Average Error: 46.4 → 2.2
Time: 3.9m
Precision: 64
Internal Precision: 4160
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{\sin k} \cdot \frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{k}{\ell}}\right) \cdot \left(\frac{\frac{\sqrt{\sqrt[3]{2}}}{\sqrt[3]{\frac{k}{\ell}}}}{\sqrt[3]{\tan k}} \cdot \frac{\frac{\sqrt{\sqrt[3]{2}}}{\sqrt[3]{\frac{k}{\ell}} \cdot \sqrt[3]{\frac{k}{\ell}}}}{t}\right)\]

Error

Bits error versus t

Bits error versus l

Bits error versus k

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Results

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Derivation

  1. Initial program 46.4

    \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]

  2. Initial simplification30.1

    \[\leadsto \frac{\frac{\frac{2}{\tan k}}{\frac{\sin k \cdot t}{\frac{\ell}{t} \cdot \frac{\ell}{t}}}}{\frac{k}{t} \cdot \frac{k}{t}}\]
  3. Using strategy rm

  4. Applied times-frac29.1

    \[\leadsto \frac{\frac{\frac{2}{\tan k}}{\color{blue}{\frac{\sin k}{\frac{\ell}{t}} \cdot \frac{t}{\frac{\ell}{t}}}}}{\frac{k}{t} \cdot \frac{k}{t}}\]

  5. Applied add-cube-cbrt29.2

    \[\leadsto \frac{\frac{\color{blue}{\left(\sqrt[3]{\frac{2}{\tan k}} \cdot \sqrt[3]{\frac{2}{\tan k}}\right) \cdot \sqrt[3]{\frac{2}{\tan k}}}}{\frac{\sin k}{\frac{\ell}{t}} \cdot \frac{t}{\frac{\ell}{t}}}}{\frac{k}{t} \cdot \frac{k}{t}}\]

  6. Applied times-frac28.7

    \[\leadsto \frac{\color{blue}{\frac{\sqrt[3]{\frac{2}{\tan k}} \cdot \sqrt[3]{\frac{2}{\tan k}}}{\frac{\sin k}{\frac{\ell}{t}}} \cdot \frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t}{\frac{\ell}{t}}}}}{\frac{k}{t} \cdot \frac{k}{t}}\]

  7. Applied times-frac18.1

    \[\leadsto \color{blue}{\frac{\frac{\sqrt[3]{\frac{2}{\tan k}} \cdot \sqrt[3]{\frac{2}{\tan k}}}{\frac{\sin k}{\frac{\ell}{t}}}}{\frac{k}{t}} \cdot \frac{\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t}{\frac{\ell}{t}}}}{\frac{k}{t}}}\]

  8. Simplified11.0

    \[\leadsto \frac{\frac{\sqrt[3]{\frac{2}{\tan k}} \cdot \sqrt[3]{\frac{2}{\tan k}}}{\frac{\sin k}{\frac{\ell}{t}}}}{\frac{k}{t}} \cdot \color{blue}{\left(\left(\frac{1}{k} \cdot \frac{\ell}{t}\right) \cdot \sqrt[3]{\frac{2}{\tan k}}\right)}\]
  9. Using strategy rm

  10. Applied div-inv11.0

    \[\leadsto \frac{\frac{\sqrt[3]{\frac{2}{\tan k}} \cdot \sqrt[3]{\frac{2}{\tan k}}}{\frac{\sin k}{\frac{\ell}{t}}}}{\color{blue}{k \cdot \frac{1}{t}}} \cdot \left(\left(\frac{1}{k} \cdot \frac{\ell}{t}\right) \cdot \sqrt[3]{\frac{2}{\tan k}}\right)\]

