Average Error: 47.4 → 1.1
Time: 5.0m
Precision: 64
Internal Precision: 128
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\frac{\frac{2}{\frac{k}{\ell} \cdot \tan k}}{\sin k \cdot \frac{t}{\frac{\ell}{k}}}\]

Error

Bits error versus t

Bits error versus l

Bits error versus k

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 47.4

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

  2. Initial simplification30.4

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

  4. Applied times-frac29.3

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

  5. Applied add-cube-cbrt29.4

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

  6. Applied times-frac29.0

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

  7. Applied times-frac18.4

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

  8. Simplified10.9

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

  10. Applied div-inv10.9

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

  11. Applied div-inv10.9

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

  12. Applied times-frac7.0

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

  13. Simplified6.9

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

  15. Applied associate-*r/6.9

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

  16. Applied associate-*l/6.8

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

  17. Simplified1.7

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

  19. Applied associate-*l/1.7

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

  20. Applied associate-/l/1.5

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

  21. Simplified1.1

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

  22. Final simplification1.1

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

Runtime

Time bar (total: 5.0m)Debug logProfile

herbie shell --seed 2018285 
(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))))