Average Error: 14.4 → 9.8
Time: 28.0s
Precision: 64
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \le -1.50657721029544602 \cdot 10^{242}:\\ \;\;\;\;w0 \cdot \sqrt{1}\\ \mathbf{elif}\;\frac{h}{\ell} \le -6.2208622639849537 \cdot 10^{-173}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \frac{h \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)}}{\ell}}\\ \end{array}\]
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 -1.50657721029544602 \cdot 10^{242}:\\
\;\;\;\;w0 \cdot \sqrt{1}\\

\mathbf{elif}\;\frac{h}{\ell} \le -6.2208622639849537 \cdot 10^{-173}:\\
\;\;\;\;w0 \cdot \sqrt{1 - {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\\

\mathbf{else}:\\
\;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \frac{h \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)}}{\ell}}\\

\end{array}
double code(double w0, double M, double D, double h, double l, double d) {
	return (w0 * sqrt((1.0 - (pow(((M * D) / (2.0 * d)), 2.0) * (h / l)))));
}
↓
double code(double w0, double M, double D, double h, double l, double d) {
	double temp;
	if (((h / l) <= -1.506577210295446e+242)) {
		temp = (w0 * sqrt(1.0));
	} else {
		double temp_1;
		if (((h / l) <= -6.220862263984954e-173)) {
			temp_1 = (w0 * sqrt((1.0 - (pow((D * (M / (2.0 * d))), 2.0) * (h / l)))));
		} else {
			temp_1 = (w0 * sqrt((1.0 - (pow(((M * D) / (2.0 * d)), (2.0 / 2.0)) * ((h * pow(((M * D) / (2.0 * d)), (2.0 / 2.0))) / l)))));
		}
		temp = temp_1;
	}
	return temp;
}

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) < -1.506577210295446e+242

    1. Initial program 50.7

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

    2. Taylor expanded around 0 33.5

      \[\leadsto w0 \cdot \color{blue}{\sqrt{1}}\]

    if -1.506577210295446e+242 < (/ h l) < -6.220862263984954e-173

    1. Initial program 14.3

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

    3. Applied *-un-lft-identity14.3

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

    4. Applied *-commutative14.3

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

    5. Applied times-frac14.2

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

    6. Simplified14.2

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

    if -6.220862263984954e-173 < (/ h l)

    1. Initial program 9.1

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

    3. Applied sqr-pow9.1

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

    4. Applied associate-*l*6.9

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

    6. Applied associate-*r/3.6

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

    7. Simplified3.6

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

  4. Final simplification9.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \le -1.50657721029544602 \cdot 10^{242}:\\ \;\;\;\;w0 \cdot \sqrt{1}\\ \mathbf{elif}\;\frac{h}{\ell} \le -6.2208622639849537 \cdot 10^{-173}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \frac{h \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)}}{\ell}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020066 
(FPCore (w0 M D h l d)
  :name "Henrywood and Agarwal, Equation (9a)"
  :precision binary64
  (* w0 (sqrt (- 1 (* (pow (/ (* M D) (* 2 d)) 2) (/ h l))))))