Average Error: 19.0 → 2.5
Time: 36.0s
Precision: binary64
\[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\]
↓
\[c0 \cdot \left(\left|\frac{\sqrt[3]{A}}{\sqrt[3]{V}}\right| \cdot \left(\sqrt{\frac{\sqrt{\sqrt[3]{\sqrt[3]{A} \cdot \sqrt[3]{A}}}}{\left(\sqrt[3]{\sqrt[3]{V}} \cdot \sqrt[3]{\sqrt[3]{V}}\right) \cdot \left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right)}} \cdot \sqrt{\frac{\sqrt{\sqrt[3]{\sqrt[3]{A} \cdot \sqrt[3]{A}}}}{\frac{\sqrt[3]{\sqrt[3]{V}}}{\frac{\sqrt[3]{\sqrt[3]{A}}}{\sqrt[3]{\ell}}}}}\right)\right)\]
c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}
↓
c0 \cdot \left(\left|\frac{\sqrt[3]{A}}{\sqrt[3]{V}}\right| \cdot \left(\sqrt{\frac{\sqrt{\sqrt[3]{\sqrt[3]{A} \cdot \sqrt[3]{A}}}}{\left(\sqrt[3]{\sqrt[3]{V}} \cdot \sqrt[3]{\sqrt[3]{V}}\right) \cdot \left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right)}} \cdot \sqrt{\frac{\sqrt{\sqrt[3]{\sqrt[3]{A} \cdot \sqrt[3]{A}}}}{\frac{\sqrt[3]{\sqrt[3]{V}}}{\frac{\sqrt[3]{\sqrt[3]{A}}}{\sqrt[3]{\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)))
   (*
    (sqrt
     (/
      (sqrt (cbrt (* (cbrt A) (cbrt A))))
      (* (* (cbrt (cbrt V)) (cbrt (cbrt V))) (* (cbrt l) (cbrt l)))))
    (sqrt
     (/
      (sqrt (cbrt (* (cbrt A) (cbrt A))))
      (/ (cbrt (cbrt V)) (/ (cbrt (cbrt A)) (cbrt 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)) * (sqrt(sqrt(cbrt(cbrt(A) * cbrt(A))) / ((cbrt(cbrt(V)) * cbrt(cbrt(V))) * (cbrt(l) * cbrt(l)))) * sqrt(sqrt(cbrt(cbrt(A) * cbrt(A))) / (cbrt(cbrt(V)) / (cbrt(cbrt(A)) / cbrt(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 19.0

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

  3. Applied add-cube-cbrt_binary64_113619.4

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

  4. Applied associate-/l*_binary64_104619.4

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

  5. Simplified17.7

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

  7. Applied div-inv_binary64_109817.7

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

  8. Applied add-cube-cbrt_binary64_113617.8

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

  9. Applied times-frac_binary64_110717.1

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

  10. Applied times-frac_binary64_110716.2

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

  11. Applied sqrt-prod_binary64_11178.0

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

  12. Simplified6.3

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

  13. Simplified6.3

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

  15. Applied add-cube-cbrt_binary64_11366.3

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

  16. Applied cbrt-prod_binary64_11326.3

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

  17. Applied associate-/l*_binary64_10466.3

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

  18. Simplified5.8

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

  20. Applied add-cube-cbrt_binary64_11365.9

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

  21. Applied *-un-lft-identity_binary64_11015.9

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

  22. Applied times-frac_binary64_11075.9

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

  23. Applied add-cube-cbrt_binary64_11366.0

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

  24. Applied times-frac_binary64_11075.9

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

  25. Applied add-sqr-sqrt_binary64_11235.9

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

  26. Applied times-frac_binary64_11075.2

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

  27. Applied sqrt-prod_binary64_11172.5

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

  28. Simplified2.5

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

  29. Final simplification2.5

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

Reproduce

herbie shell --seed 2021098 
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  :precision binary64
  (* c0 (sqrt (/ A (* V l)))))