Average Error: 47.2 → 6.5
Time: 5.2m
Precision: 64
Internal Precision: 128
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\ell \cdot \ell \le 9.301648993723362 \cdot 10^{-98}:\\ \;\;\;\;\left(\left(\sqrt[3]{\frac{2}{\tan k}} \cdot \left(\frac{1}{t} \cdot \left(\ell \cdot \frac{1}{k}\right)\right)\right) \cdot \frac{\frac{\sqrt[3]{\frac{2}{\tan k}}}{\sqrt[3]{\frac{\sin k}{\ell}} \cdot \sqrt[3]{t}}}{\frac{1}{t}}\right) \cdot \frac{\frac{\sqrt[3]{\frac{2}{\tan k}}}{\sqrt[3]{\frac{\sin k}{\frac{\ell}{t}}} \cdot \sqrt[3]{\frac{\sin k}{\frac{\ell}{t}}}}}{k}\\ \mathbf{elif}\;\ell \cdot \ell \le 1.5922660219299978 \cdot 10^{+261}:\\ \;\;\;\;\frac{\frac{\frac{2}{k}}{\sin k}}{\tan k} \cdot \frac{\ell \cdot \ell}{k \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{k}{\ell}} \cdot \left(\left(\sqrt[3]{\frac{2}{\tan k}} \cdot \left(\frac{\ell}{t} \cdot \frac{1}{k}\right)\right) \cdot \frac{\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{\sin k}{\frac{1}{t}}}}{\frac{1}{t}}\right)\\ \end{array}\]

Error

Bits error versus t

Bits error versus l

Bits error versus k

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if (* l l) < 9.301648993723362e-98

    1. Initial program 44.3

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

    2. Initial simplification26.1

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

    4. Applied times-frac25.0

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

    5. Applied add-cube-cbrt25.0

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

    6. Applied times-frac24.5

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

    7. Applied times-frac14.9

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

    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 div-inv8.8

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

    11. Applied add-cube-cbrt8.9

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

    12. Applied times-frac8.9

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

    13. Applied times-frac6.5

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

    14. Applied associate-*l*6.3

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

    16. Applied div-inv6.3

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

    17. Applied associate-*r*6.3

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

    19. Applied associate-/r/6.3

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

    20. Applied cbrt-prod6.3

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

    if 9.301648993723362e-98 < (* l l) < 1.5922660219299978e+261

    1. Initial program 44.9

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

    2. Initial simplification34.3

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

    4. Applied associate-/r/34.2

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

    5. Applied times-frac22.4

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

    6. Simplified22.4

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

    7. Simplified3.4

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

    if 1.5922660219299978e+261 < (* l l)

    1. Initial program 59.8

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

    2. Initial simplification38.2

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

    4. Applied times-frac36.5

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

    5. Applied add-cube-cbrt36.7

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

    6. Applied times-frac36.5

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

    7. Applied times-frac21.6

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

    8. Simplified14.1

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

    10. Applied div-inv14.1

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

    11. Applied div-inv14.1

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

    12. Applied *-un-lft-identity14.1

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

    13. Applied times-frac14.1

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

    14. Applied times-frac14.1

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

    15. Applied times-frac11.8

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

    16. Applied associate-*l*11.6

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

    17. Simplified11.6

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

  4. Final simplification6.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \le 9.301648993723362 \cdot 10^{-98}:\\ \;\;\;\;\left(\left(\sqrt[3]{\frac{2}{\tan k}} \cdot \left(\frac{1}{t} \cdot \left(\ell \cdot \frac{1}{k}\right)\right)\right) \cdot \frac{\frac{\sqrt[3]{\frac{2}{\tan k}}}{\sqrt[3]{\frac{\sin k}{\ell}} \cdot \sqrt[3]{t}}}{\frac{1}{t}}\right) \cdot \frac{\frac{\sqrt[3]{\frac{2}{\tan k}}}{\sqrt[3]{\frac{\sin k}{\frac{\ell}{t}}} \cdot \sqrt[3]{\frac{\sin k}{\frac{\ell}{t}}}}}{k}\\ \mathbf{elif}\;\ell \cdot \ell \le 1.5922660219299978 \cdot 10^{+261}:\\ \;\;\;\;\frac{\frac{\frac{2}{k}}{\sin k}}{\tan k} \cdot \frac{\ell \cdot \ell}{k \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{k}{\ell}} \cdot \left(\left(\sqrt[3]{\frac{2}{\tan k}} \cdot \left(\frac{\ell}{t} \cdot \frac{1}{k}\right)\right) \cdot \frac{\frac{\sqrt[3]{\frac{2}{\tan k}}}{\frac{\sin k}{\frac{1}{t}}}}{\frac{1}{t}}\right)\\ \end{array}\]

Runtime

Time bar (total: 5.2m)Debug logProfile

herbie shell --seed 2018277 
(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))))