Error

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 add-cube-cbrt47.0

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

4. Applied associate-*r*47.0

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

5. Applied simplify38.4

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

7. Applied add-sqr-sqrt38.4

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

8. Applied simplify38.4

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

9. Applied simplify27.2

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

11. Applied cbrt-prod27.2

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

Runtime

Time bar (total: 2.7m)Debug logProfile

herbie shell --seed '#(1070386091 2509006183 1430610344 1025408621 36622005 1425925650)' 
(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))))