Average Error: 19.3 → 1.5
Time: 7.4s
Precision: binary64
\[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\]
↓
\[\left(c0 \cdot \frac{\left|\frac{\sqrt[3]{A}}{\sqrt[3]{\ell}}\right|}{\sqrt{\left|\sqrt[3]{V}\right|}}\right) \cdot \frac{\sqrt{\frac{\frac{\sqrt[3]{A}}{\sqrt[3]{\ell}}}{\sqrt[3]{V}}}}{\sqrt{\left|\sqrt[3]{V}\right|}}\]
c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}
↓
\left(c0 \cdot \frac{\left|\frac{\sqrt[3]{A}}{\sqrt[3]{\ell}}\right|}{\sqrt{\left|\sqrt[3]{V}\right|}}\right) \cdot \frac{\sqrt{\frac{\frac{\sqrt[3]{A}}{\sqrt[3]{\ell}}}{\sqrt[3]{V}}}}{\sqrt{\left|\sqrt[3]{V}\right|}}
double code(double c0, double A, double V, double l) {
	return ((double) (c0 * ((double) sqrt((A / ((double) (V * l)))))));
}
↓
double code(double c0, double A, double V, double l) {
	return ((double) (((double) (c0 * (((double) fabs((((double) cbrt(A)) / ((double) cbrt(l))))) / ((double) sqrt(((double) fabs(((double) cbrt(V))))))))) * (((double) sqrt(((((double) cbrt(A)) / ((double) cbrt(l))) / ((double) cbrt(V))))) / ((double) sqrt(((double) fabs(((double) cbrt(V)))))))));
}

Error

Bits error versus c0

Bits error versus A

Bits error versus V

Bits error versus l

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 19.3

    \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\]
  2. Using strategy rm

  3. Applied *-un-lft-identity19.3

    \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{1 \cdot A}}{V \cdot \ell}}\]

  4. Applied times-frac19.1

    \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{1}{V} \cdot \frac{A}{\ell}}}\]
  5. Using strategy rm

  6. Applied add-cube-cbrt19.4

    \[\leadsto c0 \cdot \sqrt{\color{blue}{\left(\left(\sqrt[3]{\frac{1}{V}} \cdot \sqrt[3]{\frac{1}{V}}\right) \cdot \sqrt[3]{\frac{1}{V}}\right)} \cdot \frac{A}{\ell}}\]

  7. Applied associate-*l*19.4

    \[\leadsto c0 \cdot \sqrt{\color{blue}{\left(\sqrt[3]{\frac{1}{V}} \cdot \sqrt[3]{\frac{1}{V}}\right) \cdot \left(\sqrt[3]{\frac{1}{V}} \cdot \frac{A}{\ell}\right)}}\]

  8. Simplified19.4

    \[\leadsto c0 \cdot \sqrt{\left(\sqrt[3]{\frac{1}{V}} \cdot \sqrt[3]{\frac{1}{V}}\right) \cdot \color{blue}{\left(\frac{A}{\ell} \cdot \sqrt[3]{\frac{1}{V}}\right)}}\]
  9. Using strategy rm

  10. Applied cbrt-div19.4

    \[\leadsto c0 \cdot \sqrt{\left(\sqrt[3]{\frac{1}{V}} \cdot \sqrt[3]{\frac{1}{V}}\right) \cdot \left(\frac{A}{\ell} \cdot \color{blue}{\frac{\sqrt[3]{1}}{\sqrt[3]{V}}}\right)}\]

  11. Applied associate-*r/19.4

    \[\leadsto c0 \cdot \sqrt{\left(\sqrt[3]{\frac{1}{V}} \cdot \sqrt[3]{\frac{1}{V}}\right) \cdot \color{blue}{\frac{\frac{A}{\ell} \cdot \sqrt[3]{1}}{\sqrt[3]{V}}}}\]

  12. Applied cbrt-div19.4

    \[\leadsto c0 \cdot \sqrt{\left(\sqrt[3]{\frac{1}{V}} \cdot \color{blue}{\frac{\sqrt[3]{1}}{\sqrt[3]{V}}}\right) \cdot \frac{\frac{A}{\ell} \cdot \sqrt[3]{1}}{\sqrt[3]{V}}}\]

  13. Applied associate-*r/19.4

    \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\sqrt[3]{\frac{1}{V}} \cdot \sqrt[3]{1}}{\sqrt[3]{V}}} \cdot \frac{\frac{A}{\ell} \cdot \sqrt[3]{1}}{\sqrt[3]{V}}}\]

  14. Applied frac-times19.4

    \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\left(\sqrt[3]{\frac{1}{V}} \cdot \sqrt[3]{1}\right) \cdot \left(\frac{A}{\ell} \cdot \sqrt[3]{1}\right)}{\sqrt[3]{V} \cdot \sqrt[3]{V}}}}\]

  15. Applied sqrt-div13.8

    \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{\left(\sqrt[3]{\frac{1}{V}} \cdot \sqrt[3]{1}\right) \cdot \left(\frac{A}{\ell} \cdot \sqrt[3]{1}\right)}}{\sqrt{\sqrt[3]{V} \cdot \sqrt[3]{V}}}}\]

  16. Simplified13.8

    \[\leadsto c0 \cdot \frac{\color{blue}{\sqrt{\frac{\frac{A}{\ell}}{\sqrt[3]{V}}}}}{\sqrt{\sqrt[3]{V} \cdot \sqrt[3]{V}}}\]

