Average Error: 31.7 → 12.7
Time: 2.3m
Precision: 64
Internal Precision: 576
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
↓
\[\left(\frac{\frac{\ell}{t}}{\sqrt{\sqrt{\left(1 + 1\right) + \frac{k}{t} \cdot \frac{k}{t}}}} \cdot \frac{\frac{\frac{\ell}{t}}{\tan k}}{\sqrt{\sqrt[3]{\sqrt{\left(1 + 1\right) + \frac{k}{t} \cdot \frac{k}{t}}} \cdot \left(\sqrt[3]{\sqrt{\left(1 + 1\right) + \frac{k}{t} \cdot \frac{k}{t}}} \cdot \sqrt[3]{\sqrt{\left(1 + 1\right) + \frac{k}{t} \cdot \frac{k}{t}}}\right)}}\right) \cdot \frac{\frac{\frac{2}{t}}{\sin k}}{\sqrt{\left(1 + 1\right) + \frac{k}{t} \cdot \frac{k}{t}}}\]

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. Initial program 31.7

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

  2. Initial simplification24.4

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

  4. Applied add-sqr-sqrt24.5

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

  5. Applied times-frac18.5

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

  6. Applied times-frac16.6

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

  8. Applied add-sqr-sqrt16.6

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

  9. Applied sqrt-prod16.6

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

  10. Applied *-un-lft-identity16.6

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

  11. Applied times-frac13.8

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

  12. Applied times-frac12.7

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

  13. Simplified12.7

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

  15. Applied add-cube-cbrt12.7

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

  16. Final simplification12.7

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

Runtime

Time bar (total: 2.3m)Debug logProfile

herbie shell --seed 2018217 
(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))))