Average Error: 18.9 → 11.0
Time: 6.3s
Precision: 64
\[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;V \le -7.24944730302796452 \cdot 10^{-9}:\\ \;\;\;\;\left(\sqrt[3]{c0 \cdot \left(\left|\sqrt[3]{A}\right| \cdot \sqrt{\frac{\sqrt[3]{A}}{V \cdot \ell}}\right)} \cdot \sqrt[3]{c0 \cdot \left(\left|\sqrt[3]{A}\right| \cdot \sqrt{\frac{\sqrt[3]{A}}{V \cdot \ell}}\right)}\right) \cdot \sqrt[3]{c0 \cdot \left(\left|\sqrt[3]{A}\right| \cdot \sqrt{\frac{\sqrt[3]{A}}{V \cdot \ell}}\right)}\\ \mathbf{elif}\;V \le -3.2698369089058447 \cdot 10^{-220}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;V \le 8.37998221063094693 \cdot 10^{-237}:\\ \;\;\;\;c0 \cdot \left(\left|\sqrt[3]{A}\right| \cdot \sqrt{\frac{1}{\frac{V \cdot \ell}{\sqrt[3]{A}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \left(\left|\sqrt[3]{A}\right| \cdot \left(\sqrt{\frac{\sqrt[3]{\sqrt[3]{A} \cdot \sqrt[3]{A}}}{V}} \cdot \sqrt{\frac{\sqrt[3]{\sqrt[3]{A}}}{\ell}}\right)\right)\\ \end{array}\]
c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}
↓
\begin{array}{l}
\mathbf{if}\;V \le -7.24944730302796452 \cdot 10^{-9}:\\
\;\;\;\;\left(\sqrt[3]{c0 \cdot \left(\left|\sqrt[3]{A}\right| \cdot \sqrt{\frac{\sqrt[3]{A}}{V \cdot \ell}}\right)} \cdot \sqrt[3]{c0 \cdot \left(\left|\sqrt[3]{A}\right| \cdot \sqrt{\frac{\sqrt[3]{A}}{V \cdot \ell}}\right)}\right) \cdot \sqrt[3]{c0 \cdot \left(\left|\sqrt[3]{A}\right| \cdot \sqrt{\frac{\sqrt[3]{A}}{V \cdot \ell}}\right)}\\

\mathbf{elif}\;V \le -3.2698369089058447 \cdot 10^{-220}:\\
\;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\

\mathbf{elif}\;V \le 8.37998221063094693 \cdot 10^{-237}:\\
\;\;\;\;c0 \cdot \left(\left|\sqrt[3]{A}\right| \cdot \sqrt{\frac{1}{\frac{V \cdot \ell}{\sqrt[3]{A}}}}\right)\\

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

\end{array}
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) {
	double temp;
	if ((V <= -7.2494473030279645e-09)) {
		temp = ((cbrt((c0 * (fabs(cbrt(A)) * sqrt((cbrt(A) / (V * l)))))) * cbrt((c0 * (fabs(cbrt(A)) * sqrt((cbrt(A) / (V * l))))))) * cbrt((c0 * (fabs(cbrt(A)) * sqrt((cbrt(A) / (V * l)))))));
	} else {
		double temp_1;
		if ((V <= -3.2698369089058447e-220)) {
			temp_1 = (c0 * sqrt(((A / V) / l)));
		} else {
			double temp_2;
			if ((V <= 8.379982210630947e-237)) {
				temp_2 = (c0 * (fabs(cbrt(A)) * sqrt((1.0 / ((V * l) / cbrt(A))))));
			} else {
				temp_2 = (c0 * (fabs(cbrt(A)) * (sqrt((cbrt((cbrt(A) * cbrt(A))) / V)) * sqrt((cbrt(cbrt(A)) / l)))));
			}
			temp_1 = temp_2;
		}
		temp = temp_1;
	}
	return temp;
}

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 4 regimes

  2. if V < -7.2494473030279645e-09

    1. Initial program 18.1

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

    3. Applied add-cube-cbrt18.4

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

    4. Applied associate-/l*18.4

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

    6. Applied div-inv18.4

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

    7. Applied sqrt-prod13.4

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

    8. Simplified13.4

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

    9. Simplified13.1

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

    11. Applied add-cube-cbrt13.5

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

    if -7.2494473030279645e-09 < V < -3.2698369089058447e-220

    1. Initial program 17.5

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

    3. Applied associate-/r*18.1

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

    if -3.2698369089058447e-220 < V < 8.379982210630947e-237

    1. Initial program 30.0

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

    3. Applied add-cube-cbrt30.3

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

    4. Applied associate-/l*30.3

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

    6. Applied div-inv30.4

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

    7. Applied sqrt-prod25.8

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

    8. Simplified25.8

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

    9. Simplified25.8

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

    11. Applied clear-num25.8

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

    if 8.379982210630947e-237 < V

    1. Initial program 16.9

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

    3. Applied add-cube-cbrt17.3

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

    4. Applied associate-/l*17.3

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

    6. Applied div-inv17.3

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

    7. Applied sqrt-prod11.9

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

    8. Simplified11.9

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

    9. Simplified11.6

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

    11. Applied add-cube-cbrt11.7

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

    12. Applied cbrt-prod11.7

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

    13. Applied times-frac10.3

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

    14. Applied sqrt-prod3.0

      \[\leadsto c0 \cdot \left(\left|\sqrt[3]{A}\right| \cdot \color{blue}{\left(\sqrt{\frac{\sqrt[3]{\sqrt[3]{A} \cdot \sqrt[3]{A}}}{V}} \cdot \sqrt{\frac{\sqrt[3]{\sqrt[3]{A}}}{\ell}}\right)}\right)\]
  3. Recombined 4 regimes into one program.

  4. Final simplification11.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;V \le -7.24944730302796452 \cdot 10^{-9}:\\ \;\;\;\;\left(\sqrt[3]{c0 \cdot \left(\left|\sqrt[3]{A}\right| \cdot \sqrt{\frac{\sqrt[3]{A}}{V \cdot \ell}}\right)} \cdot \sqrt[3]{c0 \cdot \left(\left|\sqrt[3]{A}\right| \cdot \sqrt{\frac{\sqrt[3]{A}}{V \cdot \ell}}\right)}\right) \cdot \sqrt[3]{c0 \cdot \left(\left|\sqrt[3]{A}\right| \cdot \sqrt{\frac{\sqrt[3]{A}}{V \cdot \ell}}\right)}\\ \mathbf{elif}\;V \le -3.2698369089058447 \cdot 10^{-220}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;V \le 8.37998221063094693 \cdot 10^{-237}:\\ \;\;\;\;c0 \cdot \left(\left|\sqrt[3]{A}\right| \cdot \sqrt{\frac{1}{\frac{V \cdot \ell}{\sqrt[3]{A}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \left(\left|\sqrt[3]{A}\right| \cdot \left(\sqrt{\frac{\sqrt[3]{\sqrt[3]{A} \cdot \sqrt[3]{A}}}{V}} \cdot \sqrt{\frac{\sqrt[3]{\sqrt[3]{A}}}{\ell}}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020060 +o rules:numerics
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  :precision binary64
  (* c0 (sqrt (/ A (* V l)))))