Average Error: 47.0 → 7.4
Time: 4.3m
Precision: 64
Internal Precision: 4416
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\left(\frac{\left(\sqrt[3]{\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{\sqrt[3]{\sin k}}{\sqrt[3]{\frac{\ell}{t}}}}} \cdot \sqrt[3]{\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{\sqrt[3]{\sin k} \cdot \sqrt[3]{\sin k}}{\sqrt[3]{\frac{\ell}{t}} \cdot \sqrt[3]{\frac{\ell}{t}}}}}\right) \cdot \left(\sqrt[3]{\frac{\sqrt[3]{\frac{2}{\tan k}}}{t}} \cdot \sqrt[3]{\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{\sin k}{\ell}}}\right)}{k} \cdot \left(\sqrt[3]{\left(\frac{\ell}{\sin k} \cdot \sqrt[3]{\frac{2}{\tan k}}\right) \cdot \frac{\sqrt[3]{\frac{2}{\tan k}}}{t}} \cdot t\right)\right) \cdot \left(\sqrt[3]{\frac{2}{\tan k}} \cdot \left(\frac{1}{k} \cdot \frac{\ell}{t}\right)\right)\]

Error

Bits error versus t

Bits error versus l

Bits error versus k

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Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 47.0

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

  2. Initial simplification30.4

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

  4. Applied times-frac29.5

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

  5. Applied add-cube-cbrt29.6

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

  6. Applied times-frac29.2

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

  7. Applied times-frac18.2

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

  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 add-cube-cbrt11.1

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

  12. Applied times-frac7.1

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

  13. Simplified7.3

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

  15. Applied add-cube-cbrt7.4

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

  16. Applied add-cube-cbrt7.4

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

  17. Applied times-frac7.4

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

  18. Applied times-frac7.4

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

  19. Applied cbrt-prod7.4

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

  21. Applied associate-/r/7.4

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

  22. Applied times-frac7.4

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

  23. Applied cbrt-prod7.4

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

  24. Final simplification7.4

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

Runtime

Time bar (total: 4.3m)Debug logProfile

herbie shell --seed 2018254 
(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))))