Average Error: 14.1 → 8.4
Time: 11.0s
Precision: 64
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \le -3.5399816888251004 \cdot 10^{69}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\left(\frac{2}{2}\right)} \cdot h\right)}{\ell}}\\ \mathbf{elif}\;\frac{h}{\ell} \le -7.59548904149731453 \cdot 10^{-107}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \frac{h}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\left(\frac{2}{2}\right)} \cdot \frac{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\left(\frac{2}{2}\right)} \cdot h}{\ell}}\\ \end{array}\]
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 -3.5399816888251004 \cdot 10^{69}:\\
\;\;\;\;w0 \cdot \sqrt{1 - \frac{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\left(\frac{2}{2}\right)} \cdot h\right)}{\ell}}\\

\mathbf{elif}\;\frac{h}{\ell} \le -7.59548904149731453 \cdot 10^{-107}:\\
\;\;\;\;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)}\\

\mathbf{else}:\\
\;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\left(\frac{2}{2}\right)} \cdot \frac{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\left(\frac{2}{2}\right)} \cdot h}{\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)) <= -3.5399816888251e+69)) {
		VAR = ((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 / 2.0)) * ((double) (D / d)))), ((double) (2.0 / 2.0)))) * h)))) / l))))))));
	} else {
		double VAR_1;
		if ((((double) (h / l)) <= -7.595489041497315e-107)) {
			VAR_1 = ((double) (w0 * ((double) sqrt(((double) (1.0 - ((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)))) * ((double) (h / l))))))))))));
		} else {
			VAR_1 = ((double) (w0 * ((double) sqrt(((double) (1.0 - ((double) (((double) pow(((double) (((double) (M / 2.0)) * ((double) (D / d)))), ((double) (2.0 / 2.0)))) * ((double) (((double) (((double) pow(((double) (((double) (M / 2.0)) * ((double) (D / 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) < -3.5399816888251e+69

    1. Initial program 29.8

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

    3. Applied associate-*r/19.0

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

    5. Applied sqr-pow19.0

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

    6. Applied associate-*l*18.0

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

    8. Applied times-frac18.6

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

    if -3.5399816888251e+69 < (/ h l) < -7.595489041497315e-107

    1. Initial program 12.5

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

    3. Applied sqr-pow12.5

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

    4. Applied associate-*l*11.7

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

    if -7.595489041497315e-107 < (/ h l)

    1. Initial program 9.2

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

    3. Applied associate-*r/6.9

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

    5. Applied sqr-pow6.9

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

    6. Applied associate-*l*4.9

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

    8. Applied times-frac5.8

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

    10. Applied *-un-lft-identity5.8

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

    11. Applied times-frac5.2

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

    12. Simplified3.9

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

  4. Final simplification8.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \le -3.5399816888251004 \cdot 10^{69}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\left(\frac{2}{2}\right)} \cdot h\right)}{\ell}}\\ \mathbf{elif}\;\frac{h}{\ell} \le -7.59548904149731453 \cdot 10^{-107}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \frac{h}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\left(\frac{2}{2}\right)} \cdot \frac{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\left(\frac{2}{2}\right)} \cdot h}{\ell}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020122 
(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))))))