Average Error: 46.6 → 2.8
Time: 3.6m
Precision: 64
Internal Precision: 4160
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\left(\left(\frac{\sqrt[3]{\cos k} \cdot \sqrt[3]{\frac{2}{\sin k}}}{\sin k} \cdot \sqrt[3]{\frac{2}{\tan k}}\right) \cdot \frac{\ell}{k}\right) \cdot \left(\frac{\ell \cdot \frac{1}{k}}{t} \cdot \sqrt[3]{\frac{2}{\tan k}}\right)\]

Error

Bits error versus t

Bits error versus l

Bits error versus k

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Results

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Derivation

  1. Initial program 46.6

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

  2. Initial simplification30.1

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

  4. Applied times-frac29.2

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

  5. Applied add-cube-cbrt29.2

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

  6. Applied times-frac28.8

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

  7. Applied times-frac18.1

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

  8. Simplified10.8

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

  10. Applied *-un-lft-identity10.8

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

  11. Applied associate-/r/11.0

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

  12. Applied times-frac11.0

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

  13. Simplified11.0

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

  14. Simplified7.1

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

  16. Applied associate-*r/2.8

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

  18. Applied tan-quot2.8

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

  19. Applied associate-/r/2.8

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

  20. Applied cbrt-prod2.8

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

  21. Final simplification2.8

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

Runtime

Time bar (total: 3.6m)Debug logProfile

herbie shell --seed 2018235 
(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))))