Average Error: 23.2 → 6.0
Time: 9.0m
Precision: 64
Internal Precision: 1408
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\sqrt[3]{{\left(\frac{\frac{\sqrt[3]{\sqrt[3]{\beta - \alpha} \cdot \sqrt[3]{\beta - \alpha}}}{1}}{\sqrt[3]{\sqrt[3]{\left(2.0 + \left(\alpha + \beta\right)\right) + 2 \cdot i}} \cdot \sqrt[3]{\sqrt[3]{\left(2.0 + \left(\alpha + \beta\right)\right) + 2 \cdot i}}} \cdot \left(\frac{\frac{\sqrt[3]{\sqrt[3]{\beta - \alpha}}}{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt[3]{\sqrt[3]{\left(2.0 + \left(\alpha + \beta\right)\right) + 2 \cdot i}}} \cdot \left(\frac{\alpha + \beta}{\sqrt[3]{\left(2.0 + \left(\alpha + \beta\right)\right) + 2 \cdot i}} \cdot \frac{\sqrt[3]{\beta - \alpha} \cdot \sqrt[3]{\beta - \alpha}}{\sqrt[3]{\left(2.0 + \left(\alpha + \beta\right)\right) + 2 \cdot i}}\right)\right) + 1.0\right)}^{3}}}{2.0} \le 1.243172231824019 \cdot 10^{-12}:\\ \;\;\;\;\frac{\frac{2.0}{\alpha} + \frac{\frac{8.0}{\alpha} - 4.0}{\alpha \cdot \alpha}}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\log \left(1.0 + \frac{\frac{\frac{\sqrt[3]{\beta - \alpha}}{i \cdot 2 + \left(\alpha + \beta\right)} \cdot \left(\alpha + \beta\right)}{\sqrt[3]{\left(\left(\alpha + \beta\right) + 2.0\right) + i \cdot 2}}}{\frac{\sqrt[3]{\left(\left(\alpha + \beta\right) + 2.0\right) + i \cdot 2}}{\sqrt[3]{\beta - \alpha}} \cdot \frac{\sqrt[3]{\left(\left(\alpha + \beta\right) + 2.0\right) + i \cdot 2}}{\sqrt[3]{\beta - \alpha}}}\right)}}{2.0}\\ \end{array}\]

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

  1. Split input into 2 regimes

  2. if (/ (cbrt (pow (+ (* (/ (/ (cbrt (* (cbrt (- beta alpha)) (cbrt (- beta alpha)))) 1) (* (cbrt (cbrt (+ (+ 2.0 (+ alpha beta)) (* 2 i)))) (cbrt (cbrt (+ (+ 2.0 (+ alpha beta)) (* 2 i)))))) (* (/ (/ (cbrt (cbrt (- beta alpha))) (+ (+ alpha beta) (* 2 i))) (cbrt (cbrt (+ (+ 2.0 (+ alpha beta)) (* 2 i))))) (* (/ (+ alpha beta) (cbrt (+ (+ 2.0 (+ alpha beta)) (* 2 i)))) (/ (* (cbrt (- beta alpha)) (cbrt (- beta alpha))) (cbrt (+ (+ 2.0 (+ alpha beta)) (* 2 i))))))) 1.0) 3)) 2.0) < 1.243172231824019e-12

    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 29.8

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

    3. Applied simplify29.8

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

    if 1.243172231824019e-12 < (/ (cbrt (pow (+ (* (/ (/ (cbrt (* (cbrt (- beta alpha)) (cbrt (- beta alpha)))) 1) (* (cbrt (cbrt (+ (+ 2.0 (+ alpha beta)) (* 2 i)))) (cbrt (cbrt (+ (+ 2.0 (+ alpha beta)) (* 2 i)))))) (* (/ (/ (cbrt (cbrt (- beta alpha))) (+ (+ alpha beta) (* 2 i))) (cbrt (cbrt (+ (+ 2.0 (+ alpha beta)) (* 2 i))))) (* (/ (+ alpha beta) (cbrt (+ (+ 2.0 (+ alpha beta)) (* 2 i)))) (/ (* (cbrt (- beta alpha)) (cbrt (- beta alpha))) (cbrt (+ (+ 2.0 (+ alpha beta)) (* 2 i))))))) 1.0) 3)) 2.0)

    1. Initial program 13.7

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

    3. Applied *-un-lft-identity13.7

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

    4. Applied *-un-lft-identity13.7

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

    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-frac0.3

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

    7. Applied simplify0.3

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

    9. Applied add-cube-cbrt0.5

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

    10. Applied *-un-lft-identity0.5

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

    11. Applied add-cube-cbrt0.3

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

    12. Applied times-frac0.3

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

    13. Applied times-frac0.3

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

    14. Applied associate-*r*0.3

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

    15. Applied simplify0.3

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

    17. Applied add-cbrt-cube0.3

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

    18. Applied simplify0.3

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

    20. Applied add-exp-log0.3

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

    21. Applied simplify0.3

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

Runtime

Time bar (total: 9.0m)Debug logProfile

herbie shell --seed '#(1070609872 3456127585 2380521889 2328837196 1765472538 734540918)' 
(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))