Average Error: 47.1 → 2.1
Time: 6.2m
Precision: 64
Internal Precision: 4416
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;k \le -1.5587762695504066 \cdot 10^{-145} \lor \neg \left(k \le 2.3556239302553756 \cdot 10^{-66}\right):\\ \;\;\;\;\left(\left(\sqrt[3]{\frac{2}{\tan k}} \cdot \frac{\sqrt[3]{\frac{2}{\tan k}}}{\sin k}\right) \cdot \frac{\ell}{k}\right) \cdot \left(\frac{\ell \cdot \frac{1}{k}}{t} \cdot \sqrt[3]{\frac{2}{\tan k}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\sqrt[3]{2} \cdot \sqrt[3]{\frac{2}{\tan k}}\right) \cdot \left(\sqrt[3]{\frac{2}{\tan k}} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)\right)}{\left(\sqrt[3]{\tan k} \cdot \sin k\right) \cdot t}\\ \end{array}\]

Error

Bits error versus t

Bits error versus l

Bits error versus k

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if k < -1.5587762695504066e-145 or 2.3556239302553756e-66 < k

    1. Initial program 45.8

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

    2. Initial simplification28.3

      \[\leadsto \frac{\frac{\frac{2}{\tan k}}{\frac{\sin k \cdot t}{\frac{\ell}{t} \cdot \frac{\ell}{t}}}}{\frac{k}{t} \cdot \frac{k}{t}}\]
    3. Using strategy rm

    4. Applied times-frac27.6

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

    5. Applied add-cube-cbrt27.7

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

    6. Applied times-frac27.4

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

    7. Applied times-frac15.9

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

    8. Simplified8.8

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

    10. Applied *-un-lft-identity8.8

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

    11. Applied associate-/r/8.8

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

    12. Applied times-frac8.8

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

    13. Simplified8.8

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

    14. Simplified5.1

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

    16. Applied associate-*r/1.1

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

    if -1.5587762695504066e-145 < k < 2.3556239302553756e-66

    1. Initial program 61.8

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

    2. Initial simplification54.6

      \[\leadsto \frac{\frac{\frac{2}{\tan k}}{\frac{\sin k \cdot t}{\frac{\ell}{t} \cdot \frac{\ell}{t}}}}{\frac{k}{t} \cdot \frac{k}{t}}\]
    3. Using strategy rm

    4. Applied times-frac50.6

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

    5. Applied add-cube-cbrt50.7

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

    6. Applied times-frac48.7

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

    7. Applied times-frac41.3

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

    8. Simplified32.6

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

    10. Applied *-un-lft-identity32.6

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

    11. Applied associate-/r/34.4

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

    12. Applied times-frac34.4

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

    13. Simplified34.4

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

    14. Simplified31.3

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

    16. Applied associate-*r/21.0

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

    17. Applied associate-*l/19.3

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

    18. Applied cbrt-div19.5

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

    19. Applied frac-times19.5

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

    20. Applied associate-*l/14.2

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

    21. Applied frac-times10.7

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

    22. Simplified12.7

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

  4. Final simplification2.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;k \le -1.5587762695504066 \cdot 10^{-145} \lor \neg \left(k \le 2.3556239302553756 \cdot 10^{-66}\right):\\ \;\;\;\;\left(\left(\sqrt[3]{\frac{2}{\tan k}} \cdot \frac{\sqrt[3]{\frac{2}{\tan k}}}{\sin k}\right) \cdot \frac{\ell}{k}\right) \cdot \left(\frac{\ell \cdot \frac{1}{k}}{t} \cdot \sqrt[3]{\frac{2}{\tan k}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\sqrt[3]{2} \cdot \sqrt[3]{\frac{2}{\tan k}}\right) \cdot \left(\sqrt[3]{\frac{2}{\tan k}} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)\right)}{\left(\sqrt[3]{\tan k} \cdot \sin k\right) \cdot t}\\ \end{array}\]

Runtime

Time bar (total: 6.2m)Debug logProfile

herbie shell --seed 2018251 
(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))))