Average Error: 47.2 → 1.1
Time: 5.5m
Precision: 64
Internal Precision: 4160
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\frac{\left(\left(\sqrt{\sqrt[3]{2}} \cdot \sqrt{\sqrt[3]{2}}\right) \cdot \frac{\ell}{k}\right) \cdot \sqrt[3]{\frac{2}{\tan k}}}{\left(t \cdot \sqrt[3]{\tan k}\right) \cdot \left(\frac{\frac{\sin k}{\ell}}{\sqrt[3]{\frac{2}{\tan k}}} \cdot k\right)}\]

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 47.2

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

  2. Initial simplification30.3

    \[\leadsto \frac{\frac{\frac{2}{\tan k}}{\frac{\sin k \cdot t}{\frac{\ell}{t} \cdot \frac{\ell}{t}}}}{\frac{k}{t} \cdot \frac{k}{t}}\]
  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-frac18.3

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

  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 *-un-lft-identity11.0

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

  12. Applied times-frac7.2

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

  13. Simplified7.1

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

  15. Applied cbrt-div7.2

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

  16. Applied associate-*r/2.2

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

  17. Applied frac-times1.9

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

  18. Applied frac-times1.9

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

  19. Applied frac-times1.2

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

  20. Simplified1.2

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

  22. Applied add-sqr-sqrt1.1

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

  23. Final simplification1.1

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

Runtime

Time bar (total: 5.5m)Debug logProfile

herbie shell --seed 2018256 
(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))))