Error

Derivation

1. Split input into 4 regimes.

2. if l < -5.237492387957813e+116

   1. Initial program 56.3

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

   3. Applied add-cube-cbrt 56.3

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

   4. Applied add-cube-cbrt 56.3

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

   5. Applied add-cube-cbrt 56.3

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

   6. Applied cube-unprod 56.3

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

   7. Applied cube-unprod 56.3

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

   8. Applied simplify 26.9

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

   10. Applied cbrt-unprod 26.9

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

   11. Applied cbrt-unprod 26.7

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

   if -5.237492387957813e+116 < l < -1.4952202440978445e-167 or 9.352918442096625e-212 < l < 3.683028962283538e+81

   1. Initial program 24.9

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

   3. Applied add-cube-cbrt 25.0

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

   4. Applied add-cube-cbrt 25.0

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

   5. Applied add-cube-cbrt 25.1

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

   6. Applied add-cube-cbrt 25.1

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

   7. Applied add-cube-cbrt 25.1

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

   8. Applied cube-undiv 25.1

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

   9. Applied cube-unprod 24.0

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

   10. Applied cube-unprod 23.6

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

   11. Applied cube-unprod 23.3

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

   12. Applied simplify 13.0

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

   if -1.4952202440978445e-167 < l < 9.352918442096625e-212

   1. Initial program 26.3

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

   3. Applied add-cube-cbrt 26.3

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

   4. Applied add-cube-cbrt 26.3

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

   5. Applied add-cube-cbrt 26.3

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

   6. Applied cube-unprod 26.3

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

   7. Applied cube-unprod 26.3

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

   8. Applied simplify 12.7

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

   10. Applied square-mult 12.7

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

   11. Applied times-frac 7.7

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

   12. Applied cbrt-prod 5.6

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

   if 3.683028962283538e+81 < l

   1. Initial program 54.2

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

   3. Applied add-cube-cbrt 54.3

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

   4. Applied add-cube-cbrt 54.3

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

   5. Applied add-cube-cbrt 54.3

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

   6. Applied cube-unprod 54.3

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

   7. Applied cube-unprod 54.3

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

   8. Applied simplify 26.8

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

   10. Applied div-inv 26.8

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

   11. Applied cbrt-prod 25.3

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

   13. Applied add-sqr-sqrt 25.3

       \[\leadsto \frac{2}{{\left(\sqrt[3]{2 + {\left(\frac{k}{t}\right)}^2} \cdot \left(\sqrt[3]{\tan k} \cdot \left(\sqrt[3]{\sin k \cdot t} \cdot \color{blue}{{\left(\sqrt{\sqrt[3]{\frac{1}{{\left(\frac{\ell}{t}\right)}^2}}}\right)}^2}\right)\right)\right)}^3}\]
3. Recombined 4 regimes into one program.
4. Removed slow pow expressions

Runtime

Time bar (total: 42.3s) Debug logProfile

Please include this information when filing a bug report:

herbie shell --seed '#(1064524629 4159152179 2999149171 575749698 4006532819 692958815)'
(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))))