Average Error: 47.5 → 0.9
Time: 2.6m
Precision: 64
Internal Precision: 4480
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\frac{\frac{2}{\sin k} \cdot \left(\frac{\frac{\ell}{k}}{t} \cdot \cos k\right)}{\frac{\sin k}{\ell} \cdot k}\]

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

  1. Initial program 47.5

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

  3. Applied associate-*l/47.5

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

  4. Applied associate-*l/47.6

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

  5. Applied associate-*l/47.6

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

  6. Applied simplify39.8

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

  8. Applied associate-/r*38.1

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

  9. Applied simplify16.0

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

  10. Taylor expanded around inf 16.1

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

  11. Applied simplify4.1

    \[\leadsto \color{blue}{\frac{\frac{2}{\sin k}}{\frac{\sin k}{\ell}} \cdot \left(\frac{\frac{\ell}{k}}{t} \cdot \frac{\cos k}{k}\right)}\]
  12. Using strategy rm

  13. Applied associate-*r/4.1

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

  14. Applied frac-times0.9

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

Runtime

Time bar (total: 2.6m)Debug logProfile

herbie shell --seed '#(1070258749 1877548225 2229079127 1588002776 3179087814 1886870650)' +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))))