Average Error: 46.8 → 13.7
Time: 4.7m
Precision: 64
Internal Precision: 4416
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -3.70746225421608 \cdot 10^{-206}:\\ \;\;\;\;\frac{\frac{\frac{2}{t}}{\sin k}}{\left|\frac{k}{t}\right|} \cdot \left(\frac{\ell}{t} \cdot \left(\frac{\sqrt[3]{\frac{\frac{\ell}{t}}{\tan k}} \cdot \sqrt[3]{\frac{\frac{\ell}{t}}{\tan k}}}{\sqrt[3]{\left|\frac{k}{t}\right|} \cdot \sqrt[3]{\left|\frac{k}{t}\right|}} \cdot \frac{\sqrt[3]{\frac{\frac{\ell}{t}}{\tan k}}}{\sqrt[3]{\left|\frac{k}{t}\right|}}\right)\right)\\ \mathbf{elif}\;t \le 1.1740353769855194 \cdot 10^{-201}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot {\ell}^{2}}{\left({\left(\sin k\right)}^{2} \cdot {k}^{2}\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t} \cdot \left(\frac{\frac{\frac{\ell}{t}}{\tan k}}{\left|\frac{k}{t}\right|} \cdot \frac{\frac{\frac{2}{t}}{\sin k}}{\left|\frac{k}{t}\right|}\right)\\ \end{array}\]

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 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

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

    4. Applied add-sqr-sqrt27.0

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

    5. Applied times-frac26.8

      \[\leadsto \frac{\color{blue}{\frac{\frac{2}{t}}{\sin k} \cdot \frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\tan k}}}{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 0} \cdot \sqrt{\frac{k}{t} \cdot \frac{k}{t} + 0}}\]

    6. Applied times-frac24.9

      \[\leadsto \color{blue}{\frac{\frac{\frac{2}{t}}{\sin k}}{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 0}} \cdot \frac{\frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\tan k}}{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 0}}}\]

    7. Simplified24.9

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

    8. Simplified11.0

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

    10. Applied add-cube-cbrt11.3

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

    11. Applied add-cube-cbrt11.3

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

    12. Applied times-frac11.3

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

    if -3.70746225421608e-206 < t < 1.1740353769855194e-201

    1. Initial program 62.7

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

    2. Initial simplification54.5

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

    3. Taylor expanded around inf 29.6

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

    if 1.1740353769855194e-201 < t

    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 simplification26.4

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

    4. Applied add-sqr-sqrt26.4

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

    5. Applied times-frac26.2

      \[\leadsto \frac{\color{blue}{\frac{\frac{2}{t}}{\sin k} \cdot \frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\tan k}}}{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 0} \cdot \sqrt{\frac{k}{t} \cdot \frac{k}{t} + 0}}\]

    6. Applied times-frac24.2

      \[\leadsto \color{blue}{\frac{\frac{\frac{2}{t}}{\sin k}}{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 0}} \cdot \frac{\frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\tan k}}{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 0}}}\]

    7. Simplified24.2

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

    8. Simplified10.7

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

    10. Applied associate-*r*11.2

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

  4. Final simplification13.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -3.70746225421608 \cdot 10^{-206}:\\ \;\;\;\;\frac{\frac{\frac{2}{t}}{\sin k}}{\left|\frac{k}{t}\right|} \cdot \left(\frac{\ell}{t} \cdot \left(\frac{\sqrt[3]{\frac{\frac{\ell}{t}}{\tan k}} \cdot \sqrt[3]{\frac{\frac{\ell}{t}}{\tan k}}}{\sqrt[3]{\left|\frac{k}{t}\right|} \cdot \sqrt[3]{\left|\frac{k}{t}\right|}} \cdot \frac{\sqrt[3]{\frac{\frac{\ell}{t}}{\tan k}}}{\sqrt[3]{\left|\frac{k}{t}\right|}}\right)\right)\\ \mathbf{elif}\;t \le 1.1740353769855194 \cdot 10^{-201}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot {\ell}^{2}}{\left({\left(\sin k\right)}^{2} \cdot {k}^{2}\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t} \cdot \left(\frac{\frac{\frac{\ell}{t}}{\tan k}}{\left|\frac{k}{t}\right|} \cdot \frac{\frac{\frac{2}{t}}{\sin k}}{\left|\frac{k}{t}\right|}\right)\\ \end{array}\]

Runtime

Time bar (total: 4.7m)Debug logProfile

herbie shell --seed 2018225 
(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))))