Henrywood and Agarwal, Equation (3)

Average Error: 19.0 → 14.5

Time: 6.1s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;V \cdot \ell \le -7.3644199609459516 \cdot 10^{-178}:\\ \;\;\;\;\left(c0 \cdot \sqrt{\sqrt{\frac{A}{V \cdot \ell}}}\right) \cdot \sqrt{\sqrt{\frac{A}{V \cdot \ell}}}\\ \mathbf{elif}\;V \cdot \ell \le 0.0:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{1}{V} \cdot A}}{\sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \left(\sqrt{A} \cdot \sqrt{\frac{1}{V \cdot \ell}}\right)\\ \end{array}\]

C

double code(double c0, double A, double V, double l) {
	return ((double) (c0 * ((double) sqrt(((double) (A / ((double) (V * l))))))));
}

↓

double code(double c0, double A, double V, double l) {
	double VAR;
	if ((((double) (V * l)) <= -7.364419960945952e-178)) {
		VAR = ((double) (((double) (c0 * ((double) sqrt(((double) sqrt(((double) (A / ((double) (V * l)))))))))) * ((double) sqrt(((double) sqrt(((double) (A / ((double) (V * l))))))))));
	} else {
		double VAR_1;
		if ((((double) (V * l)) <= 0.0)) {
			VAR_1 = ((double) (c0 * ((double) (((double) sqrt(((double) (((double) (1.0 / V)) * A)))) / ((double) sqrt(l))))));
		} else {
			VAR_1 = ((double) (c0 * ((double) (((double) sqrt(A)) * ((double) sqrt(((double) (1.0 / ((double) (V * l))))))))));
		}
		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 3 regimes

2. if (* V l) < -7.3644199609459516e-178

   1. Initial program 14.4

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

   3. Applied add-sqr-sqrt 14.4

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

   4. Applied sqrt-prod 14.6

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

   5. Applied associate-*r* 14.6

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

   if -7.3644199609459516e-178 < (* V l) < 0.0

   1. Initial program 45.9

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

   3. Applied *-un-lft-identity 45.9

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

   4. Applied times-frac 31.8

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

   6. Applied associate-*r/ 31.9

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

   7. Applied sqrt-div 38.7

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

   if 0.0 < (* V l)

   1. Initial program 15.1

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

   3. Applied div-inv 15.2

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

   4. Applied sqrt-prod 7.2

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

4. Final simplification 14.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \le -7.3644199609459516 \cdot 10^{-178}:\\ \;\;\;\;\left(c0 \cdot \sqrt{\sqrt{\frac{A}{V \cdot \ell}}}\right) \cdot \sqrt{\sqrt{\frac{A}{V \cdot \ell}}}\\ \mathbf{elif}\;V \cdot \ell \le 0.0:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{1}{V} \cdot A}}{\sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \left(\sqrt{A} \cdot \sqrt{\frac{1}{V \cdot \ell}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020149 
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  :precision binary64
  (* c0 (sqrt (/ A (* V l)))))