Henrywood and Agarwal, Equation (9a)

Average Error: 14.6 → 9.5

Time: 9.6s

Precision: binary64

Math

\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]

↓

\[\begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \le -8.68496395426944278 \cdot 10^{215}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h \cdot {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2}}{\ell}}\\ \mathbf{elif}\;\frac{h}{\ell} \le -3.6399821796397183 \cdot 10^{-96}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \sqrt[3]{\frac{h}{\ell} \cdot {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2}} \cdot \left(\sqrt[3]{\frac{h}{\ell} \cdot {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2}} \cdot \sqrt[3]{\frac{h}{\ell} \cdot {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2}}\right)}\\ \mathbf{elif}\;\frac{h}{\ell} \le -1.0879697061047526 \cdot 10^{-229}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{{\left(0.5 \cdot \left(D \cdot \frac{M}{d}\right)\right)}^{\left(\frac{2}{2}\right)}}{\sqrt[3]{\ell}} \cdot \frac{h \cdot {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1}\\ \end{array}\]

C

double code(double w0, double M, double D, double h, double l, double d) {
	return ((double) (w0 * ((double) sqrt(((double) (1.0 - ((double) (((double) pow(((double) (((double) (M * D)) / ((double) (2.0 * d)))), 2.0)) * ((double) (h / l))))))))));
}

↓

double code(double w0, double M, double D, double h, double l, double d) {
	double VAR;
	if ((((double) (h / l)) <= -8.684963954269443e+215)) {
		VAR = ((double) (w0 * ((double) sqrt(((double) (1.0 - ((double) (((double) (h * ((double) pow(((double) (M * ((double) (D / ((double) (2.0 * d)))))), 2.0)))) / l))))))));
	} else {
		double VAR_1;
		if ((((double) (h / l)) <= -3.6399821796397183e-96)) {
			VAR_1 = ((double) (w0 * ((double) sqrt(((double) (1.0 - ((double) (((double) cbrt(((double) (((double) (h / l)) * ((double) pow(((double) (M * ((double) (D / ((double) (2.0 * d)))))), 2.0)))))) * ((double) (((double) cbrt(((double) (((double) (h / l)) * ((double) pow(((double) (M * ((double) (D / ((double) (2.0 * d)))))), 2.0)))))) * ((double) cbrt(((double) (((double) (h / l)) * ((double) pow(((double) (M * ((double) (D / ((double) (2.0 * d)))))), 2.0))))))))))))))));
		} else {
			double VAR_2;
			if ((((double) (h / l)) <= -1.0879697061047526e-229)) {
				VAR_2 = ((double) (w0 * ((double) sqrt(((double) (1.0 - ((double) (((double) (((double) pow(((double) (0.5 * ((double) (D * ((double) (M / d)))))), ((double) (2.0 / 2.0)))) / ((double) cbrt(l)))) * ((double) (((double) (h * ((double) pow(((double) (M * ((double) (D / ((double) (2.0 * d)))))), ((double) (2.0 / 2.0)))))) / ((double) (((double) cbrt(l)) * ((double) cbrt(l))))))))))))));
			} else {
				VAR_2 = ((double) (w0 * ((double) sqrt(1.0))));
			}
			VAR_1 = VAR_2;
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus w0

Bits error versus M

Bits error versus D

Bits error versus h

Bits error versus l

Bits error versus d

Derivation

1. Split input into 4 regimes

2. if (/ h l) < -8.68496395426944278e215

   1. Initial program 44.9

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]

   2. Simplified 44.6

      \[\leadsto \color{blue}{w0 \cdot \sqrt{1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}}\]
   3. Using strategy rm

   4. Applied associate-*r/ 23.1

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{{\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot h}{\ell}}}\]

   5. Simplified 23.1

      \[\leadsto w0 \cdot \sqrt{1 - \frac{\color{blue}{h \cdot {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2}}}{\ell}}\]

   if -8.68496395426944278e215 < (/ h l) < -3.6399821796397183e-96

   1. Initial program 14.2

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]

   2. Simplified 14.5

      \[\leadsto \color{blue}{w0 \cdot \sqrt{1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}}\]
   3. Using strategy rm

   4. Applied add-cube-cbrt 14.5

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\left(\sqrt[3]{{\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \cdot \sqrt[3]{{\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\right) \cdot \sqrt[3]{{\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}}}\]

   5. Simplified 14.5

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\left(\sqrt[3]{\frac{h}{\ell} \cdot {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2}} \cdot \sqrt[3]{\frac{h}{\ell} \cdot {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2}}\right)} \cdot \sqrt[3]{{\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}}\]

   6. Simplified 14.5

      \[\leadsto w0 \cdot \sqrt{1 - \left(\sqrt[3]{\frac{h}{\ell} \cdot {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2}} \cdot \sqrt[3]{\frac{h}{\ell} \cdot {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2}}\right) \cdot \color{blue}{\sqrt[3]{\frac{h}{\ell} \cdot {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2}}}}\]

   if -3.6399821796397183e-96 < (/ h l) < -1.0879697061047526e-229

   1. Initial program 13.2

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]

