Average Error: 48.2 → 1.9
Time: 7.1m
Precision: 64
Internal Precision: 4160
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\left(\frac{\frac{\sqrt[3]{\sqrt[3]{\frac{2}{\tan k}}}}{\sin k}}{\frac{\frac{k}{\ell}}{\sqrt[3]{\sqrt[3]{\frac{2}{\tan k}}}}} \cdot \frac{\sqrt[3]{\sqrt[3]{\frac{2}{\tan k}}}}{\frac{k}{\ell} \cdot t}\right) \cdot \left(\sqrt[3]{\frac{2}{\tan k}} \cdot \sqrt[3]{\frac{2}{\tan k}}\right)\]

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 48.2

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

  2. Initial simplification31.1

    \[\leadsto \frac{\frac{\frac{2}{\tan k}}{\frac{t \cdot \sin k}{\frac{\ell}{t} \cdot \frac{\ell}{t}}}}{\frac{k}{t} \cdot \frac{k}{t} + 0}\]
  3. Using strategy rm

  4. Applied *-un-lft-identity31.1

    \[\leadsto \frac{\frac{\frac{2}{\tan k}}{\frac{t \cdot \sin k}{\frac{\ell}{t} \cdot \frac{\ell}{t}}}}{\color{blue}{1 \cdot \left(\frac{k}{t} \cdot \frac{k}{t} + 0\right)}}\]

  5. Applied *-un-lft-identity31.1

    \[\leadsto \frac{\frac{\frac{2}{\tan k}}{\color{blue}{1 \cdot \frac{t \cdot \sin k}{\frac{\ell}{t} \cdot \frac{\ell}{t}}}}}{1 \cdot \left(\frac{k}{t} \cdot \frac{k}{t} + 0\right)}\]

  6. Applied add-cube-cbrt31.2

    \[\leadsto \frac{\frac{\color{blue}{\left(\sqrt[3]{\frac{2}{\tan k}} \cdot \sqrt[3]{\frac{2}{\tan k}}\right) \cdot \sqrt[3]{\frac{2}{\tan k}}}}{1 \cdot \frac{t \cdot \sin k}{\frac{\ell}{t} \cdot \frac{\ell}{t}}}}{1 \cdot \left(\frac{k}{t} \cdot \frac{k}{t} + 0\right)}\]

  7. Applied times-frac31.2

    \[\leadsto \frac{\color{blue}{\frac{\sqrt[3]{\frac{2}{\tan k}} \cdot \sqrt[3]{\frac{2}{\tan k}}}{1} \cdot \frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t \cdot \sin k}{\frac{\ell}{t} \cdot \frac{\ell}{t}}}}}{1 \cdot \left(\frac{k}{t} \cdot \frac{k}{t} + 0\right)}\]

  8. Applied times-frac31.1

    \[\leadsto \color{blue}{\frac{\frac{\sqrt[3]{\frac{2}{\tan k}} \cdot \sqrt[3]{\frac{2}{\tan k}}}{1}}{1} \cdot \frac{\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t \cdot \sin k}{\frac{\ell}{t} \cdot \frac{\ell}{t}}}}{\frac{k}{t} \cdot \frac{k}{t} + 0}}\]

  9. Simplified31.1

    \[\leadsto \color{blue}{\left(\sqrt[3]{\frac{2}{\tan k}} \cdot \sqrt[3]{\frac{2}{\tan k}}\right)} \cdot \frac{\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{t \cdot \sin k}{\frac{\ell}{t} \cdot \frac{\ell}{t}}}}{\frac{k}{t} \cdot \frac{k}{t} + 0}\]

  10. Simplified15.4

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

  12. Applied add-cube-cbrt15.5

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

  13. Applied times-frac15.5

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

  14. Applied times-frac11.2

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

  15. Simplified11.2

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

  16. Simplified1.9

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

  17. Final simplification1.9

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

Runtime

Time bar (total: 7.1m)Debug logProfile

herbie shell --seed 2018215 
(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))))