Average Error: 48.4 → 17.3
Time: 26.8s
Precision: 64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[2 \cdot \left({\left(\frac{1}{{\left(\sqrt[3]{k} \cdot \sqrt[3]{k}\right)}^{2} \cdot \left({\left(\sqrt[3]{k}\right)}^{2} \cdot {t}^{1}\right)}\right)}^{1} \cdot \left(\frac{\cos k \cdot \ell}{\sin k} \cdot \frac{\ell}{\sin k}\right)\right)\]
\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 \cdot \left({\left(\frac{1}{{\left(\sqrt[3]{k} \cdot \sqrt[3]{k}\right)}^{2} \cdot \left({\left(\sqrt[3]{k}\right)}^{2} \cdot {t}^{1}\right)}\right)}^{1} \cdot \left(\frac{\cos k \cdot \ell}{\sin k} \cdot \frac{\ell}{\sin k}\right)\right)
double code(double t, double l, double k) {
	return ((double) (2.0 / ((double) (((double) (((double) (((double) (((double) pow(t, 3.0)) / ((double) (l * l)))) * ((double) sin(k)))) * ((double) tan(k)))) * ((double) (((double) (1.0 + ((double) pow(((double) (k / t)), 2.0)))) - 1.0))))));
}
↓
double code(double t, double l, double k) {
	return ((double) (2.0 * ((double) (((double) pow(((double) (1.0 / ((double) (((double) pow(((double) (((double) cbrt(k)) * ((double) cbrt(k)))), 2.0)) * ((double) (((double) pow(((double) cbrt(k)), 2.0)) * ((double) pow(t, 1.0)))))))), 1.0)) * ((double) (((double) (((double) (((double) cos(k)) * l)) / ((double) sin(k)))) * ((double) (l / ((double) sin(k))))))))));
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 48.4

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

  2. Simplified40.6

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

  3. Taylor expanded around inf 21.7

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

  5. Applied add-cube-cbrt22.0

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

  6. Applied unpow-prod-down22.0

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

  7. Applied associate-*l*20.6

    \[\leadsto 2 \cdot \left({\left(\frac{1}{\color{blue}{{\left(\sqrt[3]{k} \cdot \sqrt[3]{k}\right)}^{2} \cdot \left({\left(\sqrt[3]{k}\right)}^{2} \cdot {t}^{1}\right)}}\right)}^{1} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\left(\sin k\right)}^{2}}\right)\]
  8. Using strategy rm

  9. Applied add-sqr-sqrt41.9

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

  10. Applied unpow-prod-down41.9

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

  11. Applied times-frac41.7

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

  12. Simplified41.7

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

  13. Simplified20.3

    \[\leadsto 2 \cdot \left({\left(\frac{1}{{\left(\sqrt[3]{k} \cdot \sqrt[3]{k}\right)}^{2} \cdot \left({\left(\sqrt[3]{k}\right)}^{2} \cdot {t}^{1}\right)}\right)}^{1} \cdot \left(\frac{\cos k}{\sin k} \cdot \color{blue}{\frac{{\ell}^{2}}{\sin k}}\right)\right)\]
  14. Using strategy rm

  15. Applied *-un-lft-identity20.3

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

  16. Applied unpow220.3

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

  17. Applied times-frac18.6

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

  18. Applied associate-*r*17.3

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

  19. Simplified17.3

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

  20. Final simplification17.3

    \[\leadsto 2 \cdot \left({\left(\frac{1}{{\left(\sqrt[3]{k} \cdot \sqrt[3]{k}\right)}^{2} \cdot \left({\left(\sqrt[3]{k}\right)}^{2} \cdot {t}^{1}\right)}\right)}^{1} \cdot \left(\frac{\cos k \cdot \ell}{\sin k} \cdot \frac{\ell}{\sin k}\right)\right)\]

Reproduce

herbie shell --seed 2020130 
(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))))