Average Error: 47.6 → 1.0
Time: 8.4m
Precision: 64
Internal Precision: 128
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\frac{\frac{\frac{2}{k} \cdot \ell}{\sin k \cdot t}}{\frac{\tan k}{\frac{\ell}{k}}}\]

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 47.6

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

  2. Initial simplification30.8

    \[\leadsto \frac{\frac{\frac{2}{\tan k}}{\frac{\sin k \cdot t}{\frac{\ell}{t} \cdot \frac{\ell}{t}}}}{\frac{k}{t} \cdot \frac{k}{t}}\]
  3. Using strategy rm

  4. Applied associate-*r/32.4

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

  5. Applied associate-/r/32.4

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

  6. Applied *-un-lft-identity32.4

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

  7. Applied times-frac32.2

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

  8. Applied times-frac21.2

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

  9. Simplified13.6

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

  10. Simplified12.9

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

  12. Applied frac-times10.9

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

  13. Simplified6.4

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

  15. Applied associate-/l/4.5

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

  17. Applied associate-*l/1.0

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

  18. Final simplification1.0

    \[\leadsto \frac{\frac{\frac{2}{k} \cdot \ell}{\sin k \cdot t}}{\frac{\tan k}{\frac{\ell}{k}}}\]

Runtime

Time bar (total: 8.4m)Debug logProfile

herbie shell --seed 2018336 
(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))))