Average Error: 31.9 → 9.6
Time: 3.4m
Precision: 64
Internal Precision: 576
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;k \le 1.3155893364844926 \cdot 10^{+154}:\\ \;\;\;\;\frac{\frac{\frac{\frac{\ell}{t}}{\sin k} \cdot \left(\cos k \cdot 2\right)}{\frac{k \cdot k}{t} + t \cdot 2}}{\frac{\sin k}{\frac{\ell}{t}}}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{\cos k}{2 + \frac{k}{t} \cdot \frac{k}{t}} \cdot \frac{\ell}{t}\right) \cdot \frac{\frac{2}{t}}{\sin k}\right) \cdot \frac{\frac{\ell}{t}}{\sin k}\\ \end{array}\]

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 2 regimes

  2. if k < 1.3155893364844926e+154

    1. Initial program 31.7

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

    2. Initial simplification24.8

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

    4. Applied *-un-lft-identity24.8

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

    5. Applied tan-quot24.8

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

    6. Applied associate-*r/24.8

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

    7. Applied associate-/r/24.8

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

    8. Applied times-frac24.8

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

    9. Simplified14.6

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

    11. Applied *-un-lft-identity14.6

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

    12. Applied times-frac12.4

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

    13. Applied associate-*l*10.9

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

    14. Simplified10.9

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

    16. Applied associate-*l/10.1

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

    17. Applied associate-*r/10.1

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

    19. Applied frac-times10.2

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

    20. Applied associate-*r/10.1

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

    21. Simplified8.2

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

    if 1.3155893364844926e+154 < k

    1. Initial program 33.2

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

    2. Initial simplification22.3

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

    4. Applied *-un-lft-identity22.3

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

    5. Applied tan-quot22.3

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

    6. Applied associate-*r/22.3

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

    7. Applied associate-/r/22.3

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

    8. Applied times-frac22.3

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

    9. Simplified22.2

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

    11. Applied *-un-lft-identity22.2

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

    12. Applied times-frac21.8

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

    13. Applied associate-*l*20.4

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

    14. Simplified20.4

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

    16. Applied associate-/r/20.4

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

    17. Applied associate-*l*17.2

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

  4. Final simplification9.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;k \le 1.3155893364844926 \cdot 10^{+154}:\\ \;\;\;\;\frac{\frac{\frac{\frac{\ell}{t}}{\sin k} \cdot \left(\cos k \cdot 2\right)}{\frac{k \cdot k}{t} + t \cdot 2}}{\frac{\sin k}{\frac{\ell}{t}}}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{\cos k}{2 + \frac{k}{t} \cdot \frac{k}{t}} \cdot \frac{\ell}{t}\right) \cdot \frac{\frac{2}{t}}{\sin k}\right) \cdot \frac{\frac{\ell}{t}}{\sin k}\\ \end{array}\]

Runtime

Time bar (total: 3.4m)Debug logProfile

BaselineHerbieOracleSpan%
Regimes10.59.68.42.142.2%
herbie shell --seed 2018263 
(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))))