Henrywood and Agarwal, Equation (9a)

Average Error: 14.7 → 9.4

Time: 24.4min

Precision: binary64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \le -9.3148557744348976 \cdot 10^{104}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{{\left(\frac{1}{\frac{2 \cdot d}{M \cdot D}}\right)}^{2} \cdot h}{\ell}}\\ \mathbf{elif}\;\frac{h}{\ell} \le -5.02511197815653748 \cdot 10^{-236}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(\left(\sqrt[3]{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \frac{h}{\ell}} \cdot \sqrt[3]{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \frac{h}{\ell}}\right) \cdot \sqrt[3]{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \frac{h}{\ell}}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1}\\ \end{array}\]

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) (M * D)) / ((double) (2.0 * d))), 2.0)) * (h / l)))))))));
}

↓

double code(double w0, double M, double D, double h, double l, double d) {
	double VAR;
	if (((h / l) <= -9.314855774434898e+104)) {
		VAR = ((double) (w0 * ((double) sqrt(((double) (1.0 - (((double) (((double) pow((1.0 / (((double) (2.0 * d)) / ((double) (M * D)))), 2.0)) * h)) / l)))))));
	} else {
		double VAR_1;
		if (((h / l) <= -5.0251119781565375e-236)) {
			VAR_1 = ((double) (w0 * ((double) sqrt(((double) (1.0 - ((double) (((double) pow((((double) (M * D)) / ((double) (2.0 * d))), (2.0 / 2.0))) * ((double) (((double) (((double) cbrt(((double) (((double) pow((((double) (M * D)) / ((double) (2.0 * d))), (2.0 / 2.0))) * (h / l))))) * ((double) cbrt(((double) (((double) pow((((double) (M * D)) / ((double) (2.0 * d))), (2.0 / 2.0))) * (h / l))))))) * ((double) cbrt(((double) (((double) pow((((double) (M * D)) / ((double) (2.0 * d))), (2.0 / 2.0))) * (h / l)))))))))))))));
		} else {
			VAR_1 = ((double) (w0 * ((double) sqrt(1.0))));
		}
		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 3 regimes

2. if (/ h l) < -9.3148557744348976e104

   1. Initial program 31.8

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

   3. Applied associate-*r/ 19.9

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

   5. Applied clear-num 19.9

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

   if -9.3148557744348976e104 < (/ h l) < -5.02511197815653748e-236

   1. Initial program 14.5

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

   3. Applied sqr-pow 14.5

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

   4. Applied associate-*l* 12.6

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

   6. Applied add-cube-cbrt 12.7

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

   if -5.02511197815653748e-236 < (/ h l)

   1. Initial program 8.9

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

   2. Taylor expanded around 0 4.0

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

4. Final simplification 9.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \le -9.3148557744348976 \cdot 10^{104}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{{\left(\frac{1}{\frac{2 \cdot d}{M \cdot D}}\right)}^{2} \cdot h}{\ell}}\\ \mathbf{elif}\;\frac{h}{\ell} \le -5.02511197815653748 \cdot 10^{-236}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(\left(\sqrt[3]{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \frac{h}{\ell}} \cdot \sqrt[3]{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \frac{h}{\ell}}\right) \cdot \sqrt[3]{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \frac{h}{\ell}}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020182 
(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))))))