Henrywood and Agarwal, Equation (3)

Average Error: 19.0 → 14.1

Time: 5.0s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;V \cdot \ell \le -6.04930864463111567 \cdot 10^{-251}:\\ \;\;\;\;c0 \cdot \left(\sqrt{\sqrt{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt[3]{V} \cdot \sqrt[3]{V}} \cdot \frac{\frac{A}{\ell}}{\sqrt[3]{V}}}} \cdot \sqrt{\sqrt{\frac{1}{V} \cdot \frac{A}{\ell}}}\right)\\ \mathbf{elif}\;V \cdot \ell \le 1.2426573619189175 \cdot 10^{-285}:\\ \;\;\;\;\left(c0 \cdot \sqrt{\frac{1}{V}}\right) \cdot \sqrt{\frac{A}{\ell}}\\ \mathbf{elif}\;V \cdot \ell \le 5.5957256550553615 \cdot 10^{297}:\\ \;\;\;\;c0 \cdot \left(\sqrt{A} \cdot \sqrt{\frac{1}{V \cdot \ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \end{array}\]

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 temp;
	if (((V * l) <= -6.049308644631116e-251)) {
		temp = (c0 * (sqrt(sqrt((((cbrt(1.0) * cbrt(1.0)) / (cbrt(V) * cbrt(V))) * ((A / l) / cbrt(V))))) * sqrt(sqrt(((1.0 / V) * (A / l))))));
	} else {
		double temp_1;
		if (((V * l) <= 1.2426573619189175e-285)) {
			temp_1 = ((c0 * sqrt((1.0 / V))) * sqrt((A / l)));
		} else {
			double temp_2;
			if (((V * l) <= 5.595725655055362e+297)) {
				temp_2 = (c0 * (sqrt(A) * sqrt((1.0 / (V * l)))));
			} else {
				temp_2 = (c0 * sqrt(((A / V) / l)));
			}
			temp_1 = temp_2;
		}
		temp = temp_1;
	}
	return temp;
}

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) < -6.049308644631116e-251

   1. Initial program 14.4

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

   3. Applied *-un-lft-identity 14.4

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

   4. Applied times-frac 17.2

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

   6. Applied add-sqr-sqrt 17.4

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

   8. Applied add-cube-cbrt 17.4

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

   9. Applied add-cube-cbrt 17.4

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

   10. Applied times-frac 17.4

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

   11. Applied associate-*l* 17.4

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

   12. Simplified 17.4

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

   if -6.049308644631116e-251 < (* V l) < 1.2426573619189175e-285

   1. Initial program 51.7

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

   3. Applied *-un-lft-identity 51.7

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

   4. Applied times-frac 33.3

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

   5. Applied sqrt-prod 38.0

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

   6. Applied associate-*r* 38.6

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

   if 1.2426573619189175e-285 < (* V l) < 5.595725655055362e+297

   1. Initial program 9.4

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

   3. Applied div-inv 9.4

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

   4. Applied sqrt-prod 0.4

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

   if 5.595725655055362e+297 < (* V l)

   1. Initial program 38.6

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

   3. Applied associate-/r* 22.2

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

4. Final simplification 14.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \le -6.04930864463111567 \cdot 10^{-251}:\\ \;\;\;\;c0 \cdot \left(\sqrt{\sqrt{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt[3]{V} \cdot \sqrt[3]{V}} \cdot \frac{\frac{A}{\ell}}{\sqrt[3]{V}}}} \cdot \sqrt{\sqrt{\frac{1}{V} \cdot \frac{A}{\ell}}}\right)\\ \mathbf{elif}\;V \cdot \ell \le 1.2426573619189175 \cdot 10^{-285}:\\ \;\;\;\;\left(c0 \cdot \sqrt{\frac{1}{V}}\right) \cdot \sqrt{\frac{A}{\ell}}\\ \mathbf{elif}\;V \cdot \ell \le 5.5957256550553615 \cdot 10^{297}:\\ \;\;\;\;c0 \cdot \left(\sqrt{A} \cdot \sqrt{\frac{1}{V \cdot \ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020053 +o rules:numerics
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  :precision binary64
  (* c0 (sqrt (/ A (* V l)))))