Error

Derivation

1. Initial program 32.2

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

2. Applied simplify 31.8

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

4. Applied add-cube-cbrt 31.9

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

5. Applied add-cube-cbrt 31.9

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

6. Applied add-cube-cbrt 31.9

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

7. Applied cube-undiv 31.9

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

8. Applied add-cube-cbrt 31.9

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

9. Applied cube-unprod 30.9

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

10. Applied cube-undiv 23.8

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

11. Applied cube-unprod 22.0

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

13. Applied *-un-lft-identity 22.0

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

14. Applied cbrt-prod 22.0

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

15. Applied cbrt-prod 11.8

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

16. Applied times-frac 11.8

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

17. Applied times-frac 11.7

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

18. Applied associate-*l* 11.8

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

20. Applied associate-*l/ 11.8

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

21. Applied associate-*r/ 11.8

    \[\leadsto \frac{2}{{\color{blue}{\left(\frac{\frac{t}{\frac{\sqrt[3]{\ell}}{\sqrt[3]{1}}} \cdot \left(\sqrt[3]{\sin k} \cdot \sqrt[3]{2 + {\left(\frac{k}{t}\right)}^2}\right)}{\frac{\sqrt[3]{\ell}}{\sqrt[3]{\tan k}}}\right)}}^3}\]
22. Removed slow pow expressions

Runtime

Time bar (total: 3.0m) Debug log

Please include this information when filing a bug report:

herbie --seed '#(225978912 3065894405 366316365 1627425336 2243770834 1419592875)'
(FPCore (t l k)
  :name "Toniolo and Linder, Equation (10+)"
  (/ 2 (* (* (* (/ (pow t 3) (sqr l)) (sin k)) (tan k)) (+ (+ 1 (sqr (/ k t))) 1))))