Average Error: 48.5 → 8.3
Time: 29.3s
Precision: binary64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\ell \cdot \left(2 \cdot \left({\left({\left({k}^{\left(\frac{-2}{2}\right)}\right)}^{1}\right)}^{1} \cdot \left(\left(\ell \cdot \frac{\cos k}{{\left(\sin k\right)}^{2}}\right) \cdot {\left({\left({k}^{\left(\frac{-2}{2}\right)}\right)}^{1} \cdot {\left({t}^{\left(-1\right)}\right)}^{1}\right)}^{1}\right)\right)\right)\]

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

  1. Initial program 48.5

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

  2. Simplified38.9

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

  3. Taylor expanded around inf 52.2

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

  4. Simplified16.3

    \[\leadsto \ell \cdot \color{blue}{\left(2 \cdot \left({\left({\left({k}^{\left(-2\right)}\right)}^{1} \cdot {\left({t}^{\left(-1\right)}\right)}^{1}\right)}^{1} \cdot \left(\frac{\cos k}{{\left(\sin k\right)}^{2}} \cdot \ell\right)\right)\right)}\]
  5. Using strategy rm

  6. Applied sqr-pow16.3

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

  7. Applied unpow-prod-down16.3

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

  8. Applied associate-*l*13.0

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

  9. Simplified13.0

    \[\leadsto \ell \cdot \left(2 \cdot \left({\left({\left({k}^{\left(\frac{-2}{2}\right)}\right)}^{1} \cdot \color{blue}{\left({\left({t}^{\left(-1\right)}\right)}^{1} \cdot {\left({k}^{\left(\frac{-2}{2}\right)}\right)}^{1}\right)}\right)}^{1} \cdot \left(\frac{\cos k}{{\left(\sin k\right)}^{2}} \cdot \ell\right)\right)\right)\]
  10. Using strategy rm

  11. Applied unpow-prod-down13.0

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

  12. Applied associate-*l*8.3

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

  13. Simplified9.3

    \[\leadsto \ell \cdot \left(2 \cdot \left({\left({\left({k}^{\left(\frac{-2}{2}\right)}\right)}^{1}\right)}^{1} \cdot \color{blue}{\left(\ell \cdot \left(\frac{\cos k}{{\left(\sin k\right)}^{2}} \cdot {\left({\left({k}^{\left(\frac{-2}{2}\right)}\right)}^{1} \cdot {\left({t}^{\left(-1\right)}\right)}^{1}\right)}^{1}\right)\right)}\right)\right)\]
  14. Using strategy rm

  15. Applied associate-*r*8.3

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

  16. Final simplification8.3

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

Reproduce

herbie shell --seed 2020185 
(FPCore (t l k)
  :name "Toniolo and Linder, Equation (10-)"
  :precision binary64
  (/ 2.0 (* (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k)) (- (+ 1.0 (pow (/ k t) 2.0)) 1.0))))