Average Error: 46.8 → 12.8
Time: 3.4m
Precision: 64
Internal Precision: 4416
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -3.70746225421608 \cdot 10^{-206}:\\ \;\;\;\;\frac{\frac{\frac{2}{t}}{\sin k}}{\left|\frac{k}{t}\right|} \cdot \left(\frac{\frac{\ell}{t}}{\tan k} \cdot \left(\ell \cdot \frac{\frac{1}{t}}{\left|\frac{k}{t}\right|}\right)\right)\\ \mathbf{elif}\;t \le 1.0027537407786816 \cdot 10^{-174}:\\ \;\;\;\;\frac{\cos k \cdot {\ell}^{2}}{t \cdot \left({\left(\sin k\right)}^{2} \cdot {k}^{2}\right)} \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{\ell}{t}}{\tan k} \cdot \frac{\frac{\frac{2}{t}}{\sin k}}{\left|\frac{k}{t}\right|}\right) \cdot \left(\frac{\sqrt[3]{\frac{\ell}{t}} \cdot \sqrt[3]{\frac{\ell}{t}}}{\sqrt{\left|\frac{k}{t}\right|}} \cdot \frac{\sqrt[3]{\frac{\ell}{t}}}{\sqrt{\left|\frac{k}{t}\right|}}\right)\\ \end{array}\]

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 t < -3.70746225421608e-206

    1. Initial program 44.4

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

    2. Initial simplification27.0

      \[\leadsto \frac{\frac{\frac{2}{t} \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{t}\right)}{\sin k \cdot \tan k}}{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 0)_*}\]
    3. Using strategy rm

    4. Applied add-sqr-sqrt27.0

      \[\leadsto \frac{\frac{\frac{2}{t} \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{t}\right)}{\sin k \cdot \tan k}}{\color{blue}{\sqrt{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 0)_*} \cdot \sqrt{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 0)_*}}}\]

    5. Applied times-frac26.8

      \[\leadsto \frac{\color{blue}{\frac{\frac{2}{t}}{\sin k} \cdot \frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\tan k}}}{\sqrt{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 0)_*} \cdot \sqrt{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 0)_*}}\]

    6. Applied times-frac24.9

      \[\leadsto \color{blue}{\frac{\frac{\frac{2}{t}}{\sin k}}{\sqrt{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 0)_*}} \cdot \frac{\frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\tan k}}{\sqrt{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 0)_*}}}\]

    7. Simplified24.9

      \[\leadsto \color{blue}{\frac{\frac{\frac{2}{t}}{\sin k}}{\left|\frac{k}{t}\right|}} \cdot \frac{\frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\tan k}}{\sqrt{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 0)_*}}\]

    8. Simplified11.0

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

    10. Applied *-un-lft-identity11.0

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

    11. Applied div-inv11.0

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

    12. Applied times-frac11.0

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

    13. Simplified11.0

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

    if -3.70746225421608e-206 < t < 1.0027537407786816e-174

    1. Initial program 62.6

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

    2. Initial simplification53.0

      \[\leadsto \frac{\frac{\frac{2}{t} \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{t}\right)}{\sin k \cdot \tan k}}{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 0)_*}\]

    3. Taylor expanded around inf 29.1

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

    if 1.0027537407786816e-174 < t

    1. Initial program 43.7

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

    2. Initial simplification25.7

      \[\leadsto \frac{\frac{\frac{2}{t} \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{t}\right)}{\sin k \cdot \tan k}}{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 0)_*}\]
    3. Using strategy rm

    4. Applied add-sqr-sqrt25.7

      \[\leadsto \frac{\frac{\frac{2}{t} \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{t}\right)}{\sin k \cdot \tan k}}{\color{blue}{\sqrt{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 0)_*} \cdot \sqrt{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 0)_*}}}\]

    5. Applied times-frac25.5

      \[\leadsto \frac{\color{blue}{\frac{\frac{2}{t}}{\sin k} \cdot \frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\tan k}}}{\sqrt{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 0)_*} \cdot \sqrt{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 0)_*}}\]

    6. Applied times-frac23.7

      \[\leadsto \color{blue}{\frac{\frac{\frac{2}{t}}{\sin k}}{\sqrt{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 0)_*}} \cdot \frac{\frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\tan k}}{\sqrt{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 0)_*}}}\]

    7. Simplified23.7

      \[\leadsto \color{blue}{\frac{\frac{\frac{2}{t}}{\sin k}}{\left|\frac{k}{t}\right|}} \cdot \frac{\frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\tan k}}{\sqrt{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 0)_*}}\]

    8. Simplified10.2

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

    10. Applied associate-*r*8.5

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

    12. Applied add-sqr-sqrt8.5

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

    13. Applied add-cube-cbrt8.8

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

    14. Applied times-frac8.8

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

  4. Final simplification12.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -3.70746225421608 \cdot 10^{-206}:\\ \;\;\;\;\frac{\frac{\frac{2}{t}}{\sin k}}{\left|\frac{k}{t}\right|} \cdot \left(\frac{\frac{\ell}{t}}{\tan k} \cdot \left(\ell \cdot \frac{\frac{1}{t}}{\left|\frac{k}{t}\right|}\right)\right)\\ \mathbf{elif}\;t \le 1.0027537407786816 \cdot 10^{-174}:\\ \;\;\;\;\frac{\cos k \cdot {\ell}^{2}}{t \cdot \left({\left(\sin k\right)}^{2} \cdot {k}^{2}\right)} \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{\ell}{t}}{\tan k} \cdot \frac{\frac{\frac{2}{t}}{\sin k}}{\left|\frac{k}{t}\right|}\right) \cdot \left(\frac{\sqrt[3]{\frac{\ell}{t}} \cdot \sqrt[3]{\frac{\ell}{t}}}{\sqrt{\left|\frac{k}{t}\right|}} \cdot \frac{\sqrt[3]{\frac{\ell}{t}}}{\sqrt{\left|\frac{k}{t}\right|}}\right)\\ \end{array}\]

Runtime

Time bar (total: 3.4m)Debug logProfile

herbie shell --seed 2018225 +o rules:numerics
(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))))