Average Error: 31.1 → 11.2
Time: 3.8m
Precision: 64
Internal Precision: 576
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
↓
\[\left(\left(\frac{\sqrt[3]{\sqrt[3]{\frac{2}{t}} \cdot \sqrt[3]{\frac{2}{t}}}}{\sin k} \cdot \left(\frac{\cos k}{\frac{k}{t} \cdot \frac{k}{t} + 2} \cdot \frac{\sqrt[3]{\sqrt[3]{\frac{2}{t}}}}{\frac{1}{\frac{\ell}{t}}}\right)\right) \cdot \frac{\sqrt[3]{\frac{2}{t}}}{\sqrt[3]{\frac{\sin k}{\frac{\ell}{t}}}}\right) \cdot \frac{\sqrt[3]{\frac{2}{t}}}{\sqrt[3]{\frac{\sin k}{\frac{\ell}{t}}} \cdot \sqrt[3]{\frac{\sin k}{\frac{\ell}{t}}}}\]

Error

Bits error versus t

Bits error versus l

Bits error versus k

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Derivation

  1. Initial program 31.1

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

  2. Initial simplification24.3

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

  4. Applied *-un-lft-identity24.3

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

  5. Applied tan-quot24.3

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

  6. Applied associate-*r/24.3

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

  7. Applied associate-/r/24.3

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

  8. Applied times-frac24.3

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

  9. Simplified15.5

    \[\leadsto \color{blue}{\frac{\frac{2}{t}}{\frac{\sin k}{\frac{\ell}{t}} \cdot \frac{\sin k}{\frac{\ell}{t}}}} \cdot \frac{\cos k}{\frac{k}{t} \cdot \frac{k}{t} + 2}\]
  10. Using strategy rm

  11. Applied add-cube-cbrt15.6

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

  12. Applied times-frac14.0

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

  13. Applied associate-*l*11.4

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

  15. Applied add-cube-cbrt11.5

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

  16. Applied times-frac11.5

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

  17. Applied associate-*l*11.1

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

  19. Applied div-inv11.1

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

  20. Applied add-cube-cbrt11.1

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

  21. Applied cbrt-prod11.2

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

  22. Applied times-frac11.2

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

  23. Applied associate-*l*11.2

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

  24. Final simplification11.2

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

Runtime

Time bar (total: 3.8m)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))))