Average Error: 31.8 → 11.4
Time: 3.7m
Precision: 64
Internal Precision: 576
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\ell \cdot \ell \le 1.7591987776287867 \cdot 10^{+187}:\\ \;\;\;\;\frac{\sqrt[3]{\frac{2}{t}} \cdot \sqrt[3]{\frac{2}{t}}}{\frac{\sin k}{\frac{\ell}{t}}} \cdot \left(t \cdot \left(\frac{\cos k}{\left(k \cdot \frac{k}{t}\right) \cdot \left(1 + \left(1 - 1\right)\right) + t \cdot \left({1}^{3} + {1}^{3}\right)} \cdot \frac{\sqrt[3]{\frac{2}{t}}}{\frac{\sin k}{\frac{\ell}{t}}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t}}{\sin k} \cdot \left(\frac{\frac{2}{t}}{\frac{\sin k}{\frac{\ell}{t}}} \cdot \frac{\cos k}{\left(1 + 1\right) + \frac{k}{t} \cdot \frac{k}{t}}\right)\\ \end{array}\]

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 l) < 1.7591987776287867e+187

    1. Initial program 24.5

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

    2. Initial simplification22.6

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

    4. Applied *-un-lft-identity22.6

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

    5. Applied tan-quot22.6

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

    6. Applied associate-*r/22.6

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

    7. Applied associate-/r/22.6

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

    8. Applied times-frac22.6

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

    9. Simplified12.2

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

    11. Applied add-cube-cbrt12.3

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

    12. Applied times-frac12.0

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

    13. Applied associate-*l*9.5

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

    15. Applied flip3-+9.5

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

    16. Applied associate-*r/9.5

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

    17. Applied frac-add9.5

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

    18. Applied associate-/r/9.4

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

    19. Applied associate-*r*8.6

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

    20. Simplified8.6

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

    if 1.7591987776287867e+187 < (* l l)

    1. Initial program 53.0

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

    2. Initial simplification30.3

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

    4. Applied *-un-lft-identity30.3

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

    5. Applied tan-quot30.3

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

    6. Applied associate-*r/30.3

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

    7. Applied associate-/r/30.3

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

    8. Applied times-frac30.3

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

    9. Simplified26.8

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

    11. Applied *-un-lft-identity26.8

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

    12. Applied times-frac21.2

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

    13. Applied associate-*l*19.4

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

    14. Simplified19.4

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

  4. Final simplification11.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \le 1.7591987776287867 \cdot 10^{+187}:\\ \;\;\;\;\frac{\sqrt[3]{\frac{2}{t}} \cdot \sqrt[3]{\frac{2}{t}}}{\frac{\sin k}{\frac{\ell}{t}}} \cdot \left(t \cdot \left(\frac{\cos k}{\left(k \cdot \frac{k}{t}\right) \cdot \left(1 + \left(1 - 1\right)\right) + t \cdot \left({1}^{3} + {1}^{3}\right)} \cdot \frac{\sqrt[3]{\frac{2}{t}}}{\frac{\sin k}{\frac{\ell}{t}}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t}}{\sin k} \cdot \left(\frac{\frac{2}{t}}{\frac{\sin k}{\frac{\ell}{t}}} \cdot \frac{\cos k}{\left(1 + 1\right) + \frac{k}{t} \cdot \frac{k}{t}}\right)\\ \end{array}\]

Runtime

Time bar (total: 3.7m)Debug logProfile

herbie shell --seed 2018219 
(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))))