Average Error: 47.6 → 1.0
Time: 5.9m
Precision: 64
Internal Precision: 4160
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\left(\frac{-\ell}{k} \cdot \frac{2}{\sin k}\right) \cdot \frac{\frac{\frac{\cos k}{t}}{\frac{k}{-\ell}}}{\sin k}\]

Error

Bits error versus t

Bits error versus l

Bits error versus k

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. Using strategy rm

  3. Applied add-cbrt-cube49.1

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

  4. Applied simplify36.0

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

  5. Taylor expanded around -inf 59.8

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

  6. Applied simplify10.1

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

  8. Applied *-un-lft-identity10.1

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

  9. Applied times-frac4.6

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

  10. Applied times-frac1.2

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

  11. Applied simplify1.2

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

  12. Applied simplify1.0

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

  13. Applied simplify1.0

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

Runtime

Time bar (total: 5.9m)Debug logProfile

herbie shell --seed '#(1072840222 1305617769 1692503039 1353360431 4178980589 1488672652)' +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))))