Average Error: 14.3 → 9.5
Time: 10.0s
Precision: binary64
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \le -2.5571171732421548 \cdot 10^{266}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\sqrt[3]{{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h\right)}^{3}}}{\ell}}\\ \mathbf{elif}\;\frac{h}{\ell} \le -1.7270570331742067 \cdot 10^{-119}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot h\right)}{\ell}}\\ \end{array}\]
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 -2.5571171732421548 \cdot 10^{266}:\\
\;\;\;\;w0 \cdot \sqrt{1 - \frac{\sqrt[3]{{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h\right)}^{3}}}{\ell}}\\

\mathbf{elif}\;\frac{h}{\ell} \le -1.7270570331742067 \cdot 10^{-119}:\\
\;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\\

\mathbf{else}:\\
\;\;\;\;w0 \cdot \sqrt{1 - \frac{{\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 h\right)}{\ell}}\\

\end{array}
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)) <= -2.5571171732421548e+266)) {
		VAR = ((double) (w0 * ((double) sqrt(((double) (1.0 - ((double) (((double) cbrt(((double) pow(((double) (((double) pow(((double) (((double) (M * D)) / ((double) (2.0 * d)))), 2.0)) * h)), 3.0)))) / l))))))));
	} else {
		double VAR_1;
		if ((((double) (h / l)) <= -1.7270570331742067e-119)) {
			VAR_1 = ((double) (w0 * ((double) sqrt(((double) (1.0 - ((double) (((double) pow(((double) (((double) (M / 2.0)) * ((double) (D / d)))), 2.0)) * ((double) (h / l))))))))));
		} else {
			VAR_1 = ((double) (w0 * ((double) sqrt(((double) (1.0 - ((double) (((double) (((double) pow(((double) (((double) (M * D)) / ((double) (2.0 * d)))), ((double) (2.0 / 2.0)))) * ((double) (((double) pow(((double) (((double) (M * D)) / ((double) (2.0 * d)))), ((double) (2.0 / 2.0)))) * h)))) / l))))))));
		}
		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

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if (/ h l) < -2.5571171732421548e266

    1. Initial program 52.9

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

    3. Applied associate-*r/23.5

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

    5. Applied add-cbrt-cube55.4

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

    6. Applied add-cbrt-cube57.3

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

    7. Applied cbrt-unprod57.3

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

    8. Simplified28.0

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

    if -2.5571171732421548e266 < (/ h l) < -1.7270570331742067e-119

    1. Initial program 13.9

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

    3. Applied times-frac13.9

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

    if -1.7270570331742067e-119 < (/ h l)

    1. Initial program 9.5

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

    3. Applied associate-*r/6.8

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

    5. Applied sqr-pow6.8

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

    6. Applied associate-*l*4.9

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

  4. Final simplification9.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \le -2.5571171732421548 \cdot 10^{266}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\sqrt[3]{{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h\right)}^{3}}}{\ell}}\\ \mathbf{elif}\;\frac{h}{\ell} \le -1.7270570331742067 \cdot 10^{-119}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot h\right)}{\ell}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020150 
(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))))))