Average Error: 19.3 → 8.7
Time: 6.8s
Precision: binary64
\[[V, l] = \mathsf{sort}([V, l]) \\]
\[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
↓
\[\begin{array}{l} t_0 := \sqrt{\frac{A}{V}}\\ \mathbf{if}\;V \cdot \ell \leq -7.52144986366842 \cdot 10^{+124}:\\ \;\;\;\;\frac{c0 \cdot t_0}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -1.3546975018692183 \cdot 10^{-146}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;c0 \cdot \frac{t_0}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 5.970120430153575 \cdot 10^{+246}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \end{array} \]
c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}
↓
\begin{array}{l}
t_0 := \sqrt{\frac{A}{V}}\\
\mathbf{if}\;V \cdot \ell \leq -7.52144986366842 \cdot 10^{+124}:\\
\;\;\;\;\frac{c0 \cdot t_0}{\sqrt{\ell}}\\

\mathbf{elif}\;V \cdot \ell \leq -1.3546975018692183 \cdot 10^{-146}:\\
\;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\

\mathbf{elif}\;V \cdot \ell \leq 0:\\
\;\;\;\;c0 \cdot \frac{t_0}{\sqrt{\ell}}\\

\mathbf{elif}\;V \cdot \ell \leq 5.970120430153575 \cdot 10^{+246}:\\
\;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\

\mathbf{else}:\\
\;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\


\end{array}

(FPCore (c0 A V l) :precision binary64 (* c0 (sqrt (/ A (* V l)))))
↓
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (sqrt (/ A V))))
   (if (<= (* V l) -7.52144986366842e+124)
     (/ (* c0 t_0) (sqrt l))
     (if (<= (* V l) -1.3546975018692183e-146)
       (* c0 (sqrt (/ A (* V l))))
       (if (<= (* V l) 0.0)
         (* c0 (/ t_0 (sqrt l)))
         (if (<= (* V l) 5.970120430153575e+246)
           (* c0 (/ (sqrt A) (sqrt (* V l))))
           (* c0 (sqrt (/ (/ A l) V)))))))))
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 = sqrt(A / V);
	double tmp;
	if ((V * l) <= -7.52144986366842e+124) {
		tmp = (c0 * t_0) / sqrt(l);
	} else if ((V * l) <= -1.3546975018692183e-146) {
		tmp = c0 * sqrt(A / (V * l));
	} else if ((V * l) <= 0.0) {
		tmp = c0 * (t_0 / sqrt(l));
	} else if ((V * l) <= 5.970120430153575e+246) {
		tmp = c0 * (sqrt(A) / sqrt(V * l));
	} else {
		tmp = c0 * sqrt((A / l) / V);
	}
	return tmp;
}

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 5 regimes

  2. if (*.f64 V l) < -7.52144986366842e124

    1. Initial program 24.8

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

    2. Applied add-cube-cbrt_binary6425.0

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

    3. Applied times-frac_binary6418.5

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

    4. Applied associate-*r/_binary6419.2

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

    5. Applied sqrt-div_binary6410.0

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

    6. Applied associate-*r/_binary6412.1

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

    7. Simplified11.8

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

    if -7.52144986366842e124 < (*.f64 V l) < -1.3546975018692183e-146

    1. Initial program 5.1

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

    2. Applied add-cube-cbrt_binary645.6

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

    3. Applied times-frac_binary6413.8

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

    4. Applied pow1_binary6413.8

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

    5. Applied pow1_binary6413.8

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

    6. Applied pow-prod-down_binary6413.8

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

    7. Simplified5.1

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

    if -1.3546975018692183e-146 < (*.f64 V l) < 0.0

    1. Initial program 41.8

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

    2. Applied add-cube-cbrt_binary6442.0

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

    3. Applied times-frac_binary6430.0

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

    4. Applied associate-*r/_binary6430.2

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

    5. Applied sqrt-div_binary6420.8

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

    6. Simplified20.5

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

    if 0.0 < (*.f64 V l) < 5.9701204301535753e246

    1. Initial program 9.8

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

    2. Applied sqrt-div_binary640.7

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

    if 5.9701204301535753e246 < (*.f64 V l)

    1. Initial program 33.8

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

    2. Applied add-cube-cbrt_binary6433.9

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

    3. Applied times-frac_binary6420.0

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

    4. Applied associate-*l/_binary6420.0

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

    5. Simplified19.8

      \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\frac{A}{\ell}}}{V}} \]
  3. Recombined 5 regimes into one program.

  4. Final simplification8.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -7.52144986366842 \cdot 10^{+124}:\\ \;\;\;\;\frac{c0 \cdot \sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -1.3546975018692183 \cdot 10^{-146}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 5.970120430153575 \cdot 10^{+246}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022125 
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  :precision binary64
  (* c0 (sqrt (/ A (* V l)))))