Average Error: 32.2 → 12.7
Time: 2.2m
Precision: 64
Internal Precision: 576
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
↓
\[\left(\frac{\frac{\ell}{t}}{\sqrt{\sqrt{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*}}} \cdot \frac{\frac{\frac{\ell}{t}}{\tan k}}{\sqrt{\sqrt{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*}}}\right) \cdot \frac{\frac{\frac{2}{t}}{\sin k}}{\sqrt{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*}}\]

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

  1. Initial program 32.2

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

  2. Initial simplification25.0

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

  4. Applied add-sqr-sqrt25.0

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

  5. Applied times-frac18.5

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

  6. Applied times-frac16.5

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

  8. Applied add-sqr-sqrt16.5

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

  9. Applied sqrt-prod16.4

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

  10. Applied *-un-lft-identity16.4

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

  11. Applied times-frac13.6

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

  12. Applied times-frac12.7

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

  13. Simplified12.7

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

  14. Final simplification12.7

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

Runtime

Time bar (total: 2.2m)Debug logProfile

herbie shell --seed 2018250 +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))))