Average Error: 14.6 → 8.5
Time: 18.1s
Precision: binary64
\[[M, D]=\mathsf{sort}([M, D])\]
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
↓
\[\begin{array}{l} t_0 := \frac{M \cdot D}{2 \cdot d}\\ \mathbf{if}\;t_0 \leq -1.7001134354665608 \cdot 10^{+142}:\\ \;\;\;\;\sqrt{\left(\frac{h}{\ell} \cdot {\left(\frac{D}{d}\right)}^{2}\right) \cdot -0.25} \cdot \left(-M \cdot w0\right)\\ \mathbf{else}:\\ \;\;\;\;\begin{array}{l} t_1 := \frac{t_0}{\sqrt[3]{\ell}}\\ w0 \cdot \sqrt{1 - t_1 \cdot \frac{h \cdot t_1}{\sqrt[3]{\ell}}} \end{array}\\ \end{array} \]
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}\\
\mathbf{if}\;t_0 \leq -1.7001134354665608 \cdot 10^{+142}:\\
\;\;\;\;\sqrt{\left(\frac{h}{\ell} \cdot {\left(\frac{D}{d}\right)}^{2}\right) \cdot -0.25} \cdot \left(-M \cdot w0\right)\\

\mathbf{else}:\\
\;\;\;\;\begin{array}{l}
t_1 := \frac{t_0}{\sqrt[3]{\ell}}\\
w0 \cdot \sqrt{1 - t_1 \cdot \frac{h \cdot t_1}{\sqrt[3]{\ell}}}
\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))))
   (if (<= t_0 -1.7001134354665608e+142)
     (* (sqrt (* (* (/ h l) (pow (/ D d) 2.0)) -0.25)) (- (* M w0)))
     (let* ((t_1 (/ t_0 (cbrt l))))
       (* w0 (sqrt (- 1.0 (* t_1 (/ (* h t_1) (cbrt l))))))))))
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 tmp;
	if (t_0 <= -1.7001134354665608e+142) {
		tmp = sqrt(((h / l) * pow((D / d), 2.0)) * -0.25) * -(M * w0);
	} else {
		double t_1 = t_0 / cbrt(l);
		tmp = w0 * sqrt(1.0 - (t_1 * ((h * t_1) / cbrt(l))));
	}
	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 2 regimes

  2. if (/.f64 (*.f64 M D) (*.f64 2 d)) < -1.7001134354665608e142

    1. Initial program 61.8

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

    2. Taylor expanded in M around -inf 57.1

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

    3. Simplified47.3

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

    if -1.7001134354665608e142 < (/.f64 (*.f64 M D) (*.f64 2 d))

    1. Initial program 11.1

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

    2. Applied add-cube-cbrt_binary6411.2

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

    3. Applied *-un-lft-identity_binary6411.2

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

    4. Applied times-frac_binary6411.2

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

    5. Applied associate-*r*_binary648.2

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

    6. Simplified8.2

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

    7. Applied unpow2_binary648.2

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

    8. Applied times-frac_binary647.6

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

    9. Applied associate-*l*_binary647.0

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

    10. Applied associate-*l/_binary647.0

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

    11. Simplified5.6

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

  4. Final simplification8.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{M \cdot D}{2 \cdot d} \leq -1.7001134354665608 \cdot 10^{+142}:\\ \;\;\;\;\sqrt{\left(\frac{h}{\ell} \cdot {\left(\frac{D}{d}\right)}^{2}\right) \cdot -0.25} \cdot \left(-M \cdot w0\right)\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\frac{M \cdot D}{2 \cdot d}}{\sqrt[3]{\ell}} \cdot \frac{h \cdot \frac{\frac{M \cdot D}{2 \cdot d}}{\sqrt[3]{\ell}}}{\sqrt[3]{\ell}}}\\ \end{array} \]

Reproduce

herbie shell --seed 2021313 
(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))))))