Average Error: 32.1 → 12.6
Time: 3.2m
Precision: 64
Internal Precision: 128
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
↓
\[\frac{\frac{2}{\frac{t}{\ell}}}{\frac{\sqrt{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*}}{\frac{1}{\tan k \cdot \frac{t}{\ell}}}} \cdot \left(\frac{\frac{1}{\sin k}}{\left|\sqrt[3]{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*}\right|} \cdot \frac{\frac{1}{t}}{\sqrt{\sqrt[3]{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*}}}\right)\]

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

  1. Initial program 32.1

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

  2. Simplified18.6

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

  4. Applied add-sqr-sqrt18.7

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

  5. Applied associate-*l/21.0

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

  6. Applied associate-/r/20.6

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

  7. Applied div-inv20.6

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

  8. Applied times-frac19.2

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

  9. Applied times-frac17.4

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

  10. Simplified13.8

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

  12. Applied div-inv13.8

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

  13. Applied associate-/l*12.8

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

  15. Applied add-cube-cbrt12.9

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

  16. Applied sqrt-prod12.9

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

  17. Applied div-inv12.9

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

  18. Applied times-frac12.6

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

  19. Simplified12.6

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

  20. Final simplification12.6

    \[\leadsto \frac{\frac{2}{\frac{t}{\ell}}}{\frac{\sqrt{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*}}{\frac{1}{\tan k \cdot \frac{t}{\ell}}}} \cdot \left(\frac{\frac{1}{\sin k}}{\left|\sqrt[3]{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*}\right|} \cdot \frac{\frac{1}{t}}{\sqrt{\sqrt[3]{(\left(\frac{k}{t}\right) \cdot \left(\frac{k}{t}\right) + 2)_*}}}\right)\]

Reproduce

herbie shell --seed 2019068 +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))))