Average Error: 24.3 → 6.9
Time: 3.1m
Precision: 64
Internal Precision: 1408
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\sqrt[3]{{\left(\left(\sqrt[3]{\frac{\alpha + \beta}{1}} \cdot \sqrt[3]{\frac{\alpha + \beta}{1}}\right) \cdot \left(\sqrt[3]{\frac{\alpha + \beta}{1}} \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + \left(i + i\right)}}{\left(\alpha + \beta\right) + \left(2.0 + \left(i + i\right)\right)}\right) + 1.0\right)}^{3}}}{2.0} \le -1.1102230246251564 \cdot 10^{-16}:\\ \;\;\;\;\frac{\frac{2}{\frac{\alpha}{\beta}} \cdot \left(\frac{1.0}{\beta} - \frac{3.0}{\alpha}\right) + \frac{2}{\frac{\alpha}{\beta}}}{2.0}\\ \mathbf{if}\;\frac{\sqrt[3]{{\left(\left(\sqrt[3]{\frac{\alpha + \beta}{1}} \cdot \sqrt[3]{\frac{\alpha + \beta}{1}}\right) \cdot \left(\sqrt[3]{\frac{\alpha + \beta}{1}} \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + \left(i + i\right)}}{\left(\alpha + \beta\right) + \left(2.0 + \left(i + i\right)\right)}\right) + 1.0\right)}^{3}}}{2.0} \le 6.817046827034544 \cdot 10^{-08}:\\ \;\;\;\;\frac{\frac{\frac{8.0}{\alpha}}{\alpha \cdot \alpha} + \frac{2.0 - \frac{4.0}{\alpha}}{\alpha}}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{{\left(\left(\frac{\alpha + \beta}{1} \cdot \frac{\sqrt[3]{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + \left(i + i\right)}} \cdot \sqrt[3]{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + \left(i + i\right)}}}{\sqrt{\left(\alpha + \beta\right) + \left(2.0 + \left(i + i\right)\right)}}\right) \cdot \frac{\sqrt[3]{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + \left(i + i\right)}}}{\sqrt{\left(\alpha + \beta\right) + \left(2.0 + \left(i + i\right)\right)}} + 1.0\right)}^{3}}}{2.0}\\ \end{array}\]

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) 1)) (cbrt (/ (+ alpha beta) 1))) (* (cbrt (/ (+ alpha beta) 1)) (/ (/ (- beta alpha) (+ (+ alpha beta) (+ i i))) (+ (+ alpha beta) (+ 2.0 (+ i i)))))) 1.0) 3)) 2.0) < -1.1102230246251564e-16

    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

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

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

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

    5. Applied times-frac60.8

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

    6. Applied times-frac60.8

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

    7. Applied simplify60.8

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

    9. Applied add-cube-cbrt60.8

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

    11. Applied add-cbrt-cube60.8

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

    12. Applied simplify60.8

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

    13. Taylor expanded around inf 44.7

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

    14. Applied simplify34.7

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

    if -1.1102230246251564e-16 < (/ (cbrt (pow (+ (* (* (cbrt (/ (+ alpha beta) 1)) (cbrt (/ (+ alpha beta) 1))) (* (cbrt (/ (+ alpha beta) 1)) (/ (/ (- beta alpha) (+ (+ alpha beta) (+ i i))) (+ (+ alpha beta) (+ 2.0 (+ i i)))))) 1.0) 3)) 2.0) < 6.817046827034544e-08

    1. Initial program 60.8

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

    2. Taylor expanded around inf 30.5

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

    3. Applied simplify30.5

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

    if 6.817046827034544e-08 < (/ (cbrt (pow (+ (* (* (cbrt (/ (+ alpha beta) 1)) (cbrt (/ (+ alpha beta) 1))) (* (cbrt (/ (+ alpha beta) 1)) (/ (/ (- beta alpha) (+ (+ alpha beta) (+ i i))) (+ (+ alpha beta) (+ 2.0 (+ i i)))))) 1.0) 3)) 2.0)

    1. Initial program 14.4

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

    3. Applied *-un-lft-identity14.4

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

    4. Applied *-un-lft-identity14.4

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

    5. Applied times-frac0.1

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

    6. Applied times-frac0.1

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

    7. Applied simplify0.1

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

    9. Applied add-cube-cbrt0.4

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

    11. Applied add-cbrt-cube0.3

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

    12. Applied simplify0.1

      \[\leadsto \frac{\sqrt[3]{\color{blue}{{\left(\frac{\alpha + \beta}{1} \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + \left(i + i\right)}}{\left(\alpha + \beta\right) + \left(2.0 + \left(i + i\right)\right)} + 1.0\right)}^{3}}}}{2.0}\]
    13. Using strategy rm

    14. Applied add-sqr-sqrt0.1

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

    15. Applied add-cube-cbrt0.2

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

    16. Applied times-frac0.2

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

    17. Applied associate-*r*0.2

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

Runtime

Time bar (total: 3.1m)Debug logProfile

herbie shell --seed '#(1070258749 1877548225 2229079127 1588002776 3179087814 1886870650)' 
(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))