Average Error: 14.3 → 9.3
Time: 16.9s
Precision: binary64
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{h}{\ell} = -inf.0:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left({\left(\frac{M}{2}\right)}^{2} \cdot \left({\left(\frac{D}{d}\right)}^{2} \cdot h\right)\right) \cdot \frac{1}{\ell}}\\ \mathbf{elif}\;\frac{h}{\ell} \le 4.4903910910776164 \cdot 10^{213}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\left(\frac{2}{2}\right)} \cdot \frac{h}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1}\\ \end{array}\]
w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}
↓
\begin{array}{l}
\mathbf{if}\;\frac{h}{\ell} = -inf.0:\\
\;\;\;\;w0 \cdot \sqrt{1 - \left({\left(\frac{M}{2}\right)}^{2} \cdot \left({\left(\frac{D}{d}\right)}^{2} \cdot h\right)\right) \cdot \frac{1}{\ell}}\\

\mathbf{elif}\;\frac{h}{\ell} \le 4.4903910910776164 \cdot 10^{213}:\\
\;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\left(\frac{2}{2}\right)} \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) (h / l)) <= -inf.0)) {
		VAR = ((double) (w0 * ((double) sqrt(((double) (1.0 - ((double) (((double) (((double) pow(((double) (M / 2.0)), 2.0)) * ((double) (((double) pow(((double) (D / d)), 2.0)) * h)))) * ((double) (1.0 / l))))))))));
	} else {
		double VAR_1;
		if ((((double) (h / l)) <= 4.4903910910776164e+213)) {
			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) pow(((double) (((double) (M / 2.0)) * ((double) (D / d)))), ((double) (2.0 / 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 (/ h l) < -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 div-inv64.0

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

    4. Applied associate-*r*27.2

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

    6. Applied times-frac26.2

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

    7. Applied unpow-prod-down34.4

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

    8. Applied associate-*l*34.9

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

    if -inf.0 < (/ h l) < 4.4903910910776164e+213

    1. Initial program 9.6

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

    3. Applied times-frac9.6

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

    5. Applied sqr-pow9.6

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

    6. Applied associate-*l*7.9

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

    if 4.4903910910776164e+213 < (/ h l)

    1. Initial program 36.6

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

    2. Taylor expanded around 0 0.5

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

  4. Final simplification9.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} = -inf.0:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left({\left(\frac{M}{2}\right)}^{2} \cdot \left({\left(\frac{D}{d}\right)}^{2} \cdot h\right)\right) \cdot \frac{1}{\ell}}\\ \mathbf{elif}\;\frac{h}{\ell} \le 4.4903910910776164 \cdot 10^{213}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\left(\frac{2}{2}\right)} \cdot \frac{h}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1}\\ \end{array}\]

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))))))