Average Error: 14.4 → 8.7
Time: 13.9s
Precision: binary64
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
↓
\[\begin{array}{l} \mathbf{if}\;2 \cdot d \leq -1.4801122808780888 \cdot 10^{+72}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{{\left(\frac{M}{\frac{2 \cdot d}{D}}\right)}^{2} \cdot h}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\begin{array}{l} t_0 := \frac{M \cdot D}{2 \cdot d}\\ \mathbf{if}\;2 \cdot d \leq -1.2928189090676206 \cdot 10^{-136}:\\ \;\;\;\;w0 \cdot \sqrt{1 - t_0 \cdot \left(\left(t_0 \cdot \frac{\sqrt[3]{h} \cdot \sqrt[3]{h}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}\right) \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - t_0 \cdot \left(\left(h \cdot t_0\right) \cdot \frac{1}{\ell}\right)}\\ \end{array}\\ \end{array} \]
w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}
↓
\begin{array}{l}
\mathbf{if}\;2 \cdot d \leq -1.4801122808780888 \cdot 10^{+72}:\\
\;\;\;\;w0 \cdot \sqrt{1 - \frac{{\left(\frac{M}{\frac{2 \cdot d}{D}}\right)}^{2} \cdot h}{\ell}}\\

\mathbf{else}:\\
\;\;\;\;\begin{array}{l}
t_0 := \frac{M \cdot D}{2 \cdot d}\\
\mathbf{if}\;2 \cdot d \leq -1.2928189090676206 \cdot 10^{-136}:\\
\;\;\;\;w0 \cdot \sqrt{1 - t_0 \cdot \left(\left(t_0 \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}}\right)}\\

\mathbf{else}:\\
\;\;\;\;w0 \cdot \sqrt{1 - t_0 \cdot \left(\left(h \cdot t_0\right) \cdot \frac{1}{\ell}\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
 (if (<= (* 2.0 d) -1.4801122808780888e+72)
   (* w0 (sqrt (- 1.0 (/ (* (pow (/ M (/ (* 2.0 d) D)) 2.0) h) l))))
   (let* ((t_0 (/ (* M D) (* 2.0 d))))
     (if (<= (* 2.0 d) -1.2928189090676206e-136)
       (*
        w0
        (sqrt
         (-
          1.0
          (*
           t_0
           (*
            (* t_0 (/ (* (cbrt h) (cbrt h)) (* (cbrt l) (cbrt l))))
            (/ (cbrt h) (cbrt l)))))))
       (* w0 (sqrt (- 1.0 (* t_0 (* (* h t_0) (/ 1.0 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 tmp;
	if ((2.0 * d) <= -1.4801122808780888e+72) {
		tmp = w0 * sqrt(1.0 - ((pow((M / ((2.0 * d) / D)), 2.0) * h) / l));
	} else {
		double t_0 = (M * D) / (2.0 * d);
		double tmp_1;
		if ((2.0 * d) <= -1.2928189090676206e-136) {
			tmp_1 = w0 * sqrt(1.0 - (t_0 * ((t_0 * ((cbrt(h) * cbrt(h)) / (cbrt(l) * cbrt(l)))) * (cbrt(h) / cbrt(l)))));
		} else {
			tmp_1 = w0 * sqrt(1.0 - (t_0 * ((h * t_0) * (1.0 / l))));
		}
		tmp = tmp_1;
	}
	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 2 d) < -1.4801122808780888e72

    1. Initial program 10.9

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

    2. Applied associate-*r/_binary646.0

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

    3. Applied associate-/l*_binary645.3

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

    if -1.4801122808780888e72 < (*.f64 2 d) < -1.29281890906762062e-136

    1. Initial program 12.4

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

    2. Applied unpow2_binary6412.4

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

    3. Applied associate-*l*_binary6410.5

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

    4. Applied add-cube-cbrt_binary6410.6

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

    5. Applied add-cube-cbrt_binary6410.6

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

    6. Applied times-frac_binary6410.6

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

    7. Applied associate-*r*_binary648.2

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

    8. Simplified8.2

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

    if -1.29281890906762062e-136 < (*.f64 2 d)

    1. Initial program 16.3

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

    2. Applied unpow2_binary6416.3

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

    3. Applied associate-*l*_binary6414.3

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

    4. Applied div-inv_binary6414.3

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

    5. Applied associate-*r*_binary6410.2

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

    6. Simplified10.2

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

  4. Final simplification8.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;2 \cdot d \leq -1.4801122808780888 \cdot 10^{+72}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{{\left(\frac{M}{\frac{2 \cdot d}{D}}\right)}^{2} \cdot h}{\ell}}\\ \mathbf{elif}\;2 \cdot d \leq -1.2928189090676206 \cdot 10^{-136}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{M \cdot D}{2 \cdot d} \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{\sqrt[3]{h} \cdot \sqrt[3]{h}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}\right) \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{M \cdot D}{2 \cdot d} \cdot \left(\left(h \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \frac{1}{\ell}\right)}\\ \end{array} \]

Reproduce

herbie shell --seed 2022077 
(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))))))