Henrywood and Agarwal, Equation (3)

Average Error: 19.4 → 8.6

Time: 5.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;V \cdot \ell \le -1.69984137771513769 \cdot 10^{303}:\\ \;\;\;\;\frac{c0}{\sqrt{\sqrt{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}} \cdot \frac{\sqrt{\frac{1}{V} \cdot \frac{A}{\sqrt[3]{\ell}}}}{\sqrt{\sqrt{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}}\\ \mathbf{elif}\;V \cdot \ell \le -1.1396964155309059 \cdot 10^{-95}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{1}{\frac{V \cdot \ell}{A}}}\\ \mathbf{elif}\;V \cdot \ell \le -0.0:\\ \;\;\;\;\frac{c0 \cdot \sqrt{\frac{\frac{\sqrt[3]{A} \cdot \sqrt[3]{A}}{\sqrt[3]{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}}{V} \cdot \frac{\sqrt[3]{A}}{\sqrt[3]{\sqrt[3]{\ell}}}}}{\sqrt{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \sqrt[3]{\ell}}}}{\sqrt{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}\\ \end{array}\]

C

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 VAR;
	if (((V * l) <= -1.6998413777151377e+303)) {
		VAR = ((c0 / sqrt(sqrt((cbrt(l) * cbrt(l))))) * (sqrt(((1.0 / V) * (A / cbrt(l)))) / sqrt(sqrt((cbrt(l) * cbrt(l))))));
	} else {
		double VAR_1;
		if (((V * l) <= -1.1396964155309059e-95)) {
			VAR_1 = (c0 * sqrt((1.0 / ((V * l) / A))));
		} else {
			double VAR_2;
			if (((V * l) <= -0.0)) {
				VAR_2 = ((c0 * sqrt(((((cbrt(A) * cbrt(A)) / cbrt((cbrt(l) * cbrt(l)))) / V) * (cbrt(A) / cbrt(cbrt(l)))))) / sqrt((cbrt(l) * cbrt(l))));
			} else {
				VAR_2 = ((c0 * (sqrt(A) / sqrt((V * cbrt(l))))) / sqrt((cbrt(l) * cbrt(l))));
			}
			VAR_1 = VAR_2;
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus c0

Bits error versus A

Bits error versus V

Bits error versus l

Derivation

1. Split input into 4 regimes

2. if (* V l) < -1.6998413777151377e+303

   1. Initial program 42.1

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

   3. Applied *-un-lft-identity 42.1

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

   4. Applied times-frac 21.9

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{1}{V} \cdot \frac{A}{\ell}}}\]
   5. Using strategy rm

   6. Applied add-cube-cbrt 22.0

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

   7. Applied *-un-lft-identity 22.0

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

   8. Applied times-frac 22.0

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

   9. Applied associate-*r* 30.6

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

   10. Simplified 30.6

       \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{1}{V}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}} \cdot \frac{A}{\sqrt[3]{\ell}}}\]
   11. Using strategy rm

   12. Applied associate-*l/ 22.1

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

   13. Applied sqrt-div 13.0

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

   14. Applied associate-*r/ 13.1

       \[\leadsto \color{blue}{\frac{c0 \cdot \sqrt{\frac{1}{V} \cdot \frac{A}{\sqrt[3]{\ell}}}}{\sqrt{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}}\]
   15. Using strategy rm

   16. Applied add-sqr-sqrt 13.2

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

   17. Applied times-frac 13.1

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

   if -1.6998413777151377e+303 < (* V l) < -1.1396964155309059e-95

   1. Initial program 7.7

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

   3. Applied clear-num 8.2

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

   if -1.1396964155309059e-95 < (* V l) < -0.0

   1. Initial program 36.9

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

   3. Applied *-un-lft-identity 36.9

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

   4. Applied times-frac 28.0

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{1}{V} \cdot \frac{A}{\ell}}}\]
   5. Using strategy rm

   6. Applied add-cube-cbrt 28.3

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

   7. Applied *-un-lft-identity 28.3

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

   8. Applied times-frac 28.3

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

   9. Applied associate-*r* 32.9

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

   10. Simplified 32.9

       \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{1}{V}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}} \cdot \frac{A}{\sqrt[3]{\ell}}}\]
   11. Using strategy rm

   12. Applied associate-*l/ 26.9

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

   13. Applied sqrt-div 17.2

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

   14. Applied associate-*r/ 18.0

       \[\leadsto \color{blue}{\frac{c0 \cdot \sqrt{\frac{1}{V} \cdot \frac{A}{\sqrt[3]{\ell}}}}{\sqrt{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}}\]
   15. Using strategy rm

   16. Applied add-cube-cbrt 18.0

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

   17. Applied cbrt-prod 18.1

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

   18. Applied add-cube-cbrt 18.2

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

   19. Applied times-frac 18.2

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

   20. Applied associate-*r* 17.1

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

   21. Simplified 17.1

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

   if -0.0 < (* V l)

   1. Initial program 15.4

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

   3. Applied *-un-lft-identity 15.4

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

   4. Applied times-frac 17.9

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{1}{V} \cdot \frac{A}{\ell}}}\]
   5. Using strategy rm

   6. Applied add-cube-cbrt 18.2

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

   7. Applied *-un-lft-identity 18.2

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

   8. Applied times-frac 18.2

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

   9. Applied associate-*r* 15.2

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

   10. Simplified 15.2

       \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{1}{V}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}} \cdot \frac{A}{\sqrt[3]{\ell}}}\]
   11. Using strategy rm

   12. Applied associate-*l/ 16.6

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

   13. Applied sqrt-div 12.7

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

   14. Applied associate-*r/ 13.9

       \[\leadsto \color{blue}{\frac{c0 \cdot \sqrt{\frac{1}{V} \cdot \frac{A}{\sqrt[3]{\ell}}}}{\sqrt{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}}\]
   15. Using strategy rm

   16. Applied frac-times 13.2

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

   17. Applied sqrt-div 4.4

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

   18. Simplified 4.4

       \[\leadsto \frac{c0 \cdot \frac{\color{blue}{\sqrt{A}}}{\sqrt{V \cdot \sqrt[3]{\ell}}}}{\sqrt{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}\]
3. Recombined 4 regimes into one program.

4. Final simplification 8.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \le -1.69984137771513769 \cdot 10^{303}:\\ \;\;\;\;\frac{c0}{\sqrt{\sqrt{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}} \cdot \frac{\sqrt{\frac{1}{V} \cdot \frac{A}{\sqrt[3]{\ell}}}}{\sqrt{\sqrt{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}}\\ \mathbf{elif}\;V \cdot \ell \le -1.1396964155309059 \cdot 10^{-95}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{1}{\frac{V \cdot \ell}{A}}}\\ \mathbf{elif}\;V \cdot \ell \le -0.0:\\ \;\;\;\;\frac{c0 \cdot \sqrt{\frac{\frac{\sqrt[3]{A} \cdot \sqrt[3]{A}}{\sqrt[3]{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}}{V} \cdot \frac{\sqrt[3]{A}}{\sqrt[3]{\sqrt[3]{\ell}}}}}{\sqrt{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \sqrt[3]{\ell}}}}{\sqrt{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020079 +o rules:numerics
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  :precision binary64
  (* c0 (sqrt (/ A (* V l)))))