Average Error: 47.5 → 1.4
Time: 1.8m
Precision: 64
Internal Precision: 128
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\left(\frac{\cos k}{\sin k} \cdot \frac{\frac{\ell}{k}}{\frac{\sin k}{\frac{\frac{\ell}{k}}{t}}}\right) \cdot 2\]

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. Simplified30.9

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

  3. Taylor expanded around inf 23.7

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

  4. Simplified19.7

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

  6. Applied associate-*l/20.8

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

  7. Applied associate-/r/20.7

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

  8. Applied *-un-lft-identity20.7

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

  9. Applied times-frac20.8

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

  10. Simplified6.9

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

  11. Taylor expanded around -inf 22.8

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

  12. Simplified2.6

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

  14. Applied associate-/l*1.4

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

  15. Final simplification1.4

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

Reproduce

herbie shell --seed 2019093 
(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))))