  11. Applied div-inv11.0

    \[\leadsto \frac{\frac{\sqrt[3]{\frac{2}{\tan k}} \cdot \sqrt[3]{\frac{2}{\tan k}}}{\frac{\sin k}{\color{blue}{\ell \cdot \frac{1}{t}}}}}{k \cdot \frac{1}{t}} \cdot \left(\left(\frac{1}{k} \cdot \frac{\ell}{t}\right) \cdot \sqrt[3]{\frac{2}{\tan k}}\right)\]

  12. Applied *-un-lft-identity11.0

    \[\leadsto \frac{\frac{\sqrt[3]{\frac{2}{\tan k}} \cdot \sqrt[3]{\frac{2}{\tan k}}}{\frac{\color{blue}{1 \cdot \sin k}}{\ell \cdot \frac{1}{t}}}}{k \cdot \frac{1}{t}} \cdot \left(\left(\frac{1}{k} \cdot \frac{\ell}{t}\right) \cdot \sqrt[3]{\frac{2}{\tan k}}\right)\]

  13. Applied times-frac11.6

    \[\leadsto \frac{\frac{\sqrt[3]{\frac{2}{\tan k}} \cdot \sqrt[3]{\frac{2}{\tan k}}}{\color{blue}{\frac{1}{\ell} \cdot \frac{\sin k}{\frac{1}{t}}}}}{k \cdot \frac{1}{t}} \cdot \left(\left(\frac{1}{k} \cdot \frac{\ell}{t}\right) \cdot \sqrt[3]{\frac{2}{\tan k}}\right)\]

  14. Applied times-frac12.1

    \[\leadsto \frac{\color{blue}{\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{1}{\ell}} \cdot \frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{\sin k}{\frac{1}{t}}}}}{k \cdot \frac{1}{t}} \cdot \left(\left(\frac{1}{k} \cdot \frac{\ell}{t}\right) \cdot \sqrt[3]{\frac{2}{\tan k}}\right)\]

  15. Applied times-frac8.2

    \[\leadsto \color{blue}{\left(\frac{\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{1}{\ell}}}{k} \cdot \frac{\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{\sin k}{\frac{1}{t}}}}{\frac{1}{t}}\right)} \cdot \left(\left(\frac{1}{k} \cdot \frac{\ell}{t}\right) \cdot \sqrt[3]{\frac{2}{\tan k}}\right)\]

  16. Simplified8.1

    \[\leadsto \left(\color{blue}{\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{k}{\ell}}} \cdot \frac{\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{\sin k}{\frac{1}{t}}}}{\frac{1}{t}}\right) \cdot \left(\left(\frac{1}{k} \cdot \frac{\ell}{t}\right) \cdot \sqrt[3]{\frac{2}{\tan k}}\right)\]

  17. Simplified7.0

    \[\leadsto \left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{k}{\ell}} \cdot \color{blue}{\frac{\sqrt[3]{\frac{2}{\tan k}}}{\sin k}}\right) \cdot \left(\left(\frac{1}{k} \cdot \frac{\ell}{t}\right) \cdot \sqrt[3]{\frac{2}{\tan k}}\right)\]
  18. Using strategy rm

  19. Applied cbrt-div7.0

    \[\leadsto \left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{k}{\ell}} \cdot \frac{\sqrt[3]{\frac{2}{\tan k}}}{\sin k}\right) \cdot \left(\left(\frac{1}{k} \cdot \frac{\ell}{t}\right) \cdot \color{blue}{\frac{\sqrt[3]{2}}{\sqrt[3]{\tan k}}}\right)\]

  20. Applied associate-*r/2.4

    \[\leadsto \left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{k}{\ell}} \cdot \frac{\sqrt[3]{\frac{2}{\tan k}}}{\sin k}\right) \cdot \left(\color{blue}{\frac{\frac{1}{k} \cdot \ell}{t}} \cdot \frac{\sqrt[3]{2}}{\sqrt[3]{\tan k}}\right)\]

