Average Error: 14.3 → 8.9
Time: 16.2s
Precision: binary64
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{M \cdot D}{2 \cdot d} = -inf.0:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M}{\frac{2 \cdot d}{D}}\right)}^{2} \cdot \frac{h}{\ell}}\\ \mathbf{elif}\;\frac{M \cdot D}{2 \cdot d} \le -1.45904916529408281 \cdot 10^{32}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \frac{h}{\ell}\right)}\\ \mathbf{elif}\;\frac{M \cdot D}{2 \cdot d} \le 4.53816255015036534 \cdot 10^{-111}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{{\left(M \cdot \frac{1}{\frac{2 \cdot d}{D}}\right)}^{2} \cdot h}{\ell}}\\ \mathbf{elif}\;\frac{M \cdot D}{2 \cdot d} \le 3.36159024633077472 \cdot 10^{113}:\\ \;\;\;\;\left(w0 \cdot \sqrt{\sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}}\right) \cdot \sqrt{\sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{{\left(\sqrt{\frac{M}{\frac{2 \cdot d}{D}}}\right)}^{2} \cdot \left({\left(\sqrt{\frac{M}{\frac{2 \cdot d}{D}}}\right)}^{2} \cdot h\right)}{\ell}}\\ \end{array}\]
w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}
↓
\begin{array}{l}
\mathbf{if}\;\frac{M \cdot D}{2 \cdot d} = -inf.0:\\
\;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M}{\frac{2 \cdot d}{D}}\right)}^{2} \cdot \frac{h}{\ell}}\\

\mathbf{elif}\;\frac{M \cdot D}{2 \cdot d} \le -1.45904916529408281 \cdot 10^{32}:\\
\;\;\;\;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{elif}\;\frac{M \cdot D}{2 \cdot d} \le 4.53816255015036534 \cdot 10^{-111}:\\
\;\;\;\;w0 \cdot \sqrt{1 - \frac{{\left(M \cdot \frac{1}{\frac{2 \cdot d}{D}}\right)}^{2} \cdot h}{\ell}}\\

\mathbf{elif}\;\frac{M \cdot D}{2 \cdot d} \le 3.36159024633077472 \cdot 10^{113}:\\
\;\;\;\;\left(w0 \cdot \sqrt{\sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}}\right) \cdot \sqrt{\sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}}\\

\mathbf{else}:\\
\;\;\;\;w0 \cdot \sqrt{1 - \frac{{\left(\sqrt{\frac{M}{\frac{2 \cdot d}{D}}}\right)}^{2} \cdot \left({\left(\sqrt{\frac{M}{\frac{2 \cdot d}{D}}}\right)}^{2} \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) (((double) (M * D)) / ((double) (2.0 * d)))) <= -inf.0)) {
		VAR = ((double) (w0 * ((double) sqrt(((double) (1.0 - ((double) (((double) pow(((double) (M / ((double) (((double) (2.0 * d)) / D)))), 2.0)) * ((double) (h / l))))))))));
	} else {
		double VAR_1;
		if ((((double) (((double) (M * D)) / ((double) (2.0 * d)))) <= -1.4590491652940828e+32)) {
			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 {
			double VAR_2;
			if ((((double) (((double) (M * D)) / ((double) (2.0 * d)))) <= 4.538162550150365e-111)) {
				VAR_2 = ((double) (w0 * ((double) sqrt(((double) (1.0 - ((double) (((double) (((double) pow(((double) (M * ((double) (1.0 / ((double) (((double) (2.0 * d)) / D)))))), 2.0)) * h)) / l))))))));
			} else {
				double VAR_3;
				if ((((double) (((double) (M * D)) / ((double) (2.0 * d)))) <= 3.361590246330775e+113)) {
					VAR_3 = ((double) (((double) (w0 * ((double) sqrt(((double) sqrt(((double) (1.0 - ((double) (((double) pow(((double) (((double) (M * D)) / ((double) (2.0 * d)))), 2.0)) * ((double) (h / l)))))))))))) * ((double) sqrt(((double) sqrt(((double) (1.0 - ((double) (((double) pow(((double) (((double) (M * D)) / ((double) (2.0 * d)))), 2.0)) * ((double) (h / l))))))))))));
				} else {
					VAR_3 = ((double) (w0 * ((double) sqrt(((double) (1.0 - ((double) (((double) (((double) pow(((double) sqrt(((double) (M / ((double) (((double) (2.0 * d)) / D)))))), 2.0)) * ((double) (((double) pow(((double) sqrt(((double) (M / ((double) (((double) (2.0 * d)) / D)))))), 2.0)) * h)))) / l))))))));
				}
				VAR_2 = VAR_3;
			}
			VAR_1 = VAR_2;
		}
		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 5 regimes

  2. if (/ (* M D) (* 2.0 d)) < -inf.0

    1. Initial program 64.0

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

    3. Applied associate-/l*56.0

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

    if -inf.0 < (/ (* M D) (* 2.0 d)) < -1.4590491652940828e+32

    1. Initial program 30.6

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

    3. Applied sqr-pow30.6

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

    4. Applied associate-*l*22.4

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

    if -1.4590491652940828e+32 < (/ (* M D) (* 2.0 d)) < 4.538162550150365e-111

    1. Initial program 6.7

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

    3. Applied associate-/l*6.7

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

    5. Applied associate-*r/1.1

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

    7. Applied div-inv1.1

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

    if 4.538162550150365e-111 < (/ (* M D) (* 2.0 d)) < 3.361590246330775e+113

    1. Initial program 6.1

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

    3. Applied add-sqr-sqrt6.1

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

    4. Applied sqrt-prod6.1

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

    5. Applied associate-*r*6.1

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

    if 3.361590246330775e+113 < (/ (* M D) (* 2.0 d))

    1. Initial program 54.2

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

    3. Applied associate-/l*51.8

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

    5. Applied associate-*r/52.8

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

    7. Applied add-sqr-sqrt52.8

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

    8. Applied unpow-prod-down52.8

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

    9. Applied associate-*l*45.2

      \[\leadsto w0 \cdot \sqrt{1 - \frac{\color{blue}{{\left(\sqrt{\frac{M}{\frac{2 \cdot d}{D}}}\right)}^{2} \cdot \left({\left(\sqrt{\frac{M}{\frac{2 \cdot d}{D}}}\right)}^{2} \cdot h\right)}}{\ell}}\]
  3. Recombined 5 regimes into one program.

  4. Final simplification8.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{M \cdot D}{2 \cdot d} = -inf.0:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M}{\frac{2 \cdot d}{D}}\right)}^{2} \cdot \frac{h}{\ell}}\\ \mathbf{elif}\;\frac{M \cdot D}{2 \cdot d} \le -1.45904916529408281 \cdot 10^{32}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \frac{h}{\ell}\right)}\\ \mathbf{elif}\;\frac{M \cdot D}{2 \cdot d} \le 4.53816255015036534 \cdot 10^{-111}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{{\left(M \cdot \frac{1}{\frac{2 \cdot d}{D}}\right)}^{2} \cdot h}{\ell}}\\ \mathbf{elif}\;\frac{M \cdot D}{2 \cdot d} \le 3.36159024633077472 \cdot 10^{113}:\\ \;\;\;\;\left(w0 \cdot \sqrt{\sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}}\right) \cdot \sqrt{\sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{{\left(\sqrt{\frac{M}{\frac{2 \cdot d}{D}}}\right)}^{2} \cdot \left({\left(\sqrt{\frac{M}{\frac{2 \cdot d}{D}}}\right)}^{2} \cdot h\right)}{\ell}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020148 
(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))))))