Average Error: 32.2 → 11.9
Time: 2.6m
Precision: 64
Internal Precision: 128
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
↓
\[\left(\frac{\frac{2}{t}}{\sin k} \cdot \left(\frac{\frac{\ell}{t}}{\sqrt[3]{\sqrt{2 + \frac{k}{t} \cdot \frac{k}{t}}} \cdot \sqrt[3]{\sqrt{2 + \frac{k}{t} \cdot \frac{k}{t}}}} \cdot \frac{1}{\sqrt[3]{\sqrt{2 + \frac{k}{t} \cdot \frac{k}{t}}}}\right)\right) \cdot \frac{\frac{\frac{\ell}{t}}{\tan k}}{\sqrt{2 + \frac{k}{t} \cdot \frac{k}{t}}}\]

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. Initial program 32.2

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

  2. Initial simplification24.9

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

  4. Applied *-un-lft-identity24.9

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

  5. Applied times-frac18.9

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

  6. Applied times-frac17.2

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

  7. Simplified17.2

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

  9. Applied add-sqr-sqrt17.3

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

  10. Applied *-un-lft-identity17.3

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

  11. Applied times-frac14.5

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

  12. Applied times-frac13.2

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

  13. Applied associate-*r*11.8

    \[\leadsto \color{blue}{\left(\frac{\frac{2}{t}}{\sin k} \cdot \frac{\frac{\frac{\ell}{t}}{1}}{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}}\right) \cdot \frac{\frac{\frac{\ell}{t}}{\tan k}}{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}}}\]
  14. Using strategy rm

  15. Applied add-cube-cbrt11.9

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

  16. Applied div-inv11.9

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

  17. Applied times-frac11.9

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

  18. Simplified11.9

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

  19. Final simplification11.9

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

Runtime

Time bar (total: 2.6m)Debug logProfile

herbie shell --seed 2018285 
(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))))