   2. Simplified 13.3

      \[\leadsto \color{blue}{w0 \cdot \sqrt{1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}}\]
   3. Using strategy rm

   4. Applied add-cube-cbrt 13.3

      \[\leadsto w0 \cdot \sqrt{1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\color{blue}{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \sqrt[3]{\ell}}}}\]

   5. Applied *-un-lft-identity 13.3

      \[\leadsto w0 \cdot \sqrt{1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{\color{blue}{1 \cdot h}}{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \sqrt[3]{\ell}}}\]

   6. Applied times-frac 13.3

      \[\leadsto w0 \cdot \sqrt{1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \color{blue}{\left(\frac{1}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \frac{h}{\sqrt[3]{\ell}}\right)}}\]

   7. Applied associate-*r* 14.0

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\left({\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}\right) \cdot \frac{h}{\sqrt[3]{\ell}}}}\]

   8. Simplified 14.0

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{{\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}} \cdot \frac{h}{\sqrt[3]{\ell}}}\]
   9. Using strategy rm

   10. Applied sqr-pow 14.0

       \[\leadsto w0 \cdot \sqrt{1 - \frac{\color{blue}{{\left(M \cdot \frac{D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)}}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \frac{h}{\sqrt[3]{\ell}}}\]

   11. Applied times-frac 12.0

       \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\left(\frac{{\left(M \cdot \frac{D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)}}{\sqrt[3]{\ell}} \cdot \frac{{\left(M \cdot \frac{D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)}}{\sqrt[3]{\ell}}\right)} \cdot \frac{h}{\sqrt[3]{\ell}}}\]

   12. Applied associate-*l* 8.8

       \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{{\left(M \cdot \frac{D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)}}{\sqrt[3]{\ell}} \cdot \left(\frac{{\left(M \cdot \frac{D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)}}{\sqrt[3]{\ell}} \cdot \frac{h}{\sqrt[3]{\ell}}\right)}}\]

   13. Simplified 8.8

       \[\leadsto w0 \cdot \sqrt{1 - \frac{{\left(M \cdot \frac{D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)}}{\sqrt[3]{\ell}} \cdot \color{blue}{\left(\frac{h}{\sqrt[3]{\ell}} \cdot \frac{{\left(M \cdot \frac{D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)}}{\sqrt[3]{\ell}}\right)}}\]
   14. Using strategy rm

   15. Applied frac-times 9.0

       \[\leadsto w0 \cdot \sqrt{1 - \frac{{\left(M \cdot \frac{D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)}}{\sqrt[3]{\ell}} \cdot \color{blue}{\frac{h \cdot {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}}\]

   16. Taylor expanded around 0 10.6

       \[\leadsto w0 \cdot \sqrt{1 - \frac{{\color{blue}{\left(0.5 \cdot \frac{M \cdot D}{d}\right)}}^{\left(\frac{2}{2}\right)}}{\sqrt[3]{\ell}} \cdot \frac{h \cdot {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}\]

   17. Simplified 10.3

       \[\leadsto w0 \cdot \sqrt{1 - \frac{{\color{blue}{\left(0.5 \cdot \left(\frac{M}{d} \cdot D\right)\right)}}^{\left(\frac{2}{2}\right)}}{\sqrt[3]{\ell}} \cdot \frac{h \cdot {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}\]

   if -1.0879697061047526e-229 < (/ h l)

   1. Initial program 8.8

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]

   2. Simplified 8.7

      \[\leadsto \color{blue}{w0 \cdot \sqrt{1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}}\]

   3. Taylor expanded around 0 4.0

      \[\leadsto w0 \cdot \color{blue}{\sqrt{1}}\]
3. Recombined 4 regimes into one program.

4. Final simplification 9.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \le -8.68496395426944278 \cdot 10^{215}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h \cdot {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2}}{\ell}}\\ \mathbf{elif}\;\frac{h}{\ell} \le -3.6399821796397183 \cdot 10^{-96}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \sqrt[3]{\frac{h}{\ell} \cdot {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2}} \cdot \left(\sqrt[3]{\frac{h}{\ell} \cdot {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2}} \cdot \sqrt[3]{\frac{h}{\ell} \cdot {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{2}}\right)}\\ \mathbf{elif}\;\frac{h}{\ell} \le -1.0879697061047526 \cdot 10^{-229}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{{\left(0.5 \cdot \left(D \cdot \frac{M}{d}\right)\right)}^{\left(\frac{2}{2}\right)}}{\sqrt[3]{\ell}} \cdot \frac{h \cdot {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020185 
(FPCore (w0 M D h l d)
  :name "Henrywood and Agarwal, Equation (9a)"
  :precision binary64
  (* w0 (sqrt (- 1.0 (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l))))))