Average Error: 18.9 → 3.6
Time: 20.2s
Precision: 64
\[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\]
↓
\[\left(\frac{\sqrt{\frac{\sqrt[3]{A} \cdot \sqrt[3]{A}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}}{\sqrt{\left|\sqrt[3]{V}\right|}} \cdot c0\right) \cdot \frac{\sqrt{\frac{\frac{\sqrt[3]{A}}{\sqrt[3]{V}}}{\sqrt[3]{\ell}}}}{\sqrt{\left|\sqrt[3]{V}\right|}}\]
c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}
↓
\left(\frac{\sqrt{\frac{\sqrt[3]{A} \cdot \sqrt[3]{A}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}}{\sqrt{\left|\sqrt[3]{V}\right|}} \cdot c0\right) \cdot \frac{\sqrt{\frac{\frac{\sqrt[3]{A}}{\sqrt[3]{V}}}{\sqrt[3]{\ell}}}}{\sqrt{\left|\sqrt[3]{V}\right|}}
double f(double c0, double A, double V, double l) {
        double r1613492 = c0;
        double r1613493 = A;
        double r1613494 = V;
        double r1613495 = l;
        double r1613496 = r1613494 * r1613495;
        double r1613497 = r1613493 / r1613496;
        double r1613498 = sqrt(r1613497);
        double r1613499 = r1613492 * r1613498;
        return r1613499;
}
↓
double f(double c0, double A, double V, double l) {
        double r1613500 = A;
        double r1613501 = cbrt(r1613500);
        double r1613502 = r1613501 * r1613501;
        double r1613503 = l;
        double r1613504 = cbrt(r1613503);
        double r1613505 = r1613504 * r1613504;
        double r1613506 = r1613502 / r1613505;
        double r1613507 = sqrt(r1613506);
        double r1613508 = V;
        double r1613509 = cbrt(r1613508);
        double r1613510 = fabs(r1613509);
        double r1613511 = sqrt(r1613510);
        double r1613512 = r1613507 / r1613511;
        double r1613513 = c0;
        double r1613514 = r1613512 * r1613513;
        double r1613515 = r1613501 / r1613509;
        double r1613516 = r1613515 / r1613504;
        double r1613517 = sqrt(r1613516);
        double r1613518 = r1613517 / r1613511;
        double r1613519 = r1613514 * r1613518;
        return r1613519;
}

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 18.9

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

  3. Applied *-un-lft-identity18.9

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

  4. Applied times-frac19.0

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

  6. Applied add-cube-cbrt19.3

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

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

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

  8. Applied times-frac19.3

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

  9. Applied associate-*l*19.3

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

  10. Simplified17.9

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

  12. Applied associate-*l/17.9

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

  13. Applied sqrt-div13.2

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

  14. Simplified13.2

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

  15. Simplified13.2

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

  17. Applied add-sqr-sqrt13.2

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

  18. Applied add-cube-cbrt13.4

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

  19. Applied *-un-lft-identity13.4

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

  20. Applied add-cube-cbrt13.5

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

  21. Applied times-frac13.4

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

  22. Applied times-frac11.7

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

  23. Applied sqrt-prod4.0

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

  24. Applied times-frac4.0

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

  25. Applied associate-*r*3.6

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

  26. Simplified3.6

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

  27. Final simplification3.6

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

Reproduce

herbie shell --seed 2019156 +o rules:numerics
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  (* c0 (sqrt (/ A (* V l)))))