Average Error: 18.9 → 2.7
Time: 15.3s
Precision: binary64
\[[V, l]=\mathsf{sort}([V, l])\]
\[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\ell \leq 5.601672525454924 \cdot 10^{-309}:\\ \;\;\;\;c0 \cdot \left(\sqrt{\frac{\sqrt[3]{A}}{\frac{\sqrt[3]{V} \cdot \sqrt[3]{V}}{\frac{1}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}}} \cdot \sqrt{\frac{\sqrt[3]{A}}{\frac{\sqrt[3]{V}}{\frac{\sqrt[3]{A}}{\sqrt[3]{\ell}}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \left(\left|\frac{\sqrt[3]{A}}{\sqrt[3]{V}}\right| \cdot \left(\sqrt{\frac{1}{\ell}} \cdot \sqrt{\frac{\sqrt[3]{A}}{\sqrt[3]{V}}}\right)\right)\\ \end{array}\]
c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}
↓
\begin{array}{l}
\mathbf{if}\;\ell \leq 5.601672525454924 \cdot 10^{-309}:\\
\;\;\;\;c0 \cdot \left(\sqrt{\frac{\sqrt[3]{A}}{\frac{\sqrt[3]{V} \cdot \sqrt[3]{V}}{\frac{1}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}}} \cdot \sqrt{\frac{\sqrt[3]{A}}{\frac{\sqrt[3]{V}}{\frac{\sqrt[3]{A}}{\sqrt[3]{\ell}}}}}\right)\\

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

\end{array}
(FPCore (c0 A V l) :precision binary64 (* c0 (sqrt (/ A (* V l)))))
↓
(FPCore (c0 A V l)
 :precision binary64
 (if (<= l 5.601672525454924e-309)
   (*
    c0
    (*
     (sqrt
      (/ (cbrt A) (/ (* (cbrt V) (cbrt V)) (/ 1.0 (* (cbrt l) (cbrt l))))))
     (sqrt (/ (cbrt A) (/ (cbrt V) (/ (cbrt A) (cbrt l)))))))
   (*
    c0
    (*
     (fabs (/ (cbrt A) (cbrt V)))
     (* (sqrt (/ 1.0 l)) (sqrt (/ (cbrt A) (cbrt 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 tmp;
	if (l <= 5.601672525454924e-309) {
		tmp = c0 * (sqrt(cbrt(A) / ((cbrt(V) * cbrt(V)) / (1.0 / (cbrt(l) * cbrt(l))))) * sqrt(cbrt(A) / (cbrt(V) / (cbrt(A) / cbrt(l)))));
	} else {
		tmp = c0 * (fabs(cbrt(A) / cbrt(V)) * (sqrt(1.0 / l) * sqrt(cbrt(A) / cbrt(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 2 regimes

  2. if l < 5.6016725254549244e-309

    1. Initial program 19.2

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

    3. Applied add-cube-cbrt_binary6419.6

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

    4. Applied associate-/l*_binary6419.6

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

    5. Simplified18.3

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

    7. Applied add-cube-cbrt_binary6418.4

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

    8. Applied *-un-lft-identity_binary6418.4

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

    9. Applied times-frac_binary6418.4

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

    10. Applied add-cube-cbrt_binary6418.5

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

    11. Applied times-frac_binary6417.6

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

    12. Applied times-frac_binary6415.7

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

    13. Applied sqrt-prod_binary644.7

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

    if 5.6016725254549244e-309 < l

    1. Initial program 18.9

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

    3. Applied add-cube-cbrt_binary6419.2

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

    4. Applied associate-/l*_binary6419.2

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

    5. Simplified18.0

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

    7. Applied div-inv_binary6418.0

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

    8. Applied add-cube-cbrt_binary6418.1

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

    9. Applied times-frac_binary6417.3

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

    10. Applied times-frac_binary6416.1

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

    11. Applied sqrt-prod_binary647.7

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

    12. Simplified6.1

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

    13. Simplified6.1

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

    15. Applied *-un-lft-identity_binary646.1

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

    16. Applied times-frac_binary644.6

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

    17. Applied sqrt-prod_binary642.1

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

  4. Final simplification2.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 5.601672525454924 \cdot 10^{-309}:\\ \;\;\;\;c0 \cdot \left(\sqrt{\frac{\sqrt[3]{A}}{\frac{\sqrt[3]{V} \cdot \sqrt[3]{V}}{\frac{1}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}}} \cdot \sqrt{\frac{\sqrt[3]{A}}{\frac{\sqrt[3]{V}}{\frac{\sqrt[3]{A}}{\sqrt[3]{\ell}}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \left(\left|\frac{\sqrt[3]{A}}{\sqrt[3]{V}}\right| \cdot \left(\sqrt{\frac{1}{\ell}} \cdot \sqrt{\frac{\sqrt[3]{A}}{\sqrt[3]{V}}}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2021174 
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  :precision binary64
  (* c0 (sqrt (/ A (* V l)))))