Average Error: 13.8 → 8.1
Time: 13.0s
Precision: binary64
\[[M, D] = \mathsf{sort}([M, D]) \\]
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
↓
\[\begin{array}{l} t_0 := \frac{M \cdot D}{2 \cdot d}\\ \mathbf{if}\;t_0 \leq 6.911518493043259 \cdot 10^{+166}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{t_0}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \frac{t_0 \cdot h}{\sqrt[3]{\ell}}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(\sqrt{\left(\frac{h}{\ell} \cdot {\left(\frac{D}{d}\right)}^{2}\right) \cdot -0.25} \cdot \left(-M\right)\right)\\ \end{array} \]
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 6.911518493043259 \cdot 10^{+166}:\\
\;\;\;\;w0 \cdot \sqrt{1 - \frac{t_0}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \frac{t_0 \cdot h}{\sqrt[3]{\ell}}}\\

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


\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 6.911518493043259e+166)
     (*
      w0
      (sqrt (- 1.0 (* (/ t_0 (* (cbrt l) (cbrt l))) (/ (* t_0 h) (cbrt l))))))
     (* w0 (* (sqrt (* (* (/ h l) (pow (/ D d) 2.0)) -0.25)) (- M))))))
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 <= 6.911518493043259e+166) {
		tmp = w0 * sqrt((1.0 - ((t_0 / (cbrt(l) * cbrt(l))) * ((t_0 * h) / cbrt(l)))));
	} else {
		tmp = w0 * (sqrt((((h / l) * pow((D / d), 2.0)) * -0.25)) * -M);
	}
	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)) < 6.9115184930432585e166

    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/_binary647.5

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

    3. Applied unpow2_binary647.5

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

    4. Applied associate-*l*_binary646.3

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

    5. Applied add-cube-cbrt_binary646.4

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

    6. Applied times-frac_binary645.7

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

    7. Simplified5.7

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

    8. Simplified5.7

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

    if 6.9115184930432585e166 < (/.f64 (*.f64 M D) (*.f64 2 d))

    1. Initial program 64.0

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

    2. Taylor expanded in M around -inf 57.8

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

    3. Simplified49.5

      \[\leadsto w0 \cdot \color{blue}{\left(\sqrt{\left(\frac{h}{\ell} \cdot {\left(\frac{D}{d}\right)}^{2}\right) \cdot -0.25} \cdot \left(-M\right)\right)} \]
  3. Recombined 2 regimes into one program.

  4. Final simplification8.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{M \cdot D}{2 \cdot d} \leq 6.911518493043259 \cdot 10^{+166}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\frac{M \cdot D}{2 \cdot d}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot h}{\sqrt[3]{\ell}}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(\sqrt{\left(\frac{h}{\ell} \cdot {\left(\frac{D}{d}\right)}^{2}\right) \cdot -0.25} \cdot \left(-M\right)\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022131 
(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))))))