Henrywood and Agarwal, Equation (3)

Average Error: 19.4 → 11.1

Time: 5.9s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;V \cdot \ell \le -1.3027803198688765 \cdot 10^{132}:\\ \;\;\;\;\left(\left|\frac{\sqrt[3]{A}}{\sqrt[3]{V \cdot \ell}}\right| \cdot c0\right) \cdot \sqrt{\sqrt[3]{\frac{\sqrt[3]{A} \cdot \sqrt[3]{A}}{V}} \cdot \sqrt[3]{\frac{\sqrt[3]{A}}{\ell}}}\\ \mathbf{elif}\;V \cdot \ell \le -2.76350186409490512 \cdot 10^{-133}:\\ \;\;\;\;\left(c0 \cdot \sqrt{\sqrt{\frac{A}{V \cdot \ell}}}\right) \cdot \sqrt{\sqrt{\frac{A}{V \cdot \ell}}}\\ \mathbf{elif}\;V \cdot \ell \le -9.2539290659964874 \cdot 10^{-275}:\\ \;\;\;\;\left(\left|\frac{\sqrt[3]{A}}{\sqrt[3]{V \cdot \ell}}\right| \cdot c0\right) \cdot \sqrt{\sqrt[3]{\frac{1}{V}} \cdot \sqrt[3]{\frac{A}{\ell}}}\\ \mathbf{elif}\;V \cdot \ell \le -0.0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;V \cdot \ell \le 5.28837167649122347 \cdot 10^{273}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\sqrt{A}}{V} \cdot \frac{\sqrt{A}}{\ell}}\\ \end{array}\]

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)) <= -1.3027803198688765e+132)) {
		VAR = ((double) (((double) (((double) fabs(((double) (((double) cbrt(A)) / ((double) cbrt(((double) (V * l)))))))) * c0)) * ((double) sqrt(((double) (((double) cbrt(((double) (((double) (((double) cbrt(A)) * ((double) cbrt(A)))) / V)))) * ((double) cbrt(((double) (((double) cbrt(A)) / l))))))))));
	} else {
		double VAR_1;
		if ((((double) (V * l)) <= -2.763501864094905e-133)) {
			VAR_1 = ((double) (((double) (c0 * ((double) sqrt(((double) sqrt(((double) (A / ((double) (V * l)))))))))) * ((double) sqrt(((double) sqrt(((double) (A / ((double) (V * l))))))))));
		} else {
			double VAR_2;
			if ((((double) (V * l)) <= -9.253929065996487e-275)) {
				VAR_2 = ((double) (((double) (((double) fabs(((double) (((double) cbrt(A)) / ((double) cbrt(((double) (V * l)))))))) * c0)) * ((double) sqrt(((double) (((double) cbrt(((double) (1.0 / V)))) * ((double) cbrt(((double) (A / l))))))))));
			} else {
				double VAR_3;
				if ((((double) (V * l)) <= -0.0)) {
					VAR_3 = ((double) (c0 * ((double) sqrt(((double) (((double) (A / V)) / l))))));
				} else {
					double VAR_4;
					if ((((double) (V * l)) <= 5.288371676491223e+273)) {
						VAR_4 = ((double) (c0 * ((double) (((double) sqrt(A)) / ((double) sqrt(((double) (V * l))))))));
					} else {
						VAR_4 = ((double) (c0 * ((double) sqrt(((double) (((double) (((double) sqrt(A)) / V)) * ((double) (((double) sqrt(A)) / l))))))));
					}
					VAR_3 = VAR_4;
				}
				VAR_2 = VAR_3;
			}
			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 6 regimes

2. if (* V l) < -1.3027803198688765e132

   1. Initial program 25.1

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

   3. Applied add-cube-cbrt 25.4

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

   4. Applied sqrt-prod 25.3

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

   5. Applied associate-*r* 25.3

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

   6. Simplified 25.3

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

   8. Applied cbrt-div 25.3

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

   10. Applied add-cube-cbrt 25.4

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

   11. Applied times-frac 25.9

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

   12. Applied cbrt-prod 21.1

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

   if -1.3027803198688765e132 < (* V l) < -2.76350186409490512e-133

   1. Initial program 5.0

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

   3. Applied add-sqr-sqrt 5.0

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

   4. Applied sqrt-prod 5.3

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

   5. Applied associate-*r* 5.3

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

   if -2.76350186409490512e-133 < (* V l) < -9.2539290659964874e-275

   1. Initial program 16.5

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

   3. Applied add-cube-cbrt 16.8

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

   4. Applied sqrt-prod 16.8

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

   5. Applied associate-*r* 16.9

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

   6. Simplified 16.9

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

   8. Applied cbrt-div 16.9

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

   10. Applied *-un-lft-identity 16.9

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

   11. Applied times-frac 19.1

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

   12. Applied cbrt-prod 10.8

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

   if -9.2539290659964874e-275 < (* V l) < -0.0

   1. Initial program 58.7

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

   3. Applied associate-/r* 33.9

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

   if -0.0 < (* V l) < 5.28837167649122347e273

   1. Initial program 10.9

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

   3. Applied sqrt-div 0.7

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

   if 5.28837167649122347e273 < (* V l)

   1. Initial program 36.9

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

   3. Applied add-sqr-sqrt 36.9

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

   4. Applied times-frac 23.5

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\sqrt{A}}{V} \cdot \frac{\sqrt{A}}{\ell}}}\]
3. Recombined 6 regimes into one program.

4. Final simplification 11.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \le -1.3027803198688765 \cdot 10^{132}:\\ \;\;\;\;\left(\left|\frac{\sqrt[3]{A}}{\sqrt[3]{V \cdot \ell}}\right| \cdot c0\right) \cdot \sqrt{\sqrt[3]{\frac{\sqrt[3]{A} \cdot \sqrt[3]{A}}{V}} \cdot \sqrt[3]{\frac{\sqrt[3]{A}}{\ell}}}\\ \mathbf{elif}\;V \cdot \ell \le -2.76350186409490512 \cdot 10^{-133}:\\ \;\;\;\;\left(c0 \cdot \sqrt{\sqrt{\frac{A}{V \cdot \ell}}}\right) \cdot \sqrt{\sqrt{\frac{A}{V \cdot \ell}}}\\ \mathbf{elif}\;V \cdot \ell \le -9.2539290659964874 \cdot 10^{-275}:\\ \;\;\;\;\left(\left|\frac{\sqrt[3]{A}}{\sqrt[3]{V \cdot \ell}}\right| \cdot c0\right) \cdot \sqrt{\sqrt[3]{\frac{1}{V}} \cdot \sqrt[3]{\frac{A}{\ell}}}\\ \mathbf{elif}\;V \cdot \ell \le -0.0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;V \cdot \ell \le 5.28837167649122347 \cdot 10^{273}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\sqrt{A}}{V} \cdot \frac{\sqrt{A}}{\ell}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020174 
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  :precision binary64
  (* c0 (sqrt (/ A (* V l)))))