Average Error: 14.8 → 9.7
Time: 11.0s
Precision: 64
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \le -2.4720984130882936 \cdot 10^{300} \lor \neg \left(\frac{h}{\ell} \le -7.7434133461310193 \cdot 10^{-235}\right):\\ \;\;\;\;w0 \cdot \sqrt{1}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \frac{h}{\ell}\right)}\\ \end{array}\]
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 -2.4720984130882936 \cdot 10^{300} \lor \neg \left(\frac{h}{\ell} \le -7.7434133461310193 \cdot 10^{-235}\right):\\
\;\;\;\;w0 \cdot \sqrt{1}\\

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

\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 VAR;
	if ((((h / l) <= -2.4720984130882936e+300) || !((h / l) <= -7.74341334613102e-235))) {
		VAR = (w0 * sqrt(1.0));
	} else {
		VAR = (w0 * sqrt((1.0 - (pow(((M * D) / (2.0 * d)), (2.0 / 2.0)) * (pow(((M * D) / (2.0 * d)), (2.0 / 2.0)) * (h / l))))));
	}
	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 2 regimes

  2. if (/ h l) < -2.4720984130882936e+300 or -7.74341334613102e-235 < (/ h l)

    1. Initial program 14.9

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

    2. Taylor expanded around 0 7.4

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

    if -2.4720984130882936e+300 < (/ h l) < -7.74341334613102e-235

    1. Initial program 14.7

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

    3. Applied sqr-pow14.7

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

    4. Applied associate-*l*13.1

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

  4. Final simplification9.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \le -2.4720984130882936 \cdot 10^{300} \lor \neg \left(\frac{h}{\ell} \le -7.7434133461310193 \cdot 10^{-235}\right):\\ \;\;\;\;w0 \cdot \sqrt{1}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \frac{h}{\ell}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020079 +o rules:numerics
(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))))))