Average Error: 47.3 → 8.5
Time: 7.7m
Precision: 64
Internal Precision: 4416
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\ell \le -2.2161289926066393 \cdot 10^{-132} \lor \neg \left(\ell \le 4.3054598034245887 \cdot 10^{-274}\right):\\ \;\;\;\;\frac{2}{\frac{\sin k \cdot \sin k}{\frac{\cos k \cdot \ell}{\frac{k}{\ell} \cdot \left(t \cdot k\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(\left(\frac{t \cdot k}{\frac{\ell}{k}} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \sin k}{t \cdot \cos k}}\\ \end{array}\]

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 l < -2.2161289926066393e-132 or 4.3054598034245887e-274 < l

    1. Initial program 48.2

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

    3. Applied add-cube-cbrt48.2

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

    4. Applied simplify48.2

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

    5. Applied simplify41.3

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

    7. Applied associate-*l/41.3

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

    8. Applied cbrt-div41.3

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

    9. Applied associate-*l/41.3

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

    10. Applied cbrt-div41.4

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

    11. Applied associate-*r/41.4

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

    12. Applied cbrt-div41.4

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

    13. Applied frac-times41.4

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

    14. Applied frac-times41.4

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

    15. Applied tan-quot41.4

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

    16. Applied associate-*r/41.4

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

    17. Applied frac-times39.0

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

    18. Applied simplify27.4

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

    19. Applied simplify27.3

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

    21. Applied associate-*l*23.5

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

    22. Applied simplify17.3

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

    24. Applied associate-/l*16.6

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

    25. Applied simplify7.9

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

    if -2.2161289926066393e-132 < l < 4.3054598034245887e-274

    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. Using strategy rm

    3. Applied add-cube-cbrt44.3

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

    4. Applied simplify44.3

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

    5. Applied simplify34.8

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

    7. Applied associate-*l/34.8

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

    8. Applied cbrt-div34.8

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

    9. Applied associate-*l/34.8

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

    10. Applied cbrt-div34.8

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

    11. Applied associate-*r/34.8

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

    12. Applied cbrt-div34.8

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

    13. Applied frac-times34.8

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

    14. Applied frac-times34.8

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

    15. Applied tan-quot34.8

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

    16. Applied associate-*r/34.8

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

    17. Applied frac-times30.4

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

    18. Applied simplify14.9

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

    19. Applied simplify14.9

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

    21. Applied associate-*l*12.7

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

    22. Applied simplify11.2

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

    24. Applied associate-*l*10.5

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

  4. Applied simplify8.5

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;\ell \le -2.2161289926066393 \cdot 10^{-132} \lor \neg \left(\ell \le 4.3054598034245887 \cdot 10^{-274}\right):\\ \;\;\;\;\frac{2}{\frac{\sin k \cdot \sin k}{\frac{\cos k \cdot \ell}{\frac{k}{\ell} \cdot \left(t \cdot k\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(\left(\frac{t \cdot k}{\frac{\ell}{k}} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \sin k}{t \cdot \cos k}}\\ \end{array}}\]

Runtime

Time bar (total: 7.7m)Debug logProfile

herbie shell --seed 2018167 
(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))))