  17. Simplified13.8

    \[\leadsto c0 \cdot \frac{\sqrt{\frac{\frac{A}{\ell}}{\sqrt[3]{V}}}}{\color{blue}{\left|\sqrt[3]{V}\right|}}\]
  18. Using strategy rm

  19. Applied add-sqr-sqrt13.9

    \[\leadsto c0 \cdot \frac{\sqrt{\frac{\frac{A}{\ell}}{\sqrt[3]{V}}}}{\color{blue}{\sqrt{\left|\sqrt[3]{V}\right|} \cdot \sqrt{\left|\sqrt[3]{V}\right|}}}\]

  20. Applied *-un-lft-identity13.9

    \[\leadsto c0 \cdot \frac{\sqrt{\frac{\frac{A}{\ell}}{\color{blue}{1 \cdot \sqrt[3]{V}}}}}{\sqrt{\left|\sqrt[3]{V}\right|} \cdot \sqrt{\left|\sqrt[3]{V}\right|}}\]

  21. Applied add-cube-cbrt14.0

    \[\leadsto c0 \cdot \frac{\sqrt{\frac{\frac{A}{\color{blue}{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \sqrt[3]{\ell}}}}{1 \cdot \sqrt[3]{V}}}}{\sqrt{\left|\sqrt[3]{V}\right|} \cdot \sqrt{\left|\sqrt[3]{V}\right|}}\]

  22. Applied add-cube-cbrt14.1

    \[\leadsto c0 \cdot \frac{\sqrt{\frac{\frac{\color{blue}{\left(\sqrt[3]{A} \cdot \sqrt[3]{A}\right) \cdot \sqrt[3]{A}}}{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \sqrt[3]{\ell}}}{1 \cdot \sqrt[3]{V}}}}{\sqrt{\left|\sqrt[3]{V}\right|} \cdot \sqrt{\left|\sqrt[3]{V}\right|}}\]

  23. Applied times-frac14.1

    \[\leadsto c0 \cdot \frac{\sqrt{\frac{\color{blue}{\frac{\sqrt[3]{A} \cdot \sqrt[3]{A}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \frac{\sqrt[3]{A}}{\sqrt[3]{\ell}}}}{1 \cdot \sqrt[3]{V}}}}{\sqrt{\left|\sqrt[3]{V}\right|} \cdot \sqrt{\left|\sqrt[3]{V}\right|}}\]

  24. Applied times-frac12.0

    \[\leadsto c0 \cdot \frac{\sqrt{\color{blue}{\frac{\frac{\sqrt[3]{A} \cdot \sqrt[3]{A}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}{1} \cdot \frac{\frac{\sqrt[3]{A}}{\sqrt[3]{\ell}}}{\sqrt[3]{V}}}}}{\sqrt{\left|\sqrt[3]{V}\right|} \cdot \sqrt{\left|\sqrt[3]{V}\right|}}\]

  25. Applied sqrt-prod4.0

    \[\leadsto c0 \cdot \frac{\color{blue}{\sqrt{\frac{\frac{\sqrt[3]{A} \cdot \sqrt[3]{A}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}{1}} \cdot \sqrt{\frac{\frac{\sqrt[3]{A}}{\sqrt[3]{\ell}}}{\sqrt[3]{V}}}}}{\sqrt{\left|\sqrt[3]{V}\right|} \cdot \sqrt{\left|\sqrt[3]{V}\right|}}\]

  26. Applied times-frac4.0

    \[\leadsto c0 \cdot \color{blue}{\left(\frac{\sqrt{\frac{\frac{\sqrt[3]{A} \cdot \sqrt[3]{A}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}{1}}}{\sqrt{\left|\sqrt[3]{V}\right|}} \cdot \frac{\sqrt{\frac{\frac{\sqrt[3]{A}}{\sqrt[3]{\ell}}}{\sqrt[3]{V}}}}{\sqrt{\left|\sqrt[3]{V}\right|}}\right)}\]

  27. Applied associate-*r*3.8

    \[\leadsto \color{blue}{\left(c0 \cdot \frac{\sqrt{\frac{\frac{\sqrt[3]{A} \cdot \sqrt[3]{A}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}{1}}}{\sqrt{\left|\sqrt[3]{V}\right|}}\right) \cdot \frac{\sqrt{\frac{\frac{\sqrt[3]{A}}{\sqrt[3]{\ell}}}{\sqrt[3]{V}}}}{\sqrt{\left|\sqrt[3]{V}\right|}}}\]

  28. Simplified1.5

    \[\leadsto \color{blue}{\left(c0 \cdot \frac{\left|\frac{\sqrt[3]{A}}{\sqrt[3]{\ell}}\right|}{\sqrt{\left|\sqrt[3]{V}\right|}}\right)} \cdot \frac{\sqrt{\frac{\frac{\sqrt[3]{A}}{\sqrt[3]{\ell}}}{\sqrt[3]{V}}}}{\sqrt{\left|\sqrt[3]{V}\right|}}\]

  29. Final simplification1.5

    \[\leadsto \left(c0 \cdot \frac{\left|\frac{\sqrt[3]{A}}{\sqrt[3]{\ell}}\right|}{\sqrt{\left|\sqrt[3]{V}\right|}}\right) \cdot \frac{\sqrt{\frac{\frac{\sqrt[3]{A}}{\sqrt[3]{\ell}}}{\sqrt[3]{V}}}}{\sqrt{\left|\sqrt[3]{V}\right|}}\]

Reproduce

herbie shell --seed 2020196 
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  :precision binary64
  (* c0 (sqrt (/ A (* V l)))))