Average Error: 47.0 → 1.3
Time: 6.2m
Precision: 64
Internal Precision: 4160
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\frac{2}{-\sin k} \cdot \frac{\frac{\cos k \cdot \frac{-1}{t}}{\frac{-k}{-\ell}}}{\frac{k}{-\ell} \cdot \left(-\sin k\right)}\]

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 47.0

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

  3. Applied add-cbrt-cube48.5

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

  4. Applied simplify35.2

    \[\leadsto \frac{2}{\sqrt[3]{\color{blue}{{\left(\left((\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 0)_* \cdot \sin k\right) \cdot \frac{t \cdot \tan k}{\frac{\ell}{t} \cdot \frac{\ell}{t}}\right)}^{3}}}}\]

  5. Taylor expanded around -inf 60.1

    \[\leadsto \frac{2}{\color{blue}{e^{\left(\log \left(-1 \cdot \frac{{\left(\sin k\right)}^{2}}{\cos k}\right) + 2 \cdot \log \left(\frac{-1}{\ell}\right)\right) - \left(\log \left(\frac{-1}{t}\right) + 2 \cdot \log \left(\frac{-1}{k}\right)\right)}}}\]

  6. Applied simplify9.3

    \[\leadsto \color{blue}{\frac{2}{\frac{-\sin k}{\frac{\cos k}{\sin k}}} \cdot \frac{\frac{-1}{t}}{\frac{\frac{-1}{\ell}}{\frac{-1}{k}} \cdot \frac{\frac{-1}{\ell}}{\frac{-1}{k}}}}\]
  7. Using strategy rm

  8. Applied associate-/r/9.3

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

  9. Applied associate-*l*7.8

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

  10. Applied simplify1.3

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

  11. Applied simplify1.3

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

  12. Applied simplify1.3

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

Runtime

Time bar (total: 6.2m)Debug logProfile

herbie shell --seed 2019053 +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))))