Average Error: 18.8 → 1.6
Time: 17.6s
Precision: binary64
\[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{A}{V \cdot \ell} \leq 1.5344944025491 \cdot 10^{-310}:\\ \;\;\;\;\left(c0 \cdot \frac{\sqrt{\left|\sqrt[3]{A}\right|}}{\left|\frac{\sqrt[3]{V}}{\frac{\sqrt[3]{1}}{\sqrt[3]{\ell}}}\right|}\right) \cdot \frac{\sqrt{\left|\sqrt[3]{A}\right|}}{\sqrt{\frac{\sqrt[3]{V}}{\frac{\sqrt[3]{A}}{\sqrt[3]{\ell}}}}}\\ \mathbf{elif}\;\frac{A}{V \cdot \ell} \leq 1.3731182887056827 \cdot 10^{+290}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{\sqrt[3]{V} \cdot \sqrt[3]{V}}{\frac{\sqrt[3]{1}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}}} \cdot \frac{\left|\sqrt[3]{A}\right|}{\sqrt{\frac{\sqrt[3]{V}}{\frac{\sqrt[3]{A}}{\sqrt[3]{\ell}}}}}\\ \end{array}\]
c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}
↓
\begin{array}{l}
\mathbf{if}\;\frac{A}{V \cdot \ell} \leq 1.5344944025491 \cdot 10^{-310}:\\
\;\;\;\;\left(c0 \cdot \frac{\sqrt{\left|\sqrt[3]{A}\right|}}{\left|\frac{\sqrt[3]{V}}{\frac{\sqrt[3]{1}}{\sqrt[3]{\ell}}}\right|}\right) \cdot \frac{\sqrt{\left|\sqrt[3]{A}\right|}}{\sqrt{\frac{\sqrt[3]{V}}{\frac{\sqrt[3]{A}}{\sqrt[3]{\ell}}}}}\\

\mathbf{elif}\;\frac{A}{V \cdot \ell} \leq 1.3731182887056827 \cdot 10^{+290}:\\
\;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\

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

\end{array}
(FPCore (c0 A V l) :precision binary64 (* c0 (sqrt (/ A (* V l)))))
↓
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (/ A (* V l)) 1.5344944025491e-310)
   (*
    (*
     c0
     (/ (sqrt (fabs (cbrt A))) (fabs (/ (cbrt V) (/ (cbrt 1.0) (cbrt l))))))
    (/ (sqrt (fabs (cbrt A))) (sqrt (/ (cbrt V) (/ (cbrt A) (cbrt l))))))
   (if (<= (/ A (* V l)) 1.3731182887056827e+290)
     (* c0 (sqrt (/ A (* V l))))
     (*
      (/
       c0
       (sqrt (/ (* (cbrt V) (cbrt V)) (/ (cbrt 1.0) (* (cbrt l) (cbrt l))))))
      (/ (fabs (cbrt A)) (sqrt (/ (cbrt V) (/ (cbrt A) (cbrt l)))))))))
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 ((A / (V * l)) <= 1.5344944025491e-310) {
		tmp = (c0 * (sqrt(fabs(cbrt(A))) / fabs(cbrt(V) / (cbrt(1.0) / cbrt(l))))) * (sqrt(fabs(cbrt(A))) / sqrt(cbrt(V) / (cbrt(A) / cbrt(l))));
	} else if ((A / (V * l)) <= 1.3731182887056827e+290) {
		tmp = c0 * sqrt(A / (V * l));
	} else {
		tmp = (c0 / sqrt((cbrt(V) * cbrt(V)) / (cbrt(1.0) / (cbrt(l) * cbrt(l))))) * (fabs(cbrt(A)) / sqrt(cbrt(V) / (cbrt(A) / cbrt(l))));
	}
	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 3 regimes

  2. if (/.f64 A (*.f64 V l)) < 1.534494402549084e-310

    1. Initial program 40.0

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

    3. Applied add-cube-cbrt_binary64_113640.0

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

    4. Applied associate-/l*_binary64_104640.0

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

    5. Simplified34.3

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

    7. Applied sqrt-div_binary64_111824.3

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

    8. Simplified24.3

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

    10. Applied add-cube-cbrt_binary64_113624.4

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

    11. Applied *-un-lft-identity_binary64_110124.4

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

    12. Applied cbrt-prod_binary64_113224.4

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

    13. Applied times-frac_binary64_110724.4

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

    14. Applied add-cube-cbrt_binary64_113624.4

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

    15. Applied times-frac_binary64_110722.5

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

    16. Applied sqrt-prod_binary64_11176.3

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

    17. Applied add-sqr-sqrt_binary64_11236.4

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

    18. Applied times-frac_binary64_11076.4

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

    19. Applied associate-*r*_binary64_10415.5

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

    21. Applied add-sqr-sqrt_binary64_11235.5

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

    22. Applied cbrt-prod_binary64_11325.5

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

    23. Applied times-frac_binary64_11075.5

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

    24. Applied times-frac_binary64_11075.5

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

    25. Applied rem-sqrt-square_binary64_11141.0

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

    if 1.534494402549084e-310 < (/.f64 A (*.f64 V l)) < 1.37311828870568274e290

    1. Initial program 0.5

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

    if 1.37311828870568274e290 < (/.f64 A (*.f64 V l))

    1. Initial program 59.9

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

    3. Applied add-cube-cbrt_binary64_113660.0

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

    4. Applied associate-/l*_binary64_104660.0

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

    5. Simplified54.1

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

    7. Applied sqrt-div_binary64_111837.3

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

    8. Simplified37.3

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

    10. Applied add-cube-cbrt_binary64_113637.4

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

    11. Applied *-un-lft-identity_binary64_110137.4

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

    12. Applied cbrt-prod_binary64_113237.4

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

    13. Applied times-frac_binary64_110737.4

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

    14. Applied add-cube-cbrt_binary64_113637.5

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

    15. Applied times-frac_binary64_110734.8

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

    16. Applied sqrt-prod_binary64_11178.7

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

    17. Applied *-un-lft-identity_binary64_11018.7

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

    18. Applied times-frac_binary64_11078.7

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

    19. Applied associate-*r*_binary64_10417.1

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

    20. Simplified7.1

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

  4. Final simplification1.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{V \cdot \ell} \leq 1.5344944025491 \cdot 10^{-310}:\\ \;\;\;\;\left(c0 \cdot \frac{\sqrt{\left|\sqrt[3]{A}\right|}}{\left|\frac{\sqrt[3]{V}}{\frac{\sqrt[3]{1}}{\sqrt[3]{\ell}}}\right|}\right) \cdot \frac{\sqrt{\left|\sqrt[3]{A}\right|}}{\sqrt{\frac{\sqrt[3]{V}}{\frac{\sqrt[3]{A}}{\sqrt[3]{\ell}}}}}\\ \mathbf{elif}\;\frac{A}{V \cdot \ell} \leq 1.3731182887056827 \cdot 10^{+290}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{\sqrt[3]{V} \cdot \sqrt[3]{V}}{\frac{\sqrt[3]{1}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}}} \cdot \frac{\left|\sqrt[3]{A}\right|}{\sqrt{\frac{\sqrt[3]{V}}{\frac{\sqrt[3]{A}}{\sqrt[3]{\ell}}}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020353 
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  :precision binary64
  (* c0 (sqrt (/ A (* V l)))))