Average Error: 18.9 → 1.2
Time: 6.9s
Precision: binary64
\[[V, l] = \mathsf{sort}([V, l]) \\]
\[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
↓
\[\begin{array}{l} t_0 := \frac{\frac{\sqrt[3]{A}}{\sqrt[3]{V}}}{\sqrt[3]{\ell}}\\ \left(c0 \cdot \left|t_0\right|\right) \cdot \sqrt{t_0} \end{array} \]
c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}
↓
\begin{array}{l}
t_0 := \frac{\frac{\sqrt[3]{A}}{\sqrt[3]{V}}}{\sqrt[3]{\ell}}\\
\left(c0 \cdot \left|t_0\right|\right) \cdot \sqrt{t_0}
\end{array}

(FPCore (c0 A V l) :precision binary64 (* c0 (sqrt (/ A (* V l)))))
↓
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (/ (/ (cbrt A) (cbrt V)) (cbrt l))))
   (* (* c0 (fabs t_0)) (sqrt t_0))))
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 t_0 = (cbrt(A) / cbrt(V)) / cbrt(l);
	return (c0 * fabs(t_0)) * sqrt(t_0);
}

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. Applied associate-/r*_binary6418.9

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

  3. Applied add-cube-cbrt_binary6419.2

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

  4. Applied add-cube-cbrt_binary6419.4

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

  5. Applied add-cube-cbrt_binary6419.4

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

  6. Applied times-frac_binary6419.4

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

  7. Applied times-frac_binary6415.1

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

  8. Applied sqrt-prod_binary647.0

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

  9. Applied associate-*r*_binary647.0

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

  10. Simplified1.2

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

  11. Applied *-un-lft-identity_binary641.2

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

  12. Applied cbrt-prod_binary641.2

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

  13. Simplified1.2

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

  14. Final simplification1.2

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

Reproduce

herbie shell --seed 2022103 
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  :precision binary64
  (* c0 (sqrt (/ A (* V l)))))