}}} + 1.0\right) \cdot \log \left(e^{\left(\frac{\sqrt[3]{\beta + \alpha}}{\sqrt[3]{2 \cdot i + \left(\beta + \alpha\right)}} \cdot \frac{\sqrt[3]{\beta + \alpha}}{\sqrt[3]{2 \cdot i + \left(\beta + \alpha\right)}}\right) \cdot \frac{\sqrt[3]{\beta + \alpha}}{\frac{2.0 + \left(2 \cdot i + \left(\beta + \alpha\right)\right)}{\frac{\beta - \alpha}{\sqrt[3]{2 \cdot i + \left(\beta + \alpha\right)}}}} + 1.0}\right)\right) \cdot \left(\left(\frac{\sqrt[3]{\beta + \alpha}}{\sqrt[3]{2 \cdot i + \left(\beta + \alpha\right)}} \cdot \frac{\sqrt[3]{\beta + \alpha}}{\sqrt[3]{2 \cdot i + \left(\beta + \alpha\right)}}\right) \cdot \frac{\sqrt[3]{\beta + \alpha}}{\frac{2.0 + \left(2 \cdot i + \left(\beta + \alpha\right)\right)}{\frac{\beta - \alpha}{\sqrt[3]{2 \cdot i + \left(\beta + \alpha\right)}}}} + 1.0\right)}}{2.0}\\ \end{array}\]

Runtime

Time bar (total: 2.1m)Debug logProfile

BaselineHerbieOracleSpan%
Regimes12.27.25.56.775.2%
herbie shell --seed 2018285 
(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))