Average Error: 47.0 → 3.2
Time: 9.0m
Precision: 64
Internal Precision: 4224
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\frac{\left(t \cdot k\right) \cdot \sin k}{\frac{1}{k} \cdot \ell} \cdot \sin k}{\ell \cdot \cos k} = -\infty:\\ \;\;\;\;\frac{2}{\left(\frac{k}{\ell} \cdot k\right) \cdot \frac{\frac{\sin k \cdot t}{\frac{\ell}{\sin k}}}{\cos k}}\\ \mathbf{if}\;\frac{\frac{\left(t \cdot k\right) \cdot \sin k}{\frac{1}{k} \cdot \ell} \cdot \sin k}{\ell \cdot \cos k} \le -2.9634018502883525 \cdot 10^{-248}:\\ \;\;\;\;\frac{2}{\frac{\frac{\left(t \cdot k\right) \cdot \sin k}{\frac{1}{k} \cdot \ell} \cdot \sin k}{\ell \cdot \cos k}}\\ \mathbf{if}\;\frac{\frac{\left(t \cdot k\right) \cdot \sin k}{\frac{1}{k} \cdot \ell} \cdot \sin k}{\ell \cdot \cos k} \le 3.6667675088448644 \cdot 10^{-298}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{\ell} \cdot k\right) \cdot \frac{\frac{\sin k \cdot t}{\frac{\ell}{\sin k}}}{\cos k}}\\ \mathbf{if}\;\frac{\frac{\left(t \cdot k\right) \cdot \sin k}{\frac{1}{k} \cdot \ell} \cdot \sin k}{\ell \cdot \cos k} \le 5.149440211475309 \cdot 10^{+226}:\\ \;\;\;\;\frac{2}{\frac{\frac{\left(t \cdot k\right) \cdot \sin k}{\frac{1}{k} \cdot \ell} \cdot \sin k}{\ell \cdot \cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{\ell} \cdot k\right) \cdot \frac{\frac{\sin k \cdot t}{\frac{\ell}{\sin k}}}{\cos k}}\\ \end{array}\]

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

  1. Split input into 2 regimes

  2. if (/ (* (/ (* (* t k) (sin k)) (* (/ 1 k) l)) (sin k)) (* l (cos k))) or -2.9634018502883525e-248 < (/ (* (/ (* (* t k) (sin k)) (* (/ 1 k) l)) (sin k)) (* l (cos k))) < 3.6667675088448644e-298 or 5.149440211475309e+226 < (/ (* (/ (* (* t k) (sin k)) (* (/ 1 k) l)) (sin k)) (* l (cos k)))

    1. Initial program 40.9

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

    3. Applied add-cube-cbrt40.9

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

    4. Applied times-frac40.2

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

    5. Applied simplify40.2

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

    6. Applied simplify31.5

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

    8. Applied add-cube-cbrt31.5

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

    9. Applied simplify31.5

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

    10. Applied simplify19.7

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

    12. Applied associate-*l/20.7

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

    13. Applied cbrt-div20.6

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

    14. Applied associate-*r/20.6

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

    15. Applied cbrt-div20.6

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

    16. Applied associate-*r/20.6

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

    17. Applied cbrt-div20.7

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

    18. Applied frac-times20.7

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

    19. Applied frac-times20.7

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

    20. Applied tan-quot20.7

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

    21. Applied associate-*l/20.7

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

    22. Applied associate-*l/20.4

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

    23. Applied frac-times20.6

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

    24. Applied frac-times16.6

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

    25. Applied simplify14.3

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

    26. Applied simplify14.1

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

    28. Applied times-frac12.4

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

    29. Applied simplify4.5

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

    if (/ (* (/ (* (* t k) (sin k)) (* (/ 1 k) l)) (sin k)) (* l (cos k))) < -2.9634018502883525e-248 or 3.6667675088448644e-298 < (/ (* (/ (* (* t k) (sin k)) (* (/ 1 k) l)) (sin k)) (* l (cos k))) < 5.149440211475309e+226

    1. Initial program 56.3

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

    3. Applied add-cube-cbrt56.4

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

    4. Applied times-frac54.5

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

    5. Applied simplify54.4

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

    6. Applied simplify52.0

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

    8. Applied add-cube-cbrt52.0

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

    9. Applied simplify52.0

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

    10. Applied simplify45.7

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

    12. Applied associate-*l/45.9

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

    13. Applied cbrt-div45.9

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

    14. Applied associate-*r/45.9

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

    15. Applied cbrt-div46.0

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

    16. Applied associate-*r/46.0

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

    17. Applied cbrt-div46.1

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

    18. Applied frac-times46.0

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

    19. Applied frac-times46.0

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

    20. Applied tan-quot46.0

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

    21. Applied associate-*l/46.0

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

    22. Applied associate-*l/45.4

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

    23. Applied frac-times44.8

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

    24. Applied frac-times36.6

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

    25. Applied simplify25.9

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

    26. Applied simplify23.7

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

    28. Applied associate-/r/23.8

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

    29. Applied associate-*r*23.8

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

    30. Applied simplify1.0

      \[\leadsto \frac{2}{\frac{\color{blue}{\frac{\left(t \cdot k\right) \cdot \sin k}{\frac{1}{k} \cdot \ell}} \cdot \sin k}{\ell \cdot \cos k}}\]
  3. Recombined 2 regimes into one program.

Runtime

Time bar (total: 9.0m)Debug logProfile

herbie shell --seed '#(1070355188 2193211668 3977393919 3454156579 3755371326 1656365382)' 
(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))))