Average Error: 19.0 → 9.3
Time: 6.1s
Precision: binary64
\[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;V \le 5.7984807979433622 \cdot 10^{-309}:\\ \;\;\;\;c0 \cdot \left(\left|\sqrt[3]{A}\right| \cdot \left(\sqrt{\sqrt{\frac{\sqrt[3]{A}}{V \cdot \ell}}} \cdot \sqrt{\sqrt{\frac{\sqrt[3]{A}}{V \cdot \ell}}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \left(\left|\sqrt[3]{A}\right| \cdot \left(\sqrt{\frac{1}{V}} \cdot \sqrt{\frac{\sqrt[3]{A}}{\ell}}\right)\right)\\ \end{array}\]
c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}
↓
\begin{array}{l}
\mathbf{if}\;V \le 5.7984807979433622 \cdot 10^{-309}:\\
\;\;\;\;c0 \cdot \left(\left|\sqrt[3]{A}\right| \cdot \left(\sqrt{\sqrt{\frac{\sqrt[3]{A}}{V \cdot \ell}}} \cdot \sqrt{\sqrt{\frac{\sqrt[3]{A}}{V \cdot \ell}}}\right)\right)\\

\mathbf{else}:\\
\;\;\;\;c0 \cdot \left(\left|\sqrt[3]{A}\right| \cdot \left(\sqrt{\frac{1}{V}} \cdot \sqrt{\frac{\sqrt[3]{A}}{\ell}}\right)\right)\\

\end{array}
double code(double c0, double A, double V, double l) {
	return ((double) (c0 * ((double) sqrt(((double) (A / ((double) (V * l))))))));
}
↓
double code(double c0, double A, double V, double l) {
	double VAR;
	if ((V <= 5.79848079794336e-309)) {
		VAR = ((double) (c0 * ((double) (((double) fabs(((double) cbrt(A)))) * ((double) (((double) sqrt(((double) sqrt(((double) (((double) cbrt(A)) / ((double) (V * l)))))))) * ((double) sqrt(((double) sqrt(((double) (((double) cbrt(A)) / ((double) (V * l))))))))))))));
	} else {
		VAR = ((double) (c0 * ((double) (((double) fabs(((double) cbrt(A)))) * ((double) (((double) sqrt(((double) (1.0 / V)))) * ((double) sqrt(((double) (((double) cbrt(A)) / l))))))))));
	}
	return VAR;
}

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. Split input into 2 regimes

  2. if V < 5.7984807979433622e-309

    1. Initial program 18.9

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

    3. Applied add-cube-cbrt19.2

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

    4. Applied associate-/l*19.2

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

    5. Simplified18.3

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

    7. Applied div-inv18.5

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

    8. Applied sqrt-prod14.0

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

    9. Simplified14.0

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

    10. Simplified13.8

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

    12. Applied add-sqr-sqrt13.8

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

    13. Applied sqrt-prod13.8

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

    if 5.7984807979433622e-309 < V

    1. Initial program 19.1

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

    3. Applied add-cube-cbrt19.4

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

    4. Applied associate-/l*19.4

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

    5. Simplified18.1

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

    7. Applied div-inv18.2

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

    8. Applied sqrt-prod13.2

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

    9. Simplified13.2

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

    10. Simplified13.5

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

    12. Applied *-un-lft-identity13.5

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

    13. Applied cbrt-prod13.5

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

    14. Applied times-frac12.9

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

    15. Applied sqrt-prod4.7

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

    16. Simplified4.7

      \[\leadsto c0 \cdot \left(\left|\sqrt[3]{A}\right| \cdot \left(\color{blue}{\sqrt{\frac{1}{V}}} \cdot \sqrt{\frac{\sqrt[3]{A}}{\ell}}\right)\right)\]
  3. Recombined 2 regimes into one program.

  4. Final simplification9.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;V \le 5.7984807979433622 \cdot 10^{-309}:\\ \;\;\;\;c0 \cdot \left(\left|\sqrt[3]{A}\right| \cdot \left(\sqrt{\sqrt{\frac{\sqrt[3]{A}}{V \cdot \ell}}} \cdot \sqrt{\sqrt{\frac{\sqrt[3]{A}}{V \cdot \ell}}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \left(\left|\sqrt[3]{A}\right| \cdot \left(\sqrt{\frac{1}{V}} \cdot \sqrt{\frac{\sqrt[3]{A}}{\ell}}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020179 
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  :precision binary64
  (* c0 (sqrt (/ A (* V l)))))