Average Error: 19.2 → 2.4
Time: 40.3s
Precision: binary64
\[[V, l]=\mathsf{sort}([V, l])\]
\[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
↓
\[c0 \cdot \left(\left|\frac{\sqrt[3]{A}}{\sqrt[3]{V}}\right| \cdot \left(\sqrt{\frac{1}{\frac{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}{\frac{1}{\sqrt[3]{\sqrt[3]{V} \cdot \sqrt[3]{V}}}}}} \cdot \sqrt{\frac{\sqrt[3]{\sqrt[3]{A} \cdot \sqrt[3]{A}}}{\frac{\sqrt[3]{\ell}}{\frac{\sqrt[3]{\sqrt[3]{A}}}{\sqrt[3]{\sqrt[3]{V}}}}}}\right)\right) \]
c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}
↓
c0 \cdot \left(\left|\frac{\sqrt[3]{A}}{\sqrt[3]{V}}\right| \cdot \left(\sqrt{\frac{1}{\frac{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}{\frac{1}{\sqrt[3]{\sqrt[3]{V} \cdot \sqrt[3]{V}}}}}} \cdot \sqrt{\frac{\sqrt[3]{\sqrt[3]{A} \cdot \sqrt[3]{A}}}{\frac{\sqrt[3]{\ell}}{\frac{\sqrt[3]{\sqrt[3]{A}}}{\sqrt[3]{\sqrt[3]{V}}}}}}\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)))
   (*
    (sqrt
     (/ 1.0 (/ (* (cbrt l) (cbrt l)) (/ 1.0 (cbrt (* (cbrt V) (cbrt V)))))))
    (sqrt
     (/
      (cbrt (* (cbrt A) (cbrt A)))
      (/ (cbrt l) (/ (cbrt (cbrt A)) (cbrt (cbrt V))))))))))
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)) * (sqrt(1.0 / ((cbrt(l) * cbrt(l)) / (1.0 / cbrt(cbrt(V) * cbrt(V))))) * sqrt(cbrt(cbrt(A) * cbrt(A)) / (cbrt(l) / (cbrt(cbrt(A)) / cbrt(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 19.2

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

  3. Applied add-cube-cbrt_binary6419.6

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

  4. Applied associate-/l*_binary6419.6

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

  5. Simplified18.4

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

  7. Applied div-inv_binary6418.4

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

  8. Applied add-cube-cbrt_binary6418.5

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

  9. Applied times-frac_binary6417.6

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

  10. Applied times-frac_binary6416.2

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

  11. Applied sqrt-prod_binary648.2

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

  12. Simplified6.6

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

  13. Simplified6.6

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

  15. Applied add-cube-cbrt_binary646.6

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

  16. Applied cbrt-prod_binary646.7

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

  17. Applied associate-/l*_binary646.7

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

  18. Simplified5.9

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

  20. Applied add-cube-cbrt_binary646.0

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

  21. Applied cbrt-prod_binary646.0

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

  22. Applied *-un-lft-identity_binary646.0

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

  23. Applied times-frac_binary646.0

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

  24. Applied add-cube-cbrt_binary646.1

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

  25. Applied times-frac_binary646.1

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

  26. Applied *-un-lft-identity_binary646.1

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

  27. Applied times-frac_binary645.4

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

  28. Applied sqrt-prod_binary642.4

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

  29. Final simplification2.4

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

Reproduce

herbie shell --seed 2021202 
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  :precision binary64
  (* c0 (sqrt (/ A (* V l)))))