Average Error: 47.3 → 4.3
Time: 3.3m
Precision: 64
Internal Precision: 4224
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\frac{2}{\frac{\frac{k}{\ell} \cdot t}{\frac{1}{\frac{k}{\ell}}} \cdot \frac{\sin k \cdot \sin k}{\cos k}}\]

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

  1. Initial program 47.3

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

  3. Applied add-cube-cbrt47.3

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

  4. Applied simplify47.3

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

  5. Applied simplify39.8

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

  7. Applied associate-*r/39.9

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

  8. Applied cbrt-div39.9

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

  9. Applied associate-*l/39.9

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

  10. Applied cbrt-div39.9

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

  11. Applied associate-*l/39.9

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

  12. Applied cbrt-div39.9

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

  13. Applied frac-times39.9

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

  14. Applied frac-times39.9

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

  15. Applied tan-quot39.9

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

  16. Applied associate-*r/39.9

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

  17. Applied frac-times37.5

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

  18. Applied simplify24.7

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

  19. Applied simplify24.6

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

  21. Applied add-sqr-sqrt24.6

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

  22. Applied simplify24.6

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

  23. Applied simplify12.0

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

  25. Applied times-frac11.2

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

  26. Applied simplify4.3

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

Runtime

Time bar (total: 3.3m)Debug logProfile

herbie shell --seed '#(1064269945 2896236262 301053905 1701069080 1701464310 1614783279)' 
(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))))