Average Error: 19.6 → 4.1
Time: 30.7s
Precision: binary64
\[[V, l]=\mathsf{sort}([V, l])\]
\[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\]
↓
\[\left(c0 \cdot \frac{\sqrt{\frac{\sqrt[3]{A} \cdot \sqrt[3]{A}}{\sqrt[3]{V} \cdot \sqrt[3]{V}}}}{\sqrt{\left|\sqrt[3]{\ell}\right|}}\right) \cdot \frac{\sqrt{\frac{\frac{\sqrt[3]{A}}{\sqrt[3]{\ell}}}{\sqrt[3]{V}}}}{\sqrt{\left|\sqrt[3]{\ell}\right|}}\]
c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}
↓
\left(c0 \cdot \frac{\sqrt{\frac{\sqrt[3]{A} \cdot \sqrt[3]{A}}{\sqrt[3]{V} \cdot \sqrt[3]{V}}}}{\sqrt{\left|\sqrt[3]{\ell}\right|}}\right) \cdot \frac{\sqrt{\frac{\frac{\sqrt[3]{A}}{\sqrt[3]{\ell}}}{\sqrt[3]{V}}}}{\sqrt{\left|\sqrt[3]{\ell}\right|}}
(FPCore (c0 A V l) :precision binary64 (* c0 (sqrt (/ A (* V l)))))
↓
(FPCore (c0 A V l)
 :precision binary64
 (*
  (*
   c0
   (/
    (sqrt (/ (* (cbrt A) (cbrt A)) (* (cbrt V) (cbrt V))))
    (sqrt (fabs (cbrt l)))))
  (/ (sqrt (/ (/ (cbrt A) (cbrt l)) (cbrt V))) (sqrt (fabs (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 * (sqrt((cbrt(A) * cbrt(A)) / (cbrt(V) * cbrt(V))) / sqrt(fabs(cbrt(l))))) * (sqrt((cbrt(A) / cbrt(l)) / cbrt(V)) / sqrt(fabs(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 19.6

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

  3. Applied *-un-lft-identity_binary64_110119.6

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

  4. Applied times-frac_binary64_110719.9

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

  6. Applied add-cube-cbrt_binary64_113620.2

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

  7. Applied *-un-lft-identity_binary64_110120.2

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

  8. Applied times-frac_binary64_110720.2

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

  9. Applied associate-*r*_binary64_104119.1

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

  10. Simplified19.1

    \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{1}{V}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}} \cdot \frac{A}{\sqrt[3]{\ell}}}\]
  11. Using strategy rm

  12. Applied associate-*l/_binary64_104419.1

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

  13. Applied sqrt-div_binary64_111813.9

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

  14. Simplified13.9

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

  15. Simplified13.9

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

  17. Applied add-sqr-sqrt_binary64_112314.0

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

  18. Applied add-cube-cbrt_binary64_113614.1

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

  19. Applied *-un-lft-identity_binary64_110114.1

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

  20. Applied add-cube-cbrt_binary64_113614.2

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

  21. Applied times-frac_binary64_110714.2

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

  22. Applied times-frac_binary64_110712.2

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

  23. Applied sqrt-prod_binary64_11174.5

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

  24. Applied times-frac_binary64_11074.5

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

  25. Applied associate-*r*_binary64_10414.1

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

  26. Simplified4.1

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

  27. Final simplification4.1

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

Reproduce

herbie shell --seed 2021077 
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  :precision binary64
  (* c0 (sqrt (/ A (* V l)))))