Average Error: 47.5 → 12.4
Time: 5.2m
Precision: 64
Internal Precision: 4160
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;k \le 1.1001068635784917 \cdot 10^{-94}:\\ \;\;\;\;\left(\frac{\frac{\frac{\ell}{t}}{\tan k}}{\left|\frac{k}{t}\right|} \cdot \frac{\ell}{t}\right) \cdot \left(\frac{\frac{\sqrt[3]{\frac{2}{t}} \cdot \sqrt[3]{\frac{2}{t}}}{\sqrt[3]{\sin k} \cdot \sqrt[3]{\sin k}}}{\sqrt{\left|\frac{k}{t}\right|}} \cdot \frac{\frac{\sqrt[3]{\frac{2}{t}}}{\sqrt[3]{\sin k}}}{\sqrt{\left|\frac{k}{t}\right|}}\right)\\ \mathbf{elif}\;k \le 4.031684069956918 \cdot 10^{+183}:\\ \;\;\;\;\left(\frac{\frac{\frac{2}{t}}{\sin k}}{\left|\frac{k}{t}\right|} \cdot \frac{\frac{\frac{\ell}{t}}{\tan k}}{\left|\frac{k}{t}\right|}\right) \cdot \frac{\ell}{t}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{\frac{\ell}{t}}{\tan k}}{\left|\frac{k}{t}\right|} \cdot \frac{\ell}{t}\right) \cdot \frac{2}{\left(\left|\frac{k}{t}\right| \cdot \sin k\right) \cdot t}\\ \end{array}\]

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 k < 1.1001068635784917e-94

    1. Initial program 49.0

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

    2. Initial simplification33.1

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

    4. Applied add-sqr-sqrt33.1

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

    5. Applied times-frac33.1

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

    6. Applied times-frac30.5

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

    7. Simplified30.5

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

    8. Simplified15.9

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

    10. Applied add-sqr-sqrt15.9

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

    11. Applied add-cube-cbrt16.1

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

    12. Applied add-cube-cbrt16.2

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

    13. Applied times-frac16.2

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

    14. Applied times-frac15.2

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

    if 1.1001068635784917e-94 < k < 4.031684069956918e+183

    1. Initial program 50.5

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

    2. Initial simplification31.3

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

    4. Applied add-sqr-sqrt31.3

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

    5. Applied times-frac31.2

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

    6. Applied times-frac28.1

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

    7. Simplified28.1

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

    8. Simplified8.7

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

    10. Applied associate-*r*7.5

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

    if 4.031684069956918e+183 < k

    1. Initial program 36.7

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

    2. Initial simplification22.1

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

    4. Applied add-sqr-sqrt22.1

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

    5. Applied times-frac22.1

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

    6. Applied times-frac19.0

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

    7. Simplified19.0

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

    8. Simplified11.9

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

    9. Taylor expanded around inf 11.9

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

  4. Final simplification12.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;k \le 1.1001068635784917 \cdot 10^{-94}:\\ \;\;\;\;\left(\frac{\frac{\frac{\ell}{t}}{\tan k}}{\left|\frac{k}{t}\right|} \cdot \frac{\ell}{t}\right) \cdot \left(\frac{\frac{\sqrt[3]{\frac{2}{t}} \cdot \sqrt[3]{\frac{2}{t}}}{\sqrt[3]{\sin k} \cdot \sqrt[3]{\sin k}}}{\sqrt{\left|\frac{k}{t}\right|}} \cdot \frac{\frac{\sqrt[3]{\frac{2}{t}}}{\sqrt[3]{\sin k}}}{\sqrt{\left|\frac{k}{t}\right|}}\right)\\ \mathbf{elif}\;k \le 4.031684069956918 \cdot 10^{+183}:\\ \;\;\;\;\left(\frac{\frac{\frac{2}{t}}{\sin k}}{\left|\frac{k}{t}\right|} \cdot \frac{\frac{\frac{\ell}{t}}{\tan k}}{\left|\frac{k}{t}\right|}\right) \cdot \frac{\ell}{t}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{\frac{\ell}{t}}{\tan k}}{\left|\frac{k}{t}\right|} \cdot \frac{\ell}{t}\right) \cdot \frac{2}{\left(\left|\frac{k}{t}\right| \cdot \sin k\right) \cdot t}\\ \end{array}\]

Runtime

Time bar (total: 5.2m)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))))