Average Error: 32.5 → 13.1
Time: 2.8m
Precision: 64
Internal Precision: 128
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
↓
\[\left(\frac{\frac{\ell}{t}}{\sqrt{\sqrt{2 + \frac{k}{t} \cdot \frac{k}{t}}}} \cdot \frac{\frac{\frac{\ell}{t}}{\tan k}}{\sqrt{\sqrt{2 + \frac{k}{t} \cdot \frac{k}{t}}}}\right) \cdot \frac{\frac{\frac{2}{t}}{\sin k}}{\sqrt{2 + \frac{k}{t} \cdot \frac{k}{t}}}\]

Error

Bits error versus t

Bits error versus l

Bits error versus k

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Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 32.5

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

  2. Initial simplification24.7

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

  4. Applied add-sqr-sqrt24.7

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

  5. Applied times-frac18.6

    \[\leadsto \frac{\color{blue}{\frac{\frac{2}{t}}{\sin k} \cdot \frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\tan k}}}{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2} \cdot \sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}}\]

  6. Applied times-frac16.8

    \[\leadsto \color{blue}{\frac{\frac{\frac{2}{t}}{\sin k}}{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}} \cdot \frac{\frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\tan k}}{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}}}\]
  7. Using strategy rm

  8. Applied add-sqr-sqrt16.8

    \[\leadsto \frac{\frac{\frac{2}{t}}{\sin k}}{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}} \cdot \frac{\frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\tan k}}{\color{blue}{\sqrt{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}} \cdot \sqrt{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}}}}\]

  9. Applied *-un-lft-identity16.8

    \[\leadsto \frac{\frac{\frac{2}{t}}{\sin k}}{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}} \cdot \frac{\frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\color{blue}{1 \cdot \tan k}}}{\sqrt{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}} \cdot \sqrt{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}}}\]

  10. Applied times-frac14.2

    \[\leadsto \frac{\frac{\frac{2}{t}}{\sin k}}{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}} \cdot \frac{\color{blue}{\frac{\frac{\ell}{t}}{1} \cdot \frac{\frac{\ell}{t}}{\tan k}}}{\sqrt{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}} \cdot \sqrt{\sqrt{\frac{k}{t} \cdot \frac{k}{t} + 2}}}\]

  11. Applied times-frac13.1

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

  12. Simplified13.1

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

  13. Final simplification13.1

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

Runtime

Time bar (total: 2.8m)Debug logProfile

herbie shell --seed 2018304 
(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))))