Average Error: 19.1 → 2.4
Time: 22.7s
Precision: 64
\[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\]
↓
\[c0 \cdot \left(\left(\left|\frac{\sqrt[3]{A}}{\sqrt[3]{\ell}}\right| \cdot \sqrt{\frac{\frac{\sqrt[3]{A}}{\sqrt[3]{\ell}}}{\sqrt[3]{V}}}\right) \cdot \sqrt{\frac{1}{\sqrt[3]{V} \cdot \sqrt[3]{V}}}\right)\]
c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}
↓
c0 \cdot \left(\left(\left|\frac{\sqrt[3]{A}}{\sqrt[3]{\ell}}\right| \cdot \sqrt{\frac{\frac{\sqrt[3]{A}}{\sqrt[3]{\ell}}}{\sqrt[3]{V}}}\right) \cdot \sqrt{\frac{1}{\sqrt[3]{V} \cdot \sqrt[3]{V}}}\right)
double f(double c0, double A, double V, double l) {
        double r2767456 = c0;
        double r2767457 = A;
        double r2767458 = V;
        double r2767459 = l;
        double r2767460 = r2767458 * r2767459;
        double r2767461 = r2767457 / r2767460;
        double r2767462 = sqrt(r2767461);
        double r2767463 = r2767456 * r2767462;
        return r2767463;
}
↓
double f(double c0, double A, double V, double l) {
        double r2767464 = c0;
        double r2767465 = A;
        double r2767466 = cbrt(r2767465);
        double r2767467 = l;
        double r2767468 = cbrt(r2767467);
        double r2767469 = r2767466 / r2767468;
        double r2767470 = fabs(r2767469);
        double r2767471 = V;
        double r2767472 = cbrt(r2767471);
        double r2767473 = r2767469 / r2767472;
        double r2767474 = sqrt(r2767473);
        double r2767475 = r2767470 * r2767474;
        double r2767476 = 1.0;
        double r2767477 = r2767472 * r2767472;
        double r2767478 = r2767476 / r2767477;
        double r2767479 = sqrt(r2767478);
        double r2767480 = r2767475 * r2767479;
        double r2767481 = r2767464 * r2767480;
        return r2767481;
}

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 19.1

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

  3. Applied *-un-lft-identity19.1

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

  4. Applied times-frac19.1

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

  6. Applied add-cube-cbrt19.5

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

  7. Applied *-un-lft-identity19.5

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

  8. Applied times-frac19.5

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

  9. Applied associate-*l*19.5

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

  10. Simplified19.5

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

  12. Applied sqrt-prod13.8

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

  14. Applied *-un-lft-identity13.8

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

  15. Applied cbrt-prod13.8

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

  16. Applied add-cube-cbrt14.0

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

  17. Applied add-cube-cbrt14.0

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

  18. Applied times-frac14.0

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

  19. Applied times-frac11.9

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

  20. Applied sqrt-prod4.0

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

  21. Simplified2.4

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

  22. Final simplification2.4

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

Reproduce

herbie shell --seed 2019130 +o rules:numerics
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  (* c0 (sqrt (/ A (* V l)))))