  21. Applied frac-times2.1

    \[\leadsto \left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{k}{\ell}} \cdot \frac{\sqrt[3]{\frac{2}{\tan k}}}{\sin k}\right) \cdot \color{blue}{\frac{\left(\frac{1}{k} \cdot \ell\right) \cdot \sqrt[3]{2}}{t \cdot \sqrt[3]{\tan k}}}\]

  22. Simplified2.1

    \[\leadsto \left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{k}{\ell}} \cdot \frac{\sqrt[3]{\frac{2}{\tan k}}}{\sin k}\right) \cdot \frac{\color{blue}{\frac{\sqrt[3]{2}}{\frac{k}{\ell}}}}{t \cdot \sqrt[3]{\tan k}}\]
  23. Using strategy rm

  24. Applied add-cube-cbrt2.3

    \[\leadsto \left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{k}{\ell}} \cdot \frac{\sqrt[3]{\frac{2}{\tan k}}}{\sin k}\right) \cdot \frac{\frac{\sqrt[3]{2}}{\color{blue}{\left(\sqrt[3]{\frac{k}{\ell}} \cdot \sqrt[3]{\frac{k}{\ell}}\right) \cdot \sqrt[3]{\frac{k}{\ell}}}}}{t \cdot \sqrt[3]{\tan k}}\]

  25. Applied add-sqr-sqrt2.2

    \[\leadsto \left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{k}{\ell}} \cdot \frac{\sqrt[3]{\frac{2}{\tan k}}}{\sin k}\right) \cdot \frac{\frac{\color{blue}{\sqrt{\sqrt[3]{2}} \cdot \sqrt{\sqrt[3]{2}}}}{\left(\sqrt[3]{\frac{k}{\ell}} \cdot \sqrt[3]{\frac{k}{\ell}}\right) \cdot \sqrt[3]{\frac{k}{\ell}}}}{t \cdot \sqrt[3]{\tan k}}\]

  26. Applied times-frac2.2

    \[\leadsto \left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{k}{\ell}} \cdot \frac{\sqrt[3]{\frac{2}{\tan k}}}{\sin k}\right) \cdot \frac{\color{blue}{\frac{\sqrt{\sqrt[3]{2}}}{\sqrt[3]{\frac{k}{\ell}} \cdot \sqrt[3]{\frac{k}{\ell}}} \cdot \frac{\sqrt{\sqrt[3]{2}}}{\sqrt[3]{\frac{k}{\ell}}}}}{t \cdot \sqrt[3]{\tan k}}\]

  27. Applied times-frac2.2

    \[\leadsto \left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{k}{\ell}} \cdot \frac{\sqrt[3]{\frac{2}{\tan k}}}{\sin k}\right) \cdot \color{blue}{\left(\frac{\frac{\sqrt{\sqrt[3]{2}}}{\sqrt[3]{\frac{k}{\ell}} \cdot \sqrt[3]{\frac{k}{\ell}}}}{t} \cdot \frac{\frac{\sqrt{\sqrt[3]{2}}}{\sqrt[3]{\frac{k}{\ell}}}}{\sqrt[3]{\tan k}}\right)}\]

  28. Final simplification2.2

    \[\leadsto \left(\frac{\sqrt[3]{\frac{2}{\tan k}}}{\sin k} \cdot \frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{k}{\ell}}\right) \cdot \left(\frac{\frac{\sqrt{\sqrt[3]{2}}}{\sqrt[3]{\frac{k}{\ell}}}}{\sqrt[3]{\tan k}} \cdot \frac{\frac{\sqrt{\sqrt[3]{2}}}{\sqrt[3]{\frac{k}{\ell}} \cdot \sqrt[3]{\frac{k}{\ell}}}}{t}\right)\]

Runtime

Time bar (total: 3.9m)Debug logProfile

herbie shell --seed 2018252 
(FPCore (t l k)
  :name "Toniolo and Linder, Equation (10-)"
  (/ 2 (* (* (* (/ (pow t 3) (* l l)) (sin k)) (tan k)) (- (+ 1 (pow (/ k t) 2)) 1))))