Average Error: 13.8 → 9.6
Time: 23.0s
Precision: binary64
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{M \cdot D}{2 \cdot d} \le 1.34580672114896385 \cdot 10^{39}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot h}{\ell}}\\ \mathbf{elif}\;\frac{M \cdot D}{2 \cdot d} \le 3.0137864743125973 \cdot 10^{212}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\sqrt{\frac{M \cdot D}{2 \cdot d}}\right)}^{2} \cdot \left({\left(\sqrt{\frac{M \cdot D}{2 \cdot d}}\right)}^{2} \cdot \frac{h}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1}\\ \end{array}\]
w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}
↓
\begin{array}{l}
\mathbf{if}\;\frac{M \cdot D}{2 \cdot d} \le 1.34580672114896385 \cdot 10^{39}:\\
\;\;\;\;w0 \cdot \sqrt{1 - \frac{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot h}{\ell}}\\

\mathbf{elif}\;\frac{M \cdot D}{2 \cdot d} \le 3.0137864743125973 \cdot 10^{212}:\\
\;\;\;\;w0 \cdot \sqrt{1 - {\left(\sqrt{\frac{M \cdot D}{2 \cdot d}}\right)}^{2} \cdot \left({\left(\sqrt{\frac{M \cdot D}{2 \cdot d}}\right)}^{2} \cdot \frac{h}{\ell}\right)}\\

\mathbf{else}:\\
\;\;\;\;w0 \cdot \sqrt{1}\\

\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) (((double) (M * D)) / ((double) (2.0 * d)))) <= 1.3458067211489639e+39)) {
		VAR = ((double) (w0 * ((double) sqrt(((double) (1.0 - ((double) (((double) (((double) pow(((double) (((double) (M / 2.0)) * ((double) (D / d)))), 2.0)) * h)) / l))))))));
	} else {
		double VAR_1;
		if ((((double) (((double) (M * D)) / ((double) (2.0 * d)))) <= 3.0137864743125973e+212)) {
			VAR_1 = ((double) (w0 * ((double) sqrt(((double) (1.0 - ((double) (((double) pow(((double) sqrt(((double) (((double) (M * D)) / ((double) (2.0 * d)))))), 2.0)) * ((double) (((double) pow(((double) sqrt(((double) (((double) (M * D)) / ((double) (2.0 * d)))))), 2.0)) * ((double) (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

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if (/ (* M D) (* 2.0 d)) < 1.34580672114896385e39

    1. Initial program 10.6

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
    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 times-frac6.8

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

    if 1.34580672114896385e39 < (/ (* M D) (* 2.0 d)) < 3.0137864743125973e212

    1. Initial program 24.5

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

    3. Applied add-sqr-sqrt24.6

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

    4. Applied unpow-prod-down24.6

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

    5. Applied associate-*l*17.5

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

    if 3.0137864743125973e212 < (/ (* M D) (* 2.0 d))

    1. Initial program 64.0

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

    2. Taylor expanded around 0 56.1

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

  4. Final simplification9.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{M \cdot D}{2 \cdot d} \le 1.34580672114896385 \cdot 10^{39}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot h}{\ell}}\\ \mathbf{elif}\;\frac{M \cdot D}{2 \cdot d} \le 3.0137864743125973 \cdot 10^{212}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\sqrt{\frac{M \cdot D}{2 \cdot d}}\right)}^{2} \cdot \left({\left(\sqrt{\frac{M \cdot D}{2 \cdot d}}\right)}^{2} \cdot \frac{h}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020168 
(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))))))