Average Error: 13.5 → 8.7
Time: 21.3s
Precision: binary64
\[[M, D] = \mathsf{sort}([M, D]) \\]
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
↓
\[\begin{array}{l} t_0 := \frac{M \cdot D}{2 \cdot d}\\ t_1 := w0 \cdot \sqrt{1 - {t_0}^{2} \cdot \frac{h}{\ell}}\\ \mathbf{if}\;t_1 \leq 1.0114369229749901 \cdot 10^{+300}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{M}{\frac{2 \cdot d}{D}}\right)}^{2}}\\ \mathbf{elif}\;t_1 \leq \infty:\\ \;\;\;\;w0 \cdot \left(\sqrt{\left(\frac{h}{\ell} \cdot {\left(\frac{D}{d}\right)}^{2}\right) \cdot -0.25} \cdot \left(-M\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\begin{array}{l} t_2 := \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}\\ t_3 := t_0 \cdot \left|t_2\right|\\ w0 \cdot \sqrt{1 - t_2 \cdot \left(t_3 \cdot t_3\right)} \end{array}\\ \end{array} \]
w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}
↓
\begin{array}{l}
t_0 := \frac{M \cdot D}{2 \cdot d}\\
t_1 := w0 \cdot \sqrt{1 - {t_0}^{2} \cdot \frac{h}{\ell}}\\
\mathbf{if}\;t_1 \leq 1.0114369229749901 \cdot 10^{+300}:\\
\;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{M}{\frac{2 \cdot d}{D}}\right)}^{2}}\\

\mathbf{elif}\;t_1 \leq \infty:\\
\;\;\;\;w0 \cdot \left(\sqrt{\left(\frac{h}{\ell} \cdot {\left(\frac{D}{d}\right)}^{2}\right) \cdot -0.25} \cdot \left(-M\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\begin{array}{l}
t_2 := \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}\\
t_3 := t_0 \cdot \left|t_2\right|\\
w0 \cdot \sqrt{1 - t_2 \cdot \left(t_3 \cdot t_3\right)}
\end{array}\\


\end{array}

(FPCore (w0 M D h l d)
 :precision binary64
 (* w0 (sqrt (- 1.0 (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l))))))
↓
(FPCore (w0 M D h l d)
 :precision binary64
 (let* ((t_0 (/ (* M D) (* 2.0 d)))
        (t_1 (* w0 (sqrt (- 1.0 (* (pow t_0 2.0) (/ h l)))))))
   (if (<= t_1 1.0114369229749901e+300)
     (* w0 (sqrt (- 1.0 (* (/ h l) (pow (/ M (/ (* 2.0 d) D)) 2.0)))))
     (if (<= t_1 INFINITY)
       (* w0 (* (sqrt (* (* (/ h l) (pow (/ D d) 2.0)) -0.25)) (- M)))
       (let* ((t_2 (/ (cbrt h) (cbrt l))) (t_3 (* t_0 (fabs t_2))))
         (* w0 (sqrt (- 1.0 (* t_2 (* t_3 t_3))))))))))
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 t_0 = (M * D) / (2.0 * d);
	double t_1 = w0 * sqrt(1.0 - (pow(t_0, 2.0) * (h / l)));
	double tmp;
	if (t_1 <= 1.0114369229749901e+300) {
		tmp = w0 * sqrt(1.0 - ((h / l) * pow((M / ((2.0 * d) / D)), 2.0)));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = w0 * (sqrt(((h / l) * pow((D / d), 2.0)) * -0.25) * -M);
	} else {
		double t_2 = cbrt(h) / cbrt(l);
		double t_3 = t_0 * fabs(t_2);
		tmp = w0 * sqrt(1.0 - (t_2 * (t_3 * t_3)));
	}
	return tmp;
}

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 (*.f64 w0 (sqrt.f64 (-.f64 1 (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l))))) < 1.0114369229749901e300

    1. Initial program 4.9

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

    2. Applied associate-/l*_binary645.2

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

    if 1.0114369229749901e300 < (*.f64 w0 (sqrt.f64 (-.f64 1 (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l))))) < +inf.0

    1. Initial program 59.6

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

    2. Taylor expanded in M around -inf 57.6

      \[\leadsto w0 \cdot \color{blue}{\left(-1 \cdot \left(\sqrt{-0.25 \cdot \frac{{D}^{2} \cdot h}{\ell \cdot {d}^{2}}} \cdot M\right)\right)} \]

    3. Simplified46.1

      \[\leadsto w0 \cdot \color{blue}{\left(\sqrt{\left(\frac{h}{\ell} \cdot {\left(\frac{D}{d}\right)}^{2}\right) \cdot -0.25} \cdot \left(-M\right)\right)} \]

    if +inf.0 < (*.f64 w0 (sqrt.f64 (-.f64 1 (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l)))))

    1. Initial program 64.0

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

    2. Applied add-cube-cbrt_binary6464.0

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

    3. Applied add-cube-cbrt_binary6464.0

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

    4. Applied times-frac_binary6464.0

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

    5. Applied associate-*r*_binary6434.5

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

    6. Applied add-sqr-sqrt_binary6434.5

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

    7. Applied add-sqr-sqrt_binary6439.3

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

    8. Applied unpow-prod-down_binary6439.3

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

    9. Applied unswap-sqr_binary6433.6

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

    10. Simplified33.6

      \[\leadsto w0 \cdot \sqrt{1 - \left(\color{blue}{\left(\frac{D \cdot M}{2 \cdot d} \cdot \left|\frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}\right|\right)} \cdot \left({\left(\sqrt{\frac{M \cdot D}{2 \cdot d}}\right)}^{2} \cdot \sqrt{\frac{\sqrt[3]{h} \cdot \sqrt[3]{h}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}\right)\right) \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}} \]

    11. Simplified13.1

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

  4. Final simplification8.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \leq 1.0114369229749901 \cdot 10^{+300}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{M}{\frac{2 \cdot d}{D}}\right)}^{2}}\\ \mathbf{elif}\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \leq \infty:\\ \;\;\;\;w0 \cdot \left(\sqrt{\left(\frac{h}{\ell} \cdot {\left(\frac{D}{d}\right)}^{2}\right) \cdot -0.25} \cdot \left(-M\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \left|\frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}\right|\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \left|\frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}\right|\right)\right)}\\ \end{array} \]

Reproduce

herbie shell --seed 2022088 
(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))))))