Average Error: 48.2 → 0.9
Time: 3.8m
Precision: 64
Internal Precision: 4160
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\frac{\frac{\ell}{k} \cdot 2}{t \cdot \tan k} \cdot \frac{\frac{\ell}{k}}{\sin k}\]

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 48.2

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

  2. Initial simplification31.1

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

  4. Applied *-un-lft-identity31.1

    \[\leadsto \frac{\frac{\frac{2}{\tan k}}{\frac{t \cdot \sin k}{\frac{\ell}{t} \cdot \frac{\ell}{t}}}}{\color{blue}{1 \cdot (\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 0)_*}}\]

  5. Applied associate-/r/31.3

    \[\leadsto \frac{\color{blue}{\frac{\frac{2}{\tan k}}{t \cdot \sin k} \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{t}\right)}}{1 \cdot (\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 0)_*}\]

  6. Applied times-frac29.0

    \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\tan k}}{t \cdot \sin k}}{1} \cdot \frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 0)_*}}\]

  7. Simplified29.0

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

  8. Simplified9.1

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

  10. Applied pow19.1

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

  11. Applied pow19.1

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

  12. Applied pow-prod-down9.1

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

  13. Applied pow19.1

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

  14. Applied pow-prod-down9.1

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

  15. Simplified0.9

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

  16. Final simplification0.9

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

Runtime

Time bar (total: 3.8m)Debug logProfile

herbie shell --seed 2018215 +o rules:numerics
(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))))