Average Error: 46.8 → 1.6
Time: 6.3m
Precision: 64
Internal Precision: 4416
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\frac{\frac{\frac{\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{\sqrt[3]{\sin k}}{\sqrt[3]{2} \cdot \ell}}}{\sqrt[3]{\sin k} \cdot k}}{\frac{t}{\sqrt[3]{\frac{2}{\tan k}} \cdot \frac{\ell}{k}}}}{\sqrt[3]{\sin k} \cdot \sqrt[3]{\tan k}}\]

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 46.8

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

  2. Initial simplification30.9

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

  4. Applied times-frac30.0

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

  5. Applied add-cube-cbrt30.1

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

  6. Applied times-frac29.7

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

  7. Applied times-frac18.7

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

  8. Simplified11.4

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

  10. Applied div-inv11.4

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

  11. Applied *-un-lft-identity11.4

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

  12. Applied add-cube-cbrt11.5

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

  13. Applied times-frac11.5

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

  14. Applied times-frac11.5

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

  15. Applied times-frac7.9

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

  16. Simplified7.9

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

  17. Simplified7.9

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

  19. Applied associate-*r/3.6

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

  21. Applied cbrt-div3.6

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

  22. Applied associate-*l/3.6

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

  23. Applied frac-times2.9

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

  24. Applied associate-*l/2.6

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

  25. Simplified1.6

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

  26. Final simplification1.6

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

Runtime

Time bar (total: 6.3m)Debug logProfile

herbie shell --seed 2018248 
(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))))