Average Error: 19.4 → 2.2
Time: 36.4s
Precision: binary64
\[[V, l]=\mathsf{sort}([V, l])\]
\[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\]
↓
\[c0 \cdot \left(\left|\frac{\sqrt[3]{A}}{\sqrt[3]{V}}\right| \cdot \left(\left|\sqrt[3]{\frac{1}{\ell}}\right| \cdot \sqrt{\frac{\sqrt[3]{A}}{\frac{\sqrt[3]{V}}{\sqrt[3]{\frac{1}{\ell}}}}}\right)\right)\]
c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}
↓
c0 \cdot \left(\left|\frac{\sqrt[3]{A}}{\sqrt[3]{V}}\right| \cdot \left(\left|\sqrt[3]{\frac{1}{\ell}}\right| \cdot \sqrt{\frac{\sqrt[3]{A}}{\frac{\sqrt[3]{V}}{\sqrt[3]{\frac{1}{\ell}}}}}\right)\right)
(FPCore (c0 A V l) :precision binary64 (* c0 (sqrt (/ A (* V l)))))
↓
(FPCore (c0 A V l)
 :precision binary64
 (*
  c0
  (*
   (fabs (/ (cbrt A) (cbrt V)))
   (*
    (fabs (cbrt (/ 1.0 l)))
    (sqrt (/ (cbrt A) (/ (cbrt V) (cbrt (/ 1.0 l)))))))))
double code(double c0, double A, double V, double l) {
	return c0 * sqrt(A / (V * l));
}
↓
double code(double c0, double A, double V, double l) {
	return c0 * (fabs(cbrt(A) / cbrt(V)) * (fabs(cbrt(1.0 / l)) * sqrt(cbrt(A) / (cbrt(V) / cbrt(1.0 / l)))));
}

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 19.4

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

  3. Applied add-cube-cbrt_binary6419.8

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

  4. Applied associate-/l*_binary6419.8

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

  5. Simplified18.3

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

  7. Applied div-inv_binary6418.3

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

  8. Applied add-cube-cbrt_binary6418.4

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

  9. Applied times-frac_binary6417.7

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

  10. Applied times-frac_binary6416.7

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

  11. Applied sqrt-prod_binary648.1

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

  12. Applied associate-*r*_binary649.4

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

  13. Simplified7.5

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

  15. Applied add-cube-cbrt_binary647.6

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

  16. Applied *-un-lft-identity_binary647.6

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

  17. Applied cbrt-prod_binary647.6

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

  18. Applied times-frac_binary647.6

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

  19. Applied *-un-lft-identity_binary647.6

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

  20. Applied cbrt-prod_binary647.6

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

  21. Applied times-frac_binary646.2

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

  22. Applied sqrt-prod_binary642.4

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

  23. Simplified2.4

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

  25. Applied associate-*l*_binary642.2

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

  26. Final simplification2.2

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

Alternatives

Reproduce

herbie shell --seed 2021118 
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  :precision binary64
  (* c0 (sqrt (/ A (* V l)))))