Average Error: 47.3 → 12.9
Time: 3.4m
Precision: 64
Internal Precision: 4160
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\ell \le -4.129390430874979 \cdot 10^{-119}:\\ \;\;\;\;\frac{\ell}{t} \cdot \left(\left(\frac{\sqrt[3]{\frac{\frac{2}{t}}{\sin k}}}{\sqrt{\left|\frac{k}{t}\right|}} \cdot \frac{\sqrt[3]{\frac{\frac{2}{t}}{\sin k}} \cdot \sqrt[3]{\frac{\frac{2}{t}}{\sin k}}}{\sqrt{\left|\frac{k}{t}\right|}}\right) \cdot \frac{\frac{\frac{\ell}{t}}{\tan k}}{\left|\frac{k}{t}\right|}\right)\\ \mathbf{elif}\;\ell \le 7.358982262721877 \cdot 10^{-250}:\\ \;\;\;\;\left(\frac{\frac{2}{t}}{\left|\frac{k}{t}\right|} \cdot \left(\frac{\frac{\ell}{t}}{\sin k} \cdot \frac{\frac{\ell}{t}}{\sin k}\right)\right) \cdot \frac{\cos k}{\left|\frac{k}{t}\right|}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{2}{t}}{\sin k}}{\left|\frac{k}{t}\right|} \cdot \left(\frac{\frac{\frac{\ell}{t}}{\tan k}}{\left|\frac{k}{t}\right|} \cdot \frac{\ell}{t}\right)\\ \end{array}\]

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 3 regimes

  2. if l < -4.129390430874979e-119

    1. Initial program 48.7

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

    2. Initial simplification34.1

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

    4. Applied add-sqr-sqrt34.1

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

    5. Applied times-frac34.1

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

    6. Applied times-frac31.0

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

    7. Simplified31.0

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

    8. Simplified13.8

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

    10. Applied associate-*r*14.7

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

    12. Applied add-sqr-sqrt14.8

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

    13. Applied add-cube-cbrt15.0

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

    14. Applied times-frac15.0

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

    if -4.129390430874979e-119 < l < 7.358982262721877e-250

    1. Initial program 46.2

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

    2. Initial simplification25.0

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

    4. Applied add-sqr-sqrt25.0

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

    5. Applied tan-quot25.0

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

    6. Applied associate-*r/25.0

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

    7. Applied associate-/r/25.0

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

    8. Applied times-frac25.0

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

    9. Simplified20.0

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

    10. Simplified11.4

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

    if 7.358982262721877e-250 < l

    1. Initial program 47.1

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

    2. Initial simplification30.2

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

    4. Applied add-sqr-sqrt30.2

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

    5. Applied times-frac30.0

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

    6. Applied times-frac27.1

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

    7. Simplified27.1

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

    8. Simplified12.5

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

  4. Final simplification12.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \le -4.129390430874979 \cdot 10^{-119}:\\ \;\;\;\;\frac{\ell}{t} \cdot \left(\left(\frac{\sqrt[3]{\frac{\frac{2}{t}}{\sin k}}}{\sqrt{\left|\frac{k}{t}\right|}} \cdot \frac{\sqrt[3]{\frac{\frac{2}{t}}{\sin k}} \cdot \sqrt[3]{\frac{\frac{2}{t}}{\sin k}}}{\sqrt{\left|\frac{k}{t}\right|}}\right) \cdot \frac{\frac{\frac{\ell}{t}}{\tan k}}{\left|\frac{k}{t}\right|}\right)\\ \mathbf{elif}\;\ell \le 7.358982262721877 \cdot 10^{-250}:\\ \;\;\;\;\left(\frac{\frac{2}{t}}{\left|\frac{k}{t}\right|} \cdot \left(\frac{\frac{\ell}{t}}{\sin k} \cdot \frac{\frac{\ell}{t}}{\sin k}\right)\right) \cdot \frac{\cos k}{\left|\frac{k}{t}\right|}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{2}{t}}{\sin k}}{\left|\frac{k}{t}\right|} \cdot \left(\frac{\frac{\frac{\ell}{t}}{\tan k}}{\left|\frac{k}{t}\right|} \cdot \frac{\ell}{t}\right)\\ \end{array}\]

Runtime

Time bar (total: 3.4m)Debug logProfile

herbie shell --seed 2018221 
(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))))