Average Error: 19.2 → 4.8
Time: 6.4s
Precision: binary64
\[[V, l] = \mathsf{sort}([V, l]) \\]
\[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \]
↓
\[\begin{array}{l} t_0 := \sqrt{\sqrt[3]{V} \cdot \sqrt[3]{V}}\\ \mathbf{if}\;A \leq 3.3673042949411714 \cdot 10^{-296}:\\ \;\;\;\;c0 \cdot \frac{\frac{\sqrt{\frac{A}{\sqrt[3]{V}}}}{\sqrt{\ell}}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\frac{\sqrt{A}}{\sqrt{\sqrt[3]{V} \cdot \ell}}}{t_0}\\ \end{array} \]
c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}
↓
\begin{array}{l}
t_0 := \sqrt{\sqrt[3]{V} \cdot \sqrt[3]{V}}\\
\mathbf{if}\;A \leq 3.3673042949411714 \cdot 10^{-296}:\\
\;\;\;\;c0 \cdot \frac{\frac{\sqrt{\frac{A}{\sqrt[3]{V}}}}{\sqrt{\ell}}}{t_0}\\

\mathbf{else}:\\
\;\;\;\;c0 \cdot \frac{\frac{\sqrt{A}}{\sqrt{\sqrt[3]{V} \cdot \ell}}}{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 (sqrt (* (cbrt V) (cbrt V)))))
   (if (<= A 3.3673042949411714e-296)
     (* c0 (/ (/ (sqrt (/ A (cbrt V))) (sqrt l)) t_0))
     (* c0 (/ (/ (sqrt A) (sqrt (* (cbrt V) l))) 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 = sqrt(cbrt(V) * cbrt(V));
	double tmp;
	if (A <= 3.3673042949411714e-296) {
		tmp = c0 * ((sqrt(A / cbrt(V)) / sqrt(l)) / t_0);
	} else {
		tmp = c0 * ((sqrt(A) / sqrt(cbrt(V) * l)) / t_0);
	}
	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 2 regimes

  2. if A < 3.3673042949411714e-296

    1. Initial program 19.3

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

    2. Applied add-cube-cbrt_binary6419.6

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

    3. Applied times-frac_binary6417.9

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

    4. Applied add-cube-cbrt_binary6418.0

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

    5. Applied times-frac_binary6418.0

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

    6. Applied associate-*l*_binary6415.7

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

    7. Applied associate-*l/_binary6417.5

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

    8. Applied sqrt-div_binary6412.2

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

    9. Applied associate-*r/_binary6412.1

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

    10. Applied associate-*r/_binary6414.0

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

    11. Applied sqrt-div_binary645.5

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

    12. Simplified5.3

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

    if 3.3673042949411714e-296 < A

    1. Initial program 19.1

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

    2. Applied add-cube-cbrt_binary6419.4

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

    3. Applied times-frac_binary6418.4

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

    4. Applied add-cube-cbrt_binary6418.5

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

    5. Applied times-frac_binary6418.5

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

    6. Applied associate-*l*_binary6416.1

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

    7. Applied associate-*l/_binary6417.6

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

    8. Applied sqrt-div_binary6411.7

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

    9. Applied frac-times_binary6413.2

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

    10. Applied associate-*r/_binary6413.2

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

    11. Applied sqrt-div_binary644.5

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

    12. Simplified4.3

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

    13. Simplified4.3

      \[\leadsto c0 \cdot \frac{\frac{\sqrt{A}}{\color{blue}{\sqrt{\ell \cdot \sqrt[3]{V}}}}}{\sqrt{\sqrt[3]{V} \cdot \sqrt[3]{V}}} \]
  3. Recombined 2 regimes into one program.

  4. Final simplification4.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;A \leq 3.3673042949411714 \cdot 10^{-296}:\\ \;\;\;\;c0 \cdot \frac{\frac{\sqrt{\frac{A}{\sqrt[3]{V}}}}{\sqrt{\ell}}}{\sqrt{\sqrt[3]{V} \cdot \sqrt[3]{V}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\frac{\sqrt{A}}{\sqrt{\sqrt[3]{V} \cdot \ell}}}{\sqrt{\sqrt[3]{V} \cdot \sqrt[3]{V}}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022066 
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  :precision binary64
  (* c0 (sqrt (/ A (* V l)))))