Average Error: 47.0 → 2.6
Time: 3.6m
Precision: 64
Internal Precision: 4224
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\frac{\frac{\sqrt[3]{\frac{2}{\sin k}} \cdot \sqrt[3]{\frac{2}{\sin k}}}{\frac{\frac{\sqrt[3]{\tan k} \cdot \sqrt[3]{\tan k}}{\sqrt[3]{\frac{\ell}{k}} \cdot \sqrt[3]{\frac{\ell}{k}}}}{1}}}{\frac{t}{\frac{\sqrt[3]{\frac{2}{\sin k}}}{\frac{\frac{\sqrt[3]{\tan k}}{\sqrt[3]{\frac{\ell}{k}}}}{\frac{\ell}{k}}}}}\]

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

  1. Initial program 47.0

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

  3. Applied associate-*l/47.1

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

  4. Applied associate-*l/47.1

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

  5. Applied associate-*l/47.1

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

  6. Applied simplify39.6

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

  8. Applied associate-/l*39.3

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

  9. Applied simplify10.4

    \[\leadsto \frac{2}{\frac{\tan k \cdot \sin k}{\color{blue}{\frac{\frac{\ell}{k}}{1 \cdot k} \cdot \frac{\ell}{t}}}}\]
  10. Using strategy rm

  11. Applied associate-*r/11.3

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

  12. Applied associate-/r/11.3

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

  13. Applied associate-/r*11.2

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

  14. Applied simplify8.0

    \[\leadsto \frac{\color{blue}{\frac{\frac{2}{\sin k}}{\frac{\frac{\tan k}{\frac{\ell}{k}}}{\frac{\ell}{k}}}}}{t}\]
  15. Using strategy rm

  16. Applied *-un-lft-identity8.0

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

  17. Applied add-cube-cbrt8.3

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

  18. Applied add-cube-cbrt8.4

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

  19. Applied times-frac8.4

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

  20. Applied times-frac8.4

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

  21. Applied add-cube-cbrt8.5

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

  22. Applied times-frac8.0

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

  23. Applied associate-/l*2.6

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

Runtime

Time bar (total: 3.6m)Debug log

herbie shell --seed '#(2479486159 2123901208 2662424940 349789437 14252662 202027171)' 
